The Incompatibility Operator: from Riemann’s Intrinsic View of Geometry to a New Model of Elasto-Plasticity

Abstract

The mathematical modelling in mechanics has a long-standing history as related to geometry, and significant progresses have often been achieved by the invention of new geometrical tools. Also, it happened that the elucidation of practical issues led to the invention of new scientific concepts, and possibly new paradigms, with potential impact far beyond. One such example is Riemann's intrinsic view in geometry, that offered a radically new insight in the Physics of the early 20th century. On the other hand, the rather recent intrinsic approaches in elasticity and elasto-plasticity also share this philosophical standpoint of looking from inside, i.e., from the "manifold" point of view. Of course, this approach requires smoothness, and is thus incomplete for an analyst. Nevertheless, its first aim is to highlight the concepts of metric, curvature and torsion; these notions are addressed in the first part of this survey paper. In a second part, they are given a precise functional meaning and their properties are studied systematically. Further, a novel approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying this intrinsic spirit. The main mathematical object in this theory is the incompatibility operator, i.e., a linearized version of Riemann's curvature tensor. So far, this route not only has led the authors to a new model with a solid functional foundation and proof of existence results, but also to a framework with a minimal amount of ad-hoc assumptions, and complying with both the basic principles of thermodynamics and invariance principles of Physics. The questions arising from this novel approach are complex and intriguing, but we believe that the model is now sufficiently well posed to be studied simultaneously as a problem of mathematics and of mechanics. Most of the research programme remains to be done, and this survey paper is written to present our model, with a particular care to put this approach into a historical perspective.2010 Mathematics Subject Classification. 35J48,35J58,49S05,49K20,74C05,74G99,74A05,74A15, 80A17.

A Variational Approach to Single Crystals with Dislocations

“…In the case that the dislocation cluster is associated with a single Burgers vector, by Lemma 3.11 it follows that even if δ = 0, the energy satisfies the following coercivity condition with respect to the determinant of F , 6) it can be seen that the same coercivity (5.5) holds true also in the case 6−2r 3r + 1 p = 1, by assuming 1…”

“…One of the crucial point is that the very nature of the displacement field in the presence of dislocations is multiple-valued, due to the fact that the value of the displacement field depends on the number of loops made by a circuit wrapping around the dislocation line and along which the deformation is integrated (consider the classical Michell-Cesaro formulae [6]). There are two ways to mathematically address this fact.…”

Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

“…Let us also stress that this approach, despite rarely seen today, has a long history: we date the origin of the intrinsic view to Riemann with ground-breaking applications in general relativity and later in mechanics (in particular see the Hodge and Prager approach in perfect plasticity [34]). In general, as explained in [4], Riemann's view is in contrast with Gauss' standpoint of immersions, that is a displacement or velocity-based formulation. Furthermore, as far as dislocations are involved, this geometric approach was very much developed and enhanced by the physicist E. Kröner in the second half of last century [27].…”

“…Next, we require that the intrinsic power induced by the hyperstress Ω τ • ∇d dx vanishes as soon as the deformation is compatible, i.e., that it is only due to micro-structural defects in the form of dislocations. Then it was shown in [6] (see also [4] for different arguments) that the components of D are related through a scalar , called incompatibility modulus, which eventually yields that − div τ = inc ε. Therefore the virtual power principle leads to the weak form…”

“…This approach is most probably the first historically, being the strain indeed considered to measure deformation, that is, variation in length and in mutual orientation of infinitesimal fibers within a solid body. As a matter of fact, for the geometer the strain is a metric from which all other geometric concepts (such as curvature, torsion and other sophisticated tensors, see [4,11]) are retrieved. Specifically, given a smooth strain tensor field ε, the classical Kirchhoff-Saint Venant construction (for a historical review, see [4,28,36]) in linearized elasticity basically consists in…”