Fundamental two-dimensional solutions of electroelasticity for compound piezoceramic bodies

Abstract

We construct fundamental solutions of the two-dimensional equations of electroelasticity for antiplanar strain of a piezoceramic space with an interphase defect. We study the orders of the powers of the singularities at the vertices of the defect for two cases: an interphase crack and a stiff fiber continuously joined to the upper halfspace which has flaked off the lower half-space. The solution is constructed in closed form. Bibliography: 4 titles. Kosmodamianskii et al. [ 1 ] have constructed the Green's function for a piezoceramic half-plane with elliptic holes. In the present paper we use fundamental solutions of the corresponding equations of electroelasticity in constructing integral representations of the solutions of boundary-value problems.We consider an antiplanar strain of a compound piezoceramic space under concentrated forces. Suppose the space is composed of two inhomogeneous piezoceramic half-spaces (polarized along the x3-axis ) continuously conjoined along the plane x 2 = 0 and is referred to Cartesian axes OXlX2X 3 . There is a constant shear load X 3 = -Pl (X l = X 2 = 0) along the fiber x t = Xto, x 2 = x20 > 0, -,,o < x 3 < o0 that is continuously distributed, or an electric charge of intensityp2, and at infinity the components of the stress tensor (Yi3 and the electric field intensity vector E m are zero. Under these conditions the compound space is in a state of antiplanar strain, all the field quantities being expressible in terms of four analytic functions ~r)(z) (v, r = 1, 2) as follows [2]: u~r) = Re fl(r)(z), r = Re f(r)(z), Elr)-iE~r) =-F~2r)(z),are respectively the electric potential and components of the vectors of elastic displacement and cp, u 3 and D,, electric induction; c44 =c~, el5 and ell =eSt are the shear modulus measured with zero electric field, the piezomodulus, and the dielectric permittivity measured at zero strain [3]; the subscripts 1 and 2 refer to quantities defined in the upper and lower half-spaces.The solution of this problem (the fundamental solution for a compound space) has the form [2] 2 ~(l) ~'(t) (D~)(z) A(v ') Xtxv,~ ~_m = + _ , Imz>__0(v=l, 2);

Homogeneous solutions of the electroelasticity equations for piezoceramic layers in R 3

“…The boundary-value problem of electroelasticity is reduced to a system of singular integral equations. The homogeneous solutions appearing in [37] have been used for the derivation of the fundamental solutions of special form [38] for layers with bases bounded by the air and covered by a diaphragm which is considered rigid at its plane and less rigid at the direction perpendicular at its plane. The distribution of the epicycloidal normal stress on the boundaries of a cavity in piezoceramic layers reads: σ θθ = σ 11 sin 2 ψ − σ 12 sin 2ψ + σ 22 cos 2 ψ.…”

Dynamic problems of the theory of elasticity for layers and semilayers with cavities

“…Recently, the methods of uniform solutions have been used in combination with the approach of singular integral equations [62][63][64][65][66][67][68][69][70][71][72][73][74][75]. However, it is noteworthy to say that here the following two main difficulties arise: (a) a correspondence problem appears between the boundary conditions within the theory of elasticity and the boundary conditions for the enumerable set of meta-harmonic functions involved in the uniform solutions; (b) there is a need for normalization of the integrals that diverge on the boundary of the domain.…”