Stress transfer and stiffness reduction in orthogonally cracked laminates

Mechanics of Materials 31 (1999) 303±316

Stress transfer and sti€ness reduction in orthogonally cracked laminates Wael G. Abdelrahman 1, Adnan H. Nayfeh

*

Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221-0070, USA Received 9 December 1998

Abstract A micromechanical continuum mixture model is constructed for the stress transfer and residual sti€ness in orthogonally cracked laminates. According to this technique, the discrete composite behavior is replaced by that of a higher order continuum. A rational construction of an alternative set of coupled ®eld equations that automatically satisfy all interface conditions leads to simple coupled governing equations for the total composite. This construction procedure is based on some through-thickness approximate distributions for some of the ®eld variables. Once the stress free crack surface boundary conditions are imposed and the stress ®eld components are obtained, we proceed to derive expressions for the residual sti€ness of the damaged composite. This is presented in the form of Young's and shear moduli. Treating three-dimensional problems and then identifying the results for the two-dimensional case, it will be shown that much of the earlier results in the literature are obtained as special cases of our work. Con®dence in the modeling is further enhanced by showing good agreement with experimental measurements available in the literature. Ó 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction The stress redistribution and associated sti€ness reduction that result in composites as a consequence of the initiation or perpetuation of damage have received sustained research interest in recent years. This results as a perturbation to the stress ®eld that requires satisfying the various discontinuities in the damaged regions. In laminated composites consisting of individual ®ber reinforced lamina stacked with various in-plane orientations, experiments have shown that damage develops in the form of parallel cracks along the

* 1

Corresponding author. E-mail: [email protected]. E-mail: [email protected].

®ber directions. Initially, cracking starts in laminae whose ®bers are directed normal to the loading direction. At later stages, cracks may also develop in laminae whose ®bers are not necessarily normal to the loading direction. These cracks are largely responsible for the stress redistribution and the associated sti€ness reduction in the composite system. The cracked [0°/90°]s cross-ply laminate received repeated attention because of its common use. To facilitate our discussion, we depict in Fig. 1 a schematic of this con®guration showing the geometry, cracking and typical loading situations. The study of the laminate shown in Fig. 1(a) with cracked 90° lamina was carried out experimentally by Highsmith and Reifsnider (1982). In an attempt to model the experimental data,

0167-6636/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 9 9 ) 0 0 0 0 2 - 2

304

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

Fig. 1. Schematic of a [0°/90°]s laminate with an array of transverse cracks under axial loading (b) a [0°/90°]s laminate with an orthogonal array of transverse cracks under biaxial loading.

Reifsnider and Talug (1980) constructed analytic models based on shear lag concepts. Neglecting the shear stress distribution, except in a very narrow boundary region, they were able to devise a simple method, whose relative accuracy was con®rmed by the experimental measurements. The same problem was treated later on, using the variational approach, by Hashin (1985) for both cases of normal as well as shear loadings. By using the principle of minimum complementary energy, Hashin was able to obtain approximate solutions for the stress ®eld distributions and also the lower bounds for the longitudinal and shear sti€nesses. In his work, Talreja (1985a, b) represented transverse crack type of damage, among other types, by a set of vector ®elds. He then proceeded to derive the elastic response following attainment of damage state. In a later work (1985) he used this continuum damage characterization to predict the change in the residual elastic moduli from crack initiation to crack saturation. His modeling involved unknown damage constants which were determined from experiments. For an orthotropic laminate, the method required the measuring of Young's modulus and Poisson's ratio at undamaged state and at a state for which the

crack density is known. Other available modelings of this problem include the continuum damage models of Allen et al. (1987) and the self-consistent scheme approximation of Laws et al. (1983). In all of the above studies, a two-dimensional model was required and analyzed. Another cross-ply composite con®guration that was studied is shown in Fig. 1(b). The laminate contained an orthogonal array of cracks in the 0° as well as in the 90° laminae. Modeling of the stress distribution and associated sti€ness reduction in this con®guration requires three-dimensional analysis. This has been attempted numerically by Highsmith and Reifsnider (1986), analytically by Hashin (1987) and more recently by McCartney (1992). Hashin's work constituted an extension of his earlier work (1985). McCartney, however, used a di€erent approach to the threedimensional problem by directly integrating the ®eld equations. Since he used the same assumptions used by Hashin, he was able to reproduce the variational approach results. In this paper, we introduce an alternative technique to study these classes of problems. It is based on a continuum mixture modeling of the micromechanical behavior of composites. This technique was developed in the early seventies mainly for the study of the dynamic (dispersive) behavior of laminated and ®brous composites (see for example Nayfeh and Nassar (1978), Nayfeh (1977, 1978), Nayfeh and Loh (1977), Nayfeh and Gurtman (1974), and Hegemier et al. (1973)). According to this method of analysis, the discrete composite behavior is replaced by that of a higher order continuum. Rather than solving the ®eld equations for each composite component subject to satisfying all boundary and interface conditions, a rational construction of an alternative set of coupled ®eld equations that automatically satisfy all interface conditions is presented. This leads to simple coupled governing equations for the total composite which retain the integrity of each component but allow them to coexist under some derived interfacial transfer (interaction) terms. Here, information on the distribution of displacements and stresses within each component are readily available. The construction procedure is inevitably based on some approximating assumptions. These

