Definition of a third Piola tensor based on geometric considerations

BUEZAS, F; GOICOECHEA, H.E.; R. SAMPAIO

Lyon

Conferencia; European Nonlinear Dynamics Conferences; 2022

In the development of continuum mechanics, great modern mechaniscists such as Noll, Truesdell and Coleman havepresented an ordered and rigorous formal structure of the theoretical framework. In particular, they have resumed the works of greatmathematicians and physicists of the 19 th century, like Piola, Almansi, Cauchy, Kirchoff, and Gibbs, among others, providing a rationalcontext to mechanics. The theory of Piola transformations is essential to relate spatial and material stress configurations by means of simple algebraic relations. In the previous transformations, the second Piola traction vector is converted into the Cauchy vector in the same way that tangential vectors do to fibers. The idea of this work is to investigate alternative definitions of stress tensors. A new definition of a Piola tensor following the transformation of the corresponding normal surface element is investigated. Finally, an application of this new tensor in the natural boundary conditions required for a finite deformation problem (geometrically non-linear) is shown. The problem of inflation of a tire, and inflation of a hollow prism-shaped structure with finite deformations are solved numerically.