Viscoelastic inhomogeneous

In summary, it can be seen that plastics and RPs design analysis follows the same three steps (a) to (c) as that for metals, but there are some differences of emphasis and difficulty. In particular, step (a) is usually more substantial for the newer materials, partly because a full stress/strain/deformation analysis is required and partly because of the need to take account of viscoelasticity, inhomogeneity, and/or anisotropy. For long fiber materials, the component design analysis may need to contain the associated material design analysis. 

Rivlin-Ericksen tensor of order n, for a viscoelastic liquid or solid in homogeneous deformation, is the nth time derivative of the Cauchy strain tensor at reference time, t. Note 1 For an inhomogeneous deformation the material derivatives have to be used. 

Limitations of the Tests, Especially when Applied to Viscoelastic Non-Linear Inhomogeneous Anisotropic Materials... 

It is possible to divide the problems essentially into 3 classes 1) problems associated with the material - here we must concern ourselves with the applicability of the solutions of the equations of elasticity used to calculated moduli to the actual material which may be non-linear, inhomogeneous and anisotropic as well as viscoelastic. 2) Geometrical problems of the test itself here we must be concerned with the end effects due to... 

We shall discuss separately the effect of non-linearity, inhomogeneity and anisotropy and consider how they affect individual tests. The problem of the translation of elastic solutions to the viscoelastic case has been mentioned in the previous chapter and will not be treated further. 

The basic concepts and definitions relating to sound propagation in a lossy material are reviewed. The material may be a viscoelastic polymer which converts the sound energy to heat by molecular relaxation, or the material may be a composite where sound is scattered by inhomogeneities (inclusions) in a host matrix material. 

In the quasi-static case, effective frequency dependent moduli and loss factors may be calculated from Equation 8. With respect to Equation 29, a lossy matrix material implies that k is now a complex number. The new expressions for c and a differ from Equations 31 and 32, but follow straightforwardly. Equation 30 is usually cited only for elastic matrix materials, but, of course, it need not be used to interpret a. The potential problem (also with viscoelastic inclusions) is that the derivation of Equation 30 is based on homogeneous stress waves, whereas in viscoelastic materials one should, strictly speaking, consider inhomogeneous waves. The results obtained from Equation 29 are reasonable in the sense of yielding the expected superposition of scattering and dissipation effects. 

There are many ways to fabricate and model viscoelastic polymeric materials [22-32]. Fabrication often involves nonlinear flows that are spatially inhomogeneous, nonisothermal, and temporally complex. The flows also may involve material phase changes, and/or a wide range (1-5 decades) of strains and strain rates. Rheology is often the bridge between resin design and fabrication performance, and remains an active area of research [22]. 

Craze growth at the crack tip has been qualitatively interpreted as a cooperative effect between the inhomogeneous stress field at the crack tip and the viscoelastic material behavior of PMMA, the latter leading to a decrease of creep modulus and yield stress with loading time. If a constant stress on the whole craze is assumed then time dependent material parameters can be derived by the aid of the Dugdale model. An averaged curve of the creep modulus E(t) is shown in Fig. 13 as a function of time, whilst the craze stress is shown in Fig. 24. 

Polymers showing a viscoelastic behaviour occupy the intermediate range. Out of all the existing hardness tests, the pyramid indenters are best suited for research on small specimens and microstructurally inhomogeneous samples (Tabor, 1951). Pyramid indenters provide, in addition, a contact pressure which is nearly independent of indent size and are less affected by elastic release than other indenters. 

The hierarchical structure model is generalized and applied to study the viscoelastic properties of a two-component inhomogeneous medium with chaotic, fractal structure. It is shown that just as the results obtained recently using the Hashin-Strikman model, the present model predicts the possibility of obtaining composites with an effective shear and dumping coefficient much higher than those characterizing the individual component phases. The viscoelastic properties of the fractal medium, however, differ qualitatively from the properties of the Hashin-Strikman medium. 

K, p p, denote the complex bulk modulus and the complex shear modulus of the first phase of the inhomogeneous medium, and K% = K2 and p = p2- denote the complex bulk modulus and the complex shear modulus of the second phase, respectively K, K2, pj, p2 are elastic properties for nonhomogeneous media phases). For nonbonded configurations the viscoelastic modulii and pi"11 are described by the formulae which result from the... 

The calculations were made for a two-phase (two-component) inhomogeneous medium assuming that volume strains are elastic, while shear strains are viscoelastic. The ratio of local volume modulii K[ /K 2 was set equal to 10 4. For convenience of calculations, the local shear modulii (phase shear modulii) were written in the form... 

The calculations were performed for a two-component, inhomogeneous medium. For simplicity, it is assumed that both phases are isotropic and that the first phase is purely elastic whereas the second phase is elastic from the point of view of volume deformations and viscoelastic from the point of view of shear deformations. The concentration of the purely elastic phase is denoted by p. 

However, most real systems do not comply with the presumptions made in the derivation of Eq. (17.4). Generally, more than one type of interaction force will act, and the structural elements often vary in type or size, which implies a spectrum of interaction forces we will see examples of this in the following sections. The contributions to the modulus of the various forces involved are not additive, primarily because the bonds vary in direction. Moreover, such materials are generally not fully elastic, which implies that the modulus will be complex [see Eq. (5.12)] and depend on deformation rate virtually all soft-solid foods show viscoelastic behavior of some type. Finally, some systems are quite inhomogeneous, which further complicates the relations. 

It will be clear now that the relations governing fracture of viscoelastic materials are far more complex than those for elastic solids. The discussion above gives some qualitative relations that generally hold. Quantitative prediction of the behavior from first principles is mostly not possible. It all becomes even more complex for inhomogeneous materials. 

Mechanically, the tissue is anisotropic and Inhomogenous, its moduli vary with direction and depth from the surface (10,11). Its principal mechanism for attaining stress relaxation, at strains above a critically small strain, is by exuding interstitial fluid (12). Its stress relaxation rates are therefore not only functions of the viscoelastic properties of its macromolecular network, but also the frictional resistances to fluid transport in and out of the tissue. Factors affecting fluid exudation and imbibition therefore necessarily affect the tissue s wear resistance. 

When trying to relate the probe test curves, performed at a certain probe velocity, with the linear viscoelastic properties of the adhesives, measured at a certain pulsation, the question of the equivalence between probe velocity, Vdeb and frequency always occurs. In fact a probe test imposes a highly inhomogeneous deformation on an adhesive layer and no equivalence can be rigorously made between a non-steady state experiment, which deforms the material in a... 

Due to the location and critical function of the aortic valve, it is difficult to obtain measurements of its mechanical properties in vivo however, reports are available from a small number of animal studies. This section will reference the in vivo data whenever possible and defer to the in vitro data when necessary. Since little mathematical modeling of the aortic valve s material properties has been reported, it will be sufficient to describe the known mechanical properties of the valve. Like most biological tissues, the aortic valve is anisotropic, inhomogeneous, and viscoelastic. The collagen fibers within each valve cusp are aligned along the circumferential direction. Vesely and Noseworthy [ 1992] found that both the ventricularis and fibrosa were stiffer in the circumferential direction than in the radial direction. However, the ventricularis... 

For RTSs with short fibers, such as bulk molding compounds (BMCs), there may be only a low level of viscoelasticity, anisotropy, and inhomogeneity (Chapter 4). However, these RPs with the TS resin... 

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