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

are mainly concerned with the introduction of through-thickness approximate distributions for some of the ®eld variables which automatically satisfy symmetry and interface conditions. As shall be seen below, these types of assumptions have also been adopted in the more recent treatments of Hashin and McCartney. The present work has been motivated by our recent application of the proposed technique (Nayfeh and Abdelrahman (1998)) to the concentric cylindrical model previously analyzed by McCartney (1989). In our paper, we developed a micromechanical model for the prediction of the thermomechanically induced stress distribution in the concentric cylindrical system in the presence or absence of damage. Con®dence in our modeling was gained when it identically reproduced all of the results reported by McCartney. For the laminated problem at hand, we follow the same approach we used for the concentric cylinder model. Once again, we use the averaging technique as a means to facilitate our analysis and to manipulate some of the algebraic expressions. In no where we shall ®nd it necessary to satisfy any required equation, boundary or interface condition based on its average. With reference to the illustrations of Fig. 1, we choose to treat the threedimensional problem and then identify the results for the two-dimensional case from this general solution. Damage will be simulated in the form of cracks having stress free faces. The results obtained by Hashin (1985, 1987) and McCartney (1992) will all be obtained as special cases of our work. The present work can be extended to cover cases involving interaction of delamination with transverse cracks as well as the case of dynamic loading. This, however, will be presented in a separate paper. 2. Formulation of the problem Consider, once again, the symmetric laminate shown in Fig. 1(b), intended to simulate a [0°/90°]s layup con®guration. The laminate's total thickness is 2h such that the thickness of the 0° and 90° laminae, referred to as laminae 2 and 1, are t2 and 2 t1 , respectively. This ®gure displays an orthogo-

305

nal array of periodic cracks that are normal to the x- and y-directions. Furthermore, we assume that the laminate is loaded only in the x±y plane. This is the geometric con®guration of the three-dimensional problem studied by Hashin (1987) and McCartney (1992). From the symmetry of the problem, it suces to analyze only one half of the laminate, say the one occupying the region 0 < z < h. Although it is inherently assumed that each of the laminae possess transverse isotropy in the x±y plane, in our subsequent analysis, we shall relax this condition and treat each component as having orthotropic symmetry. Results for transverse isotropy can then be obtained by merely imposing appropriate conditions on the properties. Under these conditions, the behavior of the laminate is described by the following three-dimensional equilibrium equations: orx orxy orxz ‡ ‡ ˆ 0; ox oy oz

…1†

orxy ory oryz ‡ ‡ ˆ 0; ox oy oz

…2†

orxz oryz orz ‡ ‡ ˆ 0; ox oy oz

…3†

and the constitutive relations rx ˆ C11

o ux o uy o uz ‡ C12 ‡ C13 ; ox oy oz

…4†

ry ˆ C12

o ux o uy o uz ‡ C22 ‡ C23 ; ox oy oz

…5†

o ux o uy o uz ‡ C23 ‡ C33 ; ox oy oz   o uy o uz ryz ˆ C44 ‡ ; oz oy   o ux o u z rxz ˆ C55 ‡ ; oz ox   o ux o uy ‡ : rxy ˆ C66 oy ox rz ˆ C13

…6† …7† …8† …9†

which hold for both laminae 1 and 2. Here we ®nd it more convenient to use the stress±strain in contrast with McCartney's choice of strain±stress to

306

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

describe the constitutive relations. The above ®eld equations are supplemented with the following symmetry, continuity and boundary conditions: Symmetry conditions: u…1† z …x; y; 0† ˆ 0;

…10a†

r…1† xz …x; y; 0†

ˆ 0;

…10b†

r…1† yz …x; y; 0† ˆ 0:

…10c†

Continuity conditions (on z ˆ t1 ): …2† u…1† x …x; y; t1 † ˆ ux …x; y; t1 †;

…11a†

…2† u…1† y …x; y; t1 † ˆ uy …x; y; t1 †;

…11b†

…2† u…1† z …x; y; t1 † ˆ uz …x; y; t1 †;

…11c†

…2† r…1† z …x; y; t1 † ˆ rz …x; y; t1 †;

…11d†

r…1† xz …x; y; t1 †

ˆ

r…2† xz …x; y; t1 †;

…11e†

ˆ

r…2† yz …x; y; t1 †;

…11f†

r…1† yz …x; y; t1 †

Boundary conditions (on z ˆ h): r…2† z …x; y; h† ˆ 0;

…12a†

r…2† xz …x; y; h†

ˆ 0;

…12b†

r…2† yz …x; y; h† ˆ 0;

…12c†

(5), are averaged across lamina thickness according to … †

… †

…1†

…2†

1 ˆ t1 1 ˆ t2

r…2† y …x; b; z† ˆ 0; or…2† y oy

ˆ 0:

†

…1†

dz;

…14a†

…

†

…2†

dz;

…14b†

0

Zh t1

0

1 …i† …i† or or y xy A ˆ …ÿ1†i r ; ‡ ni h @ yz oy ox

i ˆ 1; 2;

…16†

and the constitutive relations …1†

…1† r x ˆ C11

…2†

…2† r x ˆ C11

…13a† …1†

…13b†

…

and if the symmetry and boundary conditions (10a)±(10c) and (12b) and (12c) are used, one obtains the averaged equations of equilibrium 0 1 …i† …i†  o r  or xy A i ˆ …ÿ1† rxz ; …15† ni h @ x ‡ ox oy

