Phase Transformation Equation Research Articles

Investigation on Stress Field near Crack-Tip of Mode I in Shape Memory Alloy

In this paper a phase transformation equation is supposed to describe the phase transformation behaviors of the shape memory alloy (SMA) under complex stress state. The stress field near crack-tip of mode I in SMA at various temperatures is investigated based on the supposed phase transformation equation and linear elastic fracture mechanics. Results show both the martensite region and the mixed region of martensite and austenite near the crack-tip become larger with the decrease of temperature. The fracture mechanics behaviors of SMA are much influenced by the temperature.

Effect of Aging Precipitation on Properties of Cu-Ni-Si-Mg Alloy

The Cu-Ni-Si-Mg alloy was treated by aging after varying cold deformation. The effect of cold deformation, aging temperature and aging time on the microstructure and properties of the alloy were investigated. The results indicate that the alloy produce a larger number of slip bands and dislocations after cold deformation, it can accelerate the precipitation of phase during aging, improve the microhardness and electrical conductivity. The microhardness and electrical conductivity of the alloy with 80% deformation aging at 723 K for 1h are 242HV and 35.53%IACS, respectively. The volume fraction of the precipitated phase was calculated by measuring the electrical conductivity of the alloy with 80% deformation aging at 773 K, then both the equations of phase transformation and the electrical conductivity equation were obtained.

Effect of Laminar Cooling on Phase Transformation Evolution in Hot Rolling Process

Abstract The effect of temperature variation owing to the cooling pattern (CP) on the microstructural evolution was investigated by establishing a thermomechanical coupled FE (finite element) model. A set of constitutive equations of phase transformation was implanted into the commercial FE solver MARC through the user defined subroutine CR-PLAW, and the temperature field was calculated by another user defined subroutine FILM. The results show that the final microstructure is completely bainite phase for CP one, 98% of bainite phase and 2% of ferrite phase for CP two, and 55% of bainite phase, 35% of pearlite phase and 10% of ferrite phase for CP three.

A macro-mechanical constitutive model of shape memory alloys

It is of practical interest to establish a precise constitutive model which includes the equations describing the phase transformation behaviors and thermo-mechanical processes of shape memory alloy (SMA). The microscopic mechanism of super elasticity and shape memory effect of SMA is explained based on the concept of shape memory factor defined by the author of this paper. The conventional super elasticity and shape memory effect of SMA are further unified as shape memory effect. Shape memory factor is redefined in order to make clear its physical meaning. A new shape memory evolution equation is developed to predict the phase transformation behaviors of SMA based on the differential relationship between martensitic volume fraction and phase transformation free energy and the results of DSC test. It overcomes the limitations that the previous shape memory evolution equations or phase transformation equations fail to express the influences of the phase transformation peak temperatures on the phase transformation behaviors and the transformation from twinned martensite to detwinned martensite occurring in SMA. A new macro-mechanical constitutive equation is established to predict the thermo-mechanical processes realizing the shape memory effect of SMA from the expression of Gibbs free energy. It is expanded from one-dimension to three-dimension with assuming SMA as isotropic material. All material constants in the new constitutive equation can be determined from macroscopic experiments, which makes it more easily used in practical applications.

A three-dimensional constitutive model for shape memory alloy

Shape memory alloy (SMA) has a wide variety of practical applications due to its uniquesuper-elasticity and shape memory effect. It is of practical interest to establish aconstitutive model which predicts its phase transformation and mechanical behaviors. Inthis paper, a new three-dimensional phase transformation equation, which predicts thephase transformation behaviors of SMA, is developed based on the results of adifferential scanning calorimetry (DSC) test. It overcomes both limitations: thatZhou’s phase transformation equations fail to describe the phase transformationfrom twinned martensite to detwinned martensite of SMA and Brinson’s phasetransformation equation fails to express the influences of phase transformation peaktemperatures on the phase transformation behaviors of SMA. A new three-dimensionalconstitutive equation, which predicts the mechanical behaviors associated with thesuper-elasticity and shape memory effect of SMA, is developed on the basis ofthermodynamics and solid mechanics. Results of numerical simulations show that thenew constitutive model, which includes the new phase transformation equationand constitutive equation, can predict the phase transformation and mechanicalbehaviors associated with the super-elasticity and shape memory effect of SMAprecisely and comprehensively. It is proved that Brinson’s constitutive modelof SMA can be considered as one special case of the new constitutive model.

