Stretch tensor

For some purposes, it is convenient to express the constitutive equations for an inelastic material relative to the unrotated spatial configuration, i.e., one which has been stretched by the right stretch tensor U from the reference configuration, but not rotated by the rotation tensor R. The referential constitutive equations of Section 5.4.2 may be translated into unrotated terms, using the relationships given in the Appendix. 

It is also possible to find a relation between the stretching tensor d and the rate of spatial strain e. Differentiating (A. 19)... 

The stretch tensor is not indifferent but invariant under a rotation of frame. Taking the material derivative and the transpose of the first of these, and using the results in (A.23)... 

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)... 

In fig. 10.32, we indicate the type of interface we have in mind. On one side of the interface, the structure is characterized by a deformation gradient RiUi while on the opposite side of the interface the deformation is characterized by a deformation gradient U2. Note that without loss of generality, we have rewritten the compatibility condition in terms of the stretch tensors Ui and U2 and a single rotation Ri. For a given set of structures characterized by the matrices Ui and U2 our question may be posed as can we find a rotation Ri as well as vectors a and n such that the condition... 

Equations (4.11) and (4.12) are called polar decomposition of the deformation gradient. Particularly, U is called the right stretch tensor and V is called the left stretch tensor. 

It can be proved that U is a symmetric and positive definite tensor, which is a measure of the local stretching (or contraction) of material at X. V is also is a symmetric and positive definite second-order tensor called the left stretch tensor, which is a measure of the local stretching (or contraction) of the material in the deformed configuration at x. R is a proper orthogonal tensor, that is, R R = I or detR = 1, where T means transpose, I is the identity tensor, and det is the determinant. 

Similarly, it can also be rewritten in terms of the left stretch tensor V as... 

Thus Aij is the stretching tensor or rate of strain and cuy is the rotation or vorticity tensor and are illustrated below. 

R is referred to as the rotation tensor, U the right stretch tensor, V the left stretch tensor. R is orthonormal R R = /, RR = i), which gives the rotation of C and B to their principal axes. Under the polar decomposition we have... 

The velocity gradient L is decomposed into its symmetric part D, called the stretch tensor or rate-of-deformation tensor, and its anti-symmetric part W, called the spin tensor ... 

Note 2.8 (Solid and fluid). The term solid is used for the material body where the response is between the stress a and the strain e or between the stress increment da and the strain increment de. The term fluid is used for the material body where the response is between the stress a and the strain rate k (or the stretch tensor D). For a fluid we have to introduce a time-integration constant, which is referred to as the pressure ... 

The frame indifference of the stress Newtonian mechanics in that the force vector is frame indifferent. Since the stretch tensor D is frame invariant by (2.154), we use D instead of Vv. From (2.27) we have... 

Let us resolve the stress a and stretch tensor D into direct sums of volumetric and deviatoric components, respectively ... 

Thus the most fundamental constitutive law for a fluid is understood to be given as a Newtonian fluid deflning a linear relationship between the stress a and the stretch tensor D (recall that the stretch tensor D is equal to the strain rate for the solid with small strain). The constitutive law is also called Stokes law, and can be rewritten as... 

Since the stretch tensor D may be decomposed into a reversible (i.e., elastic) part and an irreversible (i.e., inelastic) part D D = D + D ), (3.26) is valid. However this decomposition is not unique. We do not discuss the details here (see Raniecki and Nguyen 2005). 

The velocity gradient L, the stretch tensor D and the spin tensor W are defined by... 

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