Free crack surface conditions: r…1† x …a; y; z† ˆ 0; or…1† x ˆ 0; ox xˆa

Zt1

…1† r y ˆ C12

…2†

…1† o u…1† y …1† o u …1† x ‡ C12 ‡ C13 uz =n1 h; ox oy

…17a†

…2† o u…2† y …2† o u …2† x ‡ C12 ‡ C13 …u0z ÿ uz †=n2 h; ox oy …17b† …1† o u…1† y …1† o u …1† x ‡ C22 ‡ C23 uz =n1 h; ox oy

…18a†

…2† o u…2† y …2† o u …2† x ‡ C22 ‡ C23 …u0z ÿ uz †=n2 h; ox oy …18b†

…13c†

…2† r y ˆ C12

…13d†

where ni ˆ ti =h, i ˆ 1, 2, and rxz , ryz and uz are the interface shear stresses and transverse displacement; namely:

yˆa

3. Method of solution 3.1. Averaging Following the procedure outlined in the references by Nayfeh et al., if Eqs. (1), (2), (4) and

…2† rxz ˆ r…1† xz …x; y; t1 † ˆ rxz …x; y; t1 †;

…19a†

…2† ryz ˆ r…1† yz …x; y; t1 † ˆ ryz …x; y; t1 †;

…19b†

…2† uz ˆ u…1† z …x; y; t1 † ˆ uz …x; y; t1 †;

…19c†

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

307

and u0z ˆ u…2† z …x; y; h† is the transverse displacement of the upper surface.

these approximations into Eqs. (15)±(17a), (17b), (18a) and (18b) leads to

3.2. Approximate distributions

ni

…i† or i x ˆ …ÿ1† n1 A1 ; ox

ni

…i† or y i ˆ …ÿ1† n1 A2 ; oy

So far we have eliminated the dependence on the z-direction of the ®eld variables and also satis®ed some of the symmetry, interface and outer boundary conditions. In return, we have introduced the four still unknown interaction terms rxz , ryz , uz and u0z . If we are able to express these interaction functions in terms of the in x-y plane …i† …i† …i† …i† stresses and displacements r x , r y , u x , u y , we would successfully reduce the problem into a quasi-two-dimensional one. This cannot, however, be done without adopting appropriate throughthickness approximations for the transverse displacement and the shear stress components. These take the forms of either linear or quadratic functions in z with their coecients adjusted to satisfy the required symmetry, interface and boundary conditions. Details of this procedure can be found in all of the references quoted above by Nayfeh et al. For the present situation, the appropriate expressions are: r…1† xz ˆ A1 …x; y†z; r…2† xz ˆ

n1 A1 …x; y†…h ÿ z†; n2

r…1† yz ˆ A2 …x; y†z; r…2† yz ˆ

n1 A2 …x; y†…h ÿ z†; n2

…2† r…1† xy ˆ rxy ˆ 0;

…20a† …20b† …21a† …21b† …22†

…23a† u…1† z ˆ B1 …x; y†z;   n1 B1 …x; y† ÿ B2 …x; y† …h ÿ z†; u…2† z ˆ t2 B2 …x; y† ‡ n2 …23b† chosen to automatically satisfy the symmetry conditions (10a)±(10c), the interface continuity conditions (11c)±(11f) and the boundary conditions (12b) and (12c). These types of approximations are also the ones adopted by Hashin (1985, 1987) and McCartney (1992). Introducing

…24† i ˆ 1; 2;

…25†

and the constitutive relations …i†

…i† r x ˆ C11

…i†

…i† r y ˆ C12

…i† o u…i† y …i† o u …1† x ‡ C12 ‡ C13 Bi ; ox oy

…26†

…i† o u…i† y …i† o u …i† x ‡ C22 ‡ C23 Bi : ox oy

…27†

We immediately notice that Eqs. (24) and (25) satisfy …1† …2† n1 r x ‡ n2 r x ˆ const ˆ Px ;

…28a†

…1† …2† n1 r y ‡ n2 r y ˆ const ˆ Py ;

…28b†

and that Eq. (9) becomes irrelevant. 3.3. Solving for A1 , A2 , B1 and B2 It now remains to solve for the functions A1 …x; y†, A2 …x; y†, B1 …x; y† and B2 …x; y† in terms of …i† …i† …i† …i† r x , r y , u x , u y . We shall attempt to do so by using whatever appropriate algebraic manipulations that guarantee satisfying the remaining ®eld equations, interface and boundary conditions. These consist of Eqs. (3), (6)±(8), (11a), (11b), (11d) and (12a). To this end, we start by satisfying the shear constitutive relations (7) and (8). After some algebraic manipulations, detailed in Appendix A, we obtain the two important relations ! h2 n1 n2 …2† …1† 2n1 ‡ …2† A1 ux ÿ ux ˆ …1† 6 C55 C55 ! o B1 o B 2 ÿ n22 ÿ n1 …2 ‡ n2 † ; …29† ox ox u…2† y

