Micro-mechanical behaviors of SMA composite materials under hygro-thermo-elastic strain fields

An analytical procedure to evaluate the behavior of shape memory alloy (SMA) composite under hygrothermal environment is presented. The SMA wires are considered as inclusions embedded in a homogeneous matrix medium of the composite. The inhomogeneity associated with the phase transformation and thermal strains in the SMA wire as well as the hygrothermal strain in the matrix is homogenized using Eshelby’s equivalent inclusion method. In the present work, a similar approach adopted for SMA composites by Marfia and Sacco [Marfia, S., Sacco, E., 2005. Micromechanics and homogenisation of SMA-wire-reinforced materials. J. Appl. Mech. 72 (2), 259–268.] is considered in order to validate the response of SMA composite subjected to thermo-elastic strain field. However, in the present approach, certain modifications and new derivations for the inelastic strain tensors is carried out. First, the constitutive laws for the SMA wire and matrix are expressed in terms of the average strain in the composite. The evolutionary equations used to characterize the pseudoelastic (PE) behavior of the SMA wire are redefined in terms of the eigen strains (phase transformation and thermal strains) occurring in the SMA wire, which are then expressed in terms of the average strain in the composite. Further, the SMA composite constitutive law under coupled hygro-thermo-elastic strain fields is proposed. The generic homogenized hygric and thermal inelastic composite tensors required for the proposed hygro-thermo-elastic constitutive law are derived. Finally, the SMA composite lamina is characterized using Eshelby’s equivalent inclusion method. Using the proposed modifications and derivations, the analytical results are validated for the case of thermo-elastic strain fields and the procedure is then extended to evaluate the SMA composite behavior under hygro-thermo-elastic strain fields. The results include the effect of thermo-elastic and hygro-thermo-elastic strains on the transformation stresses and the nature of hysteresis due to hygric and thermo-elastic strains.     

Modeling rate-independent hysteresis in large deformations of preconditioned soft tissues

A phenomenological model is proposed for characterizing rate-independent hysteresis exhibited by preconditioned soft tissues. The preconditioned tissue is modeled as an isotropic composite of a hyperelastic component and a dissipative (inelastic) component. Specifically, the constitutive equations are hyperelastic in the sense that the stress is determined by derivatives of a strain energy function. Inelasticity of the dissipative component is controlled by a yield function with different functional forms for the hardening variable during deformation loading and unloading. The constitutive equations proposed in this paper are simple. In particular, they depend on only seven material constants: three controlling the response of the elastic component and the remainder controlling the response of the dissipative component. More importantly, the material constants can be determined to match rather general loading and unloading behavior. It is observed that the hysteretic response of the model compares well with experimental data for passive uniaxial loading/unloading of Manduca muscle. Moreover, the present model treats partial loading and reloading of preconditioned tissue as elastic–plastic response, which is different from the treatment of pseudo-elastic models used in the literature.     

A combined analytical,numerical, and experimental study of shape-memory-alloy helical springs

In this paper, the pseudoelastic response of shape memory alloy (SMA) helical springs under axial force is studied both analytically and numerically. In the analytical solution two different approximations are considered. In the first approximation, both the curvature and pitch effects are assumed to be negligible. This is the case for helical springs with large ratios of mean coil radius to the cross sectional radius (spring index) and small pitch angles. Using this assumption, analysis of the helical spring is reduced to that of the pure torsion of a straight bar with circular cross section. A three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is reduced to the one-dimensional pure shear case and a closed-form solution for torsional response of SMA bars in loading and unloading is obtained. In the next step, the curvature effect is included and the SMA helical spring is analyzed using the exact solution presented for torsion of curved SMA bars. In this refined solution, the effect of the direct shear force is also considered. In the numerical analyses, the three-dimensional constitutive equations are implemented in a finite element method and using solid elements the loading–unloading of an SMA helical spring is simulated. Analytical and numerical results are compared and it is shown that the solution based on the SMA curved bar torsion gives an accurate stress analysis in the cross section of the helical SMA spring in addition to the global load–deflection response. All the results are compared with experimental data for a Nitinol helical spring. Several case studies are presented using the proposed analytical and numerical solutions and the effect of changing different parameters such as the material properties and temperature on the loading–unloading hysteretic response of SMA helical springs is studied. Finally, some practical recommendations are given for improving the performance of SMA helical springs used as energy dissipating devices, for example for seismic applications.     

