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Elastic-plastic asymptotic fields for cracks on bimaterial interfaces
Engimeerbg Fracture ~ec~ics Vol.37,NO. 3,pp. 527-538,19%
Printedin GreatBritain.
~13.~~ ~3.00+ 0.00 Q 1990PergamonPms.spk.
ELASTIC-PLASTIC ASYMPTOTIC FIELDS FOR CRACKS ON BIMATERIAL I~ERFACES T. C. WANG Division of Engineering, Brown University, Providence, Rhode Island 02912, U.S.A. Ab&ract--An exact asymptotic analysis for a crack lying on the interface of an elastic-plastic material and a linear elastic material is presented in this paper. The results obtained are free of oscillatory singularity and have the HRR type singularity in the plastic angular zone. Both the continuity of displacements and the continuity of tractions are satisfied on the interface. The crack faces open smoothly without any interpenetration. The interface behaves like a perfect elastic substrate. The asymptotic solutions of tip fields for cracks perpendicular to the interface are also obtained.
1. INTRODUCTION play an important roie in mechanical behaviour of crystalline materials, bequse inte~acial regions are general sources of defects and they are also regions where stresses and strains tend to concentrate. When local stress-strain fields reach a critical state, voids or micro-cracks will initiate on the interfaces. Much work has been published on the crack tip fields at the interfaces of isotropic linear elastic media. Early solutions have been given by Williams[l], Cherpanov[2], Erdogan[3], Sih and Rice[rl], Rich and Sih[5] and England[ci]. The associated problems for linear anisotropic materials have been investiga~d by Sih et at.[ITf and Sih and Liebowitz[8]. Willis[9]has developed a general solution for the crack tip fields at the linear anisotropic elastic media. Due to development of advanced materials as structure ceramics, composites and multi-flhase alloys, the research on interface problems has become active in the last IS years (e.g. refs [10-l 9). Recently Ting[l4] has presented an explicit solution for a general interface crack in an elastic anisotropic composite. Park and Earmme[l$ Hutchinson et uZ.[16]and Suo and Hut~hinson[l~ have obtained solutions for several elastic interfacial crack problems. Comprehensive overviews on elastic fracture mechanics concepts and some recent developments for interfacial cracks have been given by Rice[l8] and Hutchinson[ 191. The problem of an interface crack in elastic-plastic materials was originally investigated by Shih and Asaro[2&22]. They presented a full numeral solution for a crack which lies along the interface of the elastic-plastic bimaterials. Nearly separable singular tip fields have been found in the angular zone based on their numerical solutioni They have also found that the crack face opens smoothly without rapid oscillations which is a characteristic feature for the asymptotic linear elastic solution. Parks and Zywicz[23,24] analysed the elastic-perfectly plastic small scale yielding crack problem at bimate~al interfaces. Both Parks and Zywicz and Shih and Asaro i~de~ndently found that tbe elastic-plastic fields are members of a family parameterized by a new phase angle. This paper presents an exact asymptotic field analysis for a crack which lies along the interface of an elastic-plastic medium and a linear elastic medium. The elastic-plastic medium has a property which is characterized by a small strain, isotropic J2 deformation theory. A separable singular form of HRR-type fields has been found in the plastic angular zone. In the elastic angular zone, the stress fields have the same sing~a~ty as that in the plastic angular zone. The interface behaves like a perfect elastic substrate. Both the continuity of displacements and continuity of tractions are satisfied across the interfaces. The asymptotic solutions of tip fields for cracks perpendicular to the interface of elastic-plastic materials and linear elastic materials are also obtained. MERFACES
528
T. C. WANG
Material 1
Material SJ
i_
Fig. 1. Crack on bimaterialinterfaces. 2. FORMULATION OF THE PROBLEM 2.1. Constitutive equation Figure 1 shows a crack lying along the interface of medium 1 and medium 2. Medium 1 is an elastic-plastic material and occupies the region 0 < 9 < 7c, while medium 2 is a linear elastic material and occupies the region --II < 8 < 0. For the sake of simplicity, we assume that both materials have the same elastic modulus E and Poisson’s ratio v. The stress-strain relation for material 2 can be expressed as
(2-l) In uniaxial tension, material 1 deforms according to the Rambergasgood
formula,
where a,, and ~0are the yield stress and strain, a is a material constant and n is the strain hardening exponent. The constitutive equation for material 1 is given by
(2.2) where S, is the deviatoric stress component
and a, is the effective stress.
2.2. Plane strain problem For the case of plane strain, all quantities are functions of polar coordinates (r, 0). We have =kk
where A = 2,ct(a,/a,$-‘.
+
AS3318
=
0,
(2.3)
From this equation, we obtain s33
=
4;
-
vm
+
Substituting eq. (2.4) into eq. (2.2), the stress-strain
~,)/(A
+
9).
(2.4)
relations can be represented as
(2.5) where P,,#= c~,@ - its,,, 6,) c = f - v, and the Greek indices represent the numbers 1 and 2. The repeated index is summed up. The effective stress 0e can be expressed as 0, = J3k
- a$/4 + r&l.
529
Asymptotic fields for cracks on bimaterial interfaces
Introducing a stress function 4 and writing the following expressions for stress components a’+ Q,=ay23 824
(2.6)
by=%’
z
3% XY
=
-is&’
the equation of equilibrium will be automatically satisfied. The strain components are related to the displacements by the following equation:
(2.7) Equations (24, (2.5), (2.6) and (2.7) are the basic equations for the plane strain problem of an elastic-plastic solid. In the polar coordinate system, the basic equations can be rewritten as l+v E,= 7 br. - v@, + oe)] + AP,/E - d&/E, l+v &g= -
1
E
bg - ~(0, + oe)] + APB/E - dS,,/E, [
l+v 44 =Erd+
/IQ/E,
a2
l?+r
6, = -
r ar
Qe=a,29
(2.8)
r2a82’
a24
(2.9 ,
i
s
zz =
-6(o, + UeW
$1,
+
(2.10)
(2.11) ---.
ue
r
3. GOVERNING EQUATIONS FOR ASYMPTOTIC FIRLW Assume the stress fields in the whole tip zone have the same singularity, i.e., uV= r%,(0) for materials 1 and 2, so that the tractions can be continuous across the interface. The governing equations have been given by Hutchinson[25] and Rice and Rosengren[26]. Here we just cite some results. Let 4 = Kr”+2F(0),
s < 0.
