Construction of general solutions of equilibrium equations of orthotropic materials in terms of three harmonic functions

A linear mathematical model of the theory of elasticity of a three-dimensional orthotropic body is considered. The technique of integrating elastic equilibrium equations and analytical expression of elastic displacements through two functions is applied. One function satisfies a homogeneous equation in partial derivatives of the second order, and the other one of the fourth order. To solve the fourth-order equation and describe orthotropic materials, modified harmonic functions are introduced which satisfy homogeneous equations in the second-order partial derivatives. For the first time, a method of integrating elastic equilibrium equations without redundant functions and analytical expression of displacements through three modified harmonic functions was developed. The criteria for the selection of four new classes of orthotropic materials, described by three functions, were studied. Two classes contain six independent orthotropic coefficients, and the other two contain five coefficients. The other dependent orthotropic coefficients are determined from the obtained equations. The expression of deformations and stresses in an orthotropic body is recorded. It is established that there is no single representation of the general solution of the equilibrium equations of an orthotropic body.