The study of the stress-strain state of the downhole packer's jaw teeth based on the initial functions method

The polynomial solutions known in the theory of elasticity for the problem of stress-strain state of a rectangular strip are generalized to the case of a wedge-shaped strip with simultaneous application of normal and tangential loads distributed uniformly along the edges, and external concentrated forces applied to the wedge top. To solve the problem based on application of the initial functions method using a decomposition approach, calculation formulas for displacements and stresses in polar coordinates are obtained. The problem solution obtained by the operational method using the one-sided Laplace transform depends on uncertain coefficients of the corresponding homogeneous differential equation’s general solution expansion in terms of linearly independent solutions determined by the roots of the characteristic equation. For each of the problems under consideration, a method is indicated for identifying these uncertain coefficients in the presence of a load along the wedge edges, both in the presence and absence of an external force applied to the top. In the case of a problem with a load along the edges and an external force applied to the wedge top with unsupported and reinforced ribs, the general algorithm for solving the problem and the calculation formulas with well-defined constant coefficients for displacements and stresses are obtained. Based on the calculated values of displacements, stresses and strains, stress-strain diagrams were constructed and, in the case of a plane stress state that occurs in the absence of an external force, a comparison was made with the known theoretical linear dependences of stresses on strains. Keywords: packering; jaw teeth; plane stress state; symmetrically directed load; concentrated tensile force; elementary solution; stress-strain state; initial functions method; one-sided Laplace transform; decomposition approach; operational calculus; stress-strain diagram; inhomogeneous differential equation.