Preface
The most common definition of inhomogeneity1 relates to a whole composed of dissimilar or nonidentical elements or parts. In materials science examples of inhomogeneities are provided by composite materials and polycrystals. The former often calls for the notion of periodicity while the latter often relies on stochasticity. A more or less spatially extended foreign element (called an inclusion, void being a special case) in an otherwise homogeneous material also represents an inhomogeneity, but a localized one. In that respect a limit reasoning yielding the notion of cut (crack), or even point-like singularity, also brings us into the realm of material inhomogeneity. Thus the latter can simply be verbally described as a more or less rapid, sometimes abrupt, change in material properties (e.g. density, elasticity coefficients, etc.), in the absence of external stimuli, with the considered material point. This dependence on the material point in continuum mechanics is markedly exhibited when all laws, especially the differential ones such as the local basic balance laws, are expressed in the material framework, contrary to common practice. By this we do not mean the usual introduction of the first Piola-Kirchhoff stress tensor to treat nonlinear strains, but a complete projection of balance laws on the material manifold.
In the present monograph the attention is focused on the case of elastic materials with a few excursions in the realm of anelastic solids. Mostly based on recent works (1989-92) by the author and co-workers, the contents of this monograph show how the material view of continuum mechanics brings together such apparently unconnected notions as those of Hamiltonian canonical formulation of anisotropic elasticity, energy-release rate and path-independent integrals of fracture theory, geometric representation of continuous
1 We have preferred the more mathematical 'inhomogeneity' to the usual 'heterogeneity' or 'nonhomogeneous materials'.
distributions of crystal defects, momentum of electromagnetic field in matter and of quasi-particles in elastic-crystal dynamics, and conservation of total pseudomomentum for solitons in elastic structures presenting dispersion. Essential in these developments is the balance, or rather unbalance, of so-called pseudomomentum, that is, the canonical (in the sense of field theory) material momentum rather than the usual physical linear momentum, which comes to light when studying forces on the material manifold (those forces which are generated by changes in the material point) rather than physical forces (which are generated by changes in the actual physical placement).
To convey the message in a more vivid way, let us consider the following analogy between what occurs in physical space (Fig. P(a)) and what takes place in material space (Fig. P(b)). In part (a) a rigid-plastic piece of wire of length / is placed in physical space (represented by a square). If the (physical) tension force / acting on the left end of the wire is strong enough, assuming the other end fixed, then the wire may flow plastically, i.e. we have in physical space the conditions: /</c, Д/ = 0; / = /с, A/^0 possibly, but not necessarily, where /„ is the plastic threshold (see Maugin, 1992a, Chapter 1). The force / can be computed by f = dE/dl if E is the total potential energy of the system. Now consider Fig. P(b). The straight crack of length / obviously develops in material space (represented by the shaded square). If the (material) fictitious force /inh acting at the tip of the crack is high enough, the crack may start to progress in the material. We have the growth criterion: /inh</'nh, Л/ = 0; /inh = /inh, ДМ0 possibly but not necessarily, where /cnh is a threshold value characteristic of the considered material. The similarity between parts (a) and (b) is rather obvious. But in so far as the object called the crack is concerned, we may as well consider Fig. P(c), where the tip of the crack is kept fixed and the material square is moved to the right by the amount Л/ whenever the crack length grows by this amount, all other things (physical forces) being kept unchanged. We can write then finh= —дЕ/dl. The corresponding developed power all goes into dissipation as we cannot glue back the two faces of the crack! On the one hand, in part (a), the physical force is clearly generated by a variation in the actual position in physical space, the material framework being kept fixed. On the other hand, in part (b), the material force /inh is generated via a variation of the point in
