Principle of virtual work

Equating Eqs. (3.38a) and (3.38b) and couching the relation of force and charge as of Eq. (3.37), the electric field strength E may be determined as a function of the electric potential (p after auxiliary expansion with the unit vector e. of the above-mentioned differential distance vector ds  

Examining a single component Ei of the electric field strength vector E, it may be described for a linear electric potential distribution between two points with the potentials (pa and ipi, and the distance vector component Si in the regarded direction by 

The stress formula (3.134) can be put in the form equivalent to the principle of virtual work. Consider a virtual deformation which displaces the point on the material to Tq. -I- de firp in a very short time dt. In the limit of dt- 0, the velocity gradient x p = dE pldt becomes very large, so that the time evolution of V in the time interval 6t is dominated by ticpA 

Let sA be the dynamical free energy of the system si = j diRj CkgT InV+U) 

The proof is straightforward. The diange in the dynamical free energy is 

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That all actual processes are irreversible does not invalidate the results of thermodynamic reasoning with reversible processes, because the results refer to equilibrium states. This procedure is exactly analogous to the method of applying the principle of Virtual Work in analytical statics, where the conditions of equilibrium are derived from a relation between the elements of work done during virtual i.e., imaginary, displacements of the parts of the system, whereas such displacements are excluded by the condition of equilibrium of the system. 

The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

As before, we obtain the expression for the mass law by application of the principle of virtual work to the system in equilibrium. If sc) be the concentration of hydroxyl ions corresponding to a concentration X of hydrogen, viz. ... 

A quasistatic finite strain analysis is based on the rate form of the principle of virtual work,... 

In principle, eqn (11.59) is all that we need to go about computing the interaction force between an obstacle and a dislocation. However, we will find it convenient to rewrite this equation in a more transparent form for the present discussion. Using the principle of virtual work, we may rewrite the interaction energy as... 

This is known as the chemical application of the Principle of Virtual Work For an account of Virtual Work m Mechanics, for example, see A, W Porter s book, Intermediate Mechanics... 

According to Newton, equilibrium is established when the resultant forces on the masses, i.e., in the direction of the respective planes, are equal. There is another formulation of equilibrium, however, due to the principle of virtual work. Equilibrium is established in this way, if a small, allowed displacement of the masses will not cause a change in energy. 

In order to move the atoms depicted in Fig. 2.3, work has to be performed on the system. For example, if one atom is moved by an amount 8m, the work done is / 8m, using the principle of virtual work. In doing this work, the bond energy has changed by an amount A=(f) u+bu)—(u). As indicated earlier, the equilibrium condition for the deformation is given by f—dapproaches zero, A =fhu. Thus, for small displacements, the work done on the atoms is equivalent to the change in bond energy. It is now possible to define a... 

The actual formula for the stress tensor is complicated. However, a neat expression is obtained in the form of the principle of virtual work. By a straightforward calculation, it can be shown ... 

Equation (4.137) can be derived from this potential by the principle of virtual work described in Section 3.7.5. Thus we shall use eqn (4.137) for both 0 and good solvents. 

To discuss the viscoelastic properties in semidilute solutions, we first consider the expression for the stress tensor. This is obtained from the principle of virtual work as in Section 8.6.1. 

As opposed to the described MEIS with variable parameters and the mechanisms of physicochemical processes in this case we will try to determine the objective function of applied model for a dissipative system based on the equilibrium principle of conservative systems, i.e. the Lagrange principle of virtual works. Derivation will be given on the example of the closed (not exchanging the fluid flows with the environment) active (with sources of motive pressures) circuit. The simplest scheme of such a circuit is presented in Fig. 3,a. A common character of the chosen example is explained by the easiness of passing to other possible schemes. For example, if at the modeled network nodes there are external... 

Let us determine the principle of virtual works for the hydraulic circuit as follows for every branch in any infinitesimal period of time an equilibrium of all forces (pressure, friction, gravity, etc.) acting on the fluid is observed and the work expended for infinitesimal deviation of flow on the branch from its equilibrium value is equal to zero. If only three forces are available on the i -th branch the force caused by the difference in pressures between the initial and final cross-sections AfJ-, the force of motive pressure (created by pump) p ° and the force of friction, the following equation corresponds to the Lagrange principle... 

The degree of wetting is characterized by the contact angle 0 that depends on the various interfacial energies for the solid-liquid-vapor system and is usually referred to a droplet of liquid on a flat solid surface (Fig. 10.4). If the specific energies of the liquid-vapor, solid-vapor, and solid-liquid inta-faces are yiv, 7sv, and yah respectively, then by the principle of virtual work... 

The VFM is based on the fundamental equations of solid mechanics the equilibrium equation, through the principle of virtual work (PVW), the constitutive equations and the strain-displacement relationships. For an arbitrary solid in equilibrium, the PVW can be written as... 

The equations of the Lagrangian incremental description of motion can be derived from the principles of virtual work (i.e., virtual displacements, virtual forces, or mixed virtual displacements and forces). Since our ultimate objective is to develop the finite-element model of the equations governing a body, we will not actually derive the differential equations of motion but utilize the virtual work statements to develop the finite element models. 

Since bonded joints can often undergo large displacements, especially when subjected to creep-type loading, the geometrically nonlinear formulation described in References 37 and 38 is used to implement the nonlinear viscoelastic model. The principle of virtual work, in the updated Lagrangian incremental formulation, can be stated as... 

The theory was based on mechanical and hydrostatic equilibrium concepts, which were unified by C. F. Gauss using the Principle of Virtual Work. The unification introduced expressions for gravitational. 

Using the principle of virtual work, the change of equilibrium and nodal loads 6P t + St) i is calculated as... 

The finite element implementation is based on the principle of virtual work. Since the capture of the dynamic fracture process requires very small time steps, an explicit central difference time stepping scheme is used in this investigation, with the time step size A/ chosen as 4% of the limiting (Courant) value. 

The potential function V (l) has a minimum at Iq > lo, in accord with the intuitive expectation that the bond separation should increase in the presence of a tensile stress. The new activation energy (D ) required to break the stressed bond could be calculated from the principle of virtual work performed on the bond in going from Ifl to lb (Figure 15.19). 

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