Finite Volume Analysis of Nonlinear Thermo-mechanical Dynamics of Shape Memory Alloys

TL;DR

The paper addresses nonlinear coupled thermo-mechanical dynamics in Shape Memory Alloys (SMA) with martensitic transformations, modeled via conservation laws and a Landau-Ginzburg free energy. It develops a finite volume method (FVM) formulation with a differential-algebraic equation (DAE) approach, using a staggered grid and a second-order BDF time integrator, augmented by a relaxation scheme and Steklov averaging to handle strong nonlinearities. The method is applied to 1D SMA rods and 2D patches, successfully capturing mechanically and thermally induced phase transformations, hysteresis, and evolving martensitic domain patterns (e.g., square-to-rectangular transitions). This provides a conservative, scalable tool for predicting SMA dynamics in multidimensional structures, with potential impact on SMA-based devices and actuators.

Abstract

In this paper, the finite volume method is developed to analyze coupled dynamic problems of nonlinear thermoelasticity. The major focus is given to the description of martensitic phase transformations essential in the modelling of shape memory alloys. Computational experiments are carried out to study the thermo-mechanical wave interactions in a shape memory alloy rod, and a patch. Both mechanically and thermally induced phase transformations, as well as hysteresis effects, in a one-dimensional structure are successfully simulated with the developed methodology. In the two-dimensional case, the main focus is given to square-to-rectangular transformations and examples of martensitic combinations under different mechanical loadings are provided.

Figures (7)

• Figure 1: Staggered grid for space discretization using finite volume methods.
• Figure 2: Mechanical induced phase transformation and mechanical hysteresis in SMA rod.
• Figure 3: Dynamics and mechanical hysteresis of SMA at medium and high temperature
• Figure 4: Thermally induced phase transformation and thermal hysteresis in SMA rod.
• Figure 5: Thermo-mechanical waves in SMA patches caused by varying mechanical loadings.