Cauchy stress

After the substitution of Cauchy stress via Equation (3.20) and the viscous part of the extra stress in terms of rate of defonnation, the equation of motion is written as... 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

It may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

The spatial Cauchy stress tensor s is defined at time by f = sn, where t(x, t, n) is a contact force vector acting on an element of area da = n da with unit normal i and magnitude da in the current configuration. The element of area... 

Therein, T" = (T )r are the symmetric partial Cauchy stress tensors, // b represent the body forces and p" are the momentum productions, where ps + pF = o must hold due the overall conservation of momentum. 

The volumetric constitutive equations for a chemoporoelastic material can be formulated in terms of the stress S = a,p, it and the strain 8 = e, (, 9, i.e., in terms of the mean Cauchy stress a, pore pressure p, osmotic pressure it, volumetric strain e, variation of fluid content (, and relative increment of salt content 9. Note that the stress and strain are measured from a reference initial state where all the stress fields are equilibrated. The osmotic pressure it is related to the change in the solute molar fraction x according to 7r = N Ax where N = RT/v is a parameter with dimension of a stress, which is typically of 0( 102) MPa (with R = 8.31 J/K mol denoting the gas constant, T the absolute temperature, and v the molar volume of the fluid). The solute molar fraction x is defined as ms/m with m = ms + mw and ms (mw) denoting the moles of solute (solvent) per unit volume of the porous solid. The quantities ( and 9 are defined in terms of the increment Ams and Amw according to... 

Here ma is the bulk solid-fluid interaction force, T.s the partial Cauchy stress in the solid, p/ the hydrostatic pressure in the perfect fluid, IIS the second-order stress in the solid, ha the density of partial body forces, ta the partial surface tractions, ts the traction corresponding to the second-order stress tensor in the solid and dvs/dn the directional derivative of v.s. along the outward unit normal n to the boundary cXl of C. 

As a reference to something more familiar, consider the case of a fluid where incompressibility is enforced via a Lagrange multiplier. For a Stokesian fluid, it is assumed that the constitutive variables (stress, energy, heat flux) are a function of density, p, temperature, T, rate of deformation tensor, d, and possibly other variables (such as the gradients of density and temperature). Exploiting the entropy inequality in this framework produces the following constitutive restriction for the Cauchy stress tensor [10]... 

The constitutive form of the Cauchy stress tensors for the solid and liquid phases are [5]... 

Therein, A denotes the interface pressure (whole pore pressure), p l the realistic pressure, T denotes the partial effective Cauchy stress tensor and I is the unit tensor. With the expression A = ns A + nL pLR + nG pGR (Dalton s law), the constrain... 

The deformation is assumed to result in chain slippage but some chain entanglement will influence the mechanical response. This feature is assumed to be lumped into the parameters q and m so that no back stress contribution appears above Tg. Therefore, the driving stress reduces to a = o and the equivalent shear stress x in (4) is that of the Cauchy stress o. 

Broadly speaking, our description of continuum mechanics will be divided along a few traditional lines. First, we will consider the kinematic description of deformation without searching for the attributes of the forces that lead to a particular state of deformation. Here it will be shown that the displacement fields themselves do not cast a fine enough net to sufficiently distinguish between rigid body motions, which are often of little interest, and the more important relative motions that result in internal stresses. These observations call for the introduction of other kinematic measures of deformation such as the various strain tensors. Once we have settled these kinematic preliminaries, we turn to the analysis of the forces in continua that lead to such deformation, and culminate in the Cauchy stress principle. 

Fig. 2.7. Tetrahedral volume element used to illustrate the Cauchy stress principle.

The Cauchy stress principle arises through consideration of the equilibrium of body forces and surface tractions in the special case of the infinitesimal tetrahedral volume shown in fig. 2.7. Three faces of the tetrahedron are perpendicular to the Cartesian axes while the fourth face is characterized by a normal n. The idea is to insist on the equilibrium of this elementary volume, which results in the observation that the traction vector on an arbitrary plane with normal n (such as is shown in the figure) can be determined once the traction vectors on the Cartesian planes are known. In particular, it is found that = crn, where a is known as the stress tensor, and carries the information about the traction vectors associated with the Cartesian planes. The simple outcome of this argument is the claim that... 

Cauchy Tetrahedron and Equilibrium In this problem, use the balance of linear momentum to deduce the Cauchy stress by writing down the equations of continuum dynamics for the tetrahedral volume element shown in fig. 2.7. Assign an area AS to the face of the tetrahedron with normal n and further denote the distance from the origin to this plane along the direction why h. Show that the dynamical equations may be written as... 

In the simplified case of a long cylinder (plane strain), as shown in Figure 4, the Cauchy stress tensor at point M can be written as ... 

In this constitutive framework, the total deformation gradient F is decomposed into viscoplastic and viscoelastic components F = F F. The viscoelastic deformation gradient acts on both the equilibrium network A, and on the time-dependent network F = F = F. The Cauchy stress acting on network A is given by the eight-chain representa-... 

The viscoelastic deformation gradient acting on network B is decomposed into elastic and viscous parts F = F F . The Cauchy stress acting on network B is obtained from the eight-chain network representation using the same procedure that was used for network ... 

The continuum mechanics modeled by JAS3D are based on two fundamental governing equations. The kinematics is based on the conservation of momentum equation, which can be solved either for quasi-static or dynamic conditions (a quasistatic procedure was used for these analyses). The stress-strain relationships are posed in terms of the conventional Cauchy stress. JAS3D includes at least 30 different material models. 

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