Thermodynamics of Brittle Fracture 

by Michael A. Grinfeld and Tim W. Wright 

ARL-TR-3659 September 2005 

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Army Research Laboratory 

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ARL-TR-3659 September 2005 

Thermodynamics of Brittle Fracture 

Michael A. Grinfeld and Tim W. Wright 

Weapons and Materials Research Directorate, ARL 

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Thermodynamics of Brittle Fracture 

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Michael A. Grinfeld and Tim W. Wright 

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14. ABSTRACT 

This report outlines the results of the effort to suggest a theoretical method of describing brittle ceramic armor. We analyze

thermodynamics and kinetics of brittle fracture by combining the Gibbs approach to the study of heterogeneous thermodynamic 

systems, the Griffith energetic criterion for crack growth, and the Onsager irreversible thermodynamics. The central object of 

the study is the exchange between accumulated elastic energy and “chemical” energy accumulated in cohesive bonds of material 

particles. One of the main thrusts of the research is development of a continuum model describing dwell-defeat transition for a

metallic projectile hitting ceramic armor. As compared with known theories of fracture, we describe the morphology of the 

damaged zone. This makes our approach somewhat similar to the Gibbs theory of heterogeneous systems. 

15. SUBJECT TERMS

ceramic armor, brittle fracture, failure waves, morphological instability 

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Michael A. Grinfeld 

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iii

Contents 

List of Figures iv

1. Introduction 1

2. The Substance Model 2

3. Continuum Model of Fracture 4

4. Summary and Conclusions 6

5. References 7

Distribution List 8 
iv

List of Figures 

Figure 1. Damaged observation from recovered ceramics. ............................................................2 

Figure 2. The geometry of the system. ...........................................................................................2 

Figure 3. The specific energy density of the intact ( + e ) and damaged ( – e ) modifications............3 

Figure 4. Development of damage in a circular elastic plate with a cavity. ...................................5 
1

1. Introduction 

Brittle materials (i.e., ceramics) are being considered for an increasing number of applications 

ranging from armor elements to gun tube inserts to engine components, where they must 

withstand intense dynamic loads. The damage and massive failure processes that can occur in 

these conditions are not well understood. Because ceramics typically have lower symmetry than 

most metals, the basic crystal components are not only highly anisotropic but also have only 

limited slip systems for plastic deformation in polycrystalline configurations. Therefore, they 

must rely on other mechanisms such as twinning, microcracking, and void formation, as well as 

limited plasticity, to achieve required deformation. Adjacent grains of differing orientation will 

find it hard to maintain coherent boundaries; large stress concentrations within grains may build 

up; and large oscillations in stress and strain may occur from grain to grain. Homogenization 

theories can predict average material properties. However, in the circumstances just described, 

knowledge of the oscillations is also necessary. It is the purpose of this effort to develop 

techniques that will give an account of the oscillations, as well as the averages, and to apply the 

resulting theory to several important topics such as damage, incipient microcracking, and 

fracture waves. 

During several of the last decades, ceramic-based armor technologies and corresponding 

experiments and theories have shown a remarkable progress. One of the brightest achievements 

of the last decade is the discovery of the phenomena of dwell-penetration transition and of a 

defeat of a projectile targeting ceramic armor (1–3). The phenomenon of a projectile defeat is 

accompanied by the appearance of an intensively fractured zone (IFZ) within the bulk of a 

ceramic target. The shape of this zone resembles a tri-axial ellipsoid (see figure 1, where the IFZ 

zone is shown in white). In their recent publication, LaSalvia and Normandia (4) interpreted it 

on the basis of the famous Bussinesq solution of classical theory of elasticity. Similar 

observations of IFZ are known in ceramic indentation experiments (5) and in the seismology of 

earthquake epicenters (6). 

When dealing with IFZ, it is often desirable to extend the information about the overall modulae 

with the information about the shape and morphology of the IFZ. Such information can be 

obtained by using additional, specific criteria or more general principles of mechanics or 

thermodynamics. In this report, following the minimum energy approaches of Gibbs (7–8) and 

Griffith (9), we propose one such thermodynamic approach, which allows one to determine the 

shape of the IFZ simultaneously with the distributions of stresses and strains within the systems 

of interest. The general methods and results of the suggested approach are illustrated by 

considering an instructive problem of nucleation of a damaged zone in an isotropic solid space 

under sufficiently high uniform stresses at infinity. 
2

Source: Picture courtesy of J. LaSalvia and M. Normandia (4). 

