Some problems in nonlinear magnetoelasticity

In this paper we examine the influence of magnetic fields on the static response of magnetoelastic materials, such as magneto-sensitive elastomers, that are capable of large deformations. The analysis is based on a simple formulation of the mechanical equilibrium equations and constitutive law for such materials developed recently by the authors, coupled with the governing magnetic field equations. The equations are applied in the solution of some simple representative and illustrative problems, with the focus on incompressible materials. First, we consider the pure homogeneous deformation of a slab of material in the presence of a magnetic field normal to its faces. This is followed by a review of the problem of simple shear of the slab in the presence of the same magnetic field. Next we examine a problem involving non-homogeneous deformations, namely the extension and inflation of a circular cylindrical tube. In this problem the magnetic field is taken to be either axial (a uniform field) or circumferential. For each problem we give a general formulation for the case of an isotropic magnetoelastic constitutive law, and then, for illustration, specific results are derived for a prototype constitutive law. We emphasize that in general there are significant differences in the results for formulations in which the magnetic field or the magnetic induction is taken as the independent magnetic variable. This is demonstrated for one particular problem, in which restrictions are placed on the admissible class of constitutive laws if the magnetic induction is the independent variable but no restrictions if the magnetic field is the independent variable.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.     

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.     

A normal compliance contact problem in viscoelasticity: An a posteriori error analysis and computational experiments

In this work, the numerical approximation of a viscoelastic contact problem is studied. The classical Kelvin-Voigt constitutive law is employed, and contact is assumed with a deformable obstacle and modelled using the normal compliance condition. The variational formulation leads to a nonlinear parabolic variational equation. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced, by using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize time derivatives. A priori error estimates recently proved for this problem are recalled. Then, an a posteriori error analysis is provided, extending some preliminary results obtained in the study of the heat equation and other parabolic equations. Upper and lower error bounds are proved. Finally, some numerical experiments are presented to demonstrate the accuracy and the numerical behaviour of the error estimates.     

Propagation of a plane wave to a materially uniform but inhomogeneous body

We produce the equations of small deformations superimposed upon large for materially uniform but inhomogeneous bodies and specialize to an isotropic material and to a homogeneous finite elastic deformation. By assuming the small deformation to be a plane wave, a set of equations for the amplitude of the wave is produced which is accompanied by an additional set of conditions. By requiring a non-trivial solution for the amplitude, we obtain the secular equation and from it a set of necessary and sufficient conditions for having a real wave speed. The second set of conditions that have to be satisfied is due to the materials inhomogeneity. Essentially, the present analysis enhances the approach of Hayes and Rivlin for materially uniform but inhomogeneous bodies. The outcome is that for such bodies the restrictions on the constitutive law for having real wave speeds for an isotropic material subjected to a pure homogeneous deformation involves the field of the inhomogeneity as well.     

Weak solutions via bipotentials in mechanics of deformable solids

We consider a displacement-traction boundary values problem for elastic materials, under the small deformations hypothesis, for static processes. The behavior of the material is modeled by a constitutive law involving the subdifferential of a proper, convex, and lower semicontinuous map. The constitutive map and its Fenchel conjugate allow us to construct a bipotential function. Based on this construction, we propose a weak formulation of our mechanical problem. Furthermore, we prove the existence of at least one weak solution and we investigate the uniqueness of the weak solution. We also comment on the relevance of our variational approach, by considering three significant examples.     

On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney?CRivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein?CGulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.     

Problem of magnetoelasticity for a solid with the spherical cavity

The solution of a static problem of magnetoelastisity for a soft ferromagnetic elastic solid with the spherical cavity is obtained on the base of the linear theory of Brown, Pao and Yeh. It is assumed that the solid has a multi-domain structure, so the hysteresis loss and remanent magnetization are neglected. The solid is affected by a magnetic field which is uniform at infinity and determined by the magnetic induction vector. The cavity causes some distortion of the field distribution near the interface. So the field induces magnetic moments and produces stresses and deformations in the body. The problem is solved for an unperturbed strain state. An approach is discussed to find the perturbed values on the base of the solution obtained. The Fourier variable separation method is used. The stresses are presented via harmonic functions. As a result magnetoelastic stresses are obtained in the closed form. Their distribution in the body is studied and some results of numerical calculations are shown. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermal effects in orthotropic porous elastic beams

This paper is concerned with the linear theory of anisotropic porous elastic bodies. The extension and bending of orthotropic porous elastic cylinders subjected to a plane temperature field is investigated. The work is motivated by the recent interest in the using of the orthotropic porous elastic solid as model for bones and various engineering materials. First, the thermoelastic deformation of inhomogeneous beams whose constitutive coefficients are independent of the axial coordinate is studied. Then, the extension and bending effects in orthotropic cylinders reinforced by longitudinal rods are investigated. The three-dimensional problem is reduced to the study of two-dimensional problems. The method is used to solve the problem of an orthotropic porous circular cylinder with a special kind of inhomogeneity.      

