Second Strain Gradient Continuum Model for the Mechanics of Fiber-Reinforced Composites

Abstract

The mechanics of composite materials is a subject of intense study due to their versatile and tailorable mechanical properties. A composite material consists of at least two different phases; one is called the reinforcing phase and the other one in which it is embedded is called the matrix phase. Due to the presence of multi-phase and fiber-matrix interface, the characterization of local behaviors of a composite is computationally expensive. The continuum mechanics offers the necessary mathematical framework to accommodate the overall microscopic behaviors of the reinforcement phase into the model of deformations. In this thesis, the mechanics of fiber-reinforced composite materials are presented within the framework of the second strain gradient theory. A continuum-based model is developed for the analysis of elastic materials reinforced with unidirectional and bidirectional fibers and subjected to finite plane deformations. Moreover, the continuously distributed unidirectional fiber-composite system is transformed into the randomly distributed short fiber-composite system by introducing the shear lag parameter and krenchel orientation factor into the model. The mechanics of randomly distributed short fiber-composite system is also presented. The Euler equilibrium equations and the associated boundary conditions are obtained via the variational principle and iterative integration by parts. In particular, the energy density function is augmented to accommodate the first, second, and third gradient of deformations into the models of continuum deformation. The complete expressions of Piola-type triple stress and its coupled triple force arising in the third gradient of continuum deformations are formulated, which, in turn, yield the unique deformation maps in the presence of admissible boundary conditions of higher order. The solutions of the resulting systems of differential equations are obtained via the custom-built numerical scheme from which smooth and dilatational shear angle distributions are predicted throughout the entire domain of interest. It is also observed that the third gradient constitutive parameter is associated with the volume dilatation of third-gradient continua, which may be appeared in the form of shear band inclination angle. In addition, the fiber aspect ratio and the third gradient constitutive parameter are observed to be related to the effective Young's modulus of a randomly distributed short-fiber composite system. The obtained numerical results are compared with the results in the dedicated literature, which show a good agreement.