Next, the problem of a sufficiently compliant elasto-plastic adhesive between ..... is the length of the plastic zone; dr is the length of the elastic region defined in.
International Journal of Fracture 68: 55-73, 1994. © 1994 KluwerAcademic Publishers. Printed in the Netherlands.
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An asymptotic approach applied to a longitudinal crack in an adhesive layer L E S L I E B A N K S - S I L L S and R A F A E L S A L G A N I K The Eda and Jaime David Dreszer Fracture Mechanics Laboratory, Department of Solid Mechanics, Materials and Structures, The Iby and Aladar Fleishman Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel
Received 15 January 1994; accepted in revised form 6 June 1994 Abstract. The problem of a crack within an adhesive layer which is bonded to two linear elastic half-planes under tensile loading is studied. Two cases are considered. One in which the adhesive is linear elastic and, the second in which it is taken to be elasto-plastic. For the linear elastic layer, the half-planes (adherends) are assumed to be both similar and dissimilar. When the adhesive is considerably more compliant than the adherends, a method of inner and outer asymptotic expansions is employed to determine a relationship between the corresponding stress intensity factors. Expansions are determined in three regions and matched. The inner expansion relates to a region whose distance from the crack tip is much less than the adhesive thickness. The intermediate expansion relates to a region whose size is governed by the decay length of the stress in that part of the adhesive in which its compliancy is significant. The outer expansion relates to a region whose distance from the crack tip is much less than the crack length, for example, but much greater than the adhesive thickness. This method may be employed to determine all field quantities in terms of the outer stress intensity factor. For a layer which is considerably stiffer than the adherends, a similar strategy for solving the problem is sketched. In addition, for dissimilar adherends, energy considerations are employed to verify the relationship between the inner and outer stress intensity factors. It is seen that the two expressions for the stress intensity factors are identical. The problem of oscillatory stress and displacement behavior is addressed. Next, the problem of a sufficiently compliant elasto-plastic adhesive between dissimilar adherends is examined. Matched asymptotic expansions are employed to determine the plastic zone size, as well as the crack tip opening displacement. Small scale yielding is assumed. A Dugdale-Barenblatt type model is employed with the elasticity of the layer accounted for. The yield stress is taken to be constant throughout the plastic zone.
1. Introduction The problem o f cracks along and within adhesive layers has been receiving much attention in the literature. Two papers [1, 2] have addressed the difficulties involved in dealing with interface crack problems; that is, when a crack is along the interface between two dissimilar media. In particular, suggestions were given concerning a logical approach to oscillatory stress and displacement behavior. An examination o f crack path selection in an adhesive layer between two similar adherends was presented in [3]. The crack may propagate within the layer at some height, along the interface between one o f the adherends and the adhesive or alternate between the two interfaces. In [4], experiments were carded out exhibiting the various behaviors treated analytically in [3]. In this investigation, two linear elastic half-spaces, joined by either a linear elastic or an elasto-plastic adhesive o f constant thickness h, loaded in tension are considered. The straight adhesive contains a crack at its center as illustrated in Fig. 1. The crack is chosen to be sufficiently long with respect to the layer thickness h. Plane strain conditions are assumed. In order to predict decohesion of a linear elastic layer, knowledge o f the stress intensity factor is desirable. By considering various length scales, it is often possible to simplify this problem. To this end, an asymptotic approach is proposed. In Section 2.1, the method o f
Leslie Banks-Sills and Rafael Salganik
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f ~I,V1
?
~b,Vb ~t2,V2
Fig. 1. A longitudinal crack in an adhesive layer, joined by two linear elastic half-spaces. An arbitrary tensile loading is applied. For illustration, the adhesive thickness is exaggerated with respect to crack length.
