Internal Virtual Work

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))... 

The virtual work associated with convective surfaces Equation (D.22) is referred to internal... 

This formula is actually used in LMTO-ASA calculations to compute the electronic pressure as a function of volume, and from this the lattice constants and bulk moduli. The decomposition of the electronic pressure into partial-wave components is extremely useful since it allows the chemical bond to be analyzed in a new way. It should not, however, be assumed that the total energy may be decomposed in the same way. Here we are concerned with virtual work processes, which may be decomposed, but the work, once part of the internal energy, is no longer decomposable. 

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. 

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. 

Here is the virtual work of external charges, and the virtual work of internal charges. As the contained virtual electric field strength vector 8E is assembled from derivatives of the virtual electric potential Sip, the latter has to be continuously differentiable. Further on, the virtual electric potential has to comply with the actual conductive boundary conditions of Eq. (3.36). The initial axiom of Remark 3.1 may now be reformulated for the virtual electric potential. 

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 ... 

Since the forces fg and charges q A on the boundary are zero apart from their respective working surface, the surface integrals may be summarized. Then the integrands can be collated in vector form, as shown in the last line. Similarly, the virtual work of internal contributions can be formulated, where the vectors of virtual strains 6e and virtual electric field strength SE, as well as the vectors of actual stresses electric flux density D, be merged ... 

In the virtual work of internal contributions given by Eq. (3.63), the field quantities appearing in virtual and actual form are in no way connected. With the substitution of Eq. (3.64) into Eq. (3.63), this independence from... 

It is the forms of constitutive equations given by Eqs. (4.10a) that are used most often.The one on the left-hand side is suggested by the formulation of the virtual work of internal contributions in Eq. (3.63). 

The internal energy of an electrostatic system is represented by the product of the correlated field strength and flux density components, as exemplarily derived for the virtual work of internal charges, Eq. (3.53). The shear strain induced flux density components Di and D2, as given in Eq. (4.22), do not contribute by virtue of the above assumption. Thus, the assumption of unidirectional electric field strength is equivalent to the neglect of shear associated electrostatic energy contributions. 

The virtual work of internal contributions is assembled in Section 3.4.6 from the virtual strain energy and virtual work of internal charges, as supplied by the principle of virtual displacements and of virtual electric potential, respectively. In Ek[. (3.63), the virtual work of internal contributions is given for a volumetric object. The preceding analysis accomplished a reduction to two dimensions for the sheU-like wall and to one dimension for the beam. Consequently, the expression for the virtual work of internal contributions may be reformulated for the wall SLi t) and for the beam SLi t) as follows ... 

As a matter of course, the virtual work needs to be independent of the description. Thus, equating Eqs. (8.1) allows us to associate the internal mechanical as well as electric loads of beam L x,t) and wall L x, s,t) ... 

Insertion of these constitutive relations into the virtual work of internal loads of the wall SU t) and of the beam SU t) as expressed by Eqs. (8.1) results in... 

Thus, the virtual work of internal contributions may be written in terms of functions of the mechanical and electric degrees of freedom in <5W (t) and electric parameters in 5U t) such that... 

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... 

In the last line of Eq. (8.40), Eq. (8.39) is introduced and expressions are multiplied out. The first term represents the linearized virtual strain energy (t). The initial internal loads vector N x,t) can be determined in advance, while the vector N x,t) of the other internal loads needs to be substituted with the aid of a constitutive relation. Therefore, the second term is free of non-linear products, while the third term contains such products and, consequently, will be neglected. Such a second-order theory corresponds to the equilibrium of the slightly deformed system and contributes the virtual work of initial stresses Thus, the virtual work of internal mechanical... 

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 ... 

Since the virtual variants of the degrees of freedom are not time-dependent, the element vector 6i>J appearing with every term can be factored out. Those of the individual matrices of the principle stemming from the virtual work of internal loads are constructed as follows ... 

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. 

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 ... 

Using the principle of virtual work (Fig. 19) and the calculated shear force and bending moment distributions, the relative horizontal displacement can be obtained. Having the application of opposite unit forces in the virtual system, the internal virtual forces (Vim, Mint) can also be obtained. 

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