NASA/TM-2010-216692

Relating Cohesive Zone Models to Linear

Elastic Fracture Mechanics

John T Wang

Langley Research Center, Hampton, Virginia

May 2010
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NASA/TM-2010-216692

Relating Cohesive Zone Model to Linear Elastic

Fracture Mechanics

John T Wang

Langley Resear-ch Center, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

May 2010
Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802
RELATING COHESIVE ZONE MODELS TO LINEAR ELASTIC

FRACTURE MECHANICS

John T. Wang

NASA Langley Research Center

Hampton, VA

Abstract

The conditions required for a cohesive zone model (CZjW) to predict a failure

load of a cracked structure similar to that obtained by a linear elastic elastic f ,actur•e

mechanics (LEFM) analysis are investigated in this paper. This study clarifies

why many different phenomenological cohesive laws can produce similar fracture

predictions. Analytical results for five cohesive zone models are obtained, using

five different cohesive laws that have the same cohesive work rate (CWR-area

under the traction-separation curve) but different maximum tractions. The effect

of the maximum traction on the predicted cohesive zone length and the remote

applied load at fracture is presented. Similar to the small scale yielding condition

f

or an LEF.YI analysis to be valid, the cohesive zone length also needs to be much

smaller than the crack length. This is a necessary condition for a CZM to obtain

a fracture prediction equivalent to an LEFM result.

Introduction

The origin of the cohesive zone models (CZMs) can be traced back to the "Dugdale-Barrenblatt

model" [1, 2]. The cohesive zone is considered to be a fracture processing zone ahead of the

crack tip. For most cohesive laws, the traction-separation curves used to model the material

within the cohesive zone are phenomenological, and hence, are not directly related to the

physical process in the damage zone which typically is difficult to determine experimentally.

Regardless, the CZM approach has been widely accepted as a computationally convenient

fracture analysis tool. If the CZM approach is used in a finite element analysis, the crack

initiation, growth, and direction of growth can be automatically determined. Many different

cohesive laws with variances in maximum traction, maximum separation, and shape have been

proposed; such as the linear softening cohesive law by Camacho and Ortiz [3], the exponential

cohesive law by Needleman [4,5] and Xu and Needleman [6], the trapezoidal cohesive lase by

Tvergaardand and Huchinson [7], and the polynomial cohesive lase by Tvergaard[8].

Researchers found these cohesive laws generally produce results that correlated well with

experimental data such as the failure load and crack growth for cracked structures. For linear

elastic materials, these predictions are comparable to linear elastic fracture mechanics (LEFM)

results; however, the conditions under which the cohesive zone modeling approach and the

1
fracture mechanics approach are equivalent have not been systematically investigated. Here, the

equivalence of the two modeling approaches means that they can predict the same failure load

for a cracked structure.

The objective of this paper is to investigate under what conditions the cohesive zone model and

the LEFM approach are equivalent. Linear softening cohesive laws are used in this study. The

relationship between the CWR (area below the traction-separation curve of the cohesive law) and

the J-integral value [9] is discussed. Integral equations, for using CZMs to analyze the fracture of

an infinite plate with a crack under a remote tensile load, are presented [10]. An iterative

numerical procedure is implemented as a MATLAB ® M-file [11] for solving these equations to

obtain the length of the cohesive zone ahead of the original crack, the opening displacements

along the cohesive zone, and the remote applied stress at the moment of crack growth initiation.

The effect of varying the maximum traction of a cohesive law on the predicted cohesive zone

length and remote applied stress is investigated. The remote applied stress is used in an LEFM

formula to determine the energy release rate. By comparing the energy release rate with the

CWR, it is possible to determine whether the cohesive zone modeling approach is equivalent to

the LEFM approach, i.e. whether it is able to produce a similar failure load for a cracked

structure to that produced by the LEFM approach.

CWR and J-integral value

A CZM is used to study the fracture of an infinite plate with a crack length of 20 subjected to a

remote applied tensile stress 6^ as shown in Figure 1 a. Note that all the equations and analysis

results presented in this paper are related to Mode I fracture of linear elastic material. In the

cohesive zones (narrow bands) ahead of the crack tips, the prospective fracture surfaces are

assumed to be restrained by a cohesive stress that Dugdale took to be the yield stress of the

material [2]. The traction (cohesive stress) 6 and the separation, 6 = uZ — uZ , of the upper and

lower prospective crack surfaces, where u2 and uz are the opening displacements in the ydirection for the upper and lower surfaces, respectively, are shown in Figure lb.

