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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. [ABSTRACT FROM AUTHOR]

The solution to the elasticity problem in three-dimensional polyhedral domains in the vicinity of an edge around which the material properties depend on the angular angle is addressed. This asymptotic solution involves a family of eigenpairs and their shadows which are being computed by means of p-finite element methods. In particular the examples we give explicitly provide the asymptotic solution for cracks and V-notch edges and explore the eigenvalues as a function of the change in material properties in the angular direction. We demonstrate that the singular exponents may change considerably by changing the material properties variation in the angular direction. These eigenpairs are necessary to allow the extraction of the edge stress intensity functions. [ABSTRACT FROM AUTHOR]