A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite

A 3-D constitutive model for polycrystalline shape memory alloys (SMAs), based on a modified phase transformation diagram, is presented. The model takes into account both direct conversion of austenite into detwinned martensite as well as the detwinning of self-accommodated martensite. This model is suitable for performing numerical simulations on SMA materials undergoing complex thermomechanical loading paths in stress–temperature space. The model is based on thermodynamic potentials and utilizes three internal variables to predict the phase transformation and detwinning of martensite in polycrystalline SMAs. Complementing the theoretical developments, experimental data are presented showing that the phase transformation temperatures for the self-accommodated martensite to austenite and detwinned martensite to austenite transformations are different. Determination of some of the SMA material parameters from such experimental data is also discussed. The paper concludes with several numerical examples of boundary value problems with complex thermomechanical loading paths which demonstrate the capabilities of the model.     

A three-dimensional model of the thermomechanical behavior of shape memory alloys

A new macro-scale model of shape memory alloys is developed within the framework of generalized standard materials with internal constraints [Moumni, Z., 1995. Sur la modélisation du changement de phase à l’état solide. Ph.D. Thesis, École Nationale Supérieure des Ponts et Chaussées]. It is shown that the introduction of two state variables: the martensite volume fraction and the martensite orientation strain tensor, is sufficient to account for several effects exhibited by SMAs subject to thermomechanical loading, namely: self-accommodation, orientation and reorientation or martensite, as well as superelasticity and one-way shape memory. These phenomena are simulated using the same set of constitutive equations and evolution laws. A simple procedure for identifying the parameters of the model is described in detail and a validation against experimental data is conducted. The model is then used to analyze a 3D SMA structure representing a superelastic stent. Cyclic loading and other pertaining phenomena like training and two-way shape memory are considered in the second part of this paper.     

应力作用下NiTi形状记忆合金微结构演化的相场模拟及其本征应变率敏感性

基于Ginzburg-Landau动力学控制方程建立了NiTi形状记忆合金非等温相场模型,实现了对NiTi合金内应力诱导马氏体相变的数值模拟。同时将晶界能密度引入系统局部自由能密度,从而考虑多晶系统中晶界的重要作用。数值计算了单晶和多晶NiTi形状记忆合金在单轴机械载荷作用下微结构的动态演化过程和宏观力学行为,并重点研究了晶粒尺寸为60 nm的NiTi纳米多晶在低应变率下（0.000 5～15 s-1）力学行为的本征应变率敏感性。研究结果表明,单晶NiTi合金系统高温拉伸-卸载过程中马氏体相变均匀发生,未形成奥氏体-马氏体界面。而纳米多晶系统在加载阶段出现了马氏体带的形成-扩展现象,在卸载阶段出现了马氏体带的收缩-消失现象。相同外载作用过程中,NiTi单晶系统的宏观应力-应变曲线具有更大的滞回环面积,拥有更优的超弹性变形能力。计算结果显示,在中低应变率下纳米晶NiTi形状记忆合金应力-应变关系表现出较明显的应变率相关性,应变率升高导致材料相变应力提升。这一应变率相关性主要源于相场模型中外加载荷速率与马氏体空间演化速度的相互竞争关系。     

A thermodynamic finite-strain model for pseudoelastic shape memory alloys

A thermodynamic finite-strain model describing the pseudoelastic response of shape memory alloys is proposed. The model is based on a self-consistent Eulerian theory of finite deformations using the logarithmic rate. Purely elastic material response is derived from a hyperelastic potential. The mass fraction of martensite is introduced as internal state variable to indicate the thermomechanical state of the phase transforming material. The evolution of martensite is governed by a kinetic law which is derived from the Helmholtz free energy of the two-phase solid and takes the heat generated during phase transition into account. The material model is implemented into a finite element code in an updated Lagrangian scheme and calibrated to experimental data. Simulations under different loading conditions illustrate the characteristics of the model.     

A three-dimensional phenomenological model for martensite reorientation in shape memory alloys

In this work, we propose a macroscopic phenomenological model that is based on the classical framework of thermodynamics of irreversible processes and accounts for the effect of multiaxial stress states and non-proportional loading histories. The model is able to account for the evolution of both twinned and detwinned martensite. Moreover, reorientation of the product phase according to loading direction is specifically accounted for. Towards this purpose the inelastic strain is split into two contributions deriving, respectively, from creation of detwinned martensite and reorientation of previously existing martensite variants. Computational tests demonstrate the ability of the model to simulate the main aspects of the shape memory response in a one-dimensional setting and some of the features that have been experimentally found in the case of multiaxial non-proportional loading histories. Experimental non-proportional loading paths have also been simulated and a good qualitative agreement between numerical and experimental response is observed.     

