The universal solution of equations of balance of the transversely isotropic plate with initial stresses with slippery strength of the flat borders

The topic of this paper is concentrated on the problem of mechanics of a deformed solid. In the first part we solved equilibrium equations of a transversal isotropic plate with initial stresses under mixed conditions on planar faces where we applied the method of decomposition of the sought functions into Fourier series by Legendre polynomials. Normal displacement and tangent voltage were assumed to be zero. In the second part we proposed a method of representing the general analytical solution of the obtained equilibrium equations.


INTRODUCTION
Initial stresses are widely used in solving problems of a formed solid [2,3]. In [4,5], a method for constructing equations of anisotropic shells and plates with initial (residual) stresses is outlined. It is based on the method of decomposition of sought functions into Fourier series by Legendre polynomials of thickness coordinate [8]. With respect to the coefficients of expansions, a system of differential equations and corresponding boundary conditions were obtained as a function of two independent variables. On this basis, in [6] a solution to the problem of the stress state of a transversal-isotropic plate with initial stresses weakened by a circular cylindrical cavity was found.

MATERIAL AND METHODS
The cavity surface and flat faces are free of external forces, and at infinity the plate is subject to constant tensile and shear forces. In this work, by the method of decomposition of the sought functions into Fourier series by Legendre polynomials, we derive the equation of the elastic equilibrium of a transversal-isotropic plate with initial stresses at the sliding finishing of plane faces (with zero of normal displacement and tangential stresses). The method of representing the general analytical solution of the obtained system of differential equations is presented.

RESULTS AND DISCUSSION 1 Equilibrium equations
Assume that the plate is related to the Cartesian coordinate system x i ( ) And we present the components of the stress tensor as follows u , σ , a system of differential equations and corresponding boundary conditions is composed as a function of two independent variables. For a transversal isotropic plate, it splits into two independent groups of equations describing, respectively, symmetric and obliquely symmetric (relative to the median plane S) deformations of the plate. In symmetric deformation, taking into account boundary conditions (1.1), it has the form [6]   Consider a plate with a homogeneous field of initial stresses ( ) 0 ij P , and assume that ( ) const P ij = 0 for j i = and ( ) , Based on the equations [1] where ijlm cthe elastic modulus tensor, we obtain, taking into account the expansions (1.2), the relations for a transversal-isotropic plate with a homogeneous field of initial stresses, hence Multiplying (1.6) by Legendre polynomials and integrating over the plate thickness, we obtain a relation connecting the moments of the stress components and the displacement vector, i.e.     (1.14) is the Laplace operator, .

General analytical solution
We present a method for representing a general analytical solution to the system of equations (1.13), (1.14). We write equalities (1.14) in the form  From here, taking into account expression (2.2), we find We apply the operation z ∂ to equation (1.13) and in the resulting equality we consider the real part. Taking into account the formula (1.15), we obtain ( ) ( ), (2.10) − pk pk pk γ β α , , dimensionless constants whose explicit expressions are easy to write out.
To solve the system of equations (2.9), we use the operator method [7]. Consider the characteristic equation And we will assume that it has simple, non-zero roots ( ) n m k m 2 ,..., 2 , 1 = . Then, using the same method [6], we find and assume that it has simple and non-zero roots s λ . Then, by the above method, we find functions k y , i.e.