A finite deformation thermomechanical constitutive model for triple shape polymeric composites based on dual thermal transitions

Shape memory polymers (SMPs) have gained strong research interests recently due to their mechanical action that exploits their capability to fix temporary shapes and recover their permanent shape in response to an environmental stimulus such as heat, electricity, irradiation, moisture or magnetic field, among others. Along with interests in conventional “dual-shape” SMPs that can recover from one temporary shape to the permanent shape, multi-shape SMPs that can fix more than one temporary shapes and recover sequentially from one temporary shape to another and eventually to the permanent shape, have started to attract increasing attention. Two approaches have been used to achieve multi-shape shape memory effects (m-SMEs). The first approach uses polymers with a wide thermal transition temperature whilst the second method employs multiple thermal transition temperatures, most notably, uses two distinct thermal transition temperatures to obtain triple-shape memory effects (t-SMEs). Recently, one of the authors’ group reported a triple-shape polymeric composite (TSPC), which is composed of an amorphous SMP matrix (epoxy), providing the system the rubber-glass transition to fix one temporary shape, and an interpenetrating crystallizable fiber network (PCL) providing the system the melt-crystal transition to fix the other temporary shape. A one-dimensional (1D) material model developed by the authors revealed the underlying shape memory mechanism of shape memory behaviors due to dual thermal transitions. In this paper, a three-dimension (3D) finite deformation thermomechanical constitutive model is presented to enable the simulations of t-SME under more complicated deformation conditions. Simple experiments, such as uniaxial tensions, thermal expansions and stress relaxation tests were carried out to identify parameters used in the model. Using an implemented user material subroutine (UMAT), the constitutive model successfully reproduced different types of shape memory behaviors exhibited in experiments designed for shape memory behaviors. Stress distribution analyses were performed to analyze the stress distribution during those different shape memory behaviors. The model was also able to simulate complicated applications, such as a twisted sheet and a folded stick, to demonstrate t-SME.     

Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling

Shape memory polymers (SMPs) can retain a temporary shape after pre-deformation at an elevated temperature and subsequent cooling to a lower temperature. When reheated, the original shape can be recovered. Relatively little work in the literature has addressed the constitutive modeling of the unique thermomechanical coupling in SMPs. Constitutive models are critical for predicting the deformation and recovery of SMPs under a range of different constraints. In this study, the thermomechanics of shape storage and recovery of an epoxy resin is systematically investigated for small strains (within ±10%) in uniaxial tension and uniaxial compression. After initial pre-deformation at a high temperature, the strain is held constant for shape storage while the stress evolution is monitored. Three cases of heated recovery are selected: unconstrained free strain recovery, stress recovery under full constraint at the pre-deformation strain level (no low temperature unloading), and stress recovery under full constraint at a strain level fixed at a low temperature (low temperature unloading). The free strain recovery results indicate that the polymer can fully recover the original shape when reheated above its glass transition temperature (T_g). Due to the high stiffness in the glassy state (T < T_g), the evolution of the stress under strain constraint is strongly influenced by thermal expansion of the polymer. The relationship between the final recoverable stress and strain is governed by the stress–strain response of the polymer above T_g. Based on the experimental results and the molecular mechanism of shape memory, a three-dimensional small-strain internal state variable constitutive model is developed. The model quantifies the storage and release of the entropic deformation during thermomechanical processes. The fraction of the material freezing a temporary entropy state is a function of temperature, which can be determined by fitting the free strain recovery response. A free energy function for the model is formulated and thermodynamic consistency is ensured. The model can predict the stress evolution of the uniaxial experimental results. The model captures differences in the tensile and compressive recovery responses caused by thermal expansion. The model is used to explore strain and stress recovery responses under various flexible external constraints that would be encountered in applications of SMPs.     

Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers

Shape memory polymers (SMPs) are polymers that can demonstrate programmable shape memory effects. Typically, an SMP is pre-deformed from an initial shape to a deformed shape by applying a mechanical load at the temperature T_H>T_g. It will maintain this deformed shape after subsequently lowering the temperature to T_L<T_g and removing the externally mechanical load. The shape memory effect is activated by increasing the temperature to T_D>T_g, where the initial shape is recovered. In this paper, the finite deformation thermo-mechanical behaviors of amorphous SMPs are experimentally investigated. Based on the experimental observations and an understanding of the underlying physical mechanism of the shape memory behavior, a three-dimensional (3D) constitutive model is developed to describe the finite deformation thermo-mechanical response of SMPs. The model in this paper has been implemented into an ABAQUS user material subroutine (UMAT) for finite element analysis, and numerical simulations of the thermo-mechanical experiments verify the efficiency of the model. This model will serve as a modeling tool for the design of more complicated SMP-based structures and devices.     

On the evolution law for the frozen fraction in linear theories of shape memory polymers

Shape memory properties of thermally responsive polymeric materials are due mainly to a phase transition from the rubbery phase above the transition temperature (glass transition or melting temperature) to the glassy or semicrystalline phase below this temperature. Within constitutive models of shape memory polymers (SMPs), this phase transition is mathematically accounted for by the frozen volume fraction for which a suitable evolution law must be postulated or derived. In this paper, the evolution laws that have been proposed in the literature are examined both from the experimental and from the theoretical point of view. It is found that the predictive capabilities of the phenomenological laws may be improved by admitting involved material constants to depend on parameters such as pre-strain, rate of heating and cooling, and other quantities characterizing thermomechanical cyclic tests. It is next shown that for a wide class of linear constitutive models of SMPs, the evolution law for the frozen volume fraction may be derived in a systematic way from strain and stress profiles experimentally obtained in the standard thermomechanical test.     

THERMOMECHANICAL RESPONSE OF SMA FIBER COMPOSITES WITH NON-UNIFORM TEMPERATURE DISTRIBUTION

A micromechanics method based on the High-Order-Theory developed by Aboudi et al. is used to predict the thermomechanical response of composites reinforced by shape memory alloy (SMA) fibers, and the non-uniform thermal distribution in composite arising from the process of heating or cooling is considered. The numerical development based on this model was coded to predict the thermomechanical response of shape memory alloy fiber/elastomer matrix composite subjected to thermal cycle loading. When the composite is heated, two heating ways, thermal gradients and heat source by passing an electric current through the SMA fibers are imposed on the composite respectively. Upon cooling, the first thermal boundary condition and the second thermal boundary condition are subjected to the composite respectively. A series of stress distributions and temperature distributions for different instants are calculated to reveal the interaction between the SMA material and matrix. It is useful to analyze and design the SMA actuator driven by heat source or the surface temperature.     

Thermomechanical behavior of thermoset shape memory polymer programmed by cold-compression: Testing and constitutive modeling

Programming is a key process for thermally activated stress or strain recovery of shape memory polymers (SMPs). Typically, programming requires an initial heating above the glass transition temperature (T_g), subsequent cooling below T_g and removal of the applied load, in order to fix a temporary shape. This work adopted a new approach to program thermoset SMPs directly at temperatures well below T_g, which effectively simplified the shape fixing process. 1-D compression programming below T_g and free shape recovery of a thermoset SMP were experimentally investigated. Functional stability of the shape fixity under various environmental attacks was also experimentally evaluated. A mechanism-based thermoviscoelastic-thermoviscoplastic constitutive model incorporating structural and stress relaxation was then developed to predict the nonlinear shape memory behavior of the SMP trained below T_g. Comparison between the prediction and the experiment showed good agreement. The structure dependence of the thermomechanical behavior of the SMP was further discussed through a parametric study per the validated constitutive model. This study validates that programming by cold-compression is a viable alternative for thermally responsive thermoset SMPs.     

