Prandtl–Ishlinskii (PI) model has an excellent compromise to characterize an asymmetric saturated hysteresis behavior of shape-memory-alloy (SMA)-driven systems, but it cannot consider thermomechanical relations between components of SMA-driven systems. On the other hand, constitutive models are composed of these relations, but their precision needs to be improved. In this paper, PI model is proposed to boost constitutive models in two cases. In the first case, PI model is used to characterize martensite volume fraction (MVF) called hybrid model. In the second case, the model is applied as a regulator in the output of a constitutive model called PI-based output (PIO) regulator. Due to simplicity and ability of Liang–Rogers (LR) model in transformation phases, it is considered as an MVF in the original constitutive model. The performance of both proposed models is compared with the original LR-based constitutive model. Unknown parameters of all three models are identified using genetic algorithm in MATLAB Toolbox. The performance of the three models is investigated at three different frequencies of \[\frac{{2\pi }}{8}\], \[\frac{{2\pi }}{{15}}\], and \[\frac{{2\pi }}{{30}}\] Hz because the excitation frequency changes the hysteresis behavior. Results show that the proposed hybrid model keeps the precision of the original constitutive model at different frequencies. In addition, the proposed PIO model shows the best performance to predict hysteresis behavior at different frequencies.

It is of practical significance to develop a constitutive model which is able to predict the thermomechanical behaviors of the shape memory effect occurring in a shape memory polymer (SMP) accurately. The mechanism of shape memory effect of SMP is explained based on the assumption that SMP is composed by two phases, reversible phase and stationary phase. Especially the different flow elements are respectively added to the reversible phase and stationary phase in order to express the plastic behavior of SMP. There are two springs in series, one dashpot and one flow element in the reversible phase. There are two springs in parallel, one dashpot and one flow element in the stationary phase. A constitutive equation is developed to express the thermo-mechanical behaviors of shape memory effect in the SMP based on viscous-elastic mechanics and plastic theory. An internal variable, frozen ratio, is defined to follow the shape memory process in SMP, and the material properties are described as the functions of frozen ratio based on phase transition theorem. The developed constitutive model, which includes above constitutive equation and material parameter functions, is used to numerically simulate the thermo-mechanical behaviors of SMP under various load cycles. Results show that the developed constitutive model can not only predict the shape memory process of SMP accurately, but also express the rate-dependent behaviors of SMP effectively.

An analytical research is developed using the averaging technique of composites for the macroscopic behaviors of porous shape memory alloy (SMA) beam with different porosity under pure bending. The whole material is regarded as a composite beam of porous SMA and dense SMA, in which the component fractions of the porous SMA show gradient changes over geometric dimension. To get the theoretical solution of such material under pure bending, the Mises yield theory and the ideal elastoplastic model are used to describe the phase transition of the material. The macroscopic behaviors of the porous SMAs beam with different porosity are then simulated using the averaging technique of composites. Examples for a porous SMA beam with gradient porosity from 0 to 50% considering the tension compression asymmetry of the SMAs are then supplied; the results show that after transformation the stress distribution in the whole material is lower than in the case of the pure elastic gradient porous materials, and for different part of the SMA with different porosity shows different strength characters.

In rock engineering, evaluating the post-peak strength and deformation of rock is necessary. To explore the elasto-plastic behavior of sandstone in the post-peak stage, a series of strain-controlled triaxial tests were conducted under different confining pressures. According to the post-peak characteristics, a constitutive model based on nonlinear elasticity and generalized plasticity is proposed. This proposed model is characterized by the following features: (1) Nonlinear elasticity is observed under hydrostatic and shear loading; (2) the associated flow rule is followed; (3) substantial plastic deformation occurs during shear loading; and (4) post-peak softening behavior is accurately predicted. This model requires twelve material parameters, three for elasticity and nine for plasticity. The proposed model was validated by comparing the triaxial test results of Mushan sandstone at different hydrostatic pressures under dry and saturated conditions. In addition, the model is versatile; it can simulate the deformational behavior of two other sandstones. In summary, the proposed model can reasonably predict the complete stress–strain curve of sandstone.

