Stress tensor, anisotropic

The situation is more complex for rigid media (solids and glasses) and more complex fluids that is, for most materials. These materials have finite yield strengths, support shears and may be anisotropic. As samples, they usually do not relax to hydrostatic equilibrium during an experiment, even when surrounded by a hydrostatic pressure medium. For these materials, P should be replaced by a stress tensor, <3-j, and the appropriate thermodynamic equations are more complex. 

The pressure is to be identified as the component of stress in the direction of wave propagation if the stress tensor is anisotropic (nonhydrostatic). Through application of Eqs. (2.1) for various experiments, high pressure stress-volume states are directly determined, and, with assumptions on thermal properties and temperature, equations of state can be determined from data analysis. As shown in Fig. 2.3, determination of individual stress-volume states for shock-compressed solids results in a set of single end state points characterized by a line connecting the shock state to the unshocked state. Thus, the observed stress-volume points, the Hugoniot, determined do not represent a stress-volume path for a continuous loading. 

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

Rheometers are currently under development that will enable the anisotropic stress tensor of anisotropic complex fluids such as block copolymer melts and solutions to be probed, even during large amplitude shear. Here, a small amplitude probe waveform is applied orthogonal to the primary large amplitude shear flow. This could provide the linear dynamic modulus of an anisotropic system under nonlinear flow. 

In a simulation, the pressure for the primitive cell model is a nontrivial variable to compute for two reasons First, the long-range electrostatics has to be properly taken into account, and second, the system is inherently anisotropic, hence the relevant observable is the stress tensor. 

For anisotropic systems the stress tensor is the relevant observable to compute. Whereas the ideal gas contribution to the pressure still remains isotropic, the virial must be replaced by... 

The best way to determine the stresses in the scale would be a direct measurement. However, X-ray methods have usually a limited spatial resolution which makes it difficult to measure nonuniform stress fields. The application of OFS for scales consisting of a-Al203 provides a sufficient spatial resolution and permits, in principle, to examine stress variations in the scale. However, only the trace of the stress tensor can be measured for an untextured polycrystalline scale. Thus, anisotropic stress states have to be analysed in combination with a mechanical modelling of the scale loading in order to deduce the stress components from the trace of the stress tensor. 

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. 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

From Hooke s law, the six independent components of the stress tensor can be expressed as a function of the six components of the strain tensor in a symmetrical matrix of order 6 with 21 modulus components for a general anisotropic sample of material. For an isotropic body, there are only two independent components. The mode of deformation will determine which modulus will be measured. 

Ericksen proposed a theory for fluids such as nematic liquid crystals which could become anisotropic during flow. By assuming symmetry around the director, the expression for the stress tensor was somewhat simplified. We compare here briefly Ericksen s transversely isotropic fluid theory with the transient behavior observed for thermotropic copolyesters of PHB/PET. 

Significant wall force field effects are found on the velocity, density, shear stress distributions in the near-wall region that extends approximately three molecular diameters from each surface for van der Waals interactions. Within this wall force penetration depth, a density buildup with a single peak point is observed. Exactly matching the earlier static case, normal components of the stress tensor are anisotropic and... 

Leslie, F.M. Introduction to nematodynamics. In Dunmur, D., Fukuda, A., Luckhurst, G., INSPEC (eds.) Physical Properties of Liquid crystals Nematics, pp. 377-386, London (2001). Parodi, O. Stress tensor for nematic liquid crystals. J. Phys. (Paris) 31, 581-584 (1970) Miesowicz, M. The three coefficients of viscosity of anisotropic liquids. Nature 158, 27 (1946) Influence of the magnetic field on the viscosity of para-azoxyanisole. Nature 136, 261 (1936). 

When deriving the expression for the viscosity stress tensor (2.47), Leslie took into account only the first derivatives of the velocity dvifdxk and the director dL jdxx components, which could be regarded as a first approximation to the anisotropic dissipative contribution in the Navier-Stokes equations. 

The states of stress and strain in a deformed crystal being idealized as a continuum are characterized by symmetric second-rank tensors and Cjj, respectively, each comprising six independent components. Hooke s law of linear elasticity for the most general anisotropic solid expresses each component of the stress tensor linearly in terms of all components of the strain tensor in the form... 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

We next examine the response of the anisotropic medium to the applied stress. In the general case, if a homogeneous stress is applied, the resulting homogeneous strain ey—in the limit of Hooke s law—depends linearly on all nine (Ty components of the stress tensor of the same form as displayed in Eq. (5.10.2). The relation between stress and strain is encoded in the compliance tensor. As a typical example, note the following relationship ... 

In a thermally anisotropic medium, temperature variation will cause expansion or contraction and other distortions, thereby influencing the strain and stress tensors. We replace (1.8.9, 10) by... 

All parameters depend on the time and the shear rate. Steady-state conditions are obtained for t — CO. Variable (°, ) denotes the steady state values of the shear stress. The anisotropic character of the flowing solutions give rise to additional stress components, which are different in all three principal directions. This phenomenon is called the Weissenberg effect, or the normal stress phenomenon. From a physical point of view, it means that all diagonal elements of the stress tensor deviate from zero. It is convenient to express the mechanical anisotropy of the flowing solutions by the first and second normal stress difference ... 

In the absence of magnetic field, the theory exhibits viscoelastic transversally anisotropic behavior with symmetric stress tensor and orientation of director caused only by flow. Thus, this simplified approach has led to a closed set of two coupled anisotropic viscoelastic equations of quasilinear type for evolution of director and extra stress the anisotropic properties in the set being described by viscoelastic evolution equation for director. Although this theory has been developed for low enough value of Deborah number, it is still possible to compare the simulations with experimental data. 

After passing through a glass specimen, the light polarization is modified because glass becomes anisotropic when submitted to stresses (residual or transient Appendix G). The refractive index is not imique (Chapter 4, Appendix A) but depends on the stress tensor. The indices along principal directions are... 

In the genera anisotropic case, P has to be replaced by the components of the stress tensor and V by the components of the deformation tensor (Sect.3.6)]. We shall formulate the problem quantum mechanically but we often orientate ourselves by the results obtained for the classical linear chain. In particular, we use the quasi harmonic approximation, that is, we neglect the specific anharmonic term [see (5.42) for the linear chain]. Thus... 

If one allows for anisotropic frictional forces by retaining the friction tensor fin equation (51), and allowing for anisotropic Brownian motion by allowing the Maxwellian velocity distribution to be skewed (so that = — (kT/ F)[(5/5ry) f F]), then the diffusion equation and stress tensor expressions become... 

An anisotropic stress tensor means that there is non-zero dissipation if the entire fluid undergoes a rigid-body rotation, which is clearly unphysical. However, as emphasized in [28], this asymmetry is not a problem for most applications in the incompressible (or small Mach number) limit, since the form of the Navier-Stokes equation is not changed. This is in accordance with results obtained in SRD simulations of vortex shedding behind an obstacle [36], and vesicle [37] and polymer dynamics [14]. In particular, it has been shown that the linearized hydrodynamic modes are completely unaffected in two dimensions in three dimensions only the sound damping is slightly modified [28]. 

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