Modeling of electrodynamic-mechanical coupling phenomena in smart magnetoelectroelastic materials

The coupling of electric, magnetic and mechanical phenomena may have various reasons. The famous Maxwell equations of electrodynamics describe the interaction of transient magnetic and electric fields. On the constitutive level of dielectric materials, coupling mechanisms are manyfold comprising piezoelectric, magnetostrictive or magnetoelectric effects. Electromagnetically induced specific forces acting at the boundary and within the domain of a dielectric body are, within a continuum mechanics framework, commonly denoted as Maxwell stresses. In transient electromagnetic fields, the Poynting vector gives another contribution to mechanical stresses. First, a system of transient partial differential equations is presented. Introducing scalar and vector potentials for the electromagnetic fields and representing the mechanical strain by displacement fields, seven coupled differential equations govern the boundary value problem, accounting for linear constitutive equations of magnetoelectroelasticity. To reduce the effort of numerical solution, the system of equations is partly decoupled applying generalized forms of Coulomb and Lorenz gauge transformations [1,2]. A weak formulation is given to establish a basis for a finite element solution. The influence of constitutive magnetoelectric coupling on electromagnetic wave propagation is finally demonstrated with a simple one-dimensional example. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational formulation of nonlocalmaterials withmicrostructure based on dual macro- and micro-balances

We investigate a variational setting of nonlocal materials with microstructure and outline aspects of its numerical implementation. Thereby, the current state of the evolving microstructure is described by independent global degrees in addition to the macroscopic displacement field, so-called order parameters. Focussing on standard-dissipative materials, the constitutive response is governed by two fundamental functions for the energy storage and the dissipation. Based on these functions, a global dissipation postulate is introduced. Its exploitation constitutes a global variation formulation of nonlocal materials, which can be related to a minimization principle. Following this methodology, we end up with coupled macro- and microscopic field equations and corresponding boundary conditions. On the numerical side, we consider the weak counterpart of these coupled field equations and obtain after linearization a fully coupled system for increments of the displacement and the order parameters. Due to the underlying variational structure, this system of equations is symmetric. In order to show the capability of the proposed setting, we specify the above outlined scenario to a model problem of isotropic damage mechanics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed and hybrid formulations for the three‐dimensional magnetostatic problem

We propose mixed and hybrid formulations for the three‐dimensional magnetostatic problem. Such formulations are obtained by coupling finite element method inside the magnetic materials with a boundary element method. We present a formulation where the magnetic field is the state variable and the boundary approach uses a scalar Dirichlet‐Neumann map to describe the exterior domain. Also, we propose a second formulation where the magnetic induction is the state variable and a vectorial Dirichlet‐Neumann map is used to describe the outer field. Numerical discretizations with “edge” and “face” elements are proposed, and for each discrete problem we study an “inf‐sup” condition. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 85–104, 2002     

A variational framework for nonlinear viscoelastic models in finite deformation regime

This work presents a general framework for constitutive viscoelastic models in the finite deformation regime. The approach is qualified as variational since the constitutive updates consist of a minimization problem within each load increment. The set of internal variables is strain-based and uses a multiplicative decomposition of strain in elastic and viscous components. Spectral decomposition is explored in order to accommodate, into analytically tractable expressions, a wide set of specific models. Moreover, it is shown that, through appropriate choices of the constitutive potentials, the proposed formulation is able to reproduce results obtained elsewhere in the literature. Finally, numerical examples are included to illustrate the characteristics of the present formulation.     

Boundary conditions in intrinsic nonlinear elasticity

The intrinsic formulation of the displacement-traction problem of nonlinear elasticity is a system of partial differential equations and boundary conditions whose unknown is the Cauchy–Green strain tensor field instead of the deformation as is customary. We explicitly identify here the boundary conditions satisfied by the Cauchy–Green strain tensor field appearing in such intrinsic formulations.     