The predictability of modified constitutive model, based on Arrhenius type equation, for illustrating the flow behavior of Fe–36%Ni Invar alloy was investigated via isothermal hot compression tests. The hot deformation tests were carried out in a temperature range of 850–1100 °C and strain rates from 0.01 to 10 s−1. True stress-true strain curves exhibited the dependence of the flow stress on deformation temperatures and strain rates, which then described in Arrhenius-type equation by Zener–Holloman parameter. Moreover, the related material constants and hot deformation activation energy (Q) in the constitutive model were calculated by considering the effect of strain as independent function on them and employing sixth polynomial fitting. Subsequently, the performance of the modified constitutive equation was verified by correlation coefficient and average absolute relative error which were estimated in accordance with experimental and predicted data. The results showed that the modified constitutive equation possess reliable and stable ability to predict the hot flow behavior of studied material under different deformation conditions. Meanwhile, Zener–Holloman parameter map was established according to the modified constitutive equation and used to estimate the extent of dynamic recrystallization.

The strain hardening effect and dynamic recovery behavior of a Ni-based superalloy are studied by isothermal compressive tests. A new unified dislocation-density based constitutive model is developed to characterize the strain hardening effect and dynamic recovery behavior of the studied superalloy. In the developed constitutive model, some material parameters (yield stress, strain hardening coefficient, and dynamic recovery coefficient) are assumed as functions of initial grain size, deformation temperature, and strain rate. An iterative algorithm is designed to predict the high-temperature deformation behaviors under time-variant hot working conditions. The hot deformation parameters and material parameters can be updated in each strain increment. Comparisons between the experimental and calculated flow stresses indicate that the developed constitutive model can accurately describe the high-temperature deformation behavior of the studied superalloy. Furthermore, the developed constitutive model is also successfully used for analyzing time-variant hot working processes.

A fractional model generalized from the Zener model is proposed for the prediction of temperature-dependent free recovery behaviors of amorphous shape memory polymers (SMPs). This model differs from the Zener model in that it involves nonlinear differential equations of fractional, not integer, order. The theoretical solution based on this fractional model is utilized to simulate the isothermal and nonisothermal free recovery of an amorphous SMP compared with the one based on the Zener model. The results show a reasonable improvement in the prediction of the strain recovery response of SMP by the fractional calculus method.

To evaluate the influence of strain rate on mechanical behavior of nanocrystalline (NC) materials, a phase mixture constitutive model composed of ordered grain interior phase and plastically softer grain boundary dislocation pile up zone phase was built. Because of dissimilar properties and mismatch between the two phases, dislocation density evolution controlling mechanism based on statistically stored dislocations and geometrically necessary dislocations was analyzed and extended to NC regime to consider their disparate effects. Based on the composite model, a new stress–strain constitutive relation for strain rate-dependent behaviors was firstly established based on dislocation density evolution and strain gradient theory. The calculated data were then compared with corresponding experimental curves and strong strain rate-dependent behaviors were exhibited, which indicated that the predictions kept in good agreement with experiments. Further discussions were presented for calculations of strain rate sensitivity and activation volume for NC Ni through the proposed model.

The mechanical properties of a CuCr1Zr alloy were measured with the help of a Gleeble3500 machine at temperatures ranging from 250 °C to 1000 °C. Some tensile tests were alsoperformed at 20 °C. The measured data were first used to determine the thermal linearcoefficient at different temperatures. The measured curves were then simulated with anelasto-visco-plastic constitutive model assuming that the material behaviour wasisotropic. This law requires the knowledge of temperature dependent parameters that weredetermined by inverse method. This work represents a part of a PhD study on hot tearingsusceptibility during welding.