Dimitris C. Lagoudas

A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloy

Pseudoelasticity and the shape memory effect (SME) due to martensitic transformation and reorientation of polycrystalline shape memory alloy (SMA) materials are modeled using a free energy function and a dissipation potential. Three different cases are considered, based on the number of internal state variables in the free energy: (1) austenite plus a variable number of martensite variants; (2) austenite plus two types of martensite; and (3) austenite and one type of martensite. Each model accounts for three-dimensional simultaneous transformation and reorientation. The single-martensite model was chosen for detailed study because of its simplicity and its ease of experimental verification. Closed form equations are derived for the damping capacity and the actuator efficiency of converting heat into work. The first law of thermodynamics is used to demonstrate that significantly more work is required to complete the adiabatic transformation than the isothermal transformation. Also, as the hardening due to the austenite/martensite misfit stresses approaches zero, the transformation approaches the isothermal, infinite specific heat conditions of a first-order transformation. In a second paper, the single-martensite model is used in a mesomechanical derivation of the constitutive equations of an active composite with an SMA phase.

Aerospace applications of shape memory alloys

Abstract With the increased emphasis on both reliability and multi-functionality in the aerospace industry, active materials are fast becoming an enabling technology capturing the attention of an increasing number of engineers and scientists worldwide. This article reviews the class of active materials known as shape memory alloys (SMAs), especially as used in aerospace applications. To begin, a general overview of SMAs is provided. Their useful properties and engineering effects are described and the methods in which these may be utilized are discussed. A review of past and present aerospace applications is presented. The discussion addresses applications for both atmospheric earth flight as well as space flight. To complete the discussion, SMA design challenges and methodologies are addressed and the future of the field is examined.

Numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms

Previous studies by the authors and their co-workers show that the structure of equations representing shape Memory Alloy (SMA) constitutive behaviour can be very similar to those of rate-independent plasticity models. For example, the Boyd–Lagoudas polynomial hardening model has a stress-elastic strain constitutive relation that includes the transformation strain as an internal state variable, a transformation function determining the onset of phase transformation, and an evolution equation for the transformation strain. Such a structure allows techniques used in rate-independent elastoplastic behaviour to be directly applicable to SMAs. In this paper, a comprehensive study on the numerical implementation of SMA thermomechanical constitutive response using return mapping (elastic predictor-transformation corrector) algorithms is presented. The closest point projection return mapping algorithm which is an implicit scheme is given special attention together with the convex cutting plane return mapping algorithm, an explicit scheme already presented in an earlier work. The closest point algorithm involves relatively large number of tensorial operations than the cutting plane algorithm besides the evaluation of the gradient of the transformation tensor in the flow rule and the inversion of the algorithmic tangent tensor. A unified thermomechanical constitutive model, which does not take into account reorientation of martensitic variants but unifies several of the existing SMA constitutive models, is used for implementation. Remarks on numerical accuracy of both algorithms are given, and it is concluded that both algorithms are applicable for this class of SMA constitutive models and preference can only be given based on the computational cost. Copyright

Thermomechanical response of shape memory composites

A micromechanics method based on the Mori-Tanaka averaging scheme is used to predict the effective thermomechanical properties of composite materials reinforced by Shape Memory Alloy (SMA) fibers. Both elastic stiffness changes and transformation strains are taken into account in the constitutive modelling of the SMA fibers. Isothermal longitudinal and transverse stress input and stress-free thermal loading are imposed on the composite, and the composite transformation stress, the maximum transformation strain, and the hysteresis are computed. In con trast to a monolithic SMA, stress-free thermal loading of a shape memory composite is shown to produce transformation strains due to thermal stress induced phase transformation. Closed form solutions for the effective martensite and austenite start temperatures indicate that these temperatures are higher than those of the monolithic SMA material and they depend on the composite processing temperature.

Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part I: theoretical derivations

A generic form of Gibbs free energy for polycrystalline Shape Memory Alloys (SMAs) is first obtained in this work, by forming the increments of both elastic potential energy and Gibbs chemical energy over a Representative Volume Element (RVE) with respect to an infinitesimal increment of martensite. A set of internal state variables, i.e., martensitic volume fraction, macro-transformation strain, and back and drag stresses due to both martensitic phase transformation and its interaction with plastic strains, are introduced. The evolution of these internal state variables during phase transformation is proposed based on the micromechanical analysis over the RVE. Four primary mechanisms governing the transformation induced hardening effect are discussed. It is concluded that the back and drag stresses related to plastic deformation do not remain constant, but they vary during phase transformation, even though the local plastic residual stresses are assumed to be constant. The initial material heterogeneity of SMAs, which is essential for the initiation of the phase transformation, is modeled by an initial residual stress field, which can be described by a probability distribution function. In this Part I of a four-part paper, the theoretical derivations are presented. Specific cases of the thermomechanical response of SMAs predicted by the model will be presented in Part II, together with experimental results for phase transformation at constant plastic strains. Experimental results and model predictions for cyclic loading of SMAs with evolving plastic strains will be considered in Part III, while the modeling of minor hysteresis loops of SMAs will be presented in Part IV of this series of four papers on SMAs.

