Abstract We present a formulation for the extension of the stochastic finite element method with non-linear material models. Stress and stiffness fields, as well as displacements at each degree of freedom, are modeled as random processes and are expanded along the Polynomial Chaos (PC). PC terms of displacements are solved incrementally under load control by performing stochastic Galerkin projections on the equilibrium equation. PC terms of stress and stiffness are updated between each load increment from the resulting strains inside each elements, which follow a stochastic constitutive algorithm that performs stochastic Galerkin projections on the local level. The proposed approach is similar to the traditional deterministic non-linear finite element method and is consistent with the linear stochastic finite element method. The PC terms of stiffness are directly updated between increments of loading, and the stochastic tangent stiffness matrix is reconstructed at each increment. We illustrate the method to solve for the distribution of displacements caused by gravity loading in an earth dam with uncertain shear modulus and shear strength. The method gives similar results and, for the same level of accuracy, is computationally faster than the classical Monte-Carlo approach by several orders of magnitude.

中文翻译：

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非线性材料模型的随机有限元方法

摘要 我们提出了一种扩展具有非线性材料模型的随机有限元方法的公式。应力和刚度场以及每个自由度的位移被建模为随机过程并沿多项式混沌 (PC) 扩展。通过对平衡方程执行随机伽辽金投影，在载荷控制下逐步求解位移的 PC 项。应力和刚度的 PC 项在每个单元内产生的应变的每个载荷增量之间更新，这遵循在局部水平上执行随机伽辽金投影的随机本构算法。该方法类似于传统的确定性非线性有限元方法，与线性随机有限元方法一致。刚度的 PC 项在加载增量之间直接更新，随机切线刚度矩阵在每个增量处重建。我们说明了求解具有不确定剪切模量和剪切强度的土坝中重力加载引起的位移分布的方法。该方法给出了类似的结果，并且在相同的精度水平下，计算速度比经典的 Monte-Carlo 方法快几个数量级。