(3.1)
Substituting eq. (3.1) into eqs (2.9), we obtain a, = KrV@), Cg= Kr”B,g(B), TV = Kr'i
,
(3.2)
530
T. C. WANG
(3.3)
(3.4)
(3.5)
Using eqs (2.8), it follows
(3.6)
(3.7)
where R= K/a,, d = a&,,. The elastic strains have been neglected in the plastic zone. On the other hand, the term AS, has the same singularity as the stress (a, + ae) and can also be neglected. Hence, eq. (3.5) is accurate in the sense of asymptotic behavior. The strain compatibility equation takes the form (3.8) Substituting eq. (3.6) into (3.8), we arrive at E”:’ - ns(?zs + 2)E; - 2(?zs + l)E”h= 0.
(3.9)
Using eq. (3.7), eq. (3.8) can be represented as
1
-$-ns(ns+2) [a-:-1(F”-s(s+2)F)]+4(1+ns)[~:-‘(l+s)F’]’=0. [ In material 2, we have the equation F “U- [(s + 2)2 + s)F’ + s2(s + 2)2F = 0.
(3.10)
(3.11)
The general solution of this equation is F = B,
The traction-free
cos se +
B2 sin SO
+
B3COS(S
+
2)8 + B4sin(s + 2)8.
(3.12)
conditions on the crack faces require F(n) = F’(n) = 0, F(-z)=F’(-n)=O.
(3.13) (3.14)
The displacements on the plastic zone have the form u, = fiRV’+Ri?,, ( u, = o?Rr’*‘“i&,
(3.15)
where (3.16)
531
Asymptotic fields for cracks on bimaterial interfaces
As pointed out by Shih and Asaro[ZO],material 1 has lower hardening capacity, therefore near the interface we have E,,/E,z+O, where E,, and E,z are the tangent moduli of materials 1 and 2 respectively. The material system responds as that of a plastical material bonded to a perfect/elastic substrate. It means that in the asymptotic sense, we have (in the plastic region side), (3.17)
6, = ug .. -0 - ) at 0 = O+. Equation (3.17) results in I
. Q,--0-@= 0 ,
at 0 =O+.
[2( 1 + ns)fti - (6, - 6, j\,,m IL = I\ u,
(3.18)
Equation (3.18) can be expressed in terms of F, F"-s(s+2)F=O,
I
F” + [4(1 + s)(l + ns) - s(s + 2)]F’ = 0,
at 8 = O+.
(3.19)
Equations (3.10), (3.13) and (3.19) are the governing equations of the asymptotic fields in the plastic angular zone. 4. SOLUTIONS OF ASYMPTOTIC FIELDS
It is worth noting that eqs (3.10), (3.13) and (3.19) correspond to search the eigenvalue s and eigenfunction F on the region 0 < 0 < x. Using the fourth order RungeXutta method, eq. (3.10) with boundary conditions (3.13) and (3.19) can be accurately integrated. The eigenvalue s = - I/( I + n) is preferred, which is equal to the eigenvalue of the HRR fields of homogeneous media. According to Wang and Wang[27], we have the following functional equation in the plastic region,
(4.1) Equation (4.1) can be used to check our calculation results. The numerical solution of eqs (3.10), (3.13) and (3.19) satisfies eq. (4.1) with very high accuracy. The initial value of F(O+) can be taken as unity. For a given initial value of F’(O+), one can find the initial values of F”(O+) and F”(O+) from eq. (3.19). Then eq. (3.10) can be accurately integrated from 8 = 0 to 8 = x, adjusting the initial value of F’(O+)so that the boundary condition (3.13) will be exactly met. Shih[28] defined a mixity parameter MP which ranges from one for Mode I to zero for Mode II, namely MP=
2 i tan-’
-43v-0 . ( buv >
The parameter MP and the eigenfunction F(8) and its derivatives at 8 = O+ are listed in Table 1. After we obtain the accurate solution Fin the plastic region, the solution Fin the elastic region, --II < 0 < 0, can be easily derived. Noting the boundary condition (3.14), eq. (3.12) can be represented as
, (4.2) 1 iJ=e+7t.
sin@ + 2)8 - f sin sB
F = B, [cos(s + 2)8 - cos sfl+ B2
Table 1. Parameter MP and initial values of eigenfunction F(B) and its derivatives at 0 =0+
n
MP
3 10
0.96056 0.91516
F
F
F”
F”
f:8
-0.1085677 -0.2559299
-0.4375000 -0.1735537
0.1289241 0.1290225
532
T. C. WANG
Angular variations
-1.0
/ -180.0
I
, -120.0
I
, -60.0
of normalized
,
, 0.0
,
,
stresses
,
60.0
( 120.0
1 1t 1.0
8
Fig. 2. Angular variation of normalized stresses near the tip of a crack, which lies on the interface of a perfect elastic material and an elastbplastic material with hardening exponent n = 3.
