Cauchy Tetrahedron Argument applied to Higher Contact Interactions

Francesco Dell'Isola (1, 2) , Angela Madeo (3) , Pierre Seppecher (4)

1 Dipartimento di Ingegneria Strutturale e Geotecnica
2 M&MoCS - International Research Centre on Mathematics & Mechanics of Complex Systems
3 LGCIE - Laboratoire de Génie Civil et d'Ingénierie Environnementale
4 IMATH - Institut de Mathématiques de Toulon - EA 2134

Second gradient theories are nowadays used in many studies in order to describe in detail the layers in the materials where physical properties are sharply varying. Sometimes higher order theories are also evoked. Up to now these models are not based on a construction of stresses similar to the one due to Cauchy, which has been applied only for simple materials. It has been widely recognized that the fundamental assumption by Cauchy that the traction depends only on the normal of the dividing surface cannot be maintained for higher gradient theories. However, this observation did not urge any author, to our knowledge, to revisit the Cauchy construction in order to adapt it to a more general conceptual framework. This is what we do in this paper for a continuum of grade N also called N-th gradient continua. Our construction is very similar to the one due to Cauchy : based on the tetrahedron argument, it does not introduce any argument of a different nature. In particular, we avoid invoking the principle of virtual work. As one should expect, the balance assumption and the regularity hypotheses have to be adapted to the new framework and tensorial computations become more complex.

Domaines: Mécanique [physics.med-ph]

Francesco Dell'Isola, Angela Madeo, Pierre Seppecher. Cauchy Tetrahedron Argument applied to Higher Contact Interactions. 2014. ⟨hal-01060548v1⟩