Micromechanics representative volume element

Irrespective of the analysis approach, the representative volume element must be carefully defined and used. In fact, the representative volume element is crucial to the analysis and is the micromechanics analog of the free-body diagram in statics and dynamics. The representative volume element is of higher order than the free-body diagram because deformations and stresses are addressed in addition to forces. 

Our reason for stressing the concept of representative volume element is that it seems to provide a valuable dividing boundary between continuum theories and molecular or microscopic theories. For scales larger than the RVE we can use continuum mechanics (classical and large strain elasticity, linear and non-linear viscoelasticity) and derive from experiment useful and reproducible properties of the material as a whole and of the RVE in particular. Below the scale of the RVE we must consider the micromechanics if we can - which may still be analysable by continuum theories but which eventually must be studied by the consideration of the forces and displacements of polymer chains and their interactions. 

Since the assumption of uniformity in continuum mechanics may not hold at the microscale level, micromechanics methods are used to express the continuum quantities associated with an infinitesimal material element in terms of structure and properties of the micro constituents. Thus, a central theme of micromechanics models is the development of a representative volume element (RVE) to statistically represent the local continuum properties. The RVE is constracted to ensure that the length scale is consistent with the smallest constituent that has a first-order effect on the macroscopic behavior. The RVE is then used in a repeating or periodic nature in the full-scale model. The micromechanics method can account for interfaces between constituents, discontinuities, and coupled mechanical and non-mechanical properties. Their purpose is to review the micromechanics methods used for polymer nanocomposites. Thus, we only discuss here some important concepts of micromechanics as well as the Halpin-Tsai model and Mori-Tanaka model. 

The microstinctural configuration of heterogeneous materials can be correlated to the macroscopic constimtive relations within the micromechanics framework. In this approach the representative volume element (RVE) represents a specific arrangement of subphases, each of which has a specific geometry and mechanical properties. Selection of an RVE is extremely... 

Micromechanics are a study of mechanical properties of unidirectional composites in terms of those of constituent materials. In particular, the properties to be discussed are elastic modulus, hydrothermal expansion coefficients and strengths. In discussing composites properties it is important to define a volume element which is small enough to show the microscopic structural details, yet large enough to present the overall behavior of the composite. Such a volume element is called the Representative Volume Element (RVE). A simple representative volume element can consists of a fiber embedded in a matrix block, as shown in Fig. 9.3. 

FIGURE 8.4 Representative volume elements for micromechanics and macromechanics analysis of composite materials, (a) Equivalent heterogeneous element, (b) equivalent homogeneous orthotropic element, and (c) equivalent homogeneous anisotropic element. 

FIGURE 8.14 Representative volume element from a unidirectional lamina, (a) Micromechanics, representative heterogeneous element and (b) macromechanics, equivalent homogeneous orthotropic element. 

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