J. Arghavani

A constitutive model for shape memory polymers with application to torsion of prismatic bars

In this article, satisfying the second law of thermodynamics, we present a 3D constitutive model for shape memory polymers. The model is based on an additive decomposition of the strain into four parts. Also, evolution laws for internal variables during both cooling and heating processes are proposed. Since temperature has considerable effect on the shape memory polymer behavior, for simulation of a shape memory polymer–based structure, it is required to perform a heat-transfer analysis. Commonly, an experimentally observed temperature rate–dependent behavior of shape memory polymers is justified by a rate-dependent glassy temperature, but using the heat-transfer analysis, it is shown that the glassy temperature could be considered as a constant material parameter. To this end, implementing the constitutive model within a nonlinear finite element code, we simulate torsion of a shape memory polymer rectangular bar and a circular tube. Moreover, we compare the predicted results with experimental data recently reported in the literature, which shows a good agreement.

A large deformation framework for shape memory polymers: Constitutive modeling and finite element implementation

Shape memory polymers commonly experience both finite deformations and arbitrary thermomechanical loading conditions in engineering applications. This motivates the development of three-dimensional constitutive models within the finite deformation regime. In the present study, based on the principles of continuum thermodynamics with internal variables, a three-dimensional finite deformation phenomenological constitutive model is proposed taking its basis from the recent model in the small strain regime proposed by Baghani et al. (2012). In the constitutive model derivation, a multiplicative decomposition of the deformation gradient into elastic and inelastic stored parts (in each phase) is adopted. Moreover, employing the mixture rule, the Green–Lagrange strain tensor is related to the rubbery and glassy parts. In the constitutive model, the evolution laws for internal variables are derived during both cooling and heating thermomechanical loadings. Furthermore, we present the time-discrete form of the proposed constitutive model in the implicit form. Using the finite element method, we solve several boundary value problems, that is, tension and compression of bars and a three-dimensional beam made of shape memory polymers, and investigate the model capabilities as well as its numerical counterpart. The model is validated by comparing the predicted results with experimental data reported in the literature that shows a good agreement.

An improvement on the Brinson model for shape memory alloys with application to two-dimensional beam element

In this article, the one-dimensional phenomenological constitutive model originally proposed by Brinson for shape memory alloys is improved to predict asymmetric behavior in tension and compression. We propose an approach that decomposes stress-induced martensite volume fraction into two parts, one in tension and one in compression. Results of numerical examples show reasonable agreement with experimental data. Moreover, we implement the proposed model in a user-defined material subroutine in the nonlinear finite element software ABAQUS/Standard as a two-dimensional Euler–Bernoulli beam element. We simulate several beam problems and a shape memory alloy staple. Regarding the results, the proposed shape memory alloys constitutive model, employed in a two-dimensional beam element, can be used to simulate various shape memory alloys applications in the design and analysis.

A semi-analytical study on helical springs made of shape memory polymer

In this paper, the responses of shape memory polymer (SMP) helical springs under axial force are studied both analytically and numerically. In the analytical solution, we first derive the response of a cylindrical tube under torsional loadings. This solution can be used for helical springs in which both the curvature and pitch effects are negligible. This is the case for helical springs with large ratios of the mean coil radius to the cross sectional radius (spring index) and also small pitch angles. Making use of this solution simplifies the analysis of the helical springs to that of the torsion of a straight bar with circular cross section. The 3D phenomenological constitutive model recently proposed for SMPs is also reduced to the 1D shear case. Thus, an analytical solution for the torsional response of SMP tubes in a full cycle of stress-free strain recovery is derived. In addition, the curvature effect is added to the formulation and the SMP helical spring is analyzed using the exact solution presented for torsion of curved SMP tubes. In this modified solution, the effect of the direct shear force is also considered. In the numerical analysis, the 3D constitutive equations are implemented in a finite element program and a full cycle of stress-free strain recovery of an SMP (extension or compression) helical spring is simulated. Analytical and numerical results are compared and it is shown that the analytical solution gives accurate stress distributions in the cross section of the helical SMP spring besides the global load–deflection response. Some case studies are presented to show the validity of the presented analytical method.

A 3-D constitutive model for pressure-dependent phase transformation of porous shape memory alloys.