The coefficients B, and B2canbedetermined from the continuity of tractions on the interface. The angular variation of normalized stresses 5&3) are shown in Figs 24 for different hardening materials. The stresses are normalized with the requirement of (Ze)_ = 1. Figures 5-7 show the angular variations of normalized displacements ti,(@ and ii&). From these figures, it can be seen the condition (3.17) is exactly met and the crack faces obviously open. Hence the interpenetration of crack faces is ruled out. In order to compare the present results with the HRR singularity fields for homogeneous media, a plot of angular variation of normalized stresses Z&J) for homogeneous materials subjected to pure Mode I loading is shown in Fig. 8. Comparing Figs 2 and 4 with Fig. 8, it can be seen that the present results are quite similar to those with the HRR singularity fields. In homogeneous materials, S, = 0 occurs at positive angular (3,(e, = 30” for n = 3; 0, = 43” for n = 5). Meanwhile, in our calculation, 0, = 0”. Angular variations
of normalized
stresses
.z b
-1.0
II
-180.0
-120.0
-60.0
0.0
60.0
120.0
180.0
Fig. 3. Angular variation of normalized stresses near the tip of a crack, which lies on the interface of a perfect elastic material and an elastic-plastic material with hardening exponent n = 5.
533
Asymptotic fklds for cracks on bimaterial interfaces
Angular variations of normalized stresses
-1.0; -180.0
-120.0
-60.0
0.0
120.0
60.0
11 I.0
8
Fig. 4. Angular variation of normal&d stressesnear the tip of a crack, which lies on the interface of a pcrfkt elastic material and an elastic-plastic material with hardening exponent n r‘ 10.
On the other hand, the stress magnitude of the present results seems greater than that of HRR fields for homogeneous materials in the region --x < 6 6 0. The present results are also similar to the numerical results given by Shih and Asaro[20,21], but there are some differences between the two. The present results show positive shear stress ahead of the crack tip, while the numerical results of Shih and Asaro[20] show negative shear stress ahead of the crack. This may be due to the caption geometry effect. Shih and Asaro[20] obtained their results on a center-cracked panel. As pointed out by McMeeking and Parks[29] and Shih and German[30], the stress fields near the crack tip deviate significantly from HRR fields for a center-cracked panel in the case of plane strain. The deviation is associated with relaxation of the triaxial stress constraint due to the intensive plastic strain developed along the 45” inclined plane.
Angular variations of normalized displacements
E
E
50.0 -
8
3
.cn u m
.-i3
E
-50.0 -
2
I
0.0
30.0
f
I
60.0
-
1,
90.0
1
120.0
”
150.0
*
180.0
e
Fig. S. Angular variationof normal&d displacements in the plastic region near the tip of a crack, which lie8 on the intcrfnce of a perfect elastic material and an elastic-plastic material with hardening exponent n= 3.
534
T. C. WANG
Angular variations of normalized displacements
-300.0
( I 0.0
, 30.0
I
, 60.0
I
, 90.0
I
, I 120.0
,
I
150.0
180.0
8
Fig. 6. Angular variation of normalized displacements in the plastic region near the tip of a crack, which lies on the interface of a perfect elastic material and an elastic-plastic material with hardening exponent n= 5.
Hence one can imagine that the discrepancy between the present results and the numerical results of Shih and Asaro[20] are related to the geometry effect. Meanwhile, the present, solution is only suitable for crack tip stress state with specified mixity parameter W, which is listed in Table 1. For the general combined tension and shear holding, the eigenvalue may be a complex variable and the solution may be much more complex. 5. SOLUTION OF ASYMPTOTIC
FIELDS FOR CRACKS PERPENDICULAR INTERFACES
TO
As shown in Fig. 9, a crack lies only in medium 1, but its tip touches perpendicular to the interface of media 1 and 2.
Angular variations of normalized displacements
2ooo*o I 1000.0-
'. \
l.
5 t
0.0
t
I
-1ooo.o-
-2ooo.o-
-3000.0
) . 0.0
, 30.0
I
, 60.0
>
, 90.0
I
, I 120.0
, I 150.0
i
1s 8(1.0
8
Fig. 7. Angular variation of nonnaliti displacements in the plastic region near the tip of a crack, which lies on the interface of a perfect elastic material and an ebtio-plastic material with hardening exponent ?I= 10.
535
Asymptotic fieids for era&s on bimateriai interfaces
-LO
f I
-
*l&O.0
.
-
I
40.0
I
40.0
0.0 8
I ..-.a GO.0
la.0
-
Fig. 8. Angular variation of normalized stresses near the crack tip of HRR singular fields for the case of Mode I.
Assume that the external load is symmetrical remote tension. Due to symmetry, we search only the solution in the region 0 < 8 6 x. Thus, in the elastic region 0 < 8 G n/Z, we have F = BI COSSe + B3COS(S+ z)e,
o6 e G X/2.
Fig. 9. Crack which lies only in medium 1, but its tip touches perpendicular to the interface of a perfect elastic material and an elasticplastic material.
(5.1)
T. C. WANG
536
Table 2. Eigenvalues s and initial values of eigenfunction F(B) and its derivatives at e=e: n
S
F
F
3 10
-0.2125974 -0.08452362
1.0 1.O
-0.8469439 -0.8321521
Angular variations
-1.0
1 0.0
30.0
60.0
F" -0.3799972 -0.1619030
of normalized
90.0
F"
120.0
1.288041 0.6063338
stresses
150.0
1EICI.0
f3
Fig. 10. Angular variation of normaIized stresses near the tip of a crack, which is perpendicular to the interface of a perfect elastic material and an elastic-plastic material with hardening exponent n = 3.
In the plastic region a/2 Q 8 < IC,the governing equation (3.10) holds true. On the interface, we have 24,= 24,= 0,
at
e = O,+,
(5.2)
0, = a/2.
This results in
1
F” - S(S f 2)F = 0, P+[4(1 +s)(l +m)-s(2+s)]F’=O,
Angular variations
(5.3)
at e=e:*
of normalized
stresses
I
0.0 0.0
30.0
60.0
90.0
120.0
150.0
1t3C I.0
Fig. I I. Angular variation of normalized stresses near the tip of a crack, which is perpendicular to the interface of a perfect elastic material and an elasticqlastic material with hardening exponent n = 10.
53?