Variational prediction of the mechanical behavior of shape memory alloys based on thermal experiments

In this work, we present simulations of shape memory alloys which serve as first examples demonstrating the predicting character of energy-based material models. We begin with a theoretical approach for the derivation of the caloric parts of the Helmholtz free energy. Afterwards, experimental results for DSC measurements are presented. Then, we recall a micromechanical model based on the principle of the minimum of the dissipation potential for the simulation of polycrystalline shape memory alloys. The previously determined caloric parts of the Helmholtz free energy close the set of model parameters without the need of parameter fitting. All quantities are derived directly from experiments. Finally, we compare finite element results for tension tests to experimental data and show that the model identified by thermal measurements can predict mechanically induced phase transformations and thus rationalize global material behavior without any further assumptions.     

Experimental investigation on macroscopic domain formation and evolution in polycrystalline NiTi microtubing under mechanical force

This paper reports the experimental results on macroscopic deformation instability and domain morphology evolution during stress-induced austenite → martensite (A→M) phase transformation in superelastic NiTi polycrystalline shape memory alloy microtubes. High-speed data and image acquisition techniques were used to investigate the dynamic and quasi-static events which took place in a displacement-controlled quasi-static tensile loading/unloading process of the tube. These events include dynamic formation, self-merging, topology transition, convoluted front motion and front instability of a macroscopic deformation domain. The reported phenomena brought up several fundamental issues regarding the roles of macroscopic domain wall energy and kinetics as well as their interplay with the bulk strain energy of the tube in the observed morphology instability and pattern evolution under a mechanical force. These issues are believed to be essential elements in the theoretical modeling of macroscopic deformation patterns in polycrystals and need systematic examination in the future.     

Development and implementation of a beam theory model for shape memory polymers

In this paper we focus on the development of a beam theory for a small strain continuum model of thermoviscoelastic shape memory polymers (SMP). Rather than a history integral model that is common for viscoelastic materials, a thermodynamically based state evolution model developed by Ghosh and Srinivasa (2011a) is used as the basis for the beam model based on the Euler–Bernoulli beam theory. An example of a three-point bend test is simulated using the beam theory model. The numerical solution is implemented by using an operator split technique that utilizes an elastic predictor and dissipative corrector. The key idea is that the elastic predictor is based on the solution to a beam theory boundary value problem while the dissipative corrector is entirely local (and hence can be parallelized) and is applied by considering the beam as a two or three dimensional body. This enables a very rapid solution of the problem yet maintaining fidelity of the distribution of inelastic strains across the cross-section. A displacement based convergence criterion is used in each time step. This algorithm is validated by using a three-point bending experiment for three different material cases: elastic, plastic and thermoplastic response. Time step convergence and mesh density convergence studies are carried out for the thermoviscoelastic FEM model. Finally, we implement and study this model for a SMP beam undergoing three-point bending strain recovery and stress recovery thermomechanical loading.     

Theoretical analyses on hydrodynamic instability in narrow deep river with variable curvature

Bending river is the most common river type in nature, and it is also a typical example for river evolution. The transform of the flow pattern can affect the development of the riverbed form. In return, the variation in the riverbed form can affect the hydrodynamic characteristics of the flow, thereby leading to the continuous evolution of the bending river. Based on this, a theoretical model for the bending river is established. The hydrodynamic instability characteristics of the laminar flow in the channel with a variable curvature, a typical model as the meandering river, are studied, and the variations of some parameters such as the curvature, the wave number, and the wave frequency are also discussed.     

含裂纹一维欧拉梁的裂纹无效位置分析

基于集中柔度模型,建立了含裂纹一维欧拉梁的频率方程,以此为基础探讨了裂纹无效位置的求解方法。数值计算结果显示,裂纹无效位置和位移振型节点并不一致。进一步的理论推导证明裂纹无效位置就是曲率模态振型的零点位置,从曲率和力学性能基本参数的关系分析,这一结论是合理的。本文结论对于实验、测试方案设计有指导意义。     

Size effects in martensitic microstructures: Finite-strain phase field model versus sharp-interface approach

A finite-strain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharp-interface approach extended to interfacial energy effects. The model is set in the energy framework so that the kinetic equations and conditions of mechanical equilibrium are fully defined by specifying the free energy and dissipation potentials. The free energy density involves the bulk and interfacial energy contributions, the latter describing the energy of diffuse interfaces in a manner typical for phase-field approaches. To ensure volume preservation during martensite reorientation at finite deformation within a diffuse interface, it is proposed to apply linear mixing of the logarithmic transformation strains. The physically different nature of phase interfaces and twin boundaries in the martensitic phase is reflected by introducing two order-parameters in a hierarchical manner, one as the reference volume fraction of austenite, and thus of the whole martensite, and the second as the volume fraction of one variant of martensite in the martensitic phase only. The microstructure evolution problem is given a variational formulation in terms of incremental fields of displacement and order parameters, with unilateral constraints on volume fractions explicitly enforced by applying the augmented Lagrangian method. As an application, size-dependent microstructures with diffuse interfaces are calculated for the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharp-interface microstructures with interfacial energy effects.     

Bending of a thin strip by a circular tool

We present a model of transverse bending of a wide thin elastic, elastoplastic, or rigid-plastic strip by a circular tool in the case of large displacements. The bending of the initial rectilinear strip is modeled by small increments in the load on the tool with analysis of the forces, moments, curvature, and the midline equation. The final form of the strip and the residual stresses are determined after elastic unloading. The model describes technological operations of sheet slab bending in dies and on roller-bending machines.     

