Load-depth sensing of isotropic, linear viscoelastic materials using rigid axisymmetric indenters

An indentation experiment involves five variables: indenter shape, material behavior of the substrate, contact size, applied load and indentation depth. Only three variable are known afterwards, namely, indenter shape, plus load and depth as function of time. As the contact size is not measured and the determination of the material properties is the very aim of the test; two equations are needed to obtain a mathematically solvable system. For elastic materials, the contact size can always be eliminated once and for all in favor of the depth; a single relation between load, depth and material properties remains with the latter variable as unknown. For viscoelastic materials where hereditary integrals model the constitutive behavior, the relation between depth and contact size usually depends also on the (time-dependent) properties of the material. Solving the inverse problem, i.e., determining the material properties from the experimental data, therefore needs both equations. Extending Sneddon's analysis of the indentation problem for elastic materials to include viscoelastic materials, the two equations mentioned above are derived. To find the time dependence of the material properties the feasibility of Golden and Graham's method of decomposing hereditary integrals (J.M. Golden and G.A.C. Graham. Boundary value problems in linear viscoelasticity, Springer, 1988) is investigated and applied to a single load-unload process and to sinusoidally driven stationary state indentation processes.Comment: 116 pages, 29 figure

High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

Copyright @ 2014 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over inline image for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease

Finite element approximation of Maxwell’s equations with Debye memory

Copyright © 2010 Simon Shaw. All rights reserved.This article has been made available through the Brunel Open Access Publishing Fund.Maxwell’s equations in a bounded Debye medium are formulated in terms of the standard partial differential equations of electromagnetism with a Volterra-type history dependence of the polarization on the electric field intensity. This leads to Maxwell’s equations with memory. We make a correspondence between this type of constitutive law and the hereditary integral constitutive laws from linear viscoelasticity, and are then able to apply known results from viscoelasticity theory to this Maxwell system. In particular we can show long-time stability by shunning Gronwall’s lemma and estimating the history kernels more carefully by appeal to the underlying physical fading memory. We also give a fully discrete scheme for the electric field wave equation and derive stability bounds which are exactly analagous to those for the continuous problem, thus providing a foundation for long-time numerical integration. We finish by also providing error bounds for which the constant grows, at worst, linearly in time (excluding the time dependence in the norms of the exact solution). Although the first (mixed) finite element error analysis for the Debye problem was given by Jichun Li (in Comp. Meth. Appl. Mech. Eng., 196, (2007), pp. 3081–3094) this seems to be the the first time sharp constants have been given for this problem.This article is available through the Brunel Open Access Publishing Fund

A New Approach to Equations with Memory

In this work, we present a novel approach to the mathematical analysis of equations with memory based on the notion of a state, namely, the initial configuration of the system which can be unambiguously determined by the knowledge of the future dynamics. As a model, we discuss the abstract version of an equation arising from linear viscoelasticity. It is worth mentioning that our approach goes back to the heuristic derivation of the state framework, devised by L.Deseri, M.Fabrizio and M.J.Golden in "The concept of minimal state in viscoelasticity: new free energies and applications to PDEs", Arch. Ration. Mech. Anal., vol. 181 (2006) pp.43-96. Starting from their physical motivations, we develop a suitable functional formulation which, as far as we know, is completely new.Comment: 39 pages, no figur

New Insights on Free Energies and Saint- Venant’s Principle in Viscoelasticity

This work was conceived in 1999 and brought near completion by 2003. Giorgio Gentili was deeply involved in this research until his untimely death. He is greatly missed. Work pressures on the other authors forced a postponement of research on this topic, originally envisaged as lasting a few months but in the event it turned out to be nearly ten years. We now dedicate this work to the memory of Giorgio and to his Family

New Insights on Free Energies and Saint- Venant’s Principle in Viscoelasticity

This work was conceived in 1999 and brought near completion by 2003. Giorgio Gentili was deeply involved in this research until his untimely death. He is greatly missed. Work pressures on the other authors forced a postponement of research on this topic, originally envisaged as lasting a few months but in the event it turned out to be nearly ten years. We now dedicate this work to the memory of Giorgio and to his Family

Energy balance criteria for viscoelastic fracture

An energy balance criterion of the Griffith type has been used to derive conditions that are valid, in the isothermal non-inertial approximation, for the growth of cracks in viscoelastic bodies. These bodies are acted upon by general position and time-dependent load. The conditions obtained have the same form as the instability conditions obtained for the corresponding problems in elasticity theory and, in particular are independent of crack velocity. The analysis relies upon an exact calculation of the displacement and stress fields that is derived in the appendix with the aid of extensions to viscoelasticity of the Kolosov-Muskhelishvili equations of elasticity theory