Asymptotic fieids for cracks on bimattial interfaces
Angular variations
-4.0,
s
90.0
,
*
105.0
of normalized
t
1
1
’
135.0
120.0
L
displacements
’
,
150.0
1
165.0
t8O.O
Fig, 12. Angular variation of normalized displacements near the tip of a crack which is perpendicular to the interface of a perfect elastic material and an elastic-plastic material with hardening exponent n = 3.
Angular variations
-25.0 [ t 90.0
, 8 105.0
of normalized
', 1 120.0
, s 135.0
displacements
, 8 150.0
F . 165.0
180.0
Fig. 13. Angular variation of normalized displacements near the tip of a crack which is perpendicuiar to the interlace of a perfect elastic material and an elastic-plastic material with hardening exponent n = 10.
Equations (3.10), (3.13) and (5.3) are the governing equations of the asymptotic fields. Table 2 shows the eigenvalues and initial values of the eigenfunction F(B) and its derivatives at 8 =c0:. Figures 10 and 11 show the angular variations of normalized stresses for different elastic-plastic hardening materials. The stresses are normalized by (a,)-, which is just taken in the plastic region that a, oan exceed unity in the elastic region 0 6 0 G n/2. It can be seen that tractions are continuous acfuss the interface, but there is big jump in the radial stress 6, , The stress s, on the elastic region side is much greater than that on the plastic region side. This means that the interface is really a stress raiser which results in serious stress concentration. Figures 12 and 13 show the angular variation of normalized displacements for different elastic-plastic materials. From these figures, one can see that the crack faces open smoothly without overlap. Acknowledgemem-The author is greatly indebted to Prof. C. F. Shih for his important encouragement and many valuable discussions. The computations repornrd here were carried out on the VAX computer in the Computational Mechamcs Computer Facility, Division of Engineering, Brown University.
538
T. C. WANG
REFJIRENCES [l] M. L. Williams, The stress around a fault or crack in dissimilar media. Bull. Seismol. Sec. Am. 49, 199-204 (1959).
[2] P. G. Cherpanov, The stress state in a heterogeneous plate with slits. Izu. AN SSSR, OTN Mekhan. i Ma.vhin.1,131-137 (1962). [3] F. Erdogan, Stress distribution in bonded dissimilar materials with cracks. J. uppl Mech. 32, 403-410 (1965). [4] G. C. Sib and J. R. Rice, The bending of plates of dissimilar materials with cracks. J. appl. Med. X,477-482(1964). [5] J. R. Rice and G. C. Sih, Plane problems of cracks in dissimilar media. J. appl. Mech. 32, 418423 (1965). [6] A. H. England, A crack between dissimilar media. J. appl. Mech. 32, 400402 (1965). [7] G. C. Sih, P. C. Paris and G. R. Irwin, On crack in rectilinear anisotropic bodies. Inr. J. Fracture 1, 189-302 (1965). [8] G. C. Sih and H. Liebowitx, Mathematical theory of brittle fracture, In Fracture (Edited by H. Liebowitx), Vol II. pp. 115-125 (1968). [9] J. R. Willis, Fracture mechanics of interfacial cracks. J. Mech. Phys. Solids 19,353-368 (1971). [lo] D. L. Clements, A crack between dissimilar anisotropic media. Znt. J. Engng Sci. 9, 257-265 (1971). [l l] D. B. Bogy, Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. J. appi. Mech. 38, 377-386 (1971). [12] G. C. Sih and P. Chen, Cracks in CompositeMaterials, pp. 87-91 (1981). (131 S. S. Wang, Edge delamination in angle-ply composite laminates. AIAA J. 25 256-264 (1984). [14] T. C. Ting, Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. Int. J. Solidr Struct. 22, 965-983 (1986). [ls] J. H. Park and Y. Y. Earmme, Application of conservation integral to interfacial crack problems. Mech. Mater. 5, 261-276 (1986). [16] J. W. Hutchinson, M. Mear and J. R. Rice, 1987, Crack paralleling an interface between disimilar materials. J. uppi. Mech. 54, 828832 (1987). [17J Z. Suoand J. W. Hutchinson, Interface crack between two elastic layers. Harvard University Report, Mech-118 (to be published). 1181J. R. Rice, Elastic fracture mechanics concepts for interfacial cracks. J. uppf. Mech. Jb, 98-103 (1988). [19] J. W. Hutchinson, Mixed mode fracture mechanics of interfaces. Harvard University Report, Mech-139 (1989). [20] C. F. Shih and R. J. Asaro, Elastic-plastic analysis of cracks on bimaterial interfaces: Part I-small scale yielding. J. uppf. Mech. SS, 299-316 (1988). [21] C. F. Shih and R. J. Asaro, Elastic-plastic and asymptotic fields of interface crack. ht. J. Fracture (in press). [22] C. F. Shih and R. J. Asaro, Elasti*plastic analysis of cracks on bimaterial interfaces: Part H-structure of small scale yielding fields. J. appl. Med. (in press). [23] D. M. Parks and E. Zywicz,Elastic/perfectly-plastic small scale yielding at bimaterial interfaces. In Advances in Fracture Research (Edited by K. Salama et ul.), Vol. 4, pp. 3081-3088 (1989). [24] E. Zywin and D. M. Parks, Elastic yield xone around an interfacial crack tip. J. uppl. Med. (in press). [251 J. W. Hutchinson, Singular behavior at the end of a tensile crack in hardening material. J. Mech. P&s. So@ 16,13-31 (1968). [26] J. R. Rice and G. F. Rosengren, Plane strain deformation near crack tip in a power law hardening material. J. Me& Phys. Solti 16, l-12 (1968). [271 K. Wang and T. Wang, Appl. Math. Mech. 9, 791-802 (1987). [28] C. F. Shih, Small scale yielding analysis of mixed mode plane strain crack problems. Fracture analysis. ASTM STP 560, 187-210 (1974). [29] R. M. McMeeking and D. M. Parks, Elastic plastic fracture. ASTM STP 668, 175-194 (1979). [30] C. F. Shih and M. D. German, Znt.J. Fracture 17, 27-43 (1981). (Received 18 May 1989)
Printedin GreatBritain.
