Mechanics of Random and Multiscale Microstructures

In these sections we outline some basic ideas underlying the solution of statistical problems in conductivity, acoustics and elasticity. The problems we consider are both static and dynamic and the statistical nature of the problems sterns from the fact that coefficients like the heat conductivity or sound speed are random functions of position. We first point out that simple averaging procedures are inadequate to determine effective properties of a medium except in the limit of small perturbations. In general they lead to an infinite hierarchy of statistical equations. For large variations of the random coefficients different techniques are required to obtain useful information. In the static problem variational principles are used extensively to find bounds on effective constants. However, we also discuss very approximate techniques like the self-consistent scheme. In the dynamic case we consider three techniques used in different-type problems. We show how to treat the problem of scattering by a dilute collection of discrete scatterers, propagation of acoustic waves through a medium like the ocean and forward and backward scattering of acoustic waves by a one-dimensional random medium.

In this chapter, various types of models of random media are introduced. After a presentation of the basic morphological measurements that are available to quantitatively characterize the geometry of random media, some morphological models of random media are reviewed. They may be useful on two different levels: to provide a description of the heterogeneous structure, and sometimes to predict macroscopic properties of materials. This is illustrated by the calculation of third order bounds of linear properties and by various fracture statistics models.

The second section (Continuum Mechanics of Random Media), central to this chapter, focuses on the problem of Representative Volume Element (RVE) in the sense of Hill (1963). This is, in essence, the problem of coupled size and boundary condition dependence of constitutive response of microstructures described by random fields. Indeed, given three possible types of boundary conditions, we have three possible apparent responses, two of which (kinematic- and traction-controlled) hound on mesoscale the effective (in the macroscopic sense) response. The mesoscale domain involved is called a Statistical Volume Element (SVE). A review is provided here of the results obtained in this field over the past dozen years. We first study hierarchies of mesoscale bounds for elastic microstructures: both qualitative results via variational principles and quantitative results^-via computational mechanics. The same type of approach is then sketched for inelastic microstructures, although a much wider range of results is given in the elastic case. This chapter culminates in a short review of two topics in wave propagation in random media: spectral finite elements and wavefront propagation. Here the wave length, respectively wavefront thickness, again introduces an SVE in place of an RVE.

The third section (Some Computational Mechanics Methods for Random Media) directly builds on, and supports, the second one. On one hand, it is shown how the mesoscale bounds can be used as input into the finite element methods, thus leading to micromechanics-based stochastic finite elements. The semblance, as well as contradistinction, of this approach to conventional finite elements and conventional random field models of continua is discussed. On the other hand_; we provide a very brief introduction into spring network models, heavily favored by this author for rapid computation of statistics of mesoscale bounds of elastic materials, based on Monte Carlo-type generation of their random microstructures.

This chapter is divided into three main parts. In the first part, an attempt is made to show how, in practice, various sources of randomness are taken into account in structural integrity analysis. The emphasis is laid upon the aspects related to the microstructural details of the materials, using various examples, including brittle cleavage fracture and ductile rupture in steels. This part deals also with statistical aspects related to fatigue loading. Probabilistic linear fracture mechanics is briefly introduced. The second part is devoted to statistical modelling of fatigue damage. Both high strain low cycle fatigue (LCF) and high cycle fatigue (HCF) are considered. In LCF the emphasis is laid on kinetic theories describing the evolution of multiple cracks population while in HCF results obtained in the frame of weakest link theory are presented. A short account of directionality aspects in multiaxial fatigue damage is given. The relative importance of volume and surface effects is also discussed. In this part, a number of examples dealing with various materials, including steels, nodular cast iron and Ni base superalloys, are given. The third part is devoted to statistical aspects in ductile fracture which is a research field much less investigated in the literature compared to fatigue damage and brittle fracture. This part strongly relies upon recent studies on C-Mn steels and duplex stainless steels. It is shown that these materials exhibit a large scatter in strain to failure and a significant size effect when specimens of different sizes are tested. Microstructural investigations using quantitative image analysis have shown that ductile damage in these materials is highly heterogeneous. In order to predict rupture, finite element models are used. The materials are described using Gurson and Rousselier plastic potentials for damaging porous materials. In order to model size effects and scatter, it is necessary to account for the distribution of damage nucleation rates (duplex stainless steels) and for the spatial inhomogeneous distribution of nucleation sites (C-Mn steels) which were experimentally measured. Comparison of experiments with simulations shows that the models are able to describe both crack initiation and propagation. In particular, they can predict mean value and scatter observed on strain to failure (tensile bars) and on initiation and propagation energies (fracture mechanics specimens).

Composites made of n constituents, or phases, firmly bonded across interfaces, will be considered. Each phase conforms to a constitutive relation of the form