Noel N. Nemeth

Glenn Research Center, Cleveland, Ohio

Robert L. Bratton

Idaho National Laboratory, Idaho Falls, Idaho

Statistical Models of Fracture Relevant to NuclearGrade Graphite: Review and Recommendations

NASA/TM—2011-215805

March 2011
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Noel N. Nemeth

Glenn Research Center, Cleveland, Ohio

Robert L. Bratton

Idaho National Laboratory, Idaho Falls, Idaho

Statistical Models of Fracture Relevant to NuclearGrade Graphite: Review and Recommendations

NASA/TM—2011-215805

March 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The authors thank Prof. S. Leigh Phoenix and Dr. Phani Nukala for providing helpful correspondence and publications of their 

research and Andrew Walker for assistance in preparing this manuscript. This manuscript was funded by Battelle Energy Alliance,

LLC, under Contract No. DE–AC07–05ID14517 with the U.S. Department of Energy.

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://gltrs.grc.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2011-215805 iii

Contents

Summary.......................................................................................................................................................1

1.0 Introduction ..........................................................................................................................................1

1.1 Graphite and Next-Generation Nuclear Reactors .......................................................................1

1.2 Graphite Morphology and Characteristics..................................................................................7

1.3 Fracture of Graphite..................................................................................................................10

1.4 Overview of Experimental Studies, Strength Distribution, and Related Considerations..........12

2.0 Statistical Models of Fracture.............................................................................................................20

2.1 Series Systems—The Weakest Link.........................................................................................21

2.1.1 Extreme-Value and Weibull Distributions.................................................................. 21

2.1.2 The Batdorf Model and Response to Multiaxial Stresses........................................... 24

2.1.3 Strength Anisotropy and Multiaxial Stresses.............................................................. 31

2.1.4 R-Curve Effect on the Weibull Distribution ............................................................... 34

2.1.5 Ho’s Modified Weibull Distribution, Schmidt’s Rule, and Strain-Softening 

Regarding the Size Effect............................................................................................ 36

2.1.6 Graphite Microstructure Specific—The Burchell Model............................................ 38

2.1.7 The Uniform Defect Model......................................................................................... 42

2.2 Parallel Systems........................................................................................................................43

2.2.1 Two Composite Material Failure Modes .................................................................... 45

2.2.2 Equal-Load-Sharing and Local-Load-Sharing Fiber-Bundle Models......................... 45

2.2.3 Daniels’s Equal-Load-Sharing Bundle Model ............................................................ 46

2.2.4 Chain of Equal-Load-Sharing Bundles....................................................................... 46

2.2.5 Local-Load-Sharing Modeling.................................................................................... 48

2.2.6 Weakest-Link Relation for Composite Materials........................................................ 49

2.2.7 Bundle Model Ductile-Like to Brittle-Like Transition............................................... 53

2.3 Lattice Models ..........................................................................................................................54

2.3.1 Disorder and System Breakdown................................................................................ 56

2.3.2 Lattice Brittle-to-Ductile-Like Transition................................................................... 58

2.3.3 Strength Distribution and Size Effect in Lattice Models ............................................ 61

3.0 Discussion and Summary ...................................................................................................................63

3.1 Uncertainty in Experimental Data and the Weibull Distribution..............................................65

3.2 Large Intrinsic Flaws and Extreme-Value Statistics.................................................................66

3.3 Weakest-Link Theory and Asymptotic Weibull Behavior at Low Probability of Failure ........67

3.4 Non-Weibull-Distribution Models and the Role of the Brittle-to-Ductile-Like 

Transition..................................................................................................................................67

3.4.1 Bundle Models and the Brittle-to-Ductile-Like Transition......................................... 67

3.4.2 Lattice Models and the Brittle-to-Ductile-Like Transition ......................................... 68

3.5 The Physical Interpretation of the Weibull Modulus................................................................68

3.6 Strength Dispersion, Size Effect, and Material Volume ...........................................................68

3.6.1 The Functional Relationship Between Size Effect and Strength Dispersion in 

Weakest-Link Theory.................................................................................................. 70

3.6.2 Weibull Modulus, Size Effect, and Scaling Issues Versus Trends in 

Experimental Data....................................................................................................... 71

3.7 Watson-Smith-Style Modified Weibull Distribution................................................................72

3.8 The Continuum Assumption and Stress Gradient.....................................................................73

3.9 Problems in and Recommendations for Predicting the Effects of Slots or Notches.................74

3.10 Regarding Multiaxial Stresses and Anisotropic Strength .........................................................75

3.11 Regarding the Burchell Model for Graphite .............................................................................75

3.12 Regarding Probabilistic Design ................................................................................................76

3.13 Final Comments........................................................................................................................76
NASA/TM—2011-215805 iv

4.0 Conclusions........................................................................................................................................76

Appendix A.—Uniform Defect Model Approximated as a Weibull Distribution......................................79

Appendix B.—Flaw Orientation Anisotropy and Stress Intensity Anisotropy...........................................85

Appendix C.—Interchangeability of Composite Stress With Fiber Stress in the Weibull 

Distribution.........................................................................................................................................89

Appendix D.—Symbols, Definitions, and Acronyms.................................................................................91

D.1 Symbols ....................................................................................................................................91

D.2 Definitions.................................................................................................................................97

D.3 Acronyms and Initialisms.........................................................................................................98

References...................................................................................................................................................99
NASA/TM—2011-215805 1

Statistical Models of Fracture Relevant to Nuclear-Grade Graphite: 

Review and Recommendations

Noel N. Nemeth

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Robert L. Bratton

Idaho National Laboratory

Idaho Falls, Idaho 83415

Summary

The nuclear-grade (low-impurity) graphite needed for the fuel element and moderator material for 

next-generation (Gen IV) reactors displays large scatter in strength and a nonlinear stress-strain response 

from damage accumulation. This response can be characterized as quasi-brittle. In this expanded review, 

relevant statistical failure models for various brittle and quasi-brittle material systems are discussed with 

regard to strength distribution, size effect, multiaxial strength, and damage accumulation. This includes 

descriptions of the Weibull, Batdorf, and Burchell models as well as models that describe the strength 

response of composite materials, which involves distributed damage. Results from lattice simulations are 

included for a physics-based description of material breakdown. Consideration is given to the predicted 

transition between brittle and quasi-brittle damage behavior versus the density of damage (level of 

disorder) within the material system. The literature indicates that weakest-link-based failure modeling 

approaches appear to be reasonably robust in that they can be applied to materials that display distributed 

damage, provided that the level of disorder in the material is not too large. The Weibull distribution is 

argued to be the most appropriate statistical distribution to model the stochastic-strength response of 

graphite. This report is an expanded version of Nemeth and Bratton (2010).

1.0 Introduction

1.1 Graphite and Next-Generation Nuclear Reactors

At present, 10 countries, including the United States, have agreed to cooperate on the development 

of the fourth generation (Gen IV) nuclear energy system (Generation IV International Forum (2009)). 

Figure 1 shows the evolution of reactors since their introduction in the 1950s. The Gen IV system is 

expected to come into service in 2030. In the United States, Gen IV research is being conducted by the 

U.S. Department of Energy for development of the Very High Temperature Reactor design concept for 

the Next Generation Nuclear Plant Project (Idaho National Laboratory et al. (2005)). The design will have 

a graphite-moderated reactor, either a prismatic graphite-block-type core or pebble-bed core that will 

produce process heat in a highly efficient manner, reducing the need to burn fossil fuels to produce 

process heat and thereby reducing green house emissions. The application of heat is essential to nearly 

all basic material and commodity manufacturing processes, and heating processes account for about 

17 percent of all industrial energy use (U.S. Department of Energy (2008)). Figure 2 shows a schematic of 

the Gen IV Very High Temperature Reactor gas-cooled reactor. 

The Next Generation Nuclear Plant will have a projected service life of 30 to 60 years and be 

designed to ensure passive decay heat removal without fuel damage or radioactive material releases 

during accidents. The development of the design methods and validation of the graphite structures used in 

the reactor are important to the Next Generation Nuclear Plant Project because large amounts of graphite 

(up to thousands of tons) would be required for the reactor core and the individual graphite bricks that 
NASA/TM—2011-215805 2

Figure 1.—Generation I to IV development time line. LWR, light water reactor; PWR, pressurized heavy water 

reactor; BWR, boiling water reactor; CANDU, Canada Deuterium Uranium reactor; AGR, advanced gas-cooled 

reactor; ABWR, Advanced Boiling Water Reactor. Reproduced from U.S. DOE Nuclear Energy Research Advisory 

Committee and Generation IV International Forum (2002).

Figure 2.—Generation IV very high temperature gas-cooled reactor using graphite for neutron reflection and 

moderation within the reactor core. Reproduced from U.S. DOE Nuclear Energy Research Advisory Committee 

and Generation IV International Forum (2002).
NASA/TM—2011-215805 3

surround the nuclear fuel may experience significant loads. Figure 3 shows an example of in-core graphite 

components for an earlier generation II reactor design. Of particular concern is the potential for crack 

formation and even rupture in individual blocks. Therefore, failure theories—and/or effective design 

strategies that can predict and mitigate failure from fracture—are needed. NASA has complementary 

interest in this endeavor with regards to the development of Moon and Mars surface power generation and 

nuclear propulsion for interplanetary missions. The Department of Defense would have interest with 

regards to a graphite nozzle throat in a solid rocket motor.

An important characteristic of graphite is that its strength is stochastican individual specimen can 

show a large random fluctuation in strength from a population mean. Graphite also can have a nonlinear 

stress-strain response because of distributed damage and damage accumulation within the material prior 

to rupture. This behavior can be described as “quasi-brittle” or “ductile-like.” In contrast, classically 

brittle materials, such as ceramics and glasses, fail abruptly without prior damage accumulation, although 

they similarly display large scatter in strength. Other materials, such as fiber-reinforced composites, can 

accumulate significant damage prior to failure and have less scatter in strength than the individual 

constituents of the composite have. Graphite rupture behavior seemingly falls somewhere between the 

behaviors of ceramics and fiber-reinforced composites. 

This report reviews some statistical failure models that may be relevant for the design of nonirradiated nuclear-grade graphite. This report is an expanded version of Nemeth and Bratton (2010), including 

additional details and models that were not included in the original journal article. Models for various 

brittle and quasi-brittle material systems are discussed with regard to strength distribution, size effect, 

multiaxial strength, anisotropic strength, and damage accumulation. Table I summarizes the models 

discussed in this report. This includes descriptions of the Weibull, Batdorf, and Burchell models as well 

as models that describe the strength response of composite materials, which involve distributed damage. 

Results from lattice simulations are included for a physics-based description of material breakdown, and 

consideration is given to the predicted transition between brittle and quasi-brittle damage behavior versus 

the density of damage (level of disorder) within the material system. 

Figure 3.—Examples of graphite in-core reactor components. (a) Graphite components of United Kingdom advanced 

gas-cooled reactor (AGR) assembled with block-and-key construction before fuel insertion. Reproduced from Holt 

(2010). Copyright EDF Energy; used with permission. (b) Prismatic graphite fuel element from Fort St. Vrain 

research reactor (see Pinner (2000–2010)).
NASA/TM—2011-215805 4

TABLE I.—SUMMARY OF FRACTURE MODELS DESCRIBED IN THIS REVIEW

Model Section Description Material 

system

Weakest 

link

Size 

effect

Distribution 

type

Multiaxial

Weibull 2.1.1 Based on extreme-value distribution 

for classically brittle materials. Posits 

that scatter in strength and size effect is 

solely due to the presence of inherent 

flaws whereby the largest and weakest 

critically located flaw under load will 

cause rupture of the material. 

Brittle Yes Yes, based on 

the weakestlink theory 

(WLT) 

mechanism

Weibull No (Weibull’s 

polyaxial theory 

is the same as 

Batdorf’s for 

shearinsensitive 

cracks.)

Batdorf 2.1.2.1 Combines the Weibull distribution with 

linear elastic fracture mechanics 

(LEFM) to predict stochastic strength 

response under multiaxial loading. 

Originally developed for graphite and 

brittle materials in general.

Brittle Yes Yes, based on 

the WLT 

mechanism

Weibull Yes, using 

principles of 

LEFM; 

assumes the 

random orientation of flaws

Principle of 

Independent 

Action 

(PIA)

2.1.2.2 Applies the Weibull distribution to 

individual principal stresses to predict 

stochastic strength response under 

multiaxial loading. 

Brittle Yes Yes, based on 

the WLT 

mechanism

Weibull Yes, using 

individual 

principal 

stresses

Margetson 

(anisotropy)

2.1.3 Applies a PIA style of methodology 

with the Weibull distribution to predict 

stochastic strength under multiaxial 

loading for an anisotropic material 

(transversely isotropic strength).

Brittle Yes Yes, based on 

the WLT 

mechanism

Weibull Yes, using 

tensile stresses 

oriented relative 

to the material 

coordinate 

system and 

applied 

independently 

Unit Sphere 

Strength 

Anisotropy 

Model

2.1.3 and 

Appendix B

Uses a Batdorf-style approach to 

combine LEFM with preferred flaw 

orientation or fracture toughness to 

predict stochastic strength under 

multiaxial loading for an anisotropic 

material (transversely isotropic 

strength).

Brittle Yes Yes, based on 

the WLT 

mechanism

Weibull LEFM with 

preferred 

orientation of 

flaws or 

fracture 

toughness

R-curve 2.1.4 Fracture toughness changing with 

crack length affects stochastic strength 

distribution. R-curve is a characteristic 

material property.

Brittle No No, by itself 

R-curve does 

not imply a 

size effect 

with changing 

material 

volume

Deterministic 

material property unless the 

criterion of 

Planas (1995) 

and Bazant and 

Planas (1998) is 

considered

No

Ho 2.1.5 Empirical criterion to account for the 

reduction of strength due to grain size 

effects. 

Brittle Yes Yes, size 

effect 

modified by 

grain size 

relative to 

specimen size

Modified 

Weibull

No

Schmidt 2.1.5 Modified Weibull methodology to 

allow a size effect between flexural and 

tension loading but allow a size effect 

with increasing volume of a specimen. 

This is an empirical criterion.

Brittle Yes, but 

modified

Yes, but 

modified to be 

only a stress 

gradient effect

Modified 

Weibull 

methodology

Uses maximum 

deformation 

failure criterion

Li and Fok 2.1.5 Elastic damage model combined with 

the Weibull distribution for critical 

strain where damage is initiated. The 

difference in strength between tension 

and bending load is explained as being 

due to strain-softening behavior of 

quasi-brittle materials.

Quasi-brittle No Yes, size 

effect for a 

stress gradient

Weibull distribution coupled 

with nonlinear 

stress-strain 

response

No
NASA/TM—2011-215805 5

TABLE I.—SUMMARY OF FRACTURE MODELS DESCRIBED IN THIS REVIEW

Model Section Description Material 

system

Weakest 

link

Size 

effect

Distribution 

type

Multiaxial

Burchell 2.1.6 Graphite microstructure specific 

model; fracture initiates near pores and 

progresses through grains. Model 

depends on initial size distribution of 

pores and probability a crack can 

propagate through progressive rows of 

grains.

Brittle Yes Yes, but 

different from 

Weibull

Lognormal 

distribution of 

initial pore 

sizes

Yes, PIA-type 

methodology or 

Shetty (1987) 

based mixedmode criterion

Uniform 

defect 

model

2.1.7 Model is based on a critical density of 

defects as a failure criterion. Does not 

consider flaw interaction. Assumes a 

Poisson process governing the general 

equation for the probability of 

occurrence of k inhomogeneities in the 

volume V.

Brittle Yes, based 

on the 

critical 

number of 

defects 

within a 

given 

volume

Yes, but 

smaller than 

Weibull

Gamma 

distribution

No

Daniels 2.2.3 Fiber-bundle model based on equal 

load sharing. Based on an asymptotic 

analysis for a large number of fibers in 

a fiber bundle.

Composite/

quasi-brittle

No No Normal or 

Gaussian

No

Batdorf and 

Ghaffarian

2.2.5 Modeled the formation and growth of 

multiple fiber fractures as the basis of 

their failure criterion. In their model the 

Weibull relation for single fiber 

strength is used to determine the 

number of critical fiber fractures 

(singlets), double adjacent fractures 

(doublets), triple adjacent fractures 

(triplets), and so on as a function of 

stress and volume.

Composite/

quasi-brittle

Yes Yes Piecewise 

Weibull

No

Chain of 

Bundles

2.2.6 A composite is treated as a serial 

arrangement of independent bundles. 

Failure of a bundle causes failure of the 

composite.

Composite/

quasi-brittle

Yes Yes Not specified 

but likely 

asymptotically 

Weibull at 

small 

probabilities of 

failure

No

WatsonSmith

2.2.6 A modified Weibull distribution to 

decouple size effect from strength 

scatter.

Brittle Yes Yes, but not 

classically 

Weibull

Modified 

Weibull

No

Bazant and 

Pang

2.2.6 A multiscale model composed of 

mixed series and parallel systems that 

is argued to provide a physical 

underpinning to the Weibull 

distribution. In their model, quasibrittle materials will have a twoparameter Weibull distribution at the 

lower probabilities of failure.

Quasi-brittle Yes Yes, but not 

classically 

Weibull

Normal or 

gaussian at 

higher levels of 

failure 

probability and 

Weibull at 

lower 

probabilities of 

failure

No

Lattice 

Models

2.3 to 2.3.3 Methodology to simulate material 

individual inhomogeneities with a 

stochastic distribution so as to 

investigate damage evolution with load 

for quasi-brittle materials

Quasi-brittle 

(flaw 

interaction 

accounted 

for)

Depends 

on input 

parameters

Yes, but 

response 

depends on 

input 

parameters

Not clearly 

established and 

depends also on 

input 

parameters

Response not 

investigated in 

this literature 

review

WatsonSmith & 

Chain of 

Bundles

3.7 A modified Weibull distribution to 

decouple size effect from strength 

scatter. Additional terms added to 

mimic a chain-of-bundles approach.

Brittle/

quasi-brittle

Yes Yes, but not 

classically 

Weibull

Modified 

Weibull

No
NASA/TM—2011-215805 6

More specifically, Section 1.2 describes the morphology and mechanical behavior of graphite, and 

Section 1.3 provides further detail regarding the fracture of graphite. Section 1.4 describes the effects of 

strength distribution and results from size-effect studies, and Section 2.0 introduces the two broad classes 

of statistical models of fracture: weakest-link series systems and parallel systems. Section 2.1 covers 

fracture models that fall under the weakest-link series system category of brittle material failure, and 

Section 2.2 covers the category for parallel systems, describing fracture models that involve distributed 

damage. 

Section 2.1 describes the weakest-link theory (WLT) model. Section 2.1.1 describes extreme-value

distributions and introduces the Weibull extreme-value distribution and the Weibull size-effect relationship. Section 2.1.2.1 primarily discusses the Batdorf model but also provides references for other models

for predicting the effect on strength from multiaxial stress states, and Section 2.1.2.2 gives an example

multiaxial strength prediction of the Batdorf model compared with the Principle of Independent Action 

model. Strength anisotropy models with regards to multiaxial stresses are discussed in Section 2.1.3. The 

effect of R-curve on strength distribution is provided in Section 2.1.4, and the Ho and Schmidt modified 

Weibull models for diminished size effect are provided in Section 2.1.5. The Burchell model, which 

considers how the graphite microstructure affects fracture, is described in Section 2.1.6, and the uniform 

defect model, which considers the effects of defect clusters, is described in Section 2.1.7. 

Section 2.2 introduces the parallel system modeling category—specifically, composite material fiberbundle models (Secs. 2.2.1 to 2.2.7) and lattice models of disordered materials (Secs. 2.3 and 2.3.1 to 

2.3.3)—for describing fracture from distributed damage and how that is relevant to graphite. Section 2.2

discusses the important concept of the bounds of the applicability of WLT to distributed damage modeling. For composites, it describes the two extremes of composite failure modes (Sec. 2.2.1) and various 

fiber-bundle models—broadly classified as equal-load-sharing and local-load-sharing models (Sec. 2.2.2). 

The classical model of Daniels for a dry bundle of fibers is described in Section 2.2.3, and the chain-ofbundles equal-load-sharing model is described in Section 2.2.4. Local-load-sharing modeling with regards

to size effect is described in Section 2.2.5, including the Batdorf and Ghaffarian model of multiplets. The 

Harlow and Phoenix WLT relation for composite materials is provided in Section 2.2.6, along with 

Watson-Smith’s modified Weibull distribution and Bazant’s interpretation of the Weibull distribution for 

mixed-series and parallel systems. Section 2.2.7 describes the brittle-to-ductile-like transition for composite bundle systems as it relates to the stochastic strength distribution of individual fibers.

Section 2.3 introduces lattice models of quasi-brittle materials. A discussion of the role of disorder on 

system breakdown is provided in Section 2.3.1, the brittle-to-ductile-like transition in lattice models is 

described in Section 2.3.2, and results of studies of strength distribution and size effect in lattice model 

simulations are given in Section 2.3.3.

Section 3.0 is devoted to further discussion and a summary of the previous sections in the context of 

developing a more comprehensive graphite design methodology. It primarily discusses the applicability, 

limitations, and recommendations regarding the use of the Weibull distribution but also summarizes the 

main points of the report. Sections include uncertainty in experimental data and the statistics-based 

argument for using the Weibull distribution at low probabilities of failure (Sec. 3.1), the role of large-size 

intrinsic flaws on strength distribution (Sec. 3.2), a summary of the physically based argument for the 

Weibull distribution at low probabilities of failure (Sec. 3.3), non-Weibull models and the brittle-toductile-like transition summary (Secs. 3.4 and 3.4.1 for bundle models and Sec. 3.4.2 for lattice models), 

the physical interpretation of the Weibull modulus (Sec. 3.5), a discussion of strength dispersion and size 

effect versus material volume (Sec. 3.6), the functional relationship between size effect and strength 

dispersion in WLT (Sec. 3.6.1), scaling issues and the Weibull modulus—models and experimental 

data (Sec. 3.6.2), a proposed Watson-Smith-style modified Weibull distribution for uniform uniaxial 

tensile loading (Sec. 3.7), the continuum assumption and stress gradient regarding the Weibull 

distribution (Sec. 3.8), issues and recommendations for predicting the effect of slots and notches on 

graphite component strength (Sec. 3.9), a very brief treatment of multiaxial stresses and anisotropic 

strength (Sec. 3.10), some comments on the Burchell model (Sec. 3.11), some brief comments regarding 

probabilistic design (Sec. 3.12), and final comments (Sec. 3.13). The conclusions are given in Section 4.0.
NASA/TM—2011-215805 7

Appendix A shows a formulation of the uniform defect model of Section 2.1.7 approximated as a

Weibull distribution (assumed for compressive stress states), Appendix B shows further detail regarding 

the development of a mechanistic model for describing anisotropic strength from multiaxial stresses from 

Section 2.1.3, and Appendix C shows the interchangeability of using the composite stress or the fiber 

stress in probability-of-failure calculations (applies to Sec. 2.2.6). To aid the reader, symbols used in this 

report are defined in Appendix D.

This report attempts to address the important question: What is the most appropriate probabilistic 

distribution that should be used in the design of graphite components? The authors draw attention to some 

unique aspects of graphite rupture behavior that make development of a comprehensive design methodology more challenging. However, only the problem of fast-fracture—the inherent rupture strength from the 

sudden, uncontrolled crack propagation of (pristine) graphite independent of any additional time- or 

cycle-dependent material degradation effects is addressed. Also, the methodologies in this report primarily describe failure with regards to the volume of the material or the imperfections (flaws) within the 

volume of the material. For the sake of brevity, only limited descriptions are given of these methodologies 

with regards to the surface of the material—imperfections residing on the surface (surface flaws) or edges 

(edge flaws)—because the modeling approach is similar. Awareness of these modeling problems and 

some of the modeling approaches available will be useful in the eventual selection and refinement of 

methodologies for designing graphite components by organizations such as ASME International (Boiler 

and Pressure Vessel Code) and the Nuclear Regulatory Commission. 

1.2 Graphite Morphology and Characteristics

Nuclear-grade graphite refers to bulk graphite of accepted and characterized properties. One important area of investigation and subsequent design standards development such as the ASME’s Boiler and 

Pressure Vessel Code for graphite-core structures involves the testing, qualification, and design methodology for new material grades of nuclear graphite. Graphite is a brittle material or quasi-brittle material, and 

brittle materials require a design methodology that is different from ductile materials (such as metals). 

Graphite is similar to other brittle materials in that it does not exhibit plastic deformation and shows wide 

scatter in strength. However, it differs from other classically brittle materials in that it can have a nonlinear stress-strain response and large amounts of acoustic emission (damage accumulation from microcracking) prior to rupture. (This is termed “quasi-brittle” behavior.)

Graphite and its properties are discussed in Burchell (1999), Smith (1972), and Greenstreet (1968). 

Greenstreet also provides an interesting historical perspective of graphite development. As used herein, 

nuclear-grade isotropic graphite is bulk graphite of accepted and characterized properties (both nonirradiated and irradiated) with high purity (such as low boron content) and certified for use inside a nuclear 

reactor core. Graphite is a monolithic material that, in some respects, is similar to ceramic materials, 

which are not processed via melting. All graphites are nearly pure carbon and are made by mixing a coke 

solid particulate filler with a liquid binder (usually coal tar pitch). The mixture is compacted and heated to 

expel volatiles and create the desired material phases. The coke filler particles are rigid and irregularly 

shaped such that the packing is inefficient, leaving a significant fraction of interparticle voids (particles 

pack at typically less than 70-percent full density). The binder is pyrolyzed during baking—losing mass 

and decreasing in volume—producing shrinkage cracks, shrinkage cavities, connected porosity, blowholes, and macrocracks. After baking, the carbon body is graphitized (the carbon atoms arrange into the 

layered hexagonal crystal structure of graphite) by exposing the formed body to temperatures in excess of 

2700 °C. Although micrographs of graphite appear to show separate phases of particulates and binder, the 

binder does not exist as a separate phase; rather, the binder is also graphitized, just as the coke filler 

particles are. X-ray crystallography examination of various areas shows the material to be graphitic, 

regardless of where the binder or coke particles were in the original form of the article.

The degree of graphitization varies within the volume of the billet. Graphitization results in further 

shrinkage and the anisotropies of thermal contraction of individual misaligned graphite crystals (due to 

the differing thermal expansion coefficients of the a- and c-axis of the graphite’s hexagonal lattice 
NASA/TM—2011-215805 8

structure) create Mrozowski stresses (Mrozowski (1956)). These stresses may be partially relieved by the 

formation of additional microcracks. The end result is that graphite contains a variety of defect structures 

and typically has between 15- and 25-percent void volume. There tends to be wide variation in the properties of graphite as a result of processing, although the chemical composition remains the same. This can 

be true not only from one material grade to another, but between different billets of the same material 

grade or within a billet.

Different grades of graphite can show widely different textures and pore-size distributions, as well as 

the presence of subcritical cracklike formations (see Figure 4). This cracklike porosity can range from 

being fairly planar to having an “onion skin” appearance. Pears and Sanders (1970) identified two classes 

of defects: background and disparate. Background defects are generally small relative to the size of the 

filler particles and are uniformly distributed. They include small blowholes, microcracks, locations of 

weak cleavage within filler particles, shrinkage cracks and cavities, and small gas pockets (connected or 

unconnected porosity) within the binder residue. Background defects are ever present. Disparate defects 

are much larger than background defects and are most commonly blowholes, which may be elongated 

defects created during extrusion. There may also be macrocracks, formed from a variety of processes 

involving gas entrapment. Other possible disparate defects include regions of binder deficiency, where 

cohesion between filler particles is poor; spongy regions of excess binder; inclusions, usually refractory 

metal carbides; voids left by the vaporization of inclusions; and “reorganized” graphite, where an 

impurity produces a region of ordered graphite unlike that of the filler and binder residue.

Graphite can be manufactured with different average grain sizes. Coarse-grained material has grains 

larger than 4 mm; medium-grained material has grains smaller than 4 mm; fine-grained material has 

grains smaller than 100 μm; and superfine, ultrafine, and microfine materials have grain sizes smaller 

than 50, 10, and 2 μm, respectively. Nuclear graphite typically has medium to ultrafine grains. Manufactured graphite is usually extruded or molded, and the resulting grain structure has a biased orientation. 

As such, material properties are often measured with the grain (parallel to the extrusion direction and 

perpendicular to the molding axis) or against the grain (perpendicular to the extrusion direction and 

parallel to the molding axis). 

Graphite is similar to other brittle materials in that it does not exhibit plastic deformation (in the sense 

of metals) and shows wide scatter in strength. However, it differs from other classically brittle materials 

in that it can have a nonlinear stress-strain response and large amounts of acoustic emission (damage 

accumulation from microcracking) prior to rupture. This is termed “quasi-brittle” behavior. The amount 

of nonlinearity in the stress-strain response depends on the material microstructure. This nonlinearity is 

associated with the accumulation of mechanical damage within the material. When a material is stressed, 

micromechanical events such as slip, shear, cleavage, or microcracking are detected in the form of acoustic emission. In graphite, acoustic emission typically begins at the onset of loading with the frequency of 

events and energy per event increasing with load level. Kaiser (1950) found that, upon unloading, acoustic emission ceased and would not resume until the previous maximum stress was exceeded. This is 

known as the Kaiser effect, and the stress-strain effect is illustrated in Figure 5. A strain offset from 

damage accumulation under tensile load is indicated in the figure.
NASA/TM—2011-215805 9

Figure 4.—Typical graphite microstructures showing N, needle-coke particles; G, gilsocarbon 

particles; and B, binder phase. The dark regions are pores. (a) High Density Graphite 

(HDG; Morganite Co.). (b) IM1–24 (Anglo Great Lakes Corporation). (c) SM2–24 (Anglo 

Great Lakes Corporation). (d) Pile Grade A (PGA). (e) Nipple stock. (f) Unimpregnated 

electrode graphite. Reproduced from Tucker et al. (1986). Copyright Elsevier; used with 

permission.
NASA/TM—2011-215805 10

Figure 5.—Tensile stress-strain curve for repeated loading for an unspecified 

graphite. Reproduced from Losty and Orchard (1962). Copyright Elsevier; 

used with permission.

Graphite strain at failure is a few tenths of one percent in tension and 1 to 2 percent in compression; 

therefore, graphite is considerably stronger in compression than in tension. Deviation from stress-strain 

linearity is small for low stresses but more pronounced at higher stress levels. The stress-strain curve is

continuous, without any abrupt transition point. The deviation from linearity may be more pronounced in 

tension than it is in compression as reported by Arai and Oku (1979). Also, there can be significant scatter in 

the material strain-response from specimen to specimen, again as shown in Arai and Oku. In nuclear-grade 

graphite, the compression strength is 3 or 4 times the tensile strength. Bulk graphite tends to be weak in 

tensionwith strength on the order of 11 to 15 MPa, depending on whether it is measured against the grain or 

with the grain—and has low fracture toughness KIc (between 1.4 and 1.5 MPa-m1/2, as reported by General 

Atomics (1988) for H–451 graphite). Hence, graphite strength is anisotropic (transversely isotropic). Newer 

grades of nuclear graphite tend to be more isotropic (within 10 percent or less)—which is desirable. Strength 

improves with temperature (in a nonoxidizing environment) up to 2500 °C, and there is high resistance to 

thermal shock. There is wide scatter in the strength of nominally identical graphite specimens, and therefore, 

graphite strength has to be described in terms of probability distributions such as the normal, lognormal, or 

Weibull (1951). The Weibull distribution is the distribution most often cited when reporting the strength 

behavior of specimen test data (see Sec. 2.1.1 for a description).

The strength variation observed in graphite is more pronounced in tension than in compression. The 

strength varies randomly from log to log and partly randomly, partly systematically within the log where 

the systematic part reflects a general increase in strength away from the center. When adjusted for the 

systematic spatial variation, the tensile strength can be approximated by a Weibull distribution or possibly 

by a bimodal Weibull distribution.

1.3 Fracture of Graphite

The fracture of graphite can be a complex process, with different grades of graphite potentially 

having different failure behaviors. A good treatise on the fracture of graphite can be found in 

Brocklehurst (1977). The fracture of graphite is nominally brittle or quasi-brittle with little or no plasticity 
NASA/TM—2011-215805 11

prior to failure. This means that fracture is influenced by preexisting flaws or inherently weak regions in 

the material. Porosity can be important in the fracture process. Stress concentrations at, or very near, 

pores can initiate cracking or influence the crack path, and conversely, crack growth may be arrested at a 

pore until a higher stress is applied. In the binder phase, there may be regions of common basal plane 

alignment extending over 100 μm in length. Cleavage in these regions can occur at stresses well below 

the fracture stress and can act as sites for crack initiation. Finally, coke filler particles with the proper 

orientation are highly susceptible to cleavage along basal planes at low stresses.

Tensile fracture occurs when a local concentration of microcracks develops and coalesces to form an 

unstable macrocrack of critical size. Microcracks, oriented approximately normal to the applied load, 

begin forming at low strain (relative to the ultimate strain) and preferentially form in regions of stress 

concentration adjacent to the large disparate defects. These grow to a limited size and stop growing. 

These defects or pores basically act as loci for crack initiation and termination. Knibbs (1967) observed 

that well-defined systems of secondary cracking arose when the crack tip entered a pore or defect and that 

locations of low-density material impede crack growth. Increasing the strain (load) causes new microcracks to form throughout the body that similarly reach a limiting size and stop growing. This process 

continues until the density of microcracks is such that they overlap, join, coalesce, and—at some location 

in the structure—create a fracture path. 

As discussed in Tucker and McLachlan (1993), crack growth tends to be transgranular (through the 

grain), with the crack path within the individual grain corresponding to the crystallite cleavage plane 

(from Jenkins (1962) and Knibbs (1967)). Regions of similarly oriented crystallites are more likely to fail 

than collections of small randomly oriented crystals. The ultimate fracture path tends to extend from one 

large defect to another. It is interesting that the microcracks that form early in the process usually do not 

propagate directly to fracture. The fracture surface is rough, and the path is normally branching and 

irregular. In general, the fractography of fracture surfaces has not located the earliest initiating flaws. It 

can be concluded that the process of fracture in graphite falls somewhere between that of brittle 

monolithic ceramics and quasi-brittle materials such as brittle-fiber-reinforced composites.

For classically brittle materials, like ceramics and glasses, the science and art of fractography can 

often trace the source of fracture in a specimen or component to a single originating flaw (Quinn (2007)).

These originating flaws can be from individual pores, porous regions, agglomerates, compositional 

inhomogeneities, inclusions, large grains, cracks, machining damage, handling damage, surface pits, 

surface pores, and damage or chipping along sharp corners. These flaw types (or flaw populations) are 

broadly classified as either volume distributed, surface distributed, or edge distributed and are considered 

to be separate and competing failure modes. When the probability of a component surviving loading is

evaluated, it is done as a function of the component volume, the component surface (or even the component edges), or a combination of the two. This is referred to as volume-flaw reliability analysis or surfaceflaw reliability analysis (e.g., see Nemeth et al. (2003, 2005) and Gyekenyesi and Nemeth (1987)). For 

graphite, the failure process is usually considered to be a function of the material volume; therefore, the 

reliability analysis is performed over the volume of the component. This assumption is made because the 

identity and location of the earliest flaws usually cannot be established and because the subsequent 

growth and accumulation of damage through microcracking can be diffusely distributed within the 

material volume (at least for uniform uniaxial tension) prior to a final coalescence of damage and 

formation of a macrocrack.

The role surface flaws may play in the fracture of graphite is not clearly established. One would 

expect that, under flexural loading, surface flaws would play some role in the fracture of the material. 

However, the authors of this report are not aware of any studies that have examined the strength of 

nuclear-grade graphite flexural specimens as a function of the grit size of the abrasive that was used to 

grind the specimen surface. Part of the reason for this may be that the size of exposed pores on the 

specimen surface is often larger than the grit size of the abrasive, making it pointless to characterize the

surface roughness from the grinding process and indicating that the size of surface pores—rather than the 

damage caused by the grinding process—will control the strength response in flexure.
NASA/TM—2011-215805 12

1.4 Overview of Experimental Studies, Strength Distribution, and Related 

Considerations

Graphite behavior ranges from brittle to quasi-brittle, depending on the processing of the material. 

Low-void-content, fine-grained materials would likely be more brittle than high-void-content, larger

grained materials. Regardless of the processing, the strength of graphite is always stochastic—the 

strengths of nominally identical specimens fluctuate significantly from the population mean. Probability 

distributions can approximate this strength variability, but is one type of distribution more appropriate to 

use than another? 

Often distributions such as normal (gaussian), lognormal, or Weibull can fit experimental data nearly 

equally well (within the range typically sampled between 1 and 99 percent). Figure 6(a) shows a schematic example of this, comparing a two-parameter Weibull distribution and a gaussian (normal) distribution having the same mean and standard deviation. The two curves appear to be similar in this limited 

range. Danzer et al. (2008) indicates that several thousand experimental tests may be required to discern a 

difference between a normal and a Weibull distribution. 

A good example of this difficulty is shown in Kittl and Aldunate (1983), where despite a large 

amount of data (575 specimens), an appropriate distribution could not be established with high confidence 

for cement cylinders tested in compression (see Appendix A, Figure 30). Another more relevant example 

is the study by Price (1976) on nuclear graphite grade H–451 (Great Lakes Carbon Corporation) involving over 2000 tensile and four-point-bending specimens. In this study, both a normal (gaussian) and 

Weibull distribution were purported to fit equally well to the results of the various fracture experiments. 

Nemeth et al. (2011) reanalyzed these data by pooling (combining) the individual data sets into larger,

more meaningful, data sets. Two- and three-parameter Weibull distributions fit very well to the data. A 

large international study by Tennery and Ferber (1989) with over 2500 four-point-bending specimens for 

three different ceramics indicated that the Weibull distribution generally fit the data better, although for 

one material, the gaussian and Weibull distributions fit equally well.

Figure 6.—Gaussian and two-parameter Weibull distribution with a mean of 1 and a 5.2-percent coefficient of 

variation ω, where m is the Weibull parameter, ∆G is the difference in σ/σμ from the mean to the value of σ/σμ

where the probability of failure is 10–6 for the gaussian distribution, ∆W is the difference in σ/σμ from the mean 

to the value of σ/σμ where the probability of failure is 10–6 for the Weibull distribution, σ is the applied uniaxial 

stress, and σμ is the mean fracture strength. (a) Linear scale. (b) Log scale. Note the large difference in the 

abscissa for small probabilities of failure Pf. Reproduced from Bazant and Pang (2007). Copyright Elsevier; 

used with permission.
NASA/TM—2011-215805 13

The deviations from one distribution to another only become apparent in the distribution tails (as 

shown schematically in Figure 6(b) for two-parameter Weibull and gaussian distributions with identical 

means and standard deviations). However, obtaining experimental data at low probabilities of failure (the 

lower tail of the distribution and the region of highest interest) requires prohibitively large numbers of 

specimens. Hence it is difficult or impractical to determine a best distribution based solely on fit to data. 

From a design perspective, the conservatism (or nonconservatism) of one distribution relative to another 

distribution can be an important differentiator since these distributions extrapolate differently at the lower 

tail, where confirmatory data do not exist (e.g., see Liu (1997)). 

A further, and perhaps more fundamental, consideration is whether a physical basis consistent with 

graphite failure can be associated with a particular distribution and if that distribution or model can be 

used to make other predictions that can be experimentally verified, such as size effect, where average 

strength is affected by the size of the component. In fact, size effect has been investigated for graphite 

in several studies, including Brocklehurst and Darby (1974), Strizak (1991), Price (1976) (see also 

Nemeth et al. (2011)), and Department of Energy gas-cooled reactor reports from Homan and Kasten 

(1980), Kennedy (1987), and Kasten et al. (1989). Generally these investigations compared experimental 

results with predictions made using the Weibull distribution. The size effect and the Weibull distribution 

are discussed further in Section 2.1.1. The Weibull distribution predicts that average strength decreases 

with increasing volume (with the rate of decrease functionally related to the degree of strength scatter). 

However, many of the studies show inconsistent results with regard to this expectation. (Generally they 

show that flexural specimen tests follow Weibull distributions, whereas tensile specimen tests produce a 

smaller-than-expected size effect.)

Danzer et al. (2007, 2008) provide an excellent discussion regarding the applicability and problems of 

using the Weibull distribution to describe the strength response of brittle materials. Many of the same 

problems described by Danzer et al. (2007, 2008) for ceramics are relevant to graphite.

Note that the determination of graphite strength requires taking into account the nonlinear elastic 

response of the material. This was done in the following studies by using various approximations, 

including correction factors. The methods and accuracy of these approaches are not described herein.

Brocklehurst and Darby (1974) tested uniform tensile, four-point-bending, diametral compression, 

and internally pressurized specimens (the material source was not specified). Bending strength exceeded 

the tensile strength by a factor of 1.4 to 1.7, consistent with a lower Weibull modulus material; increasing 

the size of the specimen decreased strength, consistent with a higher Weibull modulus material. At any 

given strain, the elastic bend stress exceeded the tensile stress by about 35 percent, but the strain to failure 

in bending exceeded that in tension by about 10 percent. The resultant effect was that the bend strength 

was about 50 percent greater than the tensile strength. This is shown in Figure 7. Note the scatter in the 

stress-strain response as indicated by the arrows in the figure.

Brocklehurst (1977) examined tensile and bend strength as a function of the volume of cylindrical 

specimens and the inner beam span of four-point-bend flexural specimens, as shown in Figure 8. Each 

point shown represents the mean of six specimens. Standard deviations are also indicated. Figure 8(a)

shows that, for flexural specimen volumes greater than 1 cm3

, strength decreased with increasing volume,

consistent with a size effect for a Weibull modulus of m = 16. This value of Weibull modulus also was 

measured for 30 four-point flexural specimens with a constant inner span volume of 25 cm3

. Conversely, 

the tensile data showed an increasing strength with volume, although this trend leveled off at volumes 

greater than 8 cm3

. For smaller volumes, both the tensile and flexural specimens indicated decreasing 

strength with decreasing volume. In Figure 8(b) the fracture strength of the tensile specimens is shown to 

decrease with decreasing volume; the relative grain size of the material in comparison to the specimen 

dimensions likely played a role in this behavior. Ho (1979) applied an empirical correction to the Weibull 

distribution to account for grain size relative to specimen size (see Sec. 2.1.5), and this behavior may be 

due to the grain acting as a flaw or microcrack (Tucker et al. (1986)), whereby the grain size becomes an 

increasing larger fraction in comparison to a characteristic dimension of the specimen such as diameter 

(tensile specimen) or thickness (flexural specimen).
NASA/TM—2011-215805 14

Figure 7.—Averaged stress-strain to failure relation of 22 specimens 

in four-point bending and six specimens under uniaxial tensile 

loading of a nuclear-grade graphite. Arrows indicate the standard 

deviation in the data. Reproduced from Brocklehurst and Darby 

(1974). Copyright Elsevier; used with permission.
NASA/TM—2011-215805 15

Figure 8.—Fracture strength of IM1–24 grade graphite (Anglo Great Lakes Corporation) and standard deviations of 

the data. (a) Tensile strength as a function of specimen volume of tensile specimens (circles) and four-point 

flexural specimens. (b) Tensile strength as a function of specimen diameter. Reproduced from Brocklehurst (1977). 

Copyright Taylor & Francis Group; used with permission.

Price (1976) ruptured over 2000 tensile and four-point bend specimens to examine size effect, 

strength anisotropy, and strength variation versus location within the billet on H–451 graphite. As previously mentioned, normal and Weibull distributions were stated to fit the data equally well. The mean

strengths showed systematic dependence on orientation and location. Axial specimens (along the extrusion axes) were stronger than radial specimens (perpendicular to the extrusion axis). Specimens excised 

near the edge of the log (cylinder) were stronger than specimens taken from the center of the log. Strength 

fluctuations also were correlated with local density variations. The variation in strength was lower for 

axial specimens than for radial specimens and was lower for bend specimens than for tensile specimens. 

Flexural specimens averaged between 52 and 55 percent stronger than tensile specimens—consistent with 

predictions of 51 to 64 percent from the Weibull distribution, whereas the strength of small tensile specimens (724 mm3 volumes) was 3 to 8 percent higher than that of larger tensile specimens (9847 mm3

volumes)—which was significantly smaller than the expected 28- to 38-percent difference using the 

Weibull distribution. 

There were 48 data sets generated in Price (1976), with an average of 42 specimens per data set (high 

value of 48 and low value of 24 specimens in the individual data sets). The average Weibull modulus m, 

which measures scatter in strength, was 8.6 and 9.7 for the small and large tensile specimens, respectively. The average Weibull modulus for the flexural specimens was 14.7 when the fracture stresses were 

corrected for nonlinear stress-strain response. Each average value was generated from 16 data sets. The 

Weibull modulus of 15 of 16 paired data sets (consisting of a small tensile, large tensile, and a flexural 

data set in a given billet, location, and orientation) was higher for the flexural specimens than for the 

tensile specimens, likely indicating that the flexural specimens had a different and higher Weibull modulus than the tensile specimens had. It is unknown if any experimental bias existed in the testing. The large 

tensile specimens had a slightly higher average Weibull modulus than the small tensile specimens did, 

where 10 out of 16 times the Weibull modulus of the large tensile specimen was larger than that of the 

small tensile specimen within a paired data set. The difference in the Weibull modulus between the large 

and small tensile specimens was likely not significant. 

Ho (1979) comments that load eccentricity due to improper alignment of the load train may have 

contributed to the higher strength variability (and hence the lower Weibull modulus) of the tensile specimens relative to that of the bending specimens. Ho (1979) also applied an empirical correction to the 
NASA/TM—2011-215805 16

Weibull distribution scale parameter to account for grain size relative to specimen size (see Sec. 2.1.5). It 

is known that critical specimen dimensions, for example diameter in a tensile specimen, negatively affect

strength when the specimen dimension is on the order of the grain size. Ho was able to show good 

correlation to the Weibull distribution size effect when the grain size to specimen size correction factor 

was applied for the tensile specimens, although this also may also have been a case of calibrating a theory 

to data.

Nemeth et al. (2011) reanalyzed the fracture data reported by Price (1976) for H–451 grade graphite, 

which involved over 2000 specimens tested in flexure and tension. Nemeth et al. (2011) used the WeibPar 

program (Connecticut Reserve Technologies, Inc., 2009) to pool (combine) the fracture data into larger 

sets to better identify and confirm trends. They used 90-percent confidence bounds on the Weibull 

parameters to examine if one set of data was statistically similar to another set of data. Figure 9 shows a 

Weibull plot of the pooled data of the large tensile specimens, the smaller tensile specimens, and the fourpoint flexural specimens from Nemeth et al. (2011) for specimens oriented axially (Figure 9(a)) and 

radially (Figure 9(b)) relative to the material extrusion direction. The solid lines represent the predicted 

strength response based on the Weibull distribution. The flexural specimen fracture stresses were not

corrected for the nonlinear stress-strain response of the graphite. Further details and extensive Weibull 

plots of the data are provided in Nemeth et al. (2011). Similar to Price (1976), the results showed a much 

smaller size effect than would have been expected based on the Weibull distribution for the tensile 

specimens. A significant size effect was observed between the tensile and flexural specimens. The 

Weibull modulus m, which measures the scatter in the data, was consistent between the different sizes of 

tensile specimens but was higher for the flexural specimens. The difference in the Weibull moduli of 

the tensile and flexural specimens indicates that additional effects (e.g., see Li and Fok (2009) and

Sec. 2.1.5) or different failure mechanisms were operating. The diameter of the small tensile specimen 

may have played a role in the lack of a significant size effect between the two different tensile specimen 

sizes because the grain size of the material was on the same order as the diameter (see Sec. 2.1.5 for a 

further explanation).

Figure 9.—Weibull plots of large-size tensile, small-size tensile, and four-point flexural specimens of graphite-grade 

H–451. The lines represent a best fit to the pooled data considering the size effect predicted from the Weibull 

distribution. Note that the fracture stresses for the flexural specimens were not corrected for the nonlinear stressstrain response of H–451. (a) Axial orientation of 317 large-size tensile, 346 small-size tensile, and 336 four-point 

flexural specimens. (b) Radial orientation of 311 large-size tensile, 343 small-size tensile, and 346 four-point 

flexural specimens. Reproduced from Nemeth et al. (2011).
NASA/TM—2011-215805 17

Homan and Kasten (1980) used Stackpole 2020 and H–451 grade materials for their size-effect study 

of various sizes of O-ring, C-ring, and four-point-bending specimens. Good correlation was obtained for 

size effects corresponding to Weibull moduli of 16 and 6 for Stackpole 2020 and H–451, respectively.

Homan and Kasten’s study used a “95-percent stressed volume” (the volume under a stress of 95 percent

or more of the highest stress in the specimen) in the effective volume calculations, it reused portions of 

previously fractured specimens, and it corrected for variations in the density of pores in the material. Size 

effect corresponding to high density, mean density, and low density within the billet were systematically 

examined.

In Kennedy (1987), a large study performed on Stackpole 2020, Union Carbide TS1792, and Toyo 

Tanso IG11 graphites examined billet-to-billet, within-billet, and size-effect strength variations. Kennedy 

concluded that the randomness of the mean strength in the billets made it unlikely that test results from 

small sample volumes removed from the billet would be representative of the billet as a whole. Size effect 

on grades TS1792 and 2020 was studied using the methods and specimen types described in Homan and 

Kasten (1980). Size effect consistent with the trends of the Weibull distribution was obtained. For grade 

2020 this was shown to apply for up to 5 orders of magnitude difference in stressed-volume (by including 

tensile specimen results for a single size of specimen). Similar to Homan and Kasten (1980), the sizeeffect relationship for uniform uniaxial tensile specimens was not studied. It should be noted that large 

gross defects were often observed in the various billets. Figure 10 shows an acoustic scan of one end of a 

billet that had high variability. The scan shows flow lines of high-density material and the locations of 

large cracks. Density variations did not necessarily correlate with the loss of mechanical properties.

Figure 10.—Acoustic map of a section of graphite billet showing areas where no 

back-face signal was detected using a 2.25-MHz frequency probe. This 

corresponds to areas of high-density material. Reproduced from Kennedy 

(1987).
NASA/TM—2011-215805 18

Kasten et al. (1989) give further (updated) results for Union Carbide’s TS1792 with regard to size 

effect. Specimen types and testing protocols similar to Homan and Kasten (1980) were used. Size effect 

was examined relative to the location within the billet. Size effect consistent with the trends of the 

Weibull distribution was obtained, but the Weibull slope (the amount of average strength change with 

size) appeared to vary between the edge and center of the billet. Tensile strength also was correlated with 

density, with strength increasing with density. Size-effect testing of Stackpole 2020 over several orders of 

magnitude of stress volume and for various billets showed a consistent Weibull-like size effect between 

billets—where Weibull slope was little changed between billets, although mean fracture strength varied as 

much as 10 percent. Similar to Kennedy (1987), results for one size of tensile specimen were included 

with the other bending specimen results. Tensile strength results for two different specimen sizes 

(6.35 mm and 9.53 mm in diameter) were reported for H–451 and PGX grade materials. Although there 

was not a large volume difference between the two specimen sizes, the smaller diameter specimen had a 

lower average strength than the larger diameter specimen did, contrary to the trend expected from the 

Weibull distribution. These results may not have been statistically significant.

Strizak (1991) studied the effect of volume on the tensile strength of Great Lakes Carbon Corporation 

H–451 (ownership change) and Toyo Tanso Company IG–110. Specimens with gauge diameters of 6.35, 

9.35, 15.88, and 25.4 mm, and gauge volumes of 1407, 3163, 12 577, and 51 482 mm3

, respectively, were 

tested. For both materials, the average strength measured for a given specimen volume increased with 

increasing gauge volume, contrary to the trend expected with the Weibull theory. The increases were 

relatively small and may not have been statistically significant. Strizak also considered an empirical 

conjecture by Ho (1979) that grain size affects strength negatively for specimens with diameters less than 

10 to 15 times the grain size as well as the Burchell model (described in Sec. 2.1.6). Strizak concluded 

that the Weibull model correlated poorly for size effect and that Ho’s modified Weibull model had mixed 

results, especially when it indicated that strength should decrease at the largest specimen sizes, contrary to 

the experimental results. Burchell’s model showed correlation for the subset of data that was analyzed;

however, correlating with the smallest specimen size required adjusting the fracture toughness (measured 

fracture toughness was reported to decrease with decreasing specimen diameter). 

Mitchell et al. (2003) examined two billets of PPEA (UCAR International/GrafTech International) 

extruded medium-grained graphite for billet-to-billet, within billet, strength variation, and strength 

orientation anisotropy. Four-point-bending specimens and L-shaped specimens with various fillet radii 

were tested. Results from four-point testing indicated strength orientation anisotropy was small within the 

experimental margin of error but that strength varied as much as 12 percent within billets and mean 

strength between billets varied 7 percent. The Weibull modulus between individual tests varied between 

8 and 22; however, because the number of specimens for each test was small, this variation was not 

necessarily statistically significant. The Weibull modulus for L-shaped specimens varied between 21 and 

54, with the zero-fillet-radius specimens having the anomalously high Weibull modulus of 54.

The failure mode appeared to be different for the four-point and L-shaped specimens. Failure was 

abrupt and brittle-like for the four-point specimens where a single crack initiated and propagated through 

the specimen. For L-shaped specimens, failure was more gradual, with cracks starting at different locations (along the exposed face of the fillet) that joined to form a single crack that extended through the 

specimen. The time from the visible onset of cracking to when the crack extended across the face of the 

fillet varied according to the particular geometry of fillet radius and/or specimen thickness. Weibull 

predictions using the averaged properties of the four-point bend specimens and finite element analysis 

results from the L-shaped specimens did not correlate very well with the experimental data for the various 

fillet radii. A fracture mechanics analysis of the L-shaped specimen indicated that the stress intensity 

factor decreased with crack length, suggesting a stable crack growth failure mode (increasing load 

required to grow the crack), whereas analysis of the four-point specimen indicated that unstable crack 

growth would be the failure mode. It should be noted that Li and Fok (2009) reanalyzed these data by 

accounting for the nonlinear stress-strain response (see Sec. 2.1.5 for further discussion). 

The authors herein speculate that the high Weibull modulus of 54 (corresponding to a relatively

small coefficient of variation in fracture strength) for the zero-radius fillet of the L-shaped specimen in 
NASA/TM—2011-215805 19

Mitchell et al. (2003) was affected by the stress singularity at the fillet root. For example, in Choi and 

Salem (1992) scatter in strength of four-point-bending specimens decreased substantially when the 

specimens were indented. A consistently sized indentation flaw was introduced in each specimen,

decreasing the average strength and decreasing the coefficient of variation from 13 to 5 percent. This 

lowered the effect of natural flaws on the specimen strength because any natural flaw existing near the 

crack tips of the introduced flaw was substantially shorter than the introduced flaw. The combined effect 

of a relatively short natural flaw near the crack tip of an introduced flaw would make the effective 

(equivalent) length of the introduced flaw only slightly longer; hence, the specimen would have only a 

slightly smaller fracture stress than if the natural flaw had not been in the vicinity of the introduced flaw 

crack tip. The net effect is that, if each specimen had an identical-sized indentation flaw, the fracture 

strength of the specimen would always be the same (deterministic with no scatter in strength), but the 

possibility that a natural flaw could exist near the crack tips of the introduced flaw would add a relatively 

small amount of variation to the observed fracture strength. The stress singularity induced by the zero

radius of the fillet in the L-shaped specimen of Mitchell et al. (2003) had the same effect as introducing a 

large crack in the specimen and thereby decreasing the influence of the natural flaws in the specimen.

From the previous description, it can be seen that nuclear-grade graphite presents a more complicated,

and perhaps somewhat chaotic, fracture behavior than that of classically brittle materials. Traditionally,

the Weibull distribution has been used to describe the probability of failure of brittle materials under 

thermomechanical loading; however, Weibull analysis has not been uniformly successful in predicting 

graphite failure. Generally, it seems from the studies cited in this section that size effect exists for flexural 

specimens and is consistent with expected Weibull behavior. For uniform tensile loading, size effect is 

small or nonexistent—which is inconsistent with Weibull predictions. However, as noted by Ho (1979),

grain-size effects complicate this problem. It is notoriously difficult to obtain uniform stresses through the 

cross section in tensile testing, and this introduces experimental biases. Smaller-than-expected size effect 

has been reported in the literature for other materials, for example by Lu et al. (2005) for zinc oxide

electroceramics (containing significant porosity) and by Beyerlein and Phoenix (1996) for low-fibercount “minicomposites.” 

In the large study done by Price (1976) (see also Nemeth et al. (2011)), the reported Weibull modulus 

for the graphite flexural specimens was consistently higher than that for the tensile specimens. This (if we 

assume that experimental errors were not to blame), along with Figure 7 showing different stress-strain 

responses for bending and tension, raises the possibility that bending involves different failure mechanisms (or modes) than those in uniform uniaxial tension. Li and Fok (2009) explained the difference in 

strength and Weibull modulus between tension and bending load as being due to the strain-softening 

behavior of quasi-brittle materials (also see Sec. 2.1.5). Also, as mentioned before, grain size negatively 

affects the strength of small-size specimens (Ho (1979)). Tucker et al. (1986) indicated that this is due to 

the presence of small-size cracks on the order of the grain size.

The potential for serious experimental errors, such as load eccentricities in tensile specimens, and 

consistency of approach, such as accounting for nonlinear stress-strain response in failure stress calculations makes drawing firm conclusions regarding graphite behavior problematic. An example of the 

difficulty of obtaining consistent estimates of Weibull parameters can be found in Tennery and Ferber 

(1989). In that international study, 21 laboratories measured four-point-bending fracture strengths and 

estimated Weibull parameters for two grades of silicon nitride and one grade of silicon carbide. A total of 

2597 specimens were fractured, and individual data sets for each material from each laboratory ranged 

from 60 to 80 specimens. All specimens came from a single billet of each material, and Oak Ridge 

National Laboratory randomized (for each material) and distributed the specimens. Study participants 

used the same specimen and fixture dimensions and the same stressing rate. 

Despite the large size of the data sets (60 to 80 specimens per data set), estimated Weibull parameters 

(using maximum likelihood analysis) showed scatter in values beyond what was expected from natural 

statistical variation. The Weibull modulus m for the SiC material varied from 4.9 to 8.3 for a range of 

69 percent (based on the lowest value) among the 12 participating laboratories. For the ASEA-grade

(ASEA Cerama AB) silicon nitride, m varied from 12.3 to 18.2, for a range of 48 percent, from the 
NASA/TM—2011-215805 20

13 laboratories. For the GTE Products Corporation (Wesgo Division) silicon nitride, m varied from 9.7 to 

13.5, for a range of 39 percent from the 13 laboratories. The estimated characteristic strengths σθ showed 

less variation (as would be expected based on statistical uncertainty arguments). For the silicon carbide,

σθ varied from 470 to 557 MPa (an 18-percent range); for the ASEA silicon nitride, σθ varied from 624 to 

720 MPa (a 15-percent range); and for the GTE Wesgo silicon nitride, σθ varied from 611 to 714 MPa 

(a 17-percent range). The scatter in these values was beyond the ranges expected from natural statistical 

variation at the 95-percent confidence level.

Note that a discussion of expected confidence bounds on estimated parameters from natural statistical 

variation is beyond the scope of this report; however, these bounds are a function of sample size—

meaning that, for smaller size data sets (which typically use about 30 specimens), the spread in these 

bounds would be even larger. The source of these unexpectedly large differences could not be identified. 

So despite the great care taken in this round-robin exercise and the large number of specimens fractured 

in each data set, large variation in estimated Weibull parameters—larger than would be expected based on 

statistics alone—still occurred. 

Another interesting result of this study was that statistical tests performed to see if the data best fit the 

Weibull and other distributions indicated that generally the Weibull distribution was better except for the 

silicon carbide specimens, where Weibull or gaussian distributions fit the data equally well. This contrasts

with Price (1976), where a best-fit distribution could not be identified with certainty. The Nemeth et al. 

(2011) reanalysis of Price (1976) was generally more favorable regarding the Weibull distribution. The 

specimens that were oriented radially to the material extrusion direction tended to better fit a twoparameter Weibull distribution, whereas the specimens oriented axially to the extrusion direction better fit

to a three-parameter Weibull distribution or possibly some other distribution type (see Figure 9).

In the end, what can be concluded with certainty regarding graphite fracture behavior is that further 

experimental testing with the latest testing techniques using properly designed specimens and test 

matrices as well as reanalysis of historical data of good pedigree is needed to, hopefully, help clarify 

some of the apparent trends reported in the literature. In addition, identifying a best-fit distribution is 

problematic even for large-size data sets, and alternative tests, such as size-effect studies, are needed to 

help identify an appropriate distribution.

2.0 Statistical Models of Fracture

The question arises: Are design methodologies developed to describe classically brittle material 

failure suitable for graphite, or are alternative approaches needed? In the following sections, modeling 

approaches and simulation studies of brittle-constituent-controlled material failure are discussed to help 

clarify and (eventually) develop improved models describing the stochastic rupture of graphite. This 

report adds further background and updated information to reviews of graphite failure criteria by Tucker 

et al. (1986) and Tucker and McLachlan (1993).

The stochastic nature of fracture in engineered materials can be simulated by two distinct models:

series systems and parallel systems. Series systems model the abrupt failure of brittle materials, such as 

glasses and ceramics, whereas parallel systems model the more gradual or graceful failure of fiberreinforced composite materials. Series systems assume that material is composed of a set of n links 

connected in series such that the structure fails whenever any of the links fail. An electrical circuit 

analogy would be having n number of resistors or fuses connected in series so that, when a single fuse or 

resistor fails, electrical current can no longer flow. In the parallel system model, the n links are arranged 

in parallel. When one link fails, load is redistributed to the remaining n –1 links. The remaining n –1 links 

carry higher load, but the system (structure) may still survive. The structure does not fail until all n links 

fail. The electrical analog is a system of n resistors or fuses arranged in parallel. When electrical current 

flows such that one resistor or fuse element fails, the current is redistributed to the surviving elements.

Only when all the resistors or fuses in the circuit fail will the flow of current cease. The electrical circuit 

analogy is useful for random fuse lattice models as described in Section 2.3.
NASA/TM—2011-215805 21

Predictions from series system models tend to be more conservative than those from parallel system 

models. Series system models relevant to this report include the Weibull, Batdorf, and Burchell models. 

Relevant parallel system models include the uniform defect model, composite material models, and lattice 

system simulations. Overviews of these models are given in subsequent sections.

2.1 Series Systems—The Weakest Link

Series systems use the weakest-link theory (WLT) as originally proposed by Peirce (1926) to model 

failure in cotton yarns. The WLT model assumes that the structure is analogous to a chain with n links. 

Each link may have a different limiting strength. When a load is applied to the structure such that the 

weakest link fails, the structure fails. Peirce (1926) was the first to formulate the WLT model for fiber 

strength and was also the first to recognize the close relationship of this model to the asymptotic theory of 

extreme values in large samples of a statistical population. 

Consider a chain containing many links, and assume that failure is due to any number of independent 

and mutually exclusive mechanisms. Each link involves an infinitesimal probability of failure Pf. 

Discretize the component into n incremental links. The probability of survival of the i

th link (Ps)i is related 

to the probability of failure of the i

th link (Pf)i: (Ps)i = [1 – (Pf)i]. The resultant probability of survival of 

the whole structure is the product of the individual probabilities of survival: 

( ) ( ) ( ) ( ) 1 1 1 1

Π Π 1 Π exp exp 

nn n n

ss f f f i ii i i i i i

PP P P P = = = =

 

= =− ≅ − = −             ∑ (1)

The approximation arises from truncation of the first two terms of a Taylor series expansion of the 

exponential function e

x

. Equation (1) describes a series system where failure of any one element means 

failure of the whole system. This also leads to the prediction of a size effect. When more links are added 

to the chain, the probability of failure increases for a given load. The system is weaker because of the 

increased probability of having a weaker link present. For the same probability of failure to be maintained, the load would have to be decreased. The prediction of size effect is an important consideration 

when trying to determine an appropriate probabilistic distribution for modeling a material.

