Strain deviatoric stress

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)... 

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... 

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. 

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... 

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. 

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... 

As shown in Figures 7a and 7b, the differed strain increases as the deviatoric stress increases. The differed strain remains measurable whatever the deviatoric stress level. It was not possible, until now, to determine a deviatoric-stress threshold below which there would be no differed strain. 

With regard to the influence of temperature, it is shown that the time-dependent behaviour may be enhanced by an increase in temperature (Figure 7a). At low deviatoric stress, temperature effects are discreet for an rise from 20°C to 50 °C in temperature. However, from 50°C to 120°C, an acceleration of the viscoplastic strain is observed obviously, whatever the level of deviatoric stress might be. 

Figure 7. Creep tests on the argillites. 7a) Effect of temperature and deviatoric stress, 7b) Strain rate versus deviatoric stress...

Figure 3. Yield curve and plastic strain rate vector in the plane deviatoric stress vs effective isotropic stress.

The current way to characterize the mechanical behaviour of soils or rocks consists in the study of the experimental load-displacement curve (F-d) or of the deviatoric stress-axial strain one (q-e). Local internal measurement (in the confining cell) of specimen axial displacement have been performed in order to exclude bedding errors at both ends of the specimen in contact with porous discs. In addition we can also use an external axial measurement to measure the piston axial displacement. In order to have a better precision axial displacement are measured using three miniature LVDT transducers, mounted at 120° from each other. The transducers are glued on the neoprene membrane in order to measure axial displacement in the central part of the specimen (figure 2). 

An experimental set up (high pressure triaxial cell and local strain measurement system) and an experimental procedure have been developed in order to quantify the permeability evolution with deviatoric stress on clays of very low permeability. Developed techniques have been improved on concrete and sandstone. We are able to detect with precision strain localization and we are able to measure with the pulse method, permeability, at different stages of loading. A testing program on both Boom Clay and Opalinus is in progress. 

This section incorporates the unpublished work of Palmer and Weaver subsequently the fatigue analysis was included as an integral part of the FMP Shaft Design Guide which Palmer and Weaver compiled. Results are quoted, for brevity the reader is referred to references dealing with the Distortion Energy Theory of Failure (also called deviatoric stress, octahedral, von Mises, or shear strain) for a complete analysis. 

Von Mises stress is originally formulated to describe plastic response of ductile materials. It is also applicable for the analysis of plastic failure for coal undergoing high strain rate. The von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J2 reaches a critical value. In materials science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress or equivalent tensile stress, a scalar stress value that can be computed from the stress tensor ... 

Creep is the long-term continuing deformation due to sustained deviatoric stress (od =Oi - Os) conditions that occurs as a function of time after dissipation of consolidation excess pore pressures. Thus, creep behavior of sediments is a function of the type of sediment, its physical properties, stress-strain history, and time. Mitchell (1976) distinguished between creep and secondary compression by noting that the former is referred... 

Figure 8.46c presents a schematic of the effect of increasing level of deviatoric stress (Od = CTi - Os) on the level of strain as a function of time. As the applied deviatoric stress (Od = CTi - Os) is increased, the ultimate deviatoric stress strain as a function of time increases until creep rupture occurs. The proportion of the total curve each stage represents is dependent on a number of factors. These factors are the material type, the SL used during the test, and to a lesser degree the specimen shape and test conditions. 

Just as it is useful to break down strain components into dilatational and devi-atoric parts, one can perform a similar procedure with stress. The hydrostatic stress components are and the deviatoric stress components,... 

In the physically separate and distinct representation of shear and volumetric response we introduce the concepts of the deviatoric stress and the deviatoric strain which are free of volumetric response and represent only the shear response. These are defined as ... 

With these deviatoric stress and strain responses, separating the volumetric response from the regular stresses and strain elements, the linear elastic constitutive relations become particularly simple and transparent as... 

The strain and stress tensors are separated into isotropic and deviatoric parts as follow ... 

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