Basic concepts of the approach by continuum mechanics

Abstract

I have been asked, at the beginning of this meeting of scientists of various disciplines which is devoted to the understanding and to the mastering of the mechanisms of plasticity, to recall the starting points, the methods and objectives of the approach from mechanics. Revue Phys. Appl. 23 (1988) 319-323 AVRIL 1988, Classification Physics A Abstracts 62.20 I. The Laws of the interactions I.1. Kinematical description Mechanics wants to give a continuation of the geometrical representation in order to explain and to Dredict the motion -or the equilibriumof bodies and structures subjected to various physical or chemical interactions. Anv body B may be viewed as a set of element M, usually assumed to be a continuous set, provided the number of these elements is great and their mutual distances small. Mechanics, namely, always tries to start with the most simple representation in order to go as far as possible into the mredictions. At each time t , one is looking for the displacement field X(t) of the various elements ofB. Usually M is a point and X the field of the displacement vector X(M,t) from a reference configuration (Lagrange). But sometimes the element M has to be considered -sayas the schematization of a rigid body (micropolar continuum). The field X is then the field X(t) generated not only by a displacement vector but also by an orthogonal tensor which gives the orientation of the rigid bodies. As it is well known, one may recover X(t) from the field U (Euler) of the U(M,t) which defines the of the element M. 1.2. Mechanical interactions The most natural way to define mechanical actions on REVUE DE PHYSIQUE APPLIQUEE. T. 23, N° 4, AVRIL 1988 a body B -the decisive and completely new concept on which classical mechanics is basedis to say how this body reacts under this action for various kind of instantaneous small motions -one raises a little bit a suitcase if one whishes to evaluate its weiaht, one moves a little bit the belt of the ventilator of a car in order to see if it is conveniently stressed. In mathematical words, if U is a (virtual) velocity field, defined on B at time t, as an element of a vector space of virtual motions V, the mechanical actions of a system S on B may be defined, for this class of v.m. V, by its virtual power -a real number : P = L (U) ( 1 ) where L is a linear and continuous function on V. As a consequence, if B is a finite set of points these actions are defined by a vector field on B, i.e. by forces. One recovers the Newton’s description, usually adopted by physicists. Vector fields give namely a quite convenaient represent’ation ouf the mechanical actions of a system S on a body B ; but mutual mechanical interactions inside B -stressesare not represented by vector fields, but by a scalar field in a fluid in equilibrium or a field of symmetric second order tensors in classical continuum mechanics. The definition (1) provides in any case the best way to introduce the definition of stresses. It Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01988002304031900