Abstract Spacial variations of the mechanical properties have a shielding or anti-shielding effect on the crack tip by inducing an additional crack driving force term, the material inhomogeneity term, C inh . This paper explores this effect by studying the influence of a sharp bimaterial interface on the effective crack driving force in a fracture mechanics specimen. Linear elastic or elastic -ideally plastic materials are assumed with a mismatch in the elastic modulus and/or yield stress at the interface. Following a numerical stress analysis, the material inhomogeneity term, C inh , is obtained by post-processing. This parametric study is especially focused on the effect of the distance between the crack tip and the interface

F D Fischer

G X Shan

J Predan

N K Simha

O Kolednik

CiteSeerX

ECF15 
CRACK-TIP SHIELDING AND ANTI-SHIELDING BY A 
BIMATERIAL INTERFACE 
  O. Kolednik 1, J. Predan 2, G.X. Shan 3, N.K. Simha 4, F.D. Fischer 5
1 Erich Schmid Institute of Materials Science  
Austrian Academy of Sciences, A-8700 Leoben, Austria 
kolednik@unileoben.ac.at 
 2 Faculty of Mechanical Engineering, University of Maribor  
SI-2000 Maribor, Slovenia 
jozef.predan@uni-mb.si 
3 VOEST Alpine Industrieanlagenbau GmbH&Co  
A-4031 Linz, Austria 
Guoxin.Shan@vai.at  
4 Department of Mechanical Engineering, University of Miami 
Coral Gables, FL 33124-0642, USA 
nsimha@miami.edu 
5 Institute of Mechanics, Montanuniversität Leoben and  
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences  
A-8700 Leoben, Austria 
mechanik@unileoben.ac.at 
 
Abstract 
Spacial variations of the mechanical properties have a shielding or anti-shielding effect on the 
crack tip by inducing an additional crack driving force term, the material inhomogeneity term, 
Cinh. This paper explores this effect by studying the influence of a sharp bimaterial interface 
on the effective crack driving force in a fracture mechanics specimen. Linear elastic or elastic 
– ideally plastic materials are assumed with a mismatch in the elastic modulus and/or yield 
stress at the interface. Following a numerical stress analysis, the material inhomogeneity 
term, Cinh, is obtained by post-processing. This parametric study is especially focused on the 
effect of the distance between the crack tip and the interface. 
 
Introduction 
In inhomogeneous materials, the effective crack driving force becomes different from the 
applied far-field crack driving force. This well known effect has been treated mainly by 
classical fracture mechanics papers which have restrictions to elastic materials and special 
geometries, or by numerical investigations which give only specific explanations that cannot 
be easily generalized (see [1] for a literature review). The inhomogeneity effect is important 
for understanding the fracture behavior of multiphase or composite materials, brazed or 
welded components and materials with special surface treatments, such as nitrided or case-
hardened steels and any coated material. Also, the effect provides a basis for the design of 
materials and structural components where variations in material properties are intentionally 
introduced to increase the fracture resistance. 
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By using the concept of material forces [2,3], Simha et al. [1] have shown that a material 
inhomogeneity can have a shielding or anti-shielding effect on a crack tip, since it induces an 
additional crack driving force term, called the material inhomogeneity term, Cinh. The 
effective, near-tip crack driving force, Jtip, is given by the sum of the nominally applied far-
field crack driving force, Jfar, and the material inhomogeneity term,  
                                          .         (1) inhfartip CJJ +=
Post-processing methods have been developed to evaluate the material inhomogeneity term 
both for a continuous variation of the material properties [1] and for a discrete jump at a sharp 
interface [4]. Results from selected examples show that Cinh is positive and Jtip becomes larger 
than Jfar if a crack is in the  stiffer and/or higher strength material. In contrast, if the crack is in 
the more compliant and/or lower strength material Cinh is negative, and Jtip becomes smaller 
than Jfar. 
In this paper, we concentrate on the distance between the crack tip and a sharp bimaterial 
interface, and thereby explore the influence of a sharp bimaterial interface on crack growth.   
 
