Nonlinear Micropolar Beam and Plate Theories with Applications to Lattice Core Sandwich Structures and Dual Mesh Control Domain Method for Structural Elements

Abstract

In the first part of the dissertation we develop nonlinear beam and plate theories based on micropolar elasticity and formulate the corresponding finite element models. The developed non-linear beam and plate finite element models are then used to analyze the bending of lattice core sandwich beams and plates that are modeled as equivalent- ingle layer beams or plates based on micropolar elasticity. The rapid growth of manufacturing technologies has enabled the design and development of materials whose microstructure can be architected to achieve desired functionality. Lattice core sandwich structures are among such architected materials whose microstructure is the order of few centimeters. Modeling these structures with complete geometric details can be computationally expensive. Hence, efforts are made to model such structures as equivalent-single layer beams or plates with non- lassical continuum theories like micropolar elasticity. One such methodology to construct equivalent-single layer beams of web-core lattice beams is described and extended to other core structures.

The second part of this dissertation deals with formulation of a novel numerical method, named Dual Mesh Control Domain Method (DMCDM), for functionally graded structural elements; namely beams and plates. For the past few decades finite element method has been the dominant numerical method for analysis of solids and structures while finite volume method has been dominant in the field of fluid dynamics. Both the methods have their strengths and weaknesses. For example, representing a system as a collection of connected finite elements often results in a discontinuous representation of the gradients of the solution, unless so-called C-continuity is used. However, finite element method retains the concept of duality between the secondary and primary variables of the problem and thereby simplify the process of applying boundary conditions. On the other hand, although finite volume method involves fictitious nodes at the boundary control volumes and thereby complicating the application of boundary conditions, it satisfies the integrals of governing equations (with out any weight functions) on control volumes and calculates secondary variables on the interfaces of the control volume where they are uniquely defined. Considering these observations, Professor J. N. Reddy has recently proposed a novel numerical method named Dual Mesh Control Domain Method (DMCDM). It incorporates the best features of both finite element method and finite volume method by using two different meshes. A primal mesh for interpolating the primary variables and dual mesh for satisfy the governing equations in integral form without weight functions. The details of this method and its application to structural elements is discussed in detail.