Nonclassical continua, pseudobalance, and the law of action and reaction

Cauchy Tetrahedron Argument Applied to Higher Contact Interactions

“…In 1959, W. Noll [34] crystallized this faith by proving that the so-called Cauchy Postulate, the dependence of contact forces only on the normal of dividing surfaces, is indeed equivalent to the apparently weaker assumption of uniform boundedness of contact forces for all dividing surfaces. 1 The merit of Noll's results consists in pointing out the relationship between the tetrahedron argument and measure theory (see for example [17]); the drawback is in camouflaging behind a technical hypothesis the physical assumption that the contact forces depend only on the normal.…”

On classical continuum mechanics, two-scale continua, and plasticity

“…In the more fundamental approaches [Gurtin and Martins 1976;Noll 1973;Šilhavý 2008], the macroscopic balance of linear momentum (3-18) was deduced from a boundedness assumption on the system of contact actions. In the microscopic scale, this was the approach followed in my previous papers [Del Piero 2009;2014a;2014b;2017;2018a;2019], in which equations of the form (5-36) 2 were called pseudobalance equations because they were not expressing the balance of any physical quantity. Here, (5-36) 2 is a direct consequence of energy conservation.…”

“…This paper is a revisitation of some ideas developed by the present author in a series of papers [Del Piero 2009;2014a;2014b;2017;2018a;2018b;2019] concerning the principles of classical mechanics, their extension to generalized continua, and their application to plasticity. Over the years, many ideas have evolved and some became obsolete.…”

Higher Gradient Theories and Their Foundations

“…It has to be underlined that Cauchy postulate, despite its usual designation, is not a fundamental principle in the sense that balance of force or the principle of virtual work are but rather a constitutive assumption. Noll's result allows to point out the relationship between tetrahedron argument and measure theory (see, e.g., Del Piero 2014). However, one should pay special care not to forget, behind the technical hypothesis of uniform boundedness, the physical assumption that the contact forces depend only on the normal to the Cauchy cut.…”