A 1110M DVbSMD

NBSIR 85-3034

FRACTURE MECHANICS

CHARACTERIZATION OF CRACK

ARRESTAND REINITIATION IN TWO

UNCONVENTIONAL SPECIMENS

FILE COPY

TTeramoto

QQ RQJ REMOVE

R.B. King

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

June 1986

QC

100

.U56

85-3034

1986

NBSIR 85-3034

FRACTURE MECHANICS

CHARACTERIZATION OF CRACK

ARREST AND REINITIATION IN TWO

UNCONVENTIONAL SPECIMENS

T. Teramotof

D.T. Read

R.B. King ft

Fracture and Deformation Division

Institute for Materials Science and Engineering

National Bureau of Standards

Boulder, Colorado 80303

June 1986

Sponsored by

Office of Naval Research

Arlington, Virginia 22217

and

David Taylor Naval Ship Research and Development Center

Annapolis, Maryland 21402

fPresent address: Institute of Engineering Mechanics, University of Tsukuba, Sakura, Ibaraki 305, Japan

ffPresent address: IBM Research Division, San Jose, CA 95193

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

I. FRACTURE MECHANICS PARAMETERS IN CRACK ARREST SPECIMENS

1. Introduction 1

1.1 Purpose of Crack Arrest Testing 1

1.2 Need to Understand Dynamics of the Boundary 3

1.3 Need for a Method for Calculation of the

Fracture Mechanics Parameter 4

2. Previous Fracture Mechanics Analyses of Crack Arrest 5

2.1 Static Analysis 5 2.2 Kanninen Model 7 2.3 Finite Element Model (FEM) 9

3. Mass-Spring Model of Crack Arrest Specimen 10

3.1 A Crack Arrest System Under Fixed-Grip Conditions 12

3.2 Compliant Loading 13

3.3 Compliant Loading With a Viscous Spring 14

3.4 Compliant Loading of a Yielding Specimen 15

4. Extraction of the Fracture Mechanics Parameter 17

4.1 Stress Intensity Factor K From the MS Model 17

4.2 Stress Intensity Factor K From the Kanninen DCB Model 18

4.3 Stress Intensity Factor K From the FEM Model 19

4.4 Elastic-Plastic Crack-Driving Force J From the MS Model... 19

5. Comparison of the Models 21

5.1 Relation Between Load and Time 22

5.2 Relation Between Load-Point Displacement and Time 23

5.3 Relation Between K and Time 23

5.4 Conclusion for Section 5 24

6. Comparison of the MS Model With Experimental Data 24

6.1 Relation Between Load, Load-Point Displacement, and Time... 24

6.2 Relation Between J and Time 26

7. Summary and Conclusion 2 7

7.1 Validity of the MS Model for General Features of

Crack Arrest Behavior 2 7 7.2 Better Model 28

7.3 Method of Comparison With Experimental Data 29

8. Acknowledgments 29

9. References jo

Tables y> Figures

iii
Page

II. CRACK ARREST EXPERIMENTS AND ANALYSIS WITH RISING K FIELD

1. Introduction 65

2. Specimen 66

3. Analytical Methods 67

3.1 Finite Element Model 67

3.2 Mass-Spring Model 68

4. Material and Techniques 70

5 . Results 71

6. Discussion 74

7. Conclusions 75

8. Acknowledgments 76

9. References 76

Tables 78

Figures 81

IV
I. FRACTURE MECHANICS PARAMETERS IN CRACK ARREST SPECIMENS

T. Teramoto, D.T. Read, and R.B. King

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

A simple elastic-plastic-fracture-mechanics-based model of crack propagation, arrest, re-initiation, and propagation is described. This model requires

much less computing resources than dynamic, elastic-plastic finite element

calculations, and allows estimates of applied J-integral, load, and crack mouth

opening displacement during initial rapid crack propagation, re- initiation and

repropagation. A comparison of this new model to other available models and

to experimental results indicates that it can successfully reproduce the essential features of the behavior of specimens containing propagating cracks.

Keywords: crack arrest; dynamic fracture; stress intensity factor; J-integral;

mass-spring; double-cantilever-beam model; finite element model.

1. Introduction

1.1 Purpose of Crack Arrest Testing

Failure of large welded steel structures, such as ships, can occur if a

crack, once initiated, severs large sections of the structure. Such failures

have actually occurred. To prevent such failures, crack arrester plates are

often included in ships. Such plates are required to have sufficient toughness

that running cracks that enter them will be unable to propagate, and will thus

be prevented from fatally damaging the ship.

Fracture mechanics principles can be applied to the selection of materials

for structural crack arrester plates if the crack arrest toughness of candidate

materials and the driving force for dynamic crack propagation can be compared.

Exact determination of these parameters requires dynamic fracture mechanics

1
analyses of both crack arrest test specimens and crack arrester plates in structures. Such analyses are unavailable at present. Crack arrest and reinitiation

are elastic-plastic phenomena, and reinitiation can involve ductile tearing as

well as plastic deformation near the crack tip. Clearly, exact analyses of

the crack arrest-reinitiation event will be unavailable for some time. However,

analysis using simplified models has provided significant progress to date,

notably linear elastic calculations of dynamic stress intensity factors. At

present, no dynamic elastic-plastic calculations, however simplified, are available.

Tests of the crack arrest toughness of steels have been designed using

available understanding of the crack arrest process. Many large scale crack

propagation-arrest tests of brittle materials have been conducted, especially

the test described, for example, by Feely et al . [1] (Esso test) in the United

States and double-tension tests [2] in Japan. The fundamental concept used

for analysis of these tests was that a static fracture mechanics approach could

be applied, even though the crack propagation involved complicated dynamic

effects. This simple approach is strictly correct as long as reflected stress

waves from specimen boundaries do not affect the running crack tip [3]. However,

longer running cracks are apt to be affected by the boundary conditions.

Battelle Memorial Institute developed a small specimen crack arrest test,

the Double Cantilever Beam (DCB) specimen, and performed a complete dynamic

elastic analysis using a simple beam model several years ago [4]. It was concluded that dynamic fracture toughness is a function of the crack velocity and

that minimum fracture toughness should be used in structural design. That

work shed some light on dynamic fracture mechanics analysis, although new problems were discovered.

2
1.2 Need to Understand Dynamics of the Boundary

Recent experimental studies have indicated that for structural safety the

preceeding arrest concept is not necessarily sufficient because the arrested

crack may be reinitiated in the arrester when load is applied to the uncracked

ligament remaining after initial, momentary arrest. The stress redistribution

or reloading cannot take place until the running crack tip encounters the crack

arrester, because crack velocity is, in general, very high (500-1500 m/s).

The whole massive structure deforms gradually, because of inertia, reloading

the ligament and applying a large crack-driving force. If this crack-driving

force exceeds the reinitiation toughness, the structure can be further damaged

by additional, possibly unstable, crack extension.

Previous analyses have not considered this reinitiation phase, because

conventional testing procedures do not maintain the initially applied load

both during crack propagation and after initial crack arrest. Therefore, the

specimen has been regarded as being under fixed-grip conditions before crack

arrest, because of inertia.

King et al. [5] have developed a crack arrest-reinitiation test. Their

specimen configuration is similar to the DCB specimen developed at Battelle.

But their specimen consists mostly of ductile material, with a brittle crack

propagation path produced by electron-beam welding. This relatively brittle

weld in a ductile base plate simulates a low-toughness plate adjacent to a

crack arrester in a structure. The load is applied through a number of disk

springs, to prevent load drop resulting from the large compliance change during

crack extension. This spring displaces the specimen grip, producing a large

load on the uncracked ligament; thus, reinitiation of an arrested crack can

easily take place.

3
1.3 Need for a Method for Calculation of the Fracture Mechanics Parameter

The complete fracture event in steel, which consists of crack initiation,

propagation, arrest and reinitiation, has both macroscopic and microscopic

aspects. It is impossible with the use of any model to analyze such complicated behavior completely. However, it is possible to study fracture characteristics to an extent, using simple models that extract key features of the

fracture behavior. The crack arrest- rei ni tiation behavior is largely divided

into two phases. Linear Elastic Fracture Mechanics (LEFM) analysis is applicable before crack arrest because the yield point of the material is elevated

by the high strain rate at the fast-running crack tip; post-test examination

of this part of the fracture surface usually suggests a macroscopical ly brittle

failure mode. On the other hand, Elastic-Plastic Fracture Mechanics (EPFM)

analysis is applicable after crack arrest because the uncracked ligament can

yield. The dynamic stress intensity Kp, calculated by LEFM, is pertinent as a

fracture mechanics parameter before arrest. Dynamic crack propagation in a

finite body has been analyzed in terms of Kp using the Finite Element Method

(FEM) [6], Finite Difference Method (FDM) [7], and so on.

These are, in fact, calculations of the behavior of models of fracture

mechanics specimens. The models are fairly elaborate, but they are linearelastic and two-dimensional.

The linear-elastic fracture mechanics parameter Kp is not appropriate for

characterization of dynamic elastic-plastic crack tip behavior after crack

arrest. Because parameters such as the J-integral, Crack Opening Displacement

(COD) or Crack Opening Angle (COA) are found to be useful even for unstable

fractures with large plastic deformation under quasi static loading, these parameters are considered to be promising for the dynamic case, because the strain

near the crack tip after arrest, in general, changes much more slowly than

during crack propagation. These parameters could be calculated using

4
dynamic elastic-plastic finite element analyses of two-dimensional models,

which are currently under development. The computing resources required for

such analyses are significant; furthermore, given such an analysis, one would

like a way to check it. Simplified models, requiring only small computing

resources, offer insight into the crack arrest-reinitiation process and a means

of checking more exact models.

This study proposes a simple model involving masses and springs (MS model)

to analyze crack arrest-reinitiation tests. Both elastic and elastic-plastic

dynamic behavior are treated. The MS model was verified by comparing the time

dependencies of loads, load point displacements and elastic stress intensity

factors calculated using the MS model with those calculated using the Kanninen

double-cantilever-beam model and a finite element model. The values obtained

are approximately the same for all models; this agreement provides encouragement to pursue the MS model further. Elastic-plastic behavior after crack

arrest was modeled for two specimens of the one type and dynamic J-integral

values were calculated.

