A scale-free phase-field model for martensitic phase transformations (PTs) at finite strains is developed as an essential generalization of small-strain models in Levitas et al. (2004) and Idesman et al. (2005). The theory includes finite elastic and transformational strains and rotations as well as anisotropic and different elastic properties of phases. The gradient energy term is excluded, and the model is applicable for any scale greater than 100 nm. The model tracks finite-width interfaces between austenite and the mixture of martensitic variants only; volume fractions of martensitic variants are the internal variables rather than order parameters. The concept of the effective threshold for the driving force is introduced, which can be either positive (e.g., due to interface friction) or negative (e.g., due to defects and stress concentrators promoting PTs). To reproduce PT conditions obtained from experiments or atomistic simulations under general stress tensor, the effective threshold depends on the stress tensor components. Material parameters are calibrated for martensitic PT between single crystalline cubic Si I and tetragonal Si II phases, which has large transformation strains (${\mathit{\varepsilon }}_{t1}=0.1753;{\mathit{\varepsilon }}_{t2}=0.1753;{\mathit{\varepsilon }}_{t3}=-0.447$). Finite element algorithms and numerical procedures are implemented in the code deal.II. Multiple 3D problems are solved to study the effect of mesh size, holding time during quasi-static loading, and strain rate on the multivariant microstructure evolution in Si I → Si II PT under uniaxial and hydrostatic loadings. The solution exhibits significant lattice rotations both in Si I and Si II, reproducing the appearance of diffuse grain boundaries in Si I and Si II and transforming them in polycrystals, which corresponds to known experiments. While finer mesh can produce a more detailed microstructure, the solution becomes practically mesh-independent after the mesh size is 80 times smaller than the sample size. When approaching the stationary solution, rough mesh leads to convergence to the correct microstructure faster than the fine mesh, because it neglects fine details in the microstructure. In some regions, reverse PT occurs at continuous compression, despite large transformation hysteresis. For most stationary interfaces, local thermodynamic equilibrium conditions (thermodynamic driving force for the interface motion equal to the effective threshold) are satisfied.

在Levitas等人中，作为小应变模型的基本概括，开发了有限应变下马氏体相变（PTs）的无标度相场模型。（2004）和Idesman等。（2005）。该理论包括有限的弹性和相变应变和旋转，以及各相的各向异性和不同的弹性。梯度能量项不包括在内，该模型适用于任何大于100 nm的比例。该模型仅跟踪奥氏体和马氏体变体混合物之间的有限宽度界面。马氏体变体的体积分数是内部变量而不是顺序参数。引入了驱动力有效阈值的概念，该阈值可以是正的（例如由于界面摩擦）或负的（例如，由于缺陷和应力集中器促进了PT）。为了重现在一般应力张量下从实验或原子模拟获得的PT条件，有效阈值取决于应力张量分量。校准了单晶立方Si I相和四方Si II相之间马氏体PT的材料参数，这些相变应变很大（${\mathit{\varepsilon }}_{Ť\mathrm{1个}}=0.1753;{\mathit{\varepsilon }}_{Ť2}=0.1753;{\mathit{\varepsilon }}_{Ť3}=-0.447$）。在代码处理中实现了有限元算法和数值程序。解决了多个3D问题，以研究网孔尺寸，准静态载荷期间的保持时间以及应变率对单轴和静液压载荷下Si I→Si II PT中多变量微观组织演变的影响。该溶液在Si I和Si II中均表现出显着的晶格旋转，从而再现了Si I和Si II中弥散晶界的出现并将其转变为多晶，这与已知的实验相对应。虽然更细的网格可以产生更详细的微观结构，但在网格大小比样本大小小80倍之后，解决方案实际上变得与网格无关。在接近固定解时，粗糙的网格比精细的网格更快地收敛到正确的微观结构，因为它忽略了微观结构的细节。在某些区域，尽管有很大的转换滞后，但在连续压缩时会发生反向PT。对于大多数固定接口，满足局部热力学平衡条件（接口运动的热力学驱动力等于有效阈值）。