2.1.1 Extreme-Value and Weibull Distributions

Classically brittle materials such as ceramics and glasses have low fracture toughness and do not 

undergo significant plastic deformation prior to failure. Instead, when a load is applied, large stress 

concentrations occur at microscopic flaws. These flaws are unavoidable; they are the results of the 

manufacturing process or environmental factors. The microflaws or microcracks are numerous, are 

randomly located and oriented, and display a distribution of sizes. When a load acting on an individual 

microcrack reaches a critical level, catastrophic crack propagation ensues, causing the structure to fail. 

Therefore in a brittle material, fracture initiates from the combination of a tensile load and a weak flaw 

being unfavorably located and oriented. For a complex component, failure might not be initiated at the 

point of highest nominal stress. A particularly severe flaw may be located at a region of relatively low 

stress yet still be the cause of component failure. For this reason, the strength of a brittle material is 

statistical, and the entire field solution of the stresses should be considered when determining the potential 

for failure. From a statistical point of view, this problem leads to the distribution of the smallest value or 

largest value in large samples. Such distributions form the family of extreme-value distributions. 

Extreme-value distributions are the limiting distributions for the minimum or the maximum of a very 

large collection of random observations from the same arbitrary distribution. The Weibull distribution

(1951) is one of three extreme-value distributions discovered by Fisher and Tippet (1928) (the other two 

are known as the Frechet and Fisher-Tippett-Gumbel distributions). 

• Fisher-Tippett-Gumbel, or type I, extreme-value distribution—
NASA/TM—2011-215805 22

( ) exp exp u

o

x x PX x

x

    − ≤= −        (2)

• Frechet, or type II, extreme-value distribution—

( ) exp

m

u u

o

x x PX x x x

x

−       − ≤= −    ≥

     

(3)

• Reversed Weibull, or type III, extreme-value distribution—

( ) ( ) exp

m

u u

o

x x

PX x x x

x

    − − 

≤= −    ≤

     

(4)

where P denotes probability, with parameters xu > 0, xo > 0, and m > 0. Equations (2) to (4) show the 

extreme-value distributions for maxima. By substituting (–x) for x in the distribution function, one obtains

the extreme-value distributions for minima (Wikipedia (2010) and Wapedia (2009)). Of these three distributions, Bazant and Planas (1998) and Bazant and Pang (2007) argue that only the Weibull distribution 

has a sound physical and conceptual basis for modeling structural strength, but Phoenix and Beyerlein 

(2000a) argue that the Weibull and Fisher-Tippett-Gumbel (henceforth, Gumbel) distributions are 

reasonable for modeling tensile strength. The Weibull distribution is obtained if the defect size distribution is described by a power law, and the Gumbel distribution is obtained if the defect size is exponenttially distributed. Of further note, Bazant and Pang (2007) argue that neither gaussian nor lognormal 

distributions are appropriate for brittle structure strength statistics.

Weibull (1939a,b) arrived at the distribution function associated with his name by a heuristic 

argument unrelated to the asymptotic theory. He applied the WLT concept to a solid volume of a brittle 

material rather than to a fiber as was done by Peirce (1926). The Weibull equation assumes that 

catastrophic crack propagation initiates from a critically loaded flaw.

A flaw or microcrack of a specified size embedded in a material will require a critical level of applied 

stress before the crack will begin to extend. That critical level of stress is the critical strength of the given 

flaw. When there is a distribution of flaws of different sizes in a given volume, the flaws that are equal to 

or larger than a specified size (with an associated critical strength of σ) will fail at a strength σ or less. 

Weibull assumed the existence of a function ηV(σ), referred to as the crack-density function, representing 

the number of flaws (or microcracks) per unit volume having a strength equal to or less than σ. Use of V

as a subscript denotes a quantity that is a function of volume. Under a local tensile stress σi, the probability of failure of the i

th link, representing the incremental volume ∆Vi, is (PfV )i = [ηV(σi) ∆Vi], where the 

incremental volume ∆Vi is arbitrarily small such that the value of the expression within the brackets is 

much less than one (and therefore represents the probability that a flaw of strength of σi or less exists 

within ∆Vi). If a uniform tensile stress σ is applied, such that σ = σi for all incremental volumes ∆Vi, then 

from Equation (1) the resultant probability of survival for material volume V, where V is the sum of all 

∆Vi, is 

where V is the total volume. 

Note that the expression within the brackets can also be described as a function of the surface area of 

the component for flaws that reside on the material surface (e.g., flaws from machining damage). This 

subject is addressed further in Nemeth et al. (2003, 2005). For the sake of brevity, these relationships are 

not shown in this report. Equation (5) also can be obtained from the Poisson probability distribution 

sV fV 1 exp η σ V PP V       (5) 
NASA/TM—2011-215805 23

function, as shown in Section 2.1.7. If the magnitude of the stress is a function of the location within the 

component, then, again using Equation (1), the overall component failure probability becomes

This accounts for stress gradients in the component such as those experienced from flexure. 

Weibull introduced a three-parameter power function for the crack-density function ηV(σ),

( ) 1 m m V V

uV uV

V Vo o oV

   σ−σ σ−σ

η σ=    =

   σ σ (7)

where Vo represents a characteristic volume, which by convention is assumed to be a unit volume; σuV is 

the threshold stress parameter, which is the value of the applied stress below which the failure probability 

is zero. Note that from Equations (5) and (7), for a unit volume when (σ − σuV) = σoV, a value of 0.6321 is 

obtained for PfV. When σuV is zero, the two-parameter Weibull model is obtained. The two-parameter 

Weibull distribution is widely preferred because of its simplicity (one less parameter to deal with) and its 

conservatism. With the two-parameter model, the scale parameter σo corresponds to the stress level where 

63.21 percent of tensile specimens with unit volumes would fracture. The scale parameter σoV has dimensions of stress × (volume)1/ V, m where mV is the shape parameter (Weibull modulus)—a dimensionless 

parameter that measures the degree of strength variability. As mV increases, the dispersion decreases. 

These three statistical parameters are considered to be material properties, and they are temperature and 

processing dependent. 

Note that internal residual stresses and environment can affect the observed Weibull parameters

obtained from experimental data. This includes the case of environmental effects from time-dependent 

subcritical crack growth. For example, Choi and Gyekenyesi (1999) showed that ultra-fast loading rates 

of 3.3×104 MPa/s were required to eliminate the effect of slow (subcritical) crack growth from strength 

measurements so that inert strengths of various ceramics could be determined at elevated temperatures. 

The Weibull parameters in Equation (7) represent inert (fast-fracture) strength parameters. The effect of 

time- or cycle-dependent subcritical crack growth should be modeled separately, as was done in 

Nemeth et al. (1993). The combination of Equations (6) and (7) define what is known as the Weibull 

stress-volume integral.

When Equations (6) and (7) are combined with a zero threshold stress, the two-parameter Weibull 

equation becomes

(, ,) 1 exp d 

mV

fV V oV

xyz P V

    σ =− −      σ   ∫ (8)

where x, y, z is the point location in the component. This equation can be reexpressed as

( , , ) 1 exp d 

mV mV

f

fV V oV f

xyz P V

      σ     σ   =− −          σ σ        

∫ (9)

where σf is the maximum stress in the component and the term in braces { } is the effective volume Ve:

 1 exp η   σ d fV V VP V         (6) 
NASA/TM—2011-215805 24

( , , ) d

mV

e V f

xyz V V

  σ =  

σ   ∫ (10)

For two components of different sizes, but otherwise identical (except for the magnitude of loading), 

with identical stress distributions and identical survival probabilities, equating terms yields

1 2

1 2 1 exp 1 exp 

m m V V

f f

fV e e

oV oV

PV V

     σ σ  =− − =− −          σ σ    

(11)

which simplifies to

1/

1

2 1

2

 

mV

e f f

e

V

V

 

σ =σ     (12)

where subscripts 1 and 2 denote components with different sizes.

Equation (12) is the classic Weibull size-effect equation, which predicts that the strength of a component is a function of its size. As size increases, strength is predicted to decrease as a function of the 

Weibull modulus mV. This is an important relationship because it provides a testable basis from which to 

infer the applicability of a probabilistic distribution for a particular material. As mentioned in Section 1.4, 

previous size effect studies of graphite have had mixed success correlating to the Weibull distribution. 

Size effect can have important implications in the design of safe graphite material. The graphite bricks 

used in a nuclear reactor tend to be quite large in relation to the specimens used to measure strength 

properties. For example, if specimens are 100 times smaller in effective volume than the component, then 

for Weibull moduli of 5, 10, and 15, the component strength would decrease by 0.40, 0.63, and, 0.74, 

respectively. If specimens are 1000 times smaller, these numbers decrease to 0.25, 0.50, and 0.63, 

respectively.

2.1.2 The Batdorf Model and Response to Multiaxial Stresses

2.1.2.1 Batdorf Multiaxial Strength Theory

The previous section described the Weibull distribution for predicting the likelihood that a component 

will fail from an applied uniaxial stress. This section describes the Batdorf theory for predicting brittle 

material failure from applied multiaxial stress states. The Batdorf methodology is typically used in conjunction with an underlying Weibull distribution. Similar multiaxial statistical failure theories such as the 

Evans elemental strength approach (Evans (1978)) and Matsuo (1981) have also been developed. The 

Evans approach (and likely Matsuo) has been shown to be equivalent to the Batdorf theory (Chao and 

Shetty (1990)), so these approaches are not described herein. The Batdorf theory is highlighted in this 

section because it successfully describes graphite’s multiaxial strength and provides a plausible physical 

mechanism to explain this response.

Other deterministic global multiaxial failure criterion traditionally used for metals have been 

compared with graphite multiaxial rupture data (e.g., see Jortner (1972), Tang (1979), Taylor et al. 

(1967), Broutman et al. (1970), Babcock et al. (1972), Yahr et al. (1993), and Schmidt (2000)). Jayatilaka 

(1979) provides a good overview of these failure criterion as well as the fracture mechanics aspects of 

mixed-mode loading on flaws. In their general reference book, Munz and Fett (1999) provide an excellent 

treatment of global multiaxiality criteria for various defect geometries (cylindrical pore, spherical pore, 

ellipsoidal pore, and circular cracks. These failure criteria also consider the effect of compressive stresses.

The Weibull equation assumes that catastrophic crack propagation initiates from a critically loaded 

flaw; however, it does not describe the physical mechanism behind this phenomenon. Batdorf and Crose

(1974) and Batdorf and Heinisch (1978) addressed this shortcoming by combining WLT with linear 
NASA/TM—2011-215805 25

elastic fracture mechanics (LEFM). Batdorf developed his theory primarily to account for the effect of 

multiaxial stress states for aerospace-grade graphite, which is a finer grained material. (Note that nucleargrade graphite ranges from fine-grained to coarse-grained material.) Batdorf provided an improved 

physical basis for failure by incorporating an assumed crack geometry, mixed-mode fracture criterion,

and a crack-orientation function. In addition to the previously cited Batdorf references, Nemeth et al. 

(2003, 2005) provide a full treatment of the Batdorf theory and extend the theory for time- and cycledependent subcritical crack growth and for the effect of transient stresses. The Batdorf theory has been 

developed for volume-residing flaws (failure probability as a function of material volume) as well as for 

surface-residing flaws (failure probability as a function of material surface area)—again, these are

detailed in Nemeth et al. (2003, 2005). A brief summary of the development for volume-residing flaws

follows.

In the Batdorf theory, the incremental failure probability at a given location in the component can be 

described as the product of two probabilities:

1 2 ∆ =∆ ⋅ P PP fV V V (13)

where ∆P1V is the probability of the existence in ∆V of a crack having a critical strength between σIc and 

σIc + ∆σIc. Critical strength σIc is defined as the remote, uniaxial fracture strength of a given crack in 

mode I loading. The second probability, P2V, is the probability that a crack of critical strength σIc will be 

oriented in a direction such that an effective stress σIeq (which is a function of the fracture criterion, stress 

state, and crack configuration) satisfies the condition σIeq ≥ σIc. The effective stress σIeq is defined as the 

equivalent mode I stress that a flaw would experience when subjected to a multiaxial stress state that 

results in mode I, II, and III crack-surface displacements.

The strength of a component containing a flaw population is related to the critical flaw size, which is 

implicitly used in Batdorf’s theory. Batdorf and Crose (1974) describe ∆P1V as

( I )

1 I

I

d

 d

d

V c V c

c

P V η σ ∆ =∆ σ

σ

(14)

and P2V is expressed as

( I )

2

, 4

c P V

Ω ∑σ = π

(15)

where ηV(σIc) is the Batdorf crack-density function and Ω(Σ, σIc) is the area of the solid angle projected 

onto the unit radius sphere in principal stress space (see Figure 11) containing all the crack orientations 

for which σIeq ≥ σIc for the applied far-field multiaxial stress state Σ. The infinitesimal area, dA, on the 

unit sphere represents a particular flaw orientation (a direction normal to the flaw plane), and σIeq is an 

equivalent stress, which is a function of an assumed crack shape and multiaxial fracture criterion. The 

constant 4π is the surface area of a unit radius sphere and corresponds to a solid angle containing all 

possible flaw orientations.

The component probability of failure is

I ,max ( ) ( I )

I 0 I

, d 1 exp d d

4 d

eq Ic v c

fV c V c

P V

     Ω Σσ η σ  σ

=− −    σ

    π σ   ∫ ∫ (16)
NASA/TM—2011-215805 26

Figure 11.—Unit radius sphere representing all possible flaw 

orientations, where σ1, σ2, and σ3 are the principal stresses 

and α and β are angular coordinates. An infinitesimal area, 

dA, on the unit sphere represents a particular flaw 

orientation (a direction normal to the flaw plane), and σIeq is 

an equivalent stress that is a function of an assumed crack 

shape and a multiaxial fracture criterion.

where σIeq,max is the maximum effective stress that a randomly oriented flaw could experience from the 

given stress state Σ. The crack-density function ηV(σIc) is independent of stress state and has been approximated by a power function (Batdorf and Heinisch (1978)). This leads to the Batdorf crack-density 

function of the form

( ) ( ) I

I

,,, , V

V

m BV c V c m

oV

k xyz σ αβ

ησ =

σ

(17)

where x, y, z corresponds to the location and α and β are orientation angles. The normalized Batdorf 

crack-density coefficient kBV is a function of the mixed-mode fracture criterion chosen. Equation (17) is 

similar to the Weibull power-law crack-density function of Equation (7) with a zero threshold stress. The 

exponent mV is equivalent to the Weibull exponent in Equation (7). The value of the Batdorf crack-density

coefficient kBV normalizes Equation (16) to the Weibull equation (Eq. (8)) for a uniaxial stress state. It 

also means that Equation (16) takes on the characteristics of a Weibull distribution. The normalized 

Batdorf crack-density coefficient kBV and the Weibull modulus mV are evaluated from experimental inert 

strength fracture data. 

Although the Weibull (Eq. (7) with zero threshold stress) and Batdorf (Eq. (17)) crack-density 

functions are similar in form, they are not the same. The Weibull function simply depends on the applied 

uniaxial stress distribution σ and is the only term other than the volume necessary to calculate PfV. The 

Batdorf function depends on the mode I strength of the crack σIc, which has an associated probabilistic 

distribution and must be integrated over a range of values for a given stress state. Furthermore, to obtain 

PfV, a crack-orientation function, P2V, must be considered in addition to the density function and the

volume. Finally, the normalized Batdorf coefficient kBV cannot be calculated from inert strength data 

until a fracture criterion and crack shape are chosen. To determine a component probability of failure, one 

must evaluate P2V (Eq. (15)) for each elemental volume ∆Vi, within which a uniform multiaxial stress 
NASA/TM—2011-215805 27

state σ is assumed. The solid angle Ω(Σ, σIc) depends on the selected fracture criterion, the crack configuration, and the applied stress state. For multiaxial stress states, with few exceptions, Ω(Σ, σIc) must be 

determined numerically. For a sphere of unit radius (Figure 11), an elemental surface area of the sphere is 

dA = sin α dβ dα. If we project onto the spherical surface the equivalent (effective) stress σIeq(Σ,α, β), the 

solid angle Ω(Σ, σIc) will be the area of the sphere containing all the projected equivalent stresses satisfying σIeq ≥ σIc. Because of symmetry of stresses, this integration need only be performed over one-eighth

of the unit sphere; therefore,

( ) ( ) 2 2

I I I 0 0 , c H eq c , sin d d π π

Ω Σσ = σ σ ααβ ∫ ∫ (18)

where

( )

( )

II I I

II I I

, 1 

, 0 

eq c eq c

eq c eq c

H

H

σ σ = σ ≥σ

σ σ = σ <σ

Substituting into Equation (16) and integrating with respect to σIc changes the component failure 

probability to Batdorf (1978a,b) and Nemeth et al. (2003, 2005):

( ) 2 2

I 0 0

2

fV 1 exp eq sin d d d V V

P V   π π

=− − η σ ααβ     π ∫∫ ∫ (19)

where σIeq replaces σIc in Equations (17) and (19). For a given incremental volume, σIeq(x, y, z, α, β) is the 

projected equivalent stress over the unit radius sphere in principal stress space as depicted in Figure 11. 

Using the power-law crack-density function ηV of Equation (17) will characterize the Batdorf model in 

Equation (19) as a form of the Weibull distribution. Equation (19) circumvents the more involved 

numerical integration of Ω (Σ, σIc).

Batdorf (1975, 1978a) gives an example of how the form of the crack-density function is related to 

the distribution of microcrack sizes and how this affects the statistics of fracture. Later She and Landes 

(1993) investigated the Batdorf relation for a normal distribution of crack sizes. Using the modeling 

framework of Equation (16), Batdorf (1975, 1978b) derived fracture statistics for intergranular cracks. 

This was based on work done by McClintock (1973). McClintock’s model for crack-size distribution 

produced a probability-of-failure distribution that was not of the asymptotic forms of extreme-value

theory. An interesting feature of this model is that scatter in strength, and therefore by analogy the 

Weibull modulus, is not constant as a function of volume.

McClintock’s (limited) two-dimensional model assumes that each grain boundary has a small, but 

nonzero, probability q of being unbonded. Therefore, the probability that two connected grain boundaries 

are both unbonded and form a contiguous crack is q

2

, the probability that three connected grain boundaries are unbonded is q

3

, and so on. He showed that the probability of a crack (composed of connected 

unbonded grain boundaries) being longer than length a is exp(−a/aμ), where aμ is the average crack 

length. From this, Batdorf (1975) derived a crack-density function of ( ) ( ) 2 I I Vc o exp 1 , N c η σ = −σ where 

No is the number of cracks per unit volume. Applying McClintock’s conjecture for a penny-shaped crack

gives a probability of a crack being larger than radius a of exp{−ln(q−1

) π a

2

/Ao}, where Ao is the area of a 

grain. The associated crack-density function is ( ) ( ) 4 I I Vc o exp 1 . N c η σ = −σ Both crack-density functions 

have a similar form. Using these functions with Equation (16) for a uniform, uniaxial applied stress

resulted in the relative dispersion of strength decreasing as volume decreased. Fitting a three-parameter 

Weibull distribution to the predicted failure probabilities from the intergranular crack model yielded a 
NASA/TM—2011-215805 28

surprisingly good fit over a large range of volume. In the case of the penny-shaped crack, the Weibull 

modulus as a function of volume was approximated by mV = 1.7 log10(NoV). For example, when the 

volume increased by a factor of 20, the Weibull modulus would increase by 2.2. Therefore, this theory 

indicates that Weibull modulus was not constant with volume. The degree of size effect likely also 

decreased with volume, indicating a possible relationship of low strength scatter with small-size effect 

(which was visually inferred from the figures presented in the Batdorf (1975) reference).

The Weibull modulus is notoriously difficult to measure consistently (e.g., see Tennery and Ferber 

(1989)), and the ceramics community usually does not assume that the Weibull modulus varies with 

volume because of the lack of consistent evidence. However, Batdorf’s treatment of the McClintock 

model is an interesting exercise and shows that the Weibull modulus is not necessarily a true material 

constant. In the ceramics design community, the power-law crack-density function with the Weibull 

distribution is almost universally used (with no known exceptions to the best knowledge of the authors). 

The Batdorf model is usually used or described with a power-law crack-density function since it then 

becomes a form of the Weibull distribution (with all the inherent advantages of the extensive knowledge 

base and techniques that have been developed) and it has been incorporated in reliability analysis 

software such as Ceramics Analysis and Reliability Evaluation of Structures/Life (CARES/Life)

(Nemeth et al. (2003, 2005)). 

2.1.2.2 Multiaxial Strength Response of the Batdorf Model

All results shown subsequently in this section use the power-law crack-density function of 

Equation (17), which characterizes the Batdorf model as a form of the Weibull distribution. The effective stress σIeq represents an equivalent normal stress on the crack face from the combined action of the 

normal stress σn and the shear stress τ on the crack face. The microcrack orientation is defined by the 

angular coordinates of α and β, where the direction normal to the plane of the microcrack is specified by 

the radial line defined by α and β (see Figure 11). For the sake of brevity, the development of the effective stress equations is not shown (for details, see Nemeth et al. (2003, 2005)). For a penny-shaped crack 

with the Shetty mixed-mode fracture criterion (Shetty (1987)), the effective stress becomes

( )

2

2 I

1 4 2 2 eq n n C

      τ

σ = σ+ σ+           − ν  

(20)

where ν is Poisson’s ratio and C is the Shetty shear-sensitivity coefficient, with values typically in the range

0.80 ≤ C ≤ 2.0. As C increases, the response becomes progressively more shear insensitive. Shear increases

the equivalent stress as shown in Equation (20), and this has a deleterious effect on the predicted material 

strength. For a penny-shaped crack with a material having a Poisson’s ratio of about 0.22 and C = 0.80, 0.85, 

1.05, and 1.10, Equation (20) models, respectively, the following criteria: Ichikawa’s maximum energyrelease-rate approximation (Ichikawa (1991)), the maximum tangential stress (Erdogan and Sih (1963)), 

Hellen and Blackburn’s maximum strain-energy-release-rate formulation (Hellen and Blackburn (1975)), and 

colinear crack extension. The value of C also can fit empirically to experimental data—either on introduced 

cracks (as was done in Shetty (1987)) or on specimens being tested multiaxially.

The following example illustrates the multiaxial response of nuclear-grade graphite. Sookdeo et al. 

(2008) performed CARES/Life reliability analysis using the Batdorf method on graphite multiaxial tube 

specimens previously tested at Oak Ridge National Laboratory. The tubes consisted of near-isotropic 

grade IG−110 graphite material (Toyo Tanso, Inc., reference) manufactured in a tubular configuration 

whereby combined axial force and internal pressurization could be imposed. In this way, the multiaxial 

strength response was measured in the tension-tension and tension-compression quadrant of stress-space. 

CARES/Life (Nemeth et al. (1993, 2003, 2005)) in conjunction with the ANSYS finite element analysis 

program was used to predict the failure response of the specimens for the Principle of Independent Action 
NASA/TM—2011-215805 29

(PIA)

1 and Batdorf multiaxial failure criterion. The PIA criterion (Barnett, et al. (1967) and Freudenthal

(1968)) assumes that the principal stresses (σ1, σ2, and σ3) act independently, where the product of the 

reliability for each principal stress component, calculated using the Weibull equation, determines the 

reliability of the component. The PIA criterion is expressed as

( 1 23 ) 1 1 exp d VVV

V

mmm fV m V

oV

P V

 

= − − σ +σ +σ  

σ   ∫ (21)

where the effect of compressive principal stresses is ignored in the equation.

For the Batdorf criterion, a penny-shaped crack was chosen with Shetty shear-sensitivity coefficients C

of 0.82 and 1.20. The results are shown in Figure 12 at 10-, 50-, and 90-percent probability of failure levels. 

The figures clearly show that, although the PIA and Batdorf theories yield similar results in the tensiontension quadrant, the results diverge in the tension-compression quadrant. These figures show that the PIA 

criterion is nonconservative when compressive stresses are present, whereas the Batdorf criterion can be 

tuned for shear sensitivity such that good correlation to the data can be achieved. Figure 12(b) shows the 

results for C = 0.82, which is the case where the flaws are highly shear sensitive. Conversely, Figure 12(a), 

which shows the PIA criterion, is essentially identical to the graph of the Batdorf criterion with shear 

insensitive flaws (C is a large value such that the shear stress contribution is negligible in Eq. (20)). 

Figure 12(c) shows C = 1.20, which correlates well with the data. Uchimura et al. (1992) also showed good 

correlation with silicon nitride by using a Batdorf-style fracture-mechanics-based fracture criteria.

Graphite bricks are stacked in a reactor core, and compressive stress can be significant at the bottom of 

the stack. Therefore, the effect of compressive stresses should also be considered in a statistical failure 

criterion. Alpa (1984) extended the Batdorf theory to consider compressive stresses. In Alpa’s model,

compressive stress on the crack face results in friction and interlocking such that the effect of shear stresses 

on the crack face is effectively reduced. In Alpa’s model, compressive failure stems from the same flaw 

population as tensile failure, and it uses the same crack-density function. Hence in Alpa’s model, failure is 

WLT controlled and has the same Weibull modulus and, therefore, the same size effect as the tensile failure 

mode. Nemeth (1989) considered compressive failure as a separate and independent failure mode from 

tensile failure (also see Appendix A for an alternative failure criterion mechanism using the uniform defect 

model described in Sec. 2.1.7). The reliability (survival probability) of the multiaxial stress state was the 

product of the reliability of the compressive failure mode and the tensile failure mode.

PP P ss s = ⋅ ,tensile ,compressive (22)

where Ps is the survival probability. Nemeth used Batdorf’s methodology of random flaw orientation (the 

unit sphere of Figure 11) and the critical stress criterion to construct a simple effective stress relationship:

σ =τ Ieq 2 (23)

of which the value of 2 conveniently made the maximum value of effective stress σIeq in uniaxial 

compression equal to the value of σIeq in uniaxial tension on the unit sphere. This allowed the failure 

criterion to be easily normalized to the strength of uniaxial compression experimental rupture data. 

The compressive multiaxial failure response was assumed to be only a function of shear stress on the 

crack plane. This failure criterion also assumed the Weibull WLT mechanism; however, it allowed the 

Weibull modulus in compression to be independent of the Weibull modulus in tension. As indicated in 

Section 2.2.7 and in Bazant and Pang (2007), the Weibull modulus may be a function of mixed seriesparallel system modeling, and thus compressive failure is possibly still amenable to Weibull distribution 

 1

A commonly used Weibull-based multiaxial failure criterion based on principal stresses acting independently. A 

description is provided in Nemeth et al. (2003, 2005).
NASA/TM—2011-215805 30

Figure 12.—Multiaxial failure probability predictions at 10-, 50-, and 90-percent levels for IG−110 grade graphite. 

(a) Principle of Independent Action (PIA) theory and Batdorf theory for shear-insensitive cracks (lines are 

coincident). (b) Batdorf theory for penny-shaped cracks with a Shetty shear-sensitivity coefficient of C = 0.82. 

(c) Batdorf theory for penny-shaped cracks with a Shetty shear-sensitivity coefficient of C = 1.20.

modeling. A size effect is still predicted in compression, but it is a function of the Weibull modulus for 

compression. Figure 13 shows an example of a failure envelope generated with a tensile and compressive 

failure criterion. In this example, the Weibull modulus for both tension and compression is assumed to be 

10 and the material is assumed to be 5 times stronger in uniaxial compression than in uniaxial tension.
NASA/TM—2011-215805 31

Figure 13.—Example of a failure envelope where compressive failure is 

treated as a separate and independent failure mode from the tensile 

failure mode. The overall reliability is a product of the tensile mode and 

compressive mode reliabilities. The compressive mode is a function of 

mode II (shear) stress only on the flaw. In this example, the tensile and 

compressive mode have the same Weibull modulus, but in the methodology they can be independent. The dotted lines denote the transition 

between the tensile-dominated and compression-dominated failure; σx

and σy are the orthogonally applied uniaxial stresses.

2.1.3 Strength Anisotropy and Multiaxial Stresses

Graphite strength is mildly anisotropic relative to its extrusion axis. Extrusion and molding preferentially orient the graphite grains such that, for a graphite (cylinder-shaped) log, the strength response is 

transversely isotropic, with the strength stronger in the axial direction than in the radial direction 

(typically between 10 and 25 percent). Modern grades of graphite tend to be more isotropic in strength 

response (10 percent or better), and therefore, strength anisotropy becomes a less critical design issue. 

Nevertheless, strength anisotropy is a physical phenomenon that should not be ignored in a general failure 

theory for graphite. A brief discussion follows regarding work done using the Weibull distribution to 

model strength anisotropy for multiaxial stress states. 

Margetson (1976) used a PIA approach to model strength anisotropy, which was validated against 

graphite and against rocket motor components in Cooper (1988). In Margetson’s approach, the Weibull 

parameters for the uniaxial strength relative to three material axes normalize the model. A strength ellipsoid is constructed from the three material axes, and a transformation law converts the scale parameters 

for the three material axes into three equivalent scale parameters—σo,1, σo,2, and σo,3—corresponding to 

the principal stress directions from the applied multiaxial stress (see Figure 14). A PIA-type criterion is 

then constructed:

1 2 3

,1 ,2 ,3

 1 exp d d d

mmm VVV

fV VV V oo o

P VVV

          σ σ σ   =− −      + +

σσ σ          

∫∫∫ (24)

where the effect of compressive principal stresses is ignored in the equation.
NASA/TM—2011-215805 32

Figure 14.—Margetson strength ellipsoid. Weibull scale 

parameters (σο,x, σο,y, and σο,z) associated with the material 

axes are transformed into Weibull scale parameters (σο,1, σο,2, 

and σο,3) relative to the three principal stress axes for an 

applied multiaxial stress state. The parameter σo,i arbitrarily 

denotes one of the transformed Weibull scale parameters for 

a given principal stress direction denoted with direction 

cosines 1, 2, and 3.

Batdorf (1973) extended the Batdorf and Crose (1974) unit sphere approach for anisotropic multiaxial 

strength (of graphite) by assuming that flaws were randomly oriented but that critical strength σIc was 

directionally dependent relative to a material coordinate system. Rather than the unit radius sphere 

(Figure 11), a critical strength ellipse was developed where σIc became a function of angles α and β, and 

P2V in Equation (14) involved the integration of a normal stress (mode I stress on a flaw plane) σn, for the 

condition where σn ≥ σIc, over the surface of the strength ellipse (equivalent to all possible flaw 

orientations) divided by the total surface area of the ellipse as shown in Figure 15.

Buch et al. (1977) assumed that flaws were not uniformly randomly oriented but had preferential 

orientation. They used the Batdorf and Crose (1974) unit sphere approach but superimposed a cosine 

power-law flaw orientation anisotropy probability distribution function on the orientation angle α. This 

meant that a flaw was more likely to be oriented in one direction than another. In this manner they 

accounted for a transversely isotropic strength response where the exponent of the cosine function 

determined the degree of strength anisotropy. Buch et al. (1977) only considered the normal component 

of stress σn on a flaw.
NASA/TM—2011-215805 33

Figure 15.—The critical strength ellipsoid of Batdorf, where 

σx, σy, and σz are oriented relative to the material axes, σn

is the normal stress, dA is an infinitesimal area, and α and 

β are angular coordinates.

Nemeth (1989) extended the Buch et al. (1977) approach for transversely isotropic brittle materials to 

include shear sensitivity on a flaw (e.g., see Eq. (20)) as well as the anisotropic stress intensity factor KIc

versus the orientation of a flaw (more details are provided in Appendix B). Anisotropic KIc results in 

strength anisotropy modeling analogous to the Batdorf (1973) approach (and would also correspond to 

flaws being larger in one direction than another), but in this case a cosine power-law function was used as 

was done in Buch et al. (1977). This approach would allow KIc to be mapped from, for example, indentation testing with the indenter oriented at various angles to the extrusion direction (shown in Figure 16). 

