Surface and Bulk Growth Unified

Abstract

I have been puzzled for a long time by the unnatural divide between the theory of bulk growth-strikingly underdeveloped-and that for surface growth-much better developed, along apparently independent lines. Recent advances in growth mechanics (DiCarlo and Quiligotti, 2002) make it now possible to subsume growth phenomena of both kinds under one and the same format, where surface growth is obtained as an infinitely intense bulk growth confined in a layer of vanishingly small thickness. This has allowed me to recover the results collected in Gurtin (2000) from the standpoint of DiCarlo and Quiligotti (2002). In particular, I am able to construe Gurtin's technique of referential control volumes that evolve in time as a special application of the principle of virtual power. 1 This is not at all obvious, since I can change my mind. But in this respect I did not. My coauthor did (Quiligotti, 2002), on the (not minor) issue of invariance requirements, convinced by an argument intimated by Green and Naghdi (1971) and spelled out in Casey and Naghdi (1980, 1981), Casey (1987), and elsewhere. For reasons I expound elsewhere (DiCarlo, in preparation), I find that old argument faulty and misleading.

Nonlinear elasticity of incompatible surface growth

“…Despite considerable interest of compatible surface growth [13][14][15][16][17][18][19][20], here we focus on the case when the manufactured reference state is incompatible (non-Euclidean) in the sense that it cannot be realized in 3D without generating residual stresses. The underlying "geometric frustration" [21][22][23][24], which is ultimately shaped by the deposition process, may be beneficial (as in growing plants [25]) or detrimental (as in civil engineering structures [26]), as it was already exemplified in the early attempts to understand incompatible surface growth motivated by the necessity to explain the built-up of "growth stresses" in trees [27], to optimize the concrete pouring [28] and to improve the quality of industrial winding [29].…”

Configurational balance and entropy sinks

“…This point of view, however minimalist it may be, leads to whole new vistas in terms of our ability to encompass a great variety of anelastic behaviors, including viscoelasticity, growth and remodelling. Many authors (e.g., Di Carlo 2005;Di Carlo and Quiligotti 2002;Gurtin 2000;Podio-Guidugli 2002), on the other hand, have suggested that this framework can be considerably enlarged by advancing the assumption of extra balance laws to be satisfied by the material forces (such as the Eshelby stress). It seems somewhat futile to argue as to which is the right approach, since, as convincingly explained by Clifford Truesdell, Continuum Mechanics is, by its very nature, a theory of models.…”

“…Rather than hastily interpreting this as the identical vanishing of the quantities within the square brackets, one now introduces (in the terminology of Di Carlo 2005) the extra energetic responsesT andb satisfying the residual inequality:…”

Extremum principles for biological continuous bodies undergoing volumetric and surface growth

“…A unifying framework for the treatment of both bulk and surface growth phenomena has been proposed in [13], relying on the introduction of configurational forces as the internal driving forces for growth. Configurational forces for surface growth have been identified in [14], with application to bone remodeling.…”