ÿ

u…1† y

h2 ˆ 6

2n1

n1 …1†

C44

‡

n2 …2†

C44

! A2

! o B1 2 o B2 ÿ n2 ÿ n1 …2 ‡ n2 † : oy oy

…30†

308

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

Next, we turn our attention to the transverse normal stresses. Following the above procedure, we substitute from Eqs. (20a) and (21a) into Eq. (3), multiply the resulting equations by z and integrate according to Eq. (14a) and get   n21 h2 o A1 o A2  …1† z ‡ ‡ ˆ 0: …31a† rz ÿ r 3 ox oy Similarly, substituting from Eqs. (20b) and (21b) into Eq. (3), multiplying the resulting equation by …h ÿ z† and integrating as per Eq. (14b) leads to   n 1 n 2 h 2 o A 1 o A2 …2† ‡ ˆ 0; …31b† ÿ rz ÿ r z 3 ox oy where, from Eq. (11d), rz ˆ r…1† z …x; y; t1 † ˆ …x; y; t † is the common interface transverse r…2† 1 z stress. Also, by direct integration from t1 to h of Eq. (3) for the second lamina we obtain the following expression for rz :   n1 n2 h2 o A1 o A2 ‡ : …32† rz ˆ 2 ox oy In arriving at the relation (32) we also satis®ed the stress free boundary condition (12a). Upon using z from Eqs. (31a) and (31b), Eq. (32) to eliminate r we get   n1 …2 ‡ n2 † h2 o A1 o A2 …1† z ˆ ‡ ; …33a† r 6 ox oy   n1 n2 h2 o A1 o A2 …2† z ˆ r ‡ : …33b† 6 ox oy Next, if we specialize the constitutive Eq. (6) for the ®rst and second laminae, and average the resulting equations according to Eqs. (14a) and (14b) we get …i†

…i† r z ˆ C13

…i† o u…i† y …i† o u …i† x ‡ C23 ‡ C33 Bi ; ox oy

i ˆ 1; 2: …34†

We now need to eliminate the strain components. This can be done by substituting for uz from Eqs. (23a) and (23b) into Eqs. (26) and (27) and solving for the four unknown strains as o u…i† x ox

ˆ

1 …i† …i† …i† …i† ‰ C … r…i† ÿ C13 Bi † ÿ C12 … r…i† y ÿ C23 Bi † Š; E…i† 22 x …35†

o u…i† 1 y …i† …i† ˆ …i† ‰ÿ C12 … r…i† x ÿ C13 Bi † oy E …i†

…i†

r…i† ‡ C11 … y ÿ C23 Bi † Š;

…36†

where …i†

…i†2

…i†

E…i† ˆ C11 C22 ÿ C12 ; i ˆ 1; 2: Substituting these expressions into Eq. (34) and …2† …1† and r eliminating r z z , using Eqs. (33a) and (33b), we ®nally solve for B1 (x,y) and B2 (x,y) as 0 0 1 2 …1†  o r …1† 1 @ n1 …2 ‡ n2 †h2 @ o2 r y x A ‡ B1 …x; y† ˆ …1† 6 o x2 o y2 F 1 …1† …1† A …1† y ; ‡ G…1† r r x ‡H

…37†

0 0 1 2 …1†  o r …1† 1 @ n1 n2 h2 @ o2 r y A x ‡ B2 …x; y† ˆ …2† 6 o x2 o y2 F 1 …2† …2† A …2† y : r ‡ G…2† r x ‡H

…38†

Here …i†2

…i†

…i†

…i†2

…i†

…i†2

F …i† ˆ…C11 C23 ‡ C22 C13 ‡ C33 C12 …i†

…i†

…i†

ÿ C11 C22 C33 †=E…i† ; …i†

…i†

…i†

…i†

…i†

…i†

…i†

…i†

G…i† ˆ …C13 C22 ÿ C12 C23 †=E…i† and H …i† ˆ …C11 C23 ÿ C12 C13 †=E…i† ;

i ˆ 1; 2:

…i† Substituting o u…i† x =o x and o u y =o y, i ˆ 1, 2 from Eqs. (35) and (36), with B1 (x,y) and B2 (x,y) as given by Eqs. (37) and (38), into the x-derivative of Eq. (29) and the y-derivative of Eq. (30) we ®nally arrive at the following two coupled fourth  …1† …1† order partial di€erential equations in r x and r y : c1

c1

…1† …1† o4 r o2 r …1† …1† o4 r o2 r y y x x ‡ c1 2 2 ‡ c2 ‡ c3 4 2 ox ox oy ox o x2 …1† o2 r y …1† …1† ‡ c4 ‡ c5 r x ‡ c6 r y ‡ c7 ˆ 0; o y2 …1† …1† o4 r o2 r …1† …1† o4 r o2 r y y x x ‡ c ‡ c ‡ c 1 8 3 o y4 o x2 o y 2 o y2 o x2 …1† o2 r x …1† …1† ‡ c4 ‡ c9 r y ‡ c6 r x ‡ c10 ˆ 0; o y2

…39†

…40†

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

where the coecients ci , i ˆ 1; 2; . . . ; 10 are given in Appendix B. In arriving at these equations, we also used Eqs. (24) and (25) to eliminate A1 and A2 , respectively, and Eqs. (28a) and (28b) to eliminate …2† …2† r x and r y . The two coupled Eqs. (39) and (40) …1† …1† can now be solved for r x …x; y† and r y …x; y† using the free crack surface boundary conditions (13a)± (13d).

average Eq. (42b) over x between the two neighboring cracks from x ˆ ÿa to x ˆ a subject to the crack boundary conditions: d r…1† y ˆ 0; at y ˆ b; dy d r…1† x ˆ 0; at x ˆ a; dx we directly obtain c1

