Browsing by Author "Tolefson, Donald Craig"

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    Asymptotic solutions of a circular plate problem
    Tolefson, Donald Craig (Virginia Polytechnic Institute, 1967)
    The rotationally symmetric small and large finite deformations of an annular plate, clamped at its outside edge and containing a central rigid inclusion to which a normal load is applied, are considered. Perturbation techniques are applied to analyze the first approximations to the Reissner equations for the separate cases of small and large finite deformations. In the case of small finite deformations, where the first approximations to Reissner's equations correspond to the von Karman equations, we examine the limiting cases of infinitesimal and increasingly large deformations. Perturbation expansions in terms of powers of a small parameter are obtained for the dependent variables. In the limiting case of infinitesimal deformations two terms of the expansion of the dependent variable representing the bending of the plate are obtained in closed form; the second of which arises due to the coupling of the stretching and bending of the plate's middle surface. This second term of the expansion becomes important when the maximum transverse displacement of the plate begins to exceed slightly less than one-half of the thickness of the plate. For this part of the analysis of the von Karman equations the expansions are uniformly valid over the extent of the plate. In the case of increasingly large deformations the analysis of the von Karman equations leads to a singular perturbation problem. Here the edge zones, where boundary-layers have developed, and the interior region of the plate are investigated individually. Separate perturbation expansions are obtained in these regions of the plate. The theory of ''Matched Asymptotic Expansions" is utilized here in order to evaluate certain constants of integration which remain undetermined after exhausting the boundary conditions on the problem. Two and, in one case, three terms of the expansions are found in closed form. It is found that for the special case where Poisson's ratio is equal to 1/3 the results are extremely simple and here we obtain three terms in the expansions. Numerical results for the stresses and transverse displacement are compared with data obtained from the numerical integration of the Reissner equations by other authors and the agreement is very good. Lastly we examine the case of large finite deformations where the deformations have exceeded the range of validity of the von Kannan equations. A perturbation analysis of the first approximations to Reissner's equations for this case leads to a singular perturbation problem as expected. The basic differential equations for the boundary-layer zones and interior region of the plate are derived as well as the required conditions, in addition to those provided at the edges of the plate, for determining the constants of integration which arise in the analysis. Closed form solutions for one of the dependent variables can not be obtained here and numerical integration is required. For this reason numerical results are not given. In this latter part of the investigation we observe that the first approximations to the original form of Reissner's equations contain certain terms which he neglects in his more recent theory. We, therefore, conclude that these terms which are missing from his more recent theory are important if the deformations are extremely large and should be retained in this case.
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    A study of biaxial viscoelastic behavior
    Tolefson, Donald Craig (Virginia Polytechnic Institute, 1964)
    An experimental study is made to investigate the feasibility of determining a viscoelastic stress-strain law for two (or three) dimensional stress conditions by means of one dimensional tests. The conclusions are based upon comparison of theory and experiment for a creep test of a lead plate, subjected to biaxial tension. The stress-strain law that was selected is given by Sij = B σ²/σt² eij + c σ/σt eij to cover ehear effects and by σii = κεii for hydrostatic tension or compression. Sij and eij are respectively the stress deviator and σii and εii are the stress and strain tensors. Uniaxial test data was used to evaluate the constants in the above laws. The Lapplace transform technique was used to obtain the analytical solution for the strain in the plate as a function of time. Agreement between theory and experiment or the duration of the test, 1200 minutes, was quite good.