A constitutive model for shape memory alloy fibres and its applicability to prestressed composites

This contribution is concerned with a constitutive model for shape memory fibres. The 1D-constitutive model accounts for the pseudoplastic and shape memory effect (SME). The macroscopic answer of the material is determined by the evolution from a twinned martensitic lattice into a deformed and detwinned one. On the macroscopic scale these effects are responsible for the upper boundary of the hysteresis which is situated around the origin of the stress-strain-diagram. During the phase transition process inelastic strains arise. When the lattice is fully detwinned, a linear elastic branch at the end of the hysteresis is observed. The initial state of the material is recovered by unloading and heating the material subsequently. The constitutive model is derived from the Helmholtz' free energy and fulfils the 2nd law of thermodynamics. For the present model five internal state variables are employed. Two of them are used to describe the inelastic strain and a backstress. The others represent the martensitic volume fraction and are necessary to describe the SME. The latter variables are depending on the deformation state as well as on temperature. A change on temperature goes along with a reduction of the inelastic strain. The model is incorporated in a fibre matrix discretization to prestress the surrounding structure. The boundary value problem is solved for a truss element applying the finite element method. Examples will demonstrate the applicability in engineering structures. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of asymmetric effects for shape memory alloys

Experimental results of shape memory alloys show a pronounced asymmetric behaviour between tension, compression and shear. For simulation of these effects in the constitutive equations different transformation strain tensors are introduced. These are related to the different variants for the multi-variant- and detwinned-martensite as a consequence of different stress states. In the framework of plasticity the concept of stress mode dependent weighting functions is applied in order to characterize the different stress states. Verification of the proposed methodology is succeeded for simulation of the pseudoelastic behaviour of shape memory alloys with different hardening characteristics in tension, compression and shear. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of Shape Memory Alloys and Experimental Verification

Prestrained shape memory alloys change their length when heated above their transformation temperature. This effect can be used to generate high forces in a small workspace, which has particular advantages in actuator design. The optimization and control of the shape memory actuator requires a comprehensive simulation of the material behavior. However, many of the existing models are limited to specific load cases or offer rough approximations only. A material model for shape memory alloys from Seelecke [1] is examined in this paper. This model describes the behavior of a shape memory wire, which is heated by electric current. It is implemented in a simulation program to investigate the actuator output and to improve the performance. Finally, the parameters of the simulation are adapted to experimental results.     

A finite element modeling of piezoelectric materials with a superimposed stress

Based on the mechanism of domain switching, a three dimensional nonlinear finite element model for piezoelectric materials subjected to electromechancial loading is developed in this contribution. The finally considered model problem deals with differently oriented grains whereby uni-axial, quasi-static cyclic loading is applied. It is assumed that a crystal orientation switches if the reduction in free energy of the grain exceeds a critical energy barrier. The nonlinearity in the small electromechanical loading range is addresses via a polynomial probability function for domain switching. Hysteresis behavior is discussed taking the influence of a superimposed compression state into account. It is observed that the hysteresis loop flattens under the axial compression but elongates under the transverse compression. Irrespective of how the compression is applied, the remnant polarization and as well as the coercive electric field decrease. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

形状记忆合金相变过程三维大变形有限元模拟

形状记忆合金（SMA）一直被作为智能材料开发,并被用于阻尼器、促动器和智能传感器元件．形状记忆合金（SMA）的一项重要特性,是它具有恢复在机械加卸载周期下产生的大变形而不表现出永久变形的能力．该文旨在介绍一种由应力产生的相变且可以描述马氏体和奥氏体之间的超弹性滞回环现象本构方程．形状记忆合金的马氏体系数假设为应力偏张量的函数,因此形状记忆合金在相变过程中锁定体积．本构模型是在大变形有限元的基础上执行的,采用了现时构型Lagrange大变形算法．为了方便地使用Cauchy应力和线性应变本构关系,使用了与旋转无关的Jaumann应力增率计算应力．数值分析结果表明,相变引起的超弹性滞回环可以有效地通过该文提出的本构方程和大变形有限元模拟．     

Micromechanical Approach of Martensitic Transformation Problem in Steels

To determine the mechanical behavior of material involving the martensitic phase transformation (for example, steels like 100Cr6), a representative volume element (RVE) model including phase transformation criterion is desireable at micromechanical approach. A framework combining the Eshelby's inclusion theory as well as continuum mechanics with phase-transformation (PT) critical condition at RVE model is presented briefly. And application of this model to estimate the critical aspect ratio of martensitic plate or lath inside homogeneneous stress field is also included, where the RVE can be under uniaxial tension/compression or pure shear loading case. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of Asymmetric Effects in Plasticity

