Constitutive modeling of the mechanics associated with crystallizable shape memory polymers

Shape memory polymers are novel materials that can be easily formed into complex shapes, retaining memory of their original shape even after undergoing large deformations. The temporary shape is stable and return to the original shape is triggered by a suitable mechanism such as heating. In this paper, we develop constitutive equations to model the mechanical behavior of crystallizable shape memory polymers. Crystallizable shape memory polymers are called crystallizable because the temporary shape is fixed by a crystalline phase, while return to the original shape is due to the melting of this crystalline phase. The modeling is done using a framework that was developed recently for studying crystallization in polymers ([28], [25], [27], [31]) and is based on the theory of multiple natural configurations. In this paper we formulate constitutive equations for the original amorphous phase and the semi-crystalline phase that is formed after the onset of crystallization. In addition we model the melting of the crystalline phase to capture the return of the polymer to its original shape. The model has been used to simulate a typical uni-axial cycle of deformation, the results of this simulation compare very well with experimental data. In addition to this we also simulate circular shear of a hollow cylinder and present results for different cases in this geometry.     

Multi-phase modeling of shape memory polymers

In the present work, we propose a model for shape memory polymers based on the idea of the multiplicative decomposition of the deformation gradient. Evolution equations for the several deformation components are presented that provide for the storage of the entropy-elastic strain and its recovery during the transition between the frozen phase and the active phase. First characteristic shape memory cycles for small and large deformations will be shown as a last point. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Three‐dimensional solidification with two possible crystallization states: Existence of solutions with flow in the melt

In this paper we discuss the existence of solutions of a system of nonlinear and singular partial differential equations constituting a phase field model with convection for non‐isothermal solidification/melting of certain metallic alloys in the case where two different kinds of crystallization are possible. Each one of these crystallization states is described by its own phase field, while the liquid phase is described by another one. The model also allows the occurrence of fluid flow in non‐solid regions, which are a priori unknown, and then we have a free‐boundary value problem. Thus, the model relates the evolutions of these three phase fields, the temperature of the solidification/melting process and the fluid flow in non‐solid regions. Copyright © 2009 John Wiley & Sons, Ltd.     

Existence and stability results for the hyperbolic Stefan problem with relaxation

Summary The aim of this paper is to study a phase transition model, based on the Cattaneo-Fourier constitutive law for the heat flux and on a relaxed constitutive law for the phase variable. In turn, the model describes fast processes of melting and crystallization with supercooling and superheating effects. We give existence and stability results for the former phase transition problem. Uniqueness is deduced from the stability result On leave from: Lavrent'ev Institute of Hydrodynamics, 630090 Novosibirsk, Academy of Sciences of Russia, Russia.     

On some nonlinear evolution equations with the strong dissipation,II

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with the strong dissipation studied in a previous paper [5]. In particular, we give sufficient conditions under which some of the equations have global solutions and at the same time we find steady state solutions of these equations which are exponentially stable as t → ∞. In the application, we improve the existence results to the equations which describe a local statement of balance of momentum for materials for which the stress is related to strain and strain rate through some constitutive equation (cf. Greenberg et al. [6], Greenberg [7], Davis [2], Clements [1], etc.).     

Heat Conduction in a Phase Field Model for Martensitic Transformation

We study the martensitic transformation with a phase field model, where we consider the Bain transformation path in a small strain setting. For the order parameter, interpolating between an austenitic parent phase and martensitic phases, we use a Ginzburg-Landau evolution equation, assuming a constant mobility. In [1], a temperature dependent separation potential is introduced. We use this potential to extend the model in [2], by considering a transient temperature field, where the temperature is introduced as an additional degree of freedom. This leads to a coupling of both the evolution equation of the order parameter and the mechanical field equations (in terms of thermal expansion) with the heat equation. The model is implemented in FEAP as a 4-node element with bi-linear shape functions. Numerical examples are given to illustrate the influence of the temperature on the evolution of the martensitic phase. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Magneto-Visco-Elastic Model for Magnetorheological Elastomers

In this work we present the modular construction and FEM implementation of a material model for magnetorheological elastomers (MRE) that is firmly rooted in micromechanics and performs well against experimental data. Keeping in mind the composite nature at the microscale, we develop a constitutive formulation based on a multiplicative magneto-mechanical split of the deformation gradient. We consider both Lee- and Clifton-type right and left decompositions. In contrast to other recent formulations [6], this general approach allows the use of highly advanced micromechnically-based network models for polymers such as [4] and [5] in a modular format and to extend its application for coupled magnetomechanical response. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trajectory attractors for nonclassical diffusion equations with fading memory

In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7,8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.     

