基于中间构形的大变形弹塑性模型

在大变形弹塑性本构理论中,一个基本的问题是弹性变形和塑性变形的分解.通常采用两种分解方式,一是将变形率(或应变率)加法分解为弹性和塑性两部分,其中,弹性变形率与Kirchhoff应力的客观率通过弹性张量联系起来构成所谓的次弹性模型,而塑性变形率与Kirchhoff应力使用流动法则建立联系;另一种是基于中间构形将变形梯度进行乘法分解,它假定通过虚拟的卸载过程得到一个无应力的中间构形,建立所谓超弹性–塑性模型.研究了基于变形梯度乘法分解并且基于中间构形的大变形弹塑性模型所具有的若干性质,包括:在不同的构形上,塑性旋率的存在性、背应力的对称性、塑性变形率与屈服面的正交性以及它们之间的关系.首先,使用张量函数表示理论,建立了各向同性函数的若干特殊性质,并导出了张量的张量值函数在中间构形到当前构形之间进行前推后拉的简单关系式.然后,基于这些特殊性质和关系式,从热力学定律出发,建立模型在不同构形上的数学表达,包括客观率表示的率形式和连续切向刚度等,从而获得模型所具有的若干性质.最后,将模型与4种其他模型进行了比较分析.

A LARGE DEFORMATION ELASTOPLASTIC MODEL BASED ON THE INTERMEDIATE CONFIGURATION

In the large deformation elastoplastic constitutive theory, a basic problem is the decomposition of elastic deformation and plastic deformation. In the usual case, two decomposition methods are adopted. One method is to decompose deformation rate (or strain rate) into elastic and plastic parts. Among them, the elastic deformation rate and the objective rate of Kirchhoff stress are linked by elastic tensors, and by using this way, a sub-elastic model can be established. In the mean time,the plastic deformation rate is related to Kirchhoff stress by using flow law. Another method is to decompose the deformation gradient tensor based in the intermediate configuration. It is supposed that by considering the virtual unloading process, an unstressed intermediate configuration can be obtained and the so-called hyperelastic-plastic model can be established. In this paper, a large number of properties of a large deformation elastoplastic model which is based on the deformation gradient multiplicative decomposition are studied, and the model is built in the intermediate configuration. These properties include: in different configurations, the existence of the plastic spin rate; the symmetry of back stress; the orthogonality of plastic deformation rate and yield surface; and the relationships between plastic spin rate, back stress, plastic deformation rate and yield surface. First of all, by using the tensor function representation theorem which is in the appendix, some special properties of the isotropic function are obtained and some formula relationships are established, and some simple relationships of tensor value function between the intermediate configuration and the current configuration are derived. Secondly, based on these properties and relationships, and in combination with the laws of thermodynamics, the mathematical expression of the model in different configurations is established, which consists of objective rate representation and continuous tangential stiffness. Thus, some properties of the large deformation elastoplastic model based on the intermediate configuration are obtained. Finally, the model is compared and analyzed with the four models.