matched inner and outer asymptotic expansions is applied to determine a relationship between the corresponding first order stress intensity factors. The analysis is carried out when the layer is considerably more compliant than the half-spaces. The half-spaces are taken to be both similar and dissimilar. The method may be employed to determine all field quantities in terms of the outer stress intensity factor. A similar strategy for a layer which is significantly stiffer than the adherends is sketched. In order to consider the problem in which the adherends are dissimilar, various parameters employed for interface crack geometry require mention. The Dundurs' parameters [5] are given by c~
/~1 -- E2 E l -1- E 2 '
(1.1a)
where/~i = Ell(1 - u2), i = 1,2 represents the upper and lower adherends respectively, Ei are the Young's moduli and ui are the Poisson's ratios 1 #1(1 - 2 u 2 ) - #2(1 - 2Ul) , /3= ~ #1(1-u2)+#2(1-/-'1)
(1.1b)
where #i are the shear moduli, respectively. The parameter E associated with the singular oscillatory behavior is given by c=lln
1-/3
(1.2)
In Section 2.2, the energy release rate is employed to verify (to the same order of accuracy) the relationship between the inner and outer stress intensity factors for arbitrary linear elastic material properties. An identical relationship is found so that the matched asymptotic solution for other field quantities is seen to be justified.
An asymptotic approach applied to a longitudinal crack in an adhesive layer
/
~1,vi
Y
'1.2,V2
f
-~'
I -J
Ca)
+h
~Y
X
57
~"X
g2,V2
~b,Vb
1
K?
(b)
Fig. 2. Geometry considered for (a) /-length scale and (b) d-length scale problems, the adherends are composed of dissimilar materials. The/-length scale solution K~°) is the applied load for the d-length scale problem.
In Section 3, the problem of an elasto-plastic layer between two elastic, dissimilar adherends is considered. Matched asymptotic expansions are employed as in Section 2.1 for the elastic adhesive. The solution is obtained by assuming a Dugdale-Barenblatt type model with the elasticity of the layer included in the analysis. The yield stress is assumed to be constant within the plastic zone. The plastic zone size and the crack tip opening displacement are determined. Critical conditions at crack propagation are examined.
2. Elastic layer 2.1. ASYMPTOTIC APPROACH The problem of a crack within an adhesive layer as illustrated in Fig. 1 is considered. An asymptotic approach is taken. Three length scales may be distinguished. The first is the length associated with the overall problem geometry, such as crack length l. This is denoted as the/-length scale problem and is depicted in Fig. 2a. Any geometry is possible here and is incorporated into the outer stress intensity factor. The second is the decay length d of the d-length scale solution depicted in Fig. 2b. For lengths greater than d, the adhesive appears to behave rigidly in this scale. Finally, the third length scale is that of the adhesive thickness h, which gives rise to the h-length scale problem. It will be assumed that h 0 (Fig. 3b). As a first approximation, because h 0 (Fig. 4). In addition, as a result of the high adhesive compliance, (2.5) is satisfied without the arguments given and the crack faces may be assumed stress free. Further, appropriate boundary conditions are applied in the far field. After the problem has been solved with these boundary conditions, one may estimate the shear stress within the adhesive. For either a perfect or imperfect bond at the interface between the adhesive and adherends, the shear stress may be estimated through ax~ within the adhesive by means of the equation of equilibrium in the x-direction in that material. Since
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ax~ > 0(1), the adherends are essentially rigid; so that the problem is reduced to the h-length scale problem in Fig. 5. Consider next the case of an adhesive which is much stiffer than the adherends. The asymptotic method of solution to this problem is only sketched here with the restriction that the adherends be of the same material. The general scheme of the asymptotic analysis remains similar to that of the preceding examples. The l-length scale solution is the same. Referring to Fig. 3b for the d-length scale problem, some of the boundary conditions change. On the line ahead of the crack tip, the y-direction displacement v = 0; as before, the shear stress a~ v = O. Along the crack faces, the shear stress remains zero; but, the flexural rigidity of half of the adhesive must be accounted for. The normal stress will no longer be zero. Thus, a particular case of the Wiener-Hopf equation formulated in [11] may be solved. From a solution of this problem, the bending moment on the upper layer as x ~ 0 is determined. This quantity plays the role of the matching parameter for the inner or h-length scale problem. Again symmetry is employed. This final problem (the h-length scale problem) in the asymptotic analysis, is that of a symmetric double cantilever beam (see Fig. 6); the bending moment obtained from the preceding intermediate d-length scale problem is applied to the beam. Since the layer is assumed to be sufficiently stiff, for the first order approximation, deformational resistance of the adherends is not accounted for in this inner problem. It may be noted that in solving the previous problems in the h-length scale with the very compliant adhesive, the adherends were considered to be rigid. In this case, the adherends are assumed to have no stiffness. The solution for the stress intensity factor of the double cantilever beam under plane stress conditions may be found in [8, p. 231-232]. For plane strain conditions, this solution may be determined easily. Finally, the d-length scale and/-length scale solutions must be matched. In this section, methods were described for determining the field quantities by means of first order asymptotic expressions. In this manner, relations between inner and outer stress intensity factors were obtained for a sufficiently compliant adhesive. In the next section, these results are verified by means of energy considerations.