Predicting thermal shape memory of crosslinked polymer networks from linear viscoelasticity

The viscoelastic behavior of an amorphous shape memory polymer network and its dependence on time and temperature were measured by dynamic mechanical analysis. The resulting thermo-mechanical behavior was modeled and implemented in a commercial finite element code. The ability of the resulting thermomechanical model to simulate and, eventually, predict the shape storage and shape recovery of the material was evaluated against experimental shape memory thermomechanical torsion data in a large deformation regimen. The simulations showed excellent agreement with experimental shape memory thermomechanical cycle data. This demonstrates the dependence of the shape recovery on time and temperature. The results suggest that accurate predictions of the shape recovery of any amorphous polymer networks under any thermomechanical conditions combination solely depends on considering the material viscoelasticity and its time–temperature dependence.     

On phase transformations in shape memory alloy materials and large deformation generalized plasticity

A new version of rate-independent generalized plasticity, suitable for the derivation of general thermomechanical constitutive laws for materials undergoing phase transformations, is proposed within a finite deformation framework. More specifically, by assuming an additive decomposition of the finite strain tensor into elastic and inelastic (transformation induced) parts and by considering the fractions of the various material phases as internal variables, a multi-phase formulation of the theory is developed. The concepts presented are applied for the derivation of a three-dimensional thermomechanical model for shape memory alloy materials. The ability of the model in simulating several patterns of the extremely complex behavior of these materials, under both monotonic and cyclic loadings, is assessed by representative numerical examples.     

Differential and integrated form consistency in 1-D phenomenological models for shape memory alloy constitutive behavior

Shape memory alloys are being explored increasingly for developing smart structures and devices in aerospace, automotive and other application areas. The material behavior is highly nonlinear with coupled thermomechanical response involving temperature and/or stress induced phase transformations. Modeling the constitutive behavior of these materials poses several challenges and a few phenomenological models exist that provide a quick and reasonable approach to assess their behavior. Due to phenomenological approach, several assumptions are made in order to simplify the model and some of them introduce inconsistencies or anomalies into the model. In this paper, a frequently used approach, namely, Brinson [Brinson, L.C., 1993. One dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4(2), 229–242.] model, is investigated. The constitutive equation is usually expressed at the outset in the differential form and the integrated form of the same is obtained. It is shown that the two forms of equations are not consistent in the Brinson [Brinson, L.C., 1993. One dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4(2), 229–242.] model, given the assumed form of material functions. In the present work, the nature and implications of the inconsistency are highlighted. The cause of incompatibility is the inconsistent material definitions. A modified consistent constitutive model is proposed by redefining the material function which satisfies the compatibility condition. The advantages in using the proposed modified model are highlighted with numerical case studies involving hysteretic stress–strain behavior.     

A three-dimensional constitutive model for shape memory alloys

Shape memory alloys (SMAs) are materials that, among other characteristics, have the ability to present high deformation levels when subjected to mechanical loading, returning to their original form after a temperature change. Literature presents numerous constitutive models that describe the phenomenological features of the thermomechanical behavior of SMAs. The present paper introduces a novel three-dimensional constitutive model that describes the martensitic phase transformations within the scope of standard generalized materials. The model is capable of describing the main features of the thermomechanical behavior of SMAs by considering four macroscopic phases associated with austenitic phase and three variants of martensite. A numerical procedure is proposed to deal with the nonlinearities of the model. Numerical simulations are carried out dealing with uniaxial and multiaxial single-point tests showing the capability of the introduced model to describe the general behavior of SMAs. Specifically, uniaxial tests show pseudoelasticity, shape memory effect, phase transformation due to temperature change and internal subloops due to incomplete phase transformations. Concerning multiaxial tests, the pure shear stress and hydrostatic tests are discussed showing qualitatively coherent results. Moreover, other tensile–shear tests are conducted modeling the general three-dimensional behavior of SMAs. It is shown that the multiaxial results are qualitative coherent with the related data presented in the literature.     