4. Discussion and applications The quasi-two-dimensional description, represented by the two-coupled di€erential equations (39) and (40), is general. Up to this point, both of the in-plane normal stresses can be functions of x and y. We cannot a priori see a justi®cation for …1† assuming r x , for example, to be dependent only on x as adopted by Hashin, or to have a split dependence on x and y in the form adopted by McCartney. Since the present model is general, it reduces to both McCartney's and Hashin's models if we further adopt their assumptions. If we assume the special form of the normal stresses, namely

309

…1† …1† d4 r d2 r x x …1† ‡ c ‡ c5 r 2 x ‡ c7 4 dx d x2 Zb …1† ‡ c6 =2b r y ˆ 0;

…43a† …43b†

…44†

ÿb

c1

…1† …1† d4 r d2 r y y …1† ‡ c ‡ c9 r 8 y ‡ c10 d y4 d y2 Za …1† ‡ c6 =2a r x ˆ 0:

…45†

ÿa

Eqs. (44) and (45) correspond to the coupled system of equations obtained by Hashin (1987).

…1† r x ˆ X1 …x† ‡ Y1 …y†;

…41a†

5. The two-dimensional case

…1† r y ˆ X2 …x† ‡ Y2 …y†;

…41b†

The model developed in Section 3 is valid for laminates of ®nite extension in both the x and y directions, and can be subjected to biaxial loading. In order to compare with the earlier results which dealt with the dimensional laminates (in®nite in the y-direction), we specialize Eq. (39) by setting

which is the form used by McCartney (1992), we ®nd that the mixed derivative terms in Eqs. (39) and (40) disappear, and thus we recover the form of the equations in section 11.2 of McCartney (1992). Furthermore, if we make the additional assumption Y1 …y† ˆ X2 …x† ˆ 0, we further ®nd that 2 2 …1† …1† x =o y 2 ˆ 0, resulting in o2 r y =o x ˆ o r …1† d4 r x c1 d x4

‡

…1† d2 r x c2 d x2

‡ c7 ˆ 0;

…1† d2 r y …1† …1† ‡ c4 ‡ c5 r x ‡ c6 r y d y2 …42a†

…1† …1† d4 r d2 r …1† d2 r y y x …1† …1† ‡ c ‡ c ‡ c9 r c1 8 3 y ‡ c6 r x d y4 d y2 d x2 ‡ c10 ˆ 0: …42b† If we now average Eq. (42a) over y between the two neighboring cracks from y ˆ ÿb to y ˆ b, and

o=oy ˆ 0;

…1† r y ˆ 0

…46†

and get …1† …1† o4 r o2 r x x …1† ‡ c2 ‡ c5 r …47a† x ‡ c7 ˆ 0: 4 ox o x2 The coecients in Eq. (47a) resulting from the three-dimensional analysis might be more complicated algebraically from the ones obtained directly from the two-dimensional analysis as ! …2† 2 …1† ÿn1 h4 n1 …2 ‡ n2 † C11 n32 C11 ‡ …2† ; c1 ˆ 36 K …1† K c1

310

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316 …1†

n1 h2 c2 ˆ 3 …1†

c5 ˆ ÿ

…2†

ÿ…2 ‡ n2 †C13 n2 C13 n1 n2 ‡ …2† ‡ …1† ‡ …2† …1† K K C55 C55

! ;

…2†

C33 n1 C33 ÿ ; …1† K n2 K …2† …2†

c7 ˆ

Px C33 : n2 K …2†

…47b†

Despite this possible formal di€erence, the results are essentially identical and have been veri®ed numerically. It is interesting to note that these coecients can also be identi®ed from those in …1† …2† Appendix B by formally setting C12 ˆ C12 …1† …2† ˆ C23 ˆ C23 ˆ 0. Eq. (47b) is identical, at least numerically, with Eq. (2.33) in the earlier work of Hashin (1985). The relevant stress free crack surface boundary conditions in this case are o r…1† …1† x x …a† ˆ 0; ˆ 0: …47c† r ox xˆa

which satisfy the zero shear stress symmetry condition at z ˆ 0 as well as the free surface boundary condition at z ˆ h. Substituting these approximations into Eq. (48), and integrating the resulting expressions as per Eqs. (14a) and (14b), for laminae 1 and 2, respectively, we get …1† or xy ˆ ÿA3 ; ox n2

…52a†

…2† or xy ˆ n1 A3 : ox

…52b†

Similar to the normal loading case, we notice that Eqs. (52a) and (52b) satisfy the equilibrium of the internal stresses with the external loading, namely …1† …2† n1 r xy ‡ n2 r xy ˆ const ˆ s0 :

…53†

If we now specify Eq. (49) for laminae 1 and 2, and integrate the resulting equations as per Eqs. (14a) and (14b) we obtain uy ÿ u…1† y ˆ

n21 h2

…1†

3C44

A3 ;

ÿn1 n2 h2

…54a†

6. Case of shear loading

uy ÿ u…2† y ˆ

In this section, we show the applicability of the proposed technique to di€erent loading situations by considering the case of a composite laminate under shear loading. This situation was also analyzed by Hashin (1985). This type of loading implies pure in-plane shear response. The relevant equilibrium and constitutive equations for the pure shear become

where again uy is the interface value of uy . Eqs. (54a) and (54b) can be combined into a single equation by eliminating the interfacial displacement uy to yield ! n1 h2 n1 n2 …2† …1† uy ÿ uy ˆ ‡ …2† A: …55† …1† 3 C44 C44

orxy oryz ‡ ˆ 0; ox oz ryz ˆ C44 rxy ˆ C66

…48†

o uy ; oz

…49†

o uy : ox

…50†

Following the same procedure of Section 3, we assume the following approximate expressions of the shear stress distributions in the two laminae: r…1† yz ˆ A3 …x; y† z;

r…2† yz ˆ

n1 A3 …x; y†…h ÿ z†; n2

…51†

…1†

3C44

A3 ;

