Stress invariants

Thermal conductivity Mass density Isochoric specific heat Tensile yield stress Mean stress First stress invariant Principal stresses... 

Figure 3. Variations of stress invariant and mineralization degree of shale with (a) drilling fluid density and b) well inclination (after Sun et...

The mechanical model is expressed in terms of the following stress invariants and suction ... 

Where c is the cohesion, is the friction angle in compression path, po is the pre-consolidation pressure, which defines the size of the yield surface, and ni is a coefficient introduced to take into account the effect of the third stress invariant. 

This model is based on the mean features of the Mohr-Coulomb model and is expressed with stress invariants [Maleki (1999)] instead of principal stresses. Until plasticity is reached, a linear elastic behaviour is assumed. It is fully described by the drained elastic bulk and shear moduli. The yield surface of the perfectly plastic model is given by equation 7. Function 7i(0) is chosen so that the shape of the criterion in the principal stress space is close to the Lade criterion. 

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 ... 

Stresses transform from one coordinate-axis system to another according to well-defined transformation laws that utilize direction cosines of the angles of rotation between the final and initial coordinate-axis systems. Matrixes that obey such transformation laws are referred to as tensors (McClintock and Argon 1966). There are three sets of stress relations that are scalar and invariant in coordinate-axis transformations. The first such stress invariant of particular interest is the mean normal stress o- , defined as. 

Isotropic hyperelastic materials For this model, the strain energy density function is written in terms of the principal stress invariants I, h, h). Equation 1 becomes... 

The components of the stress tensor usually change with the reference coordinates, but there are functions of these components that do not change. These functions are known as stress invariants . The terms in the parentheses of Eq. (1.22d) may be rewritten in terms of such invariants as ... 

The symbol I represents the strain invariants analogous to the stress invariants given as J in Eqs. (1.22e) and (1.23). The coefficients in Eq. (1.98c) are the results of the engineering shear strain being ... 

It may be noted that n a is the first stress invariant Ij and sjjg Pg is, where J2 is the second invariant of the deviatoric stress tensor. 

The sum of the three principle stresses is known as the first stress invariant (or trace), I, ... 

The boundary of the crazed region coincided to a good approximation with contour plots showing lines of constant major principal stress 0, as shown in Figure 12.14(b) where the contour numbers are per unit of applied stress. At low applied stresses it is not possible to discriminate between the contours of constant cTi and contours showing constant values of the first stress invariant I = Oi+ a. However, the consensus of the results is in accord with a craze-stress criterion based on the former rather than on the latter and, as we have seen, the direction of the crazes is consistent with the former. 

Voorhees analysis assumes that the creep-rupture life of a vessel under complex stressing is controlled by an equivalent stress, J, termed the shear-stress invariant. This average stress is also known as the octahedral shear stress, the effective stress, the intensity of stress, and the quadratic invariant. The theory for the biaxial-stress condition was developed by Von Mises (205), and this theory was further developed to apply to the triaxial-stress condition independently by Hencky (206, 207, 208) and by Huber (209). A derivation of the relationship between the equivalent stress, /, and the three principal stresses, /i, /2, and/s where /i > /2 > /s was given by Eichinger (210). The relationship between these stresses is ... 

Voorhees analyzed the experimental data obtained on notched-bar samples tested at elevated temperatures (211, 214) and on pressure vessels tested at elevated temperatures and high pressures (204, 211), He concluded that the equivalent stress, / (shear-stress invariant) was more useful in correlating these experimental data than the maximum principal stress or the maximum shear stress. Voorhees also reviewed the lirork of other investigators in this lietd and condudefl that these studies alim indicated the u ful-ness of the equivalent stress (211). 

Define a stress invariant and give the proper expression for the first invariant of stress. 

This general expression first accounts for the principal of causality by stating that the state of stress at a time t is dependent on the strains in the past only. Secondly, by using the time dependent Finger tensor B, one extracts from the fiow fields only those properties which produce stress and eliminates motions like translations or rotations of the whole body which leave the stress invariant. Equation (7.128) thus provides us with a suitable and sound basis for further considerations. 

Fig 4. Second stress invariant distribution, at the beginning of the discharging (left), and after stabilization of the process (right), vertical cross-section through the domain. 

Frequently the high shear rate region is not observed, and is set to zero in eq. 2.4.14. Such a model fits the polymer melt data in Figure 2.4.1 quite well. This three-parameter version is often called the Ellis model. The Ellis model, however, is usually written in terms of the stress invariant... 

Stress invariant criteria are based on the value of the hydrostatic stress ajj and the second invariant of the stress deviator... 

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