Localization in NiTi tubes under bending

Nearly equiatomic NiTi can exhibit pseudoelastic behavior due to reversible solid-to-solid stress induced phase transformation at room level temperatures. In tension, the transformation leads to localized deformation of several percent that tends to spread at nearly constant stress. The deformation is recovered upon unloading while again localized deformation is exhibited. Under compression, while still pseudoelastic, the transformation strains are smaller, the stress is higher, the response is monotonic, and the deformation is essentially homogeneous. This study examines how this texture-driven, complex material asymmetry affects a simple structure: the bending of a tube. To this end, NiTi tubes are bent in a custom four-point bending facility under rotation control and isothermal conditions. The phase transformations lead to a closed moment-rotation hysteresis comprised of loading and unloading moment plateaus. During loading, localized nucleation of martensite results in a high curvature for the transformed sections of the tube and low curvature for the untransformed. Martensite, which corresponds to the higher curvature regime, spreads gradually while the moment remains nearly constant. The nucleation of martensite is in the form of bands inclined to the axis of the tube that organize themselves into diamond shaped deformation patterns on the tensioned side of the structure. The patterns are similar to those observed in bending of steel tubes with Lüders bands, however, for NiTi they develop only on the tensioned side due to the material asymmetry. A lower moment plateau is traced upon unloading with similar localized bending and the erasure of the diamond deformation patterns. This complex behavior was found to repeat for a number of temperatures in the pseudoelastic regime of NiTi with the moment-rotation hysteresis moving to higher or lower moment levels depending on the temperature.     

Time-dependent behavior of passive skeletal muscle

An isotropic three-dimensional nonlinear viscoelastic model is developed to simulate the time-dependent behavior of passive skeletal muscle. The development of the model is stimulated by experimental data that characterize the response during simple uniaxial stress cyclic loading and unloading. Of particular interest is the rate-dependent response, the recovery of muscle properties from the preconditioned to the unconditioned state and stress relaxation at constant stretch during loading and unloading. The model considers the material to be a composite of a nonlinear hyperelastic component in parallel with a nonlinear dissipative component. The strain energy and the corresponding stress measures are separated additively into hyperelastic and dissipative parts. In contrast to standard nonlinear inelastic models, here the dissipative component is modeled using an evolution equation that combines rate-independent and rate-dependent responses smoothly with no finite elastic range. Large deformation evolution equations for the distortional deformations in the elastic and in the dissipative component are presented. A robust, strongly objective numerical integration algorithm is used to model rate-dependent and rate-independent inelastic responses. The constitutive formulation is specialized to simulate the experimental data. The nonlinear viscoelastic model accurately represents the time-dependent passive response of skeletal muscle.     

Modelling large deformation behaviour under loading–unloading of semicrystalline polymers: Application to a high density polyethylene

In this work, the large deformation behaviour under monotonic loading and unloading of a high density polyethylene (HDPE) is studied. To analyze the nonlinear time-dependent response of the material, mechanical tests were conducted at room temperature under constant true strain rates and stress relaxation conditions. A physically-based inelastic model written under finite strain formulation is proposed to describe the mechanical behaviour of HDPE. In the model, the inelastic mechanisms involve two parallel elements: a visco-hyperelastic network resistance acting in parallel with a viscoelastic–viscoplastic intermolecular resistance where the amorphous and crystalline phases are explicitly taken into consideration. The semicrystalline polymer is considered as a two-phase composite. The influence of the crystallinity on the loading and unloading behaviour is investigated. Numerical results are compared to experimental data. It is shown that the model is able to accurately reproduce the experimental observations corresponding to monotonic loading, unloading and stress relaxation behaviours at different strain levels. Finally, the model capabilities to capture cyclic loading–unloading behaviour up to large strains are discussed. To demonstrate the improved modelling capabilities, simulations are also performed using the original model of Boyce et al. [Boyce, M.C., Socrate, S., Llana, P.G., 2000. Constitutive model for the finite deformation stress–strain behavior of poly(ethylene terephthalate) above the glass transition. Polymer 41, 2183–2201] modified by Ahzi et al. [Ahzi, S., Makradi, A., Gregory, R.V., Edie, D.D., 2003. Modeling of deformation behavior and strain-induced crystallization in poly(ethylene terephthalate) above the glass transition temperature. Mechanics of Materials 35, 1139–1148].     

Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals

Based on the knowledge of the anisotropy associated with the martensitic transformations obtained from tension/compression experiments with oriented CuAlNi single crystals, a simple constant stress averaging approach is employed to model the SMA polycrystal deformation behaviors. Only elastic and inelastic strains due to the martensitic transformation, variant reorientations in the martensite phase and martensite to martensite transformations in thermomechanical loads are considered. The model starts from theoretical calculation of the stress-temperature transformation conditions and their orientation dependence from basic crystallographic and material attributes of the martensitic transformations. Results of the simulations of the NiTi, NiAl, and Cu-based SMA polycrystals in stress–strain tests are shown. It follows that SMA polycrystals, even with randomly oriented grains, typically exhibit tension/compression asymmetry of the shape of the pseudoelastic σ−ε curves in transformation strain, transformation stress, hysteresis widths, character of the pseudoelastic flow and in the slope of temperature dependence of the transformation stresses. It is concluded that some macroscopic features of the SMA polycrystal behaviors originate directly from the crystallography of the undergoing MT's. The model shows clearly the crystallographic origin of these phenomena by providing a link from the crystallographic and material attributes of martensitic transformations towards the macroscopic σ−ε−T behaviors of SMA polycrystals.     

A two-level micromechanical theory for a shape-memory alloy reinforced composite

A two-level micromechanical theory is developed to study the influence of the shape and volume concentration of shape-memory alloy (SMA) inclusions on the overall stress–strain behavior of a SMA-reinforced composite. The first level exists on the smaller SMA level, in which, under the action of stress, parent austenite may transform into martensite. The second level is on the larger scale consisting of the metastable SMA inclusions and an inactive polymer matrix. The evolution of martensite microstructure is evaluated from the irreversible thermodynamics, in conjunction with the micromechanics and physics of martensitic transformation. By taking martensite to exist in the form of thin plates on the micro scale and assuming SMA inclusions to be homogeneously aligned spheroids on the macro scale, the overall stress–strain behaviors of a NiTi-reinforced composite are calculated for various SMA shapes and concentrations. The results indicate that, under a tensile axial loading, martensitic transformation is easier to take place when SMA inclusions exist in the form of long fibers, but most difficult to occur when they are in the form of flat discs. In general the levels of the applied stress at which martensite transformation commences, finishes, and austenitic transformation starts, and finishes, are found to decrease with increasing aspect ratio of the SMA inclusions while the damping capacity increases with it; these properties point to the advantage of using fibrous composites for actuators or sensors under a tensile loading.     

Coupling between martensitic phase transformations and plasticity: A microstructure-based finite element model

A microstructural finite element (MFE) model is developed to capture the interaction between martensitic transformations and plasticity in NiTi shape memory alloys (SMAs). The interaction is modeled through the grain-to-grain redistribution of stress caused by both plasticity and phase transformation, so that each mechanism affects the driving force of the other. A unique feature is that both processes are modeled at a crystallographic level and are allowed to operate simultaneously. The model is calibrated to pseudoelastic data for select single crystals of Ti–50.9at.%Ni. For polycrystals, plasticity is predicted to enhance the overall martensite volume fraction at a given applied stress. Upon unloading, residual stress can induce remnant (retained) martensite. For thermal cycling under load bias, plasticity is observed to limit the net transformation strain/cycle and increase the hysteretic width. Deformation processing, via plastic pre-straining at elevated temperature, is shown to dramatically alter subsequent pseudoelastic response, as well as induce two-way shape memory behavior during no-load thermal cycling. Overall, the model is suitable at smaller imposed strains, where martensite detwinning is not expected to dominate.     

A mechanics prediction of the behaviour of mono-crystalline silicon under nano-indentation

This paper uses a new constitutive model developed recently by the authors to analyse the multi-phase transformations in mono-crystalline silicon when subjected to nano-indentation. The finite element method is employed to the integration of the stress–strain relationship. A very good agreement is reached with the experimental measurements, with an accurate prediction of the observed pop-in and pop-out and detailed phase transformation events in loading and unloading. It was found that due to the change of microstructures, the material in the deformation zone could experience a local unloading during indentation loading or undergo a local loading during indentation unloading. The phase transformation events during indentation are closely related to the variation of both the deviatoric and hydrostatic stress components.     

