Mechanics Research Communications 37 (2010) 453–457 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom Kinetics of boundary growth Marcelo Epstein ∗ Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada article info Article history: Received 23 December 2009 Received in revised form 14 March 2010 Available online 15 June 2010 Keywords: Surface growth Volumetric growth abstract Boundary growth is defined as a particular case of surface growth. Within this restricted context, it is shown that the kinetics of boundary growth is not essentially distinguishable from that of volumetric growth and that, consequently, the a-priori notion of material particle may be devoid of an intrinsic meaning. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction A clear distinction between volumetric growth and surface growth, with many of the attendant physical and topological ram- ifications, has been drawn in the classical article by Skalak et al. (1982). In the case of volumetric growth, the full power of Contin- uum Mechanics can be brought to bear in the formulation of the field equations. The physical reason for this luxury is that the body particles remain essentially unchanged, while the mass density is permitted to evolve smoothly in time (along with other properties that may represent phenomena of remodeling and aging). Thus, for instance, the concept of reference configuration may be legiti- mately used in theories of volumetric growth in the sense that the body-manifold is a well-defined fixed entity on which a (global) coordinate chart can be imposed once and for all. In contradistinc- tion with this state of affairs, the case of surface growth presents a number of challenges to the Continuum Mechanics paradigm. Roughly speaking, surface growth consists of creation of matter instantaneously concentrated on a surface, which may be an inte- rior surface or a (part) of the instantaneous body boundary. Because of the clarity of the underlying physical picture, it is this last case that will be the focus of this note. We call this situation boundary growth. The clearest visualization of boundary growth can be gath- ered by further assuming that at each instant of time there exists a globally stress-free reference configuration. This reference config- uration is observed to grow (or wane) by accretion (or resorption) at the boundary only. Whenever new particles are added at the boundary, they may be assumed to enter the reference configura- tion at zero stress and without disturbing the existing substrate. Although not strictly necessary, one may think that the material ∗ Tel.: +1 403 2205791; fax: +1 403 2828406. E-mail address: mepstein@ucalgary.ca. is an elastic solid with a stress-free state defined uniquely up to a rotation and that the (stress-free, time-dependent) reference con- figuration is homogeneous. In this picture, it seems absolutely clear that, unlike the volumetric growth counterpart, material particles are being created or destroyed. To handle this problem, Skalak et al. (1982) proposed the introduction of a new parameter (the time elapsed from birth) for each material particle. In effect, this idea is tantamount to introducing a material space-time picture, or a body-time entity. If the deposition of new particles is in some sense smooth, the body-time entity becomes a four-dimensional smooth manifold. The instantaneous body is then a section of this manifold at constant time. As Skalak et al. (1982) point out, however, one has to distinguish between the case in which all these sections are homeomorphic (no change of topology) and the case in which they are not. The latter case describes phenomena such as the closing of a hole. This distinction is not one of detail, but one of essence. We consider here only the first case, namely, when there is no change of topology. As we address the formulation of the equations of motion of this admittedly confined class of problems, the following ques- tion arises naturally: can one legitimately say that particles have been added or removed? And if so, on what basis? 2. A one-dimensional example Consider the problem of longitudinal deformations of a bar of length 2L, and of constant cross-section, aligned with the material X-axis. The initial reference configuration consists of the segment [-L,L] with a uniform mass density  0 . The boundary growth is assumed to be given by two smooth functions X = f (t ) and X = g(t ) defined for all t ≥ 0 and representing the material positions of the left and right ends of the evolving bar, respectively, as time goes on. For consistency, we must have f (0) =-L and g(0) = L. We assume furthermore that the reference configuration at each time t ≥ 0 (consisting of the segment [f (t ),g(t )]) is stress-free and 0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.06.004