Evaluating a Stochastic Singularity Level in a V-Notch Isotropic Domain Due to Randomness in Material Properties

Abstract

The singular level of a v-notched isotropic domain is crucial as it relates to the domain’s stress intensity factor (SIF), which is essential for failure prediction. Typically, the singular level is calculated using deterministic material properties (like mean values), but this can misrepresent the severity of the singularity. The singular level’s accuracy is influenced by the stochastic nature of material properties, particularly Poisson’s ratio. This paper presents a stochastic approximation of eigenvalues for an isotropic v-notched domain with stochastic material properties. While both Young’s modulus and Poisson’s ratio are independent stochastic variables, only the stochasticity of Poisson’s ratio affects the eigenvalues. The generalized polynomial chaos method is applied to these stochastic eigenvalues, resulting in a polynomial approximation based on the domain’s stochastic Poisson ratio. The polynomial’s deterministic constants are derived from eigenvalues computed using chosen deterministic material properties according to the stochastic properties of Young’s modulus and Poisson’s ratio. A numerical example illustrates the stochastic approximation of the first two eigenvalues, showing fast convergence as the number of polynomials increases. Results are validated against the Monte Carlo method, highlighting the importance of stochastic approximation in accurately predicting the singular level, which may be more severe than expected when using deterministic approximations. These findings have significant practical implications, as they underscore the need to consider stochastic material properties in structural design and failure prediction.