Stress Tensor Components

In general, the state of stress in a flowing material in rectangular Cartesian coordinates is 

On the contrary, we have not yet discussed a constitutive equation for viscoelastic fluids, and we must resort to another method to find the stress components. Without proof it can be shown using symmetry arguments that, in general, for a viscoelastic fluid in shear flow the stress tensor must be of the form 

We note that in shear flow additional normal stresses are generated which don t appear for a Newtonian fluid. Because polymeric fluids are considered to be incompressible, the components T - - p have no direct rheological significance. Therefore, we define three independent quantities of stress of rheological significance  

VISCOELASTIC RESPONSE OF POLYMERIC FLUIDS AND FIBER SUSPENSIONS 

For shear-free flows it can be shown using symmetry arguments again that the extra stress tensor is of the form 

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Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Denote hy W = (w, w ), w horizontal and vertical displacements of the mid-surface points, respectively, and write down the formulae for strain and integrated stress tensor components y(lL), aij W) ... 

We shall consider an equilibrium problem with a constitutive law corresponding to a creep, in particular, the strain and integrated stress tensor components (IT ), ay(lT ) will depend on = (lT, w ), where (lT, w ) are connected with (IT, w) by (3.1). In this case, the equilibrium equations will be nonlocal with respect to t. 

Consider an inclined crack with the nonpenetration condition of the form (3.173), (3.176). Let % = (IL, w) be the displacement vector of the midsurface points. Introduce the strain and stress tensor components Sij =... 

Using (4) to write stress tensor Pxx components as Pyy and Pz2, matrix components for the case under consideration are expressed specifically and are substituted into expressions for stress tensor components. Thus, we obrais the following relationships (details of transformation in 29)) ... 

From the field stress tensor components, we may write the electric and magnetic field components as... 

The (jjj expression indicated above shows that at the crack tip (r = 0), the stress tensor components become infinite. Actually, for a material able to undergo plastic deformation, above some stress level yielding occurs and limits the stress to the corresponding value, ay. Thus, around the crack tip a zone exists in which the material is plastically deformed. Such a zone is called the plastic zone, and it is represented in Fig. 8 in the case of a crack across a plate thickness. 

Having all the stress tensor components, we can proceed with the equation of motion, whose components reduce to (41)... 

Another method to calculate viscoelastic quantities uses measurements of flow birefringence (see Chap. 10). In these measurements, two quantities are determined as functions of the shear rate y the birefringence An and the extinction angle y. The following relationships exist with the stress tensor components ... 

CTjj Stress tensor component Octahedral shear stress... 

The stress-strain relationship [Eq. (P4.ll)] together with Eqs. (16.119a) and (16.119b) give the stress tensor components as follows... 

Substituting the stress tensor components [Eqs. (17.214c) and (17.214d) into Eq. (17.219), we obtain... 

Accordingly, the stress function F is required to be constant along the section boundary. This constant may be chosen arbitrarily for a simply connected boundary because it has no effect on the stress tensor components. For the forthcoming discussion we assume, without loss of generality, the boundary condition... 

One can see that T is a positively determined quadratic form of the stress tensor components, while its partial derivatives with respect to those components coincide with the stress tensor components themselves. Indeed,... 

As was doiiioustratcd in Section 1.6.1, a mechanical expression for il can be derived in many cases of interest. This permits access to (absolute values of) ft from stress tensor components that can be calculated or controlled experimentally. This also holds for other thermodynamic potentials where N has been replaced by /i as a thermodynamic state variable via a Legendre transformation (see Section 1.5). 

To derive a molecular expression for the stress tensor component which is the basic quantity if one wishes to compute pseudo-experimental data F (/i) /R, we start from the thermodynamic expression for the exact differential given in Eq. (1.59), where the strain tensor [Pg.202]

Alternatively, we may derive a force expression for Txz following the derivation presented in Appendix E.3.2.2 for the stress tensor component r. It follows if we combine Eqs. (E.62) and (E.63) with Eq. (E.49) from which we obtain... 

Fig. 3.5. Description of shear and stress tensor components acting on a cube, after Landau Lifschitz (1953)...

While the stress tensor component tfor purely viscous fluids can be determined from the instantaneous values of the rate of deformation tensor 4, the past history of deformation together with the current value of 4, may become an important factor in determining t, for viscoelastic fluids. Constitutive equations to describe stress relaxation and normal stress phenomena are also needed. Unusual effects exhibited by viscoelastic fluids include rod climbing (Weis-senberg effect), die swell, recoil, tubeless siphon, drag, and heat transfer reduction in turbulent flow. 

As an alternative to the t-T superposition, a plot of the elastic stress tensor component as a function of the viscous one has been used, e.g., (o - 0 2) vs. Oj2, or G vs. G . For systems in which the t-T is obeyed, such plots provide a temperature-independent master curve, without the need for data shifting and calculating the three shift factors. Indeed, from the Doi and Edwards tube model, the following relation was derived ... 

The nine reference stress components, each of which depends on position x and time t, and referred to as the stress tensor components. In Cartesian tensor notation we may write... 

Figure 2.2.3 Cartesian components of the reference stresses (stress tensor components).

For constant-density flows where V - u = 0, the stress tensor components in rectangular Cartesian coordinates x, y, z with velocity components u, v, w are... 

The magnitude of that appears in Eq. (17.11) may also be studied in a dynamic way as described in the following Eq. (17.11) represents the time-correlation function of the stress tensor component Jxy t) in the long-time region described by the Langevin equation ... 

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