A NEW MODEL OF SHAPE MEMORY ALLOYS

A new constitutive model of shape memory alloys ( SMAs) based on Tanaka ' s martensite fraction exponential expression is produced. This new model can present recoverable shape memory strain during different phase transformation, and reflect the action of martensite reorientation . Also it can overcome the defect of Tanaka ' s Model when the SMAs ' microstructure is fully martensite . The model is very simple and suitable for using , and the correct behavior of the model is proved by test.     

纳米单晶NiTi合金单程形状记忆效应的分子动力学模拟

采用基于第二近邻修正型嵌入原子势的分子动力学方法研究了纳米单晶NiTi合金的单程形状记忆效应,详细阐明了温度诱发马氏体相变和应力诱发马氏体重定向过程中纳米单晶的变形行为和微结构演化,进一步分析了加/卸载速率对NiTi合金单程形状记忆效应的影响。结果表明,NiTi纳米单晶在应力加载过程中发生马氏体重定向,卸载后存在残余应变;当加热到奥氏体转变结束温度以上时,马氏体逆相变为奥氏体相,残余应变逐渐减小,但未完全回复;随着应力加载速率的增加,重定向临界应力和模量逐渐增加;再次降温过程中不同加载速率下的原子结构演化各不相同。     

纳米单晶NiTi合金单程形状记忆效应的分子动力学模拟

采用基于第二近邻修正型嵌入原子势的分子动力学方法研究了纳米单晶NiTi合金的单程形状记忆效应,详细阐明了温度诱发马氏体相变和应力诱发马氏体重定向过程中纳米单晶的变形行为和微结构演化,进一步分析了加/卸载速率对NiTi合金单程形状记忆效应的影响。结果表明,NiTi纳米单晶在应力加载过程中发生马氏体重定向,卸载后存在残余应变;当加热到奥氏体转变结束温度以上时,马氏体逆相变为奥氏体相,残余应变逐渐减小,但未完全回复;随着应力加载速率的增加,重定向临界应力和模量逐渐增加;再次降温过程中不同加载速率下的原子结构演化各不相同。     

Shape memory behaviour: modelling within continuum thermomechanics

A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as the transition range between pseudoelasticity and pseudoplasticity.The material model is based on a free energy function and evolution equations for internal variables. By means of the free energy function, the energy storage during thermomechanical processes is described. Evolution equations for internal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies the Clausius–Duhem inequality.Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for uniaxial processes and for non-proportional loadings as well.     

Rate-dependent domain spacing in a stretched NiTi strip

Superelastic fine-grained Nickel–Titanium (NiTi) polycrystalline shape memory alloys under tensile loading deform collectively via the nucleation and growth of macroscopic martensite domains. Recent experiments on a stretched NiTi strip showed that the number of nucleated domains (or the domain spacing) increased (decreased) with increasing applied stretching rate. It is also shown that the rate dependence of the domain formation is due to the coupling between the transfer of the locally released heat and the temperature dependence of the transformation stress. In this paper, a simple one-dimensional model is developed to quantify this effect of thermo-mechanical coupling on the observed domain spacing. Analytical relationship between the domain number, thermo-mechanical properties of the material, heat transfer boundary conditions and the externally applied strain rate is established. It is found that for the case of strong heat convection the domain spacing is inversely proportional to the applied stretching rate, while for the case of weak convection, the domain spacing is dictated by a power-law scaling with exponent ?0.5. The latter theoretical prediction agrees well quantitatively with the experimental data in stagnant air.     

Martensite Strain Memory in the Shape Memory Alloy Nickel-Titanium Under Mechanical Cycling

This paper describes an experimental study of stress-induced martensitic phase transformation in the SMA Nickel-Titanium. The rich local thermo-mechanical interactions that underlie transformation are examined using three-dimensional Digital Image Correlation (strain fields) and infrared imaging (thermal fields). We quantify the complex local interactions between released/absorbed latent heat and the extent of transformation, and explore the characteristics of the phase fronts and the evolution of martensitic volume fraction. We also quantify a strong strain memory in the martensite that forms in the wake of the phase transformation front. The accommodated strain in the martensite will remain constant during loading, even as the existing phase front propagates. There also exists a remarkable strain memory in the martensite that persists from cycle to cycle, indicating that the local elastic stress fields in the martensite are driven by a dislocation structure and martensitic nuclei that largely stabilize during the first loading cycle.     

应变硬化薄板夹持型滚弯成形过程分析

现存文献对薄板的连续滚弯成形过程的数学模拟多限于讨论三辊轮解锥型弯板过程,且基于较多的假定和板的小变形情况。而开发有更多的功能和更高的生产效率的四辊轮连续弯板机需研究夹持型滚弯成形过程。本文将「１」的工作,理想塑性材料平面应变薄板的夹持型连续滚弯成形过程的弹塑性大变形弯曲的力学分析,推广至一般应变硬化材料的情形。得出了控制微分方程的解析解。采用固支边界条件模拟刚度很大的平持系统对板的夹持。通过对线