Stress tensor three-dimensional

Intercalation-induced stresses have been modeled extensively in the Hterature. A one-dimensional model was proposed to estimate stress generation in the lithium insertion process in the spherical particles of a carbon anode [24] and an LiMn204 cathode [23]. In this model, displacement inside a particle is related to species flux by lattice velocity, and total concentration of species is related to the trace of the stress tensor by compressibihty. Species conservation equations and elasticity equations are also included. A two-dimensional porous electrode model was also proposed to predict electrochemicaUy induced stresses [30]. Following the model approach of diffusion-induced stress in metal oxidation and semiconductor doping [31-33], a model based on thermal stress analogy was proposed to simulate intercalation-induced stresses inside three-dimensional eUipsoidal particles [1]. This model was later extended to include the electrochemical kinetics at electrode particle surfaces [2]. This thermal stress analogy model was later adapted to include the effect of surface stress [34]. 

Approach to restoring of stresses SD in the three-dimensional event requires for each pixel determinations of matrix with six independent elements. Type of matrixes depends on chosen coordinate systems. It is arised a question, how to present such result for operator that he shall be able to value stresses and their SD. One of the possible ways is a calculation and a presenting in the form of image of SD of stresses tensor invariants. For three-dimensional SDS relative increase of time of spreading of US waves, polarized in directions of main axises of stresses tensor ... 

A stress that is describable by a single scalar can be identified with a hydrostatic pressure, and this can perhaps be envisioned as the isotropic effect of the (frozen) medium on the globular-like contour of an entrapped protein. Of course, transduction of the strain at the protein surface via the complex network of chemical bonds of the protein 3-D structure will result in a local strain at the metal site that is not isotropic at all. In terms of the spin Hamiltonian the local strain is just another field (or operator) to be added to our small collection of main players, B, S, and I (section 5.1). We assign it the symbol T, and we note that in three-dimensional space, contrast to B, S, and I, which are each three-component vectors. T is a symmetrical tensor with six independent elements ... 

For a three-dimensional body, discussions of elastic responses in the framework of Hooke s law become more complicated. One defines a 3 x 3 stress tensor P [12], which is the force (with emits of newtons) expressed in a Cartesian coordinate system ... 

This was an introduction to rheology. If you want to go further, you will find that it is a difficult subject. For three-dimensional problems you need sophisticated mathematics to get anywhere. You will need tensor equations the three stresses Xxy, Xyx and Xyy that you have seen are three of the nine components of the stress tensor (Figure C4-17). And that is just... 

The constitutive equation for a dry powder is a governing equation for the stress tensor, t, in terms of the time derivative of the displacement in the material, e (= v == dK/dt). This displacement often changes the density of the material, as can be followed by the continuity equation. The constitutive equation is different for each packing density of the dry ceramic powder. As a result this complex relation between the stress tensor and density complicates substantially the equation of motion. In addition, little is known in detail about the nature of the constitutive equation for the three-dimensional case for dry powders. The normal stress-strain relationship and the shear stress-strain relationship are often experimentally measured for dry ceramic powders because there are no known equations for their prediction. All this does not mean that the area is without fundamentals. In this chapter, we will not use the approach which solves the equation of motion but we will use the friction between particles to determine the force acting on a mass of dry powder. With this analysis, we can determine the force required to keep the powder in motion. 

In Sections 1.3.1 and 1.3.2, we discussed the shear stress and the extensional stress in shearing flows and extensional flows, respectively. These are components of the three-dimensional state-of-stress tensor T. The ith row of T is the force per unit area that material exterior to a unit cube exerts on a surface perpendicular to the tth coordinate axis (see Fig. 1-18). In general, if F is the force per unit area acting on a surface perpendicular to an arbitrary outward-directed unit vector, n, then... 

Regarding the use of Eq. (6.26) in practice we note that the same comments made earlier apply here as well [see discussion after Eq. (6.17)]. A detailed discussion of optimal choices for the Ewald parameters a and for dipolar systems can bo found in Rc fs. 243 and 244. Finally, readers who are interested in performing MD simulations of dipolar fluids are referred to Appendix F.2.2 where we present explicit expressions for forces and torques associated with the three-dimensional Ewald sum [see Eq. (6.26)]. Moreover, explicit expressions for various components of the stress tensor can be found in Appendix F.2.3. 

Regarding the stress tensor of the system, the (Coulomb) components corresponding to the two orthogonal directions parallel to the walls (i.e., 7 = x,y) can be calculated exactly as in the three-dimensional case (see Appendix F.1.2.2). On the other hand, the normal component (7 = z) is given by... 

Timing next to the stress tensor we realize that its normal component within the slab-adapted three-dimensional Ewald formahsm can be written as a sum of two contributions, namely... 

Since the early 1980 s, Princen s work was continued by several other authors, e.g., by Reinelt [1993]. The latter author considered theoretical aspects of shearing three-dimensional, highly concentrated foams and emulsions. Initially, the structure is an assembly of interlocked tetrakaid-ecahedra (which have six square surfaces and eight hexagonal ones). An explicit relation for stress tensor up to the elastic limit was derived. When the elastic limit is exceeded, the stress-strain dependence is discontinuous, made of a series of increasing parts of the dependence, displaced with a period of y = 2. ... 

Cauchy generalized the Hooke law to a three-dimensional elastic body and stated that the six components of stress are hnearly related to the six components of strain. The law can be written in a tensor form as... 

The formulation described above is one dimensional and expressed in terms of a shear stress. It is possible to obtain a three-dimensional representation. Indeed this was done in the original paper of Boltzmann [B26]. Equation (21) may be written in terms of a stress tensor [Pg.253]

A three-dimensional formulation of the Bingham plastic was developed by Hohenemser and Prager [H19] in 1932 using the von Mises yield criterion (see Reiner [R4] and Prager [P15]). This employs a deviatoric stress tensor T and has the form... 

The coupling factor between electrodynamics and translational mechanics is not classically used as such but as a piezoelectric voltage coefficient g (in m C ) divided by a characteristic length. In an anisotropic three-dimensional material, this coefficient is a tensor that links the stress F a to the electric field E and is equivalent to the multiplication of the coupling factor with the spatial integration of the stress (i.e., the lineic density of the force) ... 

The knowledge of the stress field enables us to characterize and optimize the use of an impeller for a defined operation. In the three dimensional case, the stress components form a second order tensor, the symmetry of which allows it to be reduced to six components. These six components are the three normal stresses x, Tjj and x and the three shear stresses, x, x and x. . Figures 7a, 7b, and 7c give, respectively, the dimensionless values of x, of x. j and x (noted x, x and x ) in the vertical plane 0 = 3°, with ... 

Purely viscous constitutive equations, which account for some of the nonlinearity in shear but not for any of the history dependence, are commonly used in process models when the deformation is such that the history dependence is expected to be unimportant. The stress in an incompressible, purely viscous liquid is of the form given in equation 2, but the viscosity is a function of one or more invariant measures of the strength of the deformation rate tensor, [Vy - - (Vy) ]. [An invariant of a tensor is a quantity that has the same value regardless of the coordinate system that is used. The second invariant of the deformation rate tensor, often denoted IId, is a three-dimensional generalization of 2(dy/dy), where dy/dy is the strain rate in a one-dimensional shear flow, and so the viscosity is often taken to be a specific function-a power law, for example-of (illu). ]... 

All the relevant interfacial quantities can be expressed as integrals over the three-dimensional pressure tensor P of the interface regarded as a three-dimensional body. The pressure tensor is simply the negative of the three-dimensional mechanical stress tensor [Pg.588]

Equations 9.8 and 9.10b are written for rectilinear flows and infinitesimal deformations. We need equations that apply to finite, three-dimensional deformations. Intuitively, one might expect simply to replace the strain rate dy/dt by the components of the symmetric deformation rate tensor (dv /dy -y dvy/dx, etc.) to obtain a three-dimensional formulation, as in Section 2.2.3, and dr/dt by the substantial derivative D/Dt of the appropriate stress components. The first substitution is correct, but intuition would lead us badly astray regarding the second. Constitutive equations must be properly invariant to changes in the frame of reference (they must satisfy the principle of material frame indifference), and the substantial derivative of a stress or deformation-rate tensor is not properly invariant. The properly invariant... 

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