Stress isotropy

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

A similar stress isotropy of the ground-state splitting is also observed for the other acceptors, but with different values of the ground state DPs bo and do-... 

Often in stress analysis we may be required to make simplified assumptions, and as a result, uneertainties or loss of aeeuraey are introdueed (Bury, 1975). The aeeuraey of ealeulation deereases as the eomplexity inereases from the simple ease, but ultimately the eomponent part will still break at its weakest seetion. Theoretieal failure formulae are devised under assumptions of ideal material homogeneity and isotropie behaviour. Homogeneous means that the materials properties are uniform throughout isotropie means that the material properties are independent of orientation or direetion. Only in the simplest of eases ean they furnish us with the eomplete solution of the stress distribution problem. In the majority of eases, engineers have to use approximate solutions and any of the real situations that arise are so eomplieated that they eannot be fully represented by a single mathematieal model (Gordon, 1991). 

For turbulent fluid-indueed stresses aeting on partieles it is neeessary to eon-sider the strueture and seale of turbulenee in relation to partiele motion in the flow field. There is as yet, however, no eompletely satisfaetory theory of turbulent flow, but a great deal has been aehieved based on the theory of isotropie turbulenee (Kolmogorov, 1941). 

If at every point of a material there is one plane in which the mechanical properties are equal in all directions, then the material is called transversely isotropic. If, for example, the 1-2 plane is the plane of isotropy, then the 1 and 2 subscripts on the stiffnesses are interchangeable. The stress-strain relations have only five independent constants ... 

We are interested in final states of stellar evolution. Therefore we can restrict ourselves to static configurations. Also, fluid-like behavior seems appropriate in the microscopical dimensions. Therefore we are looking for static configurations. Also, some fluid-like behavior is expected in the sense that stresses in the macroscopical directions freely equilibrate. Then in 3 spatial directions isotropy is expected and thence spherical symmetry. Finally, in the lack of any information so far, we may assume symmetry in the extra dimension. Then in... 

Developing the stress-strain-rate relationships is greatly facilitated in the principal coordinate directions. Since isotropy requires that the constitutive relationships be independent of coordinate orientation, the principal-direction relationships can be transformed to any other coordinate directions. At every point in a flow field the strain-rate and stress state... 

The borehole is assumed to be infinitely long and inclined with respect to the in-situ three-dimensional state of stress. The axis of the borehole is assumed to be perpendicular to the plane of isotropy of the transversely isotropic formation. Details of the problem geometry, boundary conditions and solutions for the stresses, pore pressure and temperature are available in [7], The solution is applied to assess the thermo-chemical effects on stresses and pore pressures. Both the formation pore fluid and the wellbore fluid are assumed to comprise of two chemical species, i.e., a solute fraction and solvent fraction. The formation material properties are those of a Gulf of Mexico shale [7] given as E = 1853.0 MPa u = 0.22 B = 0.92 k = 10-4 md /r = 10-9 MPa.s Ch = 8.64 x 10-5 m2/day % = 0.9 = 0.14 cn = 0.13824 m2/day asm = 6.0 x 10-6 1°C otsf = 3.0 x 10-4 /°C. A simplified example is considered wherein the in-situ stress gradients are assumed to be trivial and pore pressure gradients of the formation fluid and wellbore fluid are assumed to be = 9.8 kPa/m. The difference between the formation temperature and the wellbore fluid temperature is assumed to be 50°C. The solute concentration in the pore fluid is assumed to be more than that in the wellbore fluid such that mw — mf> = —1-8 x 10-2. 

For most particulate composites the mismatch between the particles and the matrix is more important than the anisotropy of either component (though alumina/aluminium titanate composites provide a notable exception and are described below). The main features of the stresses can therefore be understood in terms of a simple elastic model assuming thermoelastic isotropy and consisting of a spherical particle in a concentric spherical shell of matrix with dimensions chosen to give the appropriate volume fractions. The particles are predicted to be under a uniform hydrostatic stress, ap after cooling. If the particles have a larger thermal expansion coefficient than the matrix, this stress is tensile, and vice versa. For small particle volume fractions the stress... 

To close the problem, constitutive relations of powders must be introduced for the internal connections of components of the stress tensor of solids and the linkage between the stresses and velocities of solids. It is assumed that the bulk solid material behaves as a Coulomb powder so that the isotropy condition and the Mohr-Coulomb yield condition may be used. In addition, og has to be formulated with respect to the other stress components. 

The isotropy condition is that in the (R,

principal directions of stresses and the rate of deformation coincide [Jenike, 1961]. Thus, the isotropy condition is written as... 

Note that /3 and /4 are stress components in the plane of isotropy and, therefore, have the same Weibull parameters. The parameters i and /3i would be obtained from uniaxial tensile experiments along the material orientation direction, dt. The parameters a2 and /Efe would be obtained from torsional experiments of thin-walled tubular specimens where the shear stress is applied across the material orientation direction. The final two parameters, a3 and /33, would be obtained from uniaxial tensile experiments transverse to the material orientation direction. 

The requirement of isotropy permits the representation of the fourth-order tensor in terms of two material functions in such a way that the stress-strain relationship becomes... 

A near-uniform state of stress and strain within a briquette is more difficult to achieve with a roll press than with uniaxial compaction presses (either closed mold or extrusion) because of the more complicated geometry of the pressing chamber (nip plus briquette pockets). Homogeneity (but not necessarily isotropy) could be attained if either ... 

The sheet produced demonstrates enhanced superplasticity and isotropy at reduced temperatures of 650-750°C (Table 2). Elongation of 900-1000% was observed at 700-750°C and strain rate of 7xl0 4-7xl0 3/s. The value of flow stress achieved at 700-750°C and lower strain rates is typical to the... 

Substituting (2-78) into (2-76), we see that the most general constitutive equation for the total stress T that is consistent with the linear and instantaneous dependence of the deviatoric stress t on E, plus the assumption of isotropy, is... 

Given the apparent arbitrariness of the assumptions in a purely continuum-mechanics-based theory and the desire to obtain results that apply to at least some real fluids, there has been a historical tendency to either relax the Newtonian fluid assumptions one at a time (for example, to seek a constitutive equation that allows quadratic as well as linear dependence on strain rate, but to retain the other assumptions) or to make assumptions of such generality that they must apply to some real materials (for example, we might suppose that stress is a functional over past times of the strain rate, but without specifying any particular form). The former approach tends to produce very specific and reasonable-appearing constitutive models that, unfortunately, do not appear to correspond to any real fluids. The best-known example is the so-called Stokesian fluid. If it is assumed that the stress is a nonlinear function of the strain rate E, but otherwise satisfies the Newtonian fluid assumptions of isotropy and dependence on E only at the same point and at the same moment in time, it can be shown (see, e.g., Leigh29) that the most general form allowed for the constitutive model is... 

Isotropy Assumption. Isotropy of material properties implies radial symmetry of the envelope around the oq = a2 = a3 axis in principal stress space. The surface of the envelope is then entirely defined by its projection in a plane containing this particular axis and one of the principal stress axes. By... 

---