Strain-rate effects in rheological models of inelastic response

Stress–strain response under constant and variable strain-rate is studied for selected models of inelastic behavior. The derived closed-form solutions for uniaxial loading enable simple evaluation of the strain-rate effects on the material response. The effect of an abrupt change of strain-rate is also examined. Non-Newtonian viscosity which decreases with an increasing strain-rate is incorporated in the analysis. Parabolic and hyperbolic hardening are used to describe the plastic response in monotonic loading. A three-dimensional generalization of an elastic–viscoplastic model is employed to study the stress relaxation in simple shear. A combined isotropic–kinematic hardening and the concept of overstress are used in the analysis. The unloading nonlinearity of the stress–strain curve is then discussed.     

Higher-order theory for periodic multiphase materials with inelastic phases

An extension of a recently-developed linear thermoelastic theory for multiphase periodic materials is presented which admits inelastic behavior of the constituent phases. The extended theory is capable of accurately estimating both the effective inelastic response of a periodic multiphase composite and the local stress and strain fields in the individual phases. The model is presently limited to materials characterized by constituent phases that are continuous in one direction, but arbitrarily distributed within the repeating unit cell which characterizes the material's periodic microstructure. The model's analytical framework is based on the homogenization technique for periodic media, but the method of solution for the local displacement and stress fields borrows concepts previously employed by the authors in constructing the higher-order theory for functionally graded materials, in contrast with the standard finite-element solution method typically used in conjunction with the homogenization technique. The present approach produces a closed-form macroscopic constitutive equation for a periodic multiphase material valid for both uniaxial and multiaxial loading. The model's predictive accuracy in generating both the effective inelastic stress-strain response and the local stress and inelastic strain fields is demonstrated by comparison with the results of an analytical inelastic solution for the axisymmetric and axial shear response of a unidirectional composite based on the concentric cylinder model and with finite-element results for transverse loading.     

Stress–softening effect of SBR/nanocomposites by a phenomenological Gent–Zener viscoelastic model

An experimental study of a tensile loading–unloading procedure, as well as multi-cyclic response in a strain-controlled program of a Styrene-Butadiene (SBR) elastomer reinforced with four different weight fractions of carbon nanotubes (CNTs) has been performed. The Mullins effect features, namely hysteresis, damage and residual strain, exhibited by the SBR/nanocomposites were analyzed by a modified Gent–Zener rheological model, and a damage function. Especially for the multi-cyclic stress–strain curves, phenomenological equation of the model parameters evolution with strain were also introduced. The same loading procedure was applied in pre-stressed materials, revealing a different stress–strain response due to strain prehistory. The model has been proven to accurately capture the loading–unloading behavior, the residual strain, hysteresis loops as well as the multi-cyclic behavior of the SBR/CNT nanocomposites.     

Local displacements and load transfer in shape memory alloy composites

Although there has been a significant amount of research dedicated to characterizing and modeling the response of shape memory alloys (SMAs) alone, little experimental work has been done to understand the behavior of SMAs embedded in a host material. The interaction between SMA wires and a host polymer matrix was investigated by correlating local displacements and stress fields induced by the embedded wires with SMA/polymer adhesion. Most SMA composite applications require transfer of strain from the wire to the matrix. In these applications, maximum interfacial adhesion between the SMA wire and the polymer matrix is most desirable. The adhesion was varied by considering four different surface treatments: untreated, acid etched, hand sanded and sandblasted. The average interfacial bond strength of the SMA wires embedded in an epoxy matrix was measured by standard pull out tests. Sandblasting significantly increased the bond strength, whereas hand sanding and acid cleaning actually reduced interface strength. In situ displacements of embedded, surface-treated SMA wires were measured using heterodyne interferometry, whereas the resulting stresses induced in the polymer matrix were investigated using photoelasticity. Increased wire adhesion resulted in lower axial wire displacement and higher interfacial stresses due to the restraining effect of the matrix on the actuated wire. A simplified theoretical analysis was carried out to estimate the shear stress induced in the matrix due to wire actuation. The maximum shear stress predicted for the case of a perfect interfacial bond was about 7 percent larger than the value measured experimentally for the sand-blasted wire.     

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.     