Orientation stretch and the strength of crystallizable polymers

Changes in the orientation of molecules present in the amorphous and crystalline phases during orientation stretching of polymers are considered, as well as connections between these processes and the mechanical strength of solid oriented specimens. The studies were conducted on films of crystallizable polymers — polyvinyl alcohol and polyethylene. In explaining the dependence of the strength of specimens of these polymers on molecular orientation, the polymers are considered as three-component compositions.     

Local existence of solutions of a three phase‐field model for solidification

In this article we discuss the local existence and uniqueness of solutions of a system of parabolic differential partial equations modeling the process of solidification/melting of a certain kind of alloy. This model governs the evolution of the temperature field, as well as the evolution of three phase‐field functions; the first two describe two different possible solid crystallization states and the last one describes the liquid state. Copyright © 2008 John Wiley & Sons, Ltd.     

On a Sachs bound for stress-induced phase transformations in polycrystalline shape memory alloys

In earlier work [3], a Sachs and a Taylor bound on the transformation yield stress in shape memory polycrystals were derived in the context of a variational model. The aim of this article is to compare the Sachs with the Taylor bound for cubic-toorthorhombic phase transformations under biaxial loading, where the material parameters are chosen explicitly (CuAlNi). It turns out that the gap between the two bounds can be quite large depending on the underlying texture and the loading direction. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bainitic variant evolution in a low-alloyed steel including numerical aspects

Bainite formation is of particular industrial relevance especially after hot-forming and quenching of relatively bulky components. In these processes large portions of the component are subjected to appropriate cooling rates which yield a considerable fraction of the bainitic product phase. In our work, we develop a thermodynamically consistent multi-scale model for phase transformations from austenite into 24 possible bainite variants. Furthermore, the model is capable to express the macroscopic effects of volume change due to phase transformation as well as to transformation-induced plasticity (TRIP). Basic ideas for our material-model can be found among others in [1–3]. Because of the highly complex, strongly coupled model equations, the numerical implementation is a very challenging task. Therefore, we make use of a projected Newton algorithm combined with an active-set strategy, as an extension to the approach in [4] for austenite-martensite transformations in shape memory alloys. Numerical examples illustrate the quadratic convergence behaviour in a Finite-Element scheme. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of strain-induced crystallization in natural rubbers

Strain-Induced Crystallization (SIC) is a phenomenon characterized by a considerable increase in stiffness of natural rubbers. In this contribution, this phenomenon is studied in filled natural rubbers and a constitutive model of SIC is proposed. The influence of SIC on the mechanical behavior is described by means of the entropic strain energy subjected to a change due the crystallization. The distribution of crystallines is formulated on the basis of a statistical approach. Their contribution to the partial immobilization of polymer chains is accounted for by assuming the crystalline phase and the amorphous phase to be two separate networks. Finally, the model is compared with available experimental results of uniaxial tension tests. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extending a finite strain hyperelastic micro-sphere framework towards phase transformations

A finite strain micro-sphere framework for hyperelastic solids elaborated by Carol et al. is extended towards the modelling of phase transformations in order to simulate polycrystalline solids under large deformations such as, e.g., shape memory alloys and shape memory polymers. The implemented phase transformation mechanism is based on statistical physics and is not restricted in terms of the number of solid material phases that can be considered, though we restrict the provided examples to two phases for the sake of conceptual clarity. The specifically chosen non-quadratic format of the Helmholtz free energy functions considered on the micro-plane level includes Bain-type transformation strains for each of the phases considered. Following the Voigt assumption on the micro-scale, identical total micro-stretches act in each of the material phases, where a multiplicative decomposition into elastic and transformation-related contributions is applied. (© 2016 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.     

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.     

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.     

Modeling spatial adaptation of populations by a time non-local convection cross-diffusion evolution problem

In [19], Sighesada et al. presented a system of partial differential equations for modeling spatial segregation of interacting species. Apart from competitive Lotka-Volterra (reaction) and population pressure (cross-diffusion) terms, a convective term modeling the populations attraction to more favorable environmental regions is included. In this article, we introduce a modification of their convective term to take account for the notion of spatial adaptation of populations. After describing the model we briefly discuss its well-possedness and propose a numerical discretization in terms of a mass-preserving time semi-implicit finite differences scheme. Finally, we provide the results of two biologically inspired numerical experiments showing qualitative differences between the original model of [19] and the model proposed in this article.