Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours

The Laplace–Carson transform classically used for homogenization of linear viscoelastic heterogeneous media yields integral formulations of effective behaviours. These are far less convenient than internal variables formulations with respect to computational aspects as well as to theoretical extensions to closely related problems such as ageing viscoelasticity. Noticing that the collocation method is usually adopted to invert the Laplace–Carson transforms, we first remark that this approximation is equivalent to an internal variables formulation which is exact in some specific situations. This result is illustrated for a two-phase composite with phases obeying a compressible maxwellian behaviour. Next, an incremental formulation allows to extend at each time step the previous general framework to ageing viscoelasticity. Finally, with the help of a creep test of a porous viscoelastic matrix reinforced with elastic inclusions, it is shown that the method yields accurate predictions (comparing to reference results provided by periodic cell finite element computations).     

Micromechanical modeling of linear viscoelastic behavior of heterogeneous materials

Study of effective behavior of heterogeneous materials, starting from the properties of the microstructure, represents a critical step in the design and modeling of new materials. Within this framework, the aim of this work is to introduce a general internal variables approach for scale transition problem in linear viscoelastic case. A new integral formulation is established, based on the complete taking into account of field equations and differential constitutive laws of the heterogeneous problem, in which the effects of elasticity and viscosity interact in a representative volume element. Thanks to Green’s techniques applied to space convolution’s term, a new concentration relation is obtained. The step of homogenization is then carried out according to the self-consistent approximation. The results of the present model are illustrated and compared with those provided by Hashin’s and Rougier’s ones, considered as references, and by internal variables models such as those of Weng and translated fields.     

Structural morphology and relaxation spectra of viscoelastic heterogeneous materials

We analytically derive the relaxation spectra of a two-phase isotropic material whose phases are isotropic Maxwell media, according to the classical and to the generalized self-consistent schemes. Whereas these spectra are continuous in both cases, they exhibit strong differences which can be associated with the different underlying morphology, either symmetrical (polycrystal-type) in the first case or asymmetrical (composite-type) in the second case. The treatment is extended to the (N+1)-phase model which allows us to deal with coated inclusions or with an interphase between the matrix and the inclusions: the interphase is shown to strongly modify the resultant spectrum. More general cases are then considered for different kinds of constitutive behaviour as well as for coated fibre reinforced composites. As a whole, the spectral analysis method appears to be an efficient tool for the investigation of the connection between structural morphology and the overall behaviour of viscoelastic heterogeneous materials.     

Complex loading of heterogeneous materials with redistribution of internal mass

The problem of inhomogeneous material with internal friction subjected to complex loading is solved in this paper. Different complex loadings are considered by continuously rotating the principal axes of the strain tensor. A hypoplastic model with internal variables is used to describe the internal structure of the material. The model can describe loading and unloading states. It accounts for essential non-linearity of constitutive equations. The finite element algorithm reduces the problem to a system of nonlinear algebraic equations of high order which are solved by the Newton's method. The results show a good qualitative and quantitative assessment of calculated parameters when compared with data obtained from experiments. Redistribution of internal mass takes place under loading such that a fractal structure is developed in time reflecting the material discontinuities.     

Behaviour of the acceleration and shock waves in materials with internal state variables

The explicit expressions for the change in the amplitudes of one-dimensional acceleration and shock waves propagating through arbitrary homogeneous materials described by the strain and internal state variables/parameters/are derived. The existence of a critical amplitude β for the acceleration wave and a critical strain gradient λ for the shock wave is established. For an infinitesimal shock wave the general form of the solution of the governing differential equation is furnished. The differential equations for the amplitudes of these two kind of waves are applied to an elastic-viscoplastic material.     

Finite and symmetric thermomechanical waves in materials with internal state variables

The paper is devoted to the study of wave propagation through materials with internal state variables under two main assumptions: finite speed of propagation and symmetry of acceleration waves. Additional constitutive assumptions are also used: heat flux does not depend on the temperature gradient and the time-derivatives of internal state variables are linear functions of the temperature gradient. Under all these hypotheses, in the neighborhood of a strong equilibrium state, one finds four real and symmetric possible acceleration waves, at least two of them being coupled waves, and heat flux results an internal state variable. All these results are obtained in the general three-dimensional case. As an illustration, the isotropic linear theory is considered, where both acceleration and shock waves are treated.     

Integral formulation and self-consistent modelling of elastoviscoplastic behavior of heterogeneous materials

Summary Based on projection operators, an integral formulation is proposed for elastoviscoplastic heterogeneous materials. The problem requires a complete mechanical formulation, including the static equilibrium property concerning the known field σ, in addition to the classical field equations concerning the unknown fields ɛ˙ and σ˙. The formulation leads to an integral equation, in which elasticity and viscoplasticity effects interact through an homogeneous elastoviscoplastic medium with elastic moduli C and viscoplastic moduli B. To approximate the integral equation, the self-consistent scheme is followed. In order to obtain consistent approximation conditions, we introduce fluctuations of elastic and viscoplastic strain rate fields with respect to known kinematically compatible fields. It results in a strain rate concentration relation connecting the strain rate at each point to the macroscopic loading conditions and the local stress field. The results are presented and compared with other models and with experimental data in the case of a two-phase material. Received 26 August 1997; accepted for publication 2 July 1998     

