基于内变量和张量函数表示定理的本构方程

针对各向同性材料，基于张量函数表示定理，建立了本构关系的张量不变性表示，其中，3个不可约基张量取决于应力的0～2次幂，且相互正交，3个系数由塑性应变增量和应力的不变量表示。基于塑性应变增量的不变量定义内变量，本构关系归结为确定内变量的演化。使用张量函数表示定理，给出了内变量演化方程的一般表达式，它取决于应力不变量的增量，因而与主轴旋转无关。讨论了如何根据试验资料和引入适当的假定，确定具体的演化方程。通过与塑性势理论和多重屈服面理论进行比较，表明所建模型是这些理论的最一般表示，且简捷直观、使用方便。

Constitutive equations based on internal variable and tensor function representation theorem

The author presents the general invariant formulation of constitutive equations based on the representation theorem for the isotropic materials. The equations are a linear combination of three irreducible tensor function bases, which depend on the zero, first and second order power of stress tensor and are orthogonal to one another. Three coefficients depend on three invariants of stresses and plastic strain increments respectively. The internal variables are defined in terms of three invariants of the plastic strain increments. Therefore, the evolution equations of the internal variables are needed to be determined to form a closed constitutive theory. Using the representation theorem, the evolution equations are obtained in a general form. It depends on the increment of invariants of the stress, and therefore is independent of the rotation of the principal axes of stress. It is discussed how the evolution equations are specified from the experiment data in combination with some assumptions. Finally, the constitutive equations presented in this paper are compared with the classical plastic potential theory and the multi-yield surface theory. It is showed that the former is a general representation of the latter two theories, and is more simple and convenient for use.