Masters by Research
School of Mathematics and Applied Statistics - Faculty of Informatics
Padukka Withana, Himanshuki N, Two problems in finite elasticity, MScRes thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2009. http://ro.uow.edu.au/theses/869
Some materials encountered in nature and used in engineering exhibit mechanical effects which cannot be adequately explained by classical linear elastic theories. For example, rubber is an elastic material that undergoes large elastic deformations, and therefore renders a non-linear mechanical behavior. An analytical investigation dealing with the problem of static deformation of such materials therefore involves highly non-linear equations leading to arduous mathematical work. Consequently there exists only a limited number of known exact solutions for such problems in the field of finite elasticity.
This thesis is concerned with two problems of finite elastic deformations of rubber blocks. Rubber has been successfully modeled as an isotropic incompressible hyperelastic material with strain energy function given by either the neo-Hookean or Mooney forms. For this class of materials, substantial reductions of the basic un- derlying equilibrium equations can be obtained, making the problems more tractable and for plane and axially symmetric deformations of these materials, simpler stress- strain relations can be obtained. Therefore, by combining these essentially two-dimensional stress-strain relations together with the reduced equilibrium equations it is possible to obtain comparatively tractible forms of the equations.
In this thesis the following problems for axially symmetric deformations of isotropic incompressible neo-Hookean and Mooney materials are investigated:
(i) asymptotic axially symmetric deformations describing compression of rubber cylindrical tubes with bonded metal end plates;
(ii) rippling of a long rectangular rubber block bent into a sector of a solid bounded by two circular arcs.
The above mentioned reduced equllibrium equations are employed in the context of non-linear continuum mechanics to arrive at approximate solutions. The solutions are approximate in the sense that the point-wise vanishing of the stress vector on a boundary is assumed to be replaced by the vanishing of forces in an average manner.
In the first problem, for axially symmetric deformations of the perfectly elastic neo-Hookean and Mooney materials, formal asymptotic solutions are determined in terms of expansions in appropriate powers of 1/R, where R is the cylindrical polar material coordinate. Remarkably, for both the neo-Hookean and Mooney materials, the first three terms of such expansions can be completely determined analytically in terms of elementary integrals. From the incompressibility condition and the equilibrium equations, the six unknown deformation functions, that appear in the first three terms can be reduced to five formal integrations involving in total, seven arbitrary constants, and a further five integration constants, making a total of twelve integration constants for the deformation field. The solutions so obtained for the neo-Hookean material are applied to the problem of the axial compression of a cylindrical rubber tube which has bonded metal end-plates. The resulting solution is approximate in two senses; namely as an approximate solution of the governing equations and for which the stress free boundary conditions are satisfied in an average manner only. The resulting deformation and load-reflection relation are shown graphically.
The second problem examined in this thesis is that of finite elastic deformation of a long rectangular rubber block which is deformed in a perturbed cylindrical configuration. This problem is motivated from the problem of determining surface rippling that is observed in bent multi-walled carbon nano-tubes. The problem of finite elastic bending of a tube is considerably more complicated than the geometrically simpler problem of the finite elastic bending of a rectangular block. Accordingly, we examine here the simpler block problem which is assumed to be sufficiently long so that the out of plane end effects may be ignored. The general equations governing plane strain deformations of an isotropic incompressible perfectly elastic Mooney material, which models rubber like materials, are used to determine small superimposed deformations upon the well known controllable family for the deformation of rectangular blocks into a sector of a solid bounded by two circular arcs. Traction free boundary conditions are assumed to be satisfied in an average sense along the bounding circular arcs. Physically realistic rippling is found to occur and typical numerical values are used to illustrate the solution graphically.
In summary reduced equilibrium equations and simplified two-dimensional stress strain relations are used in this study to solve two problems for isotropic incompressible neo-Hookean and Mooney materials. Such deformations and the class of materials studied considerably simplify what are otherwise very complex problems from the theory of finite elasticity.
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