Delaminations in composite structures: V. V. bolotin Ba1>0,., Sam>0, rewrite equation(2)as follows ∑(G-)a≤0 Here G are the generalized driving forces, and r are the corresponding generalized resistance forces. The unper- turbed state, as well as the perturbed ones, are considered here as equilibrium states with respect to Lagrangian generalized coordinates. Applied to composites, the idea appears to assume that the crack front is given with everal very close generalized Griffith's coordinates th the front of shear damage might be a little ahead of the opening(tensile)damage front, etc. Let'pure'modes be associated with variations da,, Saz, and Sa3(hereafter Arabic figures are used for the numbers of modes) When any two damage fronts coincide, we introduce variations Sa4, Sas and Saf. When all three modes contribute almost equally to damage, we label the variation Sa7. It ought to be stressed that all the Griffiths generalized coordinates a az correspond to the material points located in the process zone whose size is small compared with a,,..., ay. It means that the generalized coordinates take very similar tudes Supposing the variations to be independent and applying equation(3), we conclude that a delamination Figure 12 Schematic presentation of the limit surface for mixed begins to propagate if one of the following conditions is attained G G2=T2,G3=T3 We use case(a) for an open delamination. Its growth is +G2=12,G1+G3=13,G2+G3=2 accompanied by the formation of longitudinal cracks. In case(b)the delamination remains closed, ' pocket-like G1+G2+G3=F (4) Delaminations in the components under compression are shown in Figure 13(c)and (d). In case (c) the Here TI, T2 and T3 are generalized resistance forces in delamination is open and subjected to buckling. In case 'pure'modes;T12,T13, T23 and T123 are those in mixed (d)the delamination is also buckled but it is to be modes. The limit surface in the space of G1, G2, G3 is considered as closed. Edge delaminations are presented piece-planar (Fi ), but it is not so bad compare in Figure 13(e)and ( f). In the last case, a secondary crack say, with equation(1). Contrary to equation(1), which appears during the buckling of the delamination. One not more than a result of interpolation, equation(4)is in dimensional bending approximation is applicable in agreement with the principles of mechanics of solids cases (a)and(c). In the other cases we have to treat although it is based on the unconventional assumption delaminations with the use of the theory of plates and of the existence of several damage fronts) Finally, we may state that the limit surface for the shells. In cases(b)and (d), the delaminations may be considered as elliptical in the plane, and in cases(e) and mixed mode fracture can be developed in the framework () as half-elliptical of analytical mechanics Various versions of the energy approach are used to predict the stability of delaminations: the strain energy release rate approach, the path-invariant integral 6 BUCKLING AND STABILITY OF NEAR approach and the strain energy SURFACE DELAMINATIONS approach- In the case of a single-parameter nation in an elastic structural component 出 If a delamination is situated near the surface of a ways produce either identical or numerically close structural component, its behavior under loading is often rediction accompanied by buckling. This is The two simplest problems are depicted in Figure 14:a components under compression, under ce heating, beam-like delamination under compression and a d sometimes for components under ter ue to circular delamination with isotropic elastic properties Poisson's effect). Examples of delaminations are shown under the uniform two-dimensional compression. These in Figure 13. Cases (a)and(b)correspond to the problems were considered by a number of authors,31-39 delaminations propagating in a component under tension. A wide variety of simplifications might be used even in 134
Delaminations in composite structures: V. V. Bolotin 6a 1 ~ 0,..., 6a m >~ O, rewrite equation (2) as follows: m ~(G~ - rj)&j ~< O. (3) .j=l Here Gj are the generalized driving forces, and F/are the corresponding generalized resistance forces. The unperturbed state, as well as the perturbed ones, are considered here as equilibrium states with respect to Lagrangian generalized coordinates. Applied to composites, the idea appears 3° to assume that the crack front is given with several very close generalized Griffith's coordinates al,..., a m with independent variations ~al, (Sam. In fact, the front of shear damage might be a little ahead of the opening (tensile) damage front, etc. Let 'pure' modes be associated with variations 6al, 6a2, and 6a3 (hereafter Arabic figures are used for the numbers of modes). When any two damage fronts coincide, we introduce variations ~5a4, ~a5 and ~Sa 6. When all three modes contribute almost equally to damage, we label the variation 6a 7. It ought to be stressed that all the Griffith's generalized coordinates al, • • •, a7 correspond to the material points located in the process zone whose size is small compared with a~,..., a 7. It means that the generalized coordinates take very similar magnitudes. Supposing the variations to be independent and applying equation (3), we conclude that a delamination begins to propagate if one of the following conditions is attained: G1 = F1, G2 ~ 1"2, G3 = 1"3 G1 + G2 = 1"12, Gl + G3 = 1"13, G2 + G3 = 1"23 G1 + G2 + G3 = 1"123' (4) Here 1"1, 1"2 and 1"3 are generalized resistance forces in 'pure' modes; F12, I~13, F23 and 1"123 are those in mixed modes. The limit surface in the space of G l, G 2, G 3 is piece-planar (Figure 12), but it is not so bad compared, say, with equation (1). Contrary to equation (1), which is not more than a result of interpolation, equation (4) is in agreement with the principles of mechanics of solids (although it is based on the unconventional assumption of the existence of several damage fronts). Finally, we may state that the limit surface for the mixed mode fracture can be developed in the framework of analytical mechanics. 