Finite deformation tensors

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

The referential constitutive equations for an inelastic material may be set into spatial terms. Casey and Naghdi [14] did so for their special case of finite deformation rigid plasticity discussed by Casey [15], Using the spatial (Almansi) strain tensor e and the relationships of the Appendix, it is possible to do so for the full inelastic referential constitutive equations of Section 5.4.2. 

One major discrepancy of the previous model can be attributed to the use of the infinitesimal strain tensor and to derivatives restricted to time changes. Indeed, in the case of large deformations, one has to refer to finite strain tensors, such as the Finger (, t (t ) or Cauchy C t(t ) strain tensors (t being the... 

The free energy of a crystal in a magnetically ordered phase or in an applied magnetic field may be presented only as an expansion in powers of displacement tensor components Uafi (see eqs. 92 and 96) the contributions of finite deformations and electron-rotation interactions to isothermal elastic constants. ... 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

Here gag and are the symmetric and antisymmetric part of the deformation tensor, respectively. It is ag = ki>ap + I pa) and u>afi = k Va -v%,). Equation (17.28) shows how the antisymmetric part a>a enters into the finite strain tensor Tjap. The tensors G p and F ys can be calculated directly from the CEF-potential (see Dohm and Fulde 1975). Similarly iTRot(J" ) is written up to second order in the deformation tensor v as... 

At finite deformations, equation 59 can be shown to be incorrect because it is not objective i.e., it predicts results which erroneously depend on the orientation of the sample with respect to laboratory coordinates. This error can be eliminated by replacing j/j in equation 59 by the components of a corotational rate-of-strain tensor or the components of one of several possible codeformational rate-of-strain tensors either of these replacements ensures that the unwanted dependence of cy on the instantaneous orientation of a fluid particle in space is removed. If the stress-strain relations are linear within the changing coordinate frame, equation 59 is modified only be replacing y,-j with a different strain rate tensor whose definition is complicated and beyond the scope of this discussion. The corresponding corotational model is that of Goddard and Miller and the codeformational models correspond to those of Lodge or Oldroyd, Walters, and Fredrickson. ... 

A consequence of finite deformations is the appearance of normal stresses in simple shearing deformations. Thus, even in steady-state simple shear flow (Fig. 1-16) where the rate of strain tensor (c/. equations 3 and 5) is... 

When there is no deformation, the strains are zero. A finite strain tensor can be defined by subtracting the identity tensor fromB... 

Just as there are various possible finite strain tensors, there are various time derivatives that can be used in place of the ordinary derivative of stress in Eq. 10.21 to satisfy the continuum mechanics requirements for a model to be able to describe large, rapid deformations in arbitrary coordinate systems. The derivative that yields a differential model equivalent to Lodge s Eq. 10.6 is the upper convected time derivative (defined in Eq. 11.19), and the resulting model is called the upper-convected Maxwell model. Other possibilities include the lower-convected derivative and the corotational derivative. Furthermore, a weighted-sum of two of these derivatives can be used to formulate a differential constitutive equation for polymeric liquids. In particular, the Gordon-Schowalter convected derivative [7] is defined in this manner. 

Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U(Ji, I2, i). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Now we have two entities that link the undeformed body to the deformed body, that is, the displacement vector u and the deformation gradient tensor F. Next, we would like to establish connections between these two entities, which will help us establish the link between the displacement and strain and the deformation gradient and strain for the case with finite strain. Let us differentiate both sides of Equation (4.2) with respect to X ... 

In precisely the same way, a spontaneously splay-deformed structure must correspond to the equilibrium condition with finite coefficient fsTi 7 0 in tensor (8.13). The corresponding term should be added to the splay term with (divn). If the molecules have, e.g., pear shape they can pack as shown in Fig. 8.7b. In this case, the local symmetry is Coov (conical) with a polar rotation axis, which is compatible with existence of the spontaneous polarization. However, such packing is unstable, as seen in sketch (b), and the conventional nematic packing (a) is more probable. The splayed stmcture similar to that pictured in Fig. 8.7b can occur close to the interface with a solid substrate or when an external electric field reduces the overall symmetry (a flexoelectric ejfecf). 

Equations 9.8 and 9.10b are written for rectilinear flows and infinitesimal deformations. We need equations that apply to finite, three-dimensional deformations. Intuitively, one might expect simply to replace the strain rate dy/dt by the components of the symmetric deformation rate tensor (dv /dy -y dvy/dx, etc.) to obtain a three-dimensional formulation, as in Section 2.2.3, and dr/dt by the substantial derivative D/Dt of the appropriate stress components. The first substitution is correct, but intuition would lead us badly astray regarding the second. Constitutive equations must be properly invariant to changes in the frame of reference (they must satisfy the principle of material frame indifference), and the substantial derivative of a stress or deformation-rate tensor is not properly invariant. The properly invariant... 

V is said to be multiplicatively decomposed into V and V . When operating with finite strains, deformation gradients must be used that, beiug tensors, are combined multiplicatively. Suppose that both V and V have the same principal directions and are expressed in diagonal form, with principal extension ratios respectively A. and A. (i = I, II, HI), then... 

The stress tensor has been introduced in Chapter 2. In small strain elasticity theory, the components of stress are defined by considering the equilibrium of an elemental cube within the body. When the strains are small, the dimensions of the body, and therefore the areas of the cube faces, are to a first approximation unaffected by the strain. It is then of no consequence whether the components of stress are defined with respect to the cube before deformation or the cube after deformation. For finite strains, however, this is not true and there are alternative definitions of stress depending on whether the deformed or undeformed state is chosen as a reference. We will choose to adopt the stress associated with the deformed state - the true stress or Cauchy stress - throughout this work. In our present axis notation, we can express this stress tensor X as... 

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