Asymptotically Correct Dimensional Reduction of Nonlinear Material Models

Abstract

This work aims at dimensional reduction of nonlinear material models in an asymptotically accurate manner. The three-dimensional(3-D) nonlinear material models considered include isotropic, orthotropic and dielectric compressible hyperelastic material models. Hyperelastic materials have potential applications in space-based inflatable structures, pneumatic membranes, replacements for soft biological tissues, prosthetic devices, compliant robots, high-altitude airships and artificial blood pumps, to name a few. Such structures have special engineering properties like high strength-to-mass ratio, low deflated volume and low inflated density. The majority of these applications imply a thin shell form-factor, rendering the problem geometrically nonlinear as well. Despite their superior engineering properties and potential uses, there are no proper analysis tools available to analyze these structures accurately yet efficiently. The development of a unified analytical model for both material and geometric nonlinearities encounters mathematical difficulties in the theory but its results have considerable scope. Therefore, a novel tool is needed to dimensionally reduce these nonlinear material models. In this thesis, Prof. Berdichevsky’s Variational Asymptotic Method(VAM) has been applied rigorously to alleviate the difficulties faced in modeling thin shell structures(made of such nonlinear materials for the first time in the history of VAM) which inherently exhibit geometric small parameters(such as the ratio of thickness to shortest wavelength of the deformation along the shell reference surface) and physical small parameters(such as moderate strains in certain applications). Saint Venant-Kirchhoff and neo-Hookean 3-D strain energy functions are considered for isotropic hyperelastic material modeling. Further, these two material models are augmented with electromechanical coupling term through Maxwell stress tensor for dielectric hyperelastic material modeling. A polyconvex 3-D strain energy function is used for the orthotropic hyperelastic model. Upon the application of VAM, in each of the above cases, the original 3-D nonlinear electroelastic problem splits into a nonlinear one-dimensional (1-D) through-the-thickness analysis and a nonlinear two-dimensional(2-D) shell analysis. This greatly reduces the computational cost compared to a full 3-D analysis. Through-the-thickness analysis provides a 2-D nonlinear constitutive law for the shell equations and a set of recovery relations that expresses the 3-D field variables (displacements, strains and stresses) through thethicknessintermsof2-D shell variables calculated in the shell analysis (2-D). Analytical expressions (asymptotically accurate) are derived for stiffness, strains, stresses and 3-D warping field for all three material types. Consistent with the three types of 2-D nonlinear constitutive laws,2-D shell theories and corresponding finite element programs have been developed. Validation of present theory is carried out with a few standard test cases for isotropic hyperelastic material model. For two additional test cases, 3-Dfinite element analysis results for isotropic hyperelastic material model are provided as further proofs of the simultaneous accuracy and computational efficiency of the current asymptotically-correct dimensionally-reduced approach. Application of the dimensionally-reduced dielectric hyperelastic material model is demonstrated through the actuation of a clamped membrane subjected to an electric field. Finally, the through-the-thickness and shell analysis procedures are outlined for the orthotropic nonlinear material model.