In this paper, dynamic J, the elastic-plastic crack driving force, will

be obtained as the fracture mechanics parameter during the crack arrest test;

we will discuss how the J-integral is calculated in several models and what

the J-integral obtained means.

2. Previous Fracture Mechanics Analyses of Crack Arrest

2. 1 Stati c Analysi

s

Dynamic crack arrest has commonly been regarded as quasistatic in conventional dynamic fracture experiments, although Kanninen [8] has proposed an

analytical solution in which a crack runs at constant velocity in infinite

material with small-scale yielding. To date, crack arrest phenomena can be

5
treated only by static analysis. Crack arrest toughness, K , a stress inten3

sity, has often been obtained by the Esso test or doubl e-tension test with a

temperature gradient. In these test specimens, the running crack is arrested

when the toughness at the crack tip exceeds the crack-driving force. It was

concluded that static analysis could be used to estimate crack arrest toughness

because dynamic effects on the crack tip were rather small just before crack

arrest. Crosley et al. [9] also insist on the validity of static analysis for

crack arrest tests using the tapered DCB specimen. But other experimental

investigations using very wide specimens have revealed that the simple

approach based on the arrest toughness concept is invalid for the case of long

arrested cracks [10].

Freund [3] analyzed a running crack in a semi-infinite body and found a

simple relation between dynamic stress intensity and static stress intensity

K as follows:

s

K

D = (i - a/c/* Ks

where Cp is Rayleigh wave velocity and a is crack velocity. On the basis of

Freund's concept, equals at crack arrest (a = 0). If the crack runs a

short distance compared with specimen size and is arrested before the reflected

stress wave significantly affects the running crack tip, the effect of dynamic

aspects, such as kinetic energy, stress wave propagation and so on may be neglected. Kanazawa et al. [11] carried out a series of dynamic fracture experiments using a doubl e-tension test with varied distance between two loading

points, where significant wave interaction occurred. Static analysis was found

to become insufficient as the crack ran farther because boundary conditions

affected the crack tip.

6
2. 2 Kanni nen Model

The DCB specimen used as a crack arrest specimen can be modeled by a simple beam configuration. Kanninen [12] attempted to model the DCB specimen as

an Eul er-Bernoull i beam with extensional springs along the uncracked ligament,

as shown in Fig. 1. The governing equation of the beam deflection of this

model is as follows:

(d 4 w/dx4

) + 4A4 H(x)w = 0 ( 1 )

where H(x) = 1 for x ^ 0

= 0 for x < 0,

and A

4 = 3k/EBh3

, a = crack length, b = remaining ligament, h = specimen halfheight, B = specimen thickness, k = spring stiffness ( = 2EB/h), w = lateral

deflection of the beam and E = Young's modulus. Equation (1) was solved using

Laplace transformation, and the compliance was expressed as

6/P = 4/EB (2)

where P = load, 6 = load point displacement, and

4 = (2/A3 h

3 ){2A3 a3 + (6X2

a

2 [sinh(Ab) cosh(Ab) + sin(Ab) cos(Ab)] + 6Aa[si nh 2 (Ab)

+ s i

n

2

( Ab ) ] + 3 [ s i nh(Ab) cosh(Ab) - sin(Ab) cos(Ab)])/[si nh 2 (Ab) -

si

n

2 (Ab)]}

.

(3)

Stress intensity is given as

K - (2V3P/ABh 3 /2 ){Aa[sinh 2 (Ab) + sin 2 Ab] +sinh(Ab) cosh(Ab) - sinAb cosAb}

/[sinh2 (Ab) - sin 2 Ab]. (4)

7
Next, Kanninen et al. [13] proposed the Timoshenko beam model with a generalized foundation, where not only extensional but rotational inertia is included,

as shown in Fig. 2. The following equation of motion was introduced under

fixed-grip conditions:

where w is average lateral deflection; £ , average rotational angle; A, cross

section area; I, moment of inertia; p, density; G, shear modulus. Extensional

(k

g

) and rotational (k^) spring stiffness are given as

H* is a modified step function as used in eq. (1). The equation of motion can

be explicitly solved using the Finite Difference Method. Crack extension is

simulated by the elimination of springs, one by one, at the crack tip position

at the specified time. If the uncracked ligament is larger than half of specimen height, compliance and stress intensity of this model are simply expressed

as follows:

(5)

k = 2EB/h

e

k = kGA/2

r

where k =_ 10(1 + v)

12 + llv , and v = Poisson's ratio.

( 6 )

( 7 )

8
Numerical and experimental investigation revealed that the recovery of

kinetic energy contributes to driving the running crack farther and that dynam

ic fracture toughness can be defined as a function of crack velocity, as

fol lows:

K

D a a

: r

= i + A * ( r) + A2 ( r) C 0 0

where K

c is static crack initiation toughness, C

q is the bar wave velocity,

and A x and A2 are material constants.

2.3 Finite Element Model (FEM)

Many numerical studies have used Finite Element Models (FEM) to investigate dynamic crack propagation and arrest phenomena [6, 14-17]. These models

include conventional triangular elements [6,14], rectangular elements [15],

and singular and isoparametric elements [16]. Any FEM model divides the speci

men into many elements. Afterwards, the equations of motion of the individual

elements are synthesized into an equation of motion of the whole specimen as

fol lows:

[M]{u] + [C]{u] + [ k ]

{

u } - {f} (8)

where [M], [C] and [k] are mass, damping and stiffness matrices, respectively;

{u}, { u} , and {u} are the acceleration, velocity and displacement vectors of

model points, respectively; and {f} is the force vector applied to nodes.

There are many ways to solve such an equation of motion numerically; care must

be used in selecting the relationship between the time step and least element

size [14]. Equation (8) will be found later to correspond with the equation of

motion of a simple model, which consists of a mass and a spring.

9
In the usual FEM, crack extension is simulated by the gradual release of

the reaction force of the node corresponding to the crack tip and by the vanishing of that force by the time the crack arrives at the next node. This numerical

technique was proved to be valid by comparison with an improved analytical

solution by Mai luck and King [6], It is thought that the energy of the whole

specimen can be dissipated near the crack tip by using this technique. On the

other hand, a singular element near the crack tip can move along with the propagating crack tip in the improved analytical solution.

FEM is a useful technique to analyze any arbitrary specimen configuration

accurately; it can enable modeling of the two-dimensional propagation of the

stress wave that radiates from the crack tip. Flowever, it requires excessive

computer time to use the FEM model even if the specimen is assumed linear

elastic.

3. Mass-Spring Model of Crack Arrest Specimen

This study describes a simple model for dynamic analysis even under largescale yielding conditions; the dynamic fracture behavior can be characterized

using the J-integral calculated from this simple model. The model is applied

to a specific DCB specimen; however, it is believed that this model can be

applied equally well to other specimen geometries. The crack arrestreinitiation test specimen treated here, the configuration of which is similar

to the DCB specimen, was developed by King et al. [5]. Figure 3a shows the

loading system of this specimen schematically. The load was applied through

the external massive spring. The MS model substitutes the simple mass and

spring system, shown in Fig. 3b, for the actual specimen and loading devices.

« ex

is the stiffness of the external loading spring and in this model k

s is

the stiffness of the specimen, which depends on crack length as given below,

from eq. (2):

10
k = EB/24>. (9)

M

gx

is the mass of both the loading spring and the associated apparatus; u is

regarded as total displacement of the loading point. It is very easy to write

and solve the equation of motion for such a simple model, and it is assumed

that total energy, displacement and load can govern the motion of this system.

The drawback is that the actual specimen and load train have many normal modes

of oscillation because the active mass and compliance are distributed throughout the specimen and load train. The simple model of Fig. 3b radically simplifies the actual mass and compliance distribution, and has only one normal mode

of osci 1 1 ati on.

Net Section Yielding (NSY), that is, complete yielding of the uncracked

ligament, is easily realized in the spring-loaded DCB specimen, because the

reloading facilitates increase of the applied stress and application of a large

moment along the short uncracked ligament. Two simple assumptions may be

adopted in this mass-spring model to express NSY and large deformation of the

specimen: limit load corresponding with NSY; and damping factor, which models

the elevation of the yielding point due to strain rate. One may question

whether this simplified model is adequate to reproduce the actual complicated

behavior that can be observed during dynamic elastic-plastic fracture. However,

it will be possible to characterize dynamic elastic-plastic fracture behavior

to some extent if load or load-point displacement, obtained numerically, can

fit those obtained by experiment. In addition, the specimen is divided into

two parts to get dynamic J using a static analogy. A spring with the same

compliance as the specimen and a limit load with a damping factor are substituted for the deformed specimen. The former yields elastic displacement of

the specimen while the latter yields plastic displacement. J is obtained using

11
these two displacements and limit load, which occurs after the applied load on

the specimen spring exceeds the limit load.

Details are given below for fixed grip, compliant loading, compliant loading

with viscosity in the spring, and compliant, viscous spring loading of a yielding

speci men.

3.1 A Crack Arrest System Under Fixed-Grip Conditions

Assume that a DCB specimen is fractured under fixed-grip conditions, which

are approximately satisfied during crack propagation for the DCB specimen under

wedge loading. It has been verified at Battel le Institute that the crack runs

at nearly constant speed during propagation [4].

For the MS model, a fixed-grip condition is achieved by fixing the displacement of the mass in Fig. 3b. The relation between crack length and time,

or crack velocity, was input to the numerical calculation. This case is static

because the mass never moves. The load P applied to the specimen is given as

P = k

s

• u. (10)

In this model, k is the spring constant of the specimen, and it decreases

with increasing crack size, that is, with time. k

& is taken to be the static

stiffness of an identical specimen with crack length equal to the instantaneous dynamic crack length.

Figure 4 shows the load change with time for three different constant

crack velocities, which encompass the experimentally observed crack velocity

[5]. Initial applied load and crack propagation distance are common to the

three cases. The load is found to drop quickly with the crack velocity because

of the large compliance change occurring during crack propagation.

12
3.2 Compliant Loading

In this section, consider that the load is applied to the specimen through

the loading spring and that the crack runs 135 mm at 714 m/s. The effect of

mass and spring on the fracture behavior of the specimen are investigated.