Sliding Mode Control of Mechanical Systems Actuated by Shape Memory Alloy

This paper presents a model-based sliding mode control law for mechanical systems, which use shape memory alloys (SMAs) as actuators. The systems under consideration are assumed to be fully actuated and represented by unconstrained equations of motion. A system model is developed by combining the equations of motion with SMA heat convection, constitutive law, and phase transformation equations, which account for hysteresis. The control law is introduced using asymptotically stable second-order sliding surfaces. Robustness is guaranteed through the inclusion of modeling uncertainties in the controller development. The control law is developed assuming only positions are available for measurement. The unmeasured states, which include velocities and SMA temperatures and stresses, are estimated using an extended Kalman filter based on the nonlinear system model. The control law is applied to a three-link planar robot for position control problem. Simulation and experimental results show good agreement and verify the robustness of the control law despite significant modeling uncertainty.

Numerical Simulation of Thermal Stress Fields during Gas Quenching

In this paper, based on the analysis of coupled action of temperature, phase transformation and stress, the phase transformation condition and equation related to the thermal physical properties are discussed, and the transient temperature distribution of a Cr12 steel cylinder workpiece during gas quenching was obtained by solving the governing equations with a nonlinear boundary. According to the theories of thermal non-elasticity, computational mechanics and phase transformation, a new constitutive equation considering effects of phase transformation with a nonlinear surface heat transfer coefficient is proposed and is solved by means of finite element method (FEM). The thermal stress field is obtained and a way of simulating the technology of the heat treatment by computer is explored.

Equations of state and phase transformation of depleted uranium DU-238 by high pressure-temperature diffraction studies

We have conducted in situ high-pressure diffraction experiments on depleted uranium (DU-238) at pressures up to $8.5\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and temperatures up to $1123\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. From the pressure $(P)$-volume $(V)$-temperature $(T)$ measurements, thermoelastic parameters were derived for $\ensuremath{\alpha}$-uranium based on a modified high-$T$ Birch-Murnaghan equation of state and a thermal-pressure approach. With the pressure derivative of the bulk modulus ${K}_{0}^{\ensuremath{'}}$ fixed at 4.0, we obtained ambient bulk modulus ${K}_{0}=117(2)\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, temperature derivative of bulk modulus at constant pressure ${(\ensuremath{\partial}K∕\ensuremath{\partial}T)}_{P}=\ensuremath{-}3.4(4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}∕\mathrm{K}$ and at constant volume ${(\ensuremath{\partial}K∕\ensuremath{\partial}T)}_{v}=\ensuremath{-}1.1(6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}∕\mathrm{K}$, volumetric thermal expansivity ${\ensuremath{\alpha}}_{T}=a+bT$, with $a=1.2(\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\phantom{\rule{0.3em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ and $b=8.0(\ifmmode\pm\else\textpm\fi{}0.7)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}\phantom{\rule{0.3em}{0ex}}{\mathrm{K}}^{\ensuremath{-}2}$, and the pressure derivative of thermal expansion ${(\ensuremath{\partial}\ensuremath{\alpha}∕\ensuremath{\partial}P)}_{T}=\ensuremath{-}2.5(5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\phantom{\rule{0.3em}{0ex}}{\mathrm{GPa}}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$. Within the experimental errors, the ambient bulk modulus and volumetric thermal expansion derived from this work are in good agreement with previous experimental results, whereas all other thermoelastic parameters represent the first determinations for the $\ensuremath{\alpha}$ phase of uranium. We also studied the $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\gamma}$ phase transformation and obtained a phase boundary described by $T$ $(\text{in}\phantom{\rule{0.3em}{0ex}}\mathrm{K})=1032+7.4P\phantom{\rule{0.3em}{0ex}}(\text{in}\phantom{\rule{0.3em}{0ex}}\mathrm{GPa})$. Although the $\ensuremath{\gamma}$-phase uranium cannot be pressure quenched to ambient conditions, it was observed to be stable upon cooling from $1123\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}300\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ at pressures of $7--8\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. These observations indicate that pressure, in addition to the commonly utilized alloying techniques, provides an alternative route for stabilizing the $\ensuremath{\gamma}$-uranium at room temperature.

