Delaminations in composite structures: V. V Bolotin far field), and the global balance of forces and energy in system cracked body-loading or loading de without going into the general statement and analytical details, let us illustrate the theory with Gr example, an open beam-like delamination in a composite specimen subjected to cyclic loading(Figure 21) As in the quasistatic crack growth, fatigue crack growth governed by equation(6). Compared with the quasistatic case, the infuence of damage accumulated in the interlayer is be taken into account. Let us introduce the damage measure(x, I)in the interlayer, at xl> a. We assume that w=0 for the undamaged interlayer, and w= l when the adjacent layer is completely debonded. Let damage be I produced with the range AT of the tangential stress T(x, t) in the interlayer, and let the equation governing the damage be assumed to be of the form ON △=△m),△2 Here Tr is a material constant characterizing resistance of he interlayer to damage accumulation, and ATth is the threshold resistance range. Power exponent m is similar to exponents of fatigue curves in standard tests. The simplest equation for the stress r(x, t) holds in the membrane approximation. It has the form exp (18) with the shear modulus Gm, the effective interlayer 4 thickness hm, and the length parameter Ao. The latter may be interpreted as a characteristic length of the boundary effect, or an ineffective length in the vicinity of he ruptured layer, or as a characteristic size of the process zone. The loading is given in equation(18)with Figure 20 Parameters of the fatigue crack growth diagram as the applied(nominal) longitudinal strain Eoc = 0oc/E functions of temperature and duration of the initial short-time thermal Using equations(17)and(18)and the initial condition action: figures 1, 2,3,4 correspond to temperature levels T=573,673, w(x, 0)=wo(x), the damage measure (m)=wa(N), N 773and873K at the tip of the delamination can be evaluated For the further discussion, the relation T=r(o) between the resistance force T and the tip damage is to be specified. We assume that Tx, N T=I0(1-°) interlayer, and a>0, e.g. a=l. This assumption on more or less realistic. The cycle number N, of the termination of the initial stage is the first positive root of the equation G(N)=T(N) at a=a0=const. Using equation 9)at E,=0 and equations(17)-(19)we obtain Figure 21 A model of delamination growth due to fatigue an equation with respect to N. △r0(ao,N) To discuss the propagation of delaminations in an wo(ao)+ analytical way, the theory of fatigue crack growth5,29,1 may be applied in a whole scale. The leading idea of the E2(N) heory is that the fatigue crack growth is a result of the (20) interaction of two different mechanisms: the accumula- tion of microdamage(both near the crack tips and in the Here E, is the critical strain defined with equation(11)Delaminations in composite structures." V. V. Bolotin ¢. t b 4~~mm 3 AGth 3 0 1 2 t, min Figure 20 Parameters of the fatigue crack growth diagram as functions of temperature and duration of the initial short-time thermal action: figures 1, 2, 3, 4 correspond to temperature levels T - 573, 673, 773 and 873 K It ,r(x,N) Figure 21 A model of delamination growth due to fatigue To discuss the propagation of delaminations in an analytical way, the theory of fatigue crack growth 5'29'51 may be applied in a whole scale. The leading idea of the theory is that the fatigue crack growth is a result of the interaction of two different mechanisms: the accumula￾tion of microdamage (both near the crack tips and in the far field), and the global balance of forces and energy in the system cracked body-loading or loading device. Without going into the general statement and analytical details, let us illustrate the theory with the simplest example, an open beam-like delamination in a composite specimen subjected to cyclic loading (Figure 21). As in the quasistatic crack growth, fatigue crack growth is governed by equation (6). Compared with the quasistatic case, the influence of damage accumulated in the interlayer is be taken into account. Let us introduce the damage measure w(x, t) in the interlayer, at Ixl >~ a. We assume that = 0 for the undamaged interlayer, and ~ = 1 when the adjacent layer is completely debonded. Let damage be produced with the range A-r of the tangential stress T(x, t) in the interlayer, and let the equation governing the damage be assumed to be of the form 0~-N = AT[ -- ATth , [AT] >~Tth. (17) Tf Here zf is a material constant characterizing resistance of the interlayer to damage accumulation, and A~-th is the threshold resistance range. Power exponent m is similar to exponents of fatigue curves in standard tests. The simplest equation for the stress r(x, t) holds in the membrane approximation. It has the form IT I GmAOI¢~[ {Ixl - a~ -- hm exp ~k~ ) (18) with the shear modulus Gm, the effective interlayer thickness hm, and the length parameter A 0. The latter may be interpreted as a characteristic length of the boundary effect, or an ineffective length in the vicinity of the ruptured layer, or as a characteristic size of the process zone. The loading is given in equation (18) with the applied (nominal) longitudinal strain e~ = a~/E x. Using equations (17) and (18) and the initial condition w(x, 0) = w0(x), the damage measure 0(N) = w[a(N), N] at the tip of the delamination can be evaluated. For the further discussion, the relation F = F(q~) between the resistance force F and the tip damage ¢ is to be specified. We assume that V = V0(1 - ¢0) (19) where F0 is the resistance force for the undamaged interlayer, and a > 0, e.g. a = 1. This assumption is more or less realistic. The cycle number N, of the termination of the initial stage is the first positive root of the equation G(N)= F(N) at a = a0 = const. Using equation (9) at e, = 0 and equations (17)-(19) we obtain an equation with respect to N,: N, -- -- -- "] m 0/a0/ Jo CLLN,).] 1/& =1 ] (20) Here et is the critical strain defined with equation (11), 138