Effective Dynamic Material Properties of Foam-like Microstructures

The effective material parameters of a microstructured material can be found using homogenization procedures based on calculations of a Representative Volume Element (RVE) of the material. In our approach the RVE is calculated in frequency domain and inertia is taken into account, leading to a frequency dependent behavior of the RVE.With the frequency response of the RVE, effective dynamic properties of the material are calculated using an optimization procedure. Due to the frequency dependent material behavior on the microscale a viscoelastic constitutive equation is applied on the macroscale. An example calculation is presented for an auxetic 2-D foam-like microstructure which is modelled as a plane frame structure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

A finite element thermo-viscoelastic creep approach for heterogeneous structures with dissipative correctors

A finite element approach is presented for three-dimensional thermo-viscoelastic macro analysis of polymer-matrix composite structures containing micro-level heterogeneities, a two-scale approach. Due to its ability to account for microstructural details, the asymptotic expansion homogenization approach is employed to first, obtain the homogenized properties for use in the macroscale problem, and second, to study the local micro-level stress distributions influenced by macro effects. The theoretical formulations are described and developed for a thermoviscoelastic solid whose time-dependent stress–strain relationship can be homogenized. Arising from homogenization of the constitutive equation in the time domain is a hereditary dissipative corrector term. The dissipative corrector is time-dependent and accounts for heterogeneous behavior across the junction of dissimilar materials at the microstructural level. The additional term is necessary for the governing constitutive equations to satisfy equilibrium at both length scales. The objectives of this paper are three-fold: (1) develop the micro and macro constitutive equations for a thermoviscoelastic Kelvin–Voight material; (2) develop a computational approach for the constitutive equations; and (3) demonstrate and verify illustrative applications using results from the theoretical developments in the literature wherever available for a viscoelastic homogeneous/heterogeneous material.     

A cohesive element approach for viscoelastic interface properties

This paper introduces a time dependent extension of the nonlinear traction separation law for an interface element. The constitutive equations of the load history dependent behaviour of the material are depicted and derived according to a generalized Maxwell‐model. This finite linear, viscoelastic approach allows the consideration of long term loading and time dependent material behaviour in thin layers. The implementation of the presented element formulation and the material approach are verified by numerical examples. The paper gives an outlook on further work and research topics. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rate‐independent processes in viscous solids at small strains

The so‐called generalized standard solids (of Halphen–Nguyen type) involving also activated typically rate‐independent processes such as plasticity, damage, or phase transformations are described as a system of a momentum equilibrium equation and a variational inequality for inelastic evolution of internal‐parameter variables. Various definitions of solutions are examined, especially from the viewpoint of the ability to combine rate‐independent processes and other rate‐dependent phenomena, as viscosity or also inertia. If those rate‐dependent phenomena are suppressed, then the system becomes fully rate‐independent. Illustrative examples are presented as well. Copyright © 2008 John Wiley & Sons, Ltd.     

On the viscoelastic properties of prestrained rubber

The material behaviour of rubber is investigated for a special type of deformations namely small amplitude vibrations superposed on finite static prestrain. By restricting the magnitude of the prestrain it is possible to derive a simplified constitutive equation which can be used to solve simple boundary value problems. On this theoretical background the stiffness of two types of sample geometries were computed and measured in a vibration experiment. The resulting expression for the dynamic sample stiffness contains besides the classical relaxation modulus three other complex, frequency dependent moduli and the static prestrain.     

A numerical homogenisation strategy for micromorphic continua

In this contribution, we apply a numerical homogenisation scheme for micromorphic continua replacing constitutive equations on the macroscale by a microscopic boundary value problem. The aim of this procedure is to describe the influence of the microtopology on the effective behaviour of microstructured materials such as biological tissues as well as polymer or metal foams. On the one hand, that allows for avoiding the numerically expensive calculation of a fully resolved microstructure. On the other hand there is no need to identify additional material parameters which are in general hard to interpret from the physical point of view. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Theoretical Model and Experimental Investigation of a Nonlinear Constitutive Equation for Elastic Porous Metal Rubbers

Starting from the theory of porous materials, the nonlinear theory of dry friction, and the model of a bent slender beam, a nonlinear constitutive equation for elastic porous metal rubbers is constructed. Static compressive experiments were carried out on hollow cylinders of such a rubber, from which relations between all coefficients of the constitutive equation and the material density were determined. Based on the data obtained, the constitutive equation of the material was predicted successfully. The effect of density of the metal rubber on its stiffness and on the nonlinear constitutive equation was revealed by experiments.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 4, pp. 449–460, July–August, 2005.     