The equation of motion for the system shown in Fig. 3b is:

M Li + ex(kv + k ) u = k u. ex s ex A

( 11 )

where u^ is the initially applied displacement at the loading end. As the

specimen stiffness changes during crack propagation, the following integral

equation ranging from time t to t + At should hold, using D'Alembert's

pri ncipl e:

>-t+At

A {[Mex

'u(t) + [« ex

+ k

s (t)]u(t)] - K ex

uA}dt = 0 ( 12 )

where At is the time increment. Therefore, eq. (11) is replaced by

M Au + (k + k )Au + Ak u = 0 ex ex s' s (13)

where Au, Au, A«

s are the increments of acceleration, displacement and spring

stiffness from t to t + At. This incremental equation of motion is explicitly

solved by Newmark's p method. The time increment is carefully chosen to be

less than one period of the natural frequency of this system as follows:

At ^ [ML /(k

ex

v ex

K

s )]

13
Therefore, At is set to 1 |js. The change of load and displacement with M

gx are shown in Figs. 5 and 6. The load P is given by eq. (10). Crack arrest is

imposed at about 190 ps, and the load drop during crack propagation is independent of mass. As the force balance that exists before crack initiation

changes during crack propagation in Fig. 3b, the force difference is inertial

force:

where F^ is the force applied to the external loading spring and F ^ is the

force applied to the specimen. If M is large enough that acceleration Li is

C A

regarded as negligible, the fixed-grip condition is achieved during crack propagation. This fact is supported by the variation of displacement with mass

shown in Fig. 6. The load is found to recover quickly after the external loading

spring begins to work, but the period depends on the mass. The displacement

increases quickly after crack arrest and has the same period as the load.

The changes of load and displacement with various stiffnesses of the external loading spring are shown in Figs. 7 and 8, respectively. Similarly,

there is no effect of the load or the displacement among three cases during

crack propagation. It is concluded from Figs. 5-8 that the external spring

stiffness affects the load and displacement much less than the mass.

3.3 Compliant Loading With a Viscous Spring

The damping term, or viscosity, is introduced in the external spring as

shown in Fig. 9, to account for the following friction effects in the actual

experiment: the friction of the loading grip; the friction in the external

spring, which consists of several disk washers; and the friction between loading

grip and spring apparatus. The equation of motion is given as

14
M (14)

ex

where C gxis the viscosity coefficient of the high spring. The incremental

equation of motion, analogous to eq. 13, is also used instead of eq. (14)

during crack propagation. The changes of the load and the displacement are

shown in Figs. 10 and 11, respectively, for arbitrarily chosen viscosity coefficients. The initial condition in the calculation is the same as used in

the last section. While the change of the load and the displacement are the

same with and without the viscosity effect during crack propagation, the recovery of the load and the displacement are found to be delayed after arrest

when viscosity is included. The result with a small viscosity coefficient

approaches the linear elastic case. It is noted that the period of the system

is unchanged in spite of the introduction of viscosity, because the range of

viscosity is such that the system is lightly damped.

3.4 Compliant Loading of a Yielding Specimen

The simple model is extended to simulate NSY behavior after crack arrest.

As shown in Fig. 12, it is assumed that the load applied to the uncracked ligament remains at the limit load after the load on the specimen reaches limit

load P,. ... Also, additional viscosity is included for elevation of the yield

point due to strain rate and internal friction of the specimen. Another mass,

M , yields the plastic displacement of the specimen. Here a value of M =

27 kg was used; this was the mass of the specimen alone. P, . .. is given as

where W is specimen width, b is remaining ligament, and ct^ is the flow stress

of the specimen material. The load is evaluated as follows:

P

1 imit

= a

f {-(2W - b) + [ ( 2W - b) 2 + b

2 ]^}

B

(15)

15
P=k(u-u)-M u ( 16 )

. s p s p

Here u is the plastic part of the displacement. This system remains elastic

r as long as the load P does not exceed P-j . The equations of motion governing

this system are

M u + C u + (i< + k ) u = k u n , for P < P. CS\S C»V v Q V C J C\\J A ’ (17a)

and

M U + C u + (k + k ) u = k u.+ku ex ex ex s' ex A s p

M U + C u + k u — k u - P, . for P t P-, • (17b)

s p s p s p s limit’ limit v J

where C

g is the viscous coefficient in the specimen. The effect of changes in

the viscosity coefficient on load and displacement is shown in Figs. 13 and

14. Flow stress ct^ is assumed to be 551 MPa. These plots present very different features after crack arrest, depending on the viscosity. When the

viscosity is rather small, the load does not increase so much, and both total

and plastic displacements increase remarkably with time. These phenomena are

considered to be applicable to an approximately elastic-plastic rate-insensitive

material. When the viscosity is too large, the Toad and total displacement

oscillate strongly. As their behaviors seem close to elastic, they can be

applied to rate sensitive material. The moderate viscosity yields a smooth

increase of the displacements and a gradual decrease of the recovered load.

Both the load and the displacements are expected to be saturated at certain

values at long times.

16
4. Extraction of the Fracture Mechanics Parameters

Since this MS model simulates the crack arrest-reinitiation test, it is

necessary to extract the appropriate fracture mechanics parameters corresponding with each aspect and to investigate their validity. To facilitate specific

comparison of the models, the extraction of fracture mechanics parameters from

Kanninen's DCB model and the FEM model are also reviewed. Then a method for

calculating the elastic-plastic crack driving force in terms of the J-integral

from the MS model is introduced.

4.1 Stress Intensity Factor K from the MS Model

K is obtained using the MS model in Fig. 3b. The potential and kinetic

energy 7i of this system is given as

71 = -x M u 2 ex

2 + !

k

s u2 +2 ex

(u

A

- u)2. (18)

The total energy release rate is obtained by differentiating eq. (18) with

crack length a, as follows:

371 _ 371 1 u

2 3k

= ± -

"TJ- ^ - [M u + (k + k )u - k U a ]- . at. • 7 3a L ex c pv ; pv A

j » ex^ (19)

a

' ' ' a

The term in square brackets vanishes because it corresponds with the equation

of motion as given in eq. (11). Then, the energy release rate is

. 371 3a

3k < 3a" ( 20 )

Equations (9) and (10) are substituted into the above equation, and eq. (20)

becomes

371 =

P£ 3^

3a EB 3a

'

17
Using the relation between stress intensity K and potential energy release

rate, K is expressed as

N

B ^9a J ’ ( 21 )

where <j) is defined in eg. (3). This K results in eg. (4) mentioned previously.

4.2 Stress Intensity Factor K From the Kanninen DCB Model

In the Kanninen model (Section 2.2), the crack propagation is simulated

by releasing the springs corresponding with the moving crack tip, which is

located discretely along the uncracked ligament. Strain energy from extending

and bending is stored in each spring. Total strain energy U and total kinetic

energy T are given:

0 i [ei (I)

2 + kga (I - c)2 + h X«

2 + m2) dx

]

(22)

and T

=J

pA + pi dx.

If the fixed-grip condition is assumed in this specimen, total energy 7i is

obtained as

7T = U + T. (23)

The energy release rate is obtained by substituting eg. (22) into eg.

(23):

971 9a e w

2 k r

t

2

) x=a

(24)

The above expression is exactly egual to the energy stored in the spring

located at the crack tip. This spring energy is dissipated with crack extension. Accordingly, K is expressed as

18
HI (k

ew2 + kr^

2

' (25)

4.3

Stress Intensity Factor K from the FEM Model

K is obtained in this study using a conventional FEM model with triangular

elements. Crack propagation is simulated as follows: the reaction force of

the node that corresponds with the crack tip is decreased gradually with crack

extension and vanishes when the crack reaches the next node (in Fig. 15).

Malluck proved [6] that the numerical result was independent of the details of

releasing the spring force. Energy is eliminated gradually from the system by

this technique because the nodal force does work. The dissipated energy D is

where v q is the crack opening displacement of the node closest behind the

crack tip when the crack reaches the next node. K is given as

where Aa is space step, which should be as small as possible. Furthermore,

the difference among element sizes should be minimized because differences can

cause unexpected stress waves.

4.4 Elastic-Plastic Crack-Dri vi ng Force J From the MS Model

The MS model, in which plasticity is incorporated, was employed in Section

2.4 to analyze elastic-plastic fracture behavior after the specimen load

exceeded the limit load. J is divided into elastic (J

g

) and plastic parts

(J ) as follows:

P

(26)

(27)

( 28 )

19
Jp is associated with COD (6 ) through the Dugdale Model

J

P

( 29 )

When it is assumed that only plastic load-point displacement can contribute to

6^ and that the DCB specimen hinges around the neutral point of the stress

along the uncracked ligament, 6

p becomes

5 = {1 + (-2W + 2b)/[(2W - b) 2 + b

2 ]^2

} u . P P (30)

Elastic load-point displacement u g is given as

J g is obtained by using K:

J = K

2 /E . e

Because K is proportional to u g

,

J oc u

2

. e e

Consequently the resulting J-integral, denoted J-^ is

J-L = (K2 /E) + a

f

6

p

.

(31)

(32)

20
Note that eq. (31) often gives unreasonably large values, because the load at

the specimen exceeds the limit load. Assume that the load applied to the elastic spring of the specimen is spurious; that is, the viscous load is spurious

and only the limit load contributes to plastic fracture. Then, J

g must be

J

e = KNSY

/E

Therefore, another J is given as

J

2 = (KNSY

/E) + a

f

6

p

* (33)

It is very difficult to obtain true J using this simple model, and it is questionable whether or 3

^ is valid.

Figure 16 shows the change of J with time for the three viscosity coefficients mentioned above. A large difference is seen between and 3

^ after

NSY. It seems that these values approach each other with time. This means

that the system becomes stable gradually. To study this question, it will be

necessary to collect detailed experimental data for a comparison with numerical

results.

5. Comparison of the Models

In this section, dependencies of load, load-point displacement and stress

intensity factor K for the three models considered are compared. Load and

load-point displacement are compared because they are considered valuable indicators of the performance of any crack arrest model , and they can be compared

with experimental results. If a model cannot produce reasonable behavior for

calculated load and load-point displacement, its ability to calculate fracture

mechanics parameters correctly must be seriously questioned. Stress inters it.