2.2. ENERGY CONSIDERATIONS The problem under consideration is shown in Fig. 1, where a characteristic length, such as crack length 1 is assumed to be large compared to the adhesive thickness h. Both adhesive and adherends are assumed to be linear elastic with the relationship between their elastic properties arbitrary. The only restriction imposed is that the characteristic length of the oscillatory behavior near the cut tip A is assumed to be small compared to h. Thus, the shear stresses everywhere throughout the adhesive may be neglected. Hence, only mode I stress intensity
An asymptotic approach applied to a longitudinal crack in an adhesive layer
65
factors will appear in both the/-length and h-length scale problems. By means of the energy release rate, a relationship between the inner and outer stress intensity factors is determined. This type of approach has been employed for other problems (see [12]). Far from the crack tip, in a region of characteristic distance R, such that h I. The derivatives of the complex displacement vector with respect to x on the upper crack face may be determined from (A.1), (A.4) and (A.9) as
(Ou + Ov+~ : 1 2m \ o~ + i o~ ] a~(,¢2+ 1) + m(,¢l + 1)
×
+ .,)F;]
(A.IO)
In order to obtain the analogous quantity at the lower crack face, it is sufficient to replace the index 1 by 2 and 2 by 1 in (A.10), so that e becomes - e , replace z with ~,, K with h', and take the complex conjugate of the left hand side of (A. 10). Subtracting this result from (A. 10) yields the derivatives of the complex crack opening displacement as
O(u+ - u-) O(v + - v-) (~1 2 1 l~2+ 1) Ox +i Ox - \ #l + #2 ×
1 47ri x ( 1 / 2 - i Q ( l - x)(1/2+ie)
fo I p(t) t(1/z-ie)(l - t) (l/2+iQ ~ dt
+-~
---
#2
p(x).
(A.11)
For the case in which the stresses are applied to the crack faces, the last term in (A. 11) produces a jump in the displacement component tangent to the crack. This results from the fact that the transverse deformation caused by equal normal stresses of opposite sign are different for dissimilar materials. For a homogeneous material, this effect does not occur. This effect is ignored here when determining the plastic zone size adjacent to the crack tip at x = I. Next the stress distribution for x > l, y = 0 is determined. From (A.1), (A.4), (A.5), (A.7) and (A.8), one may obtain this distribution as
avu - ir~u = cosh r e 7r
1
X(1/2--ie)(X -- l)(1/2+iQ
fo I p(t) tO/2-i')(l - t) (l/2+ie) t ---x d t . (A.12)
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In particular, for sufficiently small s = x - l, (A.12) yields
ayy -- iTxy = (K1 + iK2)(27rs)-(1/2)s ie
(A.13)
where
Kl + iK2 = -
~ F f coshTrCfol
l(1/2+i,)
( t ) (1~2+it) p(t) ~ dt
(A.14)
is the complex stress intensity factor. Since compressive stresses applied to the crack faces are required to open the crack, a negative p(t) corresponds to crack opening. The contribution to the stress intensity factor from the plastic zone may easily be found from (A.14) with e = 0 as
K(P)=-2ay ~ .
(A.15)
It is seen that taking e = 0 yields the same result obtained with the Dugdale model for a homogeneous body. By superposition, the total stress intensity factor is given as
KI = K~°) + K~ p) .
(A.16)
Employing (A.16) to match the/-length and d-length scale problems leads to the expression in (3.4) for the stress at the crack tip. Next, the crack tip opening displacement 6 near x = I is determined. Again, the superposition principle is employed so that
6 = 6o + 6p
(A.17)
where 6o = v + - v~- is the crack tip opening displacement resulting from the applied stress and 6p = v + - v~- is the crack tip opening displacement caused by plastic deformation. From the near crack tip asymptotic expansion ([12], Eqn. (2.27)), the resulting expression for the first case is found to be, again assuming that e = 0,
•
(AA8)
The second case may easily be determined from (A.11) by assuming that dp