The evaluation of the material inhomogeneity term 
Consider a two-dimensional body containing a crack and a sharp interface Σ. The materials 
on the left and right of the interface are homogeneous, but there is a jump in the material 
properties at the interface. Then the material inhomogeneity term is given by [4] 
                     .               (2) [ ][ ] [ ][ ]( ) dsenC jjikikinh ∫Σ −−= εσφ  
Here, φ is the stored elastic energy density, σij denotes the Cauchy stress, εij the linear strain, 
ei the unit vector in the direction of crack growth, and ni the unit normal to the interface 
Σ pointing from the left side to the right (Fig. 1). [[ ]] denotes the jump and < > the average at 
the interface. Simha et al. [4] show that the material inhomogeneity term (Eq. 2) can be 
deduced from the standard J-integral concept, see [5], by calculating the J-integral around the 
interface, Jint: 
                (3) Cinh −= intJ
We consider a compact tension specimen made of two homogeneous isotropic materials, 
separated by a sharp interface which is perpendicular to the crack plane and at a distance L in 
front of the crack tip. The specimen width is W = 50 mm, height from the crack plane to the 
upper surface is h = 30 mm, and the crack length a = 29 mm. We use a finite element 
program, ABAQUS (http//www.hks.com), to perform the stress analysis. Figure 1 shows the 
mesh of the specimen, consisting of two-dimensional 8-node elements. The minimum mesh 
dimension is 0.013 mm at the crack tip and 0.05 mm at the interface. The materials on either 
side of the interface are assumed to be perfectly bonded. Linear elastic and non-hardening 
elastic-plastic materials are considered. Elastic-plastic materials are modeled using the 
incremental plasticity model provided by ABAQUS. The computations are performed under 
plane strain conditions, using small strain formulations. The loading is controlled by 
prescribing the load-line displacement. The J-integrals, Jtip, Jfar, and Jint, are evaluated using 
the virtual crack extension method of ABAQUS; the corresponding contours are shown in 
Fig. 1. 
ECF15 
 
 
 
 
 
 
 
 
 
 
 
FIGURE 1. Finite-element mesh of the CT-specimen with a bimaterial interface.  
 
The  computations  are  performed  for  a  stationary  crack, but  the  distance  between  crack  
tip and interface is varied, -2.5 mm ≤ L ≤ 2.5 mm. Material 1 is to the left of the interface, 
while Material 2 is on the right. The following cases are examined: (1) linear elastic materials 
with modulus inhomogeneity at the interface, (2) linear elastic – ideally plastic materials with 
modulus inhomogeneity, but same yield stress, (3) linear elastic – ideally plastic materials 
with yield stress inhomogeneity, but same modulus. Both materials always have the same 
Poisson’s ratio: ν = 0.3 for the elastic regime and ν = 0.5 for the plastic regime. For elastic-
plastic materials, the stress analysis is performed using the incremental plasticity model 
provided by ABAQUS. For calculating Cinh by Eq. 2, φ is taken as the total strain energy 
density and εik as the components of the total strain. Thus, the material is treated in the post-
processing procedure as if it were non-linear elastic. 
 
Results 
Linear elastic materials – modulus inhomogeneity 
In Fig. 2, the material inhomogeneity term, Cinh, is plotted against the far-field crack driving 
force, Jfar. The upper curves belong to a stiff/compliant transition, for E1 = 210 GPa, E2 = 70 
GPa: Cinh is positive and enhances the effective crack driving force, Jtip (Eq. 1). The lower 
curves in Fig. 3 belong to the compliant/stiff transition (E1 = 70 GPa, E2 = 210 GPa): Cinh is 
negative and diminishes Jtip. Notice that for all cases, the material inhomogeneity depends 
linearly on  Jfar.  
An estimate for the slope, k = Cinh/ Jfar , has been presented in [6], 
 