A CZM model needs a cohesive law to relate the traction to the separation at the same location

along the x: -axis for describing the constitutive behavior of the cohesive zone. The cohesive

law used in this paper is a linear softening cohesive law as shown in Figure lc,

6 = cr, , (,5, — 9) l b, 0 —< b << b,

=0 cS > (S,

where cr, is the maximum traction and b, is the maximum separation. The cohesive work rate

(CWR) (D , or work of separation per unit area, is the area under the linear softening curve

(1)

2
(D ' =2UA (2)

At the moment of the crack growth initiation, the cohesive zone opening displacement at the

crack tip reaches maximum separation, 5(o) = (5^, , and the cohesive zone is fully developed as

shown in Figure 2 [7]. During the subsequent crack growth, the length of the cohesive zone and

the opening displacement along the cohesive zone length are unchanged. The J-integral [9],

taken along the boundary of a filly developed cohesive zone, the contour F, can be expressed

as,

J = f (Wn — T ` )ds

r ' ax

aZ,

(3)

where W is the strain energy per unit volume, n, is directional cosine between the positive side

(outward) of the normal N and the x -axis, T,. is the itt' component of the traction perpendicular

to F in the outward direction, ut is the i t'' component of the displacement, ds is an arc element of

F.

For contour F shown in Figure 3, n, = T, = 0 and the J -integral value for a fully developed

cohesive zone becomes [12]

^-

J,_ 

—f 

6 

au2 — z 

dx + J, — —^ Cr d x + J,_t

0 ax ax 0 ax

,5(4)

= f Td (5 + J^t ='D' + Jet

0

where J, is the critical J value immediately before the initiation of crack growth and JC , is a

J -integral value of an infinitesimal contour, not shown in the figure, around the cohesive zone

tip. For an isotropic material, J, is related to the stress intensity factor K, , if it exists, at the

cohesive zone tip

JI _t = KEt 

z

(5)

Since the J -integral is a path independent integral, the J value obtained from any contour

integral around the cohesive zone such as F, in Figure 3 must also equal to J,. If the cohesive

zone model is equivalent to an LEFM model, the stress field outside of the cohesive zone must

3
be near the same as the K-dominant stress field [13] in the LEFM model. In other words, the path

independent J, can be related to the fracture toughness K, as

K^ = EJ, (6)

and the critical energy release rate defined in LEFM as

G, = J, . (7)

For cohesive laws that assume the CWR to be the same as the critical energy release rate,

ID, — G - J (8)

the stress intensity factor at the cohesive zone tip, K, in Eq. (5), must vanish because J, must

be zero to satisfy Eq. (8). Note that G^ and K^ are material properties and are experimentally

determined, so 0, is also considered to be a material property.

Because J, = 0 in Eq. (4), then J, obtained by the contour around the cohesive zone is the CWR

cF, corresponding to the area under the traction-separation curve. Cohesive zone models with

the same CWR but different shapes will produce the same J,. If J^ is the only parameter to

determine fracture, it is expected that the same failure prediction can be obtained with different

shapes of the cohesive law. This may be the reason why CZMs with different shapes can

generate similar failure predictions.

Analytical Solutions for Cohesive Zone Models

CZM analyses of a cracked infinite plate subjected to a remote tensile stress, as shown in Figure

la, are presented in this section. Five linear softening cohesive laws shown in Figure 4, with

different maximum tractions but all having the same CWR, were used in this study. The effects

of maximum traction on the predicted cohesive zone length and remote applied stress were

investigated. The predicted remote applied stresses were used in an LEFM formula to compute

the energy release rate at failure. Good agreement between the energy release rate at failure and

the CWR determines the equivalence of a CZM and a LEFM analysis.

