Convected strain tensor

Let a convected coordinate system be denoted by f (i = 1,2, 3) and a fixed coordinate system by x (i = 1,2, 3). Then, states of a material element may be described by functions 

A deformation may be said to occur when the magnitude of the distance between any two points in a material element changes. The square of the distance between the two points in a space may then be used as a quantitative measure of deformation (strain). In terms of spatial coordinates, the distance between two material points may be represented by 

Therefore, in terms of convected coordinates, the distance between two material points may be represented, by use of Eq. (2.82) in (2.81), as 

Similarly, the convected contravariant metric vU( ,r) is related to the spatial metric by 

The change in the distance between the material points at two different times, t and f( t ), may be used as a measure of the strain, and it may be written. 

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FUNDAMENTAL PRINCIPLES OF POLYMER RHEOLOGY 2.5.1 Convected Strain Tensor... 

The definition of the convected strain tensor involves the difference between two quantities associated with a given material point at different times, and it refers to the same material point in convected coordinates. Now, we must transform the quantities i ,y( , t) and V y( , t ) (also v (, t ), and v 9(, t)) in such a manner that they both refer to the same point in a coordinate system fixed in space, because physical quantities (kinematic and dynamic variables) can only be measured relative to a frame of reference fixed in space. This can be done by making use of the transformation relations between two coordinate systems. 

Similarly, for the contravariant convected strain tensor we have... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

Similarly, the components of the convected contravariant strain tensor F i may be defined as... 

We have shown how the strain tensors in the spatial coordinates may be obtained from those in the convected coordinates,... 

Having defined strain tensors in convected coordinates, we now describe the rate-of-strain (or rate-of-deformation) tensor. This may be obtained by taking the derivative of a strain tensor with time, with the convected coordinates held constant. Such a derivative is commonly referred to as the material derivative, which may be considered as the time rate of change as seen by an observer in a convected coordinate system. Using the notation D/Dr for the substantial (material) time derivative, we have from Eq. (2.86)... 

Let us consider the upper convected Maxwell model given by Eq. (3.4). For steady-state uniaxial elongation flow, for which the rate-of-strain tensor d is defined by Eq. (2.15), we have (see Appendix 3E)... 

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. 

Consequently, the stretching tensor and the convected rate of spatial strain are indifferent, but the spin tensor is not, involving the rate of rotation of the coordinate frame. From (A.24) and (A.26)... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

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