Finite Element Analysis of Shape Memory Alloy Adaptive Trusses with Geometrical Nonlinearities

This contribution deals with the nonlinear analysis of shape memory alloy (SMA) adaptive trusses employing the finite element method. Geometrical nonlinearities are incorporated into the formulation together with a constitutive model that describes different thermomechanical behaviors of SMA. It has four macroscopic phases (three variants of martensite and an austenitic phase), and considers different material properties for austenitic and martensitic phases together with thermal expansion. An iterative numerical procedure based on the operator split technique is proposed in order to deal with the nonlinearities in the constitutive formulation. This procedure is introduced into ABAQUS as a user material routine. Numerical simulations are carried out illustrating the ability of the developed model to capture the general behavior of shape memory bars. After that, it is analyzed the behavior of some adaptive trusses built with SMA actuators subjected to different thermomechanical loadings.     

Theoretical analysis of the deformation of SMP sandwich beam in flexure

Shape memory polymers (SMPs) can have a large frozen strain but with a very small recovery stiffness in comparison with shape memory metals or ceramics. To provide more deployable stresses for the application of actuators, sandwich beams consisting of a SMP core and two thin metallic skins were considered. The packaging behaviors of two types of SMP sandwich beams, aluminum/SMP/aluminum and steel/SMP/steel, were discussed. Due to the high compliance of SMP core on packaging condition that the testing temperature is above the activation temperature of the material, buckling and post-buckling are the essential deformation mechanisms of SMP sandwich beams under bending. Theoretical solutions were derived in studying such non-linear behaviors, including the initiation of critical buckling, post-buckling response, and final failure modes. Systematic parameter’s analyses, e.g., buckling half-wavelength, amplitude, location of the neutral-strain surface in different packaging curvatures, were also presented.     

Micromechanical prediction of the two-way shape memory effect in shape memory alloy composites

The two-way shape memory effect in monolithic shape memory alloys has been widely investigated both theoretically and experimentally. In the present study, this effect is analyzed for shape memory alloy composites by employing a micromechanical model. To this end, the responses of polymeric matrix and metal matrix unidirectional composites with embedded shape memory alloy fibers are determined. For the polymeric matrix composite, the effect of axial, transverse and shear loadings as well as the fiber volume fraction on the resulting two-way shape memory behavior are studied. The local distributions of stresses among the shape memory alloy fiber and epoxy matrix in the low- and high-temperature shapes of the composite are also investigated. Two training procedures that generate the two-way shape memory effect in the metal matrix composite are offered. The present analysis shows that the two-way shape memory effect in the chosen type of metal matrix composite is not as useful as in the polymeric matrix one. Finally, for a polymeric matrix composite that is subjected to a transverse normal loading, the effect of imperfect bonding between the shape memory alloy fibers and the neighboring matrix is investigated.     

Micro–macro analysis of shape memory alloy composites

The aim of the paper is to develop a micro–macro approach for the analysis of the mechanical behavior of composites obtained embedding long fibers of Shape Memory Alloys (SMA) into an elastic matrix. In order to determine the overall constitutive response of the SMA composites, two homogenization techniques are proposed: one is based on the self-consistent method while the other on the analysis of a periodic composite. The overall response of the SMA composites is strongly influenced by the pseudo-elastic and shape memory effects occurring in the SMA material. In particular, it is assumed that the phase transformations in the SMA are governed by the wire temperature and by the average stress tensor acting in the fiber. A possible prestrain of the fibers is taken into account in the model.Numerical applications are developed in order to analyze the thermo-mechanical behavior of the SMA composite. The results obtained by the proposed procedures are compared with the ones determined through a micromechanical analysis of a periodic composite performed using suitable finite elements.Then, in order to study the macromechanical response of structural elements made of SMA composites, a three-dimensional finite element is developed implementing at each Gauss point the overall constitutive laws of the SMA composite obtained by the proposed homogenization procedures. Some numerical applications are developed in order to assess the efficiency of the proposed micro–macro model.     

