Finite element procedure and simulations for a multiphase phase field approach to martensitic phase transformations at large strains and with interfacial stresses
Finite element procedure and simulations for a multiphase phase field approach to martensitic phase transformations at large strains and with interfacial stresses
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DOI：
10.1016/j.cma.2018.08.006
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发表时间：
2019
影响因子：
7.2
通讯作者：
Anup Basak;V. Levitas
中科院分区：
文献类型：
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作者：
Anup Basak;V. Levitas
开发了一种新相场方法（Basak 和 Levitas，2018）的详细有限元程序，用于在大应变和界面应力下由温度和应力引起的多元马氏体转变。考虑具有奥氏体和氮马氏体变体的系统。使用与转变应变相关的N+1阶参数，其中之一描述奥氏体↔马氏体转变；其他 N 个阶参数描述了 N 个马氏体变体。阶次参数的演化由耦合的金兹堡-朗道和力学方程控制。假设采用牛顿迭代法求解控制方程的非整体策略，提出了一种强调切线模量推导的弱公式。值得注意的是，平衡方程的四阶切线模量不仅来自弹性应力，而且还来自结构界面应力，这是首次出现。二阶后向差分格式用于离散化 Ginzburg-Landau 方程中的时间导数。考虑自适应时间步长。有限元代码已在开源软件包协议中开发。 II 用于具有奥氏体和两种马氏体变体的系统，用于解决三个问题：（i）长方体的简单剪切变形以及奥氏体和单一马氏体变体的演变；（ii）马氏体孪晶以及样品尺寸对孪晶微观结构的影响；（iii）纳米压痕下的矩形块。前两个问题的结果描述了众所周知的解析解。使用变换变形梯度张量的两个运动学模型（KM）并比较相应的结果：KM-I表示贝恩张量中的线性变换规则，KM-II表示贝恩张量中的指数对数型变换规则。该算法自然可以扩展到多相固体中的相变、凝固、扩散相变、相变与塑性和/或断裂之间的相互作用等的研究。
A detailed finite element procedure for a new phase field approach (Basak and Levitas, 2018) to temperature-and stress induced multivariant martensitic transformations at large strains and with interfacial stresses is developed. A system with austenite and N martensitic variants is considered. N+ 1 order parameters related to the transformation strains are used, one of which describes the austenite↔ martensite transformation; the other N order parameters describe N martensitic variants. Evolution of the order parameters is governed by coupled Ginzburg–Landau and mechanics equations. Assuming a non-monolithic strategy for solving the governing equations by using Newton’s iterative method, a weak formulation with emphasis on the derivation of the tangent modulus has been presented. Notably, the fourth order tangent modulus for the equilibrium equations has a contribution not only from the elastic stresses but also from the structural interfacial stresses, which appears here for the first time. A second order backward difference scheme is used to discretize the time derivative in the Ginzburg–Landau equations. An adaptive time stepping is considered. A finite element code has been developed within an open source package deal. II for a system with austenite and two martensitic variants and used to solve three problems:(i) simple shear deformation of a rectangular parallelepiped with evolution of austenite and single martensitic variant;(ii) twinning in martensite and the effect of sample size on the twinned microstructures;(iii) a rectangular block under nanoindentation. The results for the first two problems describe the well-known analytical solutions. Two kinematic models (KMs) for the transformation deformation gradient tensor are used and the corresponding results are compared: KM-I represents a linear transformation rule in the Bain tensors and KM-II is an exponential-logarithmic type of transformation rule in the Bain tensors. The algorithm can naturally be extended for the study of phase transformations in multiphase solids, solidification, diffusive phase transitions, interaction between phase transformations and plasticity and/or fracture, etc.