Deviatoric strain

Volumetric strain First invariant of strain Magnitude of volumetric strain Deviatoric strain... 

From (A.81), /3T, = k, and this equation implies that the yield surface in stress space is a circular cylinder of radius k, shown in a FI plane projection in Fig. 5.7(a). The corresponding yield surface in strain space may be obtained by inserting the deviatoric stress relation (5.86) into the yield function (5.92)... 

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

A first-order square to rectangle transition, with deviatoric strain order parameter B, = e2 positive (negative) for rectangle along the x (y) direction. The invariant polynomials up to sixth power in the order parameter can be written in scaled form [7,8] as ... 

Figure 111 3 1. a) Single cuprate pemton greyscale plot of deviatoric strain t,(k) - in Fourier space, b) Multipemton greyscale deviatoric strain < ( ) plot in coordinate space for hole-doped mobile charges fraction x = 0.1... 

Figure 111 3 2. a) For the twinned parent compound, x=0, probability of finding a deviatoric strain (r) versus the strain, b) Similar plot forx=0.1 multipemton strain... 

Now that we have discussed the geometric interpretation of the rate of strain tensor, we can proceed with a somewhat more formal mathematical presentation. We noted earlier that the (deviatoric) stress tensor t related to the flow and deformation of the fluid. The kinematic quantity that expresses fluid flow is the velocity gradient. Velocity is a vector and in a general flow field each of its three components can change in any of the three... 

We next introduce the small strain tensor ey and its deviatoric part... 

Quested et al. [16] have conducted an extensive experimental program on the stress-strain behavior of the elastomer solithane while subjected to an ambient at high pressure. Some of their experimental results are reproduced in Fig. 13. (Note that the reported stress is the deviatoric, not the total, stress as observed from the fact that the reported stress is zero for X = 1 for the various imposed ambient pressures). For the classic ideal affine network model (all stress caused by ideal Nc Gaussian chains in a volume v with no nonbonded interactions)... 

Stress-strain experiments in an elastomer with controlled variation of p by application of high ambient pressures have been performed by Quested et al. [16]. They demonstrate clearly the sensitivity of the deviatoric stress to p and therefore testify to the importance of nonbonded interactions in its production. 

In analogy with the strain, it is possible to express the stress tensor as the sum of a dilatational component, and a deviatoric component, that is,... 

When an elastic body is under the effect of a hydrostatic pressure, both the strain and stress deviatoric tensors are zero. Owing to the fact that in this case Yii = Y22 = Y33 < 11 = < 22 = < 33, Eq. (4.85) becomes... 

The decomposition in deviatoric and dilatational components of both the stress and strain tensors are... 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

To proceed beyond the general relationship (2-69), it is necessary to make a guess of the constitutive behavior of the fluid. The simplest assumption consistent with (2-69) is that the deviatoric stress (at some point x) depends linearly on the rate of strain at the same point in space and time, that is,... 

To obtain more information, stress-whitened zones were prepared at Section B. For all the rubber-modified specimens, the size of the stress-whitened zone increased from zero at the outer surface to a maximum at the midsection (Section A). Figure 5 shows the whitened zone at Section B of specimen RF5. The whitened zone of RF series specimens can be three-dimensionally visualized, as shown in Figure 6. The shape of the whitened zone is in opposition to that of the plastic zone ahead of a crack tip in dense materials, in which the size of the plastic zone decreases to a minimum at the midsection because of the state of plane strain. At the outer surface there will always be plane stress, and hence the stress in the thickness direction, a, is zero at the surface. Concurrently, plane strain prevails in the interior, thus increasing the a in the interior. It can accordingly be seen that the maximum hydrostatic stress is found at the midsection (Section A). Thus, stress-whitening appears to be due to the hydrostatic stress components rather than the deviatoric stress components. 

The technical difference between deviatoric and extra stress, and correspondingly the differences between the pressure in equation (2-9) and the thermodynamic pressure are important for describing compressible materials at large strains. By restricting the discussion to small strains and incompressible materials, we can avoid these complications. 

A more fundamental approach is to consider the rheological properties with dynamical properties. For a given rate-of-strain tensor E and moments of the orientation vector p, Batchelor (110) derived an expression for the bulk average deviatoric stress a for a suspension of non-Brownian fibers of large aspect ratio given by... 

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