ECF15 
 
                       . 
⎥
⎥2
ν ⎦
⎤
⎢
⎢
⎣
⎡
+
−⎟
⎠
⎞
⎜
⎝
⎛−+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+
++
−
+
−
−
=
2222
22
21
21 arctan)21(4ln)43(
)1(4
1
Lh
h
L
h
hLh
hLh
EE
EE
k νν
π 
              (4) 
Thus, the slope Cinh/Jfar depends primarily on the Dundurs parameter, (E1-E2)/(E1+E2) [7], 
and on the non-dimensional distance between the interface and crack tip (L/h). 
 
-50
-40
-30
-20
-10
0
10
20
30
40
50
0 10 20 30 40 50 60 70 80 90 100
Jfar [kJ/m
2]
C
in
h 
[k
J/
m
2 ]
L=2.50 mm
L=0.15 mm
L=-0.15 mm
L=-2.50 mm
E1=70 GPa
E2=210 GPa
E1 = 210 GPa
E2 =  70 GPa
Cinh = - Jfar
Jtip = 0
 
 
 
 
 
 
 
 
 
 
 
FIGURE 2. Modulus inhomogeneity for linear elastic materials. 
 
 
-1
-0,75
-0,5
-0,25
0
0,25
0,5
0,75
1
1,25
1,5
-3 -2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5 3
L [mm]
k 
= 
C
in
h/J
fa
r M=3
M=2
M=3/2
M=2/3
M=1/2
M=1/3
M=E1/E2
 
 
 
 
 
 
 
 
 
 
FIGURE 3. Effect of the distance between interface and crack tip on the ratio of the material 
inhomogeneity term to the far-field crack driving force for linear elastic materials. 
ECF15 
In Fig. 3, the slope k = Cinh/ Jfar, is shown as a function of the distance between interface 
and crack tip, L, and the ratio of the Young’s modulus, M = E1/E2. When a crack approaches 
the interface, the absolute size of the material inhomogeneity term, |Cinh|, increases sharply. 
For the stiff/compliant transition, k seems to approach infinity for L → 0; for the 
compliant/stiff transition, the data seem to approach Cinh = -Jfar, or Jtip = 0, i.e., the effective 
crack driving force cannot become smaller than zero. This explains the asymmetry between 
the stiff/compliant and the compliant/stiff transition which is not reflected by Eq. 4. When the 
crack has penetrated the interface, L < 0, the k vs. L curves decrease in an almost symmetrical 
manner.  
 
Elastic-ideally plastic – modulus inhomogeneity 
Next, results are presented for bimaterial specimens consisting of two linear elastic – 
ideally plastic materials (strain hardening coefficient, N = 0) with constant yield stress, σy = 
500 MPa but a misfit in the elastic modulus, E = 70 and 210 GPa. At small values of Jfar, the 
slope of the Cinh vs. Jfar curves is close but not identical to that of the corresponding linear 
elastic case. However, as Jfar increases the slope continues to decrease and eventually the 
material inhomogeneity term approaches a saturation value, , at high JinhĈ far (see [7] for 
details). In Fig. 4, Cinh is plotted against the distance L for different values of Jfar between 40 
and 240 kJ/m2. When a crack approaches the interface to a more compliant material, Cinh first 
increases (at constant Jfar) with decreasing L, reaches a maximum value at a distance between 
L = 0.6 mm (for Jfar = 40 kJ/m2) and L = 1.3 mm (for Jfar = 240 kJ/m2), and then decreases. 
The asymmetry between the stiff/compliant and the compliant/stiff transition appears weaker 
than for the purely elastic bimaterials. When the crack has  penetrated  the  interface,  the  
decrease  of  Cinh slows down. It should be also noted that at a loading of Jfar = 240 kJ/m2, the 
material inhomogeneity term has reached its saturation value, , for all values of  L.  Thus, inhĈ
 
