Strain velocity tensor

Usually, investigations deal with uniaxial tension in a cylindrical sample which is easier to produce in experiments than other types of tension (see Fig. 2). In this case the strain velocity tensor is ... 

Viscous Forces In the momentum equation (Navier-Stokes equation), forces F acting on the system result from viscous stresses. It is necessary to relate these stresses to the velocity field and the fluid s viscosity. This relationship follows from the stress and strain-rate tensors, using Stokes postulates. 

Stress and Strain Rate The stress and strain-rate state of a fluid at a point are represented by tensors T and E. These tensors are composed of nine (six independent) quantities that depend on the velocity field. The strain rate describes how a fluid element deforms (i.e., dilates and shears) as a function of the local velocity field. The stress and strain-rate tensors are usually represented in some coordinate system, although the stress and strain-rate states are invariant to the coordinate-system representation. 

The relative volumetric expansion is seen to be the sum of the normal strain rates, which is the divergence of the vector velocity field. The sum of the normal strain rates is also an invariant of the strain-rate tensor, Eq. 2.95. Therefore, as might be anticipated, the relative volumetric dilatation and V V are invariant to the orientation of the coordinate system. 

The stress tensor plays a prominent role in the Navier-Stokes and the energy equations, which are at the core of all fluid-flow analyses. The purpose of the stress tensor is to define uniquely the stress state at any (every) point in a flow field. It takes nine quantities (i.e., the entries in the tensor) to represent the stress state. It is also be important to extract from the stress tensor the three quantities needed to represent the stress vector on a given surface with a particular orientation in the flow. By relating the stress tensor to the strain-rate tensor, it is possible to describe the stress state in terms of the velocity field and the fluid viscosity. 

The final objective of this chapter was to develop quantitative relationships between a fluid s strain-rate and stress fields. Expressions for the strain rates were developed in terms of velocities and velocity gradients. Then, using Stokes s postulates, the stress field was found to be proportional to the strain rates and a physical property of the fluid called viscosity. The fact that the stress tensor and strain-rate tensor share the same principal coordinates is an important factor in applying Stokes s postulates. The stress-strain-rate relationships are fundamental to the Navier-Stokes equations, which describe conservation of momentum in fluids. 

In Chapter 2 considerable effort is devoted to establishing the relationship between the stress tensor and the strain-rate tensor. The normal and shear stresses that act on the surfaces of a fluid particle are found to depend on the velocity field in a definite, but relatively complex, manner (Eqs. 2.140 and 2.180). Therefore, when these expressions for the forces are substituted into the momentum equation, Eq. 3.53, an equation emerges that has velocities (and pressure) as the dependent variables. This is a very important result. If the forces were not explicit functions of the velocity field, then more dependent variables would likely be needed and a larger, more complex system of equations would emerge. In terms of the velocity field, the Navier-Stokes equations are stated as... 

In all above- and below-cited publications in this field (e.g. 84 ) the problem was solved in order to calculate the tensors of strain velocity and stress, to prognosticate alteration of longitudinal viscosity, profile of alteration of the thickness of material over the height of the film sleeve (by coordinate on the central line of the sleeve counted from the outlet face of the extrusion head) and configuration of the sleeve ( bubble ) and also to solve thermal problems in order to determine the dependency of melt temperature upon height (or time) and to forecast the position of the crystallization... 

Here, pa,- is the bead momentum vector and u(rm. f) = iyrV is the linear streaming velocity profile, where y = dux/dy is the shear strain rate. Doll s method has now been replaced by the SLLOD algorithm (Evans and Morriss, 1984), where the Cartesian components that couple to the strain rate tensor are transposed (Equation (11)). 

A very useful and particular case of the vector gradient is the velocity vector gradient, Vu shown in eqn. (1.15). With this tensor, two very useful tensors can be defined, the strain rate tensor... 

The arithmetic mean of both angular velocities is called the strain tensor, often also called the deformation velocity tensor... 

Since viscosity can be expressed in terms of strain rate for the GNF, the stress tensor can be written in terms of the strain rate tensor and some constant parameters, as given in Equation 22.17. A power-law relationship for viscosity in terms of the strain rate has been assumed in Equation 22.17. In the case of flow between two parallel plates where one plate is fixed and the other one is moving with a given velocity. Equation 22.17 reduces to Equation... 

The only non-zero term in the strain rate tensor is Dre = (1/2)rd(ua lr)l dr. The rheometrical experiment enables the determination of the relationship between Dre and Tre, with both remaining approximately homogeneous across the material. The average value of the magnitude of the strain rate is expressed by bringing in the angular rotational velocity, Q, of the itmer cylinder ... 

Returning to the problem of the blocked pipe, consider the flow in the cylindrical coordinate system whose Oz axis, which coincides with the axis of the pipe (see Figure 1.3 of Chapter 1), is oriented in the direction of the flow. By considering the non-zero component(s) in the velocity vector, verify that the only non-zero term in the strain rate tensor is Drz. Set out the form of this term explicitly and explain why ) 5 0. Explain also why the only non-zero shear stress in the pipe is Trz, and why Vrz <0. It will be deduced there from that the rheological relation for the Bingham fluid can be written, for the flow of that fluid in a pipe, as ... 

For this velocity field, after expressing h r determine the different terms in the strain rate tensor. In the case where angle a is very small, justify that the D e term is dominant in the strain rate tensor. Any comments ... 

U - velocity vector p - hydrostatic pressure T - extra stress tensor d - strain rate tensor T/o - zero shear viscosity y - shear rate... 

The functions v,aij,Sij v) represent the velocity, components of the stress tensor and components of the rate strain tensor. The dot denotes the derivative with respect to t. The convex and continuous function describes the plasticity yield condition. It is assumed that the set... 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

Vectors are commonly used for description of many physical quantities such as force, displacement, velocity, etc. However, vectors alone are not sufficient to represent all physical quantities of interest. For example, stress, strain, and the stress-strain iaws cannot be represented by vectors, but can be represented with tensors. Tensors are an especially useful generalization of vectors. The key feature of tensors is that they transform, on rotation of coordinates, in special manners. Tsai [A-1] gives a complete treatment of the tensor theory useful in composite materials analysis. What follows are the essential fundamentals. 

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