Dilatation tensor

As discussed in Section 7.7, crystals, particularly organic crystals, usually exist in lower-symmetry nonorthogonal systems for them the off-diagonal terms of P become important. The response of a crystal to the stress tensor P is a series of fractional displacements, small compared to any dimension of the body these fractional displacements are called strains and are denoted by the strain (or dilatation) tensor s ... 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

In order to reproduce the temperature variation of the lattice constants, the anisotropy of the lattice expansion has to be taken into account. For this purpose, the tensor of thermal expansion ot is introduced instead of the scalar a , and the tensor of deformation due to the HS <- LS transition is employed instead of the dilation (Fh — Fl)/Fl. Each lattice vector x T) can now be... 

Third, the metric tensor is determined by the variables 4>, //, A. On the other hand and v never appear in Eqs.(6)-(9) (reflecting the fact that x° and x5 constant dilatations are always possible without harming the commutator relations for the Killing motions), so these equations are of first order on 4>, / and A. However, the equations can be rearranged resulting in the following symbolic structure ... 

Stress and Strain Rate The stress and strain-rate state of a fluid at a point are represented by tensors T and E. These tensors are composed of nine (six independent) quantities that depend on the velocity field. The strain rate describes how a fluid element deforms (i.e., dilates and shears) as a function of the local velocity field. The stress and strain-rate tensors are usually represented in some coordinate system, although the stress and strain-rate states are invariant to the coordinate-system representation. 

As discussed in Appendix A, symmetric tensors have properties that are important to the subsequent derivation of conservation laws. As illustrated in Fig. 2.9, there is always some orientation for the differential element in which all the shear strain rates vanish, leaving only dilatational strain rates. This behavior follows from the transformation laws... 

Fig. 2.9 Because the strain-rate tensor is symmetric, there is always an orientation of a differential element for which the strain-rates are purely dilatational.

The relative volumetric expansion is seen to be the sum of the normal strain rates, which is the divergence of the vector velocity field. The sum of the normal strain rates is also an invariant of the strain-rate tensor, Eq. 2.95. Therefore, as might be anticipated, the relative volumetric dilatation and V V are invariant to the orientation of the coordinate system. 

Poisson s61 ratio a is the change in length per unit length, usually the contraction in one direction due to the dilatation in the perpendicular direction (for isotropic elastic bodies, —1.0 < a < + 0.5). Young s62 modulus Y, also called the linear modulus of elasticity, is the 3x3 tensor of the stress P divided by the strain s ... 

There are two general types of constitutive equations for fluids Newtonian and non-Newtonian. For Newtonian fluids, the relation between the stress tensor, t, and the rate of deformation tensor or the shear stress is linear. For non-Newtonian fluids the relation between the stress tensor and the rate of deformation tensor is nonlinear. The various Newtonian and non-Newtonian rheologies of fluids are shown in Figure 12.2. There are four types of behavior (1) Newtonian, (2) pseudo-plastic, (3) Bingham plastic, and (4) dilatent. The reasons for these different rheological behaviors will also be discussed in subsequent sections of this chapter. But first it is necessary to relate the stress tensor to the rate of deformation tensor. 

Negative pressure specifically. With subscripts c, e, i, m, P, ST, TH, oo craze traction, Mises equivalent, one of three principal stresses, maximum level of craze traction where cavitation in PB begins, negative pressure in particle, negative pressure due to one of three principal stresses, negative pressure due to thermal mismatch, uniaxial applied stress at the borders With subscripts xx, yy, zz etc. for components of the local stress tensor Ratio of slope of the falling to the rising part of the traction cavitation law Craze dilatation Time constant... 

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,... 

If the viscoelastic material is under the effect of an isotropic deformation (dilatation or compression), the diagonal components of both the stress and strain tensors differ from zero. In analogy with Eq. (4.92), the relationship between the excitation and the response is given by... 

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

To avoid these mathematical details and focus on the key concepts of tablet stress analysis this discussion will examine the simplest of viscoelastic models using the method outlined by Fluggie (97). To begin the analysis, the boundary conditions which apply to tablet compaction, will be used to set up the stress and strain tensors Equations (26) and (27). Then the dilation and distortion uations (28-31) will be used lo obtain dilation and distortion tensors. After obtaining the dilational and distortional stress and strain tensors, a Kelvin viscoelastic model will be used to relate the distortional stress to distortional strain and the dilational stress to dilational strain. 

To simplify the notation, the subscript yy will be written as xx since the radial directions are equivalent. Using tensor Equations (28) and (29) the dilation becomes. 

Recalling the dilational Equationfi (28) and (29) and the distortional Equations (30) and (31). These equations can be adjusted for stress relaxation boundary conditions shown in Equations (73) and (74). Thus, using the tensor Equations (73) and (74) and Equations (28) and (29) to compute the dilational and distortional stresses and strains. Once the dilational and distortional stresses and strains are computed, the shear and bulk modulus Equations (63) and (64) can be used to relate stress to strain yielding ... 

In addition, it is by no means clear that such a structure is totally rigid. That is, it is possible that dilation and/or contraction of domains may occur as a result of the pressure tensors within the pore system [21, 22]. 

The second type of work to which wc shall restrict our di.scussion is mechanical work. As wc shall see later, confined phases can be exposed to two types of mechanical work, namely compression (dilation) and shear. In that regard, confined phases have a lot in common with bulk solids in that they are generally inhomogeneous and anisotropic in one or more spatial dimensions. Therefore, it seems sensible to cast the mechanical work term in terms of stress (t) and dimensionless strain tensors (bulk solids [12], by writing... 

Here, ay is the thermal expansion coefficient which is assumed to be equal in both lattices and 6 is the dilatation coefficient (Khs — ls)/ ls> where Fls and Fhs are the unit cell volumes of the pure LS and HS species at 0 K, respectively. In order to account for the anisotropy of the lattice, the thermal expansion v and dilatation e coefficients must be introduced as tensors instead of scalars. Similarly, an equivalent expression could be defined for pressure-induced spin conversions. 

In fact, Equation 5.281 describes an interface as a two-dimensional Newtonian fluid. On the other hand, a number of non-Newtonian interfacial rheological models have been described in the literature. Tambe and Sharma modeled the hydrodynamics of thin liquid films bounded by viscoelastic interfaces, which obey a generalized Maxwell model for the interfacial stress tensor. These authors also presented a constitutive equation to describe the rheological properties of fluid interfaces containing colloidal particles. A new constitutive equation for the total stress was proposed by Horozov et al. ° and Danov et al. who applied a local approach to the interfacial dilatation of adsorption layers. 

The viscous properties of a liquid are defined in relation to the rate of distortion and dilatation of a local region R in response to stresses applied at the surface S which encloses it. In the analysis, we consider an element SS of the surface and construct a unit normal vector n directed outwardly from the surface. The applied stress can then be resolved into components tending to stretch the vector and components tending to rotate it. The force F per unit area can then be represented in terms of a tensor P(r) operating... 

---