Stochastic shape functions and stochastic strain–displacement matrix for a stochastic finite element stiffness matrix

Summary For conventional finite element problems, element geometry is adequate to determine shape functions. However, to account for secondary effects due to material randomness, conventional shape functions need to be modified according to the spatial fluctuation of constitutive variables in each Monte Carlo sample. This paper develops a method to compute stochastic shape functions based on local equilibrium criteria when each simulated sample complies with the same order of accuracy as designated for the associated deterministic problem. The resulting stochastic stiffness matrix is then calculated via the stochastic strain–displacement matrix based on those stochastic shape functions. In order to attain high accuracy, which is the characteristic of the boundary element method, rational polynomial shape functions are used in this paper. The proposed formulation is indispensable when secondary effects (due to nano size and time scale in modern technology, fiber randomness in composites, thermodynamic interactions in biological tissues, to name a few) demand a high accuracy finite element formulation. The elasto-plastic deformation that introduces concavity motivated the numerical example elaborated here. An example of a concave quadrilateral element with spatial randomness for the modulus of elasticity is illustrated. Since isoparametric shape functions for concave quadrilaterals do not exist, the Wachspress rational polynomial shape functions with irrational terms are used. The computer algebra environment Mathematica is employed here. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday