General Theorems On Electroelasticity

Consider a piezoceramic body of volume V bounded by the surface ft in the coordinates Adhere, as before, the Greek indices assume the values 1, 2, or 3 and the Latin indices assume the values 1 or 2). For simplicity, we will only deal with the static case [82].

We write the equation for energy

Jv Ju

Here qp are the components of the bulk forces; vp are the components of the displacement vector; ap,> is the stress tensor; ep/1 is the strain tensor; Vp are the contravariant components of the electric induction vector; Ep are the covariant components of the electric strength vector; V is a volume body; and ft is the body surface. The components of the tensor np are found from the vector relation n = npep, where n is a unit vector of the normal external to the surface ft and ep are base vectors.

The electroelastic potential will be written as in elasticity theory, i.e., W=l-(ap"epll + £pVp).

We write the constitutive relations 2.10 in the Cartesian coordinates xi, xi, and x-i for a piezoceramic polarized along the v^-lines:

en ~ .sf I a,-, + s*i2(T)j + if3<7J3 + ¿31 £3 C33 = ^(<711 + 022) + i'f3cf33 + d^Ey eij ~ S(>6Crij' e,3 = 5^4(7,3 + ti{5£, T>i = £[,£■/ + d, 5 a/3

Consider two electroelastic states. One state will be marked by superscript I and the other by superscript 2. We write the expressions for the work done by the electrical and mechanical forces of the first electroelastic state on the generalized displacements of the second state, and the work done by the forces of the second state on the generalized displacements of the first state. By generalized displacements, we mean the set of the quantities v,, v2, After simple mathematics using the constitutive relations 4.2, we can show that the work of the forces of the first electroelastic state on the generalized displacements of the second state is equal to the work of the forces of the second state on the generalized displacements of the first state. This proves the Clapeyron theorem generalized to the electroelastic case:

In electroelasticity, the theorem on uniqueness will hold for appropriate boundary conditions if the integrand A in 4.1 is positive. We can show this by transforming the integrand and allowing for 4.2 to get a positive-definite quadratic form

4

b

~ _ 7T 33

T ' £33

- ¿44 T £11

-fad

5

033

The constant factors before the parentheses in 4.4 are positive numbers and can be calculated using physical constants that can be found, say, in [8],

When the general theorems are fulfilled, problems on electroelasticity can be solved by variational methods. The variational methods and extremal estimates used in elasticity theory can be generalized to electroelasticity.

Chapter 2

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