Surface Instability and Bifurcation of Elastic Materials
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Surface instabilities, such as wrinkles and creases, are often observed when elastic materials are subject to a sufficiently large compression. This thesis investigates surface instability and bifurcation of elastic materials due to their great importance to engineering application. The surface instability is studied by using the principle of minimum energy while the bifurcation is analyzed by solving the linearized equilibrium equations. In this thesis, we focus on the surface instability and bifurcation of a half-space, an infinite slab, and a rectangular block of elastic materials subject to biaxial loading. For the problem of surface instability, we use the first and second variations of the energy functional to find the necessary condition for stability. The requirement of a positive semi-definite second variation is transformed into the exploration of the minimum value of a constrained integral. Then the first variation condition of this constrained minimization problem leads to an eigenvalue problem associated with the stability. Solution of the eigenvalue problem yields a characteristic equation that determines the wavenumbers for the pattern of surface instabilities and the corresponding critical loading parameters. Subsequent to discussing the general properties of the characteristic equation, we obtain the stability region analytically. For the bifurcation problem, we linearize the boundary-value problem consisting of the Euler-Lagrange equation, the constraint of incompressibility and the corresponding boundary conditions and then solve the linearized boundary-value problem. Determination of non-trivial solutions to the linearized boundary-value problem yields a characteristic equation. Bifurcation points are obtained from the discussion of the general properties of the characteristic equation.