In this paper, a nonlinear finite element procedure is developed to solve the coupled system of phase field and elasticity equations at large strains for martensitic phase transformations at the nanoscale. The transformation is defined based on an order parameter which varies between 0 for austenite to one for martensite. The phase field equation relates the rate of change of the order parameter to the Helmholtz free energy. Both the Helmholtz free energy and the transformational strain tensor depend on the order parameter, and the coupling between the phase field and elasticity equations occurs due to the presence of elastic energy in the Helmholtz free energy and the transformational strain in the total strain. The deformation gradient tensor is introduced as a multiplicative decomposition of elastic and transformational gradient tensors. A staggered strategy is used to solve the coupled system of equations so that first, the principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion. It is then discretized and solved using the Newton–Raphson method which gives the displacements and consequently, the deformation gradient tensor. Next, since the order parameter is obtained from the phase field equation from the previous time step, the elastic deformation gradient tensor and the first Piola–Kirchhoff stress can be calculated and substituted in the phase field equation to find the order parameter for the current time step. The weighted residuals method is used to derive the finite element form of the phase field equation and the explicit method is used for its time discretization. The finite element procedure is well verified by comparing the obtained results with those from other simulations. Examples of thermal- and stress-induced phase transformations are presented, including austenite–martensite interface propagation and growth of preexisting martensitic regions. The developed algorithm and code can be advanced to solve kinetic problems coupled with mechanics at large strains for phase transformations and similar phenomena at the nanoscale.

本文提出了一种非线性有限元程序来求解大应变下的相场与弹性方程的耦合系统，以进行纳米级的马氏体相变。基于顺序参数定义转换，该顺序参数在奥氏体的0到马氏体的一个之间变化。相场方程将阶跃参数的变化率与亥姆霍兹自由能相关。亥姆霍兹自由能和相变应变张量均取决于阶数参数，并且由于亥姆霍兹自由能中存在弹性能以及总应变中存在相变应变，因此发生了相场和弹性方程之间的耦合。引入变形梯度张量作为弹性和变换梯度张量的乘法分解。使用交错策略来求解方程组的耦合系统，这样，首先，利用虚拟功的原理来获得拉格朗日运动方程的积分形式。然后使用牛顿-拉夫森方法将其离散化并求解，该方法给出位移并因此给出变形梯度张量。接下来，由于阶跃参数是从上一个时间步的相场方程获得的，因此可以计算弹性变形梯度张量和第一Piola–Kirchhoff应力，并将其代入相场方程以找到当前时间的阶跃参数。步。加权残差法用于导出相场方程的有限元形式，显式方法用于时间离散化。通过将获得的结果与其他模拟结果进行比较，可以很好地验证有限元程序。给出了由热和应力引起的相变的例子，包括奥氏体-马氏体界面的扩散和原有马氏体区的生长。可以改进已开发的算法和代码来解决动力学问题，并结合大应变力学解决纳米级的相变和类似现象。