Nicolas Charalambakis

Homogenization Techniques and Micromechanics. A Survey and Perspectives

In this paper, we present a critical survey on homogenization theory and related techniques applied to micromechanics. The validation of homogenization results, the characterization of composite materials and the optimal design of complex structures are issues of great technological importance and are viewed here as a combination of mathematical and mechanical homogenization. The mathematical tools for modeling sequentially layered composites are explained. The influence of initial and boundary conditions on the effective properties in nonlinear problems is clarified and the notion of stability by homogenization is analyzed. Multiscale micromechanics methods are outlined and the classical as well as the emerging analytical and computational techniques are presented. Computation of effective static and dynamical properties of materials with linear or nonlinear constitutive equations is closely related to the development of generalized theories such as the strain-gradient mechanics. Selected applications of these techniques are outlined. Moreover, the extension of kinetic techniques in homogenization and the related inverse imaging problem are presented.

Dynamic stress concentration around a hole in a viscoelastic plate

SummaryThe transient hoop stresses which are generated at the circumference of a circular hole in a large viscoelastic plate when a radial pressure pulse acts on the hole boundary, are determined. An analytical/numerical approach is employed which is based on the use of Laplace transform and Hankel functions, and the Dubner-Abate-Crump technique for inverting the Laplace-transformed solution. In our formulation of the problem, a general linear-viscoelastic material for the plate is considered, but numerical results are extracted only for a three-parameter model (standard linear solid). The present work accompanies recent efforts by Georgiadis on analogous transient viscoelastodynamic problems involving finite-length cracks in a stress-wave environment. The mathematically simpler problem considered here demonstrates the influence of viscoelastic effects on the dynamic-stress-concentration in marked similarity with the dynamic-stress-intensity behavior of crack-tip viscoelastic stress fields. Besides that, the present problem generalizes the classical Kromm-Selberg-Miklowitz problem in the sense that it considers viscoelastic response. The latter solutions were not utilized here through the usual elastic/viscoelastic correspondence principle but, instead, an independent solution was derived by simple means.

The stabilizing role of higher-order strain gradients in non-linear thermoviscoplasticity

SummaryThe stabilizing role of higher order strain gradients incorporated in the constitutive equation for the stress is illustrated. It is shown, in particular, that these gradients “smooth out” possible non-uniformities in strain and temperature that may develop in a thermoviscoplastic slab sheared by a constant boundary force. Moreover, it is demonstrated that the occurrence and nature of instability is qualitatively different than the localized “catastrophic” shear banding instability of the classical analysis where higher order strain gradients are not included.

Two-dimensional adiabatic Newtonian flow with temperature-dependent viscosity

We study the analytical and numerical stability of the two-dimensional adiabatic flow of an incompressible Newtonian liquid with temperature-dependent viscosity. We show that the uniform shearing is the asymptotic solution if the time becomes very large in a rate dependent on the temperature sensitivity and the referencial Eckert and Reynolds numbers. Numerical solutions for flows between parallel plates caused by steady boundary velocity are presented. The results indicate that the boundary velocity controls the thermomechanical process in complete agreement with the analytical behaviour of the flow.

Shear stability and strain, strain-rate and temperature-dependent “cold” work

Abstract We present the stability and instability conditions of a thermoviscoplastic material with the assumption that the fraction ζ of plastic work stored as energy of crystal defects (“cold” work) is a function of strain, strain-rate and temperature. The results comply with the stability and instability criteria of a material with power constitutive law and constant ζ, provided that the function ζ satisfies an additional condition, valid for many used materials. However, the maximum and minimum values of the function ζ may, respectively, affect the critical time at which shear banding is manifested and the range of bounds of the shear stress. Moreover, we present the role of the function ζ in the stability of the uniform shearing of a mild steel characterized by a Costin et al. flow stress [Inst. Phys. Conf. Ser. No. 47, Adam Hilger, Bristol, 1979, Chapter 1, p. 90]. We show that, if ζ is sufficiently increasing with respect either to strain or to temperature and, at the same time, its slope of decrease with respect to strain-rate is moderate, then the uniform shearing is stable.