 
 
 
 
 
 
 
 
 
 
 
-80
-60
-40
-20
0
20
40
60
80
100
-2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5
L [mm]
C
in
h 
[k
J/
m
2 ]
Jfar = 40
Jfar = 80
Jfar = 120
Jfar = 160
Jfar = 200
Jfar = 240
[kJ/m2]
Ε1 = 210 GPa
Ε2 = 70 GPa
Ε1 = 70 GPa
Ε2 = 210 GPa
σy = 500 MPa
    N = 0
FIGURE 4. Modulus inhomogeneity for elastic – ideally plastic materials. 
ECF15 
the maximum -values for this material combination and geometry are 78 kJ/minhĈ
2 for the 
stiff/compliant transition and 63 kJ/m2 for the compliant/stiff transition. The maximum value 
depends on the square of the constant yield stress value and difference in the elastic moduli 
(for a fixed specimen) and can be estimated using  
              . (5) ( ) hEmax
EE
E
C yinh
21
212ˆ −≈ σ
 
For a strain hardening material, the Cinh vs. Jfar curves do not saturate and the Cinh-values may 
increase appreciably above the maximum -values of the non-hardening material [7]. inhĈ
 
Elastic-ideally plastic – yield stress inhomogeneity 
The third case considered is that of linear elastic – ideally plastic bimaterials with constant 
elastic modulus, E = 210 GPa, but a mismatch in the yield stress, σy = 900 and 300 MPa. 
Figures 5 and 6 show the influence of the distance between crack tip and interface, L, on the 
material inhomogeneity term. For small loading, Cinh is negligible, since the crack tip plastic 
zones do not interact with the interface (see [8]). With increasing Jfar and for L > 0, the 
inhomogeneity effect increases in an exponential manner and converts to a line with constant 
slope; this means that the magnitude of Cinh can become comparable to that of Jfar. A strong 
asymmetry appears between the hard/soft transition with large, positive Cinh and the soft/hard 
transition where Cinh is appreciably smaller in size and negative. For the crack approaching 
the interface, Cinh seems to approach infinity for the hard/soft transition; for the soft/hard 
transition, the data seem to approach Cinh = -Jfar, or Jtip = 0, compare Figs 3 and 6.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
-20
-15
-10
-5
0
5
10
15
20
0 20 40 60 80 100 120 140
Jfar [kJ/m
2]
C
in
h 
[k
J/
m
2 ]
L=2.50 mm
L=0.60 mm
L=0.15 mm
L=-0.15 mm
L=-0.60 mm
L=-2.50 mm
E=210 GPa
     N=0σy=900/300
    L > 0
σy=900/300
    L < 0
σy=300/900
    L > 0
σy=300/900
    L < 0
FIGURE 5. Yield stress inhomogeneity for elastic – ideally plastic materials. 
ECF15 
 
 
 
 
igure 6 shows very interesting results for L < 0: For the hard/soft transition, Cinh drops 
onclusions 
rack driving force, Jtip, can be determined as the sum of the material 
i
y: Cinh scales linearly with Jfar, and the 
 
 omogeneity: Cinh saturates at large 
-60
-40
-20
0
20
40
60
80
100
-2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5
L [mm]
C
in
h [
kJ
/m
2 ]
120
Jfar= 30
Jfar= 40
Jfar= 60
Jfar= 80
Jfar=100
kJ/m2
E=210 GPa
    N=0
1
σ2=300MPa
σ1=300MPa
σ2=900MPa
σ =900MPa
 
 
 
 
 
 
 
 
 
FIGURE 6. Effect of the distance between interface and crack tip on the material 
inhomogeneity term for a bimaterial with a yield stress inhomogeneity. 
F
immediately to a very small value after the crack has penetrated the interface. For the 
soft/hard transition, the Cinh-values are appreciably larger compared to the corresponding 
value for the same Jfar and positive L. The reasons for these trends are currently being 
explored and more details will be presented in a forthcoming publication [9].  
 