The analytical procedure presented here follows the paper authored by Jin and Sun [ 10]. The

cohesive zone is treated as an extended part of the crack with the stress intensity factor being

removed at the tip of the cohesive zone. Hence,

2 J` =0 (9)

TCC ° 1 -2

/C2

4
where c is the x -coordinate of the cohesive zone tip and 6(^) is the traction at location ^ as

shown in Figure lb.

The total cohesive zone opening displacement b at location x is [14]

b(x)-4 E Fx4

^zE a Ec 

f"G(x,^)u(^)d^ (10)

where E is the Young's modulus, and G(x, 4) is an influence fiunction given by

G x =1n 1—a' /c' + 1—^ 2 /c2 11

1—x2 / c2 — 1—^2/c2

For the linear softening model shown in Figure lb, Eq. (10) can be expressed as

4ccr

rl

x2b(

1)) d

4 ` 6^(8^ —^ a

6(x) = E c2 — ;zE f G(x, )

8

^ (12)

a c

and Eq. (9) can be expressed as

cS(

2(7, d d Y

^^ (—^ c d^+ f ^ ) (13)

a

1—

IC2 a 1

— 2 I C2

Substituting 6., into Eq. (12), an integral equation is obtained for determining the cohesive zone

length, p = c — a, and cohesive zone opening displacements (CZODs) 8(x)

;TE8` (x)( 1 ` 2 x22 ` 9* ( )

— f G(x, 4),5 (^)d5 +— 1— f d^

2c6C 2

c a c ca l—` l c2

(14)

2^

_ ? 1— x2 f

d — 1 f G(x, 4)d4

C C a J1 — e/C2 Ca

where b' (x) = b(x) l bc .

Equation 14 can be solved with an iterative method shown in Figure 5, in which an initial

cohesive zone length p = c —a is given first to determine the integral interval, [a, c] , for the

5
integral terms in Eq. (14) and then the CZODs along c — a are computed. If the predicted

opening displacement at the crack tip t5(a) is not equal to the maximum separation g , an

updated cohesive length is given until the c5(a)equals,5,. The final value of c—a is the

cohesive zone length p. The remote applied stress a ., can be obtained with Eq. (13), by using

the final set of CZODs. The remote applied stress is used to compute the energy release rate

using the following LEFM formulae,

z

G(6,) E

(15)

If the ratio of the computed energy release rate G(cr j to the CWR, ^D , is close to unity, then

the CZM is considered to be equivalent to an LEFM analysis.

Results

Analytical results for five cohesive law models with different maximum tractions as shown in

Figure 4 are presented in this section. All the results are computed at the initiation of crack

growth when the cohesive zone is fiilly developed. The effect of maximum tractions on cohesive

zone length, remote applied stress, the equivalence of CZM approach and LEFM analysis is

presented. The effect of maximum traction on cohesive zone length can be found in Figure 6,

which shows that larger maximum tractions result in shorter cohesive zone lengths, p. Two

curves represent results obtained with two different crack lengths, one has a / /,h =1 and the other

one has a / h,, =10 . Note that /,,, is a characteristic length, and in this paper it is defined based on

Model B in Figure 4,

E(D`

11h = 6Z ( )16

For the rest of the paper, the term crack length means the half-crack length a in Figure 1 a. The

two curves in Figure 6 are converged for maximum tractions greater than 2o 7,, indicating that

the cohesive zone length may not depend on the crack length for a cohesive law with a large

maximum traction. However, the cohesive zone lengths depend on crack length for models with

a maximum traction less than a,. The cohesive zone length for the short crack a / /,h =1 is

longer than that of the long crack a / ^, =10 .

Cohesive zone lengths are predicted for all five cohesive law models in Figure 4. The cohesive

zone length reaches a constant value for all cohesive law models as the crack length increases.

For illustration purposes, only results of Models A, B, and D are plotted in Figure 7. Models A,

B, and D have maximum tractions of 1/2(7,, cT , and 2o7 , respectively. For the largest

6
maximum traction 2c7, model, the cohesive zone length reaches a constant value at a crack

length around a = 2/,,,, For the second largest maximum traction cr, model, the cohesive zone

length reaches a constant value at a crack length around a = 51,, . The cohesive zone length

almost reaches a constant value at a =161,,, for the model with the smallest maximum traction of

1/2 cr, .

