The 48th SNP Meeting

Invited Lecture

A Continuum Theory for Materials with Microstructures

Yi-chao Chen
Department of Mechanical Engineering
University of Houston
Engineering Building One
Houston, TX 77204-4006

Abstract. A theoretical framework, that combines the merits of the continuum approach and the micromechanics approach, is developed for material systems with microstructures in general, and for porous elastic materials in particular.

In the mechanics literature, the analysis of heterogeneous materials or materials with microstructures often follows one of the two distinct approaches. In so-called micromechanics approach, the material is replaced by a “representative volume element” (RVE). Equations that relate various average quantities over RVE are sought to represent the “effective properties” of the material. Besides the difficulties in justifying drastic simplifications made in this approach, some fundamental issues concerning kinematics and balance laws remain unaddressed. For example, it seems often to be taken for granted that the average displacement, strain, and stress would satisfy the same equations as do the local quantities, which we have found untrue. On the other hand, in various continuum theories, so-called internal variables are introduced to describe the deformation and evolution of the microstructures. These internal variables enter and thus alter the usual governing equations for continua. Additional balance laws and constitutive equations are often derived using variational arguments. However, direct links between the internal variables and the microstructures that they intend to describe are often missing.

In the present theory, the average state variables and internal variables associated with material microstructures are explicitly defined through averaging process. The concept of moving average is used in deriving the field equations for the average variables. These equations contain terms associated with pore surface effects. These surface terms are found to occur in the continuum theory developed by Nunziato and Cowin [1] for porous elastic materials. The present theory establishes the direct link between the internal variables in a continuum theory and the average variables in a micromechanical theory. The equations governing the evolution of the internal variables are also derived from the exact field equations. For example, by taking the scalar moment of the local equation of motion about the centroid of the void and integrating, an equation, termed balance of radial momentum, can be derived. This equation governs the motion of pore expansion, and has been identified as the balance of equilibrated force in [1].

To derive the constitutive equations, the total deformation is decomposed into a series of deformations of increasing orders. These deformations represent different modes of the pore surface deformations, one of which is the pore expansion mentioned above. Taking the derivatives of the average elastic energy with respect to these deformations gives the generalized forces, and thus leads to the desired constitutive equations. These generalized forces are found to be related to the equilibrated stress and the intrinsic equilibrated body force introduced in [1].

References.
1. Nunziato, J. W. and Cowin, S. C. 1979 A nonlinear theory of elastic materials with voids, Archive Rational Mech. Anal. 72, 175.

Acknowledgement. This lecture was made possible through financial support from both Pró-Reitoria de Pós-Graduação (PRPG/USP) and Coordination for the Improvement of Higher Education Personnel (CAPES).