Two stable-by-homogenization models in simple shearing of rate-dependent non-homogeneous materials

In this paper we study two models, the viscoplastic model and the thermoviscous model, of rate-dependent non-homogeneous materials with non-oscillating strain-rate sensitivity submitted to simple quasistatic shearing. We prove that the two models are stable by homogenization, i.e. that the equations in both the heterogeneous problems and the homogenized one have the same form, and we give explicit formulas for the homogenized (effective) coefficients. These formulas depend on the initial conditions, but not on the boundary conditions. Our theoretical results are illustrated by a numerical example.

Static frictional indentation of an elastic half-plane by a rigid unsymmetrical punch

Static rigid 2-D indentation of a linearly elastic half-plane in the presence of Coulomb friction which reverses its sign along the contact length is studied. The solution approach lies within the context of the mathematical theory of elastic contact mechanics. A rigid punch, having an unsymmetrical profile with respect to its apex and no concave regions, both slides over and indents slowly the surface of the deformable body. Both a normal and a tangential force may, therefore, be exerted on the punch. In such a situation, depending upon the punch profile and the relative magnitudes of the two external forces, a point in the contact zone may exist at which the surface friction changes direction. Moreover, this point of sign reversal may not coincide, in general, with the indentors apex. This position and the positions of the contact zone edges can be determined only by first constructing a solution form containing the three problems unspecified lengths, and then solving numerically a system of non-linear equations containing integrals not available in closed form.The mathematical procedure used to construct the solution deals with the Navier-Cauchy partial differential equations (plane-strain elastostatic field equations) supplied with boundary conditions of a mixed type. We succeed in formulating a second-kind Cauchy singular integral equation and solving it exactly by analytic-function theory methods.Representative numerical results are presented for two indentor profiles of practical interest—the parabola and the wedge.

An analytical/numerical approach for cracked elastic strips under concentrated loads : transient response

An analytical/numerical approach is presented for the determination of the near-tip stress field arising from the scattering of SH waves by a long crack in a strip-like elastic body. The waves are generated by a concentrated anti-plane shear force acting suddenly on each face of the crack. The problem has two characteristic lengths, i.e. the strip width, and the distance between the point of application of the concentrated forces and the crack tip. It is well-known that the second characteristic length introduces a serious difficulty in the mathematical analysis of the problem. In particular, a non-standard Wiener-Hopf (W-H) equation arises, that contains a forcing term with unbounded behaviour at infinity in the transform plane. In addition, the presence of the strips finite width results in a complicated W-H kernel introducing, therefore, further difficulties. Nevertheless, a procedure is described here which circumvents the aforementioned difficulties and holds hope for solving more complicated problems (e.g. the plane-stress/strain version of the present problem) having similar features. Our method is based on integral transform analysis, an exact kernel factorization, usage of certain theorems of analytic function theory, and numerical Laplace-transform inversion. Numerical results for the stress-intensity-factor dependence upon the ratio of characteristic lengths are presented.

Effective properties of multiphase composites made of elastic materials with hierarchical structure

In this paper, the analytical solution of the multi-step homogenization problem for multi-rank composites with generalized periodicity made of elastic materials is presented. The proposed homogenization scheme may be combined with computational homogenization for solving more complex microstructures. Three numerical examples are presented, concerning locally periodic stratified materials, matrices with wavy layers and wavy fiber-reinforced composites.

Thermoviscoplastic instabilities and double strain gradient approach in biaxial loading by effective instability analysis

Abstract In this article we study the influence of double strain gradient, reflecting microstructural inhomogeneities, on the instability regime of a thermoviscoplastic material caused by biaxial loading. A perturbation analysis proposed earlier by Dudzinski and Molinari [1991] is used. The gesults show the influence of the microstructural coefficient on the rate of growth of the instability for various values of strain hardening, strain rate sensitivity, and straining path. The role of optimal orientation is presented, and the cases of isothermal and anisothermal deformation are analysed. Our results are also compared with those predicted by the aforementioned analysis. Finally, a comparison of uniaxial and biaxial situations concerning the role of the microstructural parameter is presented.