Extracting stochastic stress intensity factors using generalized polynomial chaos

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)r^1/2φ⁽¹⁾(θ;ξ)+A₂(ξ)r^1/2φ⁽²⁾(θ;ξ)+O(r)with deterministic eigenvalues and either deterministic or stochastic eigenfunctions φ⁽ⁱ⁾(θ;ξ) and coefficients A_i. However, the stresses are represented by an asymptotic series with a stochastic behavior manifested only in the SIF: σ(r,θ;ξ)=[Formula presented]ϕ⁽¹⁾(θ)+[Formula presented]ϕ⁽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 φ_i(θ;ξ) and thereafter A_i(ξ) and K_I(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.