~13.~~ ~3.00+ 0.00 Q 1990PergamonPms.spk.
ELASTIC-PLASTIC ASYMPTOTIC FIELDS FOR CRACKS ON BIMATERIAL I~ERFACES T. C. WANG Division of Engineering, Brown University, Providence, Rhode Island 02912, U.S.A. Ab&ract--An exact asymptotic analysis for a crack lying on the interface of an elastic-plastic material and a linear elastic material is presented in this paper. The results obtained are free of oscillatory singularity and have the HRR type singularity in the plastic angular zone. Both the continuity of displacements and the continuity of tractions are satisfied on the interface. The crack faces open smoothly without any interpenetration. The interface behaves like a perfect elastic substrate. The asymptotic solutions of tip fields for cracks perpendicular to the interface are also obtained.
1. INTRODUCTION play an important roie in mechanical behaviour of crystalline materials, bequse inte~acial regions are general sources of defects and they are also regions where stresses and strains tend to concentrate. When local stress-strain fields reach a critical state, voids or micro-cracks will initiate on the interfaces. Much work has been published on the crack tip fields at the interfaces of isotropic linear elastic media. Early solutions have been given by Williams[l], Cherpanov[2], Erdogan[3], Sih and Rice[rl], Rich and Sih[5] and England[ci]. The associated problems for linear anisotropic materials have been investiga~d by Sih et at.[ITf and Sih and Liebowitz[8]. Willis[9]has developed a general solution for the crack tip fields at the linear anisotropic elastic media. Due to development of advanced materials as structure ceramics, composites and multi-flhase alloys, the research on interface problems has become active in the last IS years (e.g. refs [10-l 9). Recently Ting[l4] has presented an explicit solution for a general interface crack in an elastic anisotropic composite. Park and Earmme[l$ Hutchinson et uZ.[16]and Suo and Hut~hinson[l~ have obtained solutions for several elastic interfacial crack problems. Comprehensive overviews on elastic fracture mechanics concepts and some recent developments for interfacial cracks have been given by Rice[l8] and Hutchinson[ 191. The problem of an interface crack in elastic-plastic materials was originally investigated by Shih and Asaro[2&22]. They presented a full numeral solution for a crack which lies along the interface of the elastic-plastic bimaterials. Nearly separable singular tip fields have been found in the angular zone based on their numerical solutioni They have also found that the crack face opens smoothly without rapid oscillations which is a characteristic feature for the asymptotic linear elastic solution. Parks and Zywicz[23,24] analysed the elastic-perfectly plastic small scale yielding crack problem at bimate~al interfaces. Both Parks and Zywicz and Shih and Asaro i~de~ndently found that tbe elastic-plastic fields are members of a family parameterized by a new phase angle. This paper presents an exact asymptotic field analysis for a crack which lies along the interface of an elastic-plastic medium and a linear elastic medium. The elastic-plastic medium has a property which is characterized by a small strain, isotropic J2 deformation theory. A separable singular form of HRR-type fields has been found in the plastic angular zone. In the elastic angular zone, the stress fields have the same sing~a~ty as that in the plastic angular zone. The interface behaves like a perfect elastic substrate. Both the continuity of displacements and continuity of tractions are satisfied across the interfaces. The asymptotic solutions of tip fields for cracks perpendicular to the interface of elastic-plastic materials and linear elastic materials are also obtained. MERFACES
528
T. C. WANG
Material 1
Material SJ
i_
Fig. 1. Crack on bimaterialinterfaces. 2. FORMULATION OF THE PROBLEM 2.1. Constitutive equation Figure 1 shows a crack lying along the interface of medium 1 and medium 2. Medium 1 is an elastic-plastic material and occupies the region 0 < 9 < 7c, while medium 2 is a linear elastic material and occupies the region --II < 8 < 0. For the sake of simplicity, we assume that both materials have the same elastic modulus E and Poisson’s ratio v. The stress-strain relation for material 2 can be expressed as
(2-l) In uniaxial tension, material 1 deforms according to the Rambergasgood
formula,
where a,, and ~0are the yield stress and strain, a is a material constant and n is the strain hardening exponent. The constitutive equation for material 1 is given by
(2.2) where S, is the deviatoric stress component
and a, is the effective stress.
2.2. Plane strain problem For the case of plane strain, all quantities are functions of polar coordinates (r, 0). We have =kk
where A = 2,ct(a,/a,$-‘.
+
AS3318
=
0,
(2.3)
From this equation, we obtain s33
=
4;
-
vm
+
Substituting eq. (2.4) into eq. (2.2), the stress-strain
~,)/(A
+
9).
(2.4)
relations can be represented as
(2.5) where P,,#= c~,@ - its,,, 6,) c = f - v, and the Greek indices represent the numbers 1 and 2. The repeated index is summed up. The effective stress 0e can be expressed as 0, = J3k
- a$/4 + r&l.
529
Asymptotic fields for cracks on bimaterial interfaces
Introducing a stress function 4 and writing the following expressions for stress components a’+ Q,=ay23 824
(2.6)
by=%’
z
3% XY
=
-is&’
the equation of equilibrium will be automatically satisfied. The strain components are related to the displacements by the following equation:
(2.7) Equations (24, (2.5), (2.6) and (2.7) are the basic equations for the plane strain problem of an elastic-plastic solid. In the polar coordinate system, the basic equations can be rewritten as l+v E,= 7 br. - v@, + oe)] + AP,/E - d&/E, l+v &g= -
1
E
bg - ~(0, + oe)] + APB/E - dS,,/E, [
l+v 44 =Erd+
/IQ/E,
a2
l?+r
6, = -
r ar
Qe=a,29
(2.8)
r2a82’
a24
(2.9 ,
i
s
zz =
-6(o, + UeW
$1,
+
(2.10)
(2.11) ---.
ue
r
3. GOVERNING EQUATIONS FOR ASYMPTOTIC FIRLW Assume the stress fields in the whole tip zone have the same singularity, i.e., uV= r%,(0) for materials 1 and 2, so that the tractions can be continuous across the interface. The governing equations have been given by Hutchinson[25] and Rice and Rosengren[26]. Here we just cite some results. Let 4 = Kr”+2F(0),
s < 0.
