Fiber-reinforced Elastic Solids Research Articles

Symmetry analysis and equivalence transformations for the construction and reduction of constitutive models

A methodology based on Lie analysis is proposed to investigate the mechanical behavior of materials exhibiting experimental master curves. It is based on the idea that the mechanical response of materials is associated with hidden symmetries reflected in the form of the energy functional and the dissipation potential leading to constitutive laws written in the framework of the thermodynamics of irreversible processes. In constitutive modeling, symmetry analysis lets one formulate the response of a material in terms of so-called master curves, and construct rheological models based on a limited number of measurements. The application of symmetry methods leads to model reduction in a double sense: in treating large amounts number of measurements data to reduce them in a form exploitable for the construction of constitutive models, and by exploiting equivalence transformations extending point symmetries to efficiently reduce the number of significant parameters, and thus the computational cost of solving boundary value problems (BVPs). The symmetry framework and related conservation law analysis provide invariance properties of the constitutive models, allowing to predict the influence of a variation of the model parameters on the material response or on the solution of BVPs posed over spatial domains. The first part of the paper is devoted to the presentation of the general methodology proposed in this contribution. Examples of construction of rheological models based on experimental data are given for setting up a reduced model of the uniaxial creep and rupture behaviour of a Chrome-Molybdenum alloy (9Cr1Mo) at different temperatures and stress levels. Constitutive equations for creep and rupture master responses are identified for this alloy, and validated based on experimental data. Equivalence transformations are exemplified in the context of parameter reduction in fully nonlinear anisotropic fiber-reinforced elastic solids.

One-dimensional nonlinear elastodynamic models and their local conservation laws with applications to biological membranes

The framework of incompressible nonlinear hyperelasticity and viscoelasticity is applied to the derivation of one-dimensional models of nonlinear wave propagation in fiber-reinforced elastic solids. Equivalence transformations are used to simplify the resulting wave equations and to reduce the number of parameters. Local conservation laws and global conserved quantities of the models are systematically computed and discussed, along with other related mathematical properties. Sample numerical solutions are presented. The models considered in the paper are appropriate for the mathematical description of certain aspects of the behavior of biological membranes and similar structures.

Advanced Boundary Element Analysis of Three-Dimensional Elastic Solids with Fiber Reinforcements

An advanced boundary element methodology for the analysis of three-dimensional fiber-reinforced elastic solids using the concept of fiber elements has been presented in this paper as an extension of the earlier work of Banerjee and co-workers. The previous simplified formulation was based on the assumption that the Poisson ratios of the matrix and the fiber are equal. However, this may not be a valid assumption for all values of elastic stiffness ratios and fiber to matrix volume ratios. Moreover, such restrictions do not allow for any future extensions to nonlinear analysis. Also, because of limiting computing power available at that time, their implementation was restricted to only a small number of fiber elements in a given analysis. The new algorithm proposed in this work does not make any such assumptions. In this algorithm, the fibers are semianalytically integrated and the resulting equations are back-substituted into the equations for the composite matrix without any further approximation. The present work has been implemented in a general purpose multiregion boundary element computer program and is capable of handling very large numbers of fiber elements in a given analysis. Several numerical examples are presented to validate the proposed method of fiber composite analysis and its applicability is demonstrated via practical engineering problems.