…54b†

It remains to satisfy Eq. (50) for both laminae. To do so, we specify this equation for laminae 1 and 2, and integrate the resulting expressions as per Eqs. (14a) and (14b), to get …1† r xy ˆ C66

o u…1† y ; ox

…56a†

o u…2† y : …56b† ox Finally, If we substitute from Eqs. (52a), (56a), (56b) into the derivative of Eq. (55), and using Eq. (53) to eliminate uy we arrive at the second order di€erential equation …2† r xy ˆ C66

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

n1 h2 3 ‡

n1

n2

C44 s0

C44

‡ …1†

!

…2†

…1† o2 r 1 xy ÿ 2 ox n2

n1

n2

C66

C66

‡ …2†

!

…1†

ˆ 0;

…2†

n2 C66

…1† r xy …57a†

which completely describes the behavior of the laminated composite under shear loading. By imposing the transverse isotropy symmetry condition, this equation becomes identical with that of Hashin (1985). The relevant stress free crack boundary conditions in this case are …1† r xy …a† ˆ 0:

given by Eq. (47b). Substituting this solution back into Eq. (35) for lamina 2, and using Eq. (38) to eliminate B2 , we get the axial strain in the undamaged lamina as o u…2† x ˆ e1 cosh …ax† cos …bx† ox ‡ e2 sinh …ax† sin …bx† ‡ e3 ;

7.1. Longitudinal sti€ness reduction

e1 ˆ

n1 6n2 K …2†

e2 ˆ

a ˆ …c5 =c1 †

1=4



 1 2 arctan …4c5 c1 =c2 ÿ 1† ; cos 2

and b ˆ …c5 =c1 †

1=4

 sin

 1 2 arctan…4c5 c1 =c2 ÿ 1† ; 2

and where d1 and d2 are functions of constants that are obtained from the crack boundary conditions, namely dr…1† at x ˆ L=2 ˆ a: …59† ˆ x ˆ 0; dx Since we are dealing with a two-dimensional case, the coecients c1 , c3 , c5 c7 have the simple form

…1† r x

…2†

…2†

n1 6n2 K …2†

…2†

…2†

…1†

e3 ˆ Px C33 C33 =…n1 K …1† C33 ‡ n2 K …2† C33 †: Integrating this axial strain from x ˆ 0 to x ˆ L/ 2 ˆ a, we get the longitudinal displacement under the edge loading as u…2† x …a† ˆ e1 I1 ‡ e2 I2 ‡ e3 a;

…61†

where I1 ˆ

…a2

1 ‡ b2 †

 ‰a sinh …aa† cos …ba† ‡ b cosh…aa† sin …ba†Š;

ÿ c7 =c5 c1 ; where

…2†

 ‰n22 h2 C13 …d2 …a2 ÿ b2 † ÿ 2d1 ab† ÿ 6d2 C33 Š;

…1† r x ˆ d1 cosh …ax† cos…bx† ‡ d2 sinh …ax† sin …bx† …58†

…2†

 ‰n22 h2 C13 …d1 …a2 ÿ b2 † ‡ 2d2 ab† ÿ 6d1 C33 Š;

…1†

The continuum model developed in Section 3 for three dimensional laminates, and speci®ed in Section 5 for the two dimensional case, can be utilized to study the in¯uence of transverse cracks on the longitudinal sti€ness reduction as represented by the decrease in Young's modulus. For the two-dimensional case, standard procedures lead to the following solution of the di€erential equation (47a):

…60†

where

…57b†

7. Sti€ness reduction in the two-dimensional case

311

I2 ˆ

…a2

1 ‡ b2 †

 ‰a cosh…aa† sin …ba† ÿ b sinh …aa† cos …ba†Š: The total reduced value for Young's modulus, due to the existence of transverse cracks, is therefore Eeff ˆ

Px a : u…2† x …a†

…62†

7.2. Shear sti€ness reduction For cases involving shear loading of a composite laminate with a parallel array of cracks perpendicular to the x-axis, the system behavior is described by the second order di€erential Eq. (57a). This equation has the solution

312

…1† r xy ˆ

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316 …1†

s0 C66

…1†

…2†

…n1 C66 ‡ n2 C66 †

‰1 ÿ cosh …mx†= cosh …ma†Š; …63†

where v u u 3C …1† C …2† …n1 C …1† ‡ n2 C …2† † 44 44 66 66 mˆt : …1† …2† …2† …1† n1 n2 C66 C66 …n1 C44 ‡ n2 C44 † Using this solution together with Eq. (53) we obtain the distribution of the shear stress in the sec …2† ond lamina, r xy , substituting this distribution into Eq. (56b) and integrating the resulting equation x ˆ 0 to x ˆ a, we obtain the shear displacement at the crack surface, u…1† y …a†, as u…1† y …a† ˆ

…2†

…1†

s0 a …n2 C66 m a ‡ n1 C66 tanh …ma†† …2†

…1†

…2†

n2 C66 m a …n1 C66 ‡ n2 C66 †

;

