Finite element analysis of coupled phase-field and thermoelasticity equations at large strains for martensitic phase transformations based on implicit and explicit time discretization schemes

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.

Keywords:

• Nanoscale
• phase field
• phase transformations
• thermoelasticity
• time integrations
• total Lagrangian

Acknowledgment

The help of Isfahan University of Technology and Iran National Science Foundation is gratefully acknowledged.

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Funding

The partial support of the MIUR-PRIN project XFAST-SIMS (no. 20173C478N) is gratefully acknowledged.