APPLICATIONS OF TENSOR CALCULUS IN POINT MECHANICS, ELASTICITY THEORY, AND FRACTURE MECHANICS

Abstract

This paper presents fundamental applications of tensor calculus in point mechanics, elasticity theory, and fracture mechanics. The inertial force is formulated in vector form, and its contravariant and physical coordinates are derived in generalized curvilinear coordinate systems. The stress tensor is presented in vector and tensor forms, together with its physical, covariant, and contravariant components. The Lagrange strain tensor is also discussed in curvilinear orthogonal coordinate systems.
Special attention is devoted to the kinematics of deformable bodies. The vector of the total reduced transformation of an oriented infinitesimal line element is defined, and the corresponding vector of proper rotation is introduced together with its physical, covariant, and contravariant tensor coordinates.
The paper further outlines several basic concepts of fracture mechanics. The potential energy density, the Eshelby energy tensor, and the J and M integrals are presented in tensor form. Their physical interpretation is discussed, emphasizing the significance of the J integral as a generalized force associated with crack propagation.
Key words: Tensor calculus, Covariant coordinates, Contravariant coordinates, Material point, Deformable body, Stress tensor, Strain tensor, Rotation angle, Fracture mechanics, J integral.