Note that a more recent journal article by Quinn and Bradt (2007) recommends that indentation testing no 

longer be used for fracture toughness testing. However, in this case, only the relative difference in KIc

from one orientation to another is required rather than an accurate value for KIc, so perhaps indentation 

testing results could still be used, although further research is needed. The determined functional 

relationship in conjunction with the unit sphere approach enables the stochastic strength response to be 

predicted from the multiaxial stresses. Alternatively, specimens could be excised at various angles to the 

extrusion direction and strength properties obtained similar to that done by Margetson (1976). From that 

information, the necessary functional relationship of strength with angle would also be obtained to 

calibrate the model.

Finally, Tang (1979) performed a literature survey of triaxial failure theories for anisotropic brittle 

materials (not considering the work already mentioned) and concluded that the strength tensor theory of 

Tsai and Wu (1971) could be appropriate for graphite. A reduction of the model for transversely isotropic 

materials was provided. This was a survey of deterministic failure criteria.
NASA/TM—2011-215805 34

Figure 16.—Indentation testing used to determine the relationship between KIc and the extrusion direction. 

Experimental results from Corbin et al. (1988).

2.1.4 R-Curve Effect on the Weibull Distribution

The point of discussing R-curve is twofold. First, rising R-curve behavior tends to increase the 

apparent (experimentally observed) Weibull modulus. This decreases the scatter in measured strengths. 

Second, the subcritical (stable) crack growth that occurs prior to crack instability would cause acoustic 

emission (as the crack grows through individual grains). The implication is that a weakest-link failure 

mechanism may still be appropriate for graphite since R-curve behavior can hypothetically account for at 

least some of the acoustic emission observed prior to rupture as well as some level of distributed damage 

(more than one crack may be growing simultaneously before a single crack becomes critically unstable). 

This argument would certainly imply a Kaiser effect (1950), but how much of the acoustic emission 

profile and nonlinear stress-strain behavior can be attributed to R-curve behavior is a question that will

have to be explored further. R-curve behavior also has potential ramifications on the deviation from the 

Weibull distribution, scatter in strength versus the amount of material under tension, and size effect. Note 

that the Burchell model, discussed in Section 2.1.6, implies an R-curve behavior in graphite.

The failure criterion in the Batdorf methodology is the simple LEFM criterion for mode I loading on 

a crack

K K I I ≥ c (25)

which states that unstable catastrophic crack propagation occurs when the stress intensity KI on a crack 

equals or exceeds the critical stress intensity KIc of the material. Equation (25) directly correlates the size 

distribution of flaws in the material to the observed scatter in strength. When there is mixed-mode loading 

on a crack, an equivalent stress intensity factor KIeq replaces KI in Equation (25). If one assumes that 

material property KIc is constant with crack size, Equation (25) describes what is known as a flat R-curve 

material. This means that there is no stable subcritical crack growth or damage accumulation at all prior to 

catastrophic rupture. Hence, there would be no acoustic emission in the material, indicating crack growth, 

prior to failure. This is not consistent with what is observed with many grades of graphite. 

A rising R-curve occurs when fracture toughness or fracture resistance KR increases with crack size a. 

If the R-curve is steep enough, the crack extension is at first stable, satisfying ∂KI/∂a < ∂KR/∂a, and

becomes unstable when ∂KI/∂a ≥ ∂KR/∂a. Therefore, when a load is applied, a crack may extend some 

increment but arrest, and only if the load is increased will the crack either (1) further extend in a stable 

manner and arrest or (2) propagate uncontrollably, depending on the previously cited condition regarding 
NASA/TM—2011-215805 35

the stability of the crack. This is explained further in fracture mechanics textbooks such as Broek (1982). 

The implication for fracture statistics is that the observed scatter in strength becomes not only a function 

of the size distribution of the flaws in the material but also of the particular R-curve behavior.

Kendall et al. (1986) and Cook and Clarke (1988) experimentally demonstrated and theoretically 

showed that the strength scatter decreases, which affects the apparent Weibull modulus (observed from 

rupture experiments of specimens). Shetty and Wang (1989) and Munz and Fett (1999) pointed out that, 

for R-curve behavior modeled with a power law, the Weibull distribution for measured strengths is 

different from the hypothetical strength distribution based on the fracture toughness KR treated as a 

constant with crack size (a flat R-curve). Therefore, the apparent Weibull modulus is not only controlled 

by the statistical distribution of the sizes of flaws, but it is influenced by the physics of crack growth 

(fracture toughness KR changing with crack size). This applies for transformation-toughened ceramics and 

for ceramics that have elongated interlocking grains such that crack growth at the crack tip is impeded. 

Danzer et al. (2007) demonstrates how R-curve behavior affects the strength distribution as a function of 

volume under stress and shows that non-Weibull strength distributions can be obtained for an R-curve 

that follows an exponential law. The example provided in Danzer et al. (2007) also shows that Weibull 

behavior is asymptotically approached at low and high stress extremes and at low and high volume 

extremes. In a review of R-curve research, Munz (2007) shows how different R-curve relations affect the 

strength response and how crack-bridging parameters can be obtained from scatter in strength. Munz 

(2007) notes that extrapolation of experimental strength data to low probabilities of failure potentially 

leads to an overestimation of strength.

A statistical mechanism for R-curve development was explored by Planas (1995), and it was summarized in Bazant and Planas (1998) for transgranular fracture (fracture traveling through the grain of the 

material). Note that, for intergranular fracture, the crack front travels along the grain boundary. For a 

large crack growing under mode I, assume that the material ahead of the crack tip is divided into small 

cells (corresponding to individual material grains) with crack growth resistances uniform within each cell 

but varying randomly from cell to cell. Bazant and Planas (1998) showed that, as the crack grows through 

the individual grains, the likelihood of encountering a grain with a higher KR increases. This gives rise to 

a statistical R-curve which Bazant and Planas modeled with a Frechet distribution. From their distribution, the slope of the R-curve is steepest at the smallest crack length increments ∆a. Therefore, for an 

initial flaw or microcrack, if flaw size and grain size are not greatly different, one would expect that the 

effect of the R-curve on the Weibull modulus would be greater than if the inherent flaw was much larger 

than the grain size. In another study, Ramachandran et al. (1993) showed that, for a transformationtoughened zirconia, the scatter in strength was caused by variability in the R-curve and the instability 

crack lengths. Crack lengths at instability were much greater than the initial flaw size; thus, fracture 

strengths were insensitive to the initial flaw-size distribution. The implication for graphite is that R-curve 

behavior may make material insensitive to the size distribution of the crack-growth-initiating flaws. It is 

unclear what effect this type of behavior would have on size effect—whether extreme-value statistics 

would operate for R-curve variability and instability crack lengths or not.

From a practical point of view, it could be asserted that the Weibull distribution is still useful for 

modeling graphite strength, even when R-curve behavior is present. However, the experimentally 

observed scatter in strength represents an interplay of the distribution of the flaw sizes in the material and 

the factors related to the physics of crack instability. Danzer et al. (2007) showed that, when one starts

with a Weibull strength distribution for a flat R-curve material, the strength distribution can become nonWeibull when a rising R-curve behavior is imposed. However, WLT behavior will still be operative in 

that model. For practical purposes, if the range of scaling of material volume (between the specimen and 

component) is only a few orders of magnitude, the Weibull distribution probably gives a reasonably 

accurate approximation of the strength response. However, Munz (2007) notes that extrapolating 

experimental strength data to low probabilities of failure may lead to an overestimation of strength.

Finally Jayatilaka and Trustrum (1977), Rickerby (1980), and Diaz et al. (1999) performed work 

relating Weibull modulus, an empirical material constant, to the properties of the flaw-size distribution of 

a material. This would be for flat R-curve materials. Therefore, how the crack-density function was 
NASA/TM—2011-215805 36

functionally related to the size distribution of flaws within the material distribution was shown. In those 

approaches, the flaw-size distribution was inferred from the measured strengths. However, Chao and 

Shetty (1992) and Burchell (1986, 1996), developed the opposite approach whereby they physically 

measured flaw sizes from micrographs, established an appropriate flaw-size distribution function that fit 

the data, and then predicted the strength distribution of components. Chao and Shetty’s model was 

developed for a porous silicon nitride and is not described herein. The Burchell model was developed 

specifically for graphite and is described in Section 2.1.6.

2.1.5 Ho’s Modified Weibull Distribution, Schmidt’s Rule, and Strain-Softening Regarding the 

Size Effect

Ho (1979) introduced a correction factor to the Weibull scale parameter to empirically account for the 

effect of grain size relative to tensile specimen dimension for nuclear-grade graphite. In Ho’s relation,

specimen dimensions on the order of the grain size weaken the specimen. The correction factor was 

introduced because small-size tensile specimens were often observed to be weaker than larger-size 

specimens—contrary to the expectation based on the Weibull theory (see Strizak (1991) and Brocklehurst 

(1977), e.g., Figure 8). For instance, H–451 grade graphite has a mean grain size of 1.6 mm, so a 16-mmdiameter tensile specimen is only 10 times this size. In Strizak (1991), tensile specimen diameters between 

6 and 25 mm were studied, raising concern about the effectiveness of the study regarding comparison to the 

expected Weibull size effect. In Ho’s relation the Weibull scale parameter is modified by

( )

( ) 1

 1 exp ,

2

, cos

mV

fV

oV o

o o

P V

fhd

h fhd d

−

    σ =− −   

σ   

  =   π  

(26)

where ho is the characteristic grain size and d is the diameter of the specimen. Tucker et al. (1986)

explained that strength falls off with small diameter because the characteristic flaw size (related to grain 

size) penetrates a greater fraction of the specimen diameter.

In Schmidt (2000, 2003) the Weibull distribution was modified to eliminate the size-effect scaling of 

the specimen size but preserve the scaling effects of the stress gradient. This was developed to empirically 

fit to data from graphite-rupture experiments that did not correlate well to the traditional Weibull distribution. The equation is expressed as

,

(tensile specimen) , tot

(, ,) 1 exp

m m

e f e f V e f

x y z dV P

θ V

        σ   σ =− −        

σ σ            

∫ (27)

where σθ(tensile specimen) is the characteristic strength of a reference tensile specimen (the strength of the 

specimen at Pf = 0.6321—not normalized to the volume), Vtot is the total volume of the component for 

which failure probability is to be predicted, σe is an effective stress for multiaxial stresses, σe,f is the peak 

effective stress in the component, and σ(x, y, z) is the stress at location x, y, z in the component. The 

effective stress is computed from the maximum deformation theory (MDE), which is based on the 

hypothesis that the elastic energy per unit volume that is stored in a given material element at the moment 

of fracture is equal to the energy that is stored in the uniaxial loaded test specimen at fracture as described 

in Schmidt (2000).

Schmidt (2000) contends that multiaxial experiments on graphite show that the criterion of the 

maximum strain energy (of which MDE is a variant) does not describe the experimental results exactly 
NASA/TM—2011-215805 37

but is better than other traditional criteria and has sufficient accuracy. The MDE criterion is not 

descriptive of the physical processes of converting elastic energy to microcrack progression and 

coalescence. Then again, none of the multiaxial theories described herein can describe the unique 

qualities of graphite damage accumulation and failure completely, although the Batdorf theory probably 

comes the closest.

The Schmidt modified Weibull methodology works for uniform uniaxial tensile specimens and 

flexure specimens. The effective volume (the term in square brackets [...] in Equation (27)) is normalized 

by the volume of the component Vtot such that proportionally scaling the component dimension does not 

change the value in the square brackets, thereby leaving failure probability constant with specimen size. 

However, failure probability will differ depending on if the component is under uniform uniaxial tension 

or loaded such that a stress gradient exists. A scenario where Equation (27) fails, for example, is with a 

notched tensile rod (that has a finite fillet radius at the notch tip to avoid a stress singularity condition yet 

still be a region of high stress concentration). If the specimen diameter and notch geometry are kept 

constant, but the length of the specimen changes, then the probability of failure changes according to 

Equation (27). That result is not consistent since failure will be controlled only by the stress concentration 

at the notch root, and the length of the specimen is irrelevant. Nevertheless, Equation (27) is a useful 

attempt to preserve the strength size effect for the stress gradient while eliminating it for size scaling—

and it works at least for certain controlled situations.

Li and Fok (2009) modeled nonlinear stress-strain behavior in graphite and posit that the difference in 

strength and Weibull modulus between tension and bending load is due to the strain-softening behavior of 

quasi-brittle materials. Their model offers an alternative to the Schmidt model to account for the difference in strength between bending and tension while also eliminating size effect with increasing specimen 

size. Li and Fok (2009) cite work from previous authors who modeled strain-softening and predicted a 

nonstochastic difference in strength between flexural loading (a situation where a stress gradient is 

present) and tensile loading but they base their approach on the work of Bazant and Li (1995). Nonlinear 

stress-strain behavior is modeled as a combination of linear increase of stress with increasing strain up to 

a critical strain threshold, and beyond that threshold, the stress decreases with increasing strain until the 

ultimate strain at failure is reached. The effect of this behavior is that the strain-softening (attributed to 

damage accumulation) requires higher load to fail a beam in bending than would be required with a 

purely linear-elastic material response. If the quasi-brittle material were analyzed as though it was linear

elastic in behavior, it would appear that the material in bending was stronger than the material in pure 

tension. 

In the Li and Fok model, the critical strain is modeled with a Weibull distribution with the additional 

result that the Weibull modulus in bending would appear to be higher than the Weibull modulus in 

tension. Their model highlights the need to properly account for nonlinear stress-strain behavior in any 

analysis. It should be noted that the Li and Fok model did not indicate a size effect for increasingly larger 

specimens because the Weibull distribution for critical strain was not made a function of material volume. 

Note that there is evidence in the literature, at least for flexural loading, that strength does decrease with 

increasing specimen size (e.g., see Figure 8(a)).

A further item worth noting regarding the Li and Fok model is that, for the limiting case of a linearelastic brittle material, there is no difference in strength between flexure and tension since the failure 

criterion is simply a deterministic value of ultimate strain. For classically brittle materials like ceramics,

where the size of preexisting critical flaws controls the strength response, there is clearly a difference in 

strength between flexural and tensile loading as explained through the Weibull distribution. The Li and 

Fok model assumes that graphite has no sensitivity to preexisting flaws. If larger sized preexisting flaws 

are present in the material, then the strength response will be governed by the size distribution of the 

critical flaws and not solely by a critical strain. Their model also did not consider that graphite has 

bimodulus behavior—where the stress-strain behavior is different in tension and compression. In their 

model, the material response is initially linear elastic until the critical strain threshold is reached 

(somewhere in the neighborhood of the value of the ultimate failure strain). 
NASA/TM—2011-215805 38

For graphite the stress-strain response may be nonlinear from the very onset of loading or at relatively 

low loading levels (e.g., see Figure 5). The Li and Fok model predicts that the ratio of flexural strength to 

tensile strength increases as the ratio of ultimate strain to critical strain increases. With regard to the 

assumption that failure strain is a material constant that does not change in flexure or tension, Figure 7

shows that the failure strains in four-point flexure and tension are within about 10 percent of one another,

although it is not clear if this difference is statistically significant. Figure 7 also shows that the scatter in 

the failure strain may also need to be accounted for in modeling. Lastly, a failure criterion based on a 

critical strain criterion will give significantly different results in comparison to stress-based failure criteria 

for multiaxial loading. For example, equibiaxial loading strength will be predicted to be higher for strainbased criteria than for stress-based criteria because of the Poisson effect.

2.1.6 Graphite Microstructure Specific—The Burchell Model

The Burchell model (Burchell 1986, 1996, 1999; Burchell et al. 2006), which was developed specifically for graphite, combines fracture mechanics with a physics-based microstructural description of 

graphite failure. It directly incorporates specific graphite features such as the grain size, pore size, pore 

distribution, particle fracture toughness, graphite density, and specimen size (size effect) into the model.

Tucker and McLachlan (1993) further refined this model primarily to account for strain hysteresis and 

acoustic emission. However, the Tucker and McLachlan model, though it may provide an improved 

physical basis, requires data for 14 different input parameters, which makes its use cumbersome. 

Although these parameters were initially based on measured parameters, they were subsequently altered 

to best fit the experimental results presented in their journal article. The Tucker-McLachlan model 

appears to be best suited for explaining the mechanisms of fracture in graphite; it is not necessarily well 

suited as an engineering tool for designing a structure. In this section, the discussion focuses on the 

Burchell model and its description of fracture. 

The Burchell model is a refinement or variant of the Rose-Tucker model (Rose and Tucker (1982)

and Tucker et al. (1986)), which itself is a refinement of a model proposed by Buch (1976). In the RoseTucker model, individual graphite grains are assumed to be cubes of the dimension of the average particle 

size. The structure is composed of these grain-cubes stacked together with the layers perpendicular to the 

local stress field (see Figure 17(c)). Each grain in the structure has a randomly oriented cleavage plane as 

shown in Figure 17(a). Failure in a grain is assumed to have occurred when the tensile stress component (from the far-field applied stress) normal to the crystallite cleavage plane exceeds a critical value 

(assumed to be a deterministic value). The probability of failure of each grain under the applied stress is 

equal to the fraction of possible orientations for which cleavage would occur. Note that this is equivalent 

to calculating P2 in Equations (15) and (18) in the Batdorf methodology. Porosity is considered by 

making an appropriate fraction of grains have zero strength; however, the stress-raising properties of 

pores on the grains in the immediate vicinity of pores is not considered. The probability of specimen 

failure is the probability of a layer having sufficient contiguous failed grains equivalent in size to a single 

crack whose stress intensity factor exceeds the material KIc (shown in Figure 17 parts (b) and (c)).

In the Burchell model, fracture is assumed to initiate at or very near pores—which are stress 

concentrators. The distribution of pore sizes is modeled as a lognormal distribution (measured from 

micrographs). Crack growth extends from the pores through individual particulates. Similar to RoseTucker, the particulates are modeled as an array of cubic particles of size equal to the mean filler particle 

size. Each particulate cube has a random crystalline orientation. Figure 18 is a schematic of this idealization. Fracture initiates at or very near pores, and crack growth proceeds through the individual particulates along their cleavage planes when stress in the particulate exceeds a critical value (the fracture 

strength of the cleavage plane). Crack growth proceeds through each row of particulate blocks as shown 

in Figure 19 until overall crack length equals or exceeds KIc. Therefore, in the Burchell model two 

stochastic quantities are involved: the distribution of pore sizes and the random orientation of cleavage 

planes in individual particulates. Further details regarding the development of the Burchell model follow.
NASA/TM—2011-215805 39

Figure 17.—Failure in graphite assumed cube-shaped grains, where σ is the critical strength. (a) Cleavage planes 

randomly oriented for each grain. (b) Adjacent failed grains forming a contiguous crack. (c) Cross section of failed 

grains normal to the applied stress. Reproduced from Tucker et al. (1986). Copyright Elsevier; used with 

permission.

Figure 18.—Burchell’s idealized model of graphite, where σ is the 

applied stress. Reproduced from Burchell (1996). Copyright Elsevier; 

used with permission.
NASA/TM—2011-215805 40

Figure 19.—Crack front originating from a pore proceeds through a row of 

particulatesadding the particulate width to the overall length of the 

crack, where σ is the applied stress. Reproduced from Burchell (1996). 

Copyright Elsevier; used with permission.

The graphite grain consists of parallel hexagonal planes with weak van der Waals bonding between 

the planes and strong covalent bonding within planes (as described in Greenstreet (1968)). Because of this 

granular anisotropy, fracture within the grain is preferentially oriented along the c-axial direction. If one 

assumes that fracture is transgranular, the probability of failure of an individual graphite grain i ahead of a 

crack tip and oriented at an angle θ to it is given by

( )

1/3

4 1 I , cos c fiK P a

a

−  

σ =  

π   σ π  

(28)

where KIc is the grain fracture toughness, a is the crack size, and σ is the remote applied tensile stress. If 

there are ng grains in an entire row ahead of the crack tip, the probability that they will all fail is

( )

1/3

4 1 I , cos

g

g

n

c fn

K P a

a

−

   

σ =    

π   σ π    

(29)

Equation (29) assumes that the crack will extend from length a to a + b, where b is the average 

graphite grain size. The probability that the crack will extend from a to a + jb, fracturing j rows of grains,

may be approximated by

( ) ( )

1/3

1 I

0

4 ln , ln cos g

j

c fn g

K P a n dj

a jb

−

        σ =     π   σπ +    

∫ (30)
NASA/TM—2011-215805 41

If one assumes that the graphite microstructure contains a lognormal distribution of pores, then the 

probability that the length of a specific defect will fall between a and (a + d) is S(a) da, with S(a)

defined as

2 1 ln 2 ln

( ) c exp 2 ln d

a S

S a

S

μ

        − = −             

(31)

where c is a constant, Sμ is the mean pore size, and Sd describes the scatter in the pore distribution (similar 

to standard deviation). When N is the number of pores per unit volume, and V is the specimen volume 

under uniform tensile stress σ and the number of crack tips under stress is 2NV, then the probability of 

failure for the entire specimen is given by

( )

2

0

1 1 () , g

NV

Pf S a P a da fn

  ∞

=− − σ  

  ∫ (32)

As can be seen, Equation (32) is derived to describe Pf for specimens subjected to uniaxial, uniform 

tensile stress state due to volume-residing defects. 

Equation (32) is a WLT model. This can be shown by analogy from Chao and Shetty (1992), Epstein 

(1948), and Shih (1980). For a uniformly stressed body that contains N flaws per unit volume, fracture 

will occur within that unit volume at a stress level corresponding to the weakest strength among those N

flaws. Thus, the cumulative failure probability distribution for N flaws, where each flaw is exposed to a 

tensile stress of magnitude σ, is

1 1 ( I ) NV P P f c = − − σ ≤σ    (33)

where P(σIc ≤ σ) is the probability that a given flaw will fail at a stress level of σ or less than σ, or the 

probability that the strength of the flaw, denoted by σIc, is at a stress level of σ or less than σ. For large 

values of N, this equation is approximated by

Pf c ≈ − − σ ≤ σ  1 exp NV P( I )   (34)

When comparing Equation (32) for the Burchell model and Equation (33)—a WLT model of 

individual flaws—it is clear that they have the same functional form. Analogous to Equation (34), the 

Burchell model, therefore, could also be expressed as

( )

0

1 exp 2 ( ) , g Pf NV S a P a da fn

  ∞

≈− −  σ

  ∫ (35)

and would show a size effect consistent with the WLT model.

An interesting feature of the Burchell model is that transgranular fracture is modeled somewhat 

similar as by Planas (1995) and Bazant and Planas (1998), who describe the statistical R-curve. Indeed in

Burchell (1996) and Burchell et al. (2006), the R-curve effect can be seen for initial versus final critical 

crack size for H–451 graphite (Figure 20). Therefore, the Burchell model explicitly models a statistical 

R-curve and consequently predicts acoustic emission prior to catastrophic failure.
NASA/TM—2011-215805 42

Figure 20.—Initial defect length and final crack length at 

failure calculated with the Burchell model for H−451 

graphite. Reproduced from Burchell (1996). Copyright 

Elsevier; used with permission.

Ishihara et al. (2001) used Burchell’s model to account for stress gradients by assuming small uniformly stressed incremental volumes, which is the WLT assumption of Equation (1). Predictions made 

using the Burchell and Weibull models showed only minor differences when they were compared with

experimental results for H–451 graphite data. The effect of multiaxial stresses and anisotropic strength 

also was considered in Burchell et al. (2006), who used a PIA-style approach, and more recently by 

Battiste et al. (2010), who used the Shetty (1987) mixed-mode interaction equation.

2.1.7 The Uniform Defect Model

The uniform defect model is interesting because a probabilistic relation is developed for a critical 

density of number of defects as a failure criterion. Since load redistribution is not involved, this model is 

not a parallel model. The WLT analogy is the occurrence of a critical density of defects under a specified 

load causing catastrophic rupture. Freudenthal (1968) assumed that inhomogeneities are uniformly 

distributed over the volume of a material body and developed a fracture model where the occurrence of a 

critical number of defects within a confined volume gives rise to a statistical aspect of fracture as well as 

a size effect.

Freudenthal assumed that a Poisson process governed the general equation for the probability of 

occurrence of k inhomogeneities in the volume V. The Poisson density function is described by (e.g., see 

Hoel et al. (1971))

( ) ( )

exp( ) 0,1,2, ... !

0 elsewhere

x

x PX x f x x

λ −λ  = = = = 





(36)
NASA/TM—2011-215805 43

where λ is a positive number. The real-valued function ƒ(x) is the discrete density function of random 

variable X where P(X = x) is the probability that a discrete real-valued random variable X equals a 

possible value x. The Poisson distribution approximates the binomial distribution for large values of n, 

where n is the number of Bernoulli trials with success probability Pevent = λ/n at each trial. When P(X = 0) 

is computed for n = V and Pevent = cμ, then

( ) ( ) ( )

0 exp 0 exp exp 0!

P X c Vμ

λ −λ = = = −λ = −    (37)

where cμ, which has dimension of V–1

, denotes the mean concentration of inhomogeneities. When 

cμ= ηV(σ) is substituted from Equation (7), the Weibull equation is obtained in Equation (37). This 

represents the probability of zero defects occurring within volume V, which also represents the survival 

probability of the material under WLT. For the probability of occurrence of k inhomogeneities in the 

volume V,

( ) 1 ( ) exp !

k PX k cV c V

k μ μ = =   −  (38)

If the probability of fracture is identified with the probability of occurrence of at least n inhomogeneities, then the probability of survival is the probability of occurrence of less than n inhomogeneities,

which is identical to the sum of the probabilities of occurrence of from zero to (n – 1) inhomogeneities

1

0

 exp !

n k

sV

k

c V

P cV

k

− μ

μ

=

= −   ∑ (39)

and therefore

 exp !

k

fV

n

c V

P cV

k

∞ μ

μ = −   ∑ (40)

The probability density function of Equation (40) is a gamma distribution. Freudenthal further shows 

that 

( 1) ( , 1) 1

( 1) fV

n cV n

P

n

μ   Γ + −Γ +   = − Γ + (41)

Since cμ is the mean concentration of inhomogeneities per a specified volume, the effect of the 

volume ratio cμV can be examined. As Freudenthal reports for cμV = 1, 10, 100, the values of PfV are 

0.190, 0.990, and 1.000 for n = 3, and 0.000, 0.411, and 1.000 for n = 10, respectively. Clearly a size 

effect is predicted with this model. Freudenthal further notes that the effect of stress could be introduced 

by assuming that the critical number of inhomogeneities required for fracture increases as the applied 

stress decreases. For example n = noσcon /σ, where no and σcon are material constants. Appendix A shows 

an argument that the uniform defect model can be approximated with a Weibull distribution, which might 

be appropriate to describe the strength distribution for compressive stress states.

2.2 Parallel Systems

Nuclear graphite exhibits distributed and accumulated damage (as indicated by acoustic emission) 

and nonlinear stress-strain response. As previously described, under the action of a monotonically 
NASA/TM—2011-215805 44

increasing progressive load, microcracks form and arrest, leading to load redistribution and the formation 

of new microcracks. Eventually, a local concentration of microcracks develops and coalesces to form an 

unstable macrocrack of critical size. The Burchell model and the Weibull model with R-curve behavior 

(e.g., Danzer et al. (2007) and Munz (2007)) do not account for these interactive effects of microcracking. 

They are weakest-link series-system models, whereby the material is assumed to be composed of a set of 

n links connected in series in such a manner that the structure fails whenever any of the links fail. 

Because of the interactive nature of microcrack formation, load redistribution, and microcrack 

coalescence, it may be more appropriate to consider graphite material failure in terms of a parallel-system 

analog. Recall that in the parallel-system model, the n links are arranged in parallel. When one link fails,

load is redistributed to the remaining n – 1 links. The remaining n – 1 links carry higher load, but the 

system (structure) may still survive. The structure does not fail until all n links fail. This line of inquiry 

leads to consideration of models that describe the failure of other quasi-brittle materials, such as fiberreinforced composite materials and lattice system models, which describe the process of material breakdown in terms of connected beams, springs, bonds, or fuses. Fiber-bundle models and lattice models 

study the process of damage initiation, damage accumulation, damage clustering, and system instability 

leading to ultimate failure relative to the initial state of disorder within the material. Lattice models more 

closely approximate the structural features of graphite. 

Fiber-reinforced composite materials and graphite bear no outward resemblance to one another;

however, their processes of progressive damage leading to failure do resemble each other—at least at a 

generalized mathematical level. Phoenix and Beyerlein (2000b) have commented on the similarity 

between composite fiber-bundle models and lattice models, and an extensive body of knowledge has been 

developed regarding modeling the failure strength distribution of fiber-reinforced composite materials. 

The authors of this report believe that this body of work has relevance and is worth reviewing. Graphite 

failure behavior likely ranges between that of classically brittle materials and the damage-tolerant 

behavior of composite materials. Therefore, exploring the implications of the behaviors displayed 

between classically brittle materials and quasi-brittle materials, such as fiber-reinforced composites,

should provide useful perspective with regard to further development of appropriate graphite failure 

strength models.

Fiber-reinforced composite materials are typically engineered to fail in a damage-tolerant manner. 

That is, the material system is designed so that failure in a single fiber will redistribute the load such that 

the remaining fibers will not (in most cases) fail unless the load is sufficiently increased. Therefore, a 

composite will have significant damage accumulation and a nonlinear stress-strain response prior to 

ultimate structural failure. Similar to graphite, localized failure is arrested and (both local and global) 

stresses can redistribute. Subsequent material failure would involve accumulated damage either over the 

entire material cross section or localized material damage that reaches a critical size or density within a 

critical volume. The strength distribution of the individual fibers and the load transfer that takes place 

both locally and globally at the site of broken fibers influences the damage accumulation and ultimate 

fracture behavior of the composite; thus, fracture behavior can range from brittle (with little distributed 

damage) to quasi-brittle (with significant distributed damage).

Lattice models are a closer analog to graphite. In a lattice model a material such as graphite is 

represented as a network of elastic beams, springs, bonds, or electrical fuses with individual random 

failure thresholds. The individual discrete elements of the lattice simulate local inhomogeneities of 

mechanical properties, such as strength, within the material. Depending on the strength distribution of 

the individual elements, brittle fracture behavior or quasi-brittle distributed damage behavior can be 

simulated.

Sections 2.2.1 to 2.3.3 provide an overview of some of the work that has been done modeling and 

simulating the failure of unidirectional continuous-fiber-reinforced composites and lattice models under 

progressive uniaxial tensile loading. These models predict the consequences of distributed damage, load 

redistribution, and damage coalescence on the strength of a structure. Fiber-reinforced composite models 

are shown because the mathematical framework is more developed than for the lattice models of homogeneous materials. As pointed out in Phoenix and Beyerlein (2000b), the severe computational demands 
NASA/TM—2011-215805 45

have limited the network sizes that can be investigated with lattice simulations, whereas composite fiberbundle models in one dimension can be solved exactly or asymptotically for increasing size scale.

Emphasis is placed on describing the consequences of these models on the distribution of strength, the 

size effect, and the role of system disorder on the brittle-to-ductile (ductile-like) transition.

Sections 2.2.1 to 2.3.3 highlight the implications of local versus global load sharing, a WLT 

mechanism for composite materials, asymptotic Weibull behavior for composite materials, the WatsonSmith modified Weibull relation, extreme-value implications from the presence of large flaws in lattice 

models, and the implications of the brittle-to-ductile-like transition for both composite and lattice models. 

Unidirectional fiber-reinforced composite models are described first, followed by lattice models. In these 

sections, the evidence or the basis for using WLT and the Weibull distribution is presented. It is also 

noteworthy that other researchers such as Wetherhold (1983) and Duffy et al. (1993) have advocated the 

use of the Weibull distribution for the design of polymer matrix and ceramic matrix composite structures. 

Phoenix and Beyerlein (2000a) give an authoritative overview of statistical strength theories for 

unidirectional fiber-reinforced composite materials under tension. Alava et al. (2006) also provide an 

excellent summary of statistical strength theories for material systems (homogeneous and heterogeneous)

but are focused more toward lattice models. Mahesh and Phoenix (2004) and Phoenix and Beyerlein 

(2000b) provide useful perspective and further details. The following overview only touches upon the 

extensive body of work that has been performed regarding parallel system modeling.

