Composite beams with shape memory alloy fibres

We are concerned with the bending problem of fibrous composite beams in which fibres are made of shape memory alloys. These are alloys that may undergo a stress‐induced martensitic phase transformation. The matrix is treated as an elastic medium, and perfect bonding between matrix and fibres is supposed. In our model, the beam is decomposed into layers and the hysteretic behaviour of the shape memory fibres is taken into account. The boundary value problem is formulated in the form of an evolution variational inequality which, after finite element discretization, can be solved incrementally as a sequence of linear complementarity problems. (© 2004 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.     

Control of the thermoelastic model of a plate activated by shape memory alloy reinforcements

In the paper we study the problem of control by means of a heat source g for a thermoelastic system of equations u_tt − ρ∇· p (θ, ∇u) − νΔu_t + DΔ² u = f, c_v(θ, ∇u)θ_t − κΔθ − ρθ[ p _θ (θ, ∇u)·∇u_t] − ν∣∇u_t∣² = g, in a two-dimensional domain, where both viscosity ν and rigidity D are positive. Such a system has been considered in our former papers, and existence of solutions as well as uniqueness have been obtained. Here we prove the continuity and differentiability of solutions under somewhat stronger assumptions. An example of a control problem and necessary optimality conditions are presented. The system has an interpretation as a plate reinforced with shape memory alloy (SMA) wire mesh. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Analysis of the thermoelastic model of a plate with non-linear shape memory alloy reinforcements

In this paper we study the questions of the existence and uniqueness of the solutions for a thermoelastic system of equations in a two-dimensional domain, where both the viscosity v and the rigidity D are positive. It seems that such a system has up to now been considered in a one-dimensional setting only. The change of dimensions enforces the growth conditions with respect to θ and the additional regularity of the data. The existence of solutions in the case of the Neumann boundary conditions for θ and some weak regularity of data is proved. Under stronger regularity conditions the uniqueness is also established. The system has an interpretation as a plate reinforced by a shape memory alloy (SMA) wire mesh.     

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)     

Weak solutions for Falk's model of shape memory alloys

In this paper we discuss the system of two partial differential equations governing the dynamics of phase transitions in shape memory alloys. We consider the one‐dimensional model proposed by Falk, in which a term containing a fourth‐derivative appears. The main purpose is to show the uniqueness for weak solutions of the problem by using the approximate dual equations for the system without growth condition for the free energy function. Copyright © 2000 John Wiley & Sons, Ltd.     

A constitutive model for thermoplastic materials subjected to high strain rates

Reliable prediction of the behaviour of structures made from polymers is a topic under considerable investigation in engineering practice. Especially, if the structure is subjected to dynamic loading, constitutive models considering the mechanical behaviour properly are not available in commercial finite element codes yet. A constitutive model is derived including important phenomena like necking, strain rate dependency, unloading behaviour and damage. In particular, different yield surfaces in compression and tension and strain rate dependent failure, the latter with damage induced erosion, is taken into account. With the present formulation, standard verification tests can be simulated successfully. Also, an elastic damage model can be used to approximate the unloading behaviour of thermoplastics adequately. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global solution to the full one-dimensional Frémond model for shape memory alloys

In this paper, we prove the existence and uniqueness of the solution to the one-dimensional initial-boundary value problem resulting from the Frémond thermomechanical model of structural phase transitions in shape memory materials. In this model, the free energy is assumed to depend on temperature, macroscopic deformation and phase fractions. The resulting equilibrium equations are the balance laws of (linear) momentum and energy, coupled with an evolution variational inequality for the phase fractions. Fourth-order regularizing terms in the quasi-stationary momentum balance equation are not necessary, and, as far as we know for the first time, all the non-linear terms of the energy balance equation are taken into account.     

A note on the mechanical constitutive equations for materials with memory

Zusammenfassung Die Arbeit behandelt die Einschränkungen, welche sich für die Stoffgleichungen eines Materials mit Gedächtnis daraus ergeben, dass die simultane Rotation von Körper und Bezugssystem die Spannungen unverändert lassen muss. Zwei frühere Methoden zur Gewinnung dieser Einschränkungen wurden vertieft.     

