Piola

Piola is an Italian surname. In scientific literature, the name Piola is associated with concepts in continuum mechanics that honor an Italian 19th-century scientist with that surname. The Piola family name appears in various historical and scholarly contexts, and several important ideas in elasticity and solid mechanics bear the Piola name.

In continuum mechanics, the Piola transformation and the Piola-Kirchhoff stress tensors are central tools for relating quantities defined in different configurations of a deforming body. The Piola transformation maps tensor fields from a reference configuration to the current configuration, or vice versa, enabling the formulation of balance laws and constitutive relations in a convenient frame of reference. The first Piola-Kirchhoff stress tensor P and the second Piola-Kirchhoff stress tensor S are related to the Cauchy stress tensor σ through standard kinematic relations; for instance, P = J σ F^{-T}, where F is the deformation gradient and J is its determinant. These constructs underpin most mathematical models of finite elasticity and material response, and they are widely taught in solid mechanics and applied physics.

Beyond mathematics and physics, Piola can refer to a surname borne by various Italian figures across history, including scientists, engineers, and artists. As a result, the term Piola appears in diverse biographical and historical contexts, with the scientific use primarily linked to continuum mechanics and tensor transformation.