Diffusion in poro‐plastic media

A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial‐boundary‐value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto‐visco‐plastic type. The variational form of this problem in Hilbert space is a non‐linear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi‐static momentum equation. The essential sufficient conditions for the well‐posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid element formulation for electromechanical problems

The difficulty in the modeling of ferroelectric materials is the coverage of the complicated interactions between electrical and mechanical quantities on the macroscale, which are caused by switching processes on the microscale. In the present work we present an electric hybrid element formulation where the stresses and the electric fields are derived by constitutive relations as presented in [1]. Therefore the displacements, the electric potential and the electric displacements are approximated by bilinear ansatz functions. Applying a static condensation procedure we obtain a modified finite element formulation governed by the degrees of freedoms associated to the displacements and the electric potential. The anisotropic material behavior is modeled within a coordinate-invariant formulation [6] for an assumed transversely isotropic material [4]. In this context a general return algorithm is applied to compute the remanent quantities at the actual timestep. Resulting hysteresis loops for the ferroelectric ceramics are presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the formation of gap solitons in nonlinear electromagnetic metamaterials

A nonlinear (Kerr‐type) electromagnetic metamaterial, characterized by a double‐Lorentz model of its frequency‐dependent linear effective dielectric permittivity and magnetic permeability, is considered. The formation of gap solitons in the low‐ and high‐frequency band gaps of this metamaterial is investigated analytically. Evolution equations governing the gap solitons, of the form of a nonlinear Klein‐Gordon and a nonlinear Schrödinger equation, are obtained, and the structure of their solutions is discussed.     

The minimum free energy for incompressible viscoelastic fluids

A general closed expression is given in the frequency domain for the isothermal minimum free energy of an incompressible viscoelastic fluid, whose constitutive equation is expressed by a linear functional of the history of strain. Another equivalent form of the minimum free energy is also derived and used to study the particular case of a discrete model material response. Copyright © 2006 John Wiley & Sons, Ltd.     

Vibration analysis of Euler–Bernoulli nanobeams by using finite element method

In the present study, an efficient finite element model for vibration analysis of a nonlocal Euler–Bernoulli beam has been reported. Nonlocal constitutive equation of Eringen is proposed. Equations of motion for a nonlocal Euler–Bernoulli are derived based on varitional statement. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Vibration characteristics, such as fundamental frequencies, are illustrated in graphical and tabulated form. Numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, rotator inertia, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of nanobeams.     

On effects of discretization and the selection of constitutive equations on stability of continua

In classical continuum mechanics the set of basic equations consists of the equation of motion, the kinematic equation and the constitutive equations. The study concentrates on constitutive modeling and the effects of discretization on stability problems. The method of investigation is analytic, the spectra of linear mapping operators of continuous and discrete dynamical systems are studied. As results cases are found, when a hidden incursive nature of a material model leads to unstable behavior of the continuum. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling and numerical approach to a class of metal‐forming problems—Quasi‐steady case

A class of quasi‐steady metal‐forming problems, with rigid‐plastic, incompressible, strain and strain‐rate dependent material model and with unilateral frictionless and nonlinear, nonlocal Coulomb's frictional contact conditions is considered. A coupled variational formulation, constituted of a variational inequality, with nonlinear and nondifferentiable terms, and a strain evolution equation, is derived and under a restriction on the material characteristics and using a variable stiffness parameters method with time retardation, existence, uniqueness and convergence results are obtained and presented. An algorithm, combining this method and the finite element method, is proposed and applied for solving an example strip drawing problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Derivation of Macroscale Equations for a Class of Physical Applications

A macroscale formulation is constructed from a system of partialdifferential equations which govern the microscale dependent variables. Theconstruction is based upon transformations of equations on second ordercontact manifolds under conditions of integrability. Necessary conditions onthe structure of the macroscale equations are obtained under the requirementthat the solutions of the macroscale equations satisfy, in some approximatesense, the equations associated with the microscale. This approach offers amethod whereby one can construct only those macroscale equations that can bevalidated by a condition of consistency based on the model error. Themethodology is employed to construct a turbulence closure model forincompressible flow. It is shown that the large eddy viscosity, whichsatifies contemporary tests based on Galilean invariance, fails theconsistency condition defined here.     

A new symmetry approach to solve space–time‐dependent KdV systems

In this paper, a new method to solve space–time‐dependent non‐linear equations is proposed. After considering the variable coefficient of a non‐linear equation as a new dependent variable, some special types of space–time‐dependent equations can be solved from corresponding space–time‐independent equations by using the general classical Lie approach. The rich soliton solutions of space–time‐dependent KdV equation and mKdV equation are given with the help of the approach. Copyright © 2004 John Wiley & Sons, Ltd.