Virtual work formulation

To determine the current, a measuring instrument with low inherent resistance is used such that the caused voltage drop is low. In the ideal case, this corresponds to connected electrodes. Then the difference of electric potential ip at the opposing electrodes levels out, and thus the electric field strength E is cancelled. This permits us to draw the conclusions as of Eq. (4.37). Consequently, the mechanical state of the system can be determined with the aid of the principle of virtual work formulation of Eq. (4.38). Via the electro-mechanically coupled constitutive relation, the outcome may then be utilized to deduce the electric flux density, charge, and finally the current being proportional to the strain rate. 

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. 

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... 

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. 

Mechanical work at every particle of a continuum results from acting force and respective displacement or local stress and strain, correspondingly. So the above axiom of Remark 3.1 actually comprises two principles involving either virtual loads or virtual displacements. A brief derivation of both will be given in the following subsections. Similarly, the electric work can be treated, but we will present only one of the variants. The different formulations of the principle of virtual work are independent of a constitutive law and may be denoted as the weak forms of equilibrium, as only the equilibrium conditions have to be fulfilled in the integral mean. Weaker requirements with regard to differentiability of the involved functions have to be fulfilled, since the order of derivatives is reduced in comparison to the equilibrium formulation of Eqs. (3.f4) and (3.34). 

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. 

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 ... 

When Eq. (3.48) is manipulated as before, considering the actual and virtual boundary conditions on dAu given by Eqs. (3.16) and (3.47), the principle of virtual loads, also known as the principle of complementary virtual work, may be formulated as... 

For the rigid continumn of volume A consisting of such particles in accelerated motion, the virtual work may be formulated as given by Eq. (3.58). This extension of the principle of virtual displacements is referred to as d Alembert s principle in the Lagrangian version ... 

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 ... 

The principle of virtual work is suitable for solving a wide range of problems. There are tasks however where different but related formulations might be more useful. Thus, two prominent variational principles will be extended here to take into account materials with electromechanical couplings. This novel approach to Dirichlet s principle of minimum potential energy will be employed later in Section 6.3.2. In comparison to the principle of virtual work, the extended general Hamilton s principle is considered to be equivalent and even more versatile, but only its derivation will be demonstrated here. 

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). 

Hence, the principle of virtual work of Eqs. (3.62) and (3.63) reduces to its mechanical part. Via the dependence of mechanical stresses a = electromechanically coupled constitutive relation, see Eq. (4.17), the electric field strength E however enters the formulation as a parameter ... 

When a current source is connected to the electrodes of a piezoelectric actuator, the electric potential p is unknown. Therefore, the contributions of the principle of virtual electric potential of Eq. (3.53) need to be retained. Since the current describes the derivative trend of charge with respect to time, it prescribes the area charge qsA on the electrodes. An adequate formulation of the principle of virtual work from Eqs. (3.62) and (3.63), in consideration of Eq. (4.36), reads... 

An elastic stability analysis is presented in this paper for Timoshenko-type beams with variable cross sections taking into consideration the effects of shear deformations under the geometrically non-linear theory based on large displacements and rotations. The constitutive relationship for stresses and finite strains based on a consistent finite strain hyperelastic formulation is proposed. The generalized equilibrium equations for varying arbitrary cross-sectional beams are developed from the virtual work equation. The second variation of the Total Potentid is also derived which enables... 

We propose a simple method for the linearization of the equations, which are established in our case, based on the virtual work principle. The kinematic relations between the interconnected bodies are represented by the recursive equations. Under the small deformation assumption, the system generalized variables used in the equations are the relative joint coordinates at the connections and the deformation modal coordinates of the flexible bodies. In the linearization process, the differentiation of the kinematic terms with respect to the generalized variables must be performed. In our method, these partial derivatives are attained through the first and second order time differentiations of the body absolute angular velocities and through the first, second, and third order time differentiations of the mass center coordinates. This is the essential idea behind our method. The partial differentiation of the mechanical terms, for example, of the inertial tensors will also be presented. We have developed specific operators to perform the time differentiations. This method makes both the theoretical formulation and the programming task relatively simple, and allows fast computation. 

In order to establish the local equilibrium equation in the longitudinal direction, the calculus of variations is applied. The principle of virtual work under a total Lagrangian formulation is applied (Eq. 6). Taking into account the primary warping function s virtual components and applying the procedure analyzed in the study of Sapountzakis and Tsipiras (2010a), the following local equilibrium equation ... 

Applying the principle of virtual work under a total Lagrangian formulation (Eq. 6), taking into account the virtual quantities of u, Ox, ) (and... 

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 derivation of the equilibrium equations for SmC liquid crystals parallels that outlined in Section 2.4 for nematic and cholesteric liquid crystals, this approach being based on work by Ericksen [73, 74]. The energy density will be described in terms of the vectors a and c, and the equilibrium equations and static theory will be phrased in this formulation these vectors turn out to be particularly suitable for the mathematical description of statics and dynamics. We assume that the variation of the total energy at equihbrium satisfies a principle of virtual work for a given volume V of SmC liquid crystal of the form postulated by Leslie, Stewart and Nakagawa [173]... 

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As a high-speed single station press, mechanical compaction simulator will be able to plot compressibility profiles, Heckel graphs, calculate work of compaction, and virtually any other imaginable variable that is of interest to formulators. Tensile strength of tablets made on a Betapress and The Presster was similar. ... 

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