C
The effective c
inhomogeneity term, Cinh and the nom nally applied far-field crack driving force, Jfar. In a 
computational setting, Cinh, is evaluated by a post-processing procedure. Numerical studies of 
a fracture mechanics specimen containing a sharp interface separating two homogeneous 
materials, show that the inhomogeneity due to the interface can significantly influence the 
effective crack driving force. Specific results include  
 Linear elastic bimaterials – modulus inhomogeneit
constant slope k = Cinh/Jfar depends on the Dundurs parameter and on the distance 
between the distance between the crack tip and interface. As the crack tip approaches the 
interface, Cinh and hence Jtip approach infinity for the stiff/compliant transition, whereas 
Jtip approaches zero for the compliant/stiff transition.  
Linear elastic-ideally plastic bimaterials – modulus inh
Jfar, The magnitude of Cinh reaches a maximum value when the crack tip is at some 
distance from the interface. This maximum value scales as the square of the constant yield 
stress and linearly with the difference in elastic modulus.  
ECF15 
 Linear elastic-ideally plastic bimaterials – yield stress inhomogeneity: Cinh is comparable 
to Jfar at large values of Jfar. As the crack tip approaches the interface, Jtip approaches 
infinity for the hard/soft transition, whereas Jtip approaches zero for the soft/hard 
transition. 
The combined effects of modulus + yield stress + hardening inhomogeneity will be treated 
in a forthcoming paper [9]. The application of the material inhomogeneity effect for welded 
joints is studied in [10]. Such parametric studies identify the material combinations that 
provide either a large or negligible influence on the fracture behavior and offer possibilities 
for optimizing composite materials and structural components.  
 
Acknowledgments  
The authors thank the Materials Center, Leoben for funding (project numbers SP7, SP14). JP 
acknowledges the partial financial support by the Österreichischer Austauschdienst (ÖAD) 
and the Slovenian Ministry of Education, Science and Sport (bilateral project Sl-A12/0405).  
 
References 
 1. Simha, N.K., Fischer, F. D., Kolednik, O. and Chen, C. R., J. Mech. Phys. Solids, vol. 51, 
209-240, 2003.  
 2. Maugin, G.A., Material Inhomogeneities in Elasticity, Chapman and Hall, London, 1993. 
 3. Gurtin, M.E., Configurational Forces as Basic Concepts of Continuum Physics, Springer, 
Berlin, 2000. 
 4. Simha, N.K., Predan, J., Kolednik, O., Fischer, F. D. and Shan, G.X., J. Mech. Phys. 
Solids, submitted. 
 5. Kichuchi, M. and Miyamoto, H., In Mechanical Behavior of Materials IV, Proceedings 
of the Fourth International Conference, edited by J. Carlsson, N.G. Ohlson, N.G., 
Pergamon, Oxford, UK, 1984, Vol. 2, 1077-1083. 
 6. Kolednik, O., Predan, J., Shan, G.X., Simha, N.K. and Fischer, F. D., Int. J. Solids Struct, 
in press.  
 7. Dundurs, J., J. Appl. Mech., vol. 36, 650-652, 1969. 
 8. O. Kolednik, Int. J. Solids Struct, vol. 37, 781-808, 2000. 
 9. Kolednik, O., Predan, J., Simha, N.K. and Fischer, F. D., to be published. 
 10. Predan, J., Gubeljak, N., Kolednik, O., Fischer, F. D., In Proceedings of the 15th 
European Conference of Fracture, submitted. 
 


CRACK-TIP SHIELDING AND ANTI-SHIELDING BY A BIMATERIAL INTERFACE

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CRACK-TIP SHIELDING AND ANTI-SHIELDING BY A BIMATERIAL INTERFACE

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