On the behavior of viscoelastic plates in bending
Mase, George Edwin
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This investigation is concerned with the flexural response of linear viscoelastic plates of constant thickness. Fundamental equations for both quasi-static and dynamic response of such plates are developed and solved for important cases of each. The term quasi-static ls used to indicate that Inertia forces due to deformation are neglected. These are included, of course, in the dynamic analysis. Solutions of the quasi-static equation are compared with experimental results obtained by measuring the deflection of a test plate made of Plexiglas. The basic viscoelastic stress-strain relations used in the derivation of the fundamental plate equations are taken in the form of a differential time operator equation. Use of this equation leads to results that are In a convenient form for reduction to a particular material such as a Kelvin or a Maxwell plate. Using a generalized virtual work principle based upon irreversible thermodynamic considerations the fundamental plate equation, including shear effects, ls established. The procedure involved ls that of determining a stationary value of a certain operational invariant by means of the calculus of variations. A simplified form of this equation, omitting the shear effects, is deduced and solutions for various load conditions obtained. An extended version of this simplified form which includes inertia effects due to deformation is developed by the principle of correspondence. This is used to study free vibrations of rectangular viscoelastic plates simply supported on all edges. Solutions of the simplified form of the fundamental equation for the case of so-called proportional loading, I.e. when the load function is the product of a space function multiplied by a time function, are given in terms of the equivalent elastic solution multiplied by a function of time. For more general types of loading the deflection and the load are expanded Into suitable infinite series and these series representations are inserted directly into the previously mentioned variational expression of the generalized virtual work principle. This leads to a set of ordinary differential situations in time the unknowns of which are the coefficients of the deflection expansion. These equations, as were the similar ones arising in the case of proportional loading, are solved by the Laplace transform method of the operational calculus. As an example of such a general loading the case of a moving line load on a rectangular plate is worked out. As a means of establishing a correlation between the deflection predicted by the analytical solution and actual deflections of Inelastic plates a set of static load tests were carried out on a square plate made or Plexiglas. The results are plotted and a comparison of the theoretical and experimental values given. The problem of determining the dynamic response of viscoelastic plates is treated using the method given above for solving the case of general loading for the quasi-static deflection. Under the assumption of incompressibility of the plate material explicit solutions in terms of the physical parameters involved are presented and discussed. For compressible plate materials methods are developed to give approximate solutions the accuracy of which depends on the degree of approximation used in determining the roots of certain cubics appearing in the transformed form of the governing dynamics equation. Conditions for the dynamic solutions to be oscillatory are indicated.
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