Finite strain analysis in elastic theory
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The stability of the clamped circular thin plate is investigated beginning with v. Karman's non linear differential equations for large strains. This non-linear system is developed into two linear sequences of equations ordered in powers of a parameter. The first approximation is found to yield the elementary solution of the problem. The second approximation of the radial membrane stress is shown to decrease the compressive stress in the center of the plate and increase it at the edge. The method of removing the indeterminacy of the second approximation to the slope function is outlined but not carried out because the explicit form of the particular integral free of integrals was not obtained;A parameter expansion, due to Garabedian, is employed to order the three-dimensional equilibrium equations and surface traction conditions for the moderately thick circular plate. From these equations the differential equations for the deflection and the stretching of the middle plane of the plate are derived for large strains. A method of successive approximations is then used to solve the non-linear differential equation for the case of the clamped plate with uniform loading. The correction to the elementary solution for the deflection is found to be significant only for extremely large values of the loading. The solution for the stretching of the middle plane of the plate, using the elementary value of the deflection, is found to be possible only under the assumption that rigid clamping of the edge of the plate against lateral displacements is impossible.