Hypotheses Of The Theory Of Nonelectric Shells And The Saint Venant Principle

In the previous section, we wrote the equilibrium equations of the theory of shells. The equations directly follow from statics laws and no additional assumptions are required.

Three-dimensional constitutive relations (piezoeffect equations), strain-displacement formulas, and electrostatic equations should also be transformed so that the sought-for quantities—i.e., the vector components of the induction, electric field strength, stresses, strains, and displacements-—were referred to the middle surface of the shell.

In the theory of nonelectric shells of the Kirchhoff-Love type, some physically obvious assumptions on the dependence of the displacements and stresses on the thickness coordinate and on the smallness of some stresses relative to other stresses are a usual practice. Let us formulate the Kirchhoff-Love hypotheses.

First Hypothesis

In the constitutive relations, the normal stresses <733 acting on the surface elements parallel to the middle surface can be neglected as compared to other stresses.

Second Hypothesis

A rectilinear element that was normal to the middle surface before the deformation remains rectilinear and perpendicular to the strained surface and is not extended after deformation.

In order for the Kirchhoff-Love theory of nonelectric shells to be applicable, the shell faces should not be fixed; therefore, we can only define the stresses on the shell faces. The two-dimensional layer theory, where the displacements on the faces are given, is based on other hypotheses and described by a theory fundamentally different from the Kirchhoff-Love theory.

We assume that, of all mechanical conditions, only stresses are given on the faces of a piezoelectric shell. We will investigate the kinds of electrical conditions enumerated in Section 3. We will show that the type of two-dimensional theory depends on the kind of electrical conditions on the shell faces and that the electroelastic shell theories for every kind of electrical conditions on the surfaces are constructed separately.

The Saint Venant principle is important in deriving boundary conditions in the theory of nonelectric shells. It implies that if the stresses are arbitrarily specified on the shell edge, the nonself-balanced edge load generates a deeply penetrating stressed-strained state. This should be allowed for in the boundary conditions of the shell theory, while the part of the edge load self-balanced with respect to thickness brings about a stressed-strained state that quickly decreases with the distance from the edge. This has to be calculated by the three-dimensional theory.

Originally, the Saint Venant principle was derived for an elastic cylinder edge—loaded at the end face—and was later generalized to the case of arbitrary elastic, nonlinear, and viscoelastic bodies. Modern mathematics and mechanics employ the Saint Venant principle in the generalized form. A review on the topic can be found in [35],

When writing the boundary conditions in the theory of electroelastic shells, we will use the generalized Saint Venant principle forthe electroelastic case. As applied to piezoceramic shells, it consists in that (1) for shells with electrode-covered faces, the Saint Venant conditions are used as in elasticity theory and (2) for shells without electrode coverings of the faces the known mechanical Saint Venant conditions are used with an electrical condition. In other words, the shell theory should allow for the part of the component of the electric induction vector that is nonself-balanced over thickness and normal to the edge surface if the latter has no electrodes, or it should allow for the nonself-balanced part of the electrical potential if the edge is electrode-covered.

Shells with thickness polarization and electrode-covered faces are best studied due to their behavior, which is very much like that of nonelectric shells described by the Kirchhoff-Love type of theory [26, 73, 79, 101].

Unless specifically mentioned, the approximate theories should not differentiate between the components of the nonsymmetrical and symmetrical tensors, i.e., a, can be approximately taken to be equal to one:

We write three-dimensional equations of the electroelastic medium as the Piezoeffect equations

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