Modeling of the crystallization process of a polymer plate taking into account large deformations

Abstract

The issues of modeling the thermomechanical behavior of a polymer material during cooling accompanied by crystallization are considered. The proposed approach is applied to the problem of deformation upon cooling of an infinite plate made of low-pressure polyethylene during crystallization. Mathematical formulations of thermal kinetic and thermomechanical boundary value problems are presented. The results of the numerical solution of a coupled heat-kinetic problem, including the non-stationary heat conduction equation and the crystallization kinetics equation obtained under the assumption that the characteristics of the material depend on temperature are given. When solving the boundary value problem of thermomechanics, the previously obtained phenomenological nonlinear constitutive relations are used, which continuously describe the behavior of a polymer material in a wide temperature range, including the range of phase transformations. The construction of constitutive relations was carried out using the Peng-Landel potential. A weak variational formulation of the boundary value problem, constructed using the Galerkin method, is presented. A linearization procedure is carried out, which makes it possible to reduce the solution of an initially nonlinear boundary value problem to solving a sequence of linear problems with relatively small increments of the displacement vector components. In this case, the linearization of geometric relations is performed by superimposing small increments of deformations on finite ones, linearization of the constitutive relations - expansion in a Taylor series with subsequent retention of linear terms under the assumption that the increment of the components of the strain tensor is small. An approach is demonstrated that naturally takes into account the small increments of temperature and structural deformations arising in the material. Some aspects of the numerical implementation of the created algorithm based on the finite element technology of constructing a discrete analogue of the problem under consideration are analyzed. The results of solving the problem under discussion in a linear formulation under the assumption of small deformations and in a nonlinear formulation are compared.