ABSTRACT

The following derivation of the electromechanical fundamental relations is based on conventional formulations and notations of the theory of elasticity. These equations will be extended to get the basic relations for the coupled electromechanical problem with restrictions to linear material behaviour and small displacements. In this paper a matrix/vector description is used. In a three-dimensional continuum the mechanical equilibrium can be expressed by the differential equation

®UCT + P = pü (1)

with the stress vector a T= [an ,022^ 33,012^ 23,^31], the body force vector pT = [pl9p2,p j and the displacement vector u T = [u1?u2,u 3] described in a Cartesian co-ordinate system xt=[xi^ 2^ 3]- Furthermore p is the mass

B*D = 0 (2)

with the vector of the electrical displacements DT= P i , D2, D3] and the differential operator B$ as shown in equation (12). The corresponding static boundary conditions have the form

( t - t ) = 0 with TOTo = t (3)

(Ton - normal transformation matrix for a , F = [tn,tt ,ttJ - vector of applied surface traction). The electrical

boundary conditions have the form

(Q -Q ) = 0 with 1 ^ 1)= Q (4)

(Tdn - normal transformation matrix for D; Q - applied surface charge). In the sense of the principle of virtual work extended by the electrical part we can multiply these equations with a virtual displacement Su and with a virtual electric potential SO, respectively. Integration over the entire domain and the surface with applied loads, respectively, provides the coupled electromechanical functional

The linear coupled electromechanical constitutive equations have the form

a = C e - e E D = eTs + k E

(5)

five independent elastic constants are measured under constant (or vanishing) electric field, and the three piezoelectric constants and the two dielectric constants are measured under constant (or vanishing) deformation. Using the notation of these sets we get the following material matrices

%- J (5sTCe + 5uTpu)dV + J 8 sTeE d V + J SETeTs d V - J 5Etk EdV V V V V

-J S u Tp d V - |8 u Tq d O - JSOQdO = 0 (8) V o , o .