21
factors K from the three models are presented as further evidence of the

ability of the MS model to calculate fracture mechanics parameters during

dynamic crack propagation. The static comparison, listed in Table 1, gives

examples of the degree of agreement to be expected between different models.

Note that as b becomes equal to or less than a, the agreement deteriorates.

The values for Kanninen eq. (6) apply for the static MS model because this

equation was incorporated into the static MS model.

5.1 Relation Between Load and Time

The results from each model are compared before crack arrest under fixedgrip conditions and under compliant loading. Crack initiation, velocity, and

arrest are imposed. The comparison is confined within the elastic limit.

Time increment At and space step Ax are carefully chosen to avoid numerical

divergence as follows: At = 0.5 ps and Ax - 5 mm for the Kanninen model; At =

0.5 ps for the FEM model. The mesh is shown in Fig. 17. The Kanninen and FEM

models are explicitly solved employing the finite difference method and Newmark's

p method (p = k)

,

respectively. The change of the load with time is shown in

Fig. 18 where the crack ran for 135 mm at two speeds, and was arrested under

fixed-grip conditions. The load calculated by the MS model drops smoothly

because this model includes only the lowest normal mode. On the other hand,

the other models present complicated oscillations because higher order modes

are superposed and each element of the specimen oscillates with 1 or 2 degrees

of freedom. In the Kanninen and FEM models, the load begins to drop at about

160 ps after crack initiation, while in the MS model the load drops immediately

after crack initiation and 160 ps is the amount of time for the radiated stress

wave from the initial crack tip to propagate to the loading point. The trend

of load change in the MS model, however, is regarded as agreeing well with the

22
more elaborate models. It is concluded that the lowest normal mode is dominant

in this specimen, and is modeled accurately by the MS model.

Next, Fig. 19 shows the change of load that is applied in the mass-spring

system. Also shown is the Kanninen model that has twice the mass and external

spring stiffness because half of the specimen is treated in this model. Although the Kanninen model expresses rather large oscillations after crack arrest, its trend agrees well with that of the MS model. Thus the validity of

the MS model was supported by comparing it with the Kanninen model for the

load change.

5.2 Relation Between Load-Point Displacement and Time

Figure 20 shows the change of total load-point displacement with time in

the mass-spring system, where the crack ran for 135 mm and was arrested there.

Both the MS and the Kanninen models express a smooth increase without complicated oscillations as in load change. Excellent agreement is achieved in the

change of total load-point displacement. This suggests that an accurate prediction of the displacement may be possible for comparison with experimental

results.

5.3 Relation Between K and Time

The comparison of the fracture mechanics parameter is now made for three

different models. The change of K is shown in Fig. 21, under the same fixedgrip conditions as in Sec. 5.1. Excellent agreement is observed between the

Kanninen and the FEM models throughout the time of interest. Although there

are small differences between the MS model and the other two models just after

crack initiation, the trend of the MS model agrees well with the others. Figure

22 shows the change of K with time in the mass-spring system, as in Sec. 5.2.

No significant differences are found between the MS and the Kanninen models

23
even after crack arrest. The conclusion is that MS model is as useful for

analysis of dynamic elastic fracture behavior as are the other two models.

5.4 Conclusion for Section 5

The fracture behavior of the specimen was found to be expressed satisfactorily with the use of the MS model if the specimen was linear elastic. That

is, the fracture parameter K, which was extracted from the MS model, was consistent with that of the other models. This suggests that such a simple model

is valid from the viewpoint of fracture mechanics as expressing only the

lowest normal mode and that the fracture parameter can be estimated from this

model within engineering allowance.

6. Comparison of the MS Model With Experimental Data

6.1 Relation Between Load, Load-Point Di spl acement, and Time

In this section, the MS model is applied to analyze the crack

arrest-reinitiation test conducted at NBS [5]. The material tested is HY80,

which shows approximately el astic-perfectly-pl astic mechanical behavior with a

yield stress of 551 MPa. Young's modulus and Poisson's ratio are assumed to

be 200 GPa and 0.3, respectively. The specimen configuration and load system

are schematically shown in Fig. 3a. The load was applied to the specimen

through the external load spring. The recorded data were the load, by load

cell; loadpoint displacement, by clip-on gauge; and crack velocity, by

tripwires. The fracture surface showed a brittle failure mode before the

crack arrest and ductile failure accompanied by tearing after crack arrest.

This MS model is applied to two test specimens, where all data were completely obtained: HY80-4 and HY80-6. Momentary crack arrest was deduced from

crack velocity or strain near the crack tip, both recorded in these two tests.

24
Table 2 shows experimental conditions and results. The mass and stiffness of

the external spring are given, but this model requires two viscous coefficients

These are chosen by trial and error so that the numerical results of the MS

model fit the experimental records. The limit load is given from eq. (15):

P

1 i mi t = 181.7 kN.

The relation between load and time is shown in Fig. 23 for HY80-6. The experimental load record shows large oscillations at high frequency. It is believed

that the load calculated by the MS model with two different viscosity coefficients indicates the average variation of experimental load. The trend, that

is, the recovery of the load after crack arrest, seems to agree well with the

experimental one, although the load calculated from the MS model drops more

quickly just after crack initiation than does the experimental load, because

of stress wave propagation.

Figure 24 shows the change of load-point displacement with time for HY80-6

The experimental displacement tends to approach the numerical result, using

C = 87.6 kN*s/m and C = 63.1 kN-s/m. Dotted lines indicate the displacement

ex s r

when the specimen is assumed to be linear elastic. This is correct before

about 4 ms, but deviates greatly from experimental displacements afterwards.

This deviation indicates that elastic analysis is unsatisfactory for the actual test specimen.

Figures 25 and 26 show the change of the load and the load-point displacement with time, respectively, for HY80-4 . The displacement from the MS model

is slightly smaller overall than the experimental displacement because the

actual viscous coefficient of the external spring is smaller for HY80-4 than

for HY80-6.

25
It is important to know the fracture parameter during crack propagation.

The change of K with time is shown in Fig. 27 by the MS and the Kanninen models

for HY80-4 and HY80-6. Both results seem to be generally satisfactory. The

stress intensities at crack arrest are

K

a = 51.4 MPaVm for HY80-4 ,

K = 62.2 MPa> for HY80-6.

These stress intensities must be less than crack arrest toughness because crack

arrest certainly occurred at 190 mm.

6.2 Relation Between J and Time

There is no method available to evaluate J directly in dynamic elasticplastic fracture experiments, although J can be obtained using load vs. load

line displacement curves in the static case.

Here we describe a method for evaluating J based on the load point displacement obtained by experiment. If the load point displacement A does not

exceed A^y> then

J = K

2 /E

where K is calculated using eq. (4). A^y is the displacement evaluated at

limit load P-, . ... and is obtained usinq eq. (2): limit’ a ^

A

nsy

= 3.4 mm.

26
When A is larger than A

NSy , J is divided into the elastic component and the

plastic component J . They are expressed as in Sec. 4.3. «Nsy

1S easily

obtained by eq. (4):

« Nsy = 260 MPaVa.

Consequently, experimental J is obtained with the use of eq. (30) as

J = (K 2 sy/E) + o

f {1 + (-2W + 2b)/[(2W - b) 2 + b*f2 } (A - A[s|$y ) . (34)

Figs. 28 and 29 show the relation between J and time for HY80-6 and HY80-4,

respectively. The solid line (J-^) and dotted line t^) begin to deviate after

NSY. The experimental J is close to for HY80-4 and is between and

for HY80-6. While A corresponds with u of the MS model, A - A^y should be

equal to u . It seems, however, that the increase of u is slower than that

P P of A - A^y The relative merits of these two sets of assumptions about J are

difficult to assess because of different definitions of plastic displacement

and because of different methods to obtain J. It is likely, however, that J

calculated by either method does not deviate excessively from the actual J.

7. Summary and Conclusion

7.1 Validity of the MS Model for General Features of Crack Arrest Behavior

First, the MS model compared well with the Kanninen and the FEM models

within elastic limits under several conditions. Comparisons of calculated

values of the fracture parameter K showed the validity of the analysis of

elastic crack propagation and arrest using the MS model. Next, the introduction of viscosity made it possible to treat plasticity effects. Especially,

experimental load and load-point displacement were simulated successfully by

27
the MS model using viscosity and limit load. The methods were found attractive

for assessing J by simple assumptions, although the input parameters are not

known well enough to rigorously characterize dynamic elastic-plastic fracture.

It is satisfactory that the MS model could extract the essence of the complicated features.

7. 2 Better Model

A very simple model was proposed based on the crack arrest-rei ni ti ati on

test specimen developed at NBS. This model is a useful tool to analyze both

elastic and elastic-plastic fracture dynamically. The other models, such as

the Kanninen or the elastic FEM model, are not appropriate for the analysis of

dynamic elastic-plastic fracture. Elastic-plastic dynamic FEM models require

substantial computational effort. A simple model should be established to

extract the characteristics of fracture. Flowever, it is noted that an excessively simplified model often results in the neglect of microscopic features

of the material. Comparison is always necessary with experimental results and

with other models.

Strictly speaking, the above simple model is valid only before reinitiation, because limit load, viscosity in the specimen and the stiffness of the

specimen must vary after the crack reinitiates. It will be necessary to measure the reinitiation time and the crack velocity after reinitiation and to

change the above parameters for a more complete estimate.

A similar equation of motion could be introduced for different specimens

or different loading systems by modifying the MS model, although the stiffness

of the cracked specimen must be known. The establishment of effective mass or

stiffness can make it possible to estimate the dynamic fracture behavior for

an actual structure such as a ship section.

28
Care must be taken with the introduction of plasticity in the model.

Viscosity and limit load concepts facilitated modeling for plasticity in the

MS model, because NSY occurred after crack arrest. A similar model could be

formulated for specimens with a large plastic zone before NSY. For example, a

spring constant that varies with displacement after arrest, as in Fig. 30,

could be used.

7.3 Method of Comparison with Experimental Data

It is important to choose carefully which numerical parameters should be

compared with experiment. Dynamic measurements are often difficult to analyze.

The load tends to show a different value at each location because of stress

wave propagation. In addition, the superposed high frequency of the specimen

makes it hard to properly estimate the fracture load.

