Material stretch tensor

The deformation gradient F contains not only information about the deformation, but also about rigid-body rotations of the material. These, however, do not contribute to the deformation itself, and the two contributions thus have to be separated. This can be done by considering the deformation gradient as a composition of a deformation U, called the right stretch tensor (or, sometimes, material stretch tensor) and a subsequent rotation R. These two are multiplied using the tensor product ... 

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. 

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)... 

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. 

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 ... 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

The measurement of the work needed to increase the surface area of a solid material (e.g., an electrode metal) is more difficult. The work required to form unit area of new surface by stretching under equilibrium conditions is the surface stress (g1 ) which is a tensor because it is generally anisotropic. For an isotropic solid the work, the generalized surface parameter , or specific surface energy (ys) is the sum of two contributions ... 

As mentioned above, the tensor E contains information about both the stretching and the rotation of a material element. Yet, if a material element is rotated only and not stretched, no... 

The tensor E E, called the Piola tensor (Astarita and Marrucci 1974), is closely related to B. In an extensional deformation, E E is exactly equal to B. B, a symmetric tensor, contains information about the orientation of the three principal axes of stretch and about the magnitudes of the three principal stretch ratios, but no information about rotations of material lines that occurred during that deformation. Thus, for example, from the Finger tensor alone, one could not determine whether a deformation was a simple shear (which has rotation of material lines) or a planar extensional deformation (which does not). The Finger tensor B(r, f) describes the change in shape of a small material element between times t and t, not whether it was rotated during this time interval. 

The return mapping techniques in inelastic solutions are a natural consequence of splitting the total strain into elastic and inelastic strains. Let tensor uy, an incremental field to describe the deformation, and its gradient, Vt/,y, show the deformation rate. The solution is implemented by the following steps. Step 1 introduces a loading condition such as F = (/, 4- V ,t)Fj." where ly is the unity second-rank tensor and the superscripts n and n 4-1 represent, respectively, the previous and current load steps. In step 2 the material is elastically stretched... 

Piezoelectric Constitutive Relationships. The constitutive relationships that describe piezoelectric behavior in materials can be derived from thermodynamic principles (4). A tensor notation is adopted to identify the coupling between the various entities through mechanical and electrical coefficients. The common practice is to label directions as depicted in Figure 2. The stretch direction... 

Now let the body be deformed to a new state as shown in Figure 1.4.1. Because the points P and Q move with the material, the small displacement between them will stretch and rotate as indicated by the direction and magnitude of the new vector dx. Somehow we need to relate dx back to did. Another tensor to the rescue The change in dx with respect to dx is called the... 

These remarks about reaching a steady state apply not only to uniaxial extensional flows, data for which appear in Figure 4.2.5, but for other extensional flows as well. Besides uniaxial extension, the two most important extensional flows are equal biaxial extension and planar extension. Kinematic tensors for these extensional flows were to have been found in Exercise 2.8.1. In uniaxial extension the material is stretched in one direction and compressed equally in the other two in equal biaxial extension the material is stretched equally in two directions and compressed in the third and in planar extension the material is stretched in one direction, held... 

The upper-convected time derivative is a time derivative in a special coordinate system whose base coordinate vectors stretch and rotate with material lines. With this definition of the upper-convected time derivative, stresses are produced only when material elements are deformed mere rotation produces no stress (see Section 1.4). Because of the way it is defined, the upper-convected time (teriva-tive of the Finger tensor is identically zero (see eqs. 2.2.3S and 1.4.13) ... 

Here, b/br is called the convected derivative due to Oldroyd (1950), and it is the fixed coordinate equivalent of the material derivative of a second-order tensor referred to in convected coordinates. The physical interpretation of the right-hand side of Eq. (2.104) may be given as follows. The first two terms represent the derivative of tensor a j with time, with the fixed coordinate held constant (i.e., Da /Dr), which may be considered as the time rate of change as seen by an observer in a fixed coordinate system. The third and fourth terms represent the stretching and rotational motions of a material element referred to in a fixed coordinate system. This is because the velocity gradient dv fdx (or the velocity gradient tensor L defined by Eq. (2.59)) may be considered as a sum of the rate of pure stretching and the material derivative of the finite rotation. For this reason, the convected derivative is sometimes referred to as the codeformational derivative (Bird et al. 1987). 

Biaxial extension is another example of a nonviscometric deformation. It can be achieved, for example, by simultaneously stretching a sheet of material in its length and width directions or by blowing up a balloon. It plays a prominent role in the blow-molding process (Section 19.4). Show the components of the deviatoric stress and the rate-of-strain tensors for the biaxial extension of an incompressible material. What material function(s) would be needed to describe e equilibrium biaxial extension of a homogeneous fluid ... 

In this flow field, there is no shear deformation, and the total stress tensor as well as the extra stress tensor are diagonal. As a consequence, there are only three nonzero stress components, but, due to fluid incompressibihty, we can measure only two stress differences. Further, in uniaxial extension, the two directions that are perpendicular to the stretching direction are identical, so there is only one measurable material function the net tensile stress which is the difference 11 22 or Tn — T22-... 

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