Phase field simulation of stress evolution during solidification

Stress evolution during solidification is simulated by using the phase field model. The coupling equations for phase transformation, temperature and stress/strain are derived based on thermo-dynamical theory including elastic strain energy, which is summarized as follows: phase transformation is governed by the phase field equation in which not only temperature but also stress are introduced as parameters. In the heat conduction equation, the effect of latent heat generation due to phase change and the heat generation due to mechanical work are taken into account. The volumetric change associated with temperature variation and phase transformation is expressed in the constitutive equation, while, in this paper, the stress or flow in liquid is neglected for simplicity. In order to verify the efficiency of these equations, numerical simulations are carried out by using a simple model of the finite element method. The effects of transformation expansion coefficient for two cases of dilatation and contraction are simulated, and qualitatively adequate results are presented.

Shape Memory Alloy in Tension and Compression and its Application as Clamping Force Actuator in a Bolted Joint: Part 2 — Modeling

The mathematical modeling of a bolted joint with a Shape Memory Alloy (SMA) actuator ring is presented. In this way, phase transformation, constitutive and displacement compatibility equations are used. The obtained model is utilized to show how the application of the SMA actuator ring in the bolted joint can help the joint to recover a significant amount of its lost clamping force. The model is run using the experimental data obtained for the Ni-55.7%wt Ti SMA in the companion paper (Part 1). The mechanical properties of SMA are calculated using the mentioned equations and compared to the experimental findings. It is observed that the final clamping force obtained theoretically is close to the measured value. The proposed method can, therefore, be a starting point in order to perform the analysis and design of bolted joints with SMA actuator rings.

Stress induced martensitic transformation kinetics of polycrystalline NiTi shape memory alloy

Abstract The variation of semi-empirical volume fraction of martensite as a function of thermodynamic driving force (t.d.f.) is determined based on measurements of pseudo-elastic deformation effects appearing in tubular Ni 51 at.% –Ti specimens submitted to isothermal biaxial proportional loading. To this end a thermodynamic framework developed by one of the authors (B.R.) was employed. These new experimental results when compared with the predictions of specific formal equations for martensitic phase transformation (p.t.) proposed earlier by B.R., indicate the need for modification of the formula describing the kinetics of reverse p.t. (martensite→austenite) associated with the formation of partial hysteresis in stress–strain co-ordinates. The applied method is new and the results are of importance in macroscopic modelling (determination of appropriate sets of constitutive equations) of the pseudo-elastic behaviour of SMA under complex stress state.

Study on the rheology of subducting slabs

We calculate thermal and phase structures of subducting slabs for different subducting velocities by a modified coupling code of the kinetic phase-transformation equations and the heat-diffusion equation with latent-heat release. Whereafter, we estimate their rheology structures based on the thermal and phase structures from the mineral physical point of view. At shallow depth, the upper layer has a high effective viscosity greater than 10 34 Pa·s; while the lower layer has a relatively low effective viscosity, which is greater than 10 26 Pa·s nevertheless. The effective viscosities below the kinetic phase boundary of olivine to wadsleyite decrease obviously, and reach a minimum of 10 22 Pa·s. Small areas with higher effective viscosities exist above the depth of about 700 km in subducting slabs, which are produced by lower temperatures that are related with endothermic phase transformation of spinel to perovskite and magnesiowustite. The 1% and 99% isograds of spinel proportion delineate tortuous belts with low effective viscosities, which would affect the geodynamic behavior of subducting slabs.