Figure 1. Damaged observation from recovered ceramics. 

2. The Substance Model 

Consider a macroscopically heterogeneous configuration possessing two macrodomains, ω+ and 

ω– , occupied by the consolidated and damaged modifications, respectively. Let γ be the 

interface separating the two domains and S the external boundary of the system (see figure 2). 

Figure 2. The geometry of the system. 

The consolidated (intact) modification is characterized by the elastic energy density per unit 

volume 

1

2

ijkl

+ i| j + i| j k|l e (u )= c u u , (1) 

where i u (x) is the displacement at the point ( ) i j

i| j i x u = u/ x ∂ ∂ and ijkl

+c is the elasticity tensor of 

the consolidated modification. The absence of the free-constant and linear terms in the Taylor 

expansion of i| j e(u ) indicates that the configuration with = 0 i u is stress free, and it is chosen as 
3

the base for the calculation of the elastic energy of the substance. For the damaged modification 

of our system we can use an “effective” elastic energy density as follows: 

– –

1 () . 2

ijkl

i j i j kl e u e c uu = +° (2) 

The damaged modification is elastically weaker than the intact one, i.e., its elasticities in some 

sense are smaller than the elasticities of the intact modification. Besides, there is a nonzero 

constant ° e in the elastic energy density of the damage modification—this constant reflects the 

work required to produce defects in the intact modification. Schematically, the two energies are 

shown in figure 3. 

Figure 3. The specific energy density of 

the intact ( + e ) and damaged 

( – e ) modifications. 

The parabola of the consolidated substance passes through the origin and has the fastest growth 

with ε . It is clear that for * ε < ε , the elastic energy density of the consolidated phase is less 

than in the damaged phase (for * ε > ε , the elastic energy density of consolidated phase is 

greater than in the damaged phase). The trade-off between the opposite effects of the damage on 

“free-stress” energy – e and on the elasticity tensor ijkl

– c at various deformations is the central 

conflict under study with the help of our suggested model. 

The simplest model of the previously mentioned type is the model of an isotropic two-state 

elastic substance. For this model, the energy densities for two modifications are given by the 

following formula: 

±

±

±

+

1 – 2

i j ij

i| j ± ± .|i .| j (i| j) .|.

ν e (u )= κ +μ u u u u

ν

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

, (3) 

where ± μ are the shear Lame modules and ± ν are the Poisson’s ratios ( κ+ = 0). 
4

3. Continuum Model of Fracture 

Per classification of Grinfeld and Wright (10, 11), the model just considered is a certain twostate discrete fracture model. A continuum fracture model should include a damage parameter κ

that can assume a continuum range of values. Corresponding elastic energy e function is 

becoming a function of the elastic gradients i| j u and of the damage parameter κ : i| j e = e(u ,κ). 

For such a model, the associated equilibrium master system includes the following two bulk 

equations: 

= 0, ni|n

j

i| j

e(u ,κ)

x u

∂ ∂

∂ ∂ (4) 

and 

= 0 i| j e(u ,κ)

κ

∂

∂ . (5) 

The associated system of quasistatic evolution reads as follows: 

= 0, ni|n

j

i| j

e(u ,κ)

x u

∂ ∂

∂ ∂ (6) 

and 

i| j κ e(u ,κ) =–K

t κ

∂ ∂

∂ ∂ , (7) 

where K is a certain positive kinetic constant or function. 

The associated dynamic system reads as follows: 

,

2 i

ni|n

2 j

i| j

u e(u ,κ)

ρ = txu

∂ ∂ ∂

∂∂∂ (8) 

and 

– i| j κ e(u ,κ) = K

t κ

∂ ∂

∂ ∂ . (9) 

The important issue here is the choice of an appropriate energy function i| j e = e(u ,κ) and the 

kinetics function K. The proper choice is a hard problem of physics of ceramic materials. Some 

insight can be acquired by using approaches accepted in phenomenological damage theory (12) 
5

and the theories of proliferation of vacancies in ceramics sintering (13). Needless to say, 

cracking is not a proliferation of vacancies because of various reasons, but this might be a good 

first step. 

The quasistatic system presented in equations 6 and 7 was used to numerically explore the 

behavior of a circular plate with a thin elliptic cavity under action of external pressure (figure 4). 

