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 135Delaminations 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) beam￾like; (b) circular isotropic and isotropically strained delaminations these comparatively transparent problems, especially concerning postbuckling behavior. In particular, assum￾ing 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 delamina￾tions 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, accord￾ing 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 magni￾tudes 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