The cohesive zone opening displacements along the cohesive zone length for models with

different maximum tractions are plotted in Figure 8. There are three cohesive zone opening

displacement curves associated with three cohesive laws with maximum tractions of I/ 26C , 6, ,

and2c^ , respectively. These cohesive zone opening displacement curves along the cohesive zone

length are not linear (the shape of the cohesive zone is a cusp). Figure 8 also shows that the

smaller the maximum traction results in the longer the cohesive zone length. These opening

displacements are used in Eq. (13) to compute the remote applied stresses.

The computed remote stresses as a function of the maximum traction of all five CZM models are

plotted in Figure 9. There are two curves shown on Figure 9 representing two different crack

lengths. The bottom curve shows that the remote applied stresses for the longer crack length of

a =10 hh reaches a constant value for maximum traction greater than one c^ , but the top curve

for the shorter crack length of a =1 h,, shows that the remote applied stresses has not reached a

constant valve for the maximum traction at 4 cr, .

The computed remote applied stresses are then used in Eq. (15) to compute LEFM energy release

rates G(a-,) . The ratio of G(6^) to (D, is defined as the normalized LEFM energy release rate

(NLERR). If the NLERR is near unity, then the cohesive zone model is equivalent to an LEFM

analysis, and both models would predict the same failure load. The NLERR as a function of

maximum traction is plotted in Figure 10. The figure shows that for a long crack length,

a= I 01,h, the NLERR approaches unity for maximum tractions greater than 2 6, while for a

short crack length, a =1,,, , the NLERR reaches unity at a much slower rate (a cohesive law with

a maximum traction greater than 4 c^ needs be used).

The NLERR requires high tractions to reach unity for the shorter crack length a =1,, because the

cohesive zone length relative to the crack length is not small (<< 1). This large cohesive zone

length, similar to a large scale yielding zone, may significantly alter the original K dominant

stress field around the crack tip [13]. Hence, the results from the CZM model cannot be

equivalent to an LEFM analysis. To confirm this hypothesis, the NLERR as a function of the

ratio of cohesive zone length to crack length is plotted in Figure 11 for both the crack lengths.

This plot reveals that the two curves shown in Figure 10 can be collapsed to become a single

7
curve. This indicates that the ratio of cohesive zone length to the crack length may be an

important parameter for fracture predictions using CZM. The ratio of cohesive zone length to

crack length, p l a , must be close to zero for the CZM to be equivalent to an LEFM analysis.

Physically, this means that regions ahead of the crack tip that exibit mechanisms other than

brittle fracture (plasticity, bridging fibers, etc) must be very small relative to the crack length for

LEFM to be valid, and hence, for the CZM approach to be equivalent to LEFM.

Summary

A MATLAB® M-file was implemented for numerically solving the CZM integral equations with

an iterative procedure. Linear softening cohesive laws with a constant CWR but different

maximum tractions were used for modeling the traction-separation behavior in the cohesive

zone. Results show that a reduced cohesive zone length is predicted for a cohesive model with a

large maximum traction, and the ratio of cohesive zone length to the crack size needs be very

small for the prediction from the CZM to be equivalent to an LEFM analysis.

The conditions required for a cohesive zone model and an LEFM analysis to be equivalent were

identified from this study and other published literature. These conditions are as follows:

1. Stress intensity factor at a fully developed cohesive zone tip needs to vanish if the Jintegral value around the cohesive zone is set equal to the cohesive work rate.

2. If the J-integral value around a fully developed cohesive zone is the only parameter to

determine failure, then the maximum traction of a cohesive law can be changed while

keeping the CWR constant without affecting the failure load prediction.

3. The ratio of the cohesive zone length to the crack length needs to be small, so the

existence of the cohesive zone cannot significantly change the stress field near the crack

tip.

4. The maximum traction of a CZM needs to be high. High maximum tractions result in a

short cohesive zone length.

Acknowledgement

The author would like to acknowledge the helpful discussions with Professors C. T. Sun of

Purdue University and Z. H. Jin of The University of Maine, and the in-depth review by his

colleague Dr. T. K. O'Brien.

References

1. Barenblatt, G. I., "The mathematical theory of equilibrium cracks in brittle fracture,"

Advances in Applied Mechanics, Vol. 7. Academic Press, New York, 1962, pp. 55-129.