In the framework of plasticity the simulation of materials is addressed, which experimentally exhibit different behavior in different loading scenarios, such as tension, compression and shear. To this end an additive decomposition of the yield function and the plastic potential, is assumed into a sum of weighted stress mode related quantities. The characterization of the stress modes is obtained in the octahedral plane of the deviatoric stress space in terms of the Lode angle, such that stress mode dependent scalar weighting functions can be constructed. Verification of the proposed methodology is succeeded for simulation of the pseudoelastic behavior of shape memory alloys with different hardening characteristics in tension and shear. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlinear model for ferromagnetic shape memory alloy actuators

Ferromagnetic shape memory alloys (FSMAs) such as Ni–Mn–Ga have attracted significant attention over the last few years. As actuators, these materials offer high energy density, large stroke, and high bandwidth. These properties make FSMAs potential candidates for a new generation of actuators. The preliminary dynamic characterization of Ni–Mn–Ga illustrates evident nonlinear behaviors including hysteresis, saturation, first cycle effect, and dead zone. In this paper, in order to precisely control the position of FSMA actuators a mathematical model is developed. The Ni–Mn–Ga actuator model consists of the dynamic model of the actuator, the kinematics of the actuator, the constitutive model of the FSMA material, the reorientation kinetics of the FSMA material, and the electromagnetic model of the actuator. Furthermore, a constitutive model is proposed to take into account the elastic deformation as well as the reorientation. Simulation results are presented to demonstrate the dynamic behavior of the actuator.     

An implicit three-dimensional model for describing the inelastic response of solids undergoing finite deformation

The aim of this paper is to develop a new unified class of 3D nonlinear anisotropic finite deformation inelasticity model that (1) exhibits rate-independent or dependent hysteretic response (i.e., response wherein reversal of the external stimuli does not cause reversal of the path in state space) with or without yield surfaces. The hysteresis persists with quasistatic loading. (2) Encompasses a wide range of different types of inelasticity models (such as Mullins effect in rubber, rock and soil mechanics, traditional metal plasticity, hysteretic behavior of shape memory materials) into a simple unified framework that is relatively easy to implement in computational schemes and (3) does not require any a priori particular notion of plastic strain or yield function. The core idea behind the approach is the development of an system of implicit rate equations that allow for the continuity of the response but with different rates along different directions. The theory, which is in purely mechanical setting, subsumes and generalizes many commonly used approaches for hypoelasticity and rate-independent plasticity. We illustrate its capability by modeling the Mullins effect which is the inelastic behavior of certain rubbery materials. We are able to simulate the entire cyclic response without the use of additional internal variables, i.e., the entire response is modeled by using an implicit function of stress and strain measures and their rates.     

A variational model for the functional fatigue in polycrystalline shape memory alloys

Shape memory alloys show a very complex material behavior associated with a diffusionless solid/solid phase transformation between austenite and martensite. Due to the resulting (thermo-)mechanical properties – namely the effect of pseudoelasticity and pseudoplasticity – they are very promising materials for the current and future technical developments. However, the martensitic phase transformation comes along with a simultaneous plastic deformation and thus, the effect of functional fatigue. We present a variational material model that simulates this effect based on the principle of the minimum of the dissipation potential. We use a combined Voigt/Reuss bound and a coupled dissipation potential to predict the microstructural developments in the polycrystalline material. We present the governing evolution equations for the internal variables and yield functions. In addition, we show some numerical results to prove our model's ability to predict the shape memory alloys' complex inner processes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermomechanical behavior of pseudoelastic NiTi shape memory alloys

The thermomechanical behavior of polycrystalline Ni-rich pseudoelastic NiTi shape memory alloys is analyzed. Special focus is on regions within the stress strain diagram which are regarded as linear elastic in common phenomenological material models, i.e. the region between zero stress and the beginning of the pseudoelastic plateau as well as the region within the hysteresis. In both cases, severe temperature changes can be observed. A possible explanation for this effect is twofold: On the one hand, it might be explained by the presence of an R-phase transformation. On the other hand, unstructured martensite of the B19' phase may form. However, the assumption of a purely thermo-elastic material behavior in those regions does not seem to hold true in general. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulation of the Application of NiTi Alloys in Medical Technologies

Shape memory alloys are nowadays already established as a material which is able to solve exceptional tasks in practical applications. Particularly, its utilization in the field of medical technologies increases steadily. For example micro tools (staple, catheters) and implants (coronary stents) are made out of Nickel-Titanium well known as a basic shape memory alloy. Apart from the advantages like the avoidance of auxiliary components and joints in the system and to utilize the high volume specific work of shape memory alloys, NiTi alloys exhibit a good biocompatibility. This property is necessary with regard to either permanent or temporary implants. To optimize the use of NiTi alloys in the scope of medical technologies, the support of the development of applicable tools by numerical simulations is highly recommended. However the complex material behaviour containing a profoundly thermomechanical coupling poses indeed a big challenge to the material modeling and its implementation into a finite element code. Particularly, the material model proposed by Helm [1] proves to be a firm model containing the most common properties of shape memory alloys, as the pseudoelasticity, the shape memory effect and the two-way effect. In the present contribution the FE modelling of a medical staple used in foot surgery is presented by considering the model of Helm which was investigated by the authors to improve its performance in the finite element method [2]. The foot staple, produced by a group of members of the SFB 459 which is funded by the DFG, avails the shape memory effect to excite the desired clamping effect [3]. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A phenomenological constitutive model for magnetostrictive materials and ferroelectric ceramics

