Modeling representative volume elements

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. 

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). 

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. 

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

We now consider some models of polymer structure and ascertain their usefulness as representative volume elements. The Takayanagi48) series and parallel models are widely used as descriptive devices for viscoelastic behaviour but it is not correct to use them as RVE s for the following reasons. First, they assume homogeneous stress and displacement throughout each phase. Second, they are one-dimensional only, which means that the modulus derived from them depends upon the directions of the surface tractions. If we want to make up models such as the Takayanagi ones in three dimensions then we shall have a composite brick wall with two or more elements in each of which the stress is non-uniform. 

Jointed rock mass can be regarded as equivalent continuum media if the distribution of fractures are in uniform style, a RVE (representative volume element) can be established, and the fracture flow controls the seepage of rock masses. A hydro-mechanical model of jointed rock mass coupled with damage is developed in this paper, based on the equivalent continuum assumption. From eqations (l) (i4), a FEM formulation can be established as below... 

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. 

Microscopic modeling considers a small but statistically representative volume element of the absorber (or reactor), that is, a "point" in the equipment. Recently, Thoenes (8) grouped such considerations as "volume element modeling". It is necessary to make energy and component mass balances in the reactive liquid-phase (it is assumed that no reaction takes place in the gas phase). Fortunately for most systems, the isothermal approximation is often justified. Thus, the components mass balance yields ... 

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. 

Zeman J and Sejnoha M (2007), From random microstructures to representative volume elements , Modell Simul Mater Sci Eng, 15(4), S325-S335. 

The simulation of component parts exhibiting electromechanical coupling with the aid of commercial finite element packages is subject to some restrictions. Usually the piezoelectric effect is considered only in connection with volume elements, see Freed and Bahuska [76]. For complex structures, the modeling with volume elements often does not represent a viable procedure with respect to implementation and calculation expenditure. A prominent example for this are structures with thin walls made of multiple layers. Their mechanical behavior may be simulated efficiently with layered structural shell elements. 

The interlayer model was developed by Maurer et al. The model of the particulate-filled system is taken in which a representative volume element is assumed which contains a single particle with the interlayer surrounded by a shell of matrix material, which is itself surrounded by material with composite properties (almost the same as Kemer s model). The radii of the shell are chosen in accordance with the volume fraction of the fQler, interlayer, and matrix. Depending on the external field applied to the representative volmne element, the physical properties can be calculated on the basis of different boundary conditions. The equations for displacements and stresses in the system are derived for filler, interlayer, matrix, and composite, assuming the specific elastic constants for every phase. This theory enables one to calculate the elastic modulus of composite, depending on the properties of the matrix, interlayer, and filler. In... 

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-... 

Figure 5.5 Nanoscale representative volume elements (RVE) for modeling CNT-based nanocomposites, (a) Cylindrical, (b) Rectangular, (c) Hexagonal. Adapted from Ref [57].

The size of the micro-scale model is determined via the concept of a representative volume element (RVE). The RVE should represent the smallest sample at the micro-scale capable of capturing the behavior accurately. If the RVE is too small then a biased and unrepresentative view of the micro-structure is obtained. If the RVE is too large then computational effort is wasted. The procedure to determine the optimal RVE size is based upon physical measurement and numerical tests. Methodologies to determine the optimal RVE size have been presented by various authors (see, for example, Kouznetsova [14] and Zohdi and Wriggers [15]) and will be elaborated on further in this work. 

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