(3.1)
Substituting eq. (3.1) into eqs (2.9), we obtain a, = KrV@), Cg= Kr”B,g(B), TV = Kr'i
,
(3.2)
530
T. C. WANG
(3.3)
(3.4)
(3.5)
Using eqs (2.8), it follows
(3.6)
(3.7)
where R= K/a,, d = a&,,. The elastic strains have been neglected in the plastic zone. On the other hand, the term AS, has the same singularity as the stress (a, + ae) and can also be neglected. Hence, eq. (3.5) is accurate in the sense of asymptotic behavior. The strain compatibility equation takes the form (3.8) Substituting eq. (3.6) into (3.8), we arrive at E”:’ - ns(?zs + 2)E; - 2(?zs + l)E”h= 0.
(3.9)
Using eq. (3.7), eq. (3.8) can be represented as
1
-$-ns(ns+2) [a-:-1(F”-s(s+2)F)]+4(1+ns)[~:-‘(l+s)F’]’=0. [ In material 2, we have the equation F “U- [(s + 2)2 + s)F’ + s2(s + 2)2F = 0.
(3.10)
(3.11)
The general solution of this equation is F = B,
The traction-free
cos se +
B2 sin SO
+
B3COS(S
+
2)8 + B4sin(s + 2)8.
(3.12)
conditions on the crack faces require F(n) = F’(n) = 0, F(-z)=F’(-n)=O.
(3.13) (3.14)
The displacements on the plastic zone have the form u, = fiRV’+Ri?,, ( u, = o?Rr’*‘“i&,
(3.15)
where (3.16)
531
Asymptotic fields for cracks on bimaterial interfaces
As pointed out by Shih and Asaro[ZO],material 1 has lower hardening capacity, therefore near the interface we have E,,/E,z+O, where E,, and E,z are the tangent moduli of materials 1 and 2 respectively. The material system responds as that of a plastical material bonded to a perfect/elastic substrate. It means that in the asymptotic sense, we have (in the plastic region side), (3.17)
6, = ug .. -0 - ) at 0 = O+. Equation (3.17) results in I
. Q,--0-@= 0 ,
at 0 =O+.
[2( 1 + ns)fti - (6, - 6, j\,,m IL = I\ u,
(3.18)
Equation (3.18) can be expressed in terms of F, F"-s(s+2)F=O,
I
F” + [4(1 + s)(l + ns) - s(s + 2)]F’ = 0,
at 8 = O+.
(3.19)
Equations (3.10), (3.13) and (3.19) are the governing equations of the asymptotic fields in the plastic angular zone. 4. SOLUTIONS OF ASYMPTOTIC FIELDS
It is worth noting that eqs (3.10), (3.13) and (3.19) correspond to search the eigenvalue s and eigenfunction F on the region 0 < 0 < x. Using the fourth order RungeXutta method, eq. (3.10) with boundary conditions (3.13) and (3.19) can be accurately integrated. The eigenvalue s = - I/( I + n) is preferred, which is equal to the eigenvalue of the HRR fields of homogeneous media. According to Wang and Wang[27], we have the following functional equation in the plastic region,
(4.1) Equation (4.1) can be used to check our calculation results. The numerical solution of eqs (3.10), (3.13) and (3.19) satisfies eq. (4.1) with very high accuracy. The initial value of F(O+) can be taken as unity. For a given initial value of F’(O+), one can find the initial values of F”(O+) and F”(O+) from eq. (3.19). Then eq. (3.10) can be accurately integrated from 8 = 0 to 8 = x, adjusting the initial value of F’(O+)so that the boundary condition (3.13) will be exactly met. Shih[28] defined a mixity parameter MP which ranges from one for Mode I to zero for Mode II, namely MP=
2 i tan-’
-43v-0 . ( buv >
The parameter MP and the eigenfunction F(8) and its derivatives at 8 = O+ are listed in Table 1. After we obtain the accurate solution Fin the plastic region, the solution Fin the elastic region, --II < 0 < 0, can be easily derived. Noting the boundary condition (3.14), eq. (3.12) can be represented as
, (4.2) 1 iJ=e+7t.
sin@ + 2)8 - f sin sB
F = B, [cos(s + 2)8 - cos sfl+ B2
Table 1. Parameter MP and initial values of eigenfunction F(B) and its derivatives at 0 =0+
n
MP
3 10
0.96056 0.91516
F
F
F”
F”
f:8
-0.1085677 -0.2559299
-0.4375000 -0.1735537
0.1289241 0.1290225
532
T. C. WANG
Angular variations
-1.0
/ -180.0
I
, -120.0
I
, -60.0
of normalized
,
, 0.0
,
,
stresses
,
60.0
( 120.0
1 1t 1.0
8
Fig. 2. Angular variation of normalized stresses near the tip of a crack, which lies on the interface of a perfect elastic material and an elastbplastic material with hardening exponent n = 3.
The coefficients B, and B2canbedetermined from the continuity of tractions on the interface. The angular variation of normalized stresses 5&3) are shown in Figs 24 for different hardening materials. The stresses are normalized with the requirement of (Ze)_ = 1. Figures 5-7 show the angular variations of normalized displacements ti,(@ and ii&). From these figures, it can be seen the condition (3.17) is exactly met and the crack faces obviously open. Hence the interpenetration of crack faces is ruled out. In order to compare the present results with the HRR singularity fields for homogeneous media, a plot of angular variation of normalized stresses Z&J) for homogeneous materials subjected to pure Mode I loading is shown in Fig. 8. Comparing Figs 2 and 4 with Fig. 8, it can be seen that the present results are quite similar to those with the HRR singularity fields. In homogeneous materials, S, = 0 occurs at positive angular (3,(e, = 30” for n = 3; 0, = 43” for n = 5). Meanwhile, in our calculation, 0, = 0”. Angular variations
of normalized
stresses
.z b
-1.0
II
-180.0
-120.0
-60.0
0.0
60.0
120.0
180.0
Fig. 3. Angular variation of normalized stresses near the tip of a crack, which lies on the interface of a perfect elastic material and an elastic-plastic material with hardening exponent n = 5.