Influence of cold work and heat treatment on the shape memory effect and plastic strain development of NiTi

The influence of cold work and heat treatment on the shape memory effect and plastic strain development is investigated for the thermally induced phase transformation of NiTi shape memory alloys (SMAs) under constant applied stress. Fully annealed SMA wire specimens of identical chemical composition are cold rolled to reduce the specimen width 10, 20, 30 and 40% of the initial wire diameter and then annealed at 300, 400 or 500°C for 15 min. Thermally induced phase transformations under constant applied stress, at various levels up to 500 MPa, are performed to identify the effect of the cold work percentage and annealing temperature on the development of transformation and plastic strain in SMA specimens with one-way shape memory. Results show the maximum transformation strain is independent of cold work percentage and annealing temperature, and increased cold work, for similar annealing temperatures, raises the stress level for the onset of plastic strain and decreases the additional plastic strain development. Also, a reduction in annealing temperature, for similar cold work percentage, raises the stress level for the onset of plastic strain and decreases the additional development of plastic strain. In addition to the one-way SMAs described above, specimens with similar heat treatments are trained with two-way shape memory under a constant applied stress of 300 MPa. Increased cold work percentage is shown to reduce the plastic strain accumulation during the two-way training, as well as the amount of two-way strain developed in the specimens. A comparison between the untrained and trained specimens of identical cold work and annealing temperature shows a reduction in the maximum transformation strain.

On thermomechanics and transformation surfaces of polycrystalline NiTi shape memory alloy material

Abstract A thermomechanical description of the martenstitic phase transformation and the associated shape memory effect in polycrystalline shape memory alloys (SMAs) is presented. The rate-independent constitutive relations are derived in the stress-temperature space using a Lagrangian formulation. The Kuhn–Tucker optimality conditions, constraints on evolution equations for transformations strain and shape of transformation function in thermodynamic force space are obtained naturally through the principle of maximum transformation dissipation. Various transformation functions are investigated and a generalized type transformation function is proposed. Numerical results of the model based on different transformation functions are compared with experimental results to determine their accuracy to predict SMA characteristics like tension–compression asymmetry, negative volumetric transformation strain and pressure dependence.

Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part III: evolution of plastic strains and two-way shape memory effect

Abstract In this paper, the evolution of plastic strains and Two-Way Shape Memory effect (TWSM) of polycrystalline NiTi Shape Memory Alloys (SMAs) with respect to cyclic thermally induced transformation cycles are investigated. Experimental results of the cyclic thermally induced phase transformation under constant applied load are first presented. It is observed that the accumulation of plastic strains continues beyond a large number of cycles (2000), while TWSM is saturated and remains stable after 100–300 transformation cycles, depending on the magnitude of the applied load. Motivated by the experimental observations, and based on the general framework established in the first two parts of this series of four papers on SMAs (Z. Bo, D.C. Lagoudas, International Journal of Engineering Science, submitted; D.C. Lagoudas, Z. Bo, International Journal of Engineering Science, submitted), evolution equations for the accumulation of plastic strains and plasticity related back and drag stresses, which govern the evolution of TWSM, are proposed. Finally, model predictions are compared with experimental data, and a procedure for the determination of material constants used in the present model is discussed. Combined with the formulation developed in Parts I and II, the cyclic thermomechanical response of SMAs undergoing complete transformation cycles can be fully characterized using the present model. In Part IV of this series of four papers on SMAs, modeling of minor hysteresis loops of SMAs will be investigated.

On the numerical evaluation of Eshelby's tensor and its application to elastoplastic fibrous composites

A numerical evaluation of Eshelbys S tensor for an ellipsoidal inclusion imbedded in a general anisotropic matrix material is performed. The numerical scheme is valid for any degree of matrix anisotropy and for any aspect ratio of the ellipsoid, including the extreme cases of cracks and cylindrical inclusions. The influence of matrix anisotropy on the evaluation of S is tested extensively for cylindrical inclusions by considering plasticity induced anisotropy in the instantaneous properties of an elastic-plastic matrix material. The Mori-Tanaka averaging method is used to study the influence of the evaluation of S on the prediction of instantaneous effective properties of fibrous composites with elastic fibres and elastic-plastic matrix.

Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part IV: modeling of minor hysteresis loops

A thermomechanical model for the hysteretic response of Shape Memory Alloys (SMAs) is proposed in this paper by expanding a previous model developed by Bo and Lagoudas (Z. Bo, D.C. Lagoudas, International Journal of Engineering Science, accepted) to include minor hysteresis loops. The constitutive model for SMAs previously developed by Bo and Lagoudas is reviewed first, and a simplification for the case of fully trained SMAs with a stable major hysteresis loop is then presented. An evolution equation for the energy dissipation is proposed based on the theoretical analysis and experimental observations. A connection between the Preisach hysteresis model and the present thermomechanical model is also investigated. The memory (wiping out) property for the present model is determined in a way similar to that of the Preisach model. Comparisons between the present model predictions and the experimental results show that the present model accurately predicts the minor loop hysteresis response of SMAs, even when such minor loops are close to the transformation start and finish points. Compared with the Preisach model, the present hysteresis model follows a thermodynamic formulation, which makes it easier to account for the effects of changing loading paths and two-way memory effect induced by cyclic loading. The developed numerical implementation algorithm can also be easily incorporated in conjunction with other existing thermomechanical constitutive models, thus providing a general scheme for the modeling of hysteretic response of SMAs, based on physical parameters.