8
2. Dugdale, D. S., "Yielding of steel sheets containing slits," Journal of the Mechanics

and Physics of Solids, Vol. 8, 1960, pp. 100-104.

3. Camacho, G. T., Ortiz, M., "Computational modelling of impact damage in brittle

materials," International Journal of Solids and Structures, Vol. 33, 1996, pp. 2899-

2938.

4. Needleman, A., "A continuum model for void nucleation by inclusion debonding,"

ASME Journal of Applied Mechanics, Vol. 54, 1987, pp. 525-531.

5. Needleman, A., "An analysis of decohesion along an imperfect interface,"

International Journal of Fracture, Vol. 42, 1990, pp. 21-40.

6. Xu X. P., Needleman A., "Void nucleation by inclusions debonding in a crystal

matrix," Modelling and Simulation in Materials Science and Engineering, Vol. 1, 1993,

pp. 111-132.

7. Tvergaard V., Hutchinson J. W., "The relation between crack growth resistance and

fracture process parameters in elastic—plastic solids," Journal of the Mechanics and

Physics of Solids, Vol. 40, 1992, pp. 1377-1397.

8. Tvergaard V., "Effect of fibre debonding in a whisker-reinforced metal," Materials

Science and Engineering A, Vol. 125, 1990 pp. 203-213.

9. Rice, J. R., "A path independent integral and approximate analysis of strain

concentration by notches and cracks," ASME Journal of Applied Mechanics, Vol. 35,

1968, pp. 379-386.

10.Jin, Z. H., Sun, C. T., "Cohesive fracture model based on necking," International

Journal of Fracture, Vol. 134, 2005, pp. 91-108.

11. MATLAB ® (a high-performance technical computing environment), product of The

MathWorks Inc., http://www.mothu,-orks.com .

12. Kanninen, M. F. and Popelar, C. H., Advanced Fracture Mechanics, Oxford University

Press, New York, 1985.

13.Broek, D., Elementary Engineering Fracture Mechanics, 4th revised edition, Martinus

Nijhoff Publishers, Dordrecht, 1986.

14.Tada, H., Paris, P. C. and Irwin, G. R., The Stress Analysis of Cracks Handbook.

ASME Press, New York, 2000.

9
¢x 6

T T T

Y 6C

-C

-a a x

9C b

y y ^

a. Cohesive zone at crack tips

x

c. Linear softening cohesive law

Y

b. Cohesive traction cs and cohesive zone

opening displacement 6

Figure 1 Fracture analysis of a cracked infinite plate using a cohesive zone model.

i

Initial urloaced

crack

i

i

i

At initiation of

c -

-----

growth

i

i

d4 During

growth

a^

Figure 2 Cohesive zone fully developed at crack growth initiation and unchanged during growth

[7]

10
6

^s s

6

V L6^

s

.5, /N12

6

2,7c

,5,/2 5

6

1/2u,

s 1s,5,/4

6

00

6

U'

Figure 3 J-integral paths around the cohesive zone.

6

46,

Model A Model B Model C Model D Model E

Figure 4 Five cohesive laws with the same cohesive work rate.

11
07.

17^

Assign integral

Interval [a,c] for

Eq. (14)

( P = C — a

Phew = Pold + Op

(Set Op oc (J — c5i(a)) / c5 C C

False

eloo" if \

< 0.0001

cS

Tru e

P, 6e ' (T'

Figure 5 Solution procedure of the iterative method.

5

4

Normalized

Cohesive

zone

3

Length,

1,h

2

1

----- ------ ----------------- ------------------ I ----------------- --------------- -

10 1

----------------- ch ---- --------------- —

----------------- ------------------ I ----------------- ---------------

------------

----------------- ------------------ I ----------------- ---------------

0

0 1 2 3 4 5

Maximum Traction/6

c

Figure 6 Cohesive zone lengths as a function of maximum traction of different cohesive laws.