533
Asymptotic fklds for cracks on bimaterial interfaces
Angular variations of normalized stresses
-1.0; -180.0
-120.0
-60.0
0.0
120.0
60.0
11 I.0
8
Fig. 4. Angular variation of normal&d stressesnear the tip of a crack, which lies on the interface of a pcrfkt elastic material and an elastic-plastic material with hardening exponent n r‘ 10.
On the other hand, the stress magnitude of the present results seems greater than that of HRR fields for homogeneous materials in the region --x < 6 6 0. The present results are also similar to the numerical results given by Shih and Asaro[20,21], but there are some differences between the two. The present results show positive shear stress ahead of the crack tip, while the numerical results of Shih and Asaro[20] show negative shear stress ahead of the crack. This may be due to the caption geometry effect. Shih and Asaro[20] obtained their results on a center-cracked panel. As pointed out by McMeeking and Parks[29] and Shih and German[30], the stress fields near the crack tip deviate significantly from HRR fields for a center-cracked panel in the case of plane strain. The deviation is associated with relaxation of the triaxial stress constraint due to the intensive plastic strain developed along the 45” inclined plane.
Angular variations of normalized displacements
E
E
50.0 -
8
3
.cn u m
.-i3
E
-50.0 -
2
I
0.0
30.0
f
I
60.0
-
1,
90.0
1
120.0
”
150.0
*
180.0
e
Fig. S. Angular variationof normal&d displacements in the plastic region near the tip of a crack, which lie8 on the intcrfnce of a perfect elastic material and an elastic-plastic material with hardening exponent n= 3.
534
T. C. WANG
Angular variations of normalized displacements
-300.0
( I 0.0
, 30.0
I
, 60.0
I
, 90.0
I
, I 120.0
,
I
150.0
180.0
8
Fig. 6. Angular variation of normalized displacements in the plastic region near the tip of a crack, which lies on the interface of a perfect elastic material and an elastic-plastic material with hardening exponent n= 5.
Hence one can imagine that the discrepancy between the present results and the numerical results of Shih and Asaro[20] are related to the geometry effect. Meanwhile, the present, solution is only suitable for crack tip stress state with specified mixity parameter W, which is listed in Table 1. For the general combined tension and shear holding, the eigenvalue may be a complex variable and the solution may be much more complex. 5. SOLUTION OF ASYMPTOTIC
FIELDS FOR CRACKS PERPENDICULAR INTERFACES
TO
As shown in Fig. 9, a crack lies only in medium 1, but its tip touches perpendicular to the interface of media 1 and 2.
Angular variations of normalized displacements
2ooo*o I 1000.0-
'. \
l.
5 t
0.0
t
I
-1ooo.o-
-2ooo.o-
-3000.0
) . 0.0
, 30.0
I
, 60.0
>
, 90.0
I
, I 120.0
, I 150.0
i
1s 8(1.0
8
Fig. 7. Angular variation of nonnaliti displacements in the plastic region near the tip of a crack, which lies on the interface of a perfect elastic material and an ebtio-plastic material with hardening exponent ?I= 10.
535
Asymptotic fieids for era&s on bimateriai interfaces
-LO
f I
-
*l&O.0
.
-
I
40.0
I
40.0
0.0 8
I ..-.a GO.0
la.0
-
Fig. 8. Angular variation of normalized stresses near the crack tip of HRR singular fields for the case of Mode I.
Assume that the external load is symmetrical remote tension. Due to symmetry, we search only the solution in the region 0 < 8 6 x. Thus, in the elastic region 0 < 8 G n/Z, we have F = BI COSSe + B3COS(S+ z)e,
o6 e G X/2.
Fig. 9. Crack which lies only in medium 1, but its tip touches perpendicular to the interface of a perfect elastic material and an elasticplastic material.
(5.1)
T. C. WANG
536
Table 2. Eigenvalues s and initial values of eigenfunction F(B) and its derivatives at e=e: n
S
F
F
3 10
-0.2125974 -0.08452362
1.0 1.O
-0.8469439 -0.8321521
Angular variations
-1.0
1 0.0
30.0
60.0
F" -0.3799972 -0.1619030
of normalized
90.0
F"
120.0
1.288041 0.6063338
stresses
150.0
1EICI.0
f3
Fig. 10. Angular variation of normaIized stresses near the tip of a crack, which is perpendicular to the interface of a perfect elastic material and an elastic-plastic material with hardening exponent n = 3.
In the plastic region a/2 Q 8 < IC,the governing equation (3.10) holds true. On the interface, we have 24,= 24,= 0,
at
e = O,+,
(5.2)
0, = a/2.
This results in
1
F” - S(S f 2)F = 0, P+[4(1 +s)(l +m)-s(2+s)]F’=O,
Angular variations
(5.3)
at e=e:*
of normalized
stresses
I
0.0 0.0
30.0
60.0
90.0
120.0
150.0
1t3C I.0
Fig. I I. Angular variation of normalized stresses near the tip of a crack, which is perpendicular to the interface of a perfect elastic material and an elasticqlastic material with hardening exponent n = 10.
53?
Asymptotic fieids for cracks on bimattial interfaces
Angular variations
-4.0,
s
90.0
,
*
105.0
of normalized
t
1
1
’
135.0
120.0
L
displacements
’
,
150.0
1
165.0
t8O.O
Fig, 12. Angular variation of normalized displacements near the tip of a crack which is perpendicular to the interface of a perfect elastic material and an elastic-plastic material with hardening exponent n = 3.