…64†

The total degraded shear sti€ness is thus calculated from Geff ˆ

…2†

…1†

…2†

n2 C66 m a …n1 C66 ‡ n2 C66 † …2†

…1†

n2 C66 m a ‡ n1 C66 tanh …ma†

:

…65†

For the special case of a [0°/90°]s layup studied by Hashin (1985), the condition of transverse isotropy …1† …2† requires that C66 ˆ C66 ˆ GA . This reduces the expression of the shear modulus to Geff ˆ

GA ; 1 ‡ n1 =n2 tanh …ma†=ma

…66†

which is identical to the expression derived by Hashin. In the special case of a very small crack density ratio, a ! 1, then this expression further reduces to Geff ˆ

GA : 1 ‡ n1 =…n2 ma†

8. Numerical results In our numerical illustrations we started by comparing our results with those of Hashin. We built con®dence in our results from identically reproducing all of the ®gures presented in Hashin's paper (1985) and from comparisons with experimental measurements. In our illustrations we will use two types of damaged [0°/90°] cross-ply laminates. The ®rst is a [0°/90°]s graphite/epoxy laminate, whose properties are given in Hashin (1985), whereas the second laminate is a [0°/90°]s that was tested by Highsmith and Reifsnider (1982). The sti€ness components of the 0° degree lamina of both laminates are reproduced in Table 1. In all illustrations, the thickness of the 0° lamina is t2 ˆ 0.203 mm. All stress and sti€ness results are normalized with their respective values for the undamaged laminate case. As a ®rst numerical illustration we plot the variation of the normalized residual Young's modulus and shear modulus for the glass/epoxy damaged laminate with the crack density. The results, which are obtained from Eqs. (67) and (68), are plotted in Fig. 2. As the ®gure indicates, the low crack density limit corresponds to Table 1 Sti€ness properties of the 0° lamina of the composites used in the illustration examples Sti€ness (GPa)

C11

C33

Graphite/Epoxy Glass/Epoxy

209.7 46.17

7.875 2.843 16.98 7.444

C13

C44

C66

2.3 4.58

1.65 3.4

…67†

On the other hand, if the crack density is very high, a ! 0, then Eq. (65) reduces to Geff ˆ n2 GA ;

…68†

which re¯ects the fact that the large number of cracks has rendered the 90° lamina ine€ective. It should be pointed out that the limits (67) and (68) are identical to the ones obtained by Hashin (1985).

Fig. 2. Variation of the Young's modules and shear modulus with crack density in a [0°/90°3 ]s glass/epoxy laminate.

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

undamaged laminate, where as the high crack density limit represents the case when the 90° cracked lamina is ine€ective. Fig. 3 shows a comparison between the results of the present analysis of this case and the experimental measurements of Highsmith and Reifsnider (1982). As seen from the ®gure, there is a perfect agreement with the experimental measurements all the way from the crack initiation to the crack saturation case. Highsmith and Reifsnider also measured the variation of crack density with the applied load, and hence obtained the decrease in Young's

Fig. 3. Comparison between the predicted distribution of the residual Young's modulus and the experimental measurements of Highsmith and Reifsnider (1982).

Fig. 4. Comparison of the calculated results of the present modeling and of Talreja (1985a, b) with experimental measurements of the variation of residual Young's modulus with applied stress in a [0°/90°3 ]s glass/epoxy laminate.

313

modulus with the applied stress. Fig. 4 shows a comparison of the distributions of the normalized residual Young's modulus as measured by Highsmith and Reifsnider (1982), calculated by Talreja (1985a, b) and calculated by the present technique. The results of our micromechanical continuum modeling appear to better match the experimental data. To isolate and quantify the in¯uence of the location of the transverse crack alone on the residual Young's modulus of the composite laminate, we now treat the problem of a graphite/epoxy laminate with a single transverse crack in the 90° lamina. We consider both segments of the composite on either side of the crack. Each segment is assumed to be subject to uniform strain loading at one end, and stress free crack surface at the other. The stress distribution is then calculated as per Eq. (63), where the constants d1 and d2 have different values now. The longitudinal displacement in the 0° lamina in each segment is calculated as described in Section 7.1 and then added together to get the total laminate elongation. This is used to calculate directly Young's modulus. Fig. 5 shows the variation of the calculated normalized value of Young's modulus versus the normalized crack location. A crack very close to one edge has essentially no e€ect on the overall laminate sti€ness, where as a crack away from both edges reduces the

Fig. 5. Variation of the residual Young's Modulus with crack location in a [0°/90°3 ]s glass/epoxy laminate with a single transverse crack in the 90 lamina.

314

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

Young's modulus to a value that depends only on the laminate length.

9. Conclusion A simple technique is presented in order to study the problem of stress redistribution and the associated sti€ness reduction in laminates containing an orthogonal array of cracks. It is based on continuum mixture modeling of the micromechanical behavior of composites. The ®eld equations describing the behavior of each lamina are replaced by an alternative set of coupled ®eld equations that automatically satisfy all interface conditions. The resulting coupled governing equations for the total laminate retain the integrity of each component but allow them to coexist under some derived interaction terms. Using approximate distributions for some of the ®eld variables which automatically satisfy symmetry and interface conditions, the behavior of the three dimensional laminate is described by two coupled partial di€erential equations. By specifying these equations to the two-dimensional case, much of the stress redistribution results of damaged [0°/90°3 ]s laminates available in recent literature have been reproduced in a simple manner. Expressions for the reduction in the residual Young's modulus and shear modulus from crack initiation to crack saturation were derived from the stress ®eld solutions. The predicted e€ect of the crack density on these moduli shows good agreement with available experimental data.

