13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- WST model [17]. The rationale is that larger particles have higher probability of failure than smaller particles under identical mechanical conditions. The particles are assumed to be elastic-brittle with elastic constants equal to those of the matrix as a first approximation. The principle stresses, Σα, in a particle can be given in terms of the principal stresses, σα, and plastic strains, ε p α, in the matrix with 123 1 , , α , ε ν E Σ σ p α α α = + = + . (12) The criterion of particle failure is based on a critical value of the strain energy density in a particle associated with failure normal to the maximum principal stress. This is given by I h I c Σ Σ E ν Σ E ν ψ − + = 2 1 . (13) where ΣΙ and Σh (= σh) are the maximum principal and the hydrostatic stress in the particle, respectively. If a particle of size r ruptures normally to the maximum principal stress upon achieving some critical condition, the energy lost (or the work of rupture) will be proportional to Ψc = r3 · ψc. The survival probability, ps, of the particle must decrease with increasing work of rupture, which can be written in the form 0Ψ dΨ p dp c s s =− , (14) where Ψ0 is a scaling energy. The survival probability at a given Ψc can be determined by integrating Eq. (14) from the initial value of the work of rupture, which is zero, to the current value. The probability of particle rupture, pc, is one minus the survival probability ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ = − − ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − − 0 3 0 0 1 exp 1 exp ψ ψ r r Ψ Ψ p c c c , (15) where the rupture energy density scale ψ0 = Ψ0 / r0 3 is introduced for convenience. Note, that Eq. (15) incorporates the effects of plastic strain and stress triaxiality. Increased plastic strains will result in increased ψc and hence probability of micro-crack formation. Inversely, increased hydrostatic stress, which could lead to micro-crack blunting, will result in reduced ψc and hence probability of micro-crack formation under equal other conditions, see Eqns. (12) and (13). We use Eq. (15) and Eq. (11) into Eq. (2) to calculate individual probabilities of failure at the integration points by numerical integration. The lower limit of the integral, i.e. the critical micro-crack size, is defined with a heuristic argument leading to a new modification of the Griffith criterion. We assume that the behaviour of a micro-crack of radius r formed in the plastic matrix corresponds to a fictitious micro-crack with rf >r in an elastic matrix. The crack opening displacement of a penny-shaped crack of radius rf in an elastic material subject to normal stress σI is ( ) ( ) f f I x r r x E u x < < − − = , 0 4 1 2 2 2 π σ ν . (16) The blunting of the physical micro-crack formed after particle rupture can be approximated by r εp Ι. The fictitious micro-crack size is then defined in such a way that the opening at x = r equals the blunting of the physical micro-crack, i.e. u(r) = r εp Ι. Solving Eq. (16) with this constraint leads to the following expression for the fictitious micro-crack size ( ) 2 2 4 1 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = + I p I f E r r σ ν ε π . (17) Used in conjunction with the Griffith criterion an effective critical micro-crack size can be defined
RkJQdWJsaXNoZXIy MjM0NDE=