On the finite twisting of translating plates

Previous studies of axially moving materials have focused on systems for which a one-dimensional model adequately describes the behavior. However, one-dimensional models are not effective when considering materials in which the width is much greater than the thickness. Two-dimensional theories are necessary to model such continua which are common in material handling and processing operations.; The first contribution of this study is the development of two-dimensional models that can capture width-dependent effects in axially moving materials. From a referential form of Euler's balance laws, two, constitutively and geometrically nonlinear, axially moving plate models are derived. The general theory incorporates transverse strains and the restricted theory uses the Kirchhoff hypothesis and neglects them. The equations for each theory are decomposed into those describing large steady motions of the plate and those governing small, dynamic displacements from the steady state. In this way the free response and stability of the steady state can be investigated.; The second contribution is the analysis of the steady response of a wide axially moving material twisted out-of-plane by opposing end supports. Semi-inverse methods are used to reduce the equations of steady motion for both theories to forms that can be more easily analyzed. This semi-inverse method is compatible with a wide class of linear and nonlinear constitutive relations that include all isotropic materials and all orthotropic materials with principal directions aligned along the lateral and longitudinal axes of the rectangular plate. Using perturbation methods, verified numerically, approximate analytical solutions of the reduced equations are obtained. Solutions for the general and restricted theories are compared and contrasted. As the thickness of the plate increases the solutions from the two theories become increasingly different. The differences are largest at the free edges of the plate. Generally, the current analysis adds only small corrections to previous results obtained from weakly nonlinear plate theories. However, unlike the results obtained previously, the current analysis predicts that the lateral component of the membrane stress is negative for all non-zero twist angles and becomes more compressive as the twist angle, nominal tension, and aspect ratio are increased. Because this compression remains as the plate thickness approaches zero, lateral wrinkling, not predicted previously and observed experimentally, results for sufficiently thin plates.; The final contribution is the analysis of the free response and stability of a twisted plate. Using the methods of Frobenius and Galerkin, the equations governing small dynamic perturbations from the steady state are studied. As the plate is free to slide along the supports, a rigid body mode that exists when the plate is flat is suppressed when the plate is twisted. The fundamental transverse natural frequency is found to initially increase with increasing twist angle. Further twist causes this natural frequency to decrease and eventually reach zero at the critical twist angle. This critical twist angle decreases with decreasing plate thickness and increasing aspect ratio.