Stress tensor invariants

Approach to restoring of stresses SD in the three-dimensional event requires for each pixel determinations of matrix with six independent elements. Type of matrixes depends on chosen coordinate systems. It is arised a question, how to present such result for operator that he shall be able to value stresses and their SD. One of the possible ways is a calculation and a presenting in the form of image of SD of stresses tensor invariants. For three-dimensional SDS relative increase of time of spreading of US waves, polarized in directions of main axises of stresses tensor ... 

If the invariants are known for some arbitrary strain-rate state, then it is clear that the three equations above form a system of equations from which the principal strain rates can be uniquely determined. This analysis is explained more fully in Appendix A. Using the principal axes greatly facilitates subsequent analysis, wherein quantitative relationships are established between the strain-rate and stress tensors. 

The stress tensor represents the stress state at a point in a flow field. The nine particular numbers that comprise the tensor depend on the coordinate system in which the tensor is represented. However, the stress state itself is invariant to any particular coordinate-system representation. Thus, like all symmetric tensors, there are three certain invariants... 

Recall from the discussion in Section 2.5.4 that the stress tensor, like the strain-rate tensor, has certain invariants. For any known stress tensor, these invariant relationships can be used to determine the principal stresses. 

The sum of the diagonal elements is an invariant of the stress tensor. That is, regardless of the particular orientation of the coordinate system, or the coordinate system itself (e.g., cartesian versus cylindrical), the sum of the diagonal elements of the stress tensor is unchanged. From Eq. 2.180 it is easily seen that... 

Considering the Stokes hypothesis, evaluate the diagonal invariant of the stress tensor for this flow field. 

A close inspection of the right-hand side of eq. (2.34) reveals that it is simply equal to twice the excess free energy of the considered chain in a flowing system compared with that in a system at rest. By summing up the contributions of all chains per unit of volume, one obtains for the first invariant of the macroscopic stress tensor ... 

The stress tensor on a hydrodynamic volume element is obtained from macroscopic consideration and from rotational invariance and is given by [19]... 

IIIT Third scalar invariant of the stress tensor (2.6-4)... 

The quantities r and r] in equation (8.34) depend on the invariants of the tensor rik in accordance with equation (8.32). We ought to note that the behaviour of a non-linear viscoelastic liquid in a non-steady state would be different, if a dependence of the material parameters r and r] on the tensor velocity gradients or on the stress tensor is assumed. This is a point which is sometimes ignored. In any case, if r and r) are constant, equation (8.34) belongs to the class of equations introduced and investigated by Oldroyd (1950). 

Figure 19. The ratio of elongational to shear viscosities The theoretical dependence of the ratio of elongational to shear viscosity coefficients on the invariant of the additional stress tensor is calculated according to equation (9.71) and depicted by the dashed curve. The solid curves represent experimental data for systems listed in Table 3. Adapted from the paper of Pokrovskii and Kruchinin (1980).

Quantities C and C2 are functions of the two invariants of the stress tensor /1 and I2 for incompressible material. 

For the Maxwell field, the energy-momentum tensor Tfi(A) derived from Noether s theorem is unsymmetric, and not gauge invariant, in contrast to the symmetric stress tensor derived directly from Maxwell s equations [318], Consider the symmetric tensor 0 = T + AT, where... 

A slightly different set of invariants that corresponds to components of the stress tensor oriented to the material direction can be constructed from the above integrity basis. This new set of invariants includes... 

Similar to the ODF for texture, SODF can be subjected to a Fourier analysis by using generalized spherical harmonics. However, there are three important differences. The first is that in place of one distribution (ODF), six SODFs are analyzed simultaneously. The components of the strain, or the stress tensor can be used for analysis in the sample or in the crystal reference system. The second difference concerns the invariance to the crystal and the sample symmetry operations. The ODF is invariant to both crystal and sample symmetry operations. By contrast, the six SODFs in the sample reference system are invariant to the crystal symmetry operations but they transform similarly to Equation (65) if the sample reference system is replaced by an equivalent one. Inversely, the SODFs in the crystal reference system transform like Equation (65) if an equivalent one replaces this system and remain invariant to any rotation of the sample reference system. Consequently, for the spherical harmonics coefficients of the SODF one expects selection rules different from those of the ODF. As the third difference, the average over the crystallites in reflection (83) is structurally different from Equations (5)+ (11). In Equation (83) the products of the SODFs with the ODF are integrated, which, in comparison with Equation (5), entails a supplementary difficulty. 

Theories of yielding and flow expressed in terms of invariants of the applied stress tensor are examples of appropriate constitutive laws. The design of complex engineering components must consider the response of the component to realistic states of stress, which are inevitably multiaxial. Consequently, models of yielding and flow that accommodate such conditions must be employed in constructing the design. Descriptions of the structure of material that appear in such theories must likewise be expressed in tensorially invariant form. 

The stress tensor thus allows us to completely describe the state of stress in a continuum in terms of quantities that depend on position and time only, not on the orientation of the surface on which the stress acts. More precisely, the stress tensor should be referred to as a tensor of second order or tensor of second rank because its components transform as squares of the coordinates. We shall, however, simply use the term tensor, since tensors of order higher than second generally are not dealt with in fluid mechanics. We note in passing that a vector is a tensor of first order, its components transforming like the coordinates themselves, and a scalar is a tensor of zeroth order, a scalar being invariant under coordinate transformation. 

This is a statement of the product rule for the divergence of the vector dot product of a tensor with a vector, which is valid when the tensor is symmetric. In other words, r = r, where is the transpose of the viscous stress tensor. Synunetry of the viscous stress tensor is a controversial topic in fluid dynamics, bnt one that is invariably assumed. is short-hand notation for the scalar double-dot product of two tensors. If the viscous stress tensor is not symmetric, then r must be replaced by in the second term on the right side of the (25-29). The left side of (25-29), with a negative sign, corresponds to the rate of work done on the fluid by viscous forces. The microscopic equation of change for total energy is written in the following form ... 

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