Finite strain constitutive modeling for shape memory alloys considering transformation-induced plasticity and two-way shape memory effect

Abstract

This work presents a three-dimensional constitutive model for shape memory alloys considering the TRansformation-Induced Plasticity (TRIP) as well as the Two-Way Shape Memory Effect (TWSME) through a large deformation framework. The presented logarithmic strain based model is able to capture the large strains and rotations exhibited by SMAs under general thermomechanical cycling. By using the martensitic volume fraction, transformation strain, internal stress, and TRIP strain tensors as internal state variables, the model is capable to capture the stress-dependent TRIP generation when SMAs are subjected to a multiaxial stress state, as well as the TWSME for thermomechanically trained SMAs under load-free conditions. A detailed implementation procedure of the proposed model is presented through a user-defined material subroutine within a finite element framework allowing for solving different Boundary Value Problems (BVPs). Comprehensive instruction on calibrating the model parameters as well as the derivation of continuum tangent stiffness matrix are also provided. In the end, the simulated cyclic pseudoelastic and actuation responses by the presented model for a wide range of SMA material systems under both uniaxial and multiaxial stress states are compared against experimental results to validate the proposed modeling capabilities.

Finite Strain Constitutive Modelling of Shape Memory Alloys Considering Partial Phase Transformation with Transformation-Induced Plasticity

“…The present section focuses on the formulation of the proposed model. The extension proposed herein starts from the thermodynamically-consistent finite strain model formulation recently presented by Xu et al [47], and its continuous development considering TRIP strain and loadfree two-way SME after cyclic loading [31]. The interested reader may refer to [31,47] for details about the original model formulation and to Appendix A for modeling preliminaries.…”

“…In the framework of macroscopic modeling and of finite strain continuum mechanics, an additive decomposition of the rate of deformation tensor D into an elastic part D e , a transformation part D tr , and a TRIP part D tp , is considered as follows [31]:…”

“…To date, several modeling efforts have been dedicated to the first characteristic, i.e., to predicting the evolving properties of SMAs under cyclic thermo-mechanical loading, caused by repeated phase transformations under material yielding point [20][21][22][23][24][25][26][27][28][29][30][31][32]. However, these efforts have been restricted to the prediction of the full transformation loops and not to an accurate modeling of PT (i.e., the second characteristic).…”

“…The choice of dealing with the model by Xu et al [31] is motivated by the following reasons: (i) the model is formulated in a finite strain framework, which is well suited in case of SMA actuators experiencing 30% or higher TRIP strains during their lifetime or in the presence of cracks [45]; (ii) the model describes the two-way SME exhibited by trained SMAs at load-free conditions; (iii) the model considers stress-dependent TRIP evolution under multiaxial stress state. Despite these advantages, the model by Xu et al [31] is limited to an accurate and efficient description of only full phase transformation. Accordingly, an efficient and easy-to-implement modified hardening function [15] is introduced in the present paper to account for PT behaviour under cyclic loading.…”

A Three-Dimensional Constitutive Modeling for Shape Memory Alloys Considering Two-Way Shape Memory Effect and Transformation-Induced Plasticity

“…Based on the work of Lagoudas et al 8 and Xu et al, 17 the Gibbs free energy G is proposed to be a continuous function dependent on Cauchy stress tensor σ, temperature T , and a set of internal state variables Υ = {ε t , ε p , β, ξ}, i.e., the transformation strain tensor ε t , the TRIP strain tensor ε tp , the internal stress tensor β, and the martensitic volume fraction scalar ξ, respectively. The ε t is used to account for the inelastic strain caused by the phase transformation, ε p is used to represent the large accumulated plastic strain induced by phase transformation, β is used for the generated internal stress tensor during training process, and the martensitic volume fraction ξ (ranging 0 ξ 1) is used for differentiating the two different phases in the SMAs system.…”