On the other hand, the displacement does not oscillate as much as the

load or strain, although it is also hard to measure. Small spurious fluctuations should be neglected because they are not essential and because the simple model often fails to reproduce them. It is believed that comparison with

displacement is most reliable.

8. Acknowledgments

Financial support from the Office of Naval Research (for the first author)

and the Naval Sea Systems Command, 05R25, (for the co-authors), is gratefully

acknowledged. Technical assistance from D. P. Vigliotti and J. D. McColskey

is deeply appreciated.

29
9. References

[1] Feely, F. J. , Jr. , Northup, M. S. , Kleppe, S. R. , and Gensamer, M.

,

"Studies on the Brittle Failure of Tankage Steel Plates," Welding J. , 34,

1955, pp. 596-S-607-S.

[2] Kihara, H. , Kanzawa, T. , Ikeda, K. , Okabe, H. , Nakajima, T. , and

Yajima, H. , "Study on Welded Type Crack Arrester," IIW Doc. No. X-618-17,

1971.

[31 Freund, L. B. , "Dynamic Crack Propagation," The Mechanics of Fracture,

ed. F. Erdogan, ASME AMD, Vol. 19, ASME, 1975, pp. 105-134.

[4] Hahn, G. T. , Hoagland, R. G. , Rosenfield, A. R. and Senjoha, R. , "Rapid

Crack Propagation in a High Strength Steel," Metal. Trans., 5, 1974, pp.

475-482.

[5] King, R. B. , Teramoto, T. and Read, D. T. , "Material Selection Criteria

for Crack Arrester Strakes in Naval Vessels Second Interim Progress

Report," NBSIR 84-3012, National Bureau of Standards, 1984.

[6] Malluck, J. F. and King, W. W. , "Fast Fracture Simulated by Conventional

Finite Elements: A Comparison of Two Energy-Release Rate Algorithms,"

ASTM STP 711 , 1980, pp. 38-53.

[ 7 ] Shmuely, M. , "Effect of Poisson's Ratio on Crack Propagation and Arrest

in the Double-Cantilever-Beam Specimen," ASTM STP 627 , 1977, pp. 135-141.

[8] Kanninen, M. F. , "An Estimate of the Limiting Speed of a Propagating

Ductile Crack," J. Mech. Phys. Solids , 16, 1968, pp. 215-228.

[ 9 ] Crosley, P. B. and Ripling, E. J. , "Characteristics of a Run-Arrest

Segment of Crack Extension," ASTM STP 627 , 1977, pp. 203-227.

[10]

Kanazawa, T. , Machida, S. and Teramoto, T. , "Prel iminary Approach to

Experimental and Numerical Study on Fast Crack Propagation and Crack

Arrest," ASTM STP 627 , 1977, pp. 39-58.

[ill Kanazawa, T. , Machida, S. , Teramoto, T. , and Yoshinari, H. , "Study on Fast Fracture and Crack Arrest," Exp. Mech. , 21, 1981, pp. 78-88.

[12] Kanninen, M. F. , "An Augmented Double-Cantilever-Beam Model for Studying

Crack Propagation and Arrest," Int. J. Fracture , 9(1), 1977, pp. 83-92.

[13] Kanninen, M. F. , Popelar, C. and Gehlen, P. C. , "Dynamic Analysis of

Crack Propagation and Arrest in the Doubl e-canti 1 ever-Beam Specimen,"

ASTM STP 627 , 1977, p. 18.

[14l Keegstra, P. N. R. , "A Transient Finite Element Crack Propagation Model

for Nuclear Pressure Vessel Steels," J. Inst. Nuc. Eng. , 17(4), 1976, pp.

89-96.

30
[ 15 ] Mall, S. , Kobayashi, A. S. , and Urabe, Y. , "Dynamic Photoelastic and

Dynamic Finite Element Analysis of Polycarbonate Dynamic Tear Test

Specimens," ASTM STP 677 , 1979, pp. 498-510.

1 16l Nishioka, T. , Stonesifer, R. B. , and Atluri, S. N. , "An Evaluation of

Several Moving Singularity Finite Elements Models for Fast Fracture

Analysis," Eng. Fract. Mech. , 15(1), 1981, pp. 205-218.

[17] Kanazawa, T. , Kobayashi, A. S. , Machida, S. , and Urabe, Y. , "Fracture

Dynamic Analysis of Crack Arrest Test Specimens," Proc. 1st Inst. Conf.

of Numerical Methods in Fracture Mechanics , 1978, pp. 709-720.

31
Table 1. Comparison of Models

Stress intensity (MPa 7m) Compliance (m/N)

*

a(mm)

*

b

(mm)

FEM Kanni nen

eq. (7)

Kanni nen

eq. (4)

FEM Kanni nen

eq. (6)

Kanni nen

eq. (2)

50 255 140.6 143.0 144.4 1.797X10' 9 1.707xl0“ 9 1.603xl0' 9

100 205 200.8 183.2 210.7 3.845xl0‘ 9 3.558xl0‘ 9 3.727xl0' 9

150 155 293.3 229.1 342.0 8. 031xl0~ 9 6.501xl0' 9 8.693xl0" 9

200 105 513.4 273.5 727.7 1.892xl0' 8 1.083xl0“ 8 2.630xl0" 8

P = 0.431 MN.

K is calculated with the conversion J = K

2 /E, where the J-integral is

obtained using the isoparametric FEM model.

32
Table 2. Testing Conditions and Experimental Results

Specimen Stiffness of External Initial load Initial stress Mean crack

spri ng mass P.

l n

i ntensity vel oci ty

(MN/m) (kg) (MN) K

in (MPaVm) a (m/s) mean v '

HY80-4 13.6 120 0.431 147 750

HY80-6 9.1 136 0.372 134 365

33
Figure 1

6

i

T

Specimen configuration and Eul er-Bernoul 1 i beam model with a Winkler foundation.

34
Figure 2. Timoshenko beam model with a generalized foundation.

35
Figure 3

///////, ///

Load Cell

'A-+- External Mass (M ex)

Loading Spring

7777T7TH7

(a) Crack arrest reinitiation specimen and schematic loading

system;

36
//////////

Specimen

Part

External

Loading

Spring

Part

Figure 3. (b) MS model of the crack arrest reinitiation specimen.

37
P/Pi

Time (ms)

Figure 4. Change of load with time under fixed-grip conditions in the MS

model

.

38
P/Pi

Time (ms)

Figure 5. Changes of load v/ith tine under conditions of compliant loading for

different amounts of mass.

39
Mass-spring model (elastic)

Crack speed = 714 m/s

Time (ms)

Figure 6. Changes of displacement with time under conditions of compliant

loading for different amounts of mass.

40
P/Pi

Time (ms)

Figure 7. Changes of load with time due to spring stiffness, under conditions

of compliant loading.

41
Time (ms)

Figure 8. Changes of displacement with time due to spring stiffness, under

conditions of compliant loading.

42
Figure 9. MS model with viscosity in the loading spring.

43
Time (ms)

Figure 10. Changes of the load with time under conditions of compliant

loading with a viscous spring.

44
20

Time (ms)

Figure 11. Changes of displacement with time under conditions of compliant

loading with a viscous spring.

45
///////////

Purnit ^3

^

C« Plastic

Part

Elastic

Part

Figure 12. MS model with a viscous spring and yielding.

46
Figure 13. Changes of load with time under conditions of compliant loading

with a viscous spring and yielding.

47
Time (ms)

Figure 14. Changes of total and plastic displacements with time under

conditions of compliant loading with a viscous spring and

yielding.

48
Figure 15. Crack tip neighborhood in the finite element model.

49
Figure 16. Change of J with time under conditions of compliant loading with

a viscous spring and yielding, using eqs. (32) and (33).

50
161 Elements

106 Nodes

20 mm

Figure 17. Finite element breakdown for the crack arrest-reinitiation test

specimen

.

51
P/Pi

Figure 18. Comparison of load change with time between three different

models under fixed-grip conditions.

52
P/Pi

Time (ms)

Figure 19. Comparison of load change with time between the MS and the

Kanninen models under conditions of compliant loading.

53
7

«>

0 0.5 1.0 1.5 2.0

Time (ms)

Figure 20. Comparison of displacement change with time between the MS and

the Kanninen models under conditions of compliant loading.

54
K/Ki

Figure 21. Comparison of change of K with time between three different mode's

under fixed-grip conditions.

55
Figure 22. Comparison of change of K with time between the MS and the

Kanninen models under conditions of compliant loading.

56
Figure 23. Comparison of numerical and experimental loads as functions of

time using the MS model for HY80-6.

57
Figure 24. Compa

funct

ison of numerical

ons of time using

and experimental

the MS model for

displacements as HY80-6.

58
Figure 25. Comparison of numerical and experimental loads as functions of

time using the MS model for HY80-4.

59
u

(mm)

Figure 26. Compa

funct

ison of numerical

ons of time using

and experimental

the MS model for

displacements as HY80-4.

60
K/Ki

0 0.1 0.2 0.3 0.4

Time (ms)

Figure 27. Comparison of changes of K with time in the MS and Kanninen models

for HY80-4 and HY80-6.

61
J

(MN/m)

Figure 28. Comparison of changes of J with time obtained by the MS model,

eqs. (32) and (33), and by experiment, eq. (34), for HY80-6.

62
J

(MN/mm)

Figure 29. Comparison of changes of J with time obtained by the MS model,

eqs. (32) and (33), and by experiment, eg. (34), for HV80-4.

63
p

Figure 30. Schematic load-displacement characteristic for a specimen with a large plastic zone before net section yielding, and straight lines

corresponding to a series of spring constants.

64
II. CRACK ARREST EXPERIMENTS AND ANALYSIS WITH RISING K FIELD

T. Teramoto, D.T. Read, and R.B. King

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

Because of recent interest in crack arrest in situations where the

applied stress intensity K increases with crack length (rising K field),

experiments to allow observation of dynamic crack propagation and arrest

in a rising K field have been designed, performed, and analyzed. Both

crack arrest and complete dynamic fracture with no arrest were observed

by changing the specimen temperature and the applied load. Therefore,

this experiment was sensitive to the crack arrest toughness of the specimen material. It was concluded that using a mechanical spring in the load

train effectively simulates possible structural compliance, and can cause reinitiation of a temporarily arrested crack. Finite element analysis and

a simple mass-spring model were used to extract fracture mechanics parameters, including stress intensity for elastic behavior and J-integral for

elastic-plastic behavior.

