Field equations of two-dimensional (plane) linear piezoelectricity are written in terms of a stress function and electric potential and the solution is expressed in terms of three analytic functions \(\Phi_k(z_k)\) where \(z_k=x+i\mu_ky\), \(\mu_k\) being roots of a bicubic whose coefficients are dependent on elastic and piezoelectric moduli and dielectric constants. The authors consider the problem of infinite elastic plane subjected to a homogeneous stress with a number of curvilinear tunnel cuts made of smooth, nonintersecting Lyapunov curves. The tractions and potentials along the cuts are prescribed. The determination of complex potentials is reduced to finding the solution of a system of singular integral equations and algebraic equations by employing a standard technique. Asymptotic approximation to the solution near the tips of cuts is provided. The case of a single rectilinear crack is treated and numerical results for the stress and electric field distributions are given near the crack tips.