The Stress Constitutive Equation

Because no shear stresses have been imposed, the nonzero components of the stress tensor are [in view of Eq. (10.5.6)] as follows  

From the discussion following Eq. (10.4.6) and from Problem 10.6, we have 

Equation (10.7.8) is called a stress constitutive equation, and it relates a three-dimensional measure of strain to the three-dimensional stress. For rubbers, Eq. (10.7.8) obviously holds for the specialized case of imiaxial extension. By similar reasoning, it can be shown to hold for other idealized deformations such as biaxial extension and shear. Indeed, Eq. (10.7.8) is valid for all volumepreserving deformations [10]. The only material quantity appearing in this 

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The dependence of the stress on the strain-deformation history of macro-molecular liquids can be incorporated in two ways. The stress constitutive equation can be formulated as a differential equation, in which the extra stress r is the solution of an equation that is typically of the general form... 

The stress constitutive equation can also be formulated as an integral over the history of the deformation. The most common form used for simulations is... 

Equation 7.9 expresses the total axial stress in terms of the extra stress, so the stress constitutive equation provides the link to the kinematics. 

This problem considers a particular kind of deformation-uniaxial extension. The same procedure can be applied to other kinds of deformation, and the result is a material function or, in the case of rubber, a material constant that relates a component of the stress to a component of the strain imposed on the material. More generally, though, we can determine the relationship between an arbitrary, three-dimensional deformation and the resulting three-dimensional stress. Such a relationship is called the stress constitutive equation. We will develop such a relationship for rubbers after we review the definitions of stress and the strain in three dimensions. 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

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. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

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... 

Example 3.12 For the laminate [0/352/ - determine the elastic constants in the global directions using the Plate Constitutive Equation. When stresses of = 10 MN/m, o-y = —14 MN/m and = —5 MN/m are applied, calculate the stresses and strains in each ply in the local and global directions. If a moment of 10(X) N m/m is added, determine the new stresses, strains and curvatures in the laminate. The plies are each 1 mm thick. 

Note 4 For cases where there is a dependence of stress on strain history the following constitutive equation may be used, namely... 

In general, Hooke s law is the basic constitutive equation giving the relationship between stress and strain. Generalized Hooke s law is often expressed in the following form [20,108] ... 

The volumetric constitutive equations for a chemoporoelastic material can be formulated in terms of the stress S = a,p, it and the strain 8 = e, (, 9, i.e., in terms of the mean Cauchy stress a, pore pressure p, osmotic pressure it, volumetric strain e, variation of fluid content (, and relative increment of salt content 9. Note that the stress and strain are measured from a reference initial state where all the stress fields are equilibrated. The osmotic pressure it is related to the change in the solute molar fraction x according to 7r = N Ax where N = RT/v is a parameter with dimension of a stress, which is typically of 0( 102) MPa (with R = 8.31 J/K mol denoting the gas constant, T the absolute temperature, and v the molar volume of the fluid). The solute molar fraction x is defined as ms/m with m = ms + mw and ms (mw) denoting the moles of solute (solvent) per unit volume of the porous solid. The quantities ( and 9 are defined in terms of the increment Ams and Amw according to... 

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

The changes in the viscoelastic extra-stress components along the symmetry axis are markedly different (Fig. 22) Ty is much more important at the contraction for the GOB model on the contrary, Tvyy is more pronounced for the PTT model at the die exit, Ty x is equivalent for the two constitutive equations, but relaxes more quickly for the mPTT model. Tyyy is much more important for the GOB model, which is again consistent with the more important value of the swelling observed in Fig. 20a. Obviously, Ty y remains equal to zero along the syirunetry axis for the two constitutive equations. 

It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

Among the equations that govern a viscoelastic problem, only the constitutive equations differ formally from those corresponding to elastic relationships. In the context of an infinitesimal theory, we are interested in the formulation of adequate stress-strain relationships from some conveniently formalized experimental facts. These relationships are assumed to be linear, and field equations must be equally linear. The most convenient way to formulate the viscoelastic constitutive equations is to follow the lines of Coleman and Noll (1), who introduced the term memory by stating that the current value of the stress tensor depends upon the past history... 

As expected, three groups of undetermined terms appear in the averaged equations (3.287). The first term, V (7fc(pfcv V ) ) denotes the covariance or correlation terms. The second term, (J Uk )e, accounts for the effects of interfacial stress, heat and species mass transfer, whereas the third term, mkiJk )e, account for the interfacial transfer due to phase change. The conventional constitutive equations are discussed in chap 5. 

It has been emphasized repeatedly that continuum mechanics provides no guidance in the choice of a general constitutive hypothesis for either the heat flux vector q or the stress tensor T. On the other hand, it was noted earlier that (2 41) and (2-56), derived respectively from the law of conservation of angular momentum and the second law of thermodynamics, must be satisfied by the resulting constitutive equations. It thus behooves us to see whether... 

So far, we have simply stated the Cauchy equation of motion and the Newtonian constitutive equations as a set of nine independent equations involving u, T, and p. It is evident in this case, however, that the constitutive equation, (2-80), for the stress [or equivalently (2-86)] can be substituted directly into the Cauchy equation to provide a set of equations that involve only u and p (orp). These combined equations take the form... 

Finally, it was stated previously that fluids that satisfy the Newtonian constitutive equation for the stress are often also well approximated by the Fourier constitutive equation, (2-67), for the heat flux vector. Combining (2-67) with the thermal energy, (2-52), we obtain. 

The creep strength of AljNb is comparatively low - a stress of 10 MN/m produces 1 % strain in only 500 h and fracture in 2300 h - whereas the yield stress compares favorably with the superalloys. This illustrates the fact that the difference between the yield stress and the creep strength is much more pronounced for intermetallics than for conventional alloys. Creep of Al3Nb is controlled by dislocation climb which is accompanied by subgrain formation. The observed creep behavior corresponds to that of conventional disordered alloys and the creep rates are described by the known constitutive equations. This will be discussed in more detail with respect to NiAl (Sec. 4.3). The secondary creep rate follows the power law, i.e. Dorn equation for dislocation creep... 

It is useful to see how the Voigt linear viscoelastic models of Section 7.2 behave with a sinusoidal strain input. When the strain variation equation (Eq. 7.21) is substituted in the model constitutive equation (Eq. 7.2), the stress is given as... 

Where S J = the partial tensor of stress, = the partial tensor of strain, spherical tensor of stress, = spherical tensor of strain, G = shear modulus, K = volume modulus, ov = stress tensor, = Dirac delta function and = strain tensor. The 3D constitutive equation of viscoelastic mass undergoing static-dynamic coupling loading is derived as follows (Li et al. 2006) ... 

The transformed plane stress constitutive equation 4.4 can be inverted to give ... 

The first concept is the linear elasticity, that is, the linear relationship between the total stress and infinitesimal strain tensors for the filler and matrix as expressed by the following constitutive equations ... 

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