Representative volume elements

Figure 3-2 Representative Volume Element - Lamina with Unidirectional Fibers...

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. 

Figure 3-5 Representative Volume Element Loaded in the 1-Direction...

The average stress acts on cross-sectional area A of the representative volume element, oj acts on the cross-sectional area of the fibers Af, and acts on the cross-sectional area of the matrix A. Thus, the resultant force on the representative volume element of composite material is... 

A simple springs-in-series model represents the representative volume element loaded in the 2-direction as in Figure 3-11. There, the matrix is the soft link in the chain of stiffnesses. Thus, the spring stiffness for the matrix is quite low. We would expect, on this basis, that the matrix deformation dominates the deformation of the composite material. 

The in-plane shear modulus of a lamina, G12. is determined in the mechanics of materials approach by presuming that the shearing stresses on the fiber and on the matrix are the same (clearly, the shear deformations cannot be the samel). The loading Is shown in the representative volume element of Figure 3-15. By virtue of the basic presumption,... 

Figure 3-15 Representative Volume Element Loaded In Shear...

Use a mechanics of materials approach to determine the apparent Young s modulus for a composite material with an inclusion of arbitrary shape in a cubic element of equal unit-length sides as In the representative volume element (RVE) of Figure 3-17. Fill in the details to show that the modulus is... 

Figure 3-22 Hexagonal Array and Representative Volume Elements...

Use the bounding techniques of elasticity to determine upper and lower bounds on the shear modulus, G, of a dispersion-stiffened composite materietl. Express the results In terms of the shear moduli of the constituents (G for the matrix and G for the dispersed particles) and their respective volume fractions (V and V,j). The representative volume element of the composite material should be subjected to a macroscopically uniform shear stress t which results in a macroscopically uniform shear strain y. 

For smaller values of Vj, the behavior of the composite material might not follow Equation (3.84) because there might not be enough fibers to control the matrix elongation. That is, the matrix dominates the composite material and carries the fibers along for the ride. Thus, the fibers would be subjected to high strains with only small loads and would fracture. If all fibers break at the same strain (an occurrence that is quite unlikely from a statistical standpoint), then the composite material will fracture unless the matrix (which occupies only of the representative volume element) can take the entire load imposed on the composite material, that is. 

Figures 1 a and 1 b represent the two-phase and the three-phase models respectively in the representative volume element of the composite. In the modified model three concentric spheres were considered with each phase maintaining a constant volume 4). The novel element in this model is the introduction of the third intermediate hybrid phase, lying between the two principal phases.

Thus, in the three-layer model, with the intermediate layer having variable physical properties (and perhaps also chemical), subscripts f, i, m and c denote quantities corresponding to the filler, mesophase, matrix and composite respectively. It is easy to establish for the representative volume element (RVE) of a particulate composite, consisting of a cluster of three concentric spheres, that the following relations hold ... 

A three-layer model for fiber composites may be developed, based on the theory of self-consistent models and adapting this theory to a three-layered cylinder, delineating the representative volume element for the fiber composite. 

A better approach for the Rosen-Hashin models is to adopt models, whose representative volume element consists of three phases, which are either concentric spheres for the particulates, or co-axial cylinders for the fiber-composites, with each phase maintaining its constant volume fraction 4). 

This is because the representative volume element for a unidirectional fiber-reinforced composite consists of a cluster of three co-axial cylinders of the same height, taken equal to unity, and therefore the volume fractions of the phases are proportional to the squares of the radii of the respective cylinders. 

A series of models were introduced in this study, which take care of the existence of this boundary layer. The first model, the so-called three-layer, or N-layer model, introduces the mesophase layer as an extra pseudophase, and calculates the thickness of this layer in particulates and fiber composites by applying the self-consistent technique and the boundary- and equilibrium-conditions between phases, when the respective representative volume element of the composite is submitted to a thermal potential, concretized by an increase AT of the temperature of the model. 

Recrystallization 127, 133 Representative volume element (RVE) 159 Residual depth 138... 

The Representative Volume Element Size in Elastic Composites ... 

Pultrusion is a steady-state process in which the fiber-resin mass changes its properties as it moves from the entrance to the exit of the die. In order to track the temperature, polymer conversion, and other properties of the fiber-resin mass as it moves along the die, it is useful to define a representative volume element (RVE) that rides along the fiber at the line speed of the pultrusion process. An RVE is defined such that it will contain both the solid phase (i.e., fibers and resin), irrespective of its location in the composite. In real-life pultrusion, a thermocouple wire that passes through the pultrusion die tracks the temperature of an RVE in the composite. 

By closely examining the layout in Figure 7.7, a representative volume element can be selected for study, as shown in Figure 7.8. Region A, which comprises 25 percent of the area of the representative volume element, is in initial contact. When the consolidation pressure is applied, the deformation is initiated in region A, and the resin flows in the x-y plane to fill the gaps in regions B, C, and D. 

Figure 7.8 Representative volume element used in modeling the intimate contact achievement of a cross-ply interply interface...

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

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