The Cauchy Strain Tensor

Elasticity theory for solids is formulated with the assumption that strains remain small. One then finds the linear relations between stress and strain as they are described by Hook s law. For rubbers, deformations are generally large and the linear theory then becomes invalid. We have to ask how strain can be characterized in this general case and how it can be related to the applied stress. 

let us recall the definition of stress. In deformed rubbers the state of stress can be described in the same manner as in the case of small deformations of solids, by giving the stress tensor a = The meaning of the components 

A deformation of a body of rubber displaces all material points in the sample. There is a one-to-one correspondence between the locations of a material point in the deformed and the unstrained body. We again refer to the laboratory-fixed coordinate system and describe the relation between the locations in the deformed state 

Choosing this function rather than the reverse relation r(r ) implies that one refers to the deformed body in the description of the strain, in agreement with the description of the stress. 

As the relation between dr and dr is determined by the vector gradient of the mapping function r (r) 

Sometimes, in special cases, the stress tensor is calculated for the forces being referred to the cross-section in the undeformed state. For rubbers under a large strain, this leads to altered values and these, as mentioned earlier, are called 

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Rivlin-Ericksen tensor of order n, for a viscoelastic liquid or solid in homogeneous deformation, is the nth time derivative of the Cauchy strain tensor at reference time, t. Note 1 For an inhomogeneous deformation the material derivatives have to be used. 

Another combination of the displacement gradient tensors which are often used are the Cauchy strain tensor and the Finger strain tensor defined by B —1 = Afc A and B = EEt, respectively. 

Prediction of the second normal stress difference in shear and thermodynamic consistency obviously requires the use of a different strain measure including of the Cauchy strain tensor in the form of the K-BKZ model. With the ratio of second to first normal stress difference as a new parameter, Wagner and Demarmels [32] have shown that this is also necessary for accurate prediction of other flow situations such as equibiaxial extension, for example. 

Fig. 7.9. Notions used in the definition of the Cauchy strain tensor The material point at r in the deformed body with its neighborhood shifts on unloading to the position r. The orthogonal infinitesimal distance vectors dri and drs in the deformed state transform into the oblique pair of distance vectors dri and dr. Orthogonality is preserved for the distance vectors dra, drc oriented along the principal axes...

It is easy to see the significance of the various components of the Cauchy strain tensor, and we refer here to Fig. 7.9. First consider the infinitesimal differential vector dri parallel to the x-axis. On unloading it transforms into dr i, which has the squared length... 

Instead of employing the Cauchy strain tensor one can also utilize the Eulerian strain tensor defined as... 

As any second rank tensor, the Cauchy strain tensor possesses three invariants. These are expressions in terms of the tensor components Cii which remain invariant under all rotations of the coordinate system. The three invariants of the Cauchy strain tensor are given by the following expressions... 

We now have the ingredients to formulate Finger s constitutive equation. It relates the Cauchy strain tensor to the stress tensor in the form... 

There is an alternative form of Finger s equation which gives us a choice and is, indeed, to be preferred when dealing with rubbers. We introduce the Finger strain tensor B, being defined as the reciprocal of the Cauchy strain tensor... 

The Cauchy strain tensor is symmetric by definition. Therefore, it can be converted into a diagonal form by an appropriate rotation of the coordinate system. We deal with these conditions as indicated in Fig. 9.9, by attaching, to each selected material point, a triple of orthogonal infinitesimal distance... 

In the above description it was assumed that the boundary position does not change significantly with time which naturally restricts this discussion to infinitesimal strain theory. Hence the strain tensor in the constitutive equation will be given as the Cauchy strain tensor where... 

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