A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint

Abstract

Starting from a reformulation of the mass balance law based on the Bilby–Kröner–Lee (BKL) decomposition of the deformation gradient tensor, we study some peculiar mechanical aspects of growth in a monophasic continuum by regarding the reformulated mass balance equation as a non-holonomic and rheonomic constraint. Such constraint restricts the admissible rates of the growth tensor, i.e., one of the two factors of the BKL decomposition, to comply with a growth law provided phenomenologically. For our purposes, we put the constraint in Pfaffian form, and treat time as a fictitious, additional Lagrangian parameter, subjected to the condition that its rate must be unitary. Then, by taking some suggestions from the literature, we assume the existence of generalized forces conjugated with the virtual variations of the growth tensor, and we write a constrained version of the Principle of Virtual Work (PVW) that leads to a mixed boundary value problem whose unknowns are the motion, the growth tensor, and the Lagrange multipliers of the considered theory. This allows to extrapolate a physical interpretation of the role that the growth-conjugated forces play on the components of the growth tensor, especially on the distortional ones, i.e., those that are not directly related to the variation of mass of the body. The core message of our work is conceptual: we show that the growth laws usually encountered in the literature, which are prescribed phenomenologically, but may be difficult to justify theoretically, can be put in the framework of the PVW by regarding them as constraints. Moreover, we retrieve more particularized frameworks of growth available in the literature, while being able to switch to a theory of growth of grade one, such as a Cahn–Hilliard model of growth.

Addendum to “A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint”

“…After a further inspection into the fundamental literature on Analytical Mechanics (see, for example, [1, 2]), we have reached the conclusion that some of the results presented in our recent article A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint [3], although remaining valid, can/should be re-obtained by rephrasing them consistently with the classical approach to non-holonomic and rheonomic constraints, and with the classical definition of virtual displacements, contextualized to systems subjected to such constraints [1, 2, 4–6].…”

“…As for the original article [3], the point of departure of this work is the fact that, in several bio-mechanical problems dealing with the mechanics of volumetric growth, the mass balance law of a growing body can be put in the form of a non-holonomic and rheonomic constraint 1 on the so-called growth tensor K , i.e.,…”

“…(see equation (8) of Grillo and Di Stefano [3]). Here and in the sequel, the notation is the same as in Grillo and Di Stefano [3].…”

“…(see equation (8) of Grillo and Di Stefano [3]). Here and in the sequel, the notation is the same as in Grillo and Di Stefano [3]. However, we recall that: B is the reference placement of the body under consideration; I is the time line; F is the deformation gradient tensor; the auxiliary maps X : B × I → B and T : B × I → I are defined by X ( X , t ) = X and T ( X , t ) = t , for each pair ( X , t ) ∈ B × I ; R γ ( ph ) ≡ scriptR ⌣ γ ( ph ) ...…”

“…In Grillo and Di Stefano [3], we have studied the constraint (1) by following an approach that we have developed by taking inspiration from a paper by Nadile [8] and from a consideration on non-holonomic and rheonomic constraints given in Lanczos [7]. In particular, Lanczos [7] writes:…”