uy

ÿ

u…1† y

n2 h2 ˆ 1 3

ux

ÿ

u…1† x

n2 h 2 ˆ 1 3

A1

o B1 ÿ …1† ox C55

! ;

…A:1†

o B1 ÿ …1† ox C44

! ;

…A:2†

where ux and uy are the interface values of ux and uy , respectively. Similarly, for the second lamina, if we multiply Eqs. (7) and (8) by …h ÿ z† and integrate by parts according to Eq. (14b) we get ux

ÿ

u…2† x

n2 h2 ˆ 6

ÿ2n1 A1 …2†

C55

o B1 o B2 ‡ n2 ‡ 3n1 ox ox

! ; …A:3†

uy ÿ u…2† y

n2 h2 ˆ 6

ÿ2n1 A2 …2†

C44

o B1 o B2 ‡ n2 ‡ 3n1 oy oy

! ; …A:4†

where we have used the continuity relations required by Eqs. (11a) and (11b). By eliminating ux and uy from Eqs. (A.3) and (A.4) we get the important relations: …1† u…2† x ÿu x

h2 ˆ 6

n1

2n1

…1†

C55

‡

n2

! A1

…2†

C55

! o B1 2 o B2 ÿ n2 ÿ n1 …2 ‡ n2 † ; ox ox

u…2† y

ÿ

u…1† y

h2 ˆ 6

n1

2n1

…1†

C44

‡

n2

…A:5†

!

…2†

C44

A2

! o B1 2 o B2 ÿ n2 ÿ n1 …2 ‡ n2 † : oy oy

Appendix A. Derivation of important relations (29) and (30) Multiplying Eqs. (7) and (8) by z and integrating by parts, for the ®rst lamina, in accordance with Eq. (14a), we obtain

A2

…A:6†

Appendix B. Coecients of the two coupled di€erential equations (39) and (40)

n1 h4 c1 ˆ 36

2

n1 …2 ‡ n2 † E…1† D…1†

‡

n32 E…2† D…2†

! ;

…B:1†

W.G. Abdelrahman, A.H. Nayfeh / Mechanics of Materials 31 (1999) 303±316

n1 h2 3

c2 ˆ

‡

c3 ˆ c4 ˆ c5 ˆ

c6 ˆ

c7 ˆ

c8 ˆ

…1†

D !

n1

n2

C55

C55

‡ …1†

n1 h2 6 n1 h 6

2



D…i† ˆ F …i† E…i† ;

n2 G…2† E…2† …2†

D

…2†

…B:2†

…2 ‡ n2 †H …1† E…1† …1†

…2 ‡ n2 †G E

…1†

…1†

D

ÿ ÿ

n2 H …2† E…2† n2 G E

;

…2†

D…2†

n1 …1† C44

‡

n2

D…1† !

…2†

C44

;

ÿ

;

…B:4†

…B:7†

n2 H …2† E…2† D…2† …B:8†

 1  …1† …1† …1† …1† …1† K ÿ 2C11 C12 C23 C13 =E…1† …1† D  n1  …2† …2† …2† …2† …2† …2† K C C C =E ÿ 2C ; …B:9† ‡ 11 12 23 13 n2 D…2†

c10 ˆ

ÿ1  …2† …2† …2† …2† Py …K …2† ÿ 2C11 C12 C23 C13 =E…2† † …2† n2 D  ‡ Px M=E…2† …B:10†

and where …i†

…i†

…i†2

…i†

…i†

…i†2

K …i† ˆ …C11 C33 ÿ C13 †; L…i† ˆ …C22 C33 ÿ C23 †;

…2†

…2†

…2†

…2†2

…2†

…2†

…B:13†

References



ÿ1  …2† …2† …2† …2† Px …L…2† ÿ 2C22 C12 C23 C13 =E…2† † …2† n2 D  ‡ Py M=E…2† ; …2 ‡ n2 †H …1† E…1†

…2†

M ˆ C13 C23 …C11 C22 ‡ C12 † ÿ C12 C33 E…2† :

…B:3†

 1  …1† …1† …1† …1† …1†2 …1† …1† C13 C23 …C11 C22 ‡ C12 †=E…1† ÿ C12 C33 …1† D n1 M ; …B:6† ‡ n2 D…2† E…2†

n1 h2 3

…B:12†



D…2† …2†

i ˆ 1; 2

and

;

D…1† 

ÿ

 1  …1† …1† …1† …1† …1† …1† L C C C =E ÿ 2C 22 12 23 13 D…1†  n1  …2† …2† …2† …2† …2† …2† ‡ L C C C =E ÿ 2C ; …B:5† 22 12 23 13 n2 D…2†

‡

c9 ˆ

…2 ‡ n2 †G…1† E…1†

315

…B:11†

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Nayfeh, A.H., Gurtman, G., 1974. A continuum approach to thepropagation of shear waves in laminated wave guides. Journal of Applied Mechanics 41, 106±110. Reifsnider, K.L., Talug, A., 1980. Analysis of fatigue damage in composite laminates. International Journal of Fatigue 3.

Talreja, R., 1985a. A continuum mechanics characterization of damage in composite materials. Proceedings of the Royal Society of London 399, 195±216. Talreja, R., 1985b. Transverse cracking and sti€ness reduction in composite laminates. Journal of Composite Materials 19 (4), 355±375.