Stress rate relation

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. 

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 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... 

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... 

It is seen that (5.212) has the same form as the hypoinelastie stress rate relation (5.115), exeept that Green-Meinnis rates are used in plaee of Jaumann rates. Note that the two stress rate relations are not related by additive terms whieh are polynomials in s and linear in d since the Green-Mclnnis rate involves the rate of rotation ft. Consequently, the hypoinelastie... 

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