Cauchy strain

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. 

L /polymer extensibility smectic-layer compressive modulus E E, Finger strain tensor B , Cauchy strain tensor yriso/r, capillary number characteristic ratio, defined by R )q — Ccotib translational diffusivity-------------------------... 

FIGURE 17.3 Apparent and true deformation parameters of a test piece (of semihard cheese) under compression (as in Figure 17.Id), a is stress, e is strain. Apparent stress is force over initial cross-sectional area apparent strain is change in height over initial height (i.e., Cauchy strain). 

FIGURE 17.30 Mechanical properties of baked cellular products, (a) The Young (compression) modulus En as a function of product density (p of wafers kept in air of various water activities (indicated near the curves). (After results by G. E. Attenburrow et al. J. Cereal Sci. 9 (1989) 61.) (b) Compression and decompression (indicated by arrows) of the crumb of a model wheat bread (p/pm 0.35), giving the stress a versus the Cauchy strain c- Complete densification would presumably be reached at c 0.7. The solid line shows the first compression and decompression, the broken line the second compression. The bread was 4 hours old. (c) Same experiment, same bread, but now 7 days old. (After results by C. J. A. M. Keetels et al. J. Cereal Sci. 24 (1996) 15.)... 

The first one is the small strain definition, or Cauchy strain tensor. Equation (4.1) can be rewritten in tensor form as... 

Therefore, for small or no rotation, the Cauchy strain can also be used to measure a large strain such as in uniaxial tension or compression testing of specimens where no rotation is involved. In other words, the Cauchy strain is not limited to small stretch it is limited to small rotation. 

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... 

Cauchy strain tensor (Eq. (9.40)) stress-optical coefficient (Eq. (9.194)) fractal dimension of a polymer chain thickness of the i-layer... 

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