Angular variations
-25.0 [ t 90.0
, 8 105.0
of normalized
', 1 120.0
, s 135.0
displacements
, 8 150.0
F . 165.0
180.0
Fig. 13. Angular variation of normalized displacements near the tip of a crack which is perpendicuiar to the interlace of a perfect elastic material and an elastic-plastic material with hardening exponent n = 10.
Equations (3.10), (3.13) and (5.3) are the governing equations of the asymptotic fields. Table 2 shows the eigenvalues and initial values of the eigenfunction F(B) and its derivatives at 8 =c0:. Figures 10 and 11 show the angular variations of normalized stresses for different elastic-plastic hardening materials. The stresses are normalized by (a,)-, which is just taken in the plastic region that a, oan exceed unity in the elastic region 0 6 0 G n/2. It can be seen that tractions are continuous acfuss the interface, but there is big jump in the radial stress 6, , The stress s, on the elastic region side is much greater than that on the plastic region side. This means that the interface is really a stress raiser which results in serious stress concentration. Figures 12 and 13 show the angular variation of normalized displacements for different elastic-plastic materials. From these figures, one can see that the crack faces open smoothly without overlap. Acknowledgemem-The author is greatly indebted to Prof. C. F. Shih for his important encouragement and many valuable discussions. The computations repornrd here were carried out on the VAX computer in the Computational Mechamcs Computer Facility, Division of Engineering, Brown University.
538
T. C. WANG
REFJIRENCES [l] M. L. Williams, The stress around a fault or crack in dissimilar media. Bull. Seismol. Sec. Am. 49, 199-204 (1959).
[2] P. G. Cherpanov, The stress state in a heterogeneous plate with slits. Izu. AN SSSR, OTN Mekhan. i Ma.vhin.1,131-137 (1962). [3] F. Erdogan, Stress distribution in bonded dissimilar materials with cracks. J. uppl Mech. 32, 403-410 (1965). [4] G. C. Sib and J. R. Rice, The bending of plates of dissimilar materials with cracks. J. appl. Med. X,477-482(1964). [5] J. R. Rice and G. C. Sih, Plane problems of cracks in dissimilar media. J. appl. Mech. 32, 418423 (1965). [6] A. H. England, A crack between dissimilar media. J. appl. Mech. 32, 400402 (1965). [7] G. C. Sih, P. C. Paris and G. R. Irwin, On crack in rectilinear anisotropic bodies. Inr. J. Fracture 1, 189-302 (1965). [8] G. C. Sih and H. Liebowitx, Mathematical theory of brittle fracture, In Fracture (Edited by H. Liebowitx), Vol II. pp. 115-125 (1968). [9] J. R. Willis, Fracture mechanics of interfacial cracks. J. Mech. Phys. Solids 19,353-368 (1971). [lo] D. L. Clements, A crack between dissimilar anisotropic media. Znt. J. Engng Sci. 9, 257-265 (1971). [l l] D. B. Bogy, Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. J. appi. Mech. 38, 377-386 (1971). [12] G. C. Sih and P. Chen, Cracks in CompositeMaterials, pp. 87-91 (1981). (131 S. S. Wang, Edge delamination in angle-ply composite laminates. AIAA J. 25 256-264 (1984). [14] T. C. Ting, Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. Int. J. Solidr Struct. 22, 965-983 (1986). [ls] J. H. Park and Y. Y. Earmme, Application of conservation integral to interfacial crack problems. Mech. Mater. 5, 261-276 (1986). [16] J. W. Hutchinson, M. Mear and J. R. Rice, 1987, Crack paralleling an interface between disimilar materials. J. uppi. Mech. 54, 828832 (1987). [17J Z. Suoand J. W. Hutchinson, Interface crack between two elastic layers. Harvard University Report, Mech-118 (to be published). 1181J. R. Rice, Elastic fracture mechanics concepts for interfacial cracks. J. uppf. Mech. Jb, 98-103 (1988). [19] J. W. Hutchinson, Mixed mode fracture mechanics of interfaces. Harvard University Report, Mech-139 (1989). [20] C. F. Shih and R. J. Asaro, Elastic-plastic analysis of cracks on bimaterial interfaces: Part I-small scale yielding. J. uppf. Mech. SS, 299-316 (1988). [21] C. F. Shih and R. J. Asaro, Elastic-plastic and asymptotic fields of interface crack. ht. J. Fracture (in press). [22] C. F. Shih and R. J. Asaro, Elasti*plastic analysis of cracks on bimaterial interfaces: Part H-structure of small scale yielding fields. J. appl. Med. (in press). [23] D. M. Parks and E. Zywicz,Elastic/perfectly-plastic small scale yielding at bimaterial interfaces. In Advances in Fracture Research (Edited by K. Salama et ul.), Vol. 4, pp. 3081-3088 (1989). [24] E. Zywin and D. M. Parks, Elastic yield xone around an interfacial crack tip. J. uppl. Med. (in press). [251 J. W. Hutchinson, Singular behavior at the end of a tensile crack in hardening material. J. Mech. P&s. So@ 16,13-31 (1968). [26] J. R. Rice and G. F. Rosengren, Plane strain deformation near crack tip in a power law hardening material. J. Me& Phys. Solti 16, l-12 (1968). [271 K. Wang and T. Wang, Appl. Math. Mech. 9, 791-802 (1987). [28] C. F. Shih, Small scale yielding analysis of mixed mode plane strain crack problems. Fracture analysis. ASTM STP 560, 187-210 (1974). [29] R. M. McMeeking and D. M. Parks, Elastic plastic fracture. ASTM STP 668, 175-194 (1979). [30] C. F. Shih and M. D. German, Znt.J. Fracture 17, 27-43 (1981). (Received 18 May 1989)