Continuum Mechanics (DA060181)

(10/27/2021-12/21/2021)

Course Memo

Continuum mechanics is a section of mechanics and theoretical physics, or rather the continuation of theoretical mechanics that deals with analysis of deformable bodies. However, mathematics in continuum mechanics represents the main constructive tool. Continuum mechanics allows to demonstrate the power of logic and mathematical thinking. Based on a few fundamental postulates and principles, using the mathematical apparatus can reveal non-trivial, and even striking results. Foundation of continuum mechanics consists of: 1) material continuum model in the form of a deformable (with mechanical stresses and other macroparameters) continuous medium, described via several piecewise continuous differentiable functions. Building such a model is carried out by averaging the parameters of real materials that have a discrete atomic and molecular structure; 2) differential, integral, tensor calculus and the theory of dimensions with the fundamental idea of invariance under transformation of coordinate systems and systems dimension; 3) The laws of conservation of mass, momentum, angular momentum and energy, the laws of thermodynamics, expressed in terms of macroscopic parameters of the material continuum; 4) mechanical (rheological), thermal and electrical experiments that allow us to find connections between macroparameters of different substances at different mechanical, thermal, electromagnetic and physical-chemical processes. These representations constitute, in particular, the mathematical theory of thermo-electro-magneto-mechanical field. This course uses tensor representations in the Cartesian coordinate system of the observer. But it will one shown in detail how to to write the continuum mechanics equations in the arbitrary curvilinear coordinate system. This way the common link is not lost and the exposition becomes easier and clearer.