Viscoelasticity anisotropic

The coordinates (x, y, z) define the (velocity, gradient, vorticity) axes, respectively. For non-Newtonian viscoelastic liquids, such flow results not only in shear stress, but in anisotropic normal stresses, describable by the first and second normal stress differences (oxx-Oyy) and (o - ozz). The shear-rate dependent viscosity and normal stress coefficients are then [1]... 

The continuous chain model includes a description of the yielding phenomenon that occurs in the tensile curve of polymer fibres between a strain of 0.005 and 0.025 [ 1 ]. Up to the yield point the fibre extension is practically elastic. For larger strains, the extension is composed of an elastic, viscoelastic and plastic contribution. The yield of the tensile curve is explained by a simple yield mechanism based on Schmid s law for shear deformation of the domains. This law states that, for an anisotropic material, plastic deformation starts at a critical value of the resolved shear stress, ry =/g, along a slip plane. It has been... 

Navard and Haudin (100) examined the rheology of HPC-acetic acid solutions. The anisotropic solutions were strongly viscoelastic. 

A pseudo solid-like behavior of the T2 relaxation is also observed in i) high Mn fractionated linear polydimethylsiloxanes (PDMS), ii) crosslinked PDMS networks, with a single FID and the line shape follows the Weibull function (p = 1.5)88> and iii) in uncrosslinked c/.s-polyisoprenes with Mn > 30000, when the presence of entanglements produces a transient network structure. Irradiation crosslinking of polyisoprenes having smaller Mn leads to a similar effect91 . The non-Lorentzian free-induction decay can be a consequence of a) anisotropic molecular motion or b) residual dipolar interactions in the viscoelastic state. 

The dynamics of block copolymers melts are as intriguing as their thermodynamics leading to complex linear viscoelastic behaviour and anisotropic diffusion processes. The non-linear viscoelastic behaviour is even richer, and the study of the effect of external fields (shear, electric. ..) on the alignment and orientation of ordered structures in block copolymer melts is still in its infancy. Furthermore, these fields can influence the thermodynamics of block copolymer melts, as recent work has shown that phase transition lines shift depending on the applied shear. The theoretical understanding of dynamic processes in block copolymer melts is much less advanced than that for thermodynamics, and promises to be a particularly active area of research in the coming years. 

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... 

Laws and McLaughlin30 solve the problem of the viscoelastic ellipsoidal inclusion in anisotropic materials and then use the self consistent method to calculate the overall viscoelastic compliances for a composite. 

Im SH, Huang R (2008) Wrinkle patterns of anisotropic crystal films on viscoelastic substrates. J Mech Phys Solids 56 3315-3330... 

One can think that this situation, described by equations (3.1), can be visualised as a picture of interacting (and connected in chains) Brownian particles suspended in anisotropic viscoelastic segment liquid . Introduction of macroscopic concepts is unavoidable consequence of transition from microscopic to mesoscopic approach, or better to say, from the microscopic model of interacting Kuhn-Kramers chains to mesoscopic model of interacting chains of Brownian particles. 

Figure 14.2 Scheme of a two-step decay of the correlation function S of a unit undergoing anisotropic motion before finally full isotropisation is achieved. The dashed line indicates the case of a one-step decay by isotropic motion. For the study of viscoelastic polymers the intermediate plateau that reflects residual couplings is the... 

On a global scale, the linear viscoelastic behavior of the polymer chains in the nanocomposites, as detected by conventional rheometry, is dramatically altered when the chains are tethered to the surface of the silicate or are in close proximity to the silicate layers as in intercalated nanocomposites. Some of these systems show close analogies to other intrinsically anisotropic materials such as block copolymers and smectic liquid crystalline polymers and provide model systems to understand the dynamics of polymer brushes. Finally, the polymer melt-brushes exhibit intriguing non-linear viscoelastic behavior, which shows strainhardening with a characteric critical strain amplitude that is only a function of the interlayer distance. These results provide complementary information to that obtained for solution brushes using the SFA, and are attributed to chain stretching associated with the space-filling requirements of a melt brush. 

In this equation, viscosity is independent of the degree of dispersion. As soon as the ratio of disperse and continuous phases increases to the point where particles start to interact, the flow behavior becomes more complex. The effect of increasing the concentration of the disperse phase on the flow behavior of a disperse system is shown in Figure 8-41. The disperse phase, as well as the low solids dispersion (curves 1 and 2), shows Newtonian flow behavior. As the solids content increases, the flow behavior becomes non-Newtonian (curves 3 and 4). Especially with anisotropic particles, interaction between them will result in the formation of three-dimensional network structures. These network structures usually show non-Newtonian flow behavior and viscoelastic properties and often have a yield value. Network structure formation may occur in emulsions (Figure 8-42) as well as in particulate systems. The forces between particles that result in the formation of networks may be... 

