Multiscale Modelling of the Effective Viscoplastic Material Behaviour of Textile-Reinforced Polymers Using XFEM

In this contribution a modelling approach using numerical homogenisation techniques is applied to predict the effective nonlinear material behaviour of composites from simulations of a representative volume element (RVE). Numerical models of the heterogeneous material structure in the RVE are generated using the eXtended Finite Element Method (XFEM) which allows for a regular mesh. Suitable constitutive relations account for the material behaviour of the constituents. The influence of the nonlinear matrix material behaviour on the composite is studied in a physically nonlinear FE simulation of the local material behaviour in the RVE ­ effective stress-strain curves are computed and compared to experimental observations. The approach is currently augmented by a damage model for the fibre bundle. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of the nonlinear matrix material behaviour on the effective properties of textile-reinforced thermoplastics

For a consistent lightweight design the consideration of the nonlinear macroscopic material behaviour of composites, which is amongst others driven by damage and strain-rate effects on the mesoscale, is required. Therefore, a modelling approach using numerical homogenization techniques is applied to predict the effective nonlinear material behaviour of the composite based on the finite element simulation of a representative volume element (RVE). In this RVE suitable constitutive relations account for the material behaviour of each constituents. While the reinforcing glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law is applied to represent the strain-rate dependent, inelastic deformation of the matrix material. In order to analyse the influence of the nonlinear matrix material behaviour on the global mechanical response of the composite, effective stress-strain-curves are computed for different load cases and compared to experimental observations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Shape and Material Nonlinearities Influences about the Internal Thermal Dissipation for Elastomer-Based Vibration Isolators

This study deals with nonlinear internal dissipation inside the vibration isolators based on elastomeric composites. Due to the thermal influences on mechanical characteristics occurs during the regular exploitation regime, both conservative and dissipative components have affected and the main parameters continuously change. Beside the nonlinear behaviour of the assembly containing the isolator device, these changes also affects the thermodynamic regime inside the elastomeric material. The isolator shape also affects the internal thermal dissipation state. This complex behaviour implies nonlinear mathematical models, which have to be able to simulate the realistic viscous and elastic characteristics of such composites subjected to dynamic loads. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A class of mathematical models for alternated-current electrochemical measurements accounting for non-linear effects

The traditional approach to the modelling of alternated-current electrochemical measurements is daunted by three major drawbacks: (i) only linear behaviour of the interface investigated is accounted for; (ii) the physical meaning of many model parameters is not straightforward; and (iii) entangled inverse parameter identification problems are generated. In this paper we address these problems by defining additional mathematical models related to the Fourier expansion of a more general, physically meaningful nonlinear electrochemical kinetic reaction system. The traditional linear model is regained as the truncation at the first harmonic of the series. Analytical expressions are derived for experimental observables relating to the nonlinear electrochemical behaviour that can be measured independently during the same experiment. Eventually, we show that the higher-harmonic models can be used as a general regularization tool to reduce the number of the multiple globally optimum solutions in the nonlinear leastsquares fitting of alternated-current data, highlighted in Bozzini and Sgura [Numerical issues related to the modelling of electrochemical impedance data by nonlinear least squares, Int. J. Nonlinear Mech. 40(4) (2005) 557–570].     

Computation of the effective nonlinear material behaviour of composites using X-FEM - Analysis of the matrix material behaviour

For a consequent lightweight design the consideration of the nonlinear macroscopic material behaviour of composites, which is amongst others driven by damage and strain–rate effects on the mesoscale, is required. Therefore, the modelling approach using numerical homogenization techniques based on the simulation of representative volume elements which are modelled by the extended finite element method (X–FEM) is currently extended to nonlinear material behaviour. While the glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law accounts for strain–rate dependence and inelastic deformation of the matrix material. This paper describes the process of finding suitable constitutive relations for the polymeric matrix material Polypropylene in the small–strain regime. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison between elastohydrodynamic and dry hertzian line contacts with regard to their nonlinear vibration behaviour

The nonlinear dynamic behaviour of a simple system employing two different types of contact is compared: elastohydrodynamic line contact and Hertzian line contact with appropriate damping. The elastohydrodynamic problem is solved with finite elements. For the Hertzian line contact problem an analytical solution is known. It is found that for highly loaded cases Hertzian and elastohydrodynamic contacts lead to very similar frequency response curves. This means that instead of solving the computational expensive elastohydrodynamic problem the Hertzian contact can be used. However, considering weakly loaded contacts the two types of contact show very different behaviour. In particular the natural frequencies are different and the elastohydrodynamic contact exhibits rich nonlinear dynamic behaviour including jump phenomena as well as sub- and superharmonic bifurcations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limit behaviour of focusing solutions to nonlinear diffusions

