Analysis of viscoplastic plates subjected to rapid external heating

The solution algorithm is developed herein for modeling the transient response of a thin metallic plate with viscoplastic constitution subjected to rapid external heating. In addition to radiation boundary conditions and material inelasticity, the authors are furthermore attempting to include geometric nonlinearity. A nonlinear incremental formulation for an anisotropic plate is developed using variational methods and finite element discretization. Results of an example considered herein are significantly different from previously obtained solutions, thus demonstrating that the effects of geometric nonlinearity are significant for the plate problem addressed herein. The results of Walker's viscoplastic model are compared with Bodner and Partom's. Finally, sensitivity studies are conducted based on heating rate and plate thickness. N o m e n c l a t u r e plate surface area element surface area extensional stiffness matrix coupling stiffness matrix bending stiffness matrix elastic modulus tensor plate thickness components of effective stiffness matrix edge length of plate structure resultant bending moment components of inertia matrix resultant forces number of plate finite elements uniform surface pressure in the z direction external heat flux input displacement components in each coordinate direction element volume direction in Cartesian coordinates direction in 2-D axisymmetric coordinates stress tensor strain tensor plate curvature mass density (lbm/in2) *Research Assistant, Student hfember AIAA * * Professor, Member AIAA I inelastic contribution T thermal contribution 0 in-plane component T R transpose matrix I n t r o d u c t i o n Plates are often subjected to hostile thermal environments capable of producing highly nonlinear structural response. Examples are structural components subjected to laser heating, the skins of aerodynamic vehicles in hypersonic flight, and space structures subjected to solar radiation or other external heating. Due to the elevated temperature environment encountered in these applications it is often necessary to include thermal effects in the structural analysis. Several theoretical solutions for a heated thin plate have been reported in the literature. In most cases either classical plate theory 2-3 or large deflection plate theory 4-i was utilized. For both cases, the solutions were elastic, and the material properties were assumed to be constant. These results give engineers sufficient information to design plate structures that do not yield. With the involvement of severe thermal environment, especially rapidly applied heat load, the plate response may become highly nonlinear. The induced transient temperature field makes the inclusion of temperature dependent material properties a necessity. The nonuniformly distributed thermal strain further induces complex structural response and localized high stresses. Furthermore, the viscoplastic material response increases the degree of nonlinearity. It is thus a formidable task to extend the previous solutions to account for these nonlineari ties. Recently, attempts have been made to numerically approximate the solution for a heated elastic plate by the boundary element method. Kawakami and Sniojiro lo attempted to implement ADINAT/ADINA with the capability of performing heat transfer analysis and thermo-elastic-plastic analysis for plate/shell elements. In their research, plasticity in the element is assumed only when the cross section is totally plastic. Therefore, their solution becomes inaccurate when coping with a plate subjected to rapid high energy heating on one surface. The authors have proposed a nonlinear finite element model based on a one-way coupled assumption to predict the transient response of a plate subjected to rapidly applied concentrated heat load. In their solution, Bodner and Partom's viscoplasticity model '' is applied to account for nonlinear material behavior and classical plate theory is assumed. In this paper, the authors' previous work is improved by considering the plate bending nonlinear geometric effect. Further-