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Kinetics of boundary growth
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
a r t i c l e
i n f o
Article history: Received 23 December 2009 Received in revised form 14 March 2010 Available online 15 June 2010 Keywords: Surface growth Volumetric growth
a b s t r a c t 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 ramifications, has been drawn in the classical article by Skalak et al. (1982). In the case of volumetric growth, the full power of Continuum 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 legitimately 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 contradistinction 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 interior 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 gathered by further assuming that at each instant of time there exists a globally stress-free reference configuration. This reference configuration 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 configuration 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: [email protected]. 0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.06.004
is an elastic solid with a stress-free state defined uniquely up to a rotation and that the (stress-free, time-dependent) reference configuration 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 question 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
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M. Epstein / Mechanics Research Communications 37 (2010) 453–457
An example of such a transformation for the case depicted in Fig. 1 is the following: =
LX , L + kt
−∞ < X < ∞, t ≥ 0.
(7)
In the new coordinate plane , , the lines X = −L − kt and X = L + kt become, respectively, = −L and = +L, as can be verified by direct substitution. In this way, the problem is transformed into the standard form of the initial-boundary-value problem. In the new domain of definition, the mass density is given by: ˆ 0 (, ) = Fig. 1. Boundary growth of a supported bar.
that g(t) > f (t) for all t ≥ 0. The (elastic) constitutive equation is prescribed as:
T =E
∂x −1 ∂X
,
(1)
where T is the (relevant component of) the Piola stress, E is a (positive) modulus of elasticity and x is the longitudinal coordinate of points in the deformed bar. In the absence of body forces, the motion of the bar is governed by the law of momentum balance, whose local form is: ∂2 x ∂T − 0 2 = 0. ∂X ∂t
(2)
Taking into consideration the assumed constitutive law, the function x = x(X, t) is governed in this case by the one-dimensional wave equation: E
∂2 x ∂2 x − 0 2 = 0. ∂X 2 ∂t
(3)
The quantity c = E/0 represents the speed of sound in the bar. We assume that:
df (t) dg(t) dt , dt < c ∀t ≥ 0.
(4)
Under these conditions, the problem represented by Eq. (3) with the boundary conditions: x(f (t), t) = f (t),
x(g(t), t) = g(t)
∀t ≥ 0,
(5)
and with (consistent) initial conditions of position and velocity on the segment [−L, L] at t = 0, is a well-posed hyperbolic problem. It represents a rod supported at both ends, where the supports occupy at each instant of time the positions of the ends of the growing beam. A similar formulation can be conceived for free ends, in which case the X-derivative is made to vanish by the boundary conditions. Fig. 1 shows the particular case f (t) = −L − kt and g(t) = L + kt where k is a positive constant smaller than c. A few characteristic lines suggest schematically how this problem can be solved by the method of characteristics for the case of fixed supports. There is, on the other hand, an alternative approach to solve this problem, namely, by “straightening” its domain so as to convert it into the rectangular domain of a standard wave-equation formulation. To achieve this end, it is sufficient to provide any smooth (at least C 2 ) and smoothly invertible transformation, (X, t) → (, ), that renders f (t) and g(t) constant (−L and L, respectively). In particular, we consider transformations of the form: = (X, t) =t
(6)
∂X 0 . ∂
(8)
Clearly, this density varies with time. We may then say that, as far as the new and “nicer” configuration (nicer because it is independent of time) is concerned, there is a phenomenon of volumetric growth, not experienced by the original (boundary growing) configuration. Similarly, the constitutive Eq. (1) is transformed into:
Tˆ = E
∂ˆx/∂ −1 ∂X/∂
,
(9)
where we have denoted by x = xˆ (, ) the spatial placements in terms of the new reference configuration. Notice that in the onedimensional case under consideration the transformed Piola stress Tˆ does not need any corrective multipliers in terms of T. The physical meaning of Eq. (9) is that, as far as the new reference configuration is concerned, in addition to mass growth, the body undergoes also a process of remodeling. There is still an additional price to be paid for the new domain: a concomitant modification of the equation of motion and of the boundary conditions. The details of the transformed equation of motion can be obtained by direct substitution. For our purposes, however, what is of interest is that this equation can be cast in the form: ∂Tˆ ∂2 xˆ − ˆ 0 2 = . ∂ ∂
(10)
The term appearing on the right-hand side of this equation is a sort of reactive body force absent in the original formulation. Its exact expression is not of interest at this point. We note, however, that this is a “follower force”, in the sense that it is a function of the deformation gradient ∂ˆx/∂, the velocity gradient ∂2 xˆ /∂∂ and the second gradient of the deformation ∂2 xˆ /∂ 2 . This sort of material Coriolis force (which vanishes if is independent of t) is the result of having adopted a peculiar reference configuration, whose legitimacy, however, cannot be disputed. Recall that in the field equations of volumetric growth one usually accounts for additional terms that represent momentum sources other than the canonical source represented by the passively incoming mass (see, e.g., Cowin and Hegedus (1976), or Epstein and Maugin (2000)). Recall, moreover, that such additional terms are not necessarily subject to the principle of material frame indifference, since they are not necessarily constitutive in nature. They may represent, for example, the reaction of a colony of bacteria to the local slope or curvature of the continuum substrate. For an observer intent on keeping a fixed reference configuration, the phenomenon that we originally described as boundary growth is nothing of that sort. No particles are created or destroyed anywhere. Instead, this observer reckons a process of volumetric growth and remodeling with volumetrically distributed momentum sources. To be sure, in biological applications, the speed of the boundary growth is several orders of magnitude smaller than the speed of sound. As a result, the magnitude of the body forces is negligible. A case of particular interest because of its illuminating simplicity is given by the growth functions: f (t) = −L + kt and g(t) = L + kt. In this case, the two bounding lines of the original domain are
M. Epstein / Mechanics Research Communications 37 (2010) 453–457
455
Fig. 2. Creeping boundary growth.