Intrinsic symmetries in constitutive modeling of magneto–elastic materials

We consider a class of non-dissipative materials whose constitutive equations are derived from a suitably constructed thermodynamic potential function. The Gibbs energy relation is introduced as a function of stress, strain, magnetic field, magnetization, and temperature. By minimization with respect to the chosen subset of independent variables we derive the corresponding set of constitutive equations. The chosen form of the free energy function leads to the linear elastic and nonlinear ferromagnetic coupling where non-linearity emerges in terms associated with the strength of magnetization. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetry analysis for uniaxial compression of a hypoplastic granular material

A variety of modelling approaches currently exist to describe and predict the diverse behaviours of granular materials. One of the more sophisticated theories is hypoplasticity, which is a stress-rate theory of rational continuum mechanics with a constitutive law expressed in a single tensorial equation. In this paper, a particular version of hypoplasticity, due to Wu [2], is employed to describe a class of one-dimensional granular deformations. By combining the constitutive law with the conservation laws of continuum mechanics, a system of four nonlinear partial differential equations is derived for the axial and lateral stress, the velocity and the void ratio. Under certain restrictions, three of the governing equations may be combined to yield ordinary differential equations, whose solutions can be calculated exactly. Several new analytical results are obtained which are applicable to oedometer testing. In general this approach is not possible, and analytic progress is sought via Lie symmetry analysis. A complete set or “optimal system” of group-invariant solutions is identified using the Olver method, which involves the adjoint representation of the symmetry group on its Lie algebra. Each element in the optimal system is governed by a system of nonlinear ordinary differential equations which in general must be solved numerically. Solutions previously considered in the literature are noted, and their relation to our optimal system identified. Two illustrative examples are examined and the variation of various functions occuring in the physical variables is shown graphically.     

Modelling of predeformation- and frequency-dependent material behaviour of filled rubber under large predeformations superimposed with harmonic deformations of small amplitudes

Viscoelastic materials show a significant frequency and predeformation dependent behaviour under loadings consisting of large predeformations superimposed by small harmonic deformations. Based on further material models of Haupt & Lion [1] and Lion, Retka & Rendek [2] we introduce a recently developed constitutive approach of finite viscoelasticity in the frequency domain that is able to describe the frequency and predeformation dependent material behaviour with respect to storage and loss modulus. The constitutive equations are geometrically linearised in the neighbourhood of the predeformation and will be evaluated in the frequency domain. Furthermore a formulation for incompressible material behaviour is introduced and the corresponding dynamic modulus tensors are derived. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetry properties of two-dimensional Ciarlet–Mooney–Rivlin constitutive models in nonlinear elastodynamics

Nonlinear dynamic equations for isotropic homogeneous hyperelastic materials are considered in the Lagrangian formulation. An explicit criterion of existence of a natural state for a given constitutive law is presented, and is used to derive natural state conditions for some common constitutive relations.For two-dimensional planar motions of Ciarlet–Mooney–Rivlin solids, equivalence transformations are computed that lead to a reduction of the number parameters in the constitutive law. Point symmetries are classified in a general dynamical setting and in traveling wave coordinates. A special value of traveling wave speed is found for which the nonlinear Ciarlet–Mooney–Rivlin equations admit an additional infinite set of point symmetries. A family of essentially two-dimensional traveling wave solutions is derived for that case.     

磁弹性耦合效应引起的铁磁直杆磁场中振动频率的改变

基于磁弹性广义变分原理和Hamilton原理,对处于外加磁场中的软铁磁体,建立了磁弹性动力学理论模型．分别通过关于铁磁杆磁标势和弹性位移的变分运算,获得了包含磁场和弹性变形的所有基本方程,并给出描述磁弹性耦合作用的磁体力和磁面力．采用摄动技术和Galerkin方法,将所建立的磁弹性理论模型用于外加磁场中铁磁直杆的振动分析．结果表明,由于磁弹性耦合效应,外加磁场将对铁磁杆的振动频率产生影响:当铁磁杆的振动位移沿着磁场方向时,其频率减小并出现磁弹性屈曲失稳;当铁磁杆的振动位移垂直于磁场方向时,其频率将会增大．理论模型能够很好地解释已有实验观测的振动频率改变现象．