Representative volume

When we translate these observations into Lagrangian wave speed, the data would look like that shown in the lower diagram of Fig. 7.11. The points e and q represent volume strains at whieh elastie-perfeetly-plastie release (e) and quasi-elastie release (q) would undergo transition to large-seale, reverse plastie flow (reverse yield point). The question is the following What is responsible for quasi-elastie release from the shoeked state, and what do release-wave data tell us about the mieromeehanieal response in the shoeked state ... 

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 ... 

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