The contribution is concerned with a thermodynamic consistent constitutive model for magnetostrictive materials and ferroelectric ceramics. It captures the nonlinear phenomenological behavior which is described by hysteresis effects. Magnetostrictive alloys and ferroelectric ceramics belong to the multifunctional materials. In recent years these materials have become widely–used in actor and sensor applications. They characterize an inherent coupling between deformation and magnetic or electric field. Due to the similarities of the coupled differential equations a uniform approach is applied for both phenomena. The presented three–dimensional material model is thermodynamically motivated. It is based on the definition of a specific free energy function and a switching criterion. Furthermore an additive split of strain and the magnetic or electric field in a reversible and an irreversible part is suggested. The irreversible quantities serve as internal variables, which is analog to plasticity theory. A one–to–one–relation between the two internal variables provides conservation of volume for the irreversible strains. The presented material model can approximate the ferromagnetic or ferroelectric hysteresis curve and the related butterfly hysteresis. Furthermore an extended approach for ferrimagnetic behavior, which occurs in magnetostrictive materials, is presented. Some numerical simulations demonstrate the capability of the presented model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functional Fatigue in polycrystalline Shape Memory Alloys

Shape memory alloys show the well known effect of pseudo-elasticity associated with the formation of two stress plateaus in the stress/strain diagram for tension tests. Due to cyclic loading, the stress plateaus decrease with every load cycle, particularly the upper one. This important effect of functional fatigue results from plastic deformations that are produced during solid-solid phase transformations between the austenitic and martensitic state. Outgoing from a polycrystalline approach for shape memory alloys we develop a micromechanical material model that is based on the Principle of the Minimum of the Dissipation Potential and predicts the evolution of plastic strains. Therefore, only a small number of material parameters is necessary and additionally, only a few assumptions are sufficient to model the effect of functional fatigue. We present yield functions as well as evolution equations for the volume fractions of austenite and martensite, and the plastic strains. Furthermore, we show an exemplary calculation for Nickel Titanium and compare it with experimental measurements to demonstrate the ability of our model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulations of poly-crystalline shape-memory alloys based on a micromechanical model

Based on a micormechanical model we describe the material behaviour of specimens made of shape memory alloys by a finite element implementation. These materials undergo solid to solid phase transformations during loading and unloading which determine the characteristic pseudo-elastic or pseudo-plastic material response. Phase transitions, orientation distribution (pole figures) and the influence of pre-texture on the specimen, due to some previous treatment like rolling, are discussed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

内变量和伪弹性的热力学模型

在固体相变中经常出现滞后,需要内变量以描述滞后现象。对记忆合金单晶拉伸实验,所需内变量是奥氏体和马氏体区域的界面数。本文首先讨论有关的实验结果,然后给出以此内变量为基础的热力学模型。     

Rank-One Convexification Approach for the Modeling of Magnetic Shape Memory Response

We present an incremental energy minimization model for magnetic shape memory alloys (MSMAs) whose derivation departs from the constrained theory of magnetoelasticity [1], but additionally accounts for elastic deformations, magnetization rotation, and dissipative mechanisms. The minimization of the proposed incremental energy yields the evolution of the internal state variables. In this sense, the presented modeling concept clearly distinguishes itself from standard phenomenological approaches to MSMA modeling [4]. The extended model is applied to simulate the response of single crystalline Ni₂MnGa. It is shown to accurately capture the nonlinear, anisotropic, hysteretic, and highly stress level-dependent features of MSMA behavior, based on just a few fundamental material parameters, which is validated by comparison to experimental data. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一个新的形状记忆合金模型

借助于Tanaka用一维形核动力学方程导出的指数形式的相变百分数,建立了一个新的形状记忆合金本构模型.提出了不同相变条件下的可恢复形状记忆应变的表达式;考虑了材料在变形过程中马氏体的重定向作用;克服了Tanaka系列模型不能描述当材料为完全马氏体状态时的力学行为的缺点.本模型较现有的形状记忆合金本构模型均简单,便于应用,实验证明了模型的正确性.     

Micro‐Mechanical Modelling of the Constitutive Behaviour of NiTi Shape Memory Alloys

The superelasticity and shape memory effect in NiTi alloys are examined on the basis of micromechanics within the energy minimization framework. We describe the behaviour of polycrystalline shape‐memory alloys via orientation‐distribution of the various martensite‐variants (domains) present in the material. Stress‐strain curves are presented and special attention is payed to the volume fraction of martensite for specific NiTi alloys (Nitinol) specimen under uniaxial tension. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)