Tensor strain rate

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

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. 

We now turn to the 6-z plane, after having determined seven of the nine strain-rate elements of the strain-rate tensor. Figure 2.8 illustrates the two-dimensional projection a differential element on a 6-z surface for some r value. 

There is always a particular set of coordinates, called the principal coordinates, for which the shear components vanish the strain-rate tensor can be written as... 

Fig. 2.9 Because the strain-rate tensor is symmetric, there is always an orientation of a differential element for which the strain-rates are purely dilatational.

The principal strain rates are eigenvalues of the strain-rate tensor (matrix). As described more fully in Section A.21, the direction cosines that describe the orientation of the principal strain rates are the eigenvectors associated with the eigenvalues. In solving practical fluids problems, there is rarely a need to determine the principal strain rates or their orientations. Rather, these notions are used theoretically with the Stokes postulates to form general relationships between the strain-rate and stress tensors. It is perhaps worth noting that in solid mechanics, the principal stresses and strains have practical utility in understanding the behavior of materials and structures. 

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 basis for connecting the stress and strain-rate tensors was postulated first by G. G. Stokes in 1845 for Newtonian fluids. He presumed that a fluid is a continuous medium and that its properties are independent of direction, meaning they are isotropic. His insightful observations, itemized below, have survived without alteration, and are an essential underpinning of the Navier-Stokes equations ... 

The stress tensor T is a linear function (including a constant) of the strain-rate tensor E. 

Consider first the trivial case of a static fluid. Here there can only be normal forces on a fluid element and they must be in equilibrium. If this were not the case, then the fluid would move and deform. Certainly any valid relationship between stress and strain rate must accommodate the behavior of a static fluid. Hence, for a static fluid the strain-rate tensor must be exactly zero e(/- = 0 and the stress tensor must reduce to... 

Perhaps surprisingly, it turns out that the complex series of operations represented by Eq. 2.151 leads to a relatively simple result that is independent of the particular principal-coordinate directions. The stress tensor in a given coordinate system is related to the strain-rate tensor in the same coordinate system as... 

In the principal coordinates, of course, there are only three nonzero components of the stress and strain-rate tensors. Upon rotation, all nine (six independent) tensor components must be determined. The nine tensor components are comprised of three vector components on each of three orthogonal planes that pass through a common point. Consider that the element represented by Fig. 2.16 has been shrunk to infinitesimal dimensions and that the stress state is to be represented in some arbitrary orientation (z, r, 6), rather than one aligned with the principal-coordinate direction (Z, R, 0). We seek to find the tensor components, resolved into the (z, r, 6) coordinate directions. 

Since the principal axes are the same for the stress tensor and the strain-rate tensor, the normal strain rates are related to the principal strain rates by the same transformation rules that we just completed for the stress. Thus... 

Recall from the discussion in Section 2.5.4 that the stress tensor, like the strain-rate tensor, has certain invariants. For any known stress tensor, these invariant relationships can be used to determine the principal stresses. 

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. 

Write the strain-rate tensor (in matrix form) for these two circumstances. 

Determine the principal strain rates. Since there are so many zeros in the strain rate tensor, this eigenvalue problem can be solved exactly without too much diffficulty. 

Using the spreadsheet representation of the CVD flow field (Fig. 2.23), evaluate the strain-rate tensor at the point z = 0.045789474 m and r = 0.026666667 m. Although this is a two-dimensional axisymmetric problem, be aware that there are 0 components in the tensor. 

Explain the physical meaning of the ee component of the strain-rate tensor. 

Show that the diagonal invariant of the strain-rate tensor is preserved. 

Write out the stress tensor explicity in terms of the strain-rate components. If both the pressure and kV-V, which add equally to each diagonal component, are removed, how is the remaining part of the stress tensor related to the strain-rate tensor ... 

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... 

E Strain rate tensor (second-order tensor) 1/s... 

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)). 

The material functions, k i and k2, are called the primary and secondary normal stress coefficients, and are also functions of the magnitude of the strain rate tensor and temperature. The first and second normal stress differences do not change in sign when the direction of the strain rate changes. This is reflected in eqns. (2.51) and (2.52). Figure 2.31 [41] presents the first normal stress difference coefficient for the low density polyethylene melt of Fig. 2.30 at a reference temperature of 150°C. 

Strictly speaking, the viscosity rj, measured with shear deformation viscometers, should not be used to represent the elongational terms located on the diagonal of the stress and strain rate tensors. Elongational flows are briefly discussed later in this chapter. A rheologist s... 

Here, 7 is the magnitude of the strain rate tensor and C/ is a phenomenological coefficient which models the interactions between the fibers, usually referred to as the Folgar-Tucker interaction coefficient. The coefficient varies between 0, for a fiber without interaction with its neighbors, and 1, for a closely packed bed of fibers. For a fiber reinforced polyester resin mat with 20-50% volume fiber content, CV is usually between 0.03 and 0.06. When eqn. (8.153) is substituted into eqn. (8.152), the transient governing equation for fiber orientation distribution with fiber interaction built-in, becomes... 

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 distribution of Eq. [137] is canonical in laboratory momentum and positions for a general strain rate tensor Vu this is the expected form for a system subject to an external field. Equation [137] is the first distribution function to be derived for SLLOD-type dynamics and has provided impetus for studies concerning the nature of the distribution function in the nonequilibrium steady state. 

In formal rheology, relations between these three tensors are formulated and analyzed. Only for the two extremes of viscoelastic behaviour are such relations simple. For purely elastic materials there is a relation between the stress tensor and the strain tensor it contains the elasticity modulus and the Poisson ratio, accounting for the extent to which extension in one direction is accompamied by concomitant compression in the other two. For purely viscous fluids there is a relation between the stress tensor and the strain rate tensor. As extension in one direction is concomitant with (viscous) compression in the other two, in this case only one viscosity is required. For incompressible Newton fluids eventually an expression with only one viscosity results, see (1.6.1.131. 

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