On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations |
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Authors: | James M Hill Daniel J Arrigo |
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Institution: | (1) School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia; E-mail |
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Abstract: | In three recent papers 6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly
elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations
can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations
there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers
is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed
to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed
deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of
a second-order linear differential operator D2, which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference
to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude,
the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene
vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered
in Part II 16].
This revised version was published online in August 2006 with corrections to the Cover Date. |
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Keywords: | finite elasticity incompressible Varga strain-energy small on large deformations plane strain |
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