Bone is an anisotropic and viscoelastic ceramic matrix composite and is distinct from conventional ceramics. Its mechanical properties depend on its porosity, degree of mineralization, collagen fiber orientation, and other structural details. The data in Table 18.1 may be used to compare the physical and mechanical properties of bone, hydroxyapatite (the major mineral in bone, and hence, the most relevant material as a bioceramic), and CBPCs. 

Linear viscoelastic behavior is actually observed with polymers only in very restricted circumstances involving homogeneous, isotropic, amorphous specimens subjected to small strains at temperatures near or above Tg and under test conditions that are far removed from those in which the sample may be broken. Linear viscoelasticity theory is of limited use in predicting service behavior of polymeric articles, because such applications often involve large strains, anisotropic objects, fracture phenomena, and other effects which result in nonlinear behavior. The theory is nevertheless valuable as a reference frame for a wide range of applications, just as the thermodynamic equations for ideal solutions help organize the observed behavior of real solutions. 

Finally, there are complex fluids that are intermediate between solid and liquid in more than one of the ways listed above. Liquid crystalline polymers (LCPs) are both viscoelastic and liquid crystalline. Ordered block copolymers are viscoelastic and anisotropic. Glassy polymers possess long viscoelastic time scales both because they are glassy and because they are polymeric. Filled polymer melts possess the properties of both polymer melts and suspensions. 

As mentioned above, when the transverse dimensions of the beam are of the same order of magnitude as the length, the simple beam theory must be corrected to introduce the effects of the shear stresses, deformations, and rotary inertia. The theory becomes inadequate for the high frequency modes and for highly anisotropic materials, where large errors can be produced by neglecting shear deformations. This problem was addressed by Timoshenko et al. (7) for the elastic case starting from the balance equations of the respective moments and transverse forces on a beam element. Here the main lines of Timoshenko et al. s approach are followed to solve the viscoelastic counterpart problem. 

In recent years, experimental investigation of the depolarized Rayleigh scattering of several liquids composed of optically anisotropic molecules has confirmed the existence of a doublet-symmetric about zero frequency change and with a splitting of approximately 0.5 GHz (see Fig. 12.1.1). The existence of this doublet had been predicted on the basis of a hydrodynamic theory several years previously by Leontovich (1941). This theory assumes that local strains set up by a transverse shear wave are relieved by collective reorientation of individual molecules. Later, Rytov (1957) formulated a more general hydrodynamic theory for viscoelastic fluids that reduces to the Leontovich theory in the appropriate limit. The theories of Rytov and Leontovitch are different from the present two-variable theory, in that the primary variable is the stress tensor and not the polarizability. 

Biot, M. A., 1954.r/ieory of Stress-strain Relations in Anisotropic Viscoelasticity and Relaxation Phenomena, J. Appl. Phys., 25(11) ppl385-1391. 

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. 

Linear viscoelasticity is valid only imder conditions where structural changes in the material do not induce strain-dependent modulus. This condition is fulfilled by amorphous polymers. On the other hand, the structural changes associated with the orientation of crystalline polymers and elastomers produce anisotropic mechanical properties. Such polymers, therefore, exhibit nonlinear viscoelastic behavior. 

Although bone is a viscoelastic material, at the quasi-static strain rates in mechanical testing and even at the ultrasonic frequencies used experimentally, it is a reasonable first approximation to model cortical bone as an anisotropic, linear elastic solid with Hooke s law as the appropriate constitutive equation. Tensor notation for the equation is written as ... 

Lakes R.S. and Katz J.L. 1974. Interrelationships among the viscoelastic function for anisotropic solids Application to calcified tissues and related systems. /. Biomech. 7 259. 

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... 

These are essentially independent effects a polymer may exhibit all or any of them and they will all be temperature-dependent. Section 6.2 is concerned with the small-strain elasticity of polymers on time-scales short enough for the viscoelastic behaviour to be neglected. Sections 6.3 and 6.4 are concerned with materials that exhibit large strains and nonlinearity but (to a good approximation) none of the other departures from the behaviour of the ideal elastic solid. These are rubber-like materials or elastomers. Chapter 7 deals with materials that exhibit time-dependent effects at small strains but none of the other departures from the behaviour of the ideal elastic sohd. These are linear viscoelastic materials. Chapter 8 deals with yield, i.e. non-recoverable deformation, but this book does not deal with materials that exhibit non-linear viscoelasticity. Chapters 10 and 11 consider anisotropic materials. 

Another factor of anisotropic design analysis is greater dependence of stress distributions on materials properties. For isotropic materials, whether elastic, viscoelastic, etc., static values often result in stress fields which are independent of material stiffness properties. In part, this is due to the fact that Poisson s ratio is the only material parameter appearing in the compatibility equations for stress. This parameter does not vary widely between materials. However, the compatibility equations in stress for anisotropic materials depend on ratios of Young s moduli for different material axes, and this can introduce a strong dependence of stress on material stiffness. This approach can be used in component design, but the product and material design analysis become more closely related. 

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