Force Density

Tensile forces electrically induced in a charged-collagen membrane which supports a gradient in neutral salt concentration have also been measured [7], The observed force density of oriented collagen fiber was 1 N/cm2 within 5 min on application of an electric field with a current of 0.12 mA/cm2. This value was much smaller than the force density induced by an increase in NaCl concentration. 

Different factors contribute to the mechanical properties of plant tissue cell turgor, which is one of the most important ones, cell bonding force through middle lamella, cell wall resistance to compression or tensile forces, density of cell packaging, which defines the free spaces with gas or liquid, and some factors, also common to other products, such as sample size and shape, temperature, and strain rate (Vincent, 1994). Depending on the sample properties (mainly turgor and resistance of middle lamella), two failure modes have been described (Pitt, 1992) cell debonding and cell rupture. 

Examples of p(r) are energy density, charge density, current density (see Section 4.6), difference density (difference between a final density and an initial density), electric moment density, magnetic moment density, local reactivity functions (see Section 4.5.2), force density, etc. Note that, for ensuring the stability of matter, the net force density must vanish everywhere in space. The concept of a PDF has generated many significant developments in interpretative quantum chemistry. 

When A(r), as defined above, represents the B(r) magnetostatic field and when the relation V = 4 J/ /tB is verified, the magnetic force density is null... 

For a force density magnetic in origin, the magnetic stress tensor is written as... 

In two or three dimensions the answer is not entirely obvious. On the one hand, according to Proposition 2.1 the field induced by each particle at the surface of its counterpart is expected to saturate as the charge increases. In parallel, the surface charge density increases indefinitely, so that the electric surface force density on the particle (proportional to the field intensity multiplied by the surface charge density) may seemingly increase indefinitely, too. 

Within the gel, the force density acting on the network is given by... 

As a simple but nontrivial example we examine the dynamics of small fluctuations around anisotropic, homogeneous states by linearizing the basic equations (6.26) (6.29). From Eq. (4.39), the force density ) = — LidYlijjdXi is linearized as... 

By analogy with the standard 3D case, the 4D preonic fluid exerts force, and performs work along the four dimensions u,x,y,z, via its hydrodynamic pressure P. Thence, P is interpreted as potential energy per unit volume. The force density associated with the preons contained in a unit 3D volume is... 

Note that the spatial components of the equation of motion (69) directly represent force density in our 3D world. Also note that the right-hand side of Eq. (82) is independent of the explicit values of constants Ke and ke (as expected because charge is not a fundamental concept here). 

It is difficult to generalize concerning the flow characteristics of granular materials. Each material has its own transport characteristics based on cohesive and adhesive forces, density, size of the particles, shape and perhaps other unknown characteristics. For a material of a given composition, humidity represents a day to day or hour to hour variable which could supercede the other characteristics. For on-line sensing of granular materials, the main... 

Thus, the r and components of the velocity gradient are completely disregarded and Vq1 depends on those coordinates only through the r and dependence of the (radial) force density and the ground state shear velocity. The quality of this model increases with increasing kR. It is expected that the velocity perturbation is overestimated in this model and thus the hypothetical instability threshold is lowered, which makes the model appealing at least as a first attempt. 

Figure 1. Flotation of an elongated cylinder (fiber) by surface forces. Density of fiber greater than density of liquid. Cross sectional view of fiber at equilibrium in the air-liquid interface...

Laminar regime (Re < 102). For stirrers with very small wall clearances, such as the anchor or helical ribbon mixer, the laminar regime prevails for Rc < 100. In this regime, the viscous force dominates. The effect of inertial forces (density) is negligible, and thus the baffles are unnecessary. In this regime,... 

The first term on the right is interpreted as a force density and the second corresponds to a quantity like — f with pressure p and mass density p, or a function of an internal stress gradient rather than actual stress, as in hydrodynamics. 

The force may be equivalently expressed using Gauss s theorem as an integration of the force density — V r (r) over the basin of the atom,... 

The integration implied by dr in eqn (6,16) averages this force on the electron at r over the motions (i.e. positions) of all of the remaining particles in the system and the result is the force density F(r), the force exerted on the electron at r by the average distribution of the remaining particles in the total system. Integration of this force density over the basin of the atom 1 then yields the average electronic or Ehrenfest force exerted on the atom in the system. Even... 

The individual contributions to the electronic potential energy of an atom are obtained by determining the action of the operator — r-V on V in the expression for the basin virial, eqn (6.19). The basin virial is the viried of the Ehrenfest force and the force density F(r) of eqn (6.16) is evaluated as the first... 

Equation (8.178) has been previously obtained by Pauli (1958) and by Epstein (1975), who termed it the differential force law. The integration of this expression over the coordinates of all electrons but one by the usual recipe yields (Bader 1980) an equation governing a force density... 

As previously discussed in Chapter 6 in eqn (6.17) and following for a stationary state, the force density F(r, t) is the total instantaneous force exerted on the electron at Tj by the nuclei and the remaining electrons... 

The average of the atomic virial V,(12), eqn (8.192), as defined by the atomic variational principle, is the virial of the quantum mechanical force density as defined in the differential force law, averaged over the atomic volume. By taking the virial of F(r, t) in eqn (8.179), one obtains... 

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