Nonlinear Micropolar Models for Composite Materials: Theory and Computation

Abstract

Composites are attractive as lightweight materials for a variety of structural load bearing applications, especially in the aerospace industry. To model their behavior at the global scale (macroscale), continuum models are employed for computational efficiency. However, classical (Cauchy) continuum models often disregard the local structural effects, such as local bending and rotations of the constituents. For example, fiber reinforced composites, which are generally composed of fibers surrounded by polymer matrix material, when subjected to macroscopic loading, the fibers at the microscale undergo local bending and rotations. Similarly, in cellular materials, such as honeycomb structures and foams, when viewed as an assembly of beams or shells, there is inherent bending of cell walls introduced into the continuum deformation.

The higher order micropolar continuum theory, which is an extension of a Cauchy continuum, introduces these higher order effects with the generalization of the kinematic degrees of freedom. In addition to the displacement field, there is also an additional independent rotational field introduced into the formulation. As a consequence, there is couple-stress (moment stresses) tensor in addition to the classical force-stress tensor. These correspond to the local rotations/moments present due to the microstructure of a composite. These aspects of micropolar theory are appropriate for representing the local mechanics of fiber reinforced composites and cellular materials, which are studied in this thesis.

In literature, the challenge of micropolar theory has been two-fold: (1) the determination of the additional micropolar material constants that are introduced, and (2) the analytical and numerical implementation of finite micropolar theory. In the this thesis, physics based methods will be developed to determine the properties of fiber reinforced composites. In addition, the classical Hill-Mandel condition from classical micromechanics will be extended to a micropolar continuum to determine the constitutive relation of a structured cellular solid. Finite micropolar theory, which accounts for both geometric and material nonlinearities, is developed in this thesis. This is implemented via an updated Lagrangian finite element framework for analyzing fiber reinforced structures.

Micropolar theory is applied to boundary value problems where local rotations and moments are dominant. This includes problems where the wavelength of the deformation is comparable to the characteristic length of the microstructure. An example of this is the formation of localized deformation in fiber-reinforced composites (fiber kinking) under compression loading. Micropolar theory is not only a high fidelity model that helps to quantify the local moments and rotations, but it also prevents the loss of ellipticity of the governing equations at the onset of localization. This is useful for analyzing the post-peak response of fiber reinforced composites. The details regarding this are also explained in this thesis.