On the formulation of variational theorems involving volume constraints

S. N. Atluri, E. Reissner

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Abstract

A continued concern with variational theorems which are suitable for numerical implementation in connection with the analysis of incompressible or nearly incompressible materials has led us to the formulation of five-field, and in one case seven field, theorems for displacements, deviatoric stresses, pressure, distortional strains and volume change. In essence these theorems may be thought of as generalizations of the Hu-Washizu three-field theorem for displacements, stresses and strains and of the earlier two-field theorem for displacements and stresses. For ease of exposition, what follows is divided into three parts. The first part deals with geometrically linear elasticity. The second part deals with the effect of geometric nonlinearity in terms of Kirchhoff-Trefftz stresses and Green-Lagrange strains. The third part is concerned with results involving generalized Piola stresses and conjugate strains, as well as with results about distinguished (Biot) generalized stresses and their conjugate strains. Also for ease of exposition, attention is limited to statements about volume integral portions, omitting body force and boundary condition terms. In addition to formulating five field theorems, as well as one seven field theorem, we use these theorems, through the introduction of various constraints, for the deduction of alternate six, five, four, three, and two-field theorems for incompressible or nearly incompressible elasticity.

Original languageEnglish
Pages (from-to)337-344
Number of pages8
JournalComputational Mechanics
Volume5
Issue number5
DOIs
StatePublished - Sep 1989

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