12
5

Normalized 4

Cohesive

Zone

Length,

P

2

1

cr^

Y

C X

a P

9

6

-

--------------- ----- ---------------- -------------------- -

4

2,5,

------------------ -------

--------------- --------------------

---------------------- --------------------------- - --------------- -

6-

CC

----------------- --------------------- ------- --------------------

LOV

-- --- ----------(5- - ^--2 --------

-

- 

-

- -- -

-- -----------Sc----- --------------------

0 5 10 15 20

Ctack Length, a/ J'

Figure 7 Cohesive zone lengths vs. crack length for models with different maximum tractions.

(T

2.5

cr^

S, V 2

0.5 ............... . . ............... _. . . ............. ..... ...... . .........

CC

(5

0

0 1 2 3 4 5

Normalized Cohesive Zone Length, pllc,?

Figure 8 Cohesive opening displacements along the cohesive zone length for models with different

maximum tractions ( a

13
07^

----------- ---------------- -----------------I ----------------- ----------------- ----------------

J.

----------------- ----- ---------- ------------------L ----------------- ----------------

---------- --------- ----- ------------------r ----

- - - - - - - - - - - - - - - - - - - - - - - - - - - - —

1 7

--------------- - ----------- ch

- - - - - - - - - - - - - - - - - - - - - - - - - - - —

-Ei— a = 10 1

------------- --- ----------------- ------------------I ----------------- ----------------

---------- ----- ----------------- -----------------

- --------------- ----------------

-------- ---------------------- ----------------- ------

U.L

0

1

0.9

0.8

Normalized 17

LEFM

Energy

Release 0.6

Rate,

0.5

0.4

0.3

1 2 3 4 5

Maximum Traction /6

(T.

0-6

0.5 

Normalized

Remote 0-4

Applied

Stress,

07. 07,

0.3

0.2

01

0

110

iloo

---------------- --

- ----------- ------------------I --------------

--------------- —

------------

-----------

---------- - --------------- —

a =

10 1

------ ------- ----------------- ----------------- ----------------- --------------- —

---------------- ----------------- ----------------- ----------------- --------------- —

1 2 3 4 5

Maximum Traction/,7

Figure 9 Remote applied stress for models with different maximum tractions.

6.

0-1

Figure 10 LEFM energy release rates for cohesive laws with different maximum tractions.

14
0-

a.

x

'cl a

a.

1

0.9

0.8

0.7

Normalized

LEFM

Energy 0.6

Release

Rate, 0.5

G(q.) / (D,

0.4

0.3

0.5a

---------- - ------------- ------------- ----------------- ------------

------------- ----------------- - ----------- -----------------

---------- -- - ----------------- ----------------- --------------

Gr

------------- ----------------- ------- ------------

---------- -- - ----------------- ------- ------------

------------- ----------------- ----------------- ----------------- ------------

--------------------------------------------------------------------------------

0.2 L

0 02 0.4 0.6 0.8 1

Nondimentional Cohesive Zone Length, p l a

Figure 11 LEFM energy release rates at the moment of growth initiation as a function of the ratio of

cohesive zone length to crack length.

15
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NUMBER(S)

NASA/TM-2010-216692

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 39

Availability: NASA CASI (443) 757-5802

13. SUPPLEMENTARY NOTES

14.ABSTRACT

The conditions required for a cohesive zone model (CZM) to predict a failure load of a cracked structure similar to that

obtained by a linear elastic fracture mechanics (LEFM) analysis are investigated in this paper. This study clarifies why many

different phenomenological cohesive laws can produce similar fracture predictions. Analytical results for five cohesive zone

models are obtained, using five different cohesive laws that have the same cohesive work rate (CWR-area under the

traction-separation curve) but different maximum tractions. The effect of the maximum traction on the predicted cohesive zone

length and the remote applied load at fracture is presented. Similar to the small scale yielding condition for an LEFM analysis

to be valid. the cohesive zone length also needs to be much smaller than the crack length. This is a necessary condition for a

CZM to obtain a fracture prediction equivalent to an LEFM result.

15. SUBJECT TERMS

Cohesive zone model, Linear elastic fracture mechanics; Cohesive work rate, Energy release rate

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON

ABSTRACT a. REPORT b. ABSTRACT c. THIS PAGE PAGES STI Help Desk (email: het ^sti.nasa.aov)

19b. TELEPHONE NUMBER (Include area code)

LJ LJ U UU 20 443) 7.57-5802

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