2.2.1 Two Composite Material Failure Modes

The continuous-fiber-reinforced composite described here is a heterogeneous material consisting of 

unidirectional fibers embedded in a matrix material. The matrix usually does not carry load but does 

transfer load to the fiber. Failure of the composite involves reaching a critical level of damage 

accumulation of broken fiber segments.

The failure of composite materials falls between two distinct extremes: dispersed failure and localized 

failure. In dispersed failure, the fibers fail in a dispersed manner until the load exceeds the material’s

capacity to support load. The fracture surface of the composite appears to be brushlike. This failure mode 

is associated with things like a weak interface between the fiber and matrix, a matrix with a low yield 

strength compared with stiffness, or fibers with a large variability in strength. In localized failure, fracture 

initiates and coalesces around local clusters of fibers. This is due to the stress concentration in the 

immediate vicinity of fiber breaks. In this failure mode, the fiber-matrix interface is usually strong, the 

matrix less plastic, and/or there is lower variability in fiber strength. See Phoenix and Beyerlein (2000a)

for further details.

2.2.2 Equal-Load-Sharing and Local-Load-Sharing Fiber-Bundle Models

Fiber-bundle models fall into two classes: equal load sharing (ELS) and local load sharing (LLS). In 

the ELS model, the fiber bundles do not interact with one another (whether they are loose or embedded in 

a matrix material). When a fiber fails, the load is equally distributed to all the remaining fibers. In the 

LLS model, the fibers are embedded in a matrix and do interact with one another. If a fiber fails, adjacent

local fibers will carry an increased load and have an increased risk of failure. Breaking of a critical 

number of local fibers causes an instability point to be reached, resulting in a cascade of fiber failures 

until the whole composite catastrophically fails. Consequently, LLS shows more localized breakdown and 

flaw sensitivity, whereas ELS has a more dispersed fiber-failure character. ELS and LLS models predict 

different consequences with regard to strength distribution and size effect, which are discussed next. ELS 

and LLS models are complex and not easily amenable to closed-form solutions. Typically, they are 

studied through computer simulations or determination of asymptotic behaviors (e.g., for large numbers

of fibers or for low probabilities of failure). Also LLS models can have many variations depending on 

load-sharing rules, two-dimensional versus three-dimensional analysis, and fiber-packing arrangement. 

Although composite modeling and simulation is complex, important insights and generalizations have 

been achieved. Sections 2.2.3 to 2.2.6 describe some of this work.
NASA/TM—2011-215805 46

2.2.3 Daniels’s Equal-Load-Sharing Bundle Model

Peirce (1926) and Daniels (1945) were the first to model the failure of fiber bundles. In the bundle 

model, a structure is viewed as a bundle of parallel fibers of equal length and cross section. Each fiber is 

assumed to be governed by weakest-link strength (typically the Weibull distribution). Under a uniform 

load, failure of a single fiber redistributes load in the remaining fibers, which does not necessarily lead to 

structural failure (i.e., failure of all the fibers). The fracture strength of the “dry” bundle model (where the 

bundle is a loose collection of fibers) is determined by the forces under which a “chain reaction” process 

of consecutive filament failures occurs from the progressive overload carried by the surviving filaments. 

This ultimately leads to the final failure of all filaments. The fracture process starts at the weakest point in 

the bundle, but contrary to the WLT model, the bundle does not fail unless the load is increased sufficiently. A single fiber can cause the bundle to fail if the filaments are arranged in the order of their 

consecutive failure, which is unlikely. 

The fiber-bundle model, which was derived in Daniels’s (1945) classical paper, assumed an ELS rule,

whereby the surviving fibers share the applied load equally. For a bundle of nf filaments, where nf is large,

Daniels showed that the probability distribution of strength tends to a normal distribution, the average 

strength is independent of nf, and the variance is an inverse function of nf and thus tends toward zero for 

very large nf. There is no size effect for this model of asymptotic behavior.

For fiber strength (load per fiber) distributed according to P(x) such that limx→∞ x[1 – P(x)] = 0, then 

the strength of the fiber bundle (expressed as load per fiber) is normal with a mean of 

( ) * * σ= − μ x Px   1

  (42)

and a standard deviation of

( ) ( ) * * 1

sd

f

Px Px

n

  −  

σ = (43)

where x

* is the point where the function x*

[1 – P(x

*

)] is maximum. For Weibull-behaved fibers, the 

Daniels model shows that the bundle ultimate strength is

1/ 1 1

exp

m

o m m μ

  

σ =σ −       (44)

and the standard deviation is

1/ 1 12

exp exp

m

o

sd

f

m mm

n

      

σ      −− −         

σ = (45)

2.2.4 Chain of Equal-Load-Sharing Bundles

Gücer and Gurland (1962) (see also Smith and Phoenix (1981) and Coleman (1957)) proposed the 

chain of ELS bundles model, and Rosen (1964) and Scop and Argon (1967) applied it to fibrous composites. “Chain of bundles” refers to the fact that a broken fiber in a matrix will carry load a certain distance 

away from the break—the “ineffective length,” or “stress transfer length”—and the composite is treated 

as a serial arrangement of independent bundles, each of length 2δ (usually a few fiber diameters), which 

is shown schematically in Figure 21.
NASA/TM—2011-215805 47

Figure 21.—Schematic of chain of bundles, 

where nf is the number of fibers and nb is the 

number of bundles. Reproduced from Bazant 

and Pang (2007). Copyright Elsevier; used with 

permission.

Therefore, a composite of length L composed of nf fibers can be represented as a chain (a series) of 

nb = L/(2δ) mutually exclusive, but linked, bundles. The chain of bundles has a WLT failure mechanism

controlled by the weakest bundle, but each fiber bundle behaves as an independent ELS (or LLS) system; 

that is,

, () 1 1 () b

f b f

n

H x Gx n n n =− −     (46)

where , ( ) H x n n f b is the composite strength distribution, ( ) G x nf is the bundle strength distribution, and 

x is the load per fiber. Equation (46) implies that composite strength is determined solely from the 

statistical mechanics of the individual bundle: that is, from the load-sharing scheme and the strength 

distribution for individual fibers. Smith (1982) improved the accuracy of the model for smaller nf and 
NASA/TM—2011-215805 48

showed that, when nf is small and nb is large, asymptotic Weibull behavior is obtained. For the strength 

distribution for moderate to large nf and nb, Smith (1982) showed that the Gumbel extreme-value distribution (based on a normal distribution for link strength) is obtained. For ELS bundles with small nf, a size 

effect is present (Smith and Phoenix (1981) and Phoenix et al. (1997)), but this decreases with increasing 

nf since strength dispersion decreases as 1 . nf This is true regardless of fiber length L.

2.2.5 Local-Load-Sharing Modeling

Bundle models have been proposed with LLS rules whereby stress is concentrated in the immediate 

vicinity of failed fibers. Several variations of LLS modeling for two- and three-dimensional unidirectional 

composites are summarized in Phoenix and Beyerlein (2000a) along with appropriate references. In LLS 

models the fibers are embedded in a matrix. The fibers interact, and a critical number of breaks in a 

localized region induce a catastrophic cascade of fiber failure across the bundle, resulting in total failure.

Many factors can influence LLS modeling, including the variability of fiber strength, the yield strength of

the matrix in shear, the friction at the fiber-matrix interface after debonding, the geometrical distribution 

of fibers, and the rules used to define material failure. These factors make the modeling algorithms and 

analytical effort difficult.

A size effect exists for LLS, but it is always less pronounced than for a single fiber that is modeled 

with the Weibull distribution. Zweben and Rosen’s (1970) model prescribed composite failure as the 

occurrence of at least one cluster of failed fibers of a specified size. Harlow and Phoenix (1978a,b; 

1981a,b), Batdorf and Ghaffarian (1982), and Smith et al. (1983) extended the failure criterion to consider 

the probabilities of growing clusters up to instability. Smith (1979, 1980) indicated that, for a chain of 

bundles where k consecutive (adjacent) fiber failures within a single bundle defines failure, the composite 

strength for large system size nsys is proportional to σf ∞ c ⋅ nsys−1/km

, where σf is the failure stress, nsys is 

the product of nf and nb, m is the fiber Weibull modulus, c is a constant, and k is a constant argued to be 

between 2 and 4. The system size nsys is proportional to volume and, therefore, follows the size-effect

relation described in Equation (12) but with the modified Weibull modulus of km instead of m for the 

uniformly stressed volumes V1 and V2. This indicates that the composite size effect is less than that for the 

single fibers under LLS. Smith (1980) conjectured that the standard deviation of strength was σsd= c/ln

(nsys). This compares with Daniels (1945), where ELS indicated that the standard deviation is related by 

nf

–1/2. Therefore as the system size increases, the standard deviation decreases under both ELS and LLS.

In their analysis, Harlow and Phoenix (1978a,b; 1981a,b) indicated that scatter in strength decreases with 

volume. 

Batdorf (1982) and Batdorf and Ghaffarian (1982, 1984) simplified concepts from Harlow and 

Phoenix (1978a,b; 1981a,b) and modeled the formation and growth of multiple fiber fractures as the basis 

of their failure criterion. In their model the Weibull relation for single fiber strength is used to determine 

the number of critical fiber fractures (singlets), double adjacent fractures (doublets), triple adjacent 

fractures (triplets), and so on as a function of stress and volume. They constructed a failure envelope of 

“multiplets” consisting of line segments with slope 1/(im) where i is an integer multiple of the fiber 

Weibull modulus corresponding to the i

th multiplet where i fibers form a critical cluster of failed fibers. 

This analysis was based on a Griffith-type instability showing an inverse relationship between crack size 

(number of adjacent fiber breaks) and failure stress. In Batdorf and Ghaffarian’s theory of multiplets, the 

size effect is the same as in Equation (12) except with a modified Weibull slope of 1/m*

, where 1/m* is the 

local slope of the failure line as shown in Figure 22. As seen in the figure, the size effect is no longer 

linear with the log of σf (it is piece-wise Weibull, but overall it is non-Weibull), and it decreases with 

increasing volume.
NASA/TM—2011-215805 49

Figure 22.—Failure envelope for composite strength versus volume, 

where m is the Weibull modulus, σf is the failure stress, and nf is the 

number of fibers of length L in the composite. Note that the product of 

nfL is proportional to the composite volume. Reproduced from Batdorf, 

S.B.; and Ghaffarian, Reza; 1984: Size effect and Strength Variability 

of Unidirectional Composites. Int. J. Fracture, vol. 26, no. 2, Fig. 1, 

p. 115. Copyright Springer Publishing Company; used with kind 

permission from Springer Science + Business Media B.V.

2.2.6 Weakest-Link Relation for Composite Materials

Harlow and Phoenix (1978b) found that an LLS fiber bundle follows the WLT scaling law for 

number of fibers nf under certain conditions. This was proved rigorously by Kuo and Phoenix (1987) and 

other authors have contributed to this development (see Mahesh and Phoenix (2004)). The Harlow and 

Phoenix (1978b) model was for a two-dimensional arrangement of fibers assuming simplified loadsharing rules. For ( ) G x nf , the probability strength distribution for a fiber bundle of size nf, they found 

asymptotically that there is a WLT characteristic probability distribution W(x) independent of nf such that

( ) 1 1 ( ) as [ ] f

f

n G x Wx n n f ≅ − − → ∞ (47)

Equations (46) and (47) can be combined to express the composite strength distribution as a WLT

relation:

, ( ) 1 1 ( ) 1 exp ( ) [ ] { } f b

f b

n n H x Wx n n =− − ≅− −n nW x f b (48)

Hence a fiber-reinforced composite is predicted to fail, under assumed LLS rules, in a brittle-like 

manner. Exact analytical forms for W(x) have not been determined. Harlow and Phoenix (1978b)

determined that W(x) is independent of nsys for nsys ≥ 9 and m ≥ 5. They also found that ELS bundles do 

not follow this WLT scaling. The function W(x) is not a power law of x, so , ( ) H x n n f b

is non-Weibull. 
NASA/TM—2011-215805 50

Mahesh et al. (1999, 2002) and Landis et al. (2000) performed Monte-Carlo simulations on the 

fracture process that examined the validity of Equation (48) for various values of fiber Weibull modulus 

and system sizes. Mahesh et al. (1999) used the Hedgepeth and Van Dyke (1967) (HVDP, with P for 

periodic) LLS model for a cross section of a three-dimensional composite (square-packed and hexagonalpacked fiber arrangements) for various m. They show that m ≥ 3 was required for W(x) to be independent 

of nsys for the limited system sizes in their analysis. Mahesh et al. (2002) show that m ≥ 2 and m ≥ 1 were

required for W(x) to be independent of nsys for three-dimensional and two-dimensional bundles, respectively, using HVPD. Their results indicate that the strength distribution was neither normal nor Weibull 

distributed although it was perhaps closer to a Weibull form (Mahesh and Phoenix (2004)). 

Landis et al. (2000) performed full three-dimensional simulations of composite failure and tested the 

WLT scaling law using the relation

2 1

2 1

/

, , () 1 1 () V V P P f V f V σ= − − σ     (49)

where the probability of failure of a composite 2 Pf V, with volume V2 loaded to stress σ is related 

to probability of failure of a composite 1 Pf V, with volume V1 under identical loading. Note that 

Equation (49) is a reexpression of Equation (48) using volume instead of system size nsys = nbnf (system 

size is proportional to volume). Landis et al. (2000) found good correlation for this relation for larger 

system sizes (volumes)larger than a 25 by 25 array. This suggested that there was a minimum bundle 

size (number of fibers) for this relationship to be valid.

The Mahesh et al. (1999) simulations showed a common trend for both the particular LLS and ELS 

models and for the values of fiber Weibull moduli studied: the mean strength and standard deviation 

decreased with increasing composite size. Landis (2000) also showed similar results for their full threedimensional simulations with their LLS model. The ELS bundles closely approached Daniels’s asymptotic results as size increased. The strength distributions from the LLS simulations were not consistently 

Weibull or normal, but the ELS simulations were normally distributed. Landis et al. (2000) also found 

that the stress- (concentration-) driven failure process followed a WLT scaling, whereas the fiberstrength-dominated mode did not. 

Watson and Smith (1985) suggested a modification to the Weibull distribution to account for a 

smaller-than-expected size effect of the form

* *

( ) 1 exp

m

f

o o

L P

L

  φ    σ

σ= − −        σ  

(50)

where L is the length of the fiber, Lo is a characteristic length, and φ* is a constant between 0 and 1. This 

form is applicable both to the individual fiber and the composite if the appropriate values of φ and m are 

used for φ* and m*

, respectively, depending on the material system. When φ* = 1, the usual Weibull sizeeffect relation with exponent 1/m* is obtained (see Eq. (12)). When 0 < φ* < 1, the size effect is appropriately diminished by the exponent φ*

/m*. This factor is said to account for variations in fiber diameter 

and material changes from fiber to fiber in a yarn. Beyerlein and Phoenix (1996) (and Otani et al. (1991))

applied the Weibull distribution modified by Watson and Smith (1985) to account for the size effect of 

microcomposites (composites consisting of only a few fibers embedded in a matrix). Beyerlein and 

Phoenix (1996) required φ* = 0.6 in order to achieve good correlation to experimental results. They used 

Weibull approximations from lower-tail (asymptotic) analysis to construct a failure envelope for a fourfiber microcomposite as shown in Figure 23.

Each line (k = 1, 2, 3, 4) represents the failure probability for k critical number of failure breaks 

approximated by a Weibull distribution of the form
NASA/TM—2011-215805 51

,

,

( ) 1 exp ( ) b

km

k n b f

k

H n nφ

δ

      σ

σ= − −      σ      

(51)

where σδ,k is a Weibull scale parameter appropriate for the k

th Weibull line, as explained in Beyerlein and 

Phoenix (1996). What is interesting about Figure 23 is that it shows the failure envelope having Weibulllike failure behavior at low probability-of-failure regimes but with a modified size-effect relation. 

This approach seems to bear at least some outward similarities to the Batdorf (1982) and Batdorf and 

Ghaffarian (1984) approach. Note that for fiber-controlled failure with a Weibull-style failure distribution,

using either the fiber stress or the composite stress will not affect the results. See Appendix C for a simple 

explanation.

Figure 23.—Weibull line failure envelope construction compared with 

equal load sharing (ELS) exact solution (G4(σ)) from Harlow and 

Phoenix (1978a) for a four-fiber microcomposite. Each line (k = 

1,2,3,4) represents the failure probability for k critical number of 

failure breaks. Reproduced from Beyerlein and Phoenix (1996).

Copyright Elsevier; used with permission.
NASA/TM—2011-215805 52

Bazant and Pang (2007) developed a multiscale model composed of mixed series and parallel systems 

that they argue provides a physical underpinning to the Weibull distribution. In their model, quasi-brittle 

materials (including fiber composites, polygranular graphite, and concrete) will have a two-parameter 

Weibull distribution at the lower probabilities of failure—the lower tail of the strength distribution—

which is the region of highest interest to the design engineer. In their approach, a material is modeled with 

a representative volume element (RVE) subunit. The structure (of positive geometry2

) is modeled as a 

weakest-link-series system of RVEs, but each RVE is modeled as a parallel coupling of only two long 

subchains, each consisting of subbundles of two or three long sub-subchains of sub-subbundles, and so 

forth, until the nanoscale of the atomic lattice is reached. This is shown schematically in Figure 24.

The RVE is taken as the size of the fracture process zone (FPZ), which is roughly 3 times the size of 

the maximum inhomogeneity size (grain size for ceramics and graphite). Medium-grained graphites can,

therefore, have FPZs that are several millimeters long. Therefore, structures with cross sections much 

larger than the FPZ are brittle, and smaller structures are quasi-brittle.

Figure 24.—Model of quasi-brittle structure, where Neq is the equivalent 

number of links in a series. (a) Hierarchical model of representative volume 

element (RVE) consisting of subchains and subbundles. (b) Weakest-link 

series system of RVEs representing the structure. Reproduced from Bazant 

and Pang (2007). Copyright Elsevier; used with permission.

 2

Positive geometry means that the stress intensity factor at constant load increases with the crack length.
NASA/TM—2011-215805 53

Figure 25.—Size effect on the cumulative distribution function of structural strength with the grafting probability of 

failure Pf,gr = 0.003; Neq is the number of identical links or subunits of volume of identical size, ωo is the coefficient 

of variation, and RVE is the representative volume element. (a) Linear scale. (b) Weibull scale. Reproduced from 

Bazant and Pang (2007). Copyright Elsevier; used with permission.

At the atomic scale, the Bazant and Pang (2007) model has a power-law tail of exponent 1 of the 

cumulative distribution function of strength based on the Maxwell-Boltzmann distribution of the thermal 

energies of atoms. Bazant and Pang argue that the threshold of the power-law tail must be zero. In the 

model, the power-law tail exponent of a chain represents the lowest exponent of its individual links, and 

the power-law tail of a bundle is equal to the sum of the power-law tail exponents of all the parallel fibers 

in the bundle. Scaling this behavior up through series couplings in chains and parallel couplings in 

bundles raises this exponent to the value of the Weibull modulus observed at the level of the structure and 

is interpreted as being equal to the number of dominant cracks needed to break the RVE. At the RVE 

level, the model has a gaussian distribution (between the tails) with a power-law-tail exponent corresponding to the Weibull modulus grafted at failure probabilities in the neighborhood of 0.0001 to 0.01. 

This gives rise to a kink in the distribution. This “grafting point” moves to higher failure probabilities as 

the structure size increases and can be used to determine the size of the RVE as indicated in Figure 25.

Size effect is also indicated in the figure.

The model indicates that, on the scale of at least 500 RVEs and larger, the structure follows a Weibull 

distribution with zero threshold (a two-parameter Weibull distribution). Bazant and Pang posit that threeparameter Weibull distributions (indicating a nonzero threshold), which are often empirically fitted to 

experimental rupture data, are actually misinterpretations of the kink produced by the grafted twoparameter Weibull distribution on the core gaussian distribution of the structure. The Bazant and Pang 

model is an interesting reinterpretation of the Weibull model. 

2.2.7 Bundle Model Ductile-Like to Brittle-Like Transition

Mahesh et al. (1999) compared ELS modeling with the HVDP LLS model for fiber system sizes 

ranging from 15 by 15 to 50 by 50 arrays and fiber Weibull moduli ranging from m = 1/2 to 10. Their 

simulations showed that, for low Weibull moduli (m ≤ 1), the ELS and LLS models correlated closely. 

The particular LLS model indicated that low-Weibull-modulus fibers fail in a more dispersed manner and 

that stress concentration from load sharing is not dominant. That trend was opposite for higher Weibullmodulus fibers, where fiber failures clustered and damage accumulation was driven by stress concentration. This seems to imply that two competing failure modes exist and that the magnitude of the Weibull 
NASA/TM—2011-215805 54

modulus of the fiber controls the failure behavior of the composite—ranging between the ELS fiberstrength-dominated failure mode and the LLS fiber-strength-concentration-driven failure mode.

On the other hand, Mahesh and Phoenix (2004) indicate analytically that idealized LLS bundles 

behave in a WLT manner driven by extreme-value statistics regardless of the strength variability of the 

individual fibers and that critical defects form, although very large samples may be required—much 

larger than that used in Mahesh et al. (1999). Their result implies that there is no disorder-induced 

transition from brittle failure to a more gradual dispersed failure regime. Phoenix (2007) says that for 

practical purposes there is an LLS-like to ELS-like transition. For idealized LLS, it is m = 1; for a planar 

model, it is approximately m = 2; and for the hexagonal packing model, it is m = 4. So above these fiber 

Weibull moduli values, LLS-like behavior is predicted; and below these values, ELS-like behavior is 

expected.

To study failure behavior, Hidalgo et al. (2002) used a load-transfer function that decays as 1 , d

rκ

where rd is the distance from a broken fiber. When κ was close to zero, they obtained ELS behavior, and

for large κ, they obtained LLS expected behavior with κ ≅ 2 at the transition region. Alava (2006) notes 

the similarity of this result with the two-dimensional random fuse (lattice) model where load (current) 

is transferred in a nonlocal, but geometrically dependent, fashion with the redistribution decaying as 

2 1 d

r —which they state is the crossover point between local and global behavior.

Another aspect of failure behavior worth considering is that the variability of fiber strength affects the 

toughness of the composite. This was shown by Beyerlein and Phoenix (1997), who investigated the 

effect of a transverse notch of broken fibers on the fracture process, fracture resistance, and overall 

strength distribution for an elastic composite lamina. They showed, that for low-Weibull-modulus fibers,

the composite becomes tougher as crack growth proceeds—that is, R-curve rises, with consequently

stable crack growth. However, high-Weibull-modulus fibers resulted in a flat R-curve and, thus, unstable 

crack growth. This interesting observation seems to fall in line with Planas (1995) and Bazant and Planas 

(1998), as discussed in Section 2.1.4.

2.3 Lattice Models

Brittle materials such as monolithic ceramics and graphites appear to be homogeneous at the macroscale but are heterogeneous at the microscale. They consist of nonuniform distributions of the grain 

shapes, (possibly) multiple material grain phases, grain boundary, and flaws such as voids, inclusions, and 

microcracks. This represents a level of disorder that can be described as a characteristic of the material. In 

a material with broad disorder (quasi-brittle materials), fracture is not influenced exclusively by the weakest flaw in isolation but also by the proximity and local interaction of defects, including stress concentration and shielding effects around defects. As the material reacts to increasing load, defect clusters can 

develop, evolve, and coalesce, leading to the final breakdown of the material. 

Since the 1980s, various types of discrete element models, called lattice models, have been developed to investigate the fracture characteristics of quasi-brittle materials. As mentioned in Nukala and 

Simunovic (2004) and Phoenix and Beyerlein (2000b), there are electrical fuse (conductivity breakdown)

models (de Arcangelis and Redner (1985), de Arcangelis et al. (1986), Duxbury et al. (1986, 1987), 

Duxbury and Leath (1987), Li and Duxbury (1987, 1989), de Arcangelis and Herrmann (1989), Kahng 

et al. (1988), and Roux et al. (1988, 1991)); central-force (spring) models (Sahimi and Goddard (1986), 

Beal and Srolovitz (1988), Feng and Sen (1984), and Hansen et al. (1989)); bond-bending models (Kantor 

and Webman (1984), and Sahimi and Arbabi (1993)); and beam category models (Herrmann and Roux 

(1990), and Herrmann et al. (1989)). Alava et al. (2006) provide an excellent review of this work, and 

Nukala and Simunovic (2004) mention other comprehensive references: Hansen and Roux (2000), 

Herrmann and Roux (1990), Sahimi (1998), and Chakrabarti and Benguigui (1997). Additional useful 

references used here include Nukala et al. (2006), Zapperi and Nukala (2006), and Nukala and Simunovic 

(2004).

The following short summary highlights important results regarding failure mode, strength distribution, and size effect. However, stress-strain response, acoustic emission, toughening, and fracture 
NASA/TM—2011-215805 55

morphology are not discussed herein. The interested reader should begin with the previously mentioned 

review articles for an overview of the various material breakdown phenomena that have been 

investigated.

Lattice models are used to simulate the state of initial material disorder and the subsequent material 

breakdown process from progressive loading. Interesting questions can be investigated with these models,

including size scaling, strength distribution, nonlinear elastic response as it relates to damage, the role of 

disorder (as well as the relation to percolation), avalanches—or bursts of microfailures (which can be 

related to acoustic emission), and weakest flaw versus crack clustering. Lattice models are arrangements 

of discrete springs, beams, or electrical fuses linked together in an ordered cubic, triangular, or diamond 

lattice. Over the last 25 years, the majority of lattice model research has been with random fuse models 

(RFMs), introduced by de Arcangelis et al. (1985). These are electrical analogs to a loaded structure 

where the link is a fuse and the electrical current, voltage, and conductance are analogous to stress, strain, 

and Young’s modulus, respectively. This simplifies the model by substituting a scalar field for vectorial 

elasticity, decreasing the degrees of freedom of the system and therefore allowing larger system sizes to 

be solved for the available computational resources. Figure 26 shows a schematic of the RFM.

Nukala et al. (2005) compared the RFM with the random spring model (RSM) for a triangular lattice 

with a uniform distribution of failure threshold and found that the two models gave qualitatively similar 

results for fracture strength distribution, size effect, and damage characteristics. They concluded that the 

simplified scalar model of the RFM could be used to investigate the statistical properties of fracture in 

disordered media.

Figure 26.—Random fuse model (RFM). A triangular lattice is shown where 

each bond has a random breaking threshold, where V is the voltage and 

I is the current. Reproduced from Alava et al. (2006). Copyright Taylor & 

Francis Group; used with permission.
NASA/TM—2011-215805 56

2.3.1 Disorder and System Breakdown

Modeling disorder in a material is related to the concept of percolation. A lattice that has a randomly 

selected fraction of bonds allowing a transport process (diffusion, conduction, flow, and force) while the 

rest of the bonds are insulating (nonactive) is defined as a bond percolation network. Work on percolation 

theory is extensive and has been applied to many areas including flow phenomena in porous media, 

transport, mechanical and rheological properties of disordered materials, diffusion, and precipitation. 

Percolation is mentioned here because much of the literature regarding lattice modeling refers to this body 

of work. However, although fractured media are similar to the concept of percolation, there are distinct 

differences. Crack coalescence and macrocrack propagation is one difference, and randomly distributed 

microcracking is one similarity. Sahimi (1998) provides a good review of this body of work.

Disorder is introduced in a lattice by assigning the individual elements or links (springs, beams, or 

fuses) random values of a property, such as failure threshold, according to a prescribed statistical distribution. In this manner the level of disorder input into the system can be quantified. In its initial state, the 

lattice may be complete or have some fraction of the links removed. Applying a progressively increasing 

load or displacement causes individual bonds to break (or be diluted as in Zapperi et al. (1997)) and 

causes the internal resistance load to redistribute until a critical instability is reached and the lattice 

system breaks down. In other words, the bonds break one by one until a path of broken bonds is created 

that prevents any load (or current) from being transmitted from one end of the lattice, where load or 

displacement (or voltage) is prescribed, to the other end. One can readily visualize that there are certain 

similarities between lattice models and the fiber breakage models described in Sections 2.2.2 to 2.2.6. 

This is a “quenched” disorder problem, where the disorder is introduced prior to the breakdown process 

(the breaking threshold of each bond is assigned before loading and held constant as loading progresses). 

In contrast, an “annealed” disorder problem is where disorder is introduced stochastically as loading (and 

breakdown) progresses. 

Weak disorder systems (where the variation in bond strength is small) are dominated by stress 

concentrated around the crack tips and the early onset of unstable crack growth because of the lower 

probability of encountering strong bonds that will arrest crack growth. Consequently, WLT behavior 

would be expected in a weak disorder system. Conversely, in strong disorder systems (where variation in 

bond strength is large), a “cloud” of microcracks (blown fuses) form and eventually coalesce into a 

structure-spanning crack. Here, both the weakest and strongest bonds have important roles in the breakdown process. The stronger bonds arrest or redirect developing cracks. The weaker bonds initiate 

microcracks or allow existing cracks to propagate. The strong bonds, by arresting cracks, enable further 

microcrack nucleation at new sites. In this manner, distributed damage can develop in strong disorder 

systems. Figure 27 demonstrates the development of distributed damage and crack coalescence in an

RFM for a strong disorder system. Figure 28 shows damage progression from an initial notch in an RFM 

with strong disorder. Note that in Figure 28 diffuse damage still occurs (shown as specks in the figure) 

despite the presence of a large, centrally located macrocrack. 

A basic assumption in RFM is rapid current redistribution upon bond breakage (current redistributes 

before another bond breaks), whereby equilibration occurs before load can increase—simultaneous bond 

breakage is not allowed. Multiple bond breakage has been studied little. Duxbury et al. (1986, 1987) 

argue that, for weak disorder systems, the largest crack (connection of broken bonds) controls the 

breaking process and has current flow enhancement (stress enhancement) of a factor proportional to the 

square root of the crack length. Sahimi (1998) mentions that three-dimensional stress enhancement is 

much weaker than two-dimensional enhancement, so even a modest amount of disorder can result in 

distributed microcracking. 
NASA/TM—2011-215805 57

Figure 27.—Snapshots of damage in a two-dimensional triangular lattice of size L = 512. Number of 

broken bonds nbr at the peak load and at failure are 83 995 and 89 100, respectively. Parts (a) to 

(i) show snapshots of damage after breaking nbr bonds. (a) nbr = 25 000. (b) nbr = 50 000. (c) nbr = 

75 000. (d) nbr = 80 000. (e) nbr = 83 995. (f) nbr = 86 000. (g) nbr = 87 000. (h) nbr = 88 000. (i) nbr = 89 100 

at failure. Reproduced from Nukala and Simunovic (2004) and Alava et al. (2006). Copyright Springer 

Publishing Company and Taylor & Francis Group; used with kind permission of Springer Science+Business 

Media and Taylor & Francis Group.
NASA/TM—2011-215805 58

Figure 28.—Damage and crack formation from an initially notched random fuse model (RFM). (a) Damage 

propagation. (b) Although diffuse damage occurs, a spanning crack eventually forms from the notch. Reproduced 

from Alava et al. (2006). Copyright Taylor & Francis Group; used with permission.

2.3.2 Lattice Brittle-to-Ductile-Like Transition

Kahng et al. (1988) investigated the role that disorder plays in the transition between brittle and 

quasi-brittle, or ductile-like, failure and showed a relationship between this transition and system size L

(number of fuses along the busbar) in their model. Figure 29 is reproduced from this reference. 

In their probabilistic modeling and RFM simulations, Kahng et al. (1988) assumed that each bond had 

unit conductance and that the bonds had a uniformly distributed voltage threshold between 1 – w/2 and 

1 + w/2, so that the average threshold voltage was unity and the width of the bond-breaking distribution w 

was between 0 and 2. Upon breaking the first (the weakest) bond in the system, the vertical bonds 

horizontally adjacent to the broken bond have the largest local change in voltage. Those bonds are the 

ones most likely to break next without any additional increase in the externally applied voltage potential.

