Zhonghe Bo

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.

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.

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.

Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part II : material characterization and experimental results for a stable transformation cycle

Abstract Material characterization of polycrystalline Shape Memory Alloys (SMAs) during a stable phase transformation cycle is presented in this paper using the constitutive model established by Bo and Lagoudas (Z. Bo, D.C. Lagoudas, accepted for publication in International Journal of Engineering Science) in the first paper of this series, to be referenced here as Part I. In addition to the constitutive equations obtained in Part I, the energy balance equation describing the heat exchange during phase transformation is derived in this paper using the first law of thermodynamics. In the present study, we assume that the plastic strains, which accumulate with the number of applied transformation cycles, remain constant during a single transformation cycle. To use thermomechanical experimental results performed on SMA wires, the model developed in Part I is reduced to a 1-D form, and a procedure for the determination of the material constants is discussed in detail. A series of experiments performed on NiTi SMA wires undergoing thermally induced phase transformation under constant applied load is utilized for model simulations and subsequent comparisons with model predictions. Using the present model, thermally induced phase transformation under varying magnitude of applied load can be modeled for both untrained and trained SMAs. Finally, an application of the model to stress induced phase transformation in thin SMA specimens, where transformation strain localization occurs, is discussed. Part III of this series of four papers on SMAs will study the evolution of plastic strains accumulating in thermally induced cyclic phase transformation under constant applied load, therefore fully characterizing the thermomechanical response of SMAs undergoing multiple operating cycles. Finally, in Part IV, the thermomechanical response of SMAs under minor hysteresis loops will be investigated.

Micromechanics of Active Composites With SMA Fibers

The study of the effective thermomechanical response of active fibrous composites with shape memory alloy (SMA) fibers is the subject of this work. A 3-D constitutive response for the SMA fibers is formulated first. To model thermomechanical loading path dependence, an incremental approach is used assuming that within each stress and temperature increment the volume fraction of the martensitic phase remains constant in the SMA fibers. The Mori-Tanaka averaging scheme is then used to give an estimate of the instantaneous effective thermomechanical properties in terms of the thermomechanical properties of the two phases and martensitic volume fraction. A unit cell model for a periodic active composite with cubic and hexagonal arrangement of fibers is also developed to study the effective properties using finite element analysis. It is found that since the fibers and not the matrix undergo the martensitic phase transformation that induces eigenstrains, the Mori-Tanaka averaging scheme accurately models the thermomechanical response of the composite, relative to the finite element analysis, for different loading paths. Specific results are reported for the composite pseudoelastic and shape memory effect for an elastomeric matrix continuous SMA fiber composite.

Material Characterization of SMA Actuators Under Nonproportional Thermomechanical Loading

The application of SMA actuators in smart structures usually involves thermally induced phase transformation with a variable applied stress in SMA actuators, resulting in thermomechanical nonproportional loading of SMAs in stress-temperature space. To investigate the constitutive response of SMAs under thermomechanical nonproportional loading, experiments for thermally induced phase transformation of binary NiTi SMA wires loaded by an elastic linear spring are performed in the Active Materials Laboratory at Texas A&M University A constitutive model developed by Bo and Lagoudas (1998a, b, c) and lagoudas and Bo (1998) is used as the basis for the theoretical prediction of the response of SMA line actuators loaded by springs with different elastic constants. Model predictions are compared with experimental results.

The cylindrical bending of composite plates with piezoelectric and SMA layers

A laminated plate with orthotropic piezoelectric layers undergoing cylindrical bending under the application of electromechanical loading is considered. A known elasticity solution is extended to the coupled electroelastic field case by introducing, in addition to an Airy stress function, the electric potential for every layer. The validity of the way electric boundary conditions are applied in classical beam or plate theories is discussed. The electroelastic solution is applied to the non-linear case of a hybrid composite plate with piezoelectric layers attached to shape memory alloy (SMA) thin strips. An incremental piecewise elastic formation of the problem incorporates both stiffness changes and transformation strains in the SMA layers. The study of a laminated plate with PZT layers attached on both sides of an NiTi SMA layer shows that a full loading-unloading cycle in the applied electric field leads to energy dissipation in the laminated plate due to the dissipative process of stress induced martensitic phase transformation.

Thermodynamic constitutive model for cyclic loading of shape memory alloy materials with application to two-way training

The thermomechanical response of shape memory alloy (SMA) materials under cyclic loading is modeled in this paper. A set of evolution laws for plastic strains is first developed, based on Bodners viscoplasticity model, by replacing real time in Bodners model with an internal time variable proportional to the martensitic volume fraction. The influence of plastic residual stresses on the martensitic phase transformation is analyzed, and evolution equations for the plastic back stresses and isotropic hardening parameter during phase transformation are developed. The relationship between accumulation of plastic strains and creation of the two way shape memory effect is quantitatively explained by the present model. The changing of the stress-strain hysteresis loop and transformation start and finish stresses and temperatures are also correctly accounted by the present formulation.

Unified thermodynamic constitutive model and finite element analysis of active metal matrix composites

A unified thermodynamic constitutive model for Shape Memory Alloy (SMA) materials, derived by a proper extension of the thermodynamic model proposed by Boyd and Lagoudas is presented in this paper. The specific selections for the form of the mixing energy associated with the martensitic volume fraction are identified for several earlier models. The heat energy released or absorbed during the forward or reverse transformation predicted by the different models is compared with the experimental data from calorimetric measurements. The constitutive model is implemented in a finite element analysis scheme using a return mapping integration technique for the incremental formation of the model. Finally, the constitutive model is utilized to analyze the thermomechanical response of an active metal matrix composite with embedded SMA fibers. Both tetragonal and hexagonal periodic arrangements of SMA fibers are considered in the calculation and the results from the different arrangements of SMA fibers are compared.

Thermodynamic constitutive model for gradual phase transformation of SMA materials

A thermally induced phase transformation in a Shape Memory Alloy (SMA) polycrystal occurs gradually over a range of temperatures unlike the first-order transition (at a single temperature) of a SMA monocrystal. This spread in transformation temperatures is believed to be caused by such factors as material inhomogeneities and internal stresses in the polycrystal. Taking a recently proposed thermodynamic model by Bo and Lagoudas for the Two-Way Memory Effect of SMA materials as a starting point, we extend it to account for the presence of initial heterogeneities, by introducing a statistical distribution in the Gibbs free energy function. The extended theory is then used as a basis to correlate strain recovery vs. temperature measurements from thermally induced cyclic phase transformation in untrained polycrystalline wires.