Finite strain tensor

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... 

Here B is the standard Finger strain tensor used in continuum mechamcs, and Yfo] = 8 — B IS a relative finite strain tensor, defined in DPL, Eq. D.3-4. We note in passing that it follows from Eq. (13.5) that the quantity HfkT) QQ P, dQ is equal to oi and thus satisfies Eq. (13.6). 

It is obvious that the infinitesimal strain tensor Sy is no longer adeqrrate and it is certainly not surprising that the appropriate replacement is the finite strain tensor Vy as defined in (3.20). 

Here we have introduced the finite strain tensor 17" which is defined in terms of... 

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... 

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

Therefore, the covariant components of the finite strain tensor are... 

Using the finite strain tensor E defined by Eq. (2.74) and the memory function m(t-t )... 

Large Scale Orientation - The Need for a Finite Strain Tensor... 

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. 

Note that the x, t in the arguments of A and E serve as a particle label Le. the fluid element located at position x at time t. Next, two finite strain tensors , widely used in continuum mechanics, can be defined... 

For some purposes it is more convenient to use two closely related relative (finite) strain tensors ... 

The scalar invariants of certain kinematic tensors play important roles in continuum mechanics, constitutive equations and kinetic theories. Of particular interest are the invariants of the rate of strain tensor and of the finite-strain tensors. There are many ways to define these invariants, and we give only those definitions that are used in later sections in addition, much research has been done on the definition of joint invariants of several tensors. ... 

An example of (d) is the memory-integral expansion , which results from a Frechet expansion of the stress tensor given as a general functional of the strain history the general term in this expansion is an n-fold integral involving all various n-tuple products of the relative finite strain tensors... 

It is interesting to note that the diffusion equation contains the transpose of the velocity gradient tensor, but the solution is given in terms of one of the relative finite strain tensors. The tensor a plays an important role in the changes of the thermodynamic functions that occur when a polymer solution goes from a state of equilibrium to a state of flow. The changes in internal energy and entropy are ... 

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