An hysteretic quadrilateral plane stress element

In this work, a new plane stress element is proposed for the nonlinear static and dynamic analysis of plane stress/plane strain problems. The four node quadrilateral element formulation for the elastic case is extended by introducing a novel hysteretic constitutive relation, based on the Bouc–Wen model of hysteresis. The hysteretic model introduced is directly derived from the governing equations of classical plasticity based on the flow rule and specific hardening law. The stiffness matrix of the element is formulated using the principle of virtual displacements, where the elastic stress–strain relation is substituted by the hysteretic relation proposed. The derived stiffness matrix is expressed as a smooth function of the internal stress field both in the elastic and inelastic regime. The efficiency of the proposed element in the simulation of the cyclic behavior in plane structures is presented through illustrative examples.     

A micromechanics analysis for the microstructure design of a two-phase pseudoelastic composite

A micromechanics analysis on the possibility of designing a two-phase pseudoelastic composite is made for the case where ductile transformable shape memory alloy plastic particles are imbedded coherently in an elastic matrix. It is demonstrated that a pseudoelastic stress-strain loop in a macroscopic loading-unloading cycle can be obtained by microscopically stress induced forward and reverse martensitic transformations in the SMA particles. The relation between the macroscopic stress-strain response and the material parameters of the constituents of this composite is quantified through the micromechanics calculations, which reveals that the best ductility and thus the greatest energy absorption capacity of this novel microstructure can be obtained by the optimum material design.     

Internal loops in superelastic shape memory alloy wires under torsion – Experiments and simulations/predictions

Understanding torsional responses of shape memory alloy (SMA) specimens under partial or fully transformed cases with internal loops is of particular importance as the entire response might not be always utilized and only a portion of the entire response (internal loop) might be of significance to designers. In this work, we present experimental results of large complex loading and unloading torsional cycles which were conducted on superelastic SMA wires, under isothermal conditions with the purpose of elucidating the torsional internal loop response during loading and unloading. Such data hereto has not been available in open literature. Utilizing this data, we model the torsional response of superelastic SMA wires subjected to various loading and unloading situations that can result in different extents of transformation.A thermodynamically consistent Preisach model (Rao and Srinivasa, 2013) captures such complex internal loops with a high degree of precision by modeling driving force for phase transformation vs. volume fraction of martensite relationships. This approach is different from capturing purely phenomenological stress–strain or stress–temperature Preisach models. The thermodynamic approach utilized here has broader predictive capability. The model predictions indicate good agreement with the internal loop structures even though only the outer loop information was used for model calibration. The addition of a single inner loop information for model calibration greatly improves the predictions.     

Field fluctuations and macroscopic properties for nonlinear composites

A recently introduced nonlinear homogenization method [J. Mech. Phys. Solids 50 ( 2002) 737–757] is used to estimate the effective behavior and the associated strain and stress fluctuations in two-phase, power-law composites with aligned-fiber microstructures, subjected to anti-plane strain, or in-plane strain loading. Using the Hashin–Shtrikman estimates for the relevant “linear comparison composite,” results are generated for two-phase systems, including fiber-reinforced and fiber-weakened composites. These results, which are known to be exact to second-order in the heterogeneity contrast, are found to satisfy all known bounds. Explicit analytical expressions are obtained for the special case of rigid-ideally plastic composites, including results for arbitrary contrast and fiber concentration. The effective properties, as well as the phase averages and fluctuations predicted for these strongly nonlinear composites appear to be consistent with deformation mechanisms involving shear bands. More specifically, for the case where the fibers are stronger than the matrix, the predictions appear to be consistent with the shear bands tending to avoid the fibers, while the opposite would be true for the case where the fibers are weaker.     

A time dependent micro-mechanical fiber composite model for inelastic zone growth in viscoelastic matrices

In this work, a fiber composite model is developed to predict the time dependent stress transfer behavior due to fiber fractures, as driven by the viscoelastic behavior of the polymer matrix, and the initiation and propagation of inelastic zones. We validate this model using in situ, room temperature, micro-Raman spectroscopy fiber strain measurements. Multifiber composites were placed under constant load creep tests and the fiber strains were evaluated with time after one fiber break occurred. These composite specimens ranged in fiber volume fraction and strain level. Comparison between prediction and MRS measurements allows us to characterize key in situ material parameters, the critical matrix shear strain for inelastic zones and interfacial frictional slip shear stress. We find that the inelastic zone is predominately either shear yielding or interfacial slipping, and the type depends on the local fiber spacing.