The influence of model parameters and of the thermomechanical coupling on the behavior of shape memory devices

Shape memory materials (SMM) are receiving increasing attention for their use in applications that exploit their dynamic behavior. A thermomechanical model for devices with pseudoelastic behavior has been proposed in previous works [11] (Bernardini and Pence, 2005) [15] (Bernardini and Rega, 2005). The model takes into account several aspects of SMM behavior by means of seven model parameters.In this paper the effect of each parameter on the non-isothermal rate-dependent behavior of the device is studied, by paying particular attention to the effect of the thermomechanical coupling. Some overall synthetic indicators of the behavior of the shape memory device are defined in terms of the model parameters. By evaluating such indicators, a lot of information about the mechanical, thermal and thermomechanical effects on the device behavior can be gained before computing explicitly the response of the shape memory oscillator.The present work may provide a guide for the proper utilization of the model for the investigation of the dynamic response.     

A thermomechanical model for a 1-D shape memory alloy wire with propagating instabilities

A thermomechanical boundary value problem and constitutive model are presented for a shape memory alloy (SMA) wire under uniaxial loading. The intent is to develop a one-dimensional continuum model of an SMA element that includes all the relevant thermomechanical couplings and is suitable for inclusion in finite element analyses. Thermodynamic relations are derived from phenomenological considerations consistent with recent experimental observations and are calibrated to a typical commercially available NiTi wire material. The model includes both temperature-induced and stress-induced transformations that are necessary to exhibit the shape memory effect and pseudoelastic behaviors. The model accommodates possible unstable mechanical behavior during stress-induced transformations by allowing softening transformation paths and including strain gradient effects. This should provide a tool to study propagating transformation fronts and localized latent heat transfer with the surroundings and a variety of interesting future structural applications, such as composites with embedded SMA elements.     

Nonlinear dynamics and chaos in coupled shape memory oscillators

Shape memory and pseudoelastic effects are thermomechanical phenomena associated with martensitic phase transformations, presented by shape memory alloys. This contribution concerns with the dynamical response of coupled shape memory oscillators. Equations of motion are formulated by assuming a polynomial constitutive model to describe the restitution force of the oscillators and, since they are associated with a five-dimensional system, the analysis is performed by splitting the state space in subspaces. Free and forced vibrations are analyzed showing different kinds of responses. Periodic, quasi-periodic, chaos and hyperchaos are all possible in this system. Numerical investigations show interesting and complex behaviors. Dynamical jumps in free vibration and amplitude variation when temperature characteristics are changed are some examples. This article also shown some characteristics related to chaos–hyperchaos transition.     

Phase proportioning in CuAlBe shape memory alloys during thermomechanical loadings using electric resistance variation

The microstructure of shape memory alloys changes with the thermomechanical history of the material. During thermomechanical loading, austenite, thermally-induced martensite or stress-induced martensite can be simultaneously present in the material. In applications integrating SMA parts, utilization conditions seriously affect the microstructure and can generate macroscopic strain or stress. Consequently, during thermomechanical loadings, it is important to be able to proportion the different phases and consequently to understand the kinetic transformation. This is very useful in the development of constitutive equations. This study shows, by a series of tests, that the proposed experimental method, based on the measurement of the variation of electric resistance of CuAlBe wires, permits to determine the volume fraction of the different phases present in the material (i.e., austenite, stress-induced martensite and thermally-induced martensite). The proposed method is applied to the most common thermomechanical behavior met in engineering applications of shape memory alloys: pseudoelasticity, pseudoplasticity, recovery-stress and stress-assisted two-way shape memory effect. The proportioning method based on a mixture law integrating the resistivity of pure phases present in the SMA is first performed on different two-phase mixture cases and then applied to a three phase mixture case.