For sufficiently small w, less than the threshold wo, the initial failure of this weakest bond is always 

unstable to further cracking. This regime is the “trivial” region bounded by w = wo in Figure 29 and is 

trivially governed by extreme-value statistics. Kahng et al. (1988) argues that wo ≅ 0.24 is the lower 

bound for the true value of wo. For w > wo, there is a nonzero probability that the initial crack is stable 

without further increase of the external potential. In this region, shown in Figure 29 as the “brittle” region, 

failure occurs after one of the first few bonds broken nucleates a crack that catastrophically propagates

across the system. Therefore, fracture is controlled by the weakest, or one of the weakest, bonds in the 

system. 
NASA/TM—2011-215805 59

Figure 29.—Random fuse model (RFM) showing a brittle-to-ductile-like 

transition region as a function of system size L. The term w is the 

span over which a uniform distribution of fuse breaking threshold 

occurs. The average threshold is assumed to be 1, the uniform 

distribution is applied between 1 – w/2 and 1 + w/2, wc(L) is the critical 

transition of w between brittle-like and ductile-like behavior as a 

function of L, and wo is the threshold of w where single-bond 

breakage causes system failure. Reprinted Figure 1 with permission 

from Kahng, B.; Batrouni, G.G.; Redner, S.; de Arcangelis, L.; and 

Herrmann, H.J.: Phys. Rev. B, vol. 37, no. 13, 1988, p. 7626 (Kahng 

et al., 1988, http://link.aps.org/abstract/PRB/v37/p7625). Copyright

(1988) by the American Physical Society.

In the so-called ductile (ductile-like) regime, breakdown of the network is more gradual: individual 

bonds break over a large range of monotonically increasing voltage potential (monotonically increasing 

load). The boundary between brittle and ductile-like failure, denoted by the curve labeled wc(L) in 

Figure 29, is defined by Kahng et al. (1988) as the equality of likelihood of the unstable growth of an 

existing crack versus the forming of a new crack, which occurs after nc bond breaks, and the likelihood

that the n

th weakest vertical bond is spatially independent of all other cracks already in the system (related 

to the problem of “birthday coincidences”). If nindep is the maximum number of n where spatial independence is maintained, then if nindep > nc the system is brittle. Conversely, if nindep < nc, then crack growth is 

as likely to occur as new crack initiation when the external potential is increased, and failure of the 

network will then be more gradual than brittle fracture. Since this is a probabilistic argument, it does not 

represent an abrupt transition between brittle and ductile-like behavior. Kahng et al. (1988) indicate that 

wc is a function of system size L (proportional to volume). It is significant to note that as volume increases 

brittle behavior is expected to predominate even in high-disorder systems. In the limit of infinite system 

size, the RFM is brittle except in the special case of w = 2, which implies that the distribution of fuse 

threshold has to include the zero value for quasi-brittle behavior to exist at this limit.

Sahimi and Goddard (1986) used Hooke-type spring lattice models to investigate the role of disorder 

in brittle versus distributed damage (ductile-like) behavior. They used three basic models involving the 
NASA/TM—2011-215805 60

stochastic spring constant ks and the critical strain εc with various probability distributions. Model I 

assigned the same spring constant ks = 1 to some fraction p of randomly chosen springs; the remaining

p − 1 fraction had ks = 0, and critical strain was constant at εc. Model I simulated the initial state of 

microporosity in the material. Model II maintained εc as constant, but ks was randomly distributed to a 

prescribed distribution function. In model II, heterogeneous constituents had varying elastic response,

although they had the same strain-to-failure threshold. Model III maintained constant ks = 1 but assumed 

that εc was the distributed quantity. In other words, in model III the bonds represented individual grains 

with different strain-to-failure thresholds, or perhaps they represented small regions of material with 

stochastic strength response. Regions of varying residual stress were not considered in these models 

although this was considered by Curtin and Scher (1990a,b).

For Model I for p ≅ 1, brittle failure was observed where a broken spring caused adjacent springs to 

fail in a cascade. However, for p ≅ pce (elastic response threshold where removed members affect the 

elastic response of the system) cracks were generated at several locations and propagate. This behavior 

differentiates intrinsically flawed systems from damaged systems.

For Model II for a uniform distribution of ks between 0 and 1, brittle behavior was maintained unless 

a large fraction of springs had ks = 0 as in Model I. Quasi-brittle behavior of microcrack formation and 

propagation was observed for a power-law type distribution of ks:

fk k ( s s ) (1 01 ) −φ = −φ <φ< (52)

when φ became larger (the distribution became broader). For φ ≅ 0, brittle behavior was observed because

the distribution was very narrow.

For Model III, assuming a uniform or power-law distribution for εc (similar to Eq. (52)) showed 

behavior similar to Model II. For the uniform distribution, behavior was brittle-like with a single crack 

forming and subsequently spanning the entire lattice. Using Equation (52) caused microcracks to initiate 

at numerous distributed locations before ultimate breakdown. Sahimi and Goddard (1986) noted that 

sometimes crack growth was arrested when a relatively strong region was encountered. They proposed 

that the macroscopic response of the system depends on the form of the probability distribution functions 

(PDFs) for ks and εc, whereby if the first inverse moment of the PDF is finite, a brittle failure mode 

occurs, but if the first inverse moment of the PDF is infinite, distributed damage is the dominant 

mechanism leading to failure. That is, if f–1 is finite,

1 0

( ) d f y f y

y

∞

− = ∫ (53)

and the probability density function f (y) of either ks or εc is finite, then brittle fracture behavior is 

expected. However, if f–1 is divergent, then quasi-brittle behavior is expected.

Hansen et al. (1991) and Batrouni and Hansen (1998) used RFM with breaking thresholds uniformly 

distributed between 0 and 1, but raised to an exponent. The cumulative probability distribution was

P(σ) = σ1/|φ|

when φ > 0 and was P(σ) = 1 − σ1/|φ|when φ < 0. The smaller the value of |φ|, the smaller that 

the amount of disorder in the system was. When φ > 0, the power-law tail extended toward the weak 

bonds, and when φ < 0, the tail extended toward the strong bonds. For small values of |φ|, brittle behavior 

was observed where a macroscopic crack developed early in the fracture process. For large values of |φ|,

quasi-brittle behavior occurred where a cloud of disconnected microcracks developed before the cracks 

coalesced into the final macroscopic crack.

Although beyond the scope of this article, microfracturing has an obvious relationship to acoustic 

emission observed in real materials prior to fracturing. Strong disorder will cause significant microfracturing events prior to final breakdown. Quasi-static lattice models qualitatively indicate a power-law 

distribution of avalanche events (not including the last avalanche event), where more activity occurs as 
NASA/TM—2011-215805 61

the peak failure load is approached. The aforementioned review articles are a good starting point for

exploring this subject further.

2.3.3 Strength Distribution and Size Effect in Lattice Models

In section 6.3.1 of their journal article, Alava et al. (2006) note several studies (references not 

reproduced here) that showed in randomly diluted disorder problems (where some fraction of the links 

were randomly removed prior to loading) the defect-cluster-size distribution was exponential far away 

from the percolation threshold (the threshold where long-range connectivity first appears) but followed a 

power law close to the percolation threshold. The implication is that a Gumbel distribution more appropriately describes the fracture strength distribution far away from the percolation threshold, and a Weibull 

distribution is more appropriate close to the percolation threshold. However, Sahimi (1998), in section 

7.2.1.1 of his journal article, cites work indicating that Weibull and Gumbel worked equally well away 

from the percolation threshold but that neither distribution performed very well near the percolation 

threshold. For RFM models with weak disorder, the failure is dominated by the stress concentration near 

the crack tip, implying that WLT behavior should dominate and that fracture strength should follow the 

Weibull or Gumbel extreme-value distributions (Alava et al. (2006)). For strong disorder, a different 

situation arises because of the distributed damage, which is discussed later in this section.

The Gumbel distribution—as determined from Duxbury and Leath (1987) and Duxbury et al. (1986, 

1987) for randomly diluted RFM and subsequently used in an elastic spring model of fracture by Beale 

and Srolovitz (1988)—expresses the failure threshold stress (stress required to break the first bond in the 

system) probability distribution as

( ) 1 exp exp D G

f G u

k P c L    

σ= − − −         σ

(54)

where cG and kG are constants characteristic of microscopic properties of the system, D is the number of 

dimensions of the system (D = 2 for a two-dimensional network {∝ area}, D = 3 for a three-dimensional

network {∝ volume}), and 1 < u < 2. The defect cluster size distribution is exponential, and a size-effect 

relation is obtained by equating failure probability for different system sizes L1 and L2:

1/

2 1/

2

1 1

1

1 ln

G

u

k D

u D

L

L

 

 

σ =      

+    

σ        

(55)

In contrast, the Weibull distribution has the form

( ) 1 exp{ } D m Pf G σ= − − σ c L (56)

where the defect cluster size has a power-law distribution. The size-effect relation (also see Eq. (12)) is

1/

1

2 1

2

 

m D

D

L

L

 

σ =σ     (57)

It is not completely established which distribution (Weibull or modified Gumbel) is most appropriate 

for lattice models with weakly disordered bonds.
NASA/TM—2011-215805 62

For a strong disorder system, damage is distributed, rather than localized, and the largest defect 

cluster at peak load may not be responsible for the final material breakdown cascade. For two- and threedimensional lattice architectures, respectively, Nukala and Simunovic (2004) and Zapperi and Nukala 

(2006) investigated the effects of strong disorder on large lattice size (in comparison to previous studies) 

RFM. For the two-dimensional case, L was as large as 512, and for the three-dimensional case, it was as 

large as 64. Their systems were initially intact, with each bond being linear elastic until fracture, and they 

assumed a uniform distribution of fracture thresholds between 0 and 1. This approach differs from many 

earlier studies, where a random fraction of the links was removed prior to loading (percolation disorder). 

Nukala and Simunovic (2004) state that the defect cluster distribution evolves differently for the two 

approaches. In the initially intact RFM, the defect clusters evolve not only from the weakest bonds but

from the stress concentration effects around the defect clusters. The simulation results show broadly 

distributed damage (see Figure 27) and indicate that the fracture strength (the stress at peak load) was 

neither Weibull nor Gumbell distributed but rather best fit to the lognormal distribution. Nukala and 

Simunovic (2004) argue that this arises by way of the central limit theorem: “the product of a large 

number of independent factors, none of which dominates the product, will tend to the lognormal distribution regardless of the distribution of the individual factors involved in the product.” This contrasts with

weak disorder (narrowly distributed breaking thresholds), where the breaking of a bond significantly 

influences the further breakdown of the material and leads to WLT behavior and extreme-value theory.

It is interesting that, for the size effect, Nukala and Simunovic (2004) and Zapperi and Nukala (2006)

found that the Gumbel distribution form of Equation (55) worked well but that the Weibull size-effect

scaling did not. Specifically they found that 

1 2

1 1

c

c D

D L

L

Φ− +

μ − σ= + (58)

where mean fracture strength σμ = peak force/LD–1

, c1 and c2 are constants, and D = 2 for the twodimensional mode and D = 3 for the three-dimensional mode. The exponent Φ is 0.96 for RSM, 0.97 for 

RFM for both diamond and triangular lattice networks in two-dimensional models, and 1.95 for threedimensional RFM. For (D – 1 – Φ) << 1, Equation (58) could be approximated as

( )

3 2

1

c c

ln D L L μ ψ − σ= + (59)

where c3 is also a constant and ψ has a value of approximately 0.15. This suggests that for large lattice 

systems the mean fracture strength decreases very slowly, scaling proportional to 1/(ln L)

ψ

. However, a

word of caution has been sounded by Phoenix and Beyerlein (2000b). They note that the limited 

simulation system sizes L reported in the literature are often much too small to reveal the ultimate largescale behaviors. They recommend more studies on a larger scale of several orders of magnitude and 

analytical asymptotic studies, if possible. They state

“...regardless of their points of view, many investigators have turned to rigorous study of 

idealized, one-dimensional models of failure in an attempt to put approximate analysis 

and interpretations from simulations of more complex networks on firmer ground. Such 

models, which are often variations on the LLS models of Harlow and Phoenix, are 

analytically solvable, rich in behavior, and qualitatively show many features seen in 

simulations. In most cases, results in LLS fiber-bundle models support the logarithmic 

size scaling, but more generally, such results depend on the load-sharing scheme (LLS vs

ELS) and on the assumed form of the distribution for element failure.”
NASA/TM—2011-215805 63

Alava et al. (2006) state that for strong disorder the lognormal strength distribution is observed, but

they concede that “whether this is a general result valid at all scales or the signature of a crossover 

behavior to Gumbel or Weibull scalings, still remains to be explored.” They argue that models such as 

RFM may lie “exactly at the boundary between LLS and ELS and the lognormal distribution could then 

arise from the competition between local and global effects in a strongly disordered environment.”

Another interesting result comes from Nukala et al. (2006) regarding crack-cluster distribution for 

broad (strong) disorder systems using RFM (with an initially intact lattice with a uniform distribution of 

fracture thresholds). They found in their two- and three-dimensional models that the largest crack-cluster 

size distribution at peak load fitted a lognormal distribution rather than a power-law or exponential 

distribution, as previously indicated for randomly diluted networks (percolation disorder). They also 

found that the final spanning crack was often formed from the coalescence of smaller cracks rather than 

from the propagation of the largest crack at peak load; therefore, the largest crack at peak load and the 

fracture strength (at peak load) were not correlated (or were weakly correlated at best).

Alava et al. (2006) discussed notch sensitivity where an initial notch is introduced into a twodimensional RFM lattice with uniform threshold distribution bond strength. Figure 28 shows an example 

where the role of disorder can be seen in the wandering crack path as well as in the diffuse distributed 

damage that also developed. This illustrates the interplay between an existing crack of size a0 relative to 

the system size L and the level of disorder in the system. Figure 28 is also interesting in that it shows a

preexisting crack dominating the fracture response but that distributed damage also occurs—indicating 

that, in a real material, independent randomly distributed microfracturing events (evidenced by acoustic 

emission) could occur even if a preexisting crack was dominating the fracture behavior. 

A weak disorder leads to LEFM behavior and extreme-value statistics for heterogeneous materials 

with a population of preexisting cracklike flaws of variable (stochastic) severity. That is, the extremevalue statistics of the flaw population drive the material strength response. Strong material disorder is a 

more complicated situation, where the crack size of the preexisting flaw relative to the size of material 

microstructural heterogeneities can affect the size-effect responsegiving intermediate behaviors. 

Alava (2006) investigated the interplay between a0/L, disorder, and L on size effect. Results from 

simulations of fracture strength for various a0/L ratios versus system size L were fitted with a power law 

where fracture strength was proportional to L–c

, where c is a constant best fit for a given a0/L ratio. For a 

small a0/L, c is small and equivalent to the logarithmic relation described previously, but for a larger a0/L, 

c approaches one-half as predicted from LEFM. In other words, the presence of large intrinsic flaws in the 

strong disorder heterogeneous material will also lead to LEFM behavior and fracture strength governed 

by extreme-value statistics. Alava et al. (2006) also mention that the FPZ is typically rather small for 

RFM and decreases in relative size as a0/L increases. Curtin (1997) investigated the toughening effect that 

disorder plays in the RSM with an introduced notch. Curtin concluded that the difference in toughening 

between weak and strong disorder was only about 10 percent in the model and that, in real materials, 

other effects such as grain bridging could play a larger role.

3.0 Discussion and Summary

“... as we know, there are known knowns; there are things we know we know. We also 

know there are known unknowns; that is to say we know there are some things we do not 

know. But there are also unknown unknowns—the ones we don’t know we don’t know.” 

(D.H. Rumsfeld, former Department of Defense secretary, February 12, 2002, 

Department of Defense news briefing)

The fracture of nuclear-grade graphite is a function of the distribution, interaction, and growth 

kinetics of flaws of various forms and size scales. Graphite fracture strength is stochastic: Nominally 

identical tensile and flexural specimens show large specimen-to-specimen variation in strength. Average 

strength has been shown to vary within billet, between billets, and with the orientation relative to the 

billet (strength anisotropy relative to processing direction). Different grades of graphite can have very 
NASA/TM—2011-215805 64

different morphologies and, it would be expected, different fracture behaviors ranging from classically 

brittle to quasi-brittle. Add to this the fact that the stress-strain response is nonlinear and involves 

irreversible accumulation of damage, and it becomes clear that the fracture behavior of graphite is 

complex, falling somewhere between brittle monolithic ceramics and damage-tolerant composite 

materials. Therefore, models that access the failure potential of graphite structures will need to consider 

this rich behavior as well as account for the statistical nature of graphite failure. An ultimate goal would 

be to develop statistical models of failure that can be employed with finite-element programs, such as 

CARES/Life (see Nemeth et al. (1993, 2003, 2005)), to predict the probability of failure of a generalized 

structure under thermomechanical loading.

In experimental studies of graphite, use of the Weibull model to predict failure has met with apparent 

mixed success, and others have criticized the applicability of its physical assumptions. This has led to the 

development of alternative models. This review article describes various statistical models of brittleconstituent-controlled material failure. Some of these models were developed specifically for graphite. 

All the models assume that material breakdown begins at inherent flaws or weak regions in the material. 

However, the models differ in how material breakdown proceeds. They are broadly categorized into 

series-system or parallel-system models. Table I summarizes the models that are discussed in this report. 

From the survey of literature that was performed herein, the following four overarching observations can 

be made:

(1) WLT modeling applies over a broad range of material systems ranging from the classically brittle 

to distributed damage (bundle and lattice) models, and at low probabilities of failure this behavior may be 

approximated by a Weibull distribution. The implication is that WLT models may be appropriate for 

graphite despite the fact that the material displays distributed damage.

(2) There may be a transition where material behavior (fiber composite or monolithic) goes from 

classically brittle to quasi-brittle, which is apparently a function of the level of disorder in the material 

system. For graphite, this means that the inhomogeneity of local properties has been identified as a means 

with which to explore and possibly engineer the failure behavior of the material.

(3) For high-disorder (highly quasi-brittle) material systems, the process of microcracking leading to 

crack coalescence may not be governed by extreme-value statistics: that is, Weibull or Gumbel. The 

Weibull distribution may work empirically to approximate the distribution of strength, but for those 

grades of graphite that display a large amount of distributed damage, the actual failure mechanism may 

not necessarily be driven by WLT extreme-value statistics. This may need further investigation.

(4) However, for high-disorder material systems, intrinsic flaws, starting at some size threshold, 

behave according to LEFM; therefore, the material strength should follow an extreme-value distribution

like a Weibull distribution. The implication is that larger sized flaws (significantly larger than the individual grains in the microstructure) will control the strength response of graphite and that the distribution of 

the sizes of these flaws requires that the strength response be an extreme-value distribution such as a 

Weibull distribution.

Some of the important points that will be discussed in subsequent sections follow: 

(5) The Weibull distribution is argued to be the most appropriate probability distribution for fitting to 

experimental rupture data and extrapolating to low probabilities of failure. This is true for brittle and 

quasi-brittle materials including graphite.

(6) Size effect and strength dispersion are functionally related in WLT. This is true for any material 

whose strength response is governed by the WLT mechanism.

(7) A Watson-Smith-style modified Weibull formulation is proposed for graphite to reduce the size 

effect and better correlate with experimental data.
NASA/TM—2011-215805 65

In addition, miscellaneous issues including the physical interpretation of the Weibull modulus, the 

stress gradient, scaling problems, multiaxial stresses, microstructural models such as Burchell’s, and 

probabilistic design are discussed and summarized.

3.1 Uncertainty in Experimental Data and the Weibull Distribution

For fitting a probability distribution to experimental rupture strength data, including extrapolation to 

low probabilities of failure, the Weibull distribution is the most appropriate choice. Although the use of 

the Weibull stress-volume integral (Eq. (8)) has detractors as well as proponents, the Weibull distribution 

(expressed independently of volume; see Eq. (4) and Weibull (1951)) fits the available experimental data 

as well as or better than other commonly used distributions. For design purposes, the trends of experimental data must be extrapolated to very small probabilities of failure. The Weibull two-parameter distributions tend to be more conservative (or pessimistic) than, for example, gaussian or lognormal distributions. 

Liu (1997) systematically studied data sets of finite size (up to 100 samples per data set) generated from 

the two-parameter Weibull and lognormal distributions where distribution parameters for one distribution 

type were erroneously estimated from data generated from the other (true) distribution type. Liu concluded, “erroneously classifying data and using one or the other model is less damaging if the Weibull 

model is chosen than if the lognormal model is used.” In other words, the bias of the Weibull distribution 

tended to be conservative, whereas the bias from the lognormal distribution tended to be nonconservative. 

This level of bias increased at lower probabilities of failure. For example, if the sample is from the 

Weibull distribution but is erroneously fitted to a lognormal curve, the extrapolated failure probability at 

low levels of probability of failure (the lower tail of the distribution) will be too low and, therefore,

nonconservative. If the sample is from the lognormal distribution but is erroneously fitted to a twoparameter Weibull curve, the extrapolated failure probability will tend to be lower and, therefore, more 

conservative.

“For 20 or less failures (in data), always use the two-parameter Weibull distribution even 

if you know the underlying failure mechanism demonstrates a different distribution—the 

reason for selecting the two-parameter Weibull is both a more stable predictor and a more 

conservative predictor” (interpretation of Liu’s results by Abernethy (2008) of Weibull 

Analysis Handbook fame (Abernethy et al., 1983)). 

Therefore it seems clear that, based on the uncertainties of using experimental data and/or whenever 

there is doubt as to the true source of the data, the Weibull distribution should always be assumed. A mild 

caveat to this statement is that, for this report, the authors could not find a study of all possible distribution types—either pro or con—for this position. The Burchell model was a lognormal distribution of pore 

sizes estimated from material micrographs. In this case, extrapolation to low probabilities of failure means 

extrapolating the lognormal distribution to larger sizes of pores. It is not clear how the uncertainties 

related to this extrapolation would compare with the Weibull distribution of strength.

Another important factor to consider is that flaw populations may not be detected when specimens are 

tested. A low-Weibull-modulus flaw population is a scarce flaw population. Testing of small-size

specimens (or measuring pore-size distributions in micrographs to characterize microstructural models 

such as the Burchell model) may not sample enough material volume to detect these larger and weaker 

flaws (which are distinct from background defects). Therefore, extrapolating results from specimen 

testing to lower probabilities of failure for structures with larger volumes could result in a nonconservative estimation of structural strength if a more severe flaw population was not detected during specimen 

testing and was not accounted for. This is true regardless of the probability distribution used for the 

extrapolation. From Figure 10 it is readily apparent that large gross flaws may exist in graphite blocks,

but specimen testing may miss these flaws (specimens that contain these gross flaws might even be 

rejected before fracture testing by “quality control”).
NASA/TM—2011-215805 66

When the parameters of a probability distribution are estimated from a set of experimental data, there 

is an inherent uncertainty as to what the values of the true parameters of the underlying distribution are. 

The smaller the number of samples there are within a data set, the larger the range of uncertainty is for 

both the true values of the underlying distribution and the failure stress for a given probability of failure. 

This leads to the question of confidence bounds on parameters and confidence bands on distributions. 

Basically, the values of these intervals can be specified with a given level of statistical certainty. For 

example, the Weibull modulus can be predicted to lie within a certain range of values with a 90-percent

level of confidence for a given number of samples in a data set. A discussion of the methodology for 

determining confidence intervals, confidence bands, estimated parameters, and other statistical metrics is 

beyond the scope of this report; however, some useful references regarding the Weibull distribution 

include Thoman et al. (1969), Abernethy et al. (1983, and later revised editions), Pai and Gyekenyesi 

(1988), and Danzer et al. (2008). Meeker and Escobar (1998) provide a more generalized and comprehensive textbook regarding these subject areas. In addition, Weibull plotting software programs such as 

WeibPar (Connecticut Reserve Technologies, Inc., 2009) and SuperSMITH (Barringer & Associates, Inc., 

2009) are available for estimating Weibull parameters and determining confidence intervals and 

confidence bands. 

For materials that behave consistent with the Weibull distribution, data from separate data sets of 

various sized specimens are commonly pooled (combined) to estimate Weibull parameters instead of

obtaining these parameters from a data set of a single specimen size. This technique offers advantages in 

using information efficiently (decreasing uncertainty) and in model validation (e.g., Johnson and Tucker 

(1994)). However, for graphite these techniques must be considered carefully (or modified appropriately) 

since the Weibull modulus observed from the size effect of the different specimen sizes may not correlate 

effectively with the Weibull modulus observed from the scatter in strength data. Nevertheless, having 

rupture strength data available from two or more (significantly) different sizes of specimens is important 

for exploring the relationship between the size effect and the scatter in strength. Note that WeibPar 

(Connecticut Reserve Technologies, Inc., 2009) can estimate Weibull parameters from pooled data of 

various specimen sizes.

3.2 Large Intrinsic Flaws and Extreme-Value Statistics

The existence of gross flaws may be an important issue from another respect. As mentioned in 

Section 2.3.3, Alava et al. (2006) showed that, when embedded cracks were significantly larger than the 

background defects (the individual lattice elements), the large defect behaved according to classical

LEFM. This suggests that a population of large, relatively isolated defects should behave according to 

extreme-value statistics; that is, Weibull or Gumbel distributions. Recall that Pears and Sanders (1970)

(see Sec. 1.2) describe graphite as consisting of background and disparate defects such that the disparate 

defects are large in comparison to the background defects. If some of the disparate defects are sufficiently 

large, they will likely produce a classical WLT brittle-failure mode. Therefore, for a small-size scale,

where gross defects may not be present, graphite may be quasi-brittle with distributed microcracking, as 

lattice simulations and experimental observations suggest. However, for larger scale structures, the 

increased likelihood of a gross flaw being present shifts the failure mode toward being classically brittle. 

The presence of gross flaws is undesirable, and nondestructive evaluation (NDE) techniques are often 

used to try to identify and remove components containing the worst-of-the-worst—that is, the largest—

flaws. This truncates the flaw-size distribution and implies a threshold stress below which failure will not 

occur. Statistical techniques have been developed to determine the probability of detection, but this is

beyond the scope of this report. Proof testing is also used to screen components and provide a minimum 

assured strength. Again a discussion of this subject is beyond the scope of this report; however, multiaxial 

proof testing reliability analysis is discussed in Nemeth et al. (2003, 2005). 
NASA/TM—2011-215805 67

3.3 Weakest-Link Theory and Asymptotic Weibull Behavior at Low Probability 

of Failure

By definition, the WLT applies to series-system models, but what is interesting is that WLT modeling 

also has been applied to fiber-bundle models, as described in Sections 2.2.4 and 2.2.6. However, WLT 

may not be applicable for very high disorder systems, as discussed in Sections 2.2.7 and 2.3.2. Bazant and 

Pang (2007) argue (see Sec. 2.1.1) that WLT is always applicable at the RVE size scale and larger for 

(essentially) all brittle constituent materials. Whether it is argued that graphite behaves more like a 

composite with distributed damage or more like a classically brittle material, it seems that WLT-series 

system modeling will probably be required to physically describe the structure beginning at some 

threshold size scale. 

There are physical and mathematical arguments that can be cited to help bolster the case of using the 

Weibull distribution. Extreme-value statistics points to the Weibull distribution, with further justifications 

provided by Bazant and Planas (1998) and Bazant and Pang (2007) (see Sec. 2.1.1). The extreme-value 

argument is strongest for classically brittle materials. However, for parallel systems (or mixed seriesparallel systems), the Weibull distribution also seems appropriate, asymptotically, at the lower tail of the 

distribution. This was illustrated in Section 2.2.6 for Bazant and Pang (2007) and Beyerlein and Phoenix 

(1996). It is also supported by Batdorf (1982) and Batdorf and Ghaffarian (1984). Bazant and Pang (2007)

argue that, at the RVE level (about 3 times the size of the maximum inhomogeneity), strength is gaussian

with a Weibull lower tail and that, at the size level of about 500 RVEs, the structure has a two-parameter 

Weibull distribution. Therefore, extreme-value WLT, chain-of-bundles, and (from Bazant and Pang 

(2007)) mixed series-parallel systems argue for Weibull-like behavior at the lower tail of the strength 

distribution. This applicability covers a broad range of material systems. Regardless if one asserts that 

graphite should be modeled as a series system, a parallel system, or a mixed series-parallel system, a 

justification can be given for using the Weibull distribution, at least at low probabilities of failure. 

3.4 Non-Weibull-Distribution Models and the Role of the Brittle-to-Ductile-Like

Transition

Some of the various models reviewed predicted non-Weibull behavior. For the series-system models 

in Section 2.1.2.1, the Batdorf model can be made non-Weibull depending on the form of the crackdensity function. As discussed in Section 2.1.4, R-curve behavior can cause deviations from the Weibull 

distribution, although it is asymptotically Weibull at low probabilities of failure and for structures that 

have large volume. In Sections 2.1.6 and 2.1.7, the strength distributions are non-Weibull, but the uniform 

defect model has a strength distribution composed of Weibull-like elements. If graphite can be classified 

as a strongly disordered material system such that it is on the ductile-like end of the brittle-to-ductile-like

spectrum of material behaviors (schematically shown in Figure 29), then the strength distribution may be 

non-Weibull. This was discussed in Sections 2.2.7 and 2.3.2. 

3.4.1 Bundle Models and the Brittle-to-Ductile-Like Transition

In Section 2.2.3, with the Daniels (1945) ELS bundle model, strength is normally distributed. As 

discussed in Section 2.2.4, Smith (1982) argues that the strength distribution is Gumbel distributed, and as 

discussed in Section 2.2.6, Harlow and Phoenix (1978b) state that the strength distribution is non-Weibull 

and that ELS bundles do not follow WLT scaling. Mahesh et al. (2002) and Mahesh and Phoenix (2004)

indicate that the strength distribution of LLS bundles is neither normal nor Weibull, although it is perhaps 

closer to Weibull in form, and Mahesh et al. (1999) and Landis et al. (2000) show that ELS bundles 

approach the Daniels asymptotic results as size increases. Section 2.2.7 shows that fiber composites with 

very low Weibull modulus fibers behave more like an ELS system. This would analogously correspond to 

individual ligaments of graphite (the spacing between cracks of pores) having high scatter in strength, 

where the ligaments act as fibers, or to distributed microcracking with crack arrest. For this material 

behavior, a Daniels ELS-style model controlled by a normal distribution may be more appropriate. 
NASA/TM—2011-215805 68

However, the asymptotic modeling of Mahesh and Phoenix (2004) indicates that at some specified size 

level extreme-value statistics should dominate—although that size may be very large. 

3.4.2 Lattice Models and the Brittle-to-Ductile-Like Transition

In Section 2.3.2, lattice simulations indicate that weak disorder systems behave like a brittle material 

(governed by extremal statistics) and that moderate to strong disorder is required for the material to 

become ductile-like with distributed damage. This transition is probably not abrupt. In Section 2.3.3, 

weak disorder systems are said to follow either the Weibull or Gumbel extremal statistics. For strong 

disorder systems with distributed damage, the lattice simulations by Nukala and Simunovic (2004)

indicate that fracture strength is neither Weibull nor Gumbel but fits the lognormal distribution best. In 

contrast, Alava et al. (2006) note that it remains to be explored if a crossover to Weibull or Gumbel 

occurred at larger size scales. Alava et al. (2006) also shows that, when embedded cracks were significantly larger than the background defects (the individual lattice elements), the large defect behaved 

according to classical LEFM and, therefore, populations of large relatively isolated defects should likely 

behave according to extremal statistics—that is, Weibull or Gumbel distributions. Finally, if Bazant and 

Pang (2007) are correct, the lower tail of the strength distribution should always be Weibull, and this is 

probably regardless of the level of disorder in the system.

3.5 The Physical Interpretation of the Weibull Modulus

The Weibull modulus m calculated from fracture experiment results represents strength response 

resulting from a variety of phenomena. For classically brittle materials, such as glasses and some 

ceramics, the Weibull modulus can be directly related to the size distribution of inherent flaws by the 

rules of LEFM. For polycrystalline materials where the presence of grains may provide additional 

resistance to fracture, the Weibull modulus becomes a function of the size distribution of inherent flaws 

and the physics of crack growth (rising R-curve). This is true when porosity may arrest crack growth as 

well. For composite materials, the presence of fibers and the fiber load-sharing rules modify the Weibull 

modulus of the individual constituents (the fibers). 

