利用梯度正则化、非凸有限应变弹性理论研究三维马氏体相变解的分支与稳定性
A numerical study of branching and stability of solutions to three-dimensional martensitic phase transformations using gradient-regularized, non-convex, finite strain elasticity
AI总结：
本文通过梯度正则化非凸弹性模型研究三维马氏体相变解的分支与稳定性，分析多解分支及稳定性，并研究微结构随长度参数变化的演化。
摘要：
在连续弹性设定中，涉及马氏体变体的相变通过非凸应变空间自由能密度函数建模。本文采用现有数学模型，通过在有限应变下引入高阶梯度项正则化非凸自由能密度函数，并通过标准变分论证推导边界值问题，受Toupin梯度弹性理论启发。这些梯度项旨在防止任意细微结构的存在，同时允许总自由能的局部极小值对应多解分支，即亚稳解分支。本文的目标是通过数值方法求解三维边界值问题，观察解分支并评估每个分支的稳定性，通过数值计算总自由能的二阶变分。此外，研究这些微结构如何随着长度参数（应变梯度项系数）趋近于零而演变。
英文摘要：
In the setting of continuum elasticity, phase transformations involving martensitic variants are modeled by a free energy density function that is non-convex in strain space. Here, we adopt an existing mathematical model in which we regularize the non-convex free energy density function by higher-order gradient terms at finite strain and derive boundary value problems via the standard variational argument applied to the corresponding total free energy, inspired by Toupin's theory of gradient elasticity. These gradient terms are to preclude existence of arbitrarily fine microstructures, while still allowing for existence of multiple solution branches corresponding to local minima of the total free energy; these are classified as metastable solution branches. The goal of this work is to solve the boundary value problem numerically in three dimensions, observe solution branches, and assess stability of each branch by numerically evaluating the second variation of the total free energy. We also study how these microstructures evolve as the length-scale parameter, the coefficient of the strain gradient terms in the free energy, approaches zero.
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