Simplified representative volume element

The essential statement about the composition of the considered composite is provided by the fiber volume fraction v. As the material distribution does not 

Remark 5.2. The subsequent discussion of micro-electromechanical approaches to determine the macroscopic constitutive properties of a composite will be concentrated on properties within the following categories  

Typically, the fiber material is at most transversely isotropic, Eq. (4.17) with Eqs. (4.3), (4.8), and (4.18), while the matrix material is isotropic, Eq. (4.17) with Eqs. (4.4), (4.9), and vanishing piezoelectric moduli. The subsequent theories are not confined to such a behavior. However, it will be presumed that the distinguished axes are aligned. Deviating cases may be considered in conjunction with an appropriate transformation, see Section 3.2.5. In accordance with the considerations of Section 4.4.4, the notation with a negated electric field strength will be utilized throughout the entire chapter. 

The micro-electromechanical methodology stemming from the category of the Theory of Elasticity to be presented in this section is capable of modeling inclusions of ellipsoidal geometry. Such a description of the inclusion geometry allows us to consider fibrous and lamellar inclusions by means of one or two semiaxes approaching infinity, respectively. 

Mean stresses and electric flux densities Y as well as mean strains and electric field strengths Z are composed of the corresponding mean fields in the inclusion and matrix phase as indicated by the superscripts i and m. Crossconnecting these fields, the constitutive relations of the homogenized composite, as well as of the individual material phases, may be given as follows  

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Fig. 5.6. Simplified representative volume element in consideration of fibrous inclusions and interdigitated electrodes.

The characteristic cross-sectional dimensions are depicted together with electrode spacing 03 and electrode width 63 in Figure 5.7 by means of the simplified representative volume element. 

Fig. 5.7. Characteristic dimensions of the simplified representative volume element.

The visual inspection of the simplified representative volume element in Figure 5.7 suggests a determination of the composite s overall behavior via the examination of the stacking of constituents in axial directions transverse to the fibers, as shown on the left and in the middle of Figure 5.8. Ahead of considering the possibilities of how to combine these elementary cases, they first of all will be studied separately. In order to gain an impression of the effective... 

As the entire composite structure happens to be assembled from simplified representative volume elements by symmetric completion and repetition, the edges have to remain straight and parallel in any event. Transferred to the mechanical and electrostatic fields, this requirement may be fulfilled on the grounds of the following assumption ... 

So far, the two necessary cases of stacking of constituents have been examined for normal as well as shear modes and macroscopic constitutive relations have been obtained for each of them. In the next step, their integration with the goal to depict the simplified representative volume element needs to be considered. 

The terms zonal model and flow element are also used for the simplified characterization of the flow field in a single enclosure. There, a zone represents a partial volume of air in the enclosure, whereas in the multizone models described here, a zone represents a specific enclosure which is connected to other enclosures by air conductances (see The Airflow Network later). 

To answer the question of optimal matching between the ventricle and arterial load, we developed a framework of analysis which uses simplified models of ventricular contraction and arterial input impedance. The ventricular model consists only of a single volume (or chamber) elastance which increases to an endsystolic value with each heart beat. With this elastance, stroke volume SV is represented as a linearly decreasing function of ventricular endsystolic pressure. Arterial input impedance is represented by a 3-element Windkessel model which is in turn approximated to describe arterial end systolic pressure as a linearly increasing function of stroke volume injected per heart beat. The slope of this relationship is E. Superposition of the ventricular and arterial endsystolic pressure-stroke volume relationships yields stroke volume and stroke work expected when the ventricle and the arterial load are coupled. From theoretical consideration, a maximum energy transfer should occur from the contracting ventricle to the arterial load under the condition E = Experimental data on the external work that a ventricle performed on extensively varied arterial impedance loads supported the validity of this matched condition. The matched condition also dictated that the ventricular ejection fraction should be nearly 50%, a well-known fact under normal condition. We conclude that the ventricular contractile property, as represented by is matched to the arterial impedance property, represented by a three-element windkessel model, under normal conditions. 

A representative heterogeneous volume is chosen from the composite lamina, which is the smallest volume elanent over which stresses and strains may be considered macroscopicaUy uniform (Figure 8.4). To conceptually simplify the analysis, the fibers are shown lumped together occupying a portion of the representative element equivalent to the volume fraction of the fibers in the composite. This representation leads to equations which relate composite mechanical properties to constituent mechanical properties as a function of volume fraction. 

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