Microcracking (distributed damage) also apparently affects the Weibull modulus. Afferrante et al. 

(2006) examined an idealized array of in-plane microcracks and concluded that the Weibull modulus is 

not solely related to the defect distribution, finding that Weibull modulus depends on both the distribution 

of crack sizes and the distribution of ligaments (spacing between cracks). Li and Fok (2009) and 

Nemeth et al. (2011) indicate that the observed Weibull modulus could be affected by the nonlinear 

stress-strain response. So although the Weibull modulus can be regarded as a characteristic of a material,

it is not necessarily a function of a single physical quantity. This appears to be the case for graphite,

where multiple mechanisms—such as the defect distribution, the R-curve, crack arrest from porosity, and 

microcracking—likely play a role. What is not clear is how strong a contribution each constituent plays in 

determining the overall stochastic strength response for graphite.

Bazant and Pang (2007) assert that the observed or macroscopic Weibull modulus is a result of 

the interplay of the Maxwell-Boltzmann distribution of thermal energies modeled as a power law of 

exponent 1 at the atomic level modified by the scaling up of the material system at various levels of series 

and parallel systems. In the end, the Weibull modulus observed from fracture experiments can arise from 

multiple and synergistic phenomena whose source and limitations must be considered carefully.

3.6 Strength Dispersion, Size Effect, and Material Volume

The Weibull modulus m is a measure of scatter in strength. An important question is whether m (or 

strength dispersion) can vary with volume and whether this is related to the size effect. It is the opinion of 

the authors that the experimental studies cited do not strongly support this assertion (that m changes with 

volume—except in the case of concurrent flaw populations), but it is worth considering since some 

models described here indicate that it may. The Weibull distribution (Eqs. (8) and (9)) implies that m is 
NASA/TM—2011-215805 69

constant with volume as well as independent of specific geometry provided that the material can be 

treated as a continuum. However, Batdorf (1975) shows that physically based criterion can be constructed 

where m changes with volume. The preponderance of literature regarding fracture experiments on brittle 

ceramics and glasses does not support a general trend of m changing with volume—except in the case 

where the flaw population is changing (i.e., multiple concurrent or partially concurrent populations). In 

materials with multiple concurrent (mutually exclusive but simultaneously occurring) flaw populations 

with different m values, the flaw population with the lower m will dominate in higher volume structures. 

This is because more material is present, increasing the likelihood that a scarcer flaw type will be present

in the structure—akin to a spin of a roulette wheel, where the more times the wheel is spun, the greater 

the chance that a particular number will be selected at least once. In that situation, m will actually 

decrease with volume. An excellent discussion of multiple flaw populations and their effects on fracture 

statistics can be found in Johnson (1983).

The general situation that measuring m consistently from data sets can be problematic (e.g., Tennery 

and Ferber (1989)) plus the lack of forensic evidence from the fractography of graphite makes any assertion of Weibull modulus variability with volume more difficult to prove. Another potential problem arises

if surface-residing defects affect the failure of the material differently than volume-residing defects do. If 

surface defects are a distinct population from volume defects, then bending tests could show a different m

from tensile tests. Results from Price (1976) and the subsequent reanalysis of that data by Nemeth et al. 

(2011) show this possibility. In that case, Weibull reliability analysis has to be performed as a function 

of both the structure volume and the surface (see Nemeth et al. (2003, 2005) for a description of the 

methodology). This also accounts for the differences in stress distribution when the integration shown in 

Equation (6) is performed either over the volume of the structure or over the surface area of the structure. 

The reliability (or survival probability) of the structure is then the product of the reliability of each 

independent volume-flaw-controlled and surface-flaw-controlled failure mode. However, as previously 

indicated by Li and Fok (2009), the difference in Weibull moduli observed in flexural loading versus 

tensile loading could be attributed to a nonlinear stress-strain response. 

The examined literature shows mixed results with regards to size effect (see Sec. 1.4). Weibull 

methodology (see Eqs. (8) to (12)) seems to apply for flexural specimens (i.e., four-point- and threepoint-bending bar testing). For tensile specimens, the size effect is rather negligible—considerably less 

than that expected from Equation (12). If one assumes that the tensile testing was performed without 

systematic error, then this trend is not consistent with Weibull methodology. This has led some to 

criticize use of the Weibull methodology for graphite. Compounding the issue are fracture tests that show

lower-than-expected strength (from Eq. (12)) for small specimens. The apparently different tensile and 

flexural failure behaviors raise the suspicion that they may have different failure modes.

Several of the models in this review predict a dimunition of size effect with increasing volume and a 

simultaneous decrease of strength scatter. Of the models cited herein for predicting size effect, the 

Batdorf model (see Sec. 2.1.2.1) behaves identical to the Weibull model (Eqs. (8) and (9)) when the

crack-density function is a power law. Calculations made using the crack-density functions derived from 

the McClintock model of intergranular cracks showed that size effect and strength scatter decreased with 

increasing volume (m increased). For the Rose-Tucker, Burchell, and Tucker-McLachlan microstructural 

models (see Sec. 2.1.6), good qualitative and quantitative success in predicting size effect was reported 

(Tucker et al. (1986) and Tucker and McLachlan (1993)). This includes accounting for the dropoff in 

strength for small specimens (attributed to the fracture mechanics effects of a small crack on the order of 

grain size existing on the specimen surface). Relative scatter in strength versus volume was not reported

in these references.

For the parallel-system models, it was shown that size effect and strength distribution generally 

decrease with increasing volume. This is true for ELS and LLS bundle models spanning the more brittle 

to more ductile-like range of rupture behaviors (see Sec. 2.2 to 2.3.3 for specific details). However, for 

LLS bundle models, it may be inferred that size effect and strength dispersion will asymptotically obey a 

Weibull distribution for increasingly larger volumes. This is argued because of the WLT assumption that 

the models are built on and the asymptotic Weibull behavior predicted at low probabilities of failure (also 
NASA/TM—2011-215805 70

see Sec. 3.6.1). The authors herein believe that, for the bundle models, the LLS model is more representative of the graphite failure process than are the ELS models because of the likely role that stress 

concentration would play near nucleating microcracks. 

Somewhat similar to LLS model trends, Bazant and Pang (2007) indicate a bimodal behavior in a 

material where, at the RVE level, a Weibull distribution predominates over a normal distribution at low 

failure probabilities. As volume is increased, the Weibull distribution predominates at higher and higher 

failure probabilities, and a classic Weibull size effect occurs. For lattice models, the level of disorder in 

the system determines fracture behavior, strength dispersion, and size effect. For low to moderate 

disorder, size effect and strength dispersion behavior should follow extreme-value distributions like the 

Weibull. However, for strong disorder lattice models strength distribution appears to be lognormal and 

size effect decreases very slowly with system size. This is discussed in Section 2.3.3.

For the parallel-system models, the Daniels asymptotic model (see Sec. 2.2.3) has no size effect, the 

distribution is normal, and the standard deviation of strength varies as 1 nf . In Section 2.2.4 it was 

indicated that ELS bundles have a WLT size effect related to the length of the bundle, but the size effect 

decreases with increasing nf and this is true regardless of fiber length. In Section 2.2.5 a size effect was 

indicated, although always smaller than that for the individual (Weibull) fiber constituent. The size effect 

and standard deviation of strength decreased with system size—the standard deviation of strength being 

related by perhaps 1/ln(nf). Also Batdorf and Ghaffarian’s theory of multiplets shown in Figure 22 related

size effect to piece-wise Weibull segments, showing that size effect decreased (a function of the local 

Weibull modulus) as volume increased. As discussed in Section 2.2.6, Mahesh et al. (1999) and 

Landis et al. (2000) show in ELS simulations that mean strength and standard deviation decreased with 

increasing composite size and that stress concentration failure was driven by a process that followed 

WLT. ELS bundles approached Daniels’s asymptotic results, implying that size effect would be nonexistent at large system sizes. Bazant and Pang (2007) indicate a bimodal behavior in a material where, at 

the RVE level, a Weibull distribution predominates over a normal distribution at low failure probabilities. 

As volume increases, the Weibull distribution predominates at higher and higher failure probabilities and 

a classic Weibull size effect occurs. However, the observed effect is that strength scatter and size effect 

decrease as volume increases unless the structure size is very large relative to the RVE (see Figure 25).

As discussed in Section 2.3.3, Nukala and Simunovic and Zapperi and Nukala indicate that, for strong 

disorder and large system sizes, the size effect is small and the strength is decreasing very slowly in 

proportion to 1/(ln L)

0.15.

3.6.1 The Functional Relationship Between Size Effect and Strength Dispersion in 

Weakest-Link Theory

In fact, strength scatter and size effect are related for WLT-controlled failure-prediction methodologies (Weibull, Batdorf, Burchell, and chain of bundles). This is easily illustrated by taking the double 

logarithm of Equation (49) for two different volumes of material, V1, and V2, under tensile loads σ and 

expressing it as

2 1

2

,1 ,

1 1 ln ln ln ln ln

1 () 1 () f V f V

V

P VP

      = +  

− σ   − σ  

(60)

What this equation says is that, on a Weibull plot (i.e., ln ln[1/(1 – Pf)] versus ln σ) where an arbitrary

(Weibull or non-Weibull) WLT-based probability-of-failure function, Pf,V(σ), is ( ) 1 Pf V, σ for volume V1

and is ( ) 2 Pf V, σ for volume V2, then for volume V2, this function will displace vertically (i.e., along the 

ln ln[1/(1 – Pf)] coordinate axis) by a constant amount of ln(V2/V1) for any given value of stress σ. If V2 > 

V1, then the function will displace vertically upward on the graph.
NASA/TM—2011-215805 71

If one looks at the local slope of a small segment of the curve ( ) 1 , , Pf V σ which would be the local 

Weibull modulus mlocal (between the increments σ and σ + ∆σ), the vertically displaced portion of the 

curve will maintain the same local slope for ( ) 2 , : Pf V σ that is,

( ) ( )

( ) ( )

1 1

2 2

, ,

local

, ,

1 1 ln ln ln ln

1 ( ) 1 ()

ln ln

1 1 ln ln ln ln

1 ( ) 1 ()

ln ln

f V f V

f V f V

P P

m

P P

      − − σ + ∆σ − σ    = σ + ∆σ − σ

      − − σ + ∆σ − σ    = σ + ∆σ − σ

(61)

which is easily shown from Equation (60). Provided that mlocal is constant over a sufficient interval of 

failure probability, the local size effect will be ∆ln σlocal = (1/mlocal) ln(V2/V1), where a small mlocal

indicates a large-size effect and large mlocal indicates a small-size effect (see Eq. (12)).

The visual effect of this relation can be seen in Figure 25(b), where the curve Pf,V(σ) is displaced 

upward vertically as the volume (or system size) increases. For WLT models, the Pf,V(σ) curve that 

characterizes the model should be normalized to the volume of the RVE as per Bazant and Pang (2007).

Consider that the curve , ( ) o Pf V σ for a characteristic volume Vo spans an entire range of values of ln 

ln[1(1 – Pf)] from negative infinity to positive infinity. On a Weibull plot, the shape of the curve for any 

other value of volume Pf,V(σ) is identical to that of , ( ) o Pf V σ except that the curve is displaced vertically 

by ln(V/Vo). Therefore, the functional interpretation of Figure 25(a) and (b) appearing to show decreased 

strength scatter and size effect with increasing volume (or system size) is that it simply reflects the 

“master” curve , ( ) o Pf V σ being displaced vertically by ln(V/Vo). The appearance of change in size effect 

with increasing volume only means that a particular event that has a low probability of occurrence (a low 

probability of failure) within volume Vo has a higher probability of occurrence (a higher probability of 

failure) in a larger volume V. It also means that strength scatter and size effect are intimately related in 

WLT models: low scatter in strength translates to small-size effect and vice versa.

Another practical consideration concerns fracture specimen testing. A criticism of specimen testing is 

that it does not sample the probability-of-failure curve at low probabilities of failure, but Equation (60) 

clearly shows that this range can be sampled simply by increasing the (effective) volume of the specimens. Specimens with effective volume on the order (or larger if possible) of the effective volume of 

components will reduce the uncertainties associated with extrapolation to low probabilities of failure. In 

fact, pooling (combining) the data from a reasonable number of standard size samples with that of a 

smaller number of larger sized specimens can characterize the probability-of-failure distribution down to 

low failure probabilities (for a curve normalized to the small-size geometry). This is achieved by using 

Equation (60) to graft the large specimen data to the small specimen data. Unfortunately, specimens of an 

unreasonably large size are probably required in order to reach sufficiently low probabilities of failure.

3.6.2 Weibull Modulus, Size Effect, and Scaling Issues Versus Trends in Experimental Data

Specimen sizes tend to be small in comparison to a reactor brick—making the scaling issue an 

important consideration for a life-prediction methodology for graphite. Depending on the loading 

situation, the Weibull methodology may predict a large percentage decrease in the mean strength of a 

brick relative to measured mean specimen strength, which a designer may regard as overly conservative.

Because the Weibull moduli measured from specimen data are notoriously variable (Tennery and Ferber 

(1989)), a common practice is to determine m from size-effect studies—correlating m with the observed 

decrease in strength as size is increased. However, the negligible size effect observed from some 
NASA/TM—2011-215805 72

experimental studies implies a much larger m (not considering small-specimen fracture effects) in comparison to m estimated from fracture experiments of fixed specimen size. As explained in Section 3.6.1, 

scatter in strength and size effect are functionally related for weakest-link-series systems. If we assume 

that the reported experimental data for tensile failure are not in error, then some correction to lifeprediction methodology may be required (or one may need to test larger specimens to avoid orders of 

magnitude extrapolation). The Rose-Tucker, Burchell, and Tucker-McLachlan models have reported 

success at predicting diminution (decay) of size effect with specimen size, but as Equation (60) points out,

this decrease should also correlate with decreased scatter in fracture strength. This trend has not been 

clearly and consistently observed in experimental data.

Fracture in flexure seems to obey Weibull methodology reasonably, but fracture in tension does not 

correlate as well with regard to size effect. This raises a question of whether flexural loading and tensile 

loading have different fracture behaviors. If so, are they competing failure modes, or are they a single 

failure mode that manifests itself differently with different loading? Further study of this question is 

needed. In addition, it would be interesting to know how sensitive bending stress is to surface finish 

(surface roughness). For ceramics, strength in flexure is quite sensitive to the surface finish. However, 

this effect may be negated for graphite when open porosity at the surface is larger than grinding-induced 

damage.

Microstructure-based models have to account for the physics taking place within the RVE. The size 

of the RVE may be relatively large. If the RVE is roughly three times the size of the maximum inhomogeneity, then the RVE of nuclear-grade graphite may be 12 mm or larger (recall that grain size in nucleargrade graphite is defined as being less than 4 mm and greater than 100 μm). Therefore, typical specimen 

cross-section dimensions for tensile fracture testing could be sampling within the RVE level. It is unclear

how this may have affected experimental studies. The Bazant-Pang model assumes mixed series-parallel 

systems with the observed Weibull modulus ultimately a function of atomic force interactions. Does this 

make sense for all material systems? For classically brittle ceramics, the Weibull methodology has been 

successful, and it seems that fracture mechanics and a prescribed distribution of intrinsic flaws provide an 

adequate explanation. However, for quasi-brittle materials, the Bazant-Pang modeling approach may be 

more appropriate and certainly some of the historical roots of this methodology go back to Harlow and 

Phoenix (1978a,b).

3.7 Watson-Smith-Style Modified Weibull Distribution

It is suggested herein that the Harlow and Phoenix (1978b) WLT chain-of-bundles model (Eq. (48)),

along with the modified Weibull relation proposed by Watson and Smith (1985) (Eq. (51)), could be used 

to empirically model graphite strength and the smaller-than-expected size effect (as predicted by the 

Weibull distribution in Eq. (12)) observed in tensile loading. This would be in contrast to the modified 

Weibull model of Schmidt (2000) described in Section 2.1.5, which posits that there is no size effect with 

regard to the size of a specimen but that there is a size effect between tensile and flexural loadings. The 

proposed relation decouples the size effect from the strength scatter. On that basis, the following equation

could be considered for uniform uniaxial tensile loading of a constant material cross section 

( )

1 2

, ,

, ,

f 1 exp o o

o o

V n P VnW

V n

φ φ

δ δ

δ δ

δ δ

     =− −     σ

    

(62)

where W(σ) is some function of σ. For a power-law formulation of W(σ), a modified Weibull equation is 

obtained

1 2

, ,

, ,

1 exp

mV

f o o

o o oV

V n P V n

V n

φ φ

δ δ

δ δ

δ δ

       σ =− −         σ     

(63)
NASA/TM—2011-215805 73

It is assumed that a potential fracture path is a plane normal to the direction of loading with an 

associated thickness where microcracks may nucleate and coalesce into a macrocrack. In Equations (62) 

and (63), a cross section of a volume Vδ of height δ represents a potential fracture plane and nδ chains of 

height δ make up the total volume V = Vδ nδ. The terms Vδ,o and nδ,o are characteristic values. The exponents φ1 and φ2 are constants between 0 and 1. The height δ represents the average grain height or some 

small multiple of the average grain height. The exponents φ1 and φ2 allow decoupling from the WLT

argument. A Weibull relation is obtained as φ1 and φ2 approach 1. The assumption of Equation (62)—that 

there is a fracture plane which has a role in failure—is similar to that of Rose and Tucker (1982). Treating 

individual fracture planes as a weakest-link-series system yields φ2 =1 and 

1

,

,

1 exp

mV

f o

o oV

V P V n

V

φ

δ

δ δ

δ

      σ =− −          σ    

(64)

If φ1 and φ2 are equal (φ1 = φ2 = φ) in Equation (63), then the modified size effect is only a function of 

volume and

1 exp

V

V

m

f ox

ox o

V P V

V

  φ     σ =− −          σ   

(65)

where V is volume, Vox is some characteristic volume, and φ is a constant between 0 and 1. For a constant 

value of Pf, this equation linearizes to 

ln ln c

V

V

m

φ σ=− + (66)

where c is a constant and the Weibull slope is modified to φ/mV.

Equations (63) to (65) would accommodate the relatively negligible size effect reported for tensile 

specimens while keeping the relative amount of strength scatter constant. Prudence would also dictate that 

Equations (62) to (65) not be applied over too many orders of magnitude of size scaling because of the

potential inaccuracy of the scaling approximation or the possible presence of unaccounted-for flaw 

populations (e.g., the presence of gross flaws behaving in a classic Weibull manner). Equations (63) to 

(65) are only a starting point and further development to incorporate a methodology for the stress gradient 

and multiaxial stresses will be needed. However, the simpler route to account for flexural failure is to 

treat it as a separate competing failure mode (from tensile) that is a function of component surface area. 

This failure mode would have its own set of Weibull parameters. See Nemeth et al. (2003, 2005) for a 

further description of the methodology regarding treating volume flaws and surface flaws as separate and 

competing failure modes. 

3.8 The Continuum Assumption and Stress Gradient

The Weibull methodology (as well as the Batdorf methodology) assumes that the material is a 

continuum and that flaws are small relative to any stress gradient. For the Batdorf methodology, which 

assumes that the flaws are microcracks, this means that flaws will see no stress gradient across the length 

of the crack. This is actually a conservative assumption if the highest gradient stress value is used. 

Bruckner-Foit (2000) considered the effect of stress gradient on volume-distributed flaws for the situation 

where the stress gradient was so high (such as in thermal shock) that intrinsic microcracks would themselves experience a significant stress gradient. Their simulation showed that the Weibull stress-volume 

integral (see Eqs. (8) and (9)) overpredicted failure probability (was conservative). 
NASA/TM—2011-215805 74

For graphite-reactor-core structures, the stress gradients on flaws are a potential problem because 

flaws and microstructure can be large relative to the small fillet radii required for block-and-key construction (Figure 3). In addition, the material may not behave as a continuum in these regions because of the 

large grain size, which of course is a further complication, although this could possibly be handled with 

the Burchell model. The Bruckner-Foit analysis assumed that the material was a continuum. The handling 

of stress gradients around singularities (e.g., macroscopic cracks, sharp notches, zero-radius fillets, and 

bimaterial interfaces) is beyond the scope of this report. 

3.9 Problems in and Recommendations for Predicting the Effects of Slots or Notches

Graphite bricks for reactors may be designed with a block-and-key construction (Figure 3(a)) to 

interlock the bricks and make a stable structure. This technique also allows for individual brick expansion 

and contraction. However, the graphite bricks may have various slots and protrusions with small-radius 

fillets. Large stress gradients can occur at these fillets and are a serious concern for the design engineer. 

Mitchell et al. (2003) tested L-shaped specimens (described in Sec. 1.4) with various fillet radii and could 

not correlate well with the predicted response using the two-parameter Weibull stress-volume integral

with the PIA multiaxial theory (Eq. (21)). They used the Weibull parameters estimated from the rupture 

of bending bar specimens to predict the strength response of the L-shaped specimens. This study 

illustrates the limitations and caution that must be exercised when using statistical failure theories. 

In Mitchell et al. (2003) the flexural bars failed in the typically energetic fashion of a brittle material, 

whereas the L-shaped specimens failed in a more stable crack-growth-controlled manner. A fracture 

mechanics analysis of the L-shaped specimen indicated that the stress intensity factor decreased with 

crack length, suggesting a stable crack growth mode of failure. The Weibull modulus from the L-shaped 

specimens also was significantly higher than that estimated from the flexure specimens, potentially 

indicating a difference in the mode of failure (including also the possibility that machining damage at the 

root of the fillet may have been greater than that at the surface of the flexural specimens). One of the 

lessons that may be learned in that study is that, when comparing results from one specimen geometry to 

another, a similar mode of failure should be maintained (consistency of approach). The fact that they were 

different raises questions about the fairness of the comparison. The failure theories reviewed herein 

assume a positive geometry3

; therefore, positive geometry specimen tests should have been used 

consistently. A better comparison might have been if notched (with a finite fillet radius) flexural bars 

or notched cylindrical tensile specimens (that would likely fail in the more energetic fashion of a 

brittle material) were used rather than the L-shaped specimens. The authors herein conjecture that if

Mitchell et al. had obtained the Weibull parameters from a particular radius of the L-shaped specimen and 

then subsequently applied these parameters to predict the failure response of the other fillet radii, better 

correlation to the experimental data would likely have been achieved.

Mitchell et al. also indicated a very high Weibull modulus of 54.4 for the zero-radius fillet of the 

L-shaped specimen. The high value may have been a consequence of the stress singularity, which 

probably acted as a large crack of a deterministic size. Choi and Salem (1992) show that strength scatter 

decreases when a deterministic crack size is introduced (in this case from an indenter), as was discussed 

in Section 1.4. The Mitchell et al. (2003) study illustrates a potential shortfall of the current statistical 

failure theories, and future modeling enhancements for stable crack growth should be considered. Note 

that Li and Fok (2009) also modeled the behavior of the L-shaped specimen from Mitchell et al. (2003)

with interesting results, although it is not clear if they could fully explain the high-Weibull-modulus 

results from the zero-radius fillet of the L-shaped specimens. 

It is the opinion of the authors herein that rupture data from notched tensile specimens or notched 

bending specimens (with a nonzero fillet radius) should be used to predict the probability of failure of the 

fillet regions of reactor bricks, and data results from flexural or tensile specimens should not be used for 

this purpose. Therefore, the statistical failure of a graphite brick should be predicted from a database 

 3

Positive geometry: Stress intensity increases with increasing crack size, otherwise the crack gets arrested.
NASA/TM—2011-215805 75

consisting of a combination of tensile-specimen, flexural-specimen, and notched-specimen (tensile and/or 

flexural) data. Also, the statistical failure theories reviewed herein were not developed to analyze stress 

singularities, such as sharp notches, zero-radius fillets, or cracks; therefore, they should not be applied 

without properly accounting for the singularity condition (a subject beyond the scope of this report). A 

final note is that Weibull methodology has been used to successfully predict the failure response of 

specimens that have stress concentration from tensile specimens that have no significant stress concentration. This was shown, for example, in Sharpe et al. (2008) for silicon carbide microtensile specimens. 

In that study, 3-mm-long microtensile specimens were used to successfully predict the strength response 

of microtensile specimens with a central circular hole that was 0.1 mm in diameter. Since the material 

was single crystal, continuum-based stress analysis was still operable at this size level.

3.10 Regarding Multiaxial Stresses and Anisotropic Strength

Multiaxial strength response and anisotropic strength response were discussed briefly in 

Sections 2.1.2.2 and 2.1.3, respectively. This was not a comprehensive study of either topic. However, 

the Batdorf-style approach that was presented appears to match the experimental data satisfactorily. The 

Burchell model has also been used to model the anisotropic strength response (Burchell et al. (2006))

using a PIA modeling approach, but those results are not reported herein.

3.11 Regarding the Burchell Model for Graphite

The Burchell (and Tucker-McLachlan) model was developed specifically for nuclear-grade graphite. 

The Burchell model combines a microstructural basis with a fracture mechanics approach to failure. 

Graphite is not considered to be a continuum in the model, even though a continuum-based stress solution 

is used. Data for several physical parameters are required, and the model may be fine tuned to match 

experimental fracture data. The Burchell model directly accounts for some R-curve behavior. The model 

does not consider flaw interaction, although it is not clear how important that omission is for graphite. 

Lattice simulations of notch sensitivity (Alava (2006)) show that this sensitivity depends on the size of 

the cracklike voids relative to the background defects (and probably also the density of cracklike voids). 

The Burchell model depends on micrographs to obtain a characteristic pore-size distribution. That 

raises the possibility that rare, but ultimately strength-controlling, large flaws may not be sampled. This 

scenario was confirmed by Abe et al. (2003) in relating flaw-size populations to measured strength in 

alumina. They observed that scanning electron microscopy provided insufficient information on defects 

relevant to strength variation because only a small portion of a specimen could be examined in comparison to a transmission optical micrograph of a thinned-out section of a specimen, which was used to 

measure the size distribution of defects over a significantly larger volume. Therefore, to be useful for 

design to low probabilities of failure, a representative volume (or surface area) of material should be 

examined to characterize the distribution of the sizes of defects in the Burchell model. 

The Burchell model is intriguing and has been reported to be quite successful in matching experimental results. However, a word of caution is in order. If the Burchell model shows that size effect decreases 

with increasing volume, this means that scatter in strength also decreases at larger volumes (see Eqs. (34) 

and (35) and Sec. 3.6.1). Using this model to predict strength for larger volumes and low probabilities of 

failure may result in a nonconservative design if scatter in graphite strength does not also decrease with 

increasing volume (increasing Weibull modulus with increasing volume). 

The Burchell model should be developed further and incorporated in life-prediction codes such as 

CARES/Life (Nemeth et al. (2003, 2005)). It would be valuable to have available a software tool with the 

various statistical models of graphite failure in order to contrast and benchmark the various models.
NASA/TM—2011-215805 76

3.12 Regarding Probabilistic Design

Probabilistic design is a very important feature of any life-prediction methodology for graphite. This 

means accounting for other sources of variation that may influence the failure probability of graphite 

components over their lifetime of service, including variability in thermomechanical loading, boundary 

conditions, and material properties (as well as variation of these properties over time including the effects 

of irradiation). The methodology to achieve this was demonstrated for ceramics and microelectromechanical systems by Reh et al. (2003), Jadaan and Trethewey (2006), Nemeth et al. (2007), and Luo et al. 

(2007). These papers attempt to analyze problems for a complete probabilistic design space. For the 

ANSYS/CARES probabilistic design system (Reh et al. (2003)), this means simulating the effect of

variability in material properties, boundary conditions, and loading on component probability of failure,

including simulating variability in Weibull parameters, accounting for billet-to-billet variations or 

uncertainties in the value of estimated Weibull parameters from specimen data. This methodology 

requires performing repeated simulations (for randomly chosen sets of conditions) in an optimized 

manner (so as to run as few a number of simulations as possible) to determine a distribution of failure 

probability.

3.13 Final Comments

The simplicity of the Weibull distribution is an attractive feature for the design engineer, and now a 

substantial body of knowledge has been developed regarding the methodology (parameter estimation, 

confidence bounds, data pooling, etc.) and how physical processes manifest themselves in the strength 

response that is approximated by the Weibull distribution. It is therefore premature to dismiss this 

distribution, or its variants (such as Batdorf), as not being appropriate for graphite design. Microstructurally based models of graphite also are important because they simulate the physics of the material 

system and provide further insight into material behavior and avenues for improving the material.

Ultimately, multiple theories of material breakdown may be required to span the full range of potential 

and progressive failure modes. Also, the importance of an appropriate database (and an appropriately 

designed database) for tensile specimens, flexural specimens, notched specimens, and size-effect studies 

cannot be overemphasized.

The review of the literature cited herein indicates that significant progress has been made toward 

physically based statistical models that can explain failure across disparate material systems and that can 

accommodate the brittle-to-quasi-brittle spectrum of material failure modes. Ultimately, mathematically 

refined theories of fracture will be useful only to the extent that the associated stress analyses are equally 

accurate and that also includes considering the effect of nonlinear stress-strain response and bimodulus

(different stress-strain response between tension and compression) material behavior.

This report has only addressed the issue of stochastic material strength independent of environment 

and time of exposure. Graphite-moderated reactors are intended to have a service life of 30 to 60 years, so 

durability and damage growth methodology is required to show at least a viable design estimate. Therefore, there is also a need to develop time-, cycle-, and environment-dependent reliability prediction 

models that can assess graphite performance over these extended time periods.

4.0 Conclusions

The fracture of nuclear-grade polycrystalline graphite is a function of the distribution, interaction, and 

growth kinetics of flaws of various forms and size scales. Graphite fracture strength is stochastic, and any 

theory of graphite failure must accommodate this fact. This report described various statistical models of 

brittle-constituent-controlled material failure. All these models assumed that material breakdown begins

at inherent flaws or weak regions in the material. However, the various models differ in how material 

breakdown proceeds. This was broadly categorized into series-system or parallel-system models. Seriessystem models describe a weakest-link mode of material failure, and the Weibull, Batdorf, and Burchell 
NASA/TM—2011-215805 77

models were described. Parallel-system models describe the process of accumulating damage leading to 

ultimate material failure, and composite fiber-bundle models and lattice models were described. For 

parallel-system models, additional attention was given to the expected transition between brittle and 

quasi-brittle (ductile-like) behavior, particularly since nuclear-grade graphite is often described as quasibrittle. Composite material modeling was included for the modeling insights it provides. Results from 

lattice simulations were included for a physics-based description of material breakdown. 

The literature indicates that WLT modeling approaches cover a broad range of material systems;

however, material systems with very strong disorder may not show WLT behavior until component sizes 

are very large. The Weibull distribution was argued to be the most appropriate statistical distribution for

modeling the stochastic strength response of graphite. Consideration also was given to size effect and 

multiaxial strength response. Awareness of these modeling approaches provides additional perspective 

and potential avenues of inquiry that may be useful in selecting methodologies for designing graphite 

components by standards and regulatory organizations such as ASME International and the Nuclear 

Regulatory Commission. 

NASA/TM—2011-215805 79

Appendix A.—Uniform Defect Model Approximated

as a Weibull Distribution

Weibull statistics are typically used to describe the behavior of classically brittle materials for 

tensile stress states only. This is appropriate because the WLT is a series system where a single event 

(uncontrolled crack propagation from a flaw) determines the integrity of the whole system. For 

compressive stress states, this model does not rigorously apply. The unequal principal compressive 

stresses (σ3 ≤ σ2< σ1) produce shear stresses on arbitrarily oriented cracks that act against compressive 

normal forces on the crack face. This produces tensile stresses at the crack tips and the crack branches 

parallel to the direction of maximum compression. The crack growth is eventually arrested when the 

tension at the crack tip decreases below a critical level and an equilibrium condition is reached with the 

applied compression. A single event of crack extension, therefore, does not necessarily result in material 

failure. However, if a number of events of crack growth and arrest take place in a sufficiently confined 

region where they can join together and create what is known as a shear fault, then total failure ensues. 

