TABLE OF CONTENTS This chapter is devoted to the formulation of the complete elasticity problem. It begins with the formulation of the regular problem of thermoelasticity. Displacement (Navier) and stress (Beltrami-Michell) formulations are introduced, and the one-dimensional problem of a spherical vessel under internal and external pressure is solved as an illustration.
The general principles applicable in linear elasticity are treated next. The superposition principle and the virtual work theorem are introduced, allowing the conditions for the uniqueness of elastic solution to be established. The existence of the strain energy potential and the complementary energy potential is proven, and reciprocity theorems are presented. Saint Venant torsion is considered in detail, and the more general Saint Venant principle is introduced, together with Hoff s counterexample and the von Mises-Sternberg formulation.
The complete system of equations of elasticity consists of the equations of kinematics and dynamics, together with the linear elastic constitutive relations introduced in the previous chapter. Solution of the complete system must be found in the form of three field quantities:
vector field of displacements, u;
tensor field of small strains, ?;
tensor field of stresses, ?.
Within domain ? the following system of equations of linear thermoelasticity must be satisfied:
Kinematic equations
Constitutive equations of linear thermoelasticity:
where ? 0 is a tensor of initial stresses, C is the tensor of elastic moduli, A is a tensor of linear thermal expansion coefficients, and ? is a scalar field...
