Stress rates

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

A wide variety of nonnewtonian fluids are encountered industrially. They may exhibit Bingham-plastic, pseudoplastic, or dilatant behavior and may or may not be thixotropic. For design of equipment to handle or process nonnewtonian fluids, the properties must usually be measured experimentally, since no generahzed relationships exist to pi e-dicl the properties or behavior of the fluids. Details of handling nonnewtonian fluids are described completely by Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). The generalized shear-stress rate-of-strain relationship for nonnewtonian fluids is given as... 

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. 

From the second of these, using (5.62) and (5.I82), the inelastic contribution to the stress rate and the inelastic contribution to the strain rate are related by... 

Inserting (5.7 J and (5.22) into the stress rate relation (5.6) results in the general stress rate relation... 

In a given motion, a particular material particle will experience a strain history The stress rate relation (5.4) and flow rule (5.11), together with suitable initial conditions, may be integrated to obtain the eorresponding stress history for the particle. Conversely, using (5.16) instead of (5.4), may be obtained from by an analogous ealeulation. As before, may be represented by a continuous curve, parametrized by time, in six-dimensional symmetric stress spaee. 

When the material is at the ultimate stress point B, inelastic loading will entail a positive strain rate, and the elastic limit surface in strain space will be moving outward. On the other hand, the stress rate at this point is zero, and the elastic limit surface in stress space will be stationary. If the material is perfectly inelastic over a range of strains, then the stress rate will be zero and the elastic limit surface in stress space will be stationary on inelastic loading throughout this range. 

The direction of the stress rate in relation to the elastic limit surface in stress space, expressed in /, cannot be used as an unambiguous indicator of loading or unloading. The proper indicator of inelastic loading in stress space is =/// . 

It is possible to show from the inequality (5.55) that the inelastic contribution to the stress rate is directed along the inward normal to the elastic limit surface in strain space, i.e.,... 

Another expression for the hardening index A is obtained by inserting the stress rate relation in the form (5.4) into the expression for the loading func-... 

It is usual in the classical theory to assume that the stress rate is independent of the hardening parameters, since the elastic behavior is expected to be unaffected by plastic deformation. Consequently, the stress rate relation (5.23) reduces to... 

If the material response is entirely elastic, then A = 0. If the deformation has been elastic from an initial state in which k = k and remains so, k remains unchanged and may be omitted as a dependent variable. The stress rate relation (5.111) reduces to... 

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

In the language of Section A.4, s and d are indifferent but s is not, involving extra terms in Q. In order to render the stress rate indifferent, the extra terms must be cancelled out. This may be done using the spin tensor w defined in (A.l Ij), following the steps leading to (A.68). The result is... 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

While c in (5.112) is a linear function of d, it may be an arbitrary function of s. Truesdell considered cases where c is a polynomial in s, terming (5.112) a hypoelastic equation of grade n, where n is the power of the highest-order term in the polynomial. For a hypoelastic equation of grade zero, the elastic modulus c is independent of s and linear in dand therefore has the representation (A.89). It is convenient to nondimensionalize the stress by defining s = sjljx. Since the stress rate must vanish when d is zero, Cq = 0 and the result is... 

Jaumann s stress rate is not the only indifferent rate which could be used to render (5.117) objective. Truesdell s rate defined by (A.42) is indifferent, as shown in (A.70), and can serve just as well. Inserting Truesdell s rate instead of Jaumann s rate, (5.117) reduces to the three ordinary differential equations... 

In this case, the shear stress is linear in the shear strain. While more physically reasonable, this is not likely to provide a satisfactory representation for the large deformation shear response of many materials either, since most materials may be expected to stiffen with deformation. Note that the hypoelastic equation of grade zero (5.117) is not invariant to the choice of indifferent stress rate, the predicted response for simple shear depending on the choice which is made. 

A number of other indifferent stress rates have been used to obtain solutions to the simple shear problem, each of which provides a different shear stress-shear strain response which has no latitude, apart from the constant Lame coefficient /r, for representing nonlinearities in the response of various materials. These different solutions have prompted a discussion in the literature regarding which indifferent stress rate is the correct one to use for large deformations. 

In fact, as Atluri [17] has pointed out, the hypoelastic equation of grade zero has inadequate latitude to represent realistic nonlinear response of various materials in large deformations, and it is necessary to use a hypoelastic equation of at least grade one to do so. If the grade is one, then, continuing to use Jaumann s stress rate and nondimensionalizing the stress as before, the isotropic representation (A.92) may be used in (5.112) with d = A and s = B to obtain... 

Jaumann s stress rate, Truesdell s stress rate could have been used. Substi-... 

Since simple shear is a constant volume deformation, the solution does not depend on coefficients of terms involving tr(various values of a are shown in Fig. 5.9. The solution for a grade zero material using Jaumann s stress rate (5.120) corresponds to = Ug = Ug = 0 so that a = -1, while the solution using Truesdell s stress rate (5.122) corresponds to = 0 and Ug = 1 so that a = 0. The shear... 

Differentiating with respect to time, the stress rate relation becomes... 

Note that, since G and F are nonsingular, c is nonsingular. Similarly, if is nonsingular, b is also. Conversely, given the spatial stress rate relation (5.154)... 

The dependence of the spatial moduli c and b on f has been emphasized in writing the stress rate relation (5.154). This dependence implies that the these quantities are varying as the deformation proceeds, quite apart from their dependence on e and k. If CC and are assumed to be constant, independent of E and K, then, in component form, (5.155) becomes... 

The objectivity of the spatial stress rate relation (5.154) may be investigated by applying the coordinate transformation (A.50) representing a rotation and translation of the coordinate frame. The spatial strain and its convected rate are indifferent by (A.58) and (A.62). So are the stress and its Truesdell rate. It is readily verified from (5.151), (5.152), and the fact that K has been assumed to be invariant, that k and its Truesdell rate are also indifferent. Using these facts together with (A.53) in (5.154) with c and b given by (5.155)... 

The factors in Q on each side cancel, so that the stress rate relation is invariant to a rotation of coordinate frame, and is objective. Note that it is the special dependence of c and b on which makes the stress rate relation objective. If... 

The unrotated spatial stress rate relation may also be related to its counterpart in the current spatial description. Using (5.1792), (5.1892), (A.36), and... 

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