Extracting stochastic stress intensity factors using generalized polynomial chaos

•The elastic stochastic isotropic solution in the vicinity of a crack is addressed.•The eigen-functions & displacements are approximated using the GPC method.•The explicit stochastic approximation of the SIF’s is provided.•Four different example problems are presented.•The SIF’s computed by the GPC are compared with the SIF’s computed by Monte-Carlo.•The results show that the proposed method is accurate and efficient. Realistic material properties, as the Young modulus E and Poisson ratio ν (isotropic materials), are measured by experimental observations and are inherently stochastic. Having their stochastic representation E(ξ) or ν(ξ) where ξ is a random variable, we formulate the elastic solution of the stochastic elasticity system in the vicinity of a crack tip. We show that the stochastic asymptotic displacements are of the form.u(r,θ;ξ)=A01(ξ)φ(01)(θ;ξ)+A02(ξ)φ(02)(θ;ξ)+A1(ξ)r1/2φ(1)(θ;ξ)+A2(ξ)r1/2φ(2)(θ;ξ)+O(r)with deterministic eigenvalues and either deterministic or stochastic eigenfunctions φ(i)(θ;ξ) and coefficients Ai. However, the stresses are represented by an asymptotic series with a stochastic behavior manifested only in the SIF:σ(r,θ;ξ)=KI(ξ)2πrϕ(1)(θ)+KII(ξ)2πrϕ(2)(θ)+O(1)We present explicitly whether the expressions in series expansions are stochastic or not, depending both on the material properties and on the boundary conditions far from the crack faces. The generalized polynomial chaos (GPC) method is used thereafter to compute the stochastic expressions from deterministic finite element solutions. As an example we consider either a stochastic Young modulus or Poisson ratio to be given as random variable with a normal distribution:E(ξ)=E0+E1ξ,orν(ξ)=ν0+ν1ξ,ξ∼N(0,σ2)Numerical examples are presented in which we compute φi(θ;ξ) and thereafter Ai(ξ) and KI(ξ) from deterministic finite element analyses using the GPC. Monte Carlo simulations are used to demonstrate the efficiency of the proposed methods. Numerical examples are provided that show the efficiency and accuracy of the proposed methods.