A thermodynamically consistent constitutive model for fluid mud

In this contribution, an approach towards a thermodynamically consistent constitutive model for fluid mud is presented. Fluid mud exhibits highly non-Newtonian, thixotropic behaviour. It can be classified as a structured fluid. Typically, its viscosity is modeled using Bingham-type rheological models of different complexity [1, 2]. Here, the three-dimensional non-Newtonian constitutive behaviour will be modeled based on a visco-elasto-plastic model. At the current stage, a Drucker-Prager-like yield function has been formulated. Viscosity is assumed to be a function of shear viscosity. First results show the general ability to represent experimental data. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micromechanics model for FRC composites

A multiscale model for FRC composite structures taking into consideration the complex interactions at the scales of the fiber and microcracks is proposed. At the scale of the single fiber, a semi-analytical model characterizes the microslip behavior at the interface between the matrix and the fiber in terms of the overall composite stresses. The influence of fiber bundles on microcrack bridging and arrest is taken into account within the framework of linear elastic fracture mechanics. Upscaling to the macroscopic level using continuum micromechanics shows that the macroscopic deformation of the FRC composite is governed by a ’TERZAGHI’ like effective stress. For the finite element analyses of failure behavior at the scale of the composite structure, an ’interface solid element’ technique is used to consider localized cracking. Selected numerical and semi-analytical results together with experimental validations are provided. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A condensed microelectromechanical constitutive and damage model for tetragonal ferroelectrics

A condensed model for ferroelectric solids with tetragonal unit cells is presented. The approach is microelectromechanically and physically motivated, considering discrete switching processes on the level of unit cells and quasi-continuous evolution of inelastic fields on the domain wall level. To calculate multiple grain interactions an interaction tensor is introduced. Hysteresis loops are simulated for pure electric and electromechanical loading, demonstrating e.g. the influence of a compressive preload on the poling process and interaction between statistically arranged crystallits. The residual stresses and the corresponding principle stresses are used to simulate fatigue damage in ferroelectric materials. (© 2014 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)     

A constitutive model for continuous fiber reinforced polymer plies

In the present paper a constitutive model is reviewed which can be used to predict the non-linear behavior of continuous fiber reinforced laminates with polymeric matrix materials. The constitutive model considers stiffness degradation and plastic strain accumulation at the length scale of the individual plies (laminae). These effects are modeled via two different phenomenological approaches, however, their interaction is considered when the constitutive equations are solved by an implicit integration scheme. To demonstrate the predictive capabilities of the individual model parts, examples are given where the above mentioned effects are decoupled. This way, their impact on the laminate's response can be studied independently. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational approach towards the modeling of equilibrium twin interfaces in a single-crystalline magnetic shape memory alloy sample

In this work, a variational approach is proposed to study the magneto-mechanical response of a single-crystalline MSMA sample. By proposing a total energy functional for the whole magneto-mechanical system, the governing PDE system is derived by calculating the variations of the total energy functional with respect to the independent variables. An iterative numerical algorithm is proposed to solve the governing PDE system. As an example, a MSMA sample with cuboid shape and subject to perpendicularly applied magnetic and mechanical loadings is considered. It can be seen that the magneto-mechanical response of this sample can be predicted at a quantitative level. The whole procedure of variant reorientation in the sample can also be simulated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of the steady state for the Falk model system of shape memory alloys

In this article a stability result for the Falk model system is proven. The Falk model system describes the martensitic phase transitions in shape memory alloys. In our setting, the steady state is a nonlocal elliptic problem. We show the dynamical stability for the linearized stable critical point of the corresponding functional. Copyright © 2007 John Wiley & Sons, Ltd.     

Weak solutions for the Falk model system of shape memory alloys in energy class

We show the unique global existence of energy class solutions for the Falk model system of shape memory alloys under the general non‐linearity as well as considered in Aiki (Math. Meth. Appl. Sci. 2000; 23 : 299). Our main tools of the proofs are the Strichartz type estimate for the Boussinesq type equation and the maximal regularity estimate for the heat equation. Copyright © 2005 John Wiley & Sons, Ltd.     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the energy and solute conservation, thermodynamic and chemical potential equilibrium are adopted to calculate the temperature, solid concentration, liquid concentration and the increment of solid fraction. Compared with other models where the release of latent heat is solved in implicit or explicit form according to the solid/liquid (S/L) interface velocity, the energy diffusion and the release of latent heat in this model are solved at different scales, i.e. the macro-scale and micro-scale. The variation of solid fraction in this model is solved using several algebraic relations coming from the chemical potential equilibrium and thermodynamic equilibrium which can be cheaply solved instead of the calculation of S/L interface velocity. With the assumption of the solute conservation and energy conservation, the solid fraction can be directly obtained according to the thermodynamic data. This model is natural to be applied to multiple (< 2) spatial dimension case and multiple (< 2) component alloy. The morphologies of equiaxed dendrite are obtained in numerical experiments.