Two-dimensional theory for a transversely isotropic thin plate in nonlinear elasticity
| dc.contributor.author | Fadodun, Odunayo Olawuyi | |
| dc.date.accessioned | 2023-05-13T17:02:29Z | |
| dc.date.available | 2023-05-13T17:02:29Z | |
| dc.date.issued | 2014 | |
| dc.description | xii,125p | en_US |
| dc.description.sponsorship | This study investigated the effects of finite deformation of a thin plate made of semilinear material of John’s type, transversely-isotropic in structure and designed with the aid of a homogenized theory. This was with a view to deriving a two-dimensional theory for the thin plate from a three-dimensional finite elasticity theory. The hypothesis of hyperelasticity of Cauchy-Truesdell was invoked on the plate’s energy stored function. The Frechet’s derivative of this function with respect to the energy conjugate geometry of deformation was employed to set up nonlinear three-dimensional boundary-value problem. The three-dimensional boundary-value problem was transformed into an equivalent variational problem and divergence theorem for tensor fields was used to introduce the boundary conditions into threedimensional variational form. The Kirchhoff-Love assumptions were imposed on the space of variations and inner product of tensor fields was applied in the reduction of the three-dimensional variational problem into two-dimensional variational form. The Green’s formula was employed to obtain an equivalent two-dimensional boundaryvalue problem for plate in consideration. In the decomposition of gradient of deformation, the Polar decomposition theorem and minimum property of rotation tensor were used to factor the gradient of deformation into product of rotation tensor and stretch symmetric tensor. The flexure and von-Karman conditions were used in the development of two-dimensional flexural and membrane equations of thin plate respectively. In the finite element formulation, a dual variable was introduced to decompose the flexural equation of plate into two coupled second order equations. For the purpose of numerical computation, a four-node quadrilateral mixed plate element with two-degree of freedom per node was formulated. The result showed that plate in finite deformation resisted externally applied loads by bending moments, twisting moments, in-plane stresses and transverse shear stresses. Thus, the nonlinear plate developed in-plane and out-of-plane forces and the derived two-dimensional theory reduced to the classical Kirchhoff-Love plate theory. The existence of transverse shear forces in the plates indicated that lines originally normal to the middle surface of the plates before deformation might not remain normal to the deformed surface and the presence of in-plane forces indicated that middle surface of nonlinear plate might experience change in length. The existence and uniqueness of solutions to both the continuous and discrete flexural problems of nonlinear plate in consideration were established. Furthermore, nonlinear plate in flexure exhibited harmonic forces within its planes and the associated membrane problem was described by mid-plane displacements ζi = 0,i = 1,2,3. Among others, the transversal isotropy of the plate in consideration increased the transverse shear stresses, in-plane forces, axial and flexural rigidities. In the case of degeneracy to isotropy, the obtained flexural rigidity coincided with those of Poisson-Kirchhoff’s and Fopp-von Karman’s plate theories. The study concluded that the numerical results demonstrated the efficiency of the Galerkin’s finite element approach. It also reinforced the view that finite deformation approach to elasticity problems provided ample opportunity to reveal important effects which small deformation theory often failed to apprehend. | en_US |
| dc.identifier.citation | Fadodun,O.O.(2014). Two-dimensional theory for a transversely isotropic thin plate in nonlinear elasticity. Obafemi Awolowo University | en_US |
| dc.identifier.uri | https://ir.oauife.edu.ng/123456789/5443 | |
| dc.language.iso | en | en_US |
| dc.publisher | Mathematics,Obafemi Awolowo University | en_US |
| dc.subject | Galerkin’s finite element | en_US |
| dc.subject | Fopp-von karman | en_US |
| dc.subject | Isotropy | en_US |
| dc.subject | Nonlinear Elasticity | en_US |
| dc.title | Two-dimensional theory for a transversely isotropic thin plate in nonlinear elasticity | en_US |
| dc.type | Thesis | en_US |