Imu. 1. Non-Linear .Mechanicr, Vol. ?I. No. 5. pp. 327-337. 1986 Printed in Great Britain. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA OOZO-7462/86iS3.CI3+ .@I Pergamon Journals Ltd zyxwvutsrqpon A VARIATIONAL PRINCIPLE FOR LARGE AXISYMMETRIC STRAIN OF INCOMPRESSIBLE CIRCULAR PLATES zyxwvutsrqponmlkjihgfedcbaZYXWVU LARRY A. TABER Department of Mechanical Engineering, University of Rochester, Rochester, NY 14627, U.S.A. (Received 12 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC February 1985; received for publication 29 January 1986) Abstract-A generalized variational principle is established for large axisymmetric deformation of a circular plate composed of an incompressible, hyperelastic material. Independent variation of stresses, strains, displacements and boundary tractions provides the field equations and boundary conditions for arbitrarily large strain with transverse shear deformation neglected. Based on the principle of stationary potential energy, a Rayleigh-Ritz technique yields approximate solutions for clamped and hinged plates under uniform surface pressure. Coupling between bending and membrane action induces significant alterations in the stress behavior, especially near the edge of the clamped plate. INTRODUCTION Large axisymmetric deflection of circular plates has been studied for many years. For problems limited to moderately large rotation, the von I&man equations Cl] are accurate. On the other hand, when displacements and rotations are arbitrarily large, the Reissner equations [Z] hold as long as the strains remain small. While these theories apply to most practical problems, deformation of plates composed of rubber or soft biological tissue, for example, may require a theory which allows large elastic strains. In a series of papers, Reissner [3-51 modified his theory for shells of revolution (the flat plate is a special case) to include shear deformation and large strain effects within the assumption of constant thickness. And since large strains can produce significant changes in thickness, Cook [6] further extended Reissner’s theory to incorporate a uniform transverse normal strain. These equations are valid when stretching dominates bending. For strong bending, calculations by Libai and Simmonds [7] show that stress and moment resultants depend heavily on the asymmetric motion of points relative to the reference surface. Thus, Taber [S] recently proposed modifications to Reissner’s strain relations to allow both thickness change and a shifting reference surface in an incompressible shell material. Including the effects of transverse shear strains, Simmonds [9, lo] obtained similar relations for shells of revolution and of general geometry. This paper first presents governing equations for arbitrarily large axisymmetric defor- mation of a circular plate, which is composed of an incompressible, hyperelastic material. Using a displacement formulation, Simmonds [ll] proposed a set of field equations and boundary conditions for a shell of revolution. The present work, which is based on the strain relations derived in [8], directly extends Reissner’s plate theory to allow asymmetric stretching of normals. For small strain, the equations reduce to those of Reissner [2], and for zero bending stiffness, they become those of non-linear membrane theory [12]. Next, a generalized variational principle is established. Independent variation of stresses, strains, displacements and boundary tractions provides all of the field equations and boundary conditions for the plate problem. The formulation is guided by the fundamental principle for non-linear elastostatics that was derived independently by Hu [13] and Washizu [14]. Similar variational principles for geometrically non-linear plate and shell problems are discussed in the book by Washizu [15]. Finally, approximate solutions are computed for clamped and simply supported circular plates with uniform pressure loads. The calculations are based on the principle of stationary potential energy and a modified Rayleigh-Ritz procedure. The interaction of membrane and bending stresses is studied. NLn 11:5-1. 327