Key words : dynamic crack arrest; dynamic crack propagation; elastic- plastic

;

fracture; J-integral; mass spring model; rising K; side groove

1 . INTRODUCTION

High toughness crack arresters are commonly inserted at critical

locations in ships so that unstable (brittle) fracture, if initiated, can be

stopped. When a crack runs in brittle plate and meets a tough crack

arrester, arrest occurs because of the step in toughness. In conventional

crack-arrest testing, arrest occurs in a toughness gradient produced by a thermal gradient. For sufficiently long crack jumps, after crack arrest the

65
recovered load is applied to the short uncracked ligament, and significant

extension of the plastic zone is anticipated. Linear elastic fracture

mechanics is invalid for characterizing crack tip behavior after the initial

arrest because of plastic strain effects.

A new specimen has allowed observation of the crack run-arrest event in

a rising K (stress intensity) field. The experiments are analyzed on the

basis of a dynamic, linear elastic, finite element method (FEM) model and a simple mass-spring (MS) model. The fracture parameters relevant during and

after crack arrest are calculated using the MS model. This model takes into

account inertia, plasticity, and strain-rate-induced elevation of flow

strength

.

2. SPECIMEN

In order to achieve a rising K field, the specimen configuration of

Fig. 1 was chosen. This specimen is divided into two parts: the crack

initiation part, under jack loading, and the crack propagation and arrest

part, under the main load. Variable dimensions of the specimen are listed in

Table 1. A preliminary static, linear elastic FEM analysis was performed

using conventional techniques. Figure 2 shows the stress distributions along

the crack propagation line. The combined stress is positive everywhere,

although the stress distribution from the jack load alone is partly negative

ahead of the starting crack tip. The maximum stress appears at the initial

crack tip. Stress curves are given for various values of the average stress,

o 0

, defined as main load divided by gross cross-sectional area, and the

static intensity, Kg, at the initial crack tip.

Figure 3 shows the variation with crack length of stress intensity under

static conditions, Kg, produced by the main load and the jack load. The

differences between fixed-grip and fixed-load conditions grow with crack

length. The total K

g distribution includes contributions from the main load

and from the jack load. K

g values are given for several values of average

stress o Q

. The stress intensity at crack initiation is controlled almost

entirely by the jack load.

Electron beam (EB) melting was used in an attempt to produce a brittle

crack propagation path in the EH36 carbon-manganese steel specimen material

66
1 ]. It was expected, from a previous study of a nickel steel, that a step

in toughness would be produced between virgin material and EB-melted

material. However, this proved not to be the case; therefore, the specimens

were grooved all along one side to provide a crack path, and arrest occurred

in a thermal gradient.

3. ANALYTICAL METHODS

3.1 Finite Element Model

A conventional finite element model (FEM), consisting of constant strain

triangular elements, was employed to analyze dynamic crack propagation and

arrest behavior. Linear elastic analysis was used. The equations of motion

of each element were synthesized into an equation of motion of the whole

specimen as follows:

[M] {0} + [<] {u} = tf} , (1)

where [M] and [<] are mass and stiffness matrices, respectively, and { u }

,

{ u

}

and { f

}

are the acceleration, displacement and force vectors of model

points. Eq. (1) was solved by Newmark's 8 method using 8 = 1/4.

All four specimens were analyzed using a dynamic elastic FEM, since the

crack behavior was assumed to be elastic at least during crack propagation.

Figures 4 (a) and (b) show the finite element breakdown of the specimens. The

crack tip position vs time relation was employed as an input condition in

the FEM computations. The external mass and the loading spring stiffness

were assumed to be 120 kg and 13-6 MN/m, respectively.

Crack extension was numerically simulated as follows. A crack was propagated through the FEM idealization at a preselected velocity. The

reaction force, f, of the node corresponding to the crack tip (Fig. 5) was decreased gradually and vanished when the crack reached the next node. The

stress intensity was obtained from the work done by the reaction force. The

dissipated energy, D, is

D = ft dv (2)

67
where dv is the increment of crack opening displacement of the node behind

the crack tip. Then the dynamic stress intensity, K

D> was obtained from [2]

K

d

= [(E/BAa) / fdv] 1/2

, (3)

where Aa is the crack growth step. E and B are Young's modulus and specimen

thickness, respectively. After crack arrest, K

D

is evaluated by Keegstra's

technique [ 3 ] as follows:

K

d

= 6.414 f(2Aa/7r) 1/2 /B . (4)

3.2 Mass-Spring Model [4]

Although the dynamic, linear elastic FEM model is the best method for

treatment of rapid crack propagation and initial arrest, which occur under

predominantly linear elastic conditions, the model cannot treat reinitiation.

Dynamic, elastic-plastic FEM models are cumbersome and expensive.

Accordingly, a simplified mass spring (MS) model has been developed [ 3 ];

it can account for both linear elastic and plastic strain effects.

Since elastic behavior is dominant during initial rapid crack propagation, the simple elastic mass spring model (Fig. 6) was also used to analyze

this phase of the crack propagation. The equation of motion is

^ex^ + ^ K ex

+ K s^ u = K ex u A » ^ 5 ^

where u and u are acceleration and displacement, respectively, of the load

point in the specimen. MPX and K ex are external mass and spring stiffness,

u

A is initially applied displacement. k s means the stiffness at the load

point of the specimen, varying with crack length. k

s is given for the

specimen with side groove (side groove depth is 7.62 mm) as

k

s

= [686.76 - 46.797 (a/W) - 544.39 (a/W) 2

+ 174.72 ( a/W) 3 - 8408.4 (a/W) 4 + 10151.0 (a/W) 5

] (MN/m) (6)

68
where a is crack length and W is the specimen width (=260 mm). Stress

intensity (static) is given as

K

s

= P [69.647 + 158.99 (a/W) + 324.21 (a/W) 2

+ 446.22 (a/W)3 + 624.45 (a/W) 21 + 897.28 (a/W) 5

] (MPa/m) , (7)

where P is the magnitude of the applied load in meganewtons. Expressions (6)

and (7) for compliance and stress intensity factor for the specimen with

side groove, shown in Fig. 4b, were obtained from the results of twodimensional linear-elastic finite element analyses of a series of model

specimens with different crack lengths. The calculated compliance and stress

intensity factors were approximated by best fit polynomials in relative crack

size a/W. Equations (6) and (7) refer to this specimen geometry only.

Next, plastic deformation was introduced into this model to analyze the

crack behavior after crack arrest. As shown in Fig. 7, a limit load element

and a viscous element were included. Their displacement is the plastic part

of the total displacement. The added elements accounted not only for

friction, but also for strain-rate-induced elevation of yield stress [1],

The equations of motion used were:

where u

p

is the plastic part of the displacement, C

s and Cex are the

viscosity coefficients, M

g is specimen mass, and P is the applied load.

Limit load (for DCB specimens only) is evaluated as follows:

( 8 )

and

(9)

P limit = {E(W-b ) 2 + b

2

] 1/2 - (W-b} o f

*B

where b is uncracked ligament length and o f is the flow stress.

69
After P exceeds P

i i m j_ t » an elasto-plastic fracture mechanics parameter

is needed to characterize the crack. The modified J-integral is defined as

J kPe * 2p limit U

p /B ' b (10)

where K

g is the stress intensity evaluated at load P using eq. (7). P is given by the elastic part of the displacement as

P = k

s

• (u - Up) (11)

4. MATERIAL AND TECHNIQUES

The material used was classed as ABS-EH 36 , a ship steel. Its chemical

composition and mechanical properties are shown in Table 2. Its yield and

flow strengths are assumed to be 370 MPa and 520 MPa, respectively, at the

temperatures used for crack-arrest experiments. Its Charpy impact energy has

a step-like brittle/ductile transition at a temperature of about -70°C.

Stress-relief heat treatment and a deep side groove were used for specimens

EH36-5.-6, and -

7 , because the initiated crack was apt to turn and deviate

from the EB-melted zone as is in specimens without a side groove. The side

groove was cut on one side only, to allow installation of crack propagation

gauges on the other side of the specimen. The side-groove depth was 7.6 mm, leaving a net ligament of 5.1 mm. Specimens were cooled with liquid nitrogen

and dry ice so that the desired temperature was achieved at the EB-melted

zone tip. The initial crack tip was supercooled so that the crack was easily

initiated in a completely brittle manner, without noticeable macroscopic

plastic deformation.

After the main load was applied to the specimen through the loading

spring by the testing machine (

1 -MN capacity) and was maintained at the

specified level, a sub-load was applied to the crack-initiation part of the

specimen by two jacks. These two loads were separate; the main load was expected to be maintained during and after crack propagation by the highly

compliant loading spring. The jacks were so stiff that fixed grip conditions

were assumed at the jack loading point. However, this did not affect the

crack tip after crack arrest; the specimen was deformed because of large-scale

70
yielding of the net section, so that the jack load was relieved.

The crack velocity was measured from the change in the electrical

resistance of crack detector trip wires, which were connected in parallel,

using a resistance network, and were placed normal to the crack propagation

line. Strain gauges were placed near the EB-melted zone tip, to monitor

not only the crack tip location with time but the extension of plastic

zone after crack arrest. The main load was monitored by the test machine

load cell. The load line displacement over a guage length of 260 mm was measured by a clip-on gauge. Transient data from all instruments were

stored in digital oscilloscopes. The temperature distribution along the

crack propagation line was monitored with several thermocouples.

5. RESULTS

Table 3 shows the experimental results. Dynamic fracture toughness,

Kq, at first arrest is obtained by using dynamic linear elastic FEM analysis. For EH36-2, the crack was arrested temporarily inside the EB-melted

line. However, it was observed that the crack became slanted; that is, it

deviated slightly from EB-melted line into base plate, on one side of the

specimen although it remained in the EB-melted zone. Therefore, it is

believed that the slightly tougher base plate contributed to the crack

arrest. For EH36-5, the crack was continuous on the grooved side of the

specimen, but was discontinuous on the other side. The crack was arrested

within the EB-melted zone. Arrest may have been due to the relatively low

applied load and to toughness increase because of the temperature gradient.