parallel. In physical terms, what is lost on one end is matched by an equal growth on the other end. The transformation = X − kt clearly does the job of straightening the domain. In this particular case, therefore, the left-hand side of the equation of motion (2) remains formally unchanged, and the only price to be paid for the steady material snake-like creeping of the bar to the right, is given by the appearance of the reactive body force:
= ˆ0
k2
∂2 xˆ ∂2 xˆ − 2k 2 ∂∂ ∂
,
prescribed explicitly. More realistically, this motion may be coupled with the solution by means of an evolution equation (specifying, for example, the speed of the bar end as a function of the reaction at the corresponding support). In closing this section we notice that, in the one-dimensional case, a global stress-free configuration always exists. We can, therefore, envisage also the inverse transformation from volumetric to boundary growth of the very special kind just described.
(11) 3. The general case
which, as already pointed out, is negligible in most applications, since usually k c. The boundary conditions are: xˆ (−L, ) = −L + k and xˆ (L, ) = L + k. Fig. 2 shows the history of a material particle in the boundary-growth representation. In this case, the observer presumes to know what a “real” particle is, but pays the price of birth and mortality for each such particle. In the straightened domain of Fig. 3, on the other hand, particles are immortal and their world lines correspondingly infinite. It may be claimed that the first representation is better because it has a simpler constitutive law. But this fact cannot be known a priori in all cases. In real phenomena, involving three-dimensional bodies, there may not exist a stress-free reference configuration and the decision as to which portion of the growth is happening at the boundary and which portion occurs in the interior cannot be made canonically. Any contemplated motion of the material boundary (that is, in the material body-time manifold), whether directly or constitutively prescribed, can be transformed away by the technique just shown. The very notion of material particle is thus literally in the eyes of the beholder. If what matters in the end is the solution of the initial-boundary value problem in the sense that it delivers the mass density as a function of physical space and time, then which “particles” are responsible for the presence of such density may not be an answerable, or even relevant, question. Remark 2.1. One may reasonably claim that a particle can be identified as the carrier of a mark, such as an ink dot drawn on the instantaneous boundary of the body. If this dot disappears as it is covered by material, it is reasonable to claim that surface growth has taken place. These ideas, however, do not fall within the realm of a mechanical theory. We have considered the case in which the motion of the boundary of the time-dependent stress-free reference configuration is
The preceding example suggests that a similar point of view may be adopted for the general three-dimensional case. This generalization, however, is not a trivial one as it requires a more sophisticated conceptual framework due, in part, to the fact that, whereas the boundary of a line consists of just two points, the boundary of a body is a two-dimensional manifold. We start by assuming the existence of a material universe, namely, a three-dimensional manifold M of material particles with no a-priori mechanical properties. In this context, a material body is defined as a connected trivial three-dimensional body manifold B together with an embedding : B → M. Without this embedding, the manifold B serves mainly the purpose of establishing, once and for all, the topology of the body. Accordingly, the body manifold B is a time-independent entity consisting of points without any fixed a-priori material content, just as the line segment −L ≤ ≤ L in our one-dimensional example. The embedding can be considered as serving the function of capturing material particles, just like a cookie cutter applied to a sheet of dough. In the terminology of Segev (1996), who was the first to use this idea in the context of growth, the embedding is called a content. In the conventional treatment of Continuum Mechanics, this embedding is established ab initio and, therefore, can be incorporated into the very definition of the body, without any explicit mention. In the case of growth and remodeling, however, we may allow the capturing embedding to be a (smooth) function of the time variable t (measured with respect to some defining event). When this fact needs to be emphasized, we will use the notation t . Proceeding to the notion of configuration, we immediately see that (in a manner somewhat analogous to the Lagrangian and Eulerian formulations of the equations of balance) we have two possibilities to define a configuration. On the one hand, we may
Fig. 3. Volumetric-growth representation of the creeping bar.
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M. Epstein / Mechanics Research Communications 37 (2010) 453–457
define a configuration as an embedding: : (B) → E3 .
ward application of the chain rule yