External Virtual Work

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... 

The displacement finite element model is based on the principle of virtual displacements. The principle requires that the sum of the external virtual work done on a body and the internal virtual work stored in the body should be equal to zero (see Reddy(46))... 

In order to maintain equilibrium with the element, a system of external nodal forces F is applied which will reduce the virtual work (dW) to zero. In the... 

Taking a segment of the cap between the centres of two boilers and by means of virtual work method, the cap reinforcement is analysed at the edge of standpipe area and at the inside of the barrel wall, i.e. at the junction between the cap and the wall, subject to external hoop prestress and ultimate gas pressure. The amount of steel thus required is... 

This fundamental principle of physics is given by the axiom of Remark 3.1 in its most general formulation, where SW is the total virtual work of the system. For mechanical fields in deformable structures as well as for electrostatic fields in dielectric domains, it can be restated by the equality of internal 51A and external 6V contributions. 

This formulation of the principle of virtual work is the principle of virtual displacements, which appears in the hterature sometimes under the name of the preceding. Naturally, the virtual strain energy 6U exists only for mechanical systems with deformable parts. As the contained virtual strain tensor is assembled from derivatives of the virtual displacements, these have to be continuously differentiable. The virtual work of external impressed loads ymd includes the limiting cases of line or point loads. External reactive loads do not contribute when the virtual displacements are required to vanish at the points of action of these loads, and thus the virtual displacements have to comply with the actual geometric or displacement boimdary conditions of Eq. (3.16). With these presumptions, the initial axiom of Remark 3.1 may now be reformulated for the virtual displacements. 

Remark 3.2. A uniform mechanical system will be in equilibrium if the virtual work of the actual external and internal loads for arbitrary admissible virtual displacements vanishes. 

The other formulation of the principle of virtual work for mechanical systems requires the introduction of virtual loads instead of virtual displacements. Therefore, only those variations of external loads and stress tensor are considered admissible that are compatible with the equations of equilibrium inside the mechanical system and on the boimdary. The interior equilibrium of Eq. (3.14) for the virtual loading leads to the following form ... 

Here JV is the complementary virtual work of external loads, and (JW the complementary virtual strain energy. The initial axiom of Remark 3.1 may now be reformulated for the virtual loads. 

The criteria of admissibility for the virtual displacements have been discussed in Section 3.4.2. As rigidity has been assumed in the case at hand, the occurring displacements do not cause strains. Therefore, virtual strains do not exist and, consequently, there are no contributions of internal loads to the virtual work. As expected, the virtual work of external impressed loads is identical to the term in the static principle of virtual displacements. The accelerated motion results in the additional term representing the virtual work of the loads of inertia. In general, the principle may be formulated as follows ... 

The electric contributions from the principle of virtual potential, as derived in Section 3.4.4 and given by Eq. (3.53), still have to be incorporated. This can be achieved equivalently by the addition or subtraction of Eqs. (3.60b) and (3.53). In conformance with Allik and Hughes [4] and in view of the symmetry properties of the not yet introduced constitutive relation, the electrostatic expressions will be subtracted from the mechanical ones. The virtual work of external contributions takes the following form ... 

In the static portions of the virtual work of external contributions, the forces and charges acting on the constant volume and surface of the structure are not altered by the arbitrary variations bu of displacements and of electric potential respectively. Thus, the left-hand sides of Eqs. (3.45) and (3.53) may be written in the following form ... 

Since no volume charges qA will be specified, the corresponding term in the virtual work of external charges vanishes, see Eqs. (3.53) and (3.62) respectively. 

With the aid of the principle of virtual work, the equilibrium and boundary conditions can be obtained for the quasi-static case, where, in principle, loads may change over time but inertia effects are not considered. The contributions required for this purpose have been already obtained and will be joined together in the following. The internal virtual work SU t) is given by Eq. (8.23), while the external virtual work SV t) of Elq. (8.24) reduces for the quasi-static case to those contributions due to the applied loads... 

The underlying mechanical degrees of freedom u (x, t) are given by Eq. (8.26). The matrices G(a , t) and G(x, t) contain the initial internal loads to be determined with the aid of Eqs. (8.43). They depend on the initial external loads n x,t), which in turn are composed of the applied external loads h x,t) and those rotational effects that concern the initial state. The latter have been obtained implicitly within the derivation of the virtual work of inertia loads. They are marked in Eq. (8.33), and consequently the initial external loads are given by... 

Following from the general principle of virtual work of Eq. (3.41), the equality of internal and external virtual work is also demanded for the beam and shall serve as the basis for the derivation of the equations of motion ... 

The external virtual work SV t) is given by Eq. (8.24), and its individual contributions are specified in Eqs. (8.25) and (8.33). Again in terms of the combined vectors of mechanical and electric variables, this leads to... 

With V being the volume in the undeformed state, the symbol S denotes kinematically admissible variations, is the variation of the external virtual work and the integral represents the internal virtual work. 

Somewhat analogous to our earlier principle of virtual work, we here assume that the rate of working of external forces and moments either goes into increasing the kinetic and elastic energies, or is dissipated as viscous dissipation. Thus for a volume V of liquid crystal bounded by surface S... 

The virtual work of all the external forces acting on the body k can be also written as ... 

Methods of analytical mechanics provide the natural basis to develop such a generalized approach. Within the bounds of quasistatic problems, methods of analytical statics are sufficient. The use of the principle of virtual work, instead of the energy balance equation, permits one to generalize the theory of fracture and fatigue to multi parametric problems and to omit restrictions on the potential character of external and internal forces. In this paper, only "non-healing" cracks are considered typical for most structural materials. Therefore, we consider mechanical systems with unilateral constraints. The principle of virtual work for such systems takes the form a system with ideal unilateral constraints stays in the equilibrium state if and only if the summed virtual work of all active forces on all small displacements compatible with the constraints is equal to zero or negative ... 

In order to establish the nonlinear equations of motion, the principle of virtual work Eq. 6 under a total Lagrangian formulation is employed. It is worth here noting that in the examined general case, the expression of the external work (Eq. 7c) takes into account the change of the eccentricity... 

The liquid state of matter is intermediate in its physical properties between the solid and gaseous states. Chemists who study reactions in solution deal with the so-called normal liquids, very rarely they deal with liquid crystals, and do not virtually work with quantum liquids. In the absence of external actions, normal liquids are macroscopically uniform and isotropic. A liquid is close to a solid in many properties, especially near the melting point. As a solid, a liquid has the interface and withstands strong tensile forces without rupture. The liquid and solid have close values of density, specific thermal capacity, specific thermal conductivity, and electric conductivity. All this is a result of tight contact between molecules in the liquid and solid. 

Equation 9.6 determines the conditions of a mechanical equilibrium of the curved interface. This can be illustrated with an example of a spherical bubble of radius r. To compete the surface tension the pressure inside the bubble should exceed the external pressure with AP, which is determined from the work, W, for virtual change of r dll APdl adl. Under equilibrium, dll = 0 and APdU=erd4, thus... 

---