The kinetics function K was assumed a certain positive constant. For the energy density, we 

have chosen the following Kachanov-Lifshitz function: 

( ) ( )2

1–2 2

i j ij °

i| j .|i .| j (i| j) .|.

ν ξ e(u ,κ)= μφ κ u u +u u + κ – κ

ν

⎛ ⎞ ⎜ ⎟ ⎝ ⎠ , (10) 

with the damage function φ(κ) (10, 11) as follows: 

( ) min 1– 1– *

κ (κ)= c ,

κ

φ (11) 

 0 * ≤ ≤ κ κ , (12) 

and 

min 0 1 < c ≤ . (13) 

Figure 4. Development of damage in a circular elastic plate with a cavity. 

Thus, the suggested model depends on six constants— min and * μ ° , ν, ξ, κ , κ , c . The first two of 

them are just the shear modulus and Poisson’s ratio of the intact substance; the physical meaning 

of the remaining four constants are explained elsewhere. For a certain choice of the constants, 

the evolution of the relative damage parameter * κ/κ is shown in figure 4. The depth of gray 

reflects the magnitude of the relative damage parameter, white corresponds to a fully damaged 

material, and black to a fully undamaged. Not surprisingly, the largest damage occurs at the tips 

of the cavity. 
6

4. Summary and Conclusions 

• The proposed theoretical approach to brittle fracture allows the determination of the 

morphology of damaged zones together with the traditional fields like deformation and 

stress. 

• Similar to the Griffith model, the proposed approach is based on the trade-off between the 

elastic energy and the energy of chemical bonds. 

• In the proposed approach, the damaged and intact substances are treated as different phases 

of the same substance, and “phase transformation” damage-recovery is taken into account. 

• The equilibrium system includes coupled equations for the unknown displacement field, 

damage parameter, and morphology. 

• Because of its deep nonlinearity, even for small deformations, the equilibrium system may 

have multiple solutions. 

• The equilibrium system does not allow static solution for some boundary loads; hence, the 

system should be extended by the inclusion of inertia and dissipation mechanisms. 
7

5. References 

1. Hauver, G. E.; Netherwood, P. H.; Benck, R. F.; Kecskes, L. J. Ballistic Performance of 

Ceramic Targets. Army Symposium on Solid Mechanics, Plymouth, MA, 17–19 August 

1993. 

2. Hauver, G. E.; Netherwood, P. H.; Benck, R. F.; Kecskes, L. J. Enhanced Ballistic 

Performance of Ceramic Targets. Proceedings of the 19th Army Science Conference, 

Orlando, FL, 20–24 June 1994. 

3. Bourne, N.; Millett, J.; Pickup, I. Delayed Failure in Shocked Silicon Carbide. J. Appl. 

Phys. 1997, 81, 6019–23. 

4. LaSalvia, J. C.; Normandia, M. J. An Analytical Prediction for the Effect of Ceramic 

Thickness and Mechanical Properties on the Dwell/Penetration Transition Velocity. 20th 

International Symposium on Ballistics, Orlando, FL, 2002. 

5. Lawn, B. R. Indentation of Ceramics with Spheres: a Century After Hertz. J. Am. Ceram. 

Soc. 1998, 81, 1977–1994. 

6. Kostrov, B. V.; Das, S. Principles of Earthquake Source Mechanics. Cambridge University 

Press: New York, 1988. 

7. Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. Trans. Connect. Acad. Sci. 

1876, 3, 108–248. 

8. Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. Trans. Connect. Acad. Sci. 

1878, 3, 343–524. 

9. Griffith, A. A. The Phenomenon of Rupture and Flow in Solids. Phil. Trans. Roy. Soc. 

Lond, Ser. A. 1921, 221, 163–198. 

10. Grinfeld, M. A.; Wright, T. W. Thermodynamics of Solids: Recent Progress with 

Applications to Brittle Fracture and Nanotechnology. 23rd U.S. Army Science Conference, 

Orlando, FL, 2002. 

11. Grinfeld, M. A.; Wright, T. W. Morphology of Fractured Domains in Brittle Fracture. 

Metallurgical and Materials Transactions A 2004, 35A, 2651–2661. 

12. Kachanov, L. M. Introduction to Continuum Damage Mechanics; Dordrecht-BostonLancaster, Martinus-Nijhoff Publishers, 1986. 

13. Lifshits, I. M. Towards Theory of Diffusion-Viscous Flow of Polycrystalline Media. 

ZhETP 1963, 44, 1349–1367. 
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