For EH 36 -

6 , tested at a very low temperature, the crack severed the specimen

completely. The fracture surface indicated brittle fracture along the crack

path. For EH 36 -

7 , the crack was successfully arrested at the EB-melted zone

tip because of the difference in toughness; and then the crack was reinitiated after the load, which had dropped during crack propagation, began

to be recovered. The relation between the crack velocity and crack tip

position are discussed in Sec. 6.

Figures 8 (a), (b), (c) and (d) show the change of crack velocity with

crack length and temperature distribution for EH36-2, - 5 , -6 and - 7 . It was found that the crack velocity decreased quickly just before crack arrest.

71
These plots show that arrests occurred for temperatures between approximately

-40°C and ~60°C. No arrest occurred for a specimen cooled to -80°C along

the whole crack path. Average stress before crack propagation in the

ligament remaining after side-grooving ranged from 142 to 217 MPa in

specimens in which arrest occurred. The stress in specimen EH 36-6 for which

arrest did not occur, was 117 MPa.

Figures 9 (a), (b), (c) and (d) show the change of the load with time

for EH36-2, -

5 , ~6 & and -

7 . The experimental load data for EH36-2 were

lost, so only calculated values are shown. The load values calculated by FEM

generally agree with experimental values before crack arrest, except for

specimen EH 36 -

5 . The load calculated by the MS model was found useful in

spite of the simplicity of the model.

Figures 10 (a), (b), (c) and (d) show the change of the y-direction

strain with time. The difference between numerical and experimental

strain increases with crack extension, except for EH 36 -

6 . This difference

is regarded as intrinsic, even though the FEM mesh is somewhat coarse. It is

interpreted to mean that a different model is necessary to incorporate

plasticity effects. For EH 36 -

6 , however, elastic analysis is adequate to

analyze the running crack, because it did not arrest.

Figures 11 (a), (b), (c) and (d) show the change of the applied dynamic

stress intensity and crack velocity with time. Static, dynamic FEM (eq. 4),

and mass-spring model (eqs. 7 and 10) values of K

D are shown. It seems that

the dynamic stress intensity calculated by FEM is affected strongly by crack

velocity [5] for this material. Although K

D is not strictly correct after

crack arrest, because of plasticity effects, it seems reasonable because of

the small deviation of from the static stress intensity factor K

s

. The MS

model was found inappropriate to evaluate K

D during initial crack propagation

because the effect of the jack was not considered.

Recall that the temperature varies across each specimen; therefore the

toughness varies along the crack path. This means that a correlation between

material resistance to dynamic crack propagation and crack velocity cannot be

obtained from the data of Figs. 8 and 11.

72
Plastic zone extension occurred as load increased under nearly static

conditions after arrest, as shown in Figs. 12 (a) and (b). Quite a high

load was necessary to produce net section yield for EH36-2. In the other

specimens, the plastic zone concentrated only in the narrow net section

left by the side groove and net section yielding occurred at lower loads.

Numerical results were obtained from the MS model incorporating

plasticity. Figure 13 compares calculated and experimental loads as functions of time for EH 36 -

5 . The experimental load oscillates at a high

frequency, believed to be the natural frequency the specimen’s mouth-opening

mode of resonance oscillation. The trend, however, seems to indicate that

the MS model accurately treats the lowest normal mode of the specimen-spring

system as a whole. Two values of the viscosity parameter C

g were used.

These had been found by trial and error to bracket the range of values that

most closely simulated the actual specimen behavior [M]. Figure 1 H plots the

calculated total and plastic displacement against time. It was concluded

that the plastic displacement for EH 36-5 was so small that the elastic deformation was dominant. Figure 15 shows the relation between J and time. It was found that computed J is of the order of %SY 2/E > corresponding with the

onset of net section yielding.

Figure 16 compares results of the MS model and the experiment as a function of time for EH 36 -

7 , for three values of the friction parameter.

The experimental load results, in general, tend to approach the MS model

prediction when the middle friction value is used, although the experimental

load oscillates. In this case, the applied load is higher than limit load

after crack arrest. Plastic deformation must be accounted for in the MS

model for EH 36 - 7 . The relationship between the total and plastic displacements and time is shown in Fig. 17. The experimental load line displacement

seems to grow while oscillating. This high amplitude oscillation is believed

to result from the natural frequency of the clip gauge. Figure 18 shows the

relation between J and time. J is much higher than %S y

2 /E an(j that an elastic plastic fracture mechanics parameter is necessary after crack arrest.

The experimental J-integral, J, is introduced as follows [6]:

J = KNSY 2/e + 2 p limit * (A-AN5Y )/B*b

73
where A is load line displacement and ANSY is load line displacement at

the limit load from Eq. (6).

Experimental J in general seems to agree with the model result using

the intermediate value for the friction parameter.

6. DISCUSSION

For EH36-7, brittle crack propagation, arrest, and reinitiation

occurred. That is, the worst case in actual ship failure was simulated in

a laboratory experiment. The following discussion of crack arrest and

reinitiation is based on the behavior of the specimen in that simulation.

From Fig. 7 (d) crack arrest seems to occur at the EB-melted zone tip.

The crack velocity decreases quickly just before the EB-melted zone tip. The

variation of strains, from strain gauges placed 12 and 30 mm ahead of the

EB-melted zone tip, shows that after the measured strain increased quickly

because of the approaching crack, the strains remained almost constant

(Fig. 18). This means that the crack was arrested at least temporarily. If

the crack had passed the location of the strain gauges, the strains would

have dropped immediately, as shown in Fig. 10 (c) for EH36-6. Since the

strain exceeded the yield strain, a large plastic zone must have been formed,

indicating a high crack driving force.

It is hard to judge exactly when the arrested crack was reinitiated from

the oscilloscope photographs of the strain and load traces (Figs. 18 and 19).

The load dropped to 44 percent of the initially applied load because of crack

propagation, that is, rapid change of the specimen compliance, and then was recovered by the loading spring. Since the recovered load was as large as the initial load, it was large enough to deform the uncracked ligament after

crack arrest and produce crack reinitiation. The crack reinitiation probably

should be considered to occur at the time of maximum strain (=2 ms) or load

(=3 ms), because compliance is changed by the reinitiation. The apparent

time difference between these two maxima is due to the delay produced by

elastic wave propagation. For example, when the elastic wave velocity is

assumed to be 500 m/s (approximately the bar wave velocity) and the distance

from the arrested crack to the load cell is about 50 cm, the elastic wave

from the crack tip takes 1 ms to reach the load cell.

74
The J-integral during reinitiation is shown in Fig. 20. Although

complete ductile tearing (or slant fracture) was observed after the crack

reinitiation, we believe that the repropagation velocity was very low and

that the reinitiated crack temporarily stopped and then propagated again.

The strains appear to remain nearly constant until 8 ms in Fig. 1 8 , as the

load does in Fig. 19. The specimen seems to have deformed gradually after

reinitiation as there is little oscillation of the load or strain. The

repropagation velocity is estimated from Fig. 18 to be 18 m/s, using the time

difference (si ms) between the two strain gauge strains. This value is close

to that observed in unstable ductile fracture experiments.

7. CONCLUSIONS

From this study of crack arrest, the following conclusions about the

present specimen design and experimental results were drawn:

(i) Introduction of a deep side groove resulted in a straighter-running

crack, as desired; the cracks were apt to turn in specimens without a side groove.

(ii) The residual stress, which may have caused the crack to turn, was relieved by a conventional heat treatment. However, the toughness

difference between EH36 base plate and EB-melted zone seemed small,

both before and after heat treatment.

(iii) The applied load, having decreased during crack propagation, was recovered by the loading spring and caused reinitiation after crack

arrest

.

(iv) A crack can be arrested even in a rising K field if the temperature,

and thus the material toughness, is high enough.

(v) The MS model, with appropriately chosen viscosity parameter, successfully simulates the behavior in time of the lowest frequency component

of the load and displacement, and therefore appears to be useful for

extracting fracture mechanics parameters from test data.

(vi) It is possible that the dynamic arrest toughness changes with the depth

of the side groove, although it is known that side grooves have littl*

effect on the plane strain fracture toughness [7]. The presence

single side groove may affect the measured crack arrest toughness in

75
this specimen, because only two-dimensional analyses have been done.

However, it is known that side grooves have little effect in the static

plane strain fracture toughness [7].

8. ACKNOWLEDGMENTS

Financial support from the Office of Naval Research (for the first

author) and the Naval Sea Systems Command, 05R25, (for the coauthors), is

gratefully acknowledged. Technical assistance from D. P. Vigliotti and

J. D. McColskey is deeply appreciated.

9 . REFERENCES

[1] R. B. King, T. Teramoto and D. T. Read, "Material Selection Criteria

for Crack Arrester Strakes in Naval Vessels, Second Interim Progress

Report," NBSIR 84-3012.

[ 2\ J. F. Malluck and W. W. King, "Fast Fracture Simulated by Conventional

Finite Elements: A Comparison of Two Energy-Release Rate Algorithms."

ASTM STP 711 , American Society for Testing and Materials, Philadelphia,

1980, pp. 38-53.

[3] P. N. R. Keegstra, "A Transient Finite Element Crack Propagation Model

for Nuclear Pressure Vessel Steels", J. Inst. Nuc. Eng.

Vol . 17 - No. 14, pp. 89-96, 1976.

[4] T. Teramoto, D. T. Read and R. B. King, "Fracture Mechanics

Parameters in Crack Arrest Specimens," companion paper in this NBSIR.

[5] R. G. Hoagland, A. R. Rosenfield, P. C. Gehler and G. T. Hahn, "A Crack

Arrest Measuring Procedure for Kj m , Kjq, and Kj

a Properties," ASTM STP

627 , American Society for Testing and Materials, Philadelphia, pp. 1 7 7~

202, 1977.

76
[61 J. R. Rice, P. C. Paris, and J. G. Merkle, "Some Further Results of

J-Integral Analysis and Estimates," Progress in Flaw Growth and Fracture

Toughness Testing , ASTM STP 536 , American Society for Testing and

Materials, Philadelphia, 1973, pp. 231 -245.