6 BUCKLING AND STABILITY OF NEARSURFACE DELAMINATIONS If a delamination is situated near the surface of a structural component, its behavior under loading is often accompanied by buckling. This is typical for the components under compression, under surface heating, and sometimes for components under tension (due to Poisson's effect). Examples of delaminations are shown in Figure 13. Cases (a) and (b) correspond to the delaminations propagating in a component under tension. Gm JIIC Gn Figure 12 Schematic presentation of the limit surface for mixed interlaminar fracture mode We use case (a) for an open delamination. Its growth is accompanied by the formation of longitudinal cracks. In case (b) the delamination remains closed, 'pocket-like'. Delaminations in the components under compression are shown in Figure 13 (c) and (d). In case (c) the delamination is open and subjected to buckling. In case (d) the delamination is also buckled but it is to be considered as closed. Edge delaminations are presented in Figure 13 (e) and (f). In the last case, a secondary crack appears during the buckling of the delamination. Onedimensional bending approximation is applicable in cases (a) and (c). In the other cases we have to treat delaminations with the use of the theory of plates and shells. In cases (b) and (d), the delaminations may be considered as elliptical in the plane, and in cases (e) and (f) as half-elliptical. Various versions of the energy approach are used to predict the stability of delaminations: the strain energy release rate approach 21, the path-invariant integral approach 22'23, and the strain energy density approach 2°. In the case of a single-parameter delamination in an elastic structural component, all these ways produce either identical or numerically close predictions. The two simplest problems are depicted in Figure 14: a beam-like delamination under compression and a circular delamination with isotropic elastic properties under the uniform two-dimensional compression. These problems were considered by a number of authors 29'31-39 A wide variety of simplifications might be used even in 134
Delaminations in composite structures: V. V. Bolotin Z Figure 13 Near-surface delaminations: (a)open delamination in tension; (b)closed one in tension; (c)open buckled delamination; (d)closed buckled one:() edge buckled delamination; (f the same with a Using a half-nonlinear'approach of the theory of elastic stability, we present the energy of the systems in the form U= const E、abh 2(1-vxyvy Here a, b and h are the dimensions shown in Figure 14 (a), Ex is Youngs modulus in the x-direction, vxy and vy are Poissons ratios. It is assumed that the general loading is strain-controlled with the applied strain Eoo, and the mem brane strain in the buckled delamination Figure 14 Elementary problems of buckled delaminations:(a)beam remains equal to the Euler's critical strain ike;(b)circular isotropic and isotropically strained delaminations (a) these comparatively transparent problems, especially concerning postbuckling behavior. In particular, assum- The energy in equation (7) is related to a half of the ing that the buckling mode is given with a single component, x>0. The generalized driving force, accord parameter, say, with the maximum lateral displacement, ing to equation(5),is one can calculate the potential energy of the system Ebh U=UC, a), where a is the size of the delamination G Figure 14). The generalized driving force is aU Equalizing, according to equation(6) the right-hand (5) side of equation(9) to the resistance force r=?b The generalized resistance force is r= yb for the beam come to the equation that connects the critical magi tudes of Em and a delamination(b is the width of the beam), and r= 2ray for the circular delamination. The growth of delamina- E2+2exe.(a)-32(a)=2 tions takes place under conditions similar to those in Here the notation is used 2y(1 As an example, assume that in the case depicted in It is easy to see that E, is the critical strain for an open Figure 14(a)the buckled mode is w(x)=fcos"(Tx/2a) delamination under tension [Figure 13(a)] if the work of 135
Delaminations in composite structures." V. V. Bolotin Z Z (a) Z 2' s½ (b) (c) Z Z 2b y Y~ X Y X S~ (d) (e) (0 Figure 13 Near-surface delaminations: (a) open delamination in tension; (b) closed one in tension; (c) open buckled delamination; (d) closed buckled one; (e) edge buckled delamination; (f) the same with a secondary crack Ca) (b) Figure 14 Elementary problems of buckled delaminations: (a) beamlike; (b) circular isotropic and isotropically strained delaminations these comparatively transparent problems, especially concerning postbuckling behavior. In particular, assuming that the buckling mode is given with a single parameter, say, with the maximum lateral displacement, one can calculate the potential energy of the system U = U(f,a), where a is the size of the delamination (Figure 14). The generalized driving force is OU a - Oa " (5) The generalized resistance force is P = 7b for the beam delamination (b is the width of the beam), and F = 2rra"/ for the circular delamination. The growth of delaminations takes place under conditions similar to those in equation (4): a = F. (6) As an example, assume that in the case depicted in Figure 14 (a) the buckled mode is w(x) =fcos2(Trx/2a). Using a 'half-nonlinear' approach of the theory of elastic stability, we present the energy of the systems in the form Exabh U = const (e 2 - 2e~e, + e2). (7) 2(1 - UxyUyx) Here a, b and h are the dimensions shown in Figure 14 (a), E~ is Young's modulus in the x-direction, Uxy and Yyx are Poisson's ratios. It is assumed that the general loading is strain-controlled with the applied strain e~, and the membrane strain in the buckled delamination remains equal to the Euler's critical strain. c,(a) = i5 (8) The energy in equation (7) is related to a half of the component, x >~ 0. The generalized driving force, according to equation (5), is a Exbh (e 2 - 2e~e, + e2,). (9) 2(1 - UxUy ) Equalizing, according to equation (6) the right-hand side of equation (9) to the resistance force P =-yb, we come to the equation that connects the critical magnitudes of e~ and a: e~ 2 + 2e~e,(a) 2 (10) - 3eZ(a) = et. Here the notation is used e 2 = 2"/(1 - UxyUyx) (11) Gh It is easy to see that el is the critical strain for an open delamination under tension [Figure 13 (a)] if the work of 135