Tensor Random Fields in Continuum Mechanics

Synonyms

Conductivity; Elasticity; Random fields; Stochastics; Uncertainty quantification

Definitions

Let $(\varOmega ,\mathfrak {F},\mathsf {P})$ be a probability space, i.e., Ω is a set, $\mathfrak {F}$ is a σ-field of subsets of Ω, and P is a probability measure on $\mathfrak {F}$. Let V be a finite-dimensional linear space consisting of tensors, and let $\mathfrak {B}(\mathsf {V})$ be the σ-field of Borel sets of V. A mapping T: Ω →V is called a random tensor if it is measurable, i.e., for any Borel set B we have $\mathsf {T}^{-1}(B)\in \mathfrak {F}$. If V consists of scalars, the term random variable is then used instead of random scalar. A tensor random field on a real finite-dimensional affine space E is a function of two variables T: E × Ω →V such that for any A ∈ E the function T(A, ω) is a random tensor.

Why Tensor Random Fields in Continuum Mechanics?

In this article, by continuum physics we understand continuum mechanics and other classical (non-quantum,...