Correct analytical solutions to the thermoelasticity problems in a semi-plane

Using the method of direct integration, we found continuous and integrable solutions of the differential equations for plane elasticity and thermoelasticity in a semi-plane with the traction, displacement, or mixed type boundary conditions. We established the one-to-one relations between the tractions and displacements on the boundary. To ensure the correctness of the solutions for the problems of plane elasticity, thermoelasticity, and heat conduction, the necessary integral equilibrium conditions for tractions, integral compatibility conditions for displacements, and the condition of thermal balance were formulated. We illustrated our approach by applying it to the computation of the stress-strain state in a semi-plane exposed to stationary thermal local perturbation of the boundary.