Continuum aspects of the natural approach

Abstract

The natural finite element approach introduced by Argyris in the early sixties is marked essentially by the distinction between rigid body motion and deformation, on the one hand, and by the description of the latter in compliance with the element purpose and geometry, on the other hand. For triangular and tetrahedral elements the approach suggests strain and stress measures defined along the sides, respectively the edges as homogeneous normal quantities, free of shear. In the mechanics of continua the respective infinitesimal elements represent minimum configurations to define local deformation in two- and three dimensions. The paper presents developments of the natural approach on the continuum level within a consistent theoretical framework. It is proposed to begin with a reference system of supernumerary coordinates associated with the elementary tetrahedron in the space and with the triangle in the plane. Vectorial quantities are defined, the operations of gradient and divergence are interpreted in this system. The natural deformation rate is deduced from the velocity field, the stress is introduced as work conjugate measure. The condition for local equilibrium is presented in natural quantities as well as the stress definition in association with the resultant forces. The set up of material constitutive relations is exemplified for the viscous solid. Beyond the description of the momentary kinematics as from the velocity field, the appearance of finite deformation is considered basing on displacements. Illustration of the methodology for a plane elastic case is appended.