Abstract

In this paper, a staggered, nonlinear finite element procedure is developed to solve the large-strain based coupled system of time dependent Ginzburg-Landau (GL) and thermoelasticity equations for phase transformations at the nanoscale. Geometrical nonlinearities are included based on the total Lagrangian description where the total deformation gradient is defined as the multiplicative decomposition of elastic, transformational and thermal deformation gradient tensors. The principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion whose discretized form is solved using the iterative methods which gives the thermal and transformational deformation gradient tensors. Next, the elastic deformation gradient tensor and the first Piola-Kirchhoff stress are calculated and substituted in the GL equation which gives the phase order parameter. The weighted residual method is used to derive the corresponding finite element form which allows for the phase-dependent surface energy BCs. Explicit and different implicit methods belonging to the generalized trapezoidal family are used for time discretization of the GL equation. Various examples of phase transformations are simulated and discussed. The effect of different time integration schemes are also investigated. The current study allows for a proper modeling of various phase field problems at large and small strains.

摘要

在本文中，开发了一种交错非线性有限元程序来求解基于大应变的时变 Ginzburg-Landau (GL) 耦合系统和纳米尺度相变的热弹性方程。基于总拉格朗日描述包括几何非线性，其中总变形梯度被定义为弹性、变换和热变形梯度张量的乘法分解。利用虚功原理获得拉格朗日运动方程的积分形式，其离散形式通过迭代方法求解，得到热变形梯度张量和变换变形梯度张量。下一个，计算弹性变形梯度张量和第一 Piola-Kirchhoff 应力并将其代入给出相序参数的 GL 方程。加权残差法用于推导相应的有限元形式，该形式允许相位相关的表面能 BC。属于广义梯形族的显式和不同的隐式方法用于 GL 方程的时间离散化。模拟和讨论了各种相变示例。还研究了不同时间积分方案的效果。目前的研究允许对大应变和小应变下的各种相场问题进行适当的建模。加权残差法用于推导相应的有限元形式，该形式允许相位相关的表面能 BC。属于广义梯形族的显式和不同的隐式方法用于 GL 方程的时间离散化。模拟和讨论了各种相变示例。还研究了不同时间积分方案的效果。目前的研究允许对大应变和小应变下的各种相场问题进行适当的建模。加权残差法用于推导相应的有限元形式，该形式允许相位相关的表面能 BC。属于广义梯形族的显式和不同的隐式方法用于 GL 方程的时间离散化。模拟和讨论了各种相变示例。还研究了不同时间积分方案的效果。目前的研究允许对大应变和小应变下的各种相场问题进行适当的建模。