Representative Volume Element RVE

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... 

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 ... 

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

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. 

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. 

In 1963 Hill47) defined the Representative Volume Element (RVE) in a consideration of general properties of composite materials. The definition is more exact than Sander s, which it includes. 

In the present analyses, prismatic dislocation loops distributed on different slip planes are used as agents for dislocation generation. For copper, sources length of about 0.60 p,m are used. It is worthy to mention that the boundary conditions of the computational cell sides are different in FE and DD parts of the code. In DD, periodic boundary condition for the representative volume element RVE is used to ensure both the continuity of the dislocation curves and the conservation of dislocation flux across the boundaries, by that we take into account the periodicity of single crystals in an infinite media. In FE analysis however, the sides are constrained to move only in the z direction so that a imiaxial strain consistent with the shock experiment is achieved. In order for the boundary conditions in FE and DD to be consistent, periodic FE bormdary condition is implemented as well. The result of this implementation is discussed in the next section. 

In most polymers, a marked necking phenomenon occurs very early after the yield point. This is the reason why it is not possible, in the range of large deformations, to determine strains in a large representative volume element (RVE). Consequently none of the dilatometers utilized to date can be used, except in a very restricted strain range. The latter statement concerns dual clip gage extensometers (axial + transversal) and also hquid displacement dilatometers (23). Once plastic instability has... 

Bazant formulated a statistical theory of fracture for quasibrittle materials [5, 23, 24]. He assumed that there exist several hierarchical orders which each can be described by parallel and serial linking of so-called representative volume elements (RVEs). For large specimens (and low probability of failures) the fracture statistics is equal to the Weibull statistics, i.e. if the specimens size is larger than 500 to 1000 times of the size of one RVE. In the actual case this is similar to the diameter of the critical flaw. For smaller specimens the volume effect disappears and the fracture... 

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 9.3 A Representative Volume Element (RVE). The total volume and mass of each constituent are denoted by V and M, respectively. The subscripts m and f stand for matrix and fiber, respectively. 

Consider a composite of mass M and volume V, illustrated schematically in Fig. 9.3, V is the volume of a Representative Volume Element (RVE), since the composite is made of fibers and matrix, the mass M is the sum of the total mass M.of fibers and mass M of matrix ... 

Kulkarni et al. [83] studied the failure processes occurring at the micro-scale in heterogeneous adhesives using a multi-scale cohesive scheme. They also considered failure effect on the macroscopic cohesive response. Investigating the representative volume element (RVE) size has demonstrated that for the macroscopic response to represent the loading histories, the microscopic domain width needs to be 2 or 3 times the layer thickness. Additionally, they analyzed the effect of particle size, volume fraction and particle-matrix interfacial parameters on the failure response as well as effective... 

In modelling the physical and the mechanical properties of textiles, there are usually two choices, i.e. whether to consider a discrete or a continuum model. A continuum model assumes that the property of any small part of the material can be considered equal to that of the whole volume. In order to model a textile structure as a continuum, its volume is divided into small parts termed unit cells or representative volume elements (RVE). RVEs model the material structure at a miaoscopic level, i.e. at the level of individual fibre arrangements. The mechanical properties of RVEs are modelled and then used at a macroscopic level, which is the level of yarn or fabric, under the assumption that the whole volume of the material can be re-constructed from a number of RVEs. 

The effective properties of a composite material correspond to properties averagedover a repeating representative volume element (RVE). This element should be large enough to represent the microstructure yet sufficiently smaller than the macroscopic structural dimensions. In fibre-reinforced composites, the RVE length scale is several times the fibre diameter. If the RVE dimension is small compared with the characteristic dimensions of the structure, the material can be assumed as homogeneous. Figure 11.4 depicts a schematic example of a RVE with other examples of elements that cannot be considered RVEs. 

The Representative volume element", RVE, was originally defined by Hill [35] as a sample of model material that is structurally entirely typical of the whole mixture on average . Since its introduction in 1963, many alternative inter-... 

Keywords Epoxy-clay nanocomposites, gallery failure, representative volume element (RVE), finite element method (FEM), cohesive elements. 

In order to proper, characterize the macroscopic properties of the fibrous structure the effective properties of the nanofiber on microscale must be prior determined. The effective properties of the nanofiber can be determined by homogenization procedure using representative volume element (RVE). A concentric composite cylinder embedded with a caped carbon nanotube represents RVE as shown by Figure 2. A carbon nanotube with a length 2, radii a is embedded at the center of matrix materials with a radii R and length 2L. 

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