Strength of Materials and Theory of Elasticity in 19th Century Italy

Abstract

This book examines the theoretical foundations underpinning the field of strength of materials/theory of elasticity, beginning from the origins of the modern theory of elasticity. While the focus is on the advances made within Italy during the nineteenth century, these achievements are framed within the overall European context. The vital contributions of Italian mathematicians, mathematical physicists, and engineers in respect of the theory of elasticity, continuum mechanics, structural mechanics, the principle of least work, and graphical methods in engineering are carefully explained and discussed. The book represents a work of historical research that primarily comprises original contributions and summaries of work published in journals. It is directed at those graduates in engineering, but also in architecture, who wish to achieve a more global and critical view of the discipline and will also be invaluable for all scholars of the history of mechanics

The role of virtual work in Levi‐Civita's parallel transport

“…This law is basically the same since Lagrange's formulation, where inertia is a force to be added to the active ones, and dynamics is reduced to statics. 1 The reactions of perfect constraints spend no work on admissible displacements; if δ P i is the first-order displacement of the point where active forces plus inertia F i are applied, the law of virtual work is ( [5], p. 14)…”

“…Tullio Levi-Civita (1873-1941)'s teachers gave him a strong training in mathematical physics and mechanics, and developed absolute differential calculus, an inheritance of Beltrami's investigations on Riemannian manifolds [1,2]. In the mid-1910s, Levi-Civita investigated relativity by absolute differential calculus: he studied the curvature of four-dimensional Riemannian manifolds, modelling space-time, through the parallel transport of vectors over these manifolds.…”

On the notion of stress in classical continuum mechanics

“…The first method, henceforth called the nonvariational approach, was conceived mainly by Cauchy [1823;1827a] and assumes forces and moments as the elemental quantities. The second method, which traces back to Lagrange [1788] and Piola [1832;1848;2014;dell'Isola et al 2015a], is of a variational nature and defines forces in a generalized sense as the quantities dual to the virtual displacement field and the gradients thereof.…”

“…The first formulation of continuum mechanics can be attributed to Cauchy with his celebrated publications [Cauchy 1823;1827a]. Cauchy restricted forces to be of volume and surface nature only.…”

Beltrami's continuum mechanics in non‐Euclidean spaces

“…He published also on the mathematical aspects of potential theory [4][5][6]. His first paper coping with physical aspects (linear elasticity) dates 1882 [7]; the only link with engineering of his time is [8], where he showed he knew the Italian mechanical school not only from the point of view of mathematical physics [9].…”