Stress tensor magnitude

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... 

We can now identify the first term in (A.4) with Maxwell s stress tensor, which acts on any dielectric in an electric field. The magnitude of this force Pe is given by... 

It is interesting to examine the bead-spring models to see what flow-induced configurational changes would be required in order to develop N2 values of the proper magnitude and sign. In the Rouse model, the components of the stress tensor are related directly to averages of the internal coordinates of the beads. For the simplest case of the elastic dumbbell ... 

The CFD solution determines the pressure variation relative to a uniform thermodynamic pressure, p = pa + p. In this problem, pa = 10,000 Pa, and the sheet called relative p reports the local values of p. How does the pressure compare in magnitude to the other stress components What observations can be made about the effect of the pressure on the structure of the stress tensor ... 

Tensors that are proportional to 8 are sometimes called isotropic tensors. For an incompressible material, gradients of p, but not p itself, can affect fluid motion. Thus, a uniform isotropic tensor of arbitrary magnitude can be added to T (or or) without consequence to the flow behavior. Adding such an isotropic tensor is equivalent to adding a constant to each diagonal component of the stress tensor. Thus, if the fluid is incompressible, a is determined only up to an additive isotropic tensor, and the stress-free state is synonymous with the state of isotropic stress. 

If the cross-sectional area of the macroscopic network is doubled, then twice as large a force is required to obtain the same deformation. This leads naturally to a definition of stress as the ratio of force and cross-sectional area. Both the force and the cross-sectional are have direction and magnitude (the direction of the cross-sectional area being described by the unit vector normal to its surface), making the stress a tensor. The (/-component of the stress tensor is the force applied in the i direction per unit cross-sectional area of a network perpendicular to the j axis. For... 

The minus sign in this equation is a matter of convention t(n) is considered positive when it acts inward on a surface whereas n is the outwardly directed normal, andp is taken as always positive. The fact that the magnitude of the pressure (or surface force) is independent of n is self-evident from its molecular origin but also can be proven on purely continuum mechanical grounds, because otherwise the principle of stress equilibrium, (2 25), cannot be satisfied for an arbitrary material volume element in the fluid. The form for the stress tensor T in a stationary fluid follows immediately from (2 59) and the general relationship (2-29) between the stress vector and the stress tensor ... 

Time-dependent fluids are those for which the components of the stress tensor are a function of both the magnitude and the duration of the rate of deformation at constant temperature and pressure [4]. These fluids are usually classified into two groups—thixotropic fluids and rheopectic fluids—depending on whether the shear stress decreases or increases with time at a given shear rate. Thixotropic and rheopectic behavior are common to slurries and suspensions of solids or colloidal aggregates in liquids. Figure 10.2 shows the general behavior of these fluids. 

Consider a homogeneously stressed material and imagine a small surface area 55 somewhere within it (see fig. A.l). Let the direction of the vector 55 represent the normal to the area and let the length of 55 represent the magnitude of the area. The material on the side of 55 towards which the vector 55 points exerts a force on the material on the opposite side of 55. Let this force be 5f. Because the stress is homogeneous, 5F must be proportional to 55, but it must also depend on the direction of the normal to 55. Thus the two vectors 5F and 55 must be related by a second-rank tensor [ujfi, called the stress tensor, so that 5F, = 55,. (The symbol ct is conventionally used both for electrical... 

The magnitude of that appears in Eq. (17.11) may also be studied in a dynamic way as described in the following Eq. (17.11) represents the time-correlation function of the stress tensor component Jxy t) in the long-time region described by the Langevin equation ... 

The computed trial elastic stress is relaxed by the plastic-corrector method, as described in Table 5.1. In Table 5.1, Qg is the hardening tensor, 2 is the inelastic multiplier, which shows the magnitude of inelastic deformation, and fi is the shear modulus. The second-order tensor hy defines the direction of the inelastic flow for example, in the case of the associated formulation it is hy = dif//dffy, where the yield surface is = 0. The deviatoric stress and back stress tensors are, respectively, identified by Sy = [Pg.195]

The heat transfer rate from a flush-mounted shear stress tensor depends on the near-wall flow, i.e., the magnitude of the velocity gradient. For a laminar two-dimensional thermal boundary layer developing over the heated sensor with an approaching linear velocity profile (Fig. 2) and negligible free convectimi effect, the heat loss from the thermal element can be derived from the thermal boundary layer equation as... 

Coefficients aij T) depend explicitly on temperature T. Coefficients afj, aijki, are supposed to be temperature independent, constants giju and Vijkimn determine the magnitude of the gradient energy. Tensors gijki, Oijki and positively defined. Tensor is the surface excess elastic moduli, p, p is the surface stress tensor [81,82], is the surface piezoelectric or piezomagnetic tensor [67], qijki are the bulk striction coefficients Ciju are components of elastic stiffness tensor [83]. 

The change in stress state between 0.7 ps and 0.8 ps is caused by a change in the structure of the material. The transformation consists of a homogeneous rotation of half the SiOe octahedra that persists for the remainder of the simulation (Fig. 8). The new structure has a (—H-i-) pattern of octahedral rotation which corresponds to Pmmn symmetry. The change with respect to the Pbnm phase is more subtle than phase transformations that had been contemplated in previous theoretical and experimental work none of the three octahedral rotations vanish in the Pmmn structure and the magnitude of the octahedral rotations is similar to that in Pbnm. The increase in mean stress at the transition means that the Pmmn phase has a slightly larger volume than Pbnm at the same pressure. The anisotropy of the stress tensor reflects the differences in equilibrium axial ratios between the two phases. 

Total) stress tensor (tension +) First invariant of stress Second invariant of stress Third invariant of stress Volumetric stress tensor (tension, +) Magnitude of volumetric stress (tension, +)... 

There is only one commonly used invariant of a vector its magnitude. However there are three possible invariant scalar functions of a tensor. For the stress tensor we can give these three invariants physical meaning through the principal stresses. 

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