Weibull statistics and Batdorf methodology have been applied by Alpa (1985) to access material 

integrity in compression. This modeling scheme uses a WLT methodology to predict the first event of 

crack extension (initiation of damage). The Weibull modulus for compressive stress states and tensile 

stress states is assumed to be the same since the strength response is controlled by the same flaw 

population. Alpa’s model predicts a Weibull size effect for compressive stresses similar to the tensile 

stress response. The basis for using the WLT to describe strength in compression is justifiably open to 

question. Kittl and Aldunate (1983) tested compact cement cylinders in compression, and a best-fit 

distribution could not be determined with a high level of confidence. Figure 30 shows lines of best fit to 

the 575 specimens tested for the normal, lognormal, and three-parameter Weibull distributions. Experiments measuring the compressive strength of specimens often show smaller scatter (higher Weibull 

modulus) and size effect than the tensile strength. The approach of employing separate Weibull moduli 

for tension and compression (described in Sec. 2.1.2.2) improves the correlation to measured data;

however, the justification for doing this is somewhat arbitrary. The following is an argument that the 

uniform defect model (see Sec. 2.1.7) can be approximated as a Weibull distribution. This may help as a 

justification for using a Weibull distribution for strength in compressive stress states and for treating 

compressive failure and tensile failure as independent failure modes. 

The crack-density function ηV (σIc) is defined to represent the average number of microcracks per unit 

volume with a critical mode I strength equal to or less than σIc. A random volume flaw in a multiaxial 

stress field can be oriented such that there is a combination of a compressive normal force and a shearing 

force causing the critical mode stress intensity to be exceeded at the crack tips. The flaw is assumed to 

extend in a stable manner until its growth is eventually arrested. For an appropriately sized incremental 

volume element ∆Vd it is assumed that as the magnitude of the applied stress is increased, the number of 

critically oriented flaws that experience damage (stable crack extension) increase in accordance with the 

crack-density function. Each event of crack extension creates new surface within the incremental volume. 

When the total amount of new surface area per unit volume reaches a critical ratio, or alternatively, when 

the number of flaws nd that experience growth within ∆Vd reaches a critical number ndc, local material 

failure is assumed to occur. Therefore, a failure criterion for the RVE is constructed such that

nd < ndc indicates that damage is not critical in ∆Vd

(A1)

nd ≥ ndc indicates local material failure in ∆Vd
NASA/TM—2011-215805 80

Figure 30.—Strength failure probability distributions versus strength of 575 cement cylinders tested in compression. 

Although statistical tests do not reject one distribution over another, the lognormal distribution subjectively appears 

to be slightly better than the normal distribution. (a) Shown in normal (gaussian) probability coordinates. (b) Shown 

in lognormal probability coordinates. (c) Shown in Weibull probability coordinates with a three-parameter distribution and a threshold stress σu of 97 MPa. Reproduced from Kittl and Aldunate (1983). Copyright Springer 

Publishing Company; used with permission.
NASA/TM—2011-215805 81

for an applied compressive stress. In addition, some conservative and simplifying assumptions are made:

(1) The flaws are independently and uniformly distributed throughout the material volume

(a) The defects arise at separate sites (there are no overlaps).

(b) The number of defect occurrences in separate volumes are independent random variables.

(c) The number of defect occurrences is independent of position.

(2) Local material failure constitutes component failure (or what could be defined as a damaged or 

rejected component).

(3) Crack growth is mutually exclusive—interactions between adjacent cracks are not considered (in 

other words, how the flaws interact and how the surrounding material is affected is not 

considered).

(4) ∆Vd is small (on the order of the critical crack length that spans the volume element).

Following Freudenthal’s (1968) reasoning, assumption (1) allows for the construction of a spatialpoint Poisson process (from which the Poisson distribution derives). The Poisson distribution models the 

probabilistic distribution of random particles in space or events in time. This probability distribution is an 

approximation to the binomial distribution for so-called rare events when the probability of occurrence of 

an event is close to zero. The Poisson distribution is represented as follows:

( ) !

x e PX x

x

−λ λ = = (A2)

P(X = x) is the probability that the discrete random variable X takes on the values x = 0, 1, 2, 3, ..., and 

λ = n Pevent, where n is the sample size and Pevent is the probability of an event. As an aside, it is 

worthwhile to note the property that

0 1

!

x

x e

x

∞ −λ

=

λ ∑ = (A3)

which validates that the Poisson distribution is a probability function.

The probability of failure in the material element ∆Vd is the probability that more than one flaw (at 

least ndc number of flaws) grows within this incremental volume, and therefore for λ = Vdη(σIc),

{ } ( ) ! d dc

x

f V dc x n P PX n e

x

∞ −λ

∆ =

λ

= ≥= ∑ (A4)

or in terms of the survival probability

{ } ( ) ( 1)

0 !

dc

d

x n

s V dc x P PX n e

x

− −λ

∆ =

λ

= <= ∑ (A5)

Note that how the applied compressive stress acts to extend a crack is not considered. For λ << 1, 

then e

–λ ≅ 1 and the higher order terms in Equation (A4) can be neglected. This allows Equation (A4) to 

be simplified as

{ } ! !

dc

d dc

x n

f V x n dc

P

x n

∞

∆ =

λ λ

≅ ≅ ∑ (A6)

If λ = ∆Vdη(σIc) is substituted into Equation (A6), then the elemental failure probability can be 

approximated by
NASA/TM—2011-215805 82

{ }

I ( )

!

dc

d

n d c

f V

dc

V

P

n

∆

  ∆ ησ   ≅ (A7)

If a volume V is discretized into n incremental units such that n = V/∆Vd, the probability of survival of 

V is the product of the incremental probabilities of survival (see Eq. (1)):

( { } ) ( { } )

{ } { }

1 1

1 1

1

exp exp

d d

d d

n n

sV s V f V i i

n n

sV f V f V i i

P Pi P i

PP P

∆ ∆ = =

∆ ∆ = =

= = −

  ≈ − =−        

∏ ∏

∏ ∑

(A8)

where the natural log exponential function can be substituted if f V { d } P ∆ is small. Substituting Equation 

(A7) into (A8) yields

( )

( ) ( ) 1

I

exp !

ndc ndc

d c

sV

dc

V

P V

n

  − ∆ η σ    = − 

 

(A9)

Substituting for the crack-density function ( I I ) ( )

mV

ησ = σ σ c eq oV where σIeq describes an effective 

stress on the flaw from the applied compressive stress (see Eq. (23)) yields

( )

( )

( 1 )

I

1/

/

exp

!

V dc

V dc

dc V dc

m n

eq

sV m n oV dc

n mn d

P V

n

V −

       

σ = −     σ

 

  ∆      

(A10)

Finally substituting a constant term σocV for the expression in the braces {...} and calling this term the 

scale parameter in compression yields

exp

m n V dc

e sV

ocV

P V

    σ = −      σ  

(A11)

Equation (A11) implies that, with the uniform defect model, failure strength can be approximated by 

a Weibull distribution with a Weibull modulus term, mVndc, that is a multiple, ndc, of the Weibull modulus 

mV describing the power-law crack-density function exponent. This failure criterion posits significantly 

decreased scatter and size effect relative to the Weibull distribution with the exponent mV. 

The uniform defect model is a rudimentary model of compression failure, yet the Weibull exponent 

modified by ndc has interesting parallels with the asymptotic behavior of Batdorf and Ghaffarian (1984)

(see Figure 22) and Beyerlein and Phoenix (1996) (see Figure 23) at low probabilities of failure. The 

point of this exercise was to explore a basis where the Weibull distribution could be justified to describe 

strength in compression. On the practical side, there are several factors that may contribute to observed 

scatter in strength and, therefore, the value used for the compressive Weibull modulus mcV = mVndc should

be obtained from rupture experiments with compressive specimens. Also, strength scatter and size effect 

may not correlate well. This was reported by Kittl and Aldunate (1983), where no significant size effect 
NASA/TM—2011-215805 83

was detected (although the difference between the minimum and maximum volumes tested was not very 

large). Thus, the modified Weibull distribution of Watson and Smith (1985) to account for unexpectedly 

small-size effect is probably a good distribution to model strength in compression. The discussion to this 

point did not consider multiaxial stresses. The Batdorf unit sphere approach could be used to account for 

multiaxial stress states as described in Sections 2.1.2.1 and 2.1.2.2.

NASA/TM—2011-215805 85

Appendix B.—Flaw Orientation Anisotropy and Stress Intensity Anisotropy

Flaw orientation anisotropy refers to the situation where a flaw has a higher likelihood of being 

oriented in one direction versus another. This means that a material will be stronger on average in one 

direction versus another. An isotropic brittle material is equally strong in any direction, and thus its flaws 

are randomly oriented. However, for components made by processes such as extrusion or hot pressing,

which induce texture, a bias will exist in the distribution of processing flaws. Also, components finished 

by surface grinding will also contain machining damage in the form of cracks that are oriented parallel 

and transverse to the grinding direction. Salem et al. (1996) explored a methodology for modeling 

grinding damage. That work assumes that the machining flaws are all closely aligned in one direction. 

This was based on a more general approach developed by Buch et al. (1977) (specifically for graphite) 

and further explored by Nemeth (1989) for transversely isotropic strength response. 

For volume-distributed flaws, flaw orientation anisotropy relative to a material coordinate system is 

modeled by introducing a probability density distribution ℘ into the Batdorf unit sphere formulation for 

P2V in Equation (15). For one-half of the unit sphere,

( ) ( ) 2 2

2 I I 0 0

P H V , , sin d d eq c

π π

= ℘αβ σ σ α α β ∫ ∫ (B1)

where

( ) ( )

( ) 2 2

0 0

, ,

, sin d d π π

ζ αβ ℘αβ =

ζ αβ α α β ∫ ∫

(B2)

and

( )

( )

II I I

II I I

, 1 

, 0 

eq c eq c

eq c eq c

H

H

σ σ = σ ≥σ

σ σ = σ <σ

℘(α,β) sin α dα dβ is the probability that a normal vector to the flaw plane is oriented at α,β within the 

intervals (α – dα/2, α + dα/2) and (β – dβ/2, β + dβ/2).

For a transversely isotropic strength response, ζ is only a function of α. Buch et al. (1977) introduced 

a cosine power function for ζ(α) = [cos (ξα)]γ where ξ and γ are constants. This relation was modified by 

Nemeth (1989) to enhance the functional flexibility

( ) ( ) ( )

( ) ( )

cos 0 2

0 22

γ ζ α =  ξα  ≤ α ≤ π ξ  

ζ α = π ξ <α≤π

(B3)

or alternatively,

( ) ( )

( ) ( { ( ) }) ( )

0 0 22

sin 2 2 2 2 2

γ

ζ α = ≤α<π −π ξ

  ζ α = ξ α − π − π ξ  π −π ξ ≤ α ≤ π    

(B4)

where ξ and γ are constants that control the degree of anisotropy, and ξ ≥ 1 and γ ≥ 0. When ξ = 1 and γ = 0, 

an isotropic strength response is obtained. Equations (B3) and (B4) are defined for one-half of the unit 

sphere (the top half of the unit sphere shown in Figure 31). In reference to Figure 31, Equation (B3) 

represents the “polar cap” or longitudinal distribution of flaws and Equation (B4) represents an
NASA/TM—2011-215805 86

Figure 31.—Unit sphere with probability density distribution functions describing 

anisotropy of flaw orientation. Orientation is described with the normal to the crack 

plane. In this figure, two orientation functions are described: (1) a “polar-cap” distribution describing crack planes symmetrically distributed (centered) about a plane 

(in this case, the σy–σz plane) and (2) an “equatorial-belt” distribution, where crack 

planes are symmetrically distributed (centered along a line, in this case, the σx axis) 

and α and β are the angular coordinates.

“equatorial belt” or transverse distribution of flaws. The polar-cap distribution describes crack planes 

symmetrically distributed (centered) about a plane (in this case the σy−σz plane), and the equatorial-belt 

distribution describes crack planes symmetrically distributed (centered) along a line (in this case the σx

axis). The separate polar-cap and equatorial-belt distributions were originally introduced to describe a 

unidirectionally fiber-reinforced composite where the polar cap represented the fiber strength distribution 

and the equatorial belt represented the matrix-fiber interface.
NASA/TM—2011-215805 87

Similar to Equation (19) the strength anisotropy function can be more simply expressed as

( ) ( ) 2 2

I 0 0 fV 1 exp , eq sin d d d

V V P V π π   =− − ℘αβ η σ ααβ     ∫∫ ∫ (B5)

and it should be noted that the crack-density function ηV(σIc), as expressed in Equation (17), in this case 

has a normalized Batdorf crack-density coefficient kBV that is a function both of the mixed-mode fracture 

criterion and the flaw orientation function for an applied uniaxial stress state.

Other quantities that can be made anisotropic with orientation using the Batdorf unit-sphere approach 

include the crack-density function η, the average size of the flaw, the critical stress intensity factor KIc, 

and the critical strength σIc. Anisotropic KIc, flaw size, and critical strength σIc are functionally related 

quantities. For example anisotropic KIc can be made functionally equivalent to anisotropic flaw size, and

KIc and σIc are directly related. 

Batdorf (1973) approached strength anisotropy using the σIc strength ellipsoid approach (describing 

an ellipsoid rather than a unit sphere). Nemeth (1989) considered strength anisotropy using KIc, where KIc

was a function of the orientation angle on the unit sphere. The approach is similar to that developed for 

flaw orientation anisotropy described previously. The critical strength σIc is defined as the fracture 

strength of the crack in mode I loading and is directly proportional to KIc. Therefore, for anisotropic 

KIc(α,β) = (c σIc(α,β)) = [ ,] σ αβ I max I c c f ( ) , where c is a constant, σIc,max is the maximum value of σIc over 

the unit sphere (for all α,β), and fIc (α β, ) is a normalized function expressing the degree of anisotropy. 

For this approach, the unit sphere formulation described in Equations (13) to (19) in Section 2.1.2.2 is 

used except the Heaviside step function in Equation (18) is modified as follows

( ) ( )

( ) ( )

I

I I I ,max

I

I

I I I ,max

I

, 1 ,

, 0 ,

eq

eq c c

c

eq

eq c c

c

H

f

H

f

σ

σ σ = ≥ σ

α β

σ

σ σ = < σ

α β

(B6)

For a transversely isotropic response where anisotropy is only a function of angle α,

I ( ) ( ) 1 ( ) 1 cos 1 0 2 Lc L L

L

f

r

γ   α = − ξ α −   ≤α≤π ξ

 

fIc LT () 1 2 2 2 α = π ξ <α<π −π ξ ( ) ( )

I ( ) ( ) 1 ( ) 1 sin 1 2 2 2 2

2 2

T

c T T T

T T

f

r

γ          π π α = − ξ α− −     − π −π ξ ≤α≤π +π ξ         ξ    

(B7)

where ξL, ξ T, γL, γT, rL, and rT are constants. Equation (B7) is defined for the top half of the unit sphere as 

shown in Figure 31. The L subscript relates to the polar-cap strength anisotropy distribution for crack 

planes symmetrically distributed (centered) about a plane (in this case the σy−σz plane), and the T

subscript relates to the equatorial-belt strength anisotropy distribution for crack planes symmetrically 

distributed (centered) along a line (in this case the σx axis) (see also Figure 31 for a reference frame). 
NASA/TM—2011-215805 88

Figure 16 shows an example using Equation (B7) to fit to experimental data for anisotropic KIc

measured from a Knoop indenter. Figure 32 shows a hypothetical example of the predicted strength 

response for a uniaxial load on a unit volume rotated relative to the extrusion direction for shear-sensitive 

versus shear-insensitive flaws. Penny-shaped cracks (Eq. (20)) are assumed with a Shetty-shearsensitivity coefficient C = 1.0 for the shear-sensitive response and C = 100.0 for the shear-insensitive 

response. A value of ξL= 18.0 was used such that the cosine function only spanned a 10° increment 

between –5° ≤ α ≤ 5°, which provided a narrow distribution fIc (α) such that the difference between a 

shear-sensitive and a shear-insensitive response is easily seen. This example was developed to show an 

approximately 25-percent anisotropy in strength. For the shear-insensitive response, mV = 10.0, σoV =

100.0, C = 100.0, kBV = 21.92, ξL = 18.0, and rL = 2.215; and for the shear-sensitive response, mV = 10.0,

σoV = 100.0, C = 1.0, kBV = 3.921, ξL = 18.0, and rL = 2.693. The T population parameters are not active 

in this example (γT = 0, rT = 1.0, and ξT = 1.0 (arbitrarily)). 

Figure 32.—Strength anisotropy example of a shear-sensitive and shearinsensitive response for a uniaxial load rotated at an angle relative to the 

extrusion direction. Anisotropy of the critical mode I stress-intensity factor, 

KIc, is assumed, which is also equivalent to the microcrack size being 

anisotropic.
NASA/TM—2011-215805 89

Appendix C.—Interchangeability of Composite Stress

With Fiber Stress in the Weibull Distribution

In a unidirectional composite where fiber strength controls the material failure and a Weibull distribution is used to describe the failure response based on fiber strength, for uniaxial loading a general 

form of the distribution may be expressed as

fiber

fiber

,fiber

( ) 1 exp

m

f

o

P

      σ

σ =− −   

σ       (C1) 

If perfect bonding is assumed between the fibers and the matrix,

ε =ε =ε composite matrix fiber (C2)

If the fibers and matrix behave elastically,

fiber fiber fiber

matrix matrix matrix

E

E

σ= ε

σ= ε

(C3)

The resultant load is

Fcomposite composite fiber fiber matrix matrix = ε + (EA E A ) (C4)

where A is the cross-sectional area of the composite, Afiber is the cross-sectional area of the fiber fraction 

of the composite, and Amatrix is the cross-sectional matrix fraction of the composite. The composite 

stress is

composite composite fiber fiber matrix matrix ( )

composite

composite fiber matrix

composite composite composite c

F EA E A

A A A

ε +

σ= =

+

σ =ε

(C5)

Substituting Equations (C2) and (C3) into Equation (C1) yields

composite fiber composite fiber

fiber

,fiber fiber ,composite fiber

composite

fiber

,composite

( ) 1 exp 1 exp

( ) 1 exp

m m

f

o o

m

f

o

E E

P

E E

P

      ε ε     

σ =− −    =− −  

ε ε           

      ε

σ =− −   

ε      

(C6)

since εo,composite = εo,fiber. Substituting Equation (C5) into (C6) yields
NASA/TM—2011-215805 90

( )

( ) ( )

composite composite composite

fiber

,composite ,composite composite

composite

fiber composite

,composite

c

1 exp 1 exp

c

1 exp

m m

f

o o

m

f f

o

P

P P

        ε ε 

σ =− − =− −      

ε ε         

      σ

σ =− −    ≡ σ

σ      

(C7)

Therefore, for a composite with fiber-controlled failure, if a Weibull distribution form is used, the 

failure probability expressed in terms of the composite stress σcomposite and the failure probability 

expressed in terms of the fiber stress σfiber are equivalent: ∴ Pf (σcomposite) = Pf (σfiber).
NASA/TM—2011-215805 91

Appendix D.—Symbols, Definitions, and Acronyms

D.1 Symbols

dA sin α dα dβ infinitesimal area on the surface of a unit radius sphere where σIeq ≥ σIc

Acomposite cross-sectional area of the composite

Afiber cross-sectional area of the fiber fraction in a composite

Amatrix cross-sectional area of the matrix fraction in a composite

Ao area of the grain 

a crack length or crack radius

aμ average crack length or crack radius

a0 initial or preexisting crack size

b average graphite grain size

C Shetty shear sensitivity constant

c constant 

ccomposite constant value

cG constant

c1, c2, c3 constants

cμ mean concentration of inhomogeneities

D constant (D = 2 for a two-dimensional network {∼area}, D = 3 for a three-dimensional 

network {∝ volume})

d diameter of the specimen

E Young’s modulus of elasticity

Efiber Young’s modulus of elasticity for composite fiber

Ematrix Young’s modulus of elasticity for composite matrix

exp exponential function

Fcomposite resultant load or force on composite

f( ) function

f(x) real value function of x

f(y) probability density function (PDF) of either ks or uc

fIc (α β, ) normalized anisotropy function of KIc or σIc as a function of angles α and β

f–1 first inverse moment of the probability distribution function

( ) nf G x bundle strength distribution for nf number of fibers

G4(σ) bundle strength distribution for four fibers as a function of stress σ

H Heaviside step function

, k nb H composite strength distribution for k critical number of fiber breaks
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, ( ) n n f b H x composite strength distribution

I current

i integer 

j integer; number of rows of grains that fracture

KI mode I stress-intensity factor

KIc critical mode I stress-intensity factor 

KR fracture toughness or fracture resistance 

k critical number of failure breaks

kG constant

km modified Weibull modulus

ks stochastic spring constant

k BV Batdorf uniaxial stress state normality constant for the volume-flaw 

failure mode 

L fiber length; length of composite material; system size 

Lo characteristic length 

L1, L2 system sizes

123   , , direction cosines

m Weibull modulus

m* modified Weibull modulus

mcV Weibull modulus for the volume-flow failure mode for compressive stress states

mlocal local Weibull modulus or slope at a point of an arbitrary curve on a Weibull plot

mV Weibull modulus for the volume-flow failure mode

N number of pores per unit volume

Neq equivalent number of links in a series

No number of cracks per unit volume

n integer; number of Bernoulli trials with success probability p; number of items in a set; 

number of bonds in a lattice system

nb number of mutually exclusive but linked bundles

nbr number of broken bonds

nc critical number of bond breaks in a lattice system

nd number of flaws that experience growth within ∆Vd

ndc critical number of flaws that experience growth within ∆Vd

nf number of fibers

ng number of graphite grains in entire row ahead of crack tip

nindep maximum number of n bonds where spatial independence is maintained
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no constant

nsys product of nf and nb, which is the size of the composite system

nδ number of chains of height δ

nδ,o characteristic value of nδ

P probability

P(X = x) probability random variable X equals the value x

Pevent success probability or probability of an event

Pf probability of failure (Pf = 1– Ps) 

Pfi probability of failure in the i

th individual graphite grain

(Pf)i probability of failure of the i

th link 

Pf(σcomposite) probability of failure of composite from σcomposite stress 

Pf(σfiber) probability of failure of fiber from σfiber stress 

Pf,gr grafting probability of failure 

Pfi(σ, a) probability of failure in the i

th individual graphite grain for crack size a and uniaxial 

stress σ

g Pfn probability of failure of ng graphite grains

( , ) g P a fn σ probability of failure of ng graphite grains for crack size a and uniaxial stress σ

PfV probability of failure of material volume 

∆PfV probability of failure of a crack with a strength between σIc and σIc + ∆σIc in ∆V

1 2 , , , P P fV fV probability of failure of volume 1, probability of failure of volume 2

1 , ( ) Pf V σ probability of failure for value of σ for volume 1

2 , ( ) Pf V σ probability of failure for value of σ for volume 2

f V { d } P ∆ probability of failure for incremental volume element ∆Vd

Ps reliability or probability of survival (Ps = 1– Pf) 

Ps,compressive probability of survival from a compressive failure mode

(Ps)i probability of survival of the i

th link 

Ps,tensile probability of survival from a tensile failure mode

PsV probability of survival of material volume V

s V { d } P ∆ probability of survival for incremental volume element ∆Vd

P(x) distribution of fiber strength

∆P1V probability of existence of a crack with strength between σIc and σIc + ∆σIc in an 

incremental volume

P2V probability that a crack of critical strength will be oriented in a particular direction such 

that it will grow and cause failure

p fraction of randomly chosen springs
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pce elastic response threshold (where removed members affect the elastic response of the 

system)

℘(α, β) probability density distribution of flaw normals (oriented perpendicular to the plane of 

the flaw) and given by angles α and β

q small, but nonzero, probability

rd distance from a broken fiber

rL constant (ratio) or parameter in KIc anisotropy function

rT constant (ratio) or parameter in KIc anisotropy function

S(a) distribution of pore sizes—a function of a where a is crack size

Sd scatter (standard deviation) in pore size distribution

Sμ mean pore size

u constant or parameter; constant between 1 and 2 in value

V volume; voltage 

∆V incremental volume

∆Vd incremental volume element

Ve effective volume

Ve1, Ve2 effective volumes of body 1 and 2

Vo characteristic volume

Vox characteristic volume for modified Weibull relation

Vtot total volume of component

Vδ cross section of volume of height δ for modified Weibull relation

Vδ,o characteristic value of Vδ for modified Weibull relation

V1, V2 uniformly stressed volumes of body 1 and 2

W(x) composite material weakest-link characteristic probability distribution independent 

of nf

W(σ) characteristic probability distribution as a function of σ

w span over which a uniform distribution of fuse breaking occurs

wc(L) boundary between brittle and ductile-like failure as a function of system size

wo threshold of w where single bond breakage causes system failure 

X discrete real-valued random variable

x specific value of X; any variable; load per fiber

x, y, z location in the body of the structure; Cartesian coordinate directions

x

* the point where x*

[1 – P(x

*

)] is maximum

xo characteristic value of x

xu threshold value of x
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α orientation angle, angular coordinate

β orientation angle, angular coordinate

Γ gamma function

γ constant or parameter in flaw orientation anisotropy function 

γL constant or parameter in KIc anisotropy function

γT constant or parameter in KIc anisotropy function

∆ increment

∆G difference in σ/σμ from the mean to the value of σ/σμ where the probability of failure is 

10–6 for the gaussian distribution

∆W difference in σ/σμ from the mean to the value of σ/σμ where the probability of failure is 

10–6 for the Weibull distribution

δ incremental volume; average grain height or multiple of; ineffective length; stress 

transfer length

Σ summation function; applied far-field multiaxial stress state

ε strain

εc critical strain

εcomposite composite strain

εfiber composite fiber strain

εmatrix composite matrix strain

εo,composite Weibull scale parameter for composite strain

εo,fiber Weibull scale parameter for composite fiber strain

ζ(α) ζ as a function of angle α describing the anisotropy of flaw orientation

ζ(α, β) function describing the anisotropy of flaw orientation where the normal direction to the 

flaw plane is given by angles α and β

η(σ) crack-density function

ηV(σ) crack-density function, number of flaws per unit volume with strength equal to or less 

than σ

ηV(σIc) crack-density function for mode I strength, σIc, of a flaw

θ angle of individual graphite grain in relation to the crack tip

κ constant or parameter

λ parameter in Poisson distribution

ν Poisson’s ratio

ξ constant or parameter in flaw orientation anisotropy function

ξL constant or parameter in KIc anisotropy function

ξT constant or parameter in KIc anisotropy function C

Π product
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π pi, 3.14159

σ applied uniaxial tensile stress

∆σ increment of σ

σ(x, y, z) uniaxial stress at location x, y, z in a body

σcomposite composite stress

σcon constant

σe effective stress for multiaxial stresses

σe,f peak effective stress in component

σf maximum stress in component; maximum stress in the component at failure

σfiber composite fiber stress

σf1, σf2 σf for body 1 and 2

σIc critical mode I strength

σIc(x, y, z, α, β) mode I far-field strength of a flaw located at coordinates x, y, and z and oriented at 

angles α and β

∆σIc increment of σIc

σIc max maximum value of σIc over the unit sphere

σIeq effective stress or equivalent stress

σIeq max maximum value of σIeq over the unit sphere from the applied multiaxial stress Σ

σi local tensile strength in incremental volume ∆Vi

σmatrix composite matrix stress

σn applied far-field stress component normal to a crack face

σo Weibull scale parameter

σo,composite Weibull scale parameter for composite

σocV Weibull scale parameter for uniaxial compression

σo,fiber Weibull scale parameter for fiber

σo,i transformed Weibull scale parameter for principal stress direction denoted with 

direction cosines 1, 2, and 3

σoV Weibull scale parameter for the volume-flaw failure mode normalized to unit volume

σo1, σo2, σo3 Weibull scale parameter σo for respective principal stresses σ1, σ2, σ3

σsd standard deviation of fiber strength

σu threshold stress

σuV threshold strength parameter for volume

σx, σy, σz global coordinate system

σδ,k Weibull scale parameter for k

th Weibull line

σθ Weibull characteristic strength 
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σθ(tensile specimen) Weibull characteristic strength of reference tensile specimen

σμ mean fracture strength

σ1, σ2, σ3 principal stresses (σ1 ≥ σ2 ≥ σ3) 

τ applied far-field shear stress on a crack face; shear stress acting on the oblique plane 

whose normal is determined by angles α and β

Φ constant or exponent

φ constant or parameter

φ scaling constant between 0 and 1

φ* modified scaling constant between 0 and 1 depending on material system

φ1, φ2 scaling constants between 0 and 1

ψ lattice model size effect constant

Ω(Σ,σIc) area of a solid angle projected onto a unit radius sphere in three-dimensional stress 

space for which σIeq ≥ σIc from an applied multiaxial stress state Σ

ω coefficient of variation

ωo coefficient of variation

∞ infinity

! factorial function

Subscripts

i i

th value or ith term

j j

th value or jth term

max maximum

n normal; n

th value

V volume or a volume-based property (e.g., indicates volume-flaw analysis)

μ average or characteristic

0 initial or starting value 

D.2 Definitions

Batdorf component reliability model using (typically) the Weibull distribution and 

fracture mechanics principles to account for the effect of multiaxial stress 

states on reliability (see Sec. 2.1.2.1)

Burchell The Burchell model—combines fracture mechanics with a physics-based 

microstructural description of graphite failure (see Sec. 2.1.6)

extreme fiber stress the location (point) in the body of the component where the stress is 

maximum (see σf)

fast fracture component rupture in the absence of slow crack growth where strength is 

strictly controlled by the fracture toughness and the size, distribution, and 

orientation of inherent flaws 
NASA/TM—2011-215805 98

mode I crack opening mode

mode II crack sliding mode (in-plane shear)

mode III crack tearing mode (out-of-plane shear)

R-curve where fracture toughness KR varies with crack size—typically increasing 

with crack size (see Sec. 2.1.4)

transient reliability analysis predicting the probability of survival of a component while accounting for 

loads and temperatures that can vary over time

Weibull distribution see Equations (4) and (8)

D.3 Acronyms and Initialisms

CARES Ceramics Analysis and Reliability Evaluation of Structures

ELS equal load sharing

FPZ fracture process zone

HVDP Hedgepeth and Van Dyke (1967) periodic LLS model

LEFM linear elastic fracture mechanics

LLS local load sharing

PIA principle of independent action—a component reliability model based on the Weibull 

distribution that accounts for multiaxial stress states by using principal stresses applied 

independently of one another (see Eq. (21) in Sec. 2.1.2.2, Barnett et al. (1967), and 

Freudenthal (1968))

RFM random fuse model

RSM random spring model

RVE representative volume element

WLT weakest-link theory
NASA/TM—2011-215805 99

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Statistical Models of Fracture Relevant to Nuclear-Grade Graphite: Review and 

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This report is an expanded version of Nemeth, Noel N.; and Bratton, Robert L.; 2010: Overview of Statistical Models of Fracture for 

Nonirradiated Nuclear-Graphite Components. Nucl. Eng. Des., vol. 240, no. 1, pp. 1-29. 

14. ABSTRACT

The nuclear-grade (low-impurity) graphite needed for the fuel element and moderator material for next-generation (Gen IV) reactors 

displays large scatter in strength and a nonlinear stress-strain response from damage accumulation. This response can be characterized as 

quasi-brittle. In this expanded review, relevant statistical failure models for various brittle and quasi-brittle material systems are discussed 

with regard to strength distribution, size effect, multiaxial strength, and damage accumulation. This includes descriptions of the Weibull, 

Batdorf, and Burchell models as well as models that describe the strength response of composite materials, which involves distributed 

damage. Results from lattice simulations are included for a physics-based description of material breakdown. Consideration is given to the 

predicted transition between brittle and quasi-brittle damage behavior versus the density of damage (level of disorder) within the material 

system. The literature indicates that weakest-link-based failure modeling approaches appear to be reasonably robust in that they can be 

applied to materials that display distributed damage, provided that the level of disorder in the material is not too large. The Weibull 

distribution is argued to be the most appropriate statistical distribution to model the stochastic-strength response of graphite. 

15. SUBJECT TERMS

Graphite; Strength; Flexural strength; Nuclear electric power generation 

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