Cracks in Complex Bodies: Covariance of Tip Balances

Covariance in plasticity

“…[1]). The view can include non-conservative body forces when we evaluate the first variation of the energy balance [3], along the guidelines of the Marsden-Hughes theorem [1], or dissipative stresses, by considering a d'Alembert-Lagrange-type principle, as suggested in Mariano [4] within the general setting of the mechanics of complex materials. When dissipation plays the role that we recognize in standard viscosity and plasticity, a covariance approach based on the first principle of thermodynamics is no more entirely satisfactory for it does not furnish indications on the possible dissipative component of the stress (the one occurring in viscoelasticity) and, above all, the expression of the dissipation.…”

Mechanics of complex bodies: comments on the unified modeling of active substructures

“…As regards the morphological descriptor maps, constitutive assumptions on the structure of M are are necessary: (a) M is Riemannian with (at least) C 1 −metric g M , and (b) covariant derivatives are explicitly calculated by making use of the natural Levi-Civita connection. A metric over M has non-trivial physical meaning with respect to the representation of the (independent) substructural kinetic energy (when it exists) and a consequent influence on the representation of the microstress (see [1] and [2]). The C 1 −Riemannian structure (assumption (a) above) implies that M can be isometrically embedded in R N by Nash theorem: it is considered here as a closed submanifold in some linear space isomorphic to R N for some appropriate N .…”

Galilean relativistic fluid mechanics

“…The principle and also its formulation initiated a long and unfinished discussion among those who are interested in the fundamental aspects of continuum physics. Here we give an incomplete list of the most important looking related works [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43].…”