Review of continuum mechanics and its history. Part I: Deformation and stress. Conservation laws. Constitutive equations

This is a review of continuum mechanics and its history, citing its original sources. It "bridges"' the contributions of Bernoulli, Euler, Lagrange, Cauchy, Helmholtz, St. Venant, Stokes, Fresnel, Cesaro, and others, written in a period of two centuries in 5 languages, in a coherent and historically accurate presentation in the contemporary notation. The only prerequisite knowledge to understand the paper is advanced calculus and elementary differential equations. Some valuable, but little known, results are reviewed in detail, like the exact solution of Cesaro to the system of differential equations which every continuous medium obeys, as well as his derivation of the conditions of St. Venant for compatibility of the deformations. The last section presents the contemporary applications of continuum mechanics. The review continues with Part~II. The Mechanics of Thermoelastic Media. Perfect Fluids, reference [45]. It discusses the consequences of Navier's system of linear elasticity and approaches for its solution. It also gives a perspective of how waves propagate in continuous media. Reviewed are perfect fluids and linearly viscous fluids. At the end, Part II discusses the conditions for compatibility of the stresses.