Effective thermal conductivity of heterogeneous multi-component materials: an SPH implementation

Modeling heat transfer and fluid flow in materials with complicated micro-structures is a major challenge to numerical methods due to their multiscale and multiphysics nature. A relatively novel numerical technique—the meshless smoothed particle hydrodynamics (SPH) method has the potential of making a significant contribution to this research field. In the present SPH modeling effort, a 2D modeling system is devised for the prediction of the effective thermal conductivity in heterogeneous materials containing two or three different components. The microscopic component configuration inside the materials is constructed in the SPH methodology by randomly assigning particles as a certain component to meet the required macroscopic composition. For heterogeneous two-component materials, the effective thermal conductivity predicted by the modified effective medium theory model with the so-called “flexible” factor f equal to 4.5 agrees well with the SPH data. On the basis of a simple “step-process” concept, the effective thermal conductivity of a heterogeneous multi-component material can be derived from the corresponding “degenerate” materials which consist of fewer components.     

Linear viscoelastic behavior of the Hookean dumbbell with internal viscosity

The linear viscoelastic modulus G(t) predicted by the analytical formulations of Schieber (1993), Wedgewood (1993), Dasbach et al. (1992), and Booij and van Wiechen (1970) for the free-draining Hookean dumbbell with internal viscosity (IV) are compared with exact analytical results and exact numerical results obtained from Brownian dynamics simulations. All of these analytical formulations employ the Booij and van Wiechen expression for the IV force, thereby eliminating errors associated with linearization of the deformational velocity, however the theories differ in the approximations employed to solve configuration moment equations. Comparison with the exact G(t) results provides a means of testing these approximations. The approximate theories all correctly predict the singular part of G(t) at t = 0, providing correct predictions of , however deviations from the exact G(t) are seen in all cases for t > 0.     

A non-linear viscoelastic model with structure-dependent relaxation times: I. Basic formulation

A non-linear constitutive equation for polymer melts and concentrated solutions is presented. Based on known results of network theories, the model contains a distinctive feature: that of letting the relaxation times depend upon the existing structure.The model extends the constitutive equation of linear viscoelasticity to the non-linear region in a well-defined way, with the uncertainty of just a single adjustable parameter.Predictions of the model for common cases of non-linear response are derived and discussed.     

Measurement of the viscoelastic properties of bituminous materials using an oscillating needle technique

An embedded oscillating needle is used to measure the dynamic viscoelastic properties of a stiff bituminous material. A Micro-Fourier Rheometer was used to cause the embedded needle to undergo pseudorandom small amplitude oscillations in the axial direction with measurement of the instantaneous resistance force. The phase and magnitude of the force signal are used to calculate the storage and loss moduli. A theoretical framework for this technique is developed from the Mindlin solution coupled with slender body theory, and the correspondence principle of linear viscoelasticity. Experiments are performed on neat bitumen binders as well as mixtures of glass spheres in bitumen; the results show that the presence of the glass spheres dramatically increases the viscoelastic response functions. The results agree reasonably well with those obtained using the parallel plate squeezing mode. Received: 31 March 1999 Accepted: 14 July 1999     

Lagrangian formulation of continuum with internal long-range interactions

Based on a new definition of nonlocal variable, this paper establishes the Lagrangian formulation for continuum with internal long-range interactions. Distinguished from the existing theories, the nonlocal term in the Lagrangian formulation automatically satisfies the zero mean condition determined by the action and reaction law. By this formulation, elastic wave in a rod with the internal long-range interactions is investigated. The dispersion of the elastic wave is predicted.     

Representation of the glass-transition in mechanical and thermal properties of glass-forming materials: A three-dimensional theory based on thermodynamics with internal state variables

In the vicinity of the glass transition, glass-forming materials exhibit pronounced frequency-dependent changes in the mechanical material properties, the thermal expansion behaviour and the specific heat. The frequency dependence becomes apparent under harmonic stress, strain or temperature excitations. The Prigogine-Defay ratio is a characteristic number which connects the changes in magnitude of these quantities at the glass transition. In order to represent the thermoviscoelastic properties of glass-forming materials in continuum mechanics, a three-dimensional approach which is based on the Gibbs free energy as thermodynamic potential is developed in this article. The Gibbs free energy depends on the stress tensor, the temperature and a set of internal variables which is introduced to take history-dependent phenomena into account. In the vicinity of an equilibrium reference state, the specific Gibbs free energy is approximated up to second order terms. Evaluating the Clausius-Duhem inequality, the constitutive relations for the strain tensor, the entropy and the internal variables are derived. In comparison with other approaches, the entropy, the strain tensor and the internal variables are functionals not only of the stress tensor but also of the temperature. Applying harmonic temperature- or stress-controlled excitations, complex frequency-dependent relations for the specific heat under constant stress, for the thermal expansion coefficients as well as for the dynamic mechanical compliance are obtained. The frequency-dependence of these quantities depicts the experimentally observed behaviour of glass-forming materials as published in literature. Under the assumption of isotropic material behaviour, it is shown that the developed theory is compatible with the Prigogine-Defay inequality for arbitrary values of the material parameters.