MARKOV PROCESS MODEL FOR RANDOM GROWTH OF CRACK WITH R-CURVE

Due to the statistical nature of material properties, crack growth is a random process. This process may be described by the Markov chain model (see Fig. 11. However, model must exhibit R-curve behavior. The basic relation for the Markov chain model (Bogdanoff and Kozin. 1985) is Px = P pX (I) () In which Eo is the initial state probability v"ctor, Eo = <1(1 ' 1(2 '... I(B_I' OIT, r 1(. " I, in which 1(. = Prob(damage state j is initially occupied). We L J J aSsume 1(1 = I, with other I ( j = 0, which means the crack (or damage) always starts from state I; P is the damage state probability, p = <p (I), p (2), T-x-x x x px(B». where pxU),. Prob(damage state j is occupied at stress level X); P is the probability transition matrix, PI ql 0...... 0 0 P2 q2......