[ 7 ] V. Dantam and G. T. Hahn, "Definition of Crack Arrest Performance of

Tough Alloys," presented at the U.S. -Japan Seminar on "Fracture

Tolerance Evaluation," Honolulu, HI, Dec. 7 _ 11, 1981.

77
Table 1 . Variable dimensions of rising K crack arrest specimen.

w = 275 mm

w = 260 mm

C

i = 25 mm

d = 25 mm

c

i = 12 mm

c

2

= 18 mm

d = 1 5 mm

for EH36-2 (1 strain gauge)

for EH36-5, -6, -7 (2 strain gauges)

for EH36-2

for EH36-5, -6, -7

78
Table 2. Chemical composition in wieght per cent and mechanical

properties for EH36.

Chemical composition., weight percent

c Mn P S Si Cu Ni Cr Mo Fe

0.12 1.39 0.015 0.006 0.380 0.05 0.03 0.05 0.007 bal.

Mechanical Properties at Room Temperature

Yield Tensile Elongation

Stress Stress (per cent)

(MPa) (MPa)

365 1182 29

79
Table

3.

Experimental

results

of

crack

arrest

tests

of

rising K

specimens

of

EH-36

carbon-manganese

steel

bOC '

CTJ ^

00

03

X

a ±-> O bC Cl

CD CC

aO I I O CM I I fCD

03

4-> C

0) cc

CD

w

CD

O o CD

X 2 Z >H

03o

(0 ^

<D E

L- 03

03 D_2: o ^

CD

O

00

4-) O CD C \ <—i E

CD ^> E c <c

03

CD Z o

on

=T o

vO

CM

CO

vO

O

CM

CM

4-3

-*-3 CD

03 <D ^

J- C_) • 1- o

Q. 03

N—

E

(D

D < H CD £-

comi O

inj

4c

4c0

Ov1

CM=ri

CD

<D

x

L.

t- 00 E

03 c e

<D

CD

CM

OO

co i on o

05

L. ^ O X E

4-3 E

»H 00

—-

03

C

-h CD

C

4-> «—I *rH —' «< c

HHc

CDEO

CDa.

C0

4-3 c

CDE • 4-> >>

03 03 0) TD

t4-> 9-

4~> t- 03 O

CD Cu_ X

00 DC E C —< \

03 rH Z ^ o bO s: o J* E O vO E o m in on CD CM m i

CM CO in CM

O T— «

r—

O o vO 03 * • X o o o o C— L.3 H ^ X c

4-3 0) Cl

Cl

X •H CD

<D -o 4-3 CD *o CD >— • *o C o CO 9 CO L. rH

*— co CO CO CD 23 (D Q-,

-=T •

CM C\J > o .,-1 • • 4-> O X 4-3

O o O o CD

O • CD

CD L. 9~ cd

£_ bo to

J- i- U3 CD c

0) <D

O CD TD <*-. 4-3 03 t4-> -r~> 03 E Cl o CD Ct, CD + c o CD rH

4c 4t

X

OD 03 bO CM in vO Q 4-3 o rH C C 1 i I 4c •r-i O 03 X •«—t

vO vO vO

Q > CD •o on m m 1 4-3 03 X X X X 4c X O

LO CO Ci3 LtJ 4- 4c 4c U2 -J

80
Main Load

Figure 1

.

Specimen configuration and instrumentation.
j Ks =155 MPa^/m (141 ksi vTn)

( Reduced Jack Load

I Ks =75 MPa\/rri (71 ksi x/Tn)

Figure 2. Stress distribution along crack propagation line from main load

and/or jack load.

82

Stress,

ksi
Figure 3- Variation with crack length of stress intensity factor from main

load or jack load.

83
For EH 36-2

Figure 4a. Finite element breakdown for EH36-2.

84
For EH36-5, 6 & 7

Figure *Jb. Finite element breakdown for EH36-5, “

6 , &

-

7 .

85
Figure 5. Crack opening near crack tip in finite element model.

86
//////////

\ Specimen

Part

External

Loading

Spring

Part

Figure 6. Mass spring model for elastic behavior.

87
///////////

pLimit

E

3 V

1 C Plastic

Part

Elastic

Part

Figure 7. Mass spring model for elasto-plastic behavior.

88
brittle surface ductile surface with shear lip

Figure 8a. Variation of crack velocity with crack length and temperature

distribution for specimen EH36-2.

89

Temperature,
2000

EH36-5

Velocity

i2 1500 —

E

o

^

1000 —

o

05

O

500 —

/A

Temperature

0

-20

40

-60

0 :L£

Initial crack

Crack arrest (?)

O

o

-80

0 40 80 120 | 140

-100

Position, mm EB- Weld tip

Figure 8b. Variation of crack velocity with crack length and temperature

distribution for specimen EH36-5.

90

Temperature,
1000 0

EH36-6

800 —

cn

E

>; 600

o

o

2

o

5

o

Crack velocity Oo

-50

400

200 — Initial crack tip

o-.

0

-I—

Temperature5 ^ „ -O

0 40 80 120

Position, mm

160 180

-100

Figure 8 c. Variation of crack velocity with crack length and temperature

distribution for specimen EH36-6.

91

Temperature,
1000 0

EH36-7

co

o

O

2

o

co 6

800

600

400

200

0

Crack velocity

.-o*

/\

/ Temperature

Initial crack tip /

/ Crack arrest

“il

0 40 80 120 160 200

Position, mm

-20

-40

-60

-80

-100

Figure 8 d. Variation of crack velocity with crack length and temperature

distribution for specimen EH36-7.

92

Temperature,

°C
Load,

MN

0.5

Figure

0.4

0.3

0.2

by FEM

'Arrest

0.1 J_ L 0 0.1 0.2 0.3 0 .- Time, ms

0.5 0.6

9a. Calculated load as a function of time for specimen EH36-2.

93
Main

Load,

MN

Figure 9 b. Comparison of measured and calculated load as a function of time

for specimen EH36-5.

94
Load,

MN

Figure 9c. Comparison of measured and calculated load as a function of time

for specimen EH36-6.

95
Figure 9 d. Comparison of measured and calculated load as a function of time

for specimen EH36-7.

96
Figure 10a. Comparison of experimental and calculated strain,

specimen EH36-2.

97
Figure 10b. Comparison of experimental and calculated strain,

specimen EH36-5.

98
Time, ms

Figure 10c. Comparison of experimental and calculated strain,

specimen EH36-6.

99
Figure lOd. Comparison of experimental and calculated strain,

specimen EH36-7.

100
Dynamic

Applied

Stress

Intensity,

K

D

,

MPa\/m 140

Figure 11a. Variation of stress intensity with time, specimen EH36-2.

101

Kd,

ksi

v/m
2000

Figure 11b. Variation of stress intensity with time, specimen EH36-5.

102

Crack

velocity,

m/s
Figure 11 c. Variation of stress intensity with time, specimen EH36-6.

103
Applied

Stress

Intensity,

K,

MPa'

V

/m 1000

Figure lid. Variation of stress intensity with time, specimen EH36-7

.

104

Velocity,

m/s
Figure 12a. Calculated size of plastic zone from main load for EH36-2.

105
Figure 12b. Calculated size of plastic zone from main load for EH36-5,-6,

and -

7 .

106
Load,

MN

Time, ms

Figure 13. Variation of load with time for EH36-5, up to 5 ms after crack

initiation.

107
Displacement,

mm

0 1 2 3 4 5

Time, ms

Figure 14 . Variation of total and plastic displacement with time for EH36-5.

108
J-integral,

MN/m

Time, ms

Figure 15. Variation of J integral with time for EH36

109
Figure 16. Variation of load with time for EH 36 -

7 , up to 5 ms after crack

initiation.

110
Figure 17 . Variation of total and plastic displacement with time for EH36-7.

Ill
20ms—— —

H

Strain vs. Time

£-1 £-2

12mm

Figure 18. Oscilloscope trace of the strain gauge signal for specimen

EH36-7.

112

15mm
40 ms

Load vs. Time

Figure 19. Oscilloscope trace of the load signal for speciment EH36-7.

113
114
NBS-1 »4A I RE v. 2 -aCl U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1 . PUBLICATION OR

REPORT NO. NBSIR 85-3034

2. Performing Organ. Report No 3. Publ i cation Date

June 1986

4 . TITLE AND SUBTITLE

FRACTURE MECHANICS CHARACTERIZATION OF CRACK ARREST AND REINITIATION IN TWO

UNCONVENTIONAL SPECIMENS

5. AUTHOR(S)

T. Teramoto, D.T. Read, and R.B. King

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Office of Naval Research David Taylor Naval Ship Research

800 North Quincy Street and and Development Center

Arlington, Virginia 22217 Annapolis, Maryland 21402

10. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bl iography or literature survey, mention it here)

The fracture mechanics parameters K (the stress intensity factor) and J (the value of the J-integral) are estimated for crack arrest (K) and reinitiation (J)

in two unconventional specimen geometries: a short double-cantilever-beam or "compact" specimen; and a single-edge-cracked tensile specimen that provides a rising K field as the crack advances. These specimens both have relatively large

in-plane dimensions (t, 0.5 m) and were loaded through a large mechanical spring

that simulated structural compliance. Both of these specimens were shown to be

capable of complete dynamic fracture with no arrest as well as of crack arrest,

depending on the temperature. This indicates sensitivity to the crack arrest

toughness of the specimen material.

A simple elastic-plastic-fracture-mechanics-based model of crack propagation,

arrest, reinitiation, and propagation is described and used to estimate K at

crack arrest and J during reinitiation. This model requires much less computing

resources than dynamic, elastic-plastic finite element calculations and allows

esimates of applied J-integral, load, and crack mouth opening displacement during

initial rapid crack propagation, reinitiation, and repropagation . A comparison of

this new model to other available models and to experimental results indicates that

it can successfully reproduce the essential features of the behavior of specimens

containing propagating cracks.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semi color, \ dynamic crack arrest; dynamic crack propagation; elastic-plastic; fracture;

J-integral; mass spring model; rising K; side groove; stress-intensity factor

13. AVAILABILITY

iXX Uni imited

1 | For Official Distribution. Do Not Release to NTIS ~ Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

120

15. Price

U.S. GOVERNMENT PRINTING OFFICE:1986 —672-143/45242 USCOmm-DC «04)*P|