岩土本构模型的数学基础与广义位势理论

Mathematical foundation of constitutive models of geotechnical material and generalized potential theory

对于种类繁多的岩土材料的本构模型,认识其数学基础及假设以及相关的物理意义是深刻理解和正确使用它们的基础。在本文中,揭示了目前在岩土工程中现有的不同类型的本构模型的数学基础和假设及其联系;指出了其基本的问题是其中应力(应变)或者其增量是否为一有势场的梯度矢量。同时涉及到应力与应变及其增量的主轴是否相同;有势场的梯度矢量的旋度是否为零;刚度(柔度)矩阵是否对称及其秩是否为1等。还提出了广义位势理论:可以用唯一的势函数,也可以任意设立2个或者3个其梯度矢量线性无关的势函数,用它们的梯度矢量表示应力(应变)或者其增量及塑性应变增量,用简单的试验确定参数。对于具有极其复杂力学性质的岩土材料,广义位势理论为建立统一的岩土理论模型提供了基础。

The mathematical foundation and corresponding physical sense in large numbers of constitutive models of geotechnical material are important to understand and use them. The mathematical foundation and condition of all kinds of geotechnical constitutive models are pointed out. The keystone is whether or not there is unique potential field in which the stresses (strains) or their increments are its gradient vector. The mathematical conditions include coaxiality of principal stresses and strains and their increments; symmetry of modulus matrix; rank of the matrix equal to 1; the curl of gradient vector equal to 0. The generalized potential theory is proposed in which unique potential function or 2-3 potential functions with linearly independent gradient vectors are used to determine the stress-strain relationship. The theory provides large capacity and with wider application field for geotechnical material with complex mechanical properties.