Porous shape memory alloys (SMAs) exhibit the interesting characteristics of porous metals together with shape memory effect and pseudo-elasticity of SMAs that make them appropriate for biomedical applications. In this paper, a 3-D phenomenological constitutive model for the pseudo-elastic behavior and shape memory effect of porous SMAs is developed within the framework of irreversible thermodynamics. Comparing to micromechanical and computational models, the proposed model is computationally cost effective and predicts the behavior of porous SMAs under proportional and non-proportional multiaxial loadings. Considering the pressure dependency of phase transformation in porous SMAs, proper internal variables, free energy and limit functions are introduced. With the aim of numerical implementation, time discretization and solution algorithm for the proposed model are also presented. Due to lack of enough experimental data on multiaxial loadings of porous SMAs, we employ a computational simulation method (CSM) together with available experimental data to validate the proposed constitutive model. The method is based on a 3-D finite element model of a representative volume element (RVE) with random pores pattern. Good agreement between the numerical predictions of the model and CSM results is observed for elastic and phase transformation behaviors in various thermomechanical loadings.

Analysis of Al A359/SiCp Functionally Graded Cylinder Subjected to Internal Pressure and Temperature Gradient with Elastic-Plastic Deformation

In this article, an analytical elastic-plastic solution for thick-walled cylinders made of Functionally Graded Materials (FGMs) subjected to internal pressure and thermal loading is presented. Based on the experimental results, a mathematical model to predict the yielding through the thickness of FG AlA359/SiCp cylinder is developed. It is shown that under the temperature gradient loading, there is a point in the cylinder where the circumferential stress changes from compressive to tensile. The position of this point depends on the geometry and material properties of the FG cylinder and is independent of the temperature gradient.

A three-dimensional phenomenological constitutive model for porous shape memory alloys including plasticity effects

Porous shape memory alloys are a class of very interesting materials exhibiting features typical of porous metals and of shape memory alloys. In contrast to dense shape memory alloys, considerable plastic strain accumulates in porous shape memory alloys even during phase transformation. Moreover, due to the microstructure of porous materials, phase transformation and plasticity phenomena are significantly pressure-dependent. In this article, we propose a three-dimensional phenomenological constitutive model for the thermomechanical behavior of porous shape memory alloys able to predict shape memory effect, pseudo-elastic behavior and plastic behavior under proportional as well as non-proportional multiaxial loadings. To this end, proper internal variables together with free energy and limit functions are introduced. The material parameters are determined for several available porous shape memory alloys with a wide range of porosity (from 16% to 51%), and a good agreement between numerical predictions of the proposed model and experimental results is observed. Moreover, plastic and transformation strain evolution during increasing and decreasing cyclic loadings are demonstrated through several examples. Finally, to show model capabilities, shape memory effect and pressure-dependent behavior as well as model response under non-proportional loadings are also presented.

Implementation of Microplane Model Into Three-Dimensional Beam Element for Shape Memory Alloys

In this study, a three-dimensional (3D) beam element based on Timoshenko beam theory is introduced for shape memory alloys (SMAs). Employing the microplane approach, we use a 3D SMA constitutive model extended from a 1D model proposed by Brinson. The SMA model is implemented into a user-defined subroutine (UMAT) in the nonlinear finite element software ABAQUS/Standard. Results of numerical examples show reasonable agreement with experimental data in proportional and non-proportional loadings. Furthermore, several applications (staple, spring, structure) are simulated and the results are compared with those of continuum elements. According to the results, the 3D SMA beam element can be used in the design and analysis of various SMA applications.

An analytical study on the bending of prismatic SMA beams

In this study, an analytical solution is presented for pure bending of shape memory alloy (SMA) beams with symmetric cross section as well as symmetric behavior in tension and compression. To this end, a three-dimensional constitutive equation is reduced to one-dimensional form and employed to study the bending response of SMA beams at high (pseudo-elasticity) and low (shape memory effect) temperatures. An analytical expression for bending stress as well as polynomial approximation for shear stress and deflection are obtained. Derived equations for bending are employed to analyze an SMA beam with rectangular cross section and results are compared with those of the finite element method. The results of this work show good agreement when compared with experimental data and finite element results. Furthermore, the existence of several zero-stress fibers during unloading of SMA beams at low temperature is demonstrated.

A three-dimensional constitutive model for magnetic shape memory alloys under magneto-mechanical loadings

This paper presents a three-dimensional phenomenological constitutive model for magnetic shape memory alloys (MSMAs), developed within the framework of irreversible continuum thermodynamics. To this end, a proper set of internal variables is introduced to reflect the microstructural consequences on the material macroscopic behavior. Moreover, a stress-dependent thermodynamic force threshold for variant reorientation is introduced which improves the model accuracy. Preassumed kinetic equations for magnetic domain volume fractions, decoupled equations for magnetization unit vectors and appropriate presentation of the limit function for martensite variant reorientation lead to a simple formulation of the proposed constitutive model. To show the model capability in reproducing the main features of MSMAs, several numerical examples are solved and compared with available experimental data as well as available three-dimensional constitutive models in the literature. Demonstrating good agreement with experimental data besides possessing computational advantages, the proposed constitutive model can be used for analysis of MSMA-based smart structures.