The paper is concerned with the behaviour of focusing solutions to nonlinear diffusion problems. These solutions describe the movement of a flow filling a hole and have consequences for the qualitative theory of degenerate nonlinear parabolic equations. The general equation under study is theso-called doubly nonlinear diffusion equation a²with parameters m > 0 and p > 1 such that m(p - 1) > 1 so that the finite propagation property holds and free boundaries occur. Well-known particular cases are the Porous Medium Equation and the evolutionary p-

Laplacian Equation. We study the behaviour of the families of selfsimilar focusing solutions as the parameters m and p tend to their limiting values and identify the limit problems these limits solve. In the case m(p - 1) -+ 1 we find as appropriate asymptotic problems a family of Hamilton-Jacobi equations. When we let m + ffi we obtain in the limit the Hele-Shaw problem. When p + cc we

obtain linear travelling waves with arbitrary speed, solutions of a certain ∞-Laplacian evolution problem.     

The influence of the filler volume fraction on the mechanical behaviour of thermoplastic materials

Uniaxial experiments clarify that the mechanical behaviour of PTFE compounds depends strongly on the amount of filler particles. In order to describe these dependencies, a finite endochronic viscoplastic material model based on material isomorphisms has been applied to various glass fibre filled PTFE compounds. The model allows to characterize viscoplastic material behaviour with equilibrium hysteresis using a rate‐independent endochronic elastoplastic model in parallel connection with a nonlinear Maxwell model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of reduced electric arc models

An energy equation coupled to the eddy-current Maxwell equations with nonlinear material coefficients gives a reduced electric arc model with interesting existence and stability behaviour. Applications are arcs in industrial plasma flows. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formulation and analysis of two energy-consistent methods for nonlinear elastodynamic frictional contact problems

Energy-conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability. Furthermore, some physical phenomena (such as friction) can generate dissipation; then in this work, we present and analyse two energy-consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon. The first approach permits one to enforce, respectively, the Kuhn–Tucker and persistency conditions during each time step by combining an adapted continuation of the Newton method and a Lagrangean formulation. In addition the second method which is based on the work in [P. Hauret, P. Le Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact, Comput. Methods Appl. Mech. Eng. 195 (2006) 4890–4916] represents a specific penalisation of the unilateral contact conditions. Some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods.     

New approach for fractional Schrödinger-Boussinesq equations with Mittag-Leffler kernel

In this paper, we find the solution and analyse the behaviour of the obtained results for the nonlinear Schrödinger-Boussinesq equations using q -homotopy analysis transform method (q -HATM) within the frame of fractional order. The considered system describes the interfaces between intermediate long and short waves. The projected fractional operator is proposed with the help of Mittag-Leffler function to incorporate the nonsingular kernel to the system. The projected algorithm is a modified and accurate method with the help of Laplace transform. The convergence analysis is presented with the help of the fixed point theorem in the form existence and uniqueness. To validate and illustrate the effectiveness of the algorithm considered, we exemplified considered system with respect of arbitrary order. Further, the behaviour of achieved results is captured in contour and 3D plots for distinct arbitrary order. The results show that the projected scheme is very effective, highly methodical and easy to apply for complex and nonlinear systems and help us to captured associated behaviour diverse classes of the phenomenon.     

A nonlinear fracture differential kinetic model to depict chaotic atom motions at a fatigue crack tip based on the differentiable manifold methodology

A nonlinear differential kinetic model describing dynamical behaviours of an atom at a fatigue crack tip is developed in this paper. It is assumed that the forces acted on this atom by its surrounding atoms consist of the following three components: (1) an elastic restoring force governed by Leonard-Jones potential, which describes the elastic interaction between atoms; (2) a nonlinear damping force proportional to its velocity through a linear function of its displacement as a coefficient that empirically simulates the energy loss from the crack tip to its surroundings; (3) an external remote driving force to represent thermally activated energy supplied to the crack tip from the surroundings. Based on these assumptions of the interaction forces between the atoms around the crack tip, a nonlinear dynamic equation describing the motion of the atom at a crack tip using the Newton’s second principle is derived. For a periodic external force and a random one influenced by parameters omitted, deterministic and a stochastic analyses on the dynamic equation obtained above are completed. Based on the theories of the Hopf bifurcation, global bifurcation and stochastic bifurcation, the extent and some possible implications of the existence of atomic-scale chaotic and stochastic bifurcative motions involving the fracture behaviour of actual materials are systematically and qualitatively discussed and the extreme sensitivity of chaotic motions to minute changes in initial conditions is explored. As demonstrated in the paper, chaotic behaviour may be observed in the case of a larger amplitude of the driving force and a smaller damping constant. The white noise introduced in the atomistic motion process may leads to a drift of the divergence point of the nonlinear stochastic differential kinetic system in contrast to the homoclinic divergence of the nonlinear deterministic differential kinetic system.     

Isoparametric finite element approximation of the flow in magnetic fluid rotary shaft seals

The hydrodynamic behaviour of the magnetic liquid in a magnetic fluid rotary shaft seal can be described by a coupled system of nonlinear partial differential equations. A decoupling numerical solving strategy is proposed and applied to determine the flow in a magnetic fluid seal. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global attractors for von Karman evolutions with a nonlinear boundary dissipation

Dynamic von Karman equations with a nonlinear boundary dissipation are considered. Questions related to long time behaviour, existence and structure of global attractors are studied. It is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global (compact) attractor for all weak (finite energy) solutions. This result has been known in the case of full interior dissipation, but it is new in the case when the boundary damping is the main dissipative mechanism in the system. In addition, we prove that fractal dimension of the attractor is finite. The proofs depend critically on the infinite speed of propagation associated with the von Karman model considered.     

Micromechanical strength analysis of fiber reinforced plastics

Strength of an unidirectional lamina is computed with a representative volume element. An approximative “voxel” meshing method is used in conjunction with continuum damage mechanics to simulate crack growth in the RVE. Two material models for the nonlinear material behaviour of the epoxy resin are compared. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chaotic dynamic and control for micro-electro-mechanical systems of massive storage with harmonic base excitation

This paper explores chaotic behaviour and control of micro-electro-mechanical systems (MEMS), which consist of thousands of small read/write probe tips that access gigabytes of data stored in a non-volatile magnetic surface. The model of the system is formed by two masses connected by a nonlinear spring and a viscous damping. The paper shows that, by means of an adequate feedback law, the masses can behave as two coupled Duffing’s oscillators, which may reach chaotic behaviour when harmonic forces are applied. The chaotic motion is destroyed by applying the following control strategies: (i) static output feedback control law with constant forces and (ii) geometric nonlinear control. The aim is to drive the masses to a set point even with harmonic base excitation, by using chaotic dynamics and nonlinear control. The paper shows that it is possible to obtain a positioning time around a few ms with sub-nanometre accuracy, velocities, accelerations and forces, as it appears in the design of present MEMS devices. Numerical simulations are used to verify the mathematical discussions.     

Molecular static simulations of ferroelectric material hysteresis behaviour

The present contribution deals with molecular static modelling and the simulation of ferroelectric material hysteresis behaviour. Therefore the core-shell model is implemented in a molecular static algorithm. Moreover the algorithm is implemented as a finite element method for nonlinear trusses. Thereby the computational costs are reduced significantly compared to molecular dynamics. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the direct connection of rheological elements in nonlinear continuum mechanics

The structure of complicated phenomenological material models at finite strains is often exemplified with the help of rheological elements. Thereby, simple material behaviour, i.e. elasticity or viscous and plastic flow, are composes by components. In our approach, we directly apply this concept to obtain material models at finite strains. Towards this end, the thermodynamically consistent material behaviour of single elements is defined first. Subsequently, the elements are connected by evaluation of stress equilibria equations formulated on interconnecting configurations. The basic equations of this concept are presented using the example of nonlinear viscoelasticity of Maxwell type. The model results from a series connection of an elastic and a viscous element, whereas both are formulated in a thermodynamically consistent way within the framework on nonlinear continuum mechanics. Furthermore, an approach of numerical implementation using the stress equilibria is suggested. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Autonomous self-similar ordinary differential equations and the Painlevé connection

We demonstrate an intimate connection between nonlinear higher-order ordinary differential equations possessing the two symmetries of autonomy and self-similarity and the leading-order behaviour and resonances determined in the application of the Painlevé Test. Similar behaviour is seen for systems of first-order differential equations. Several examples illustrate the theory. In an integrable case of the ABC system the singularity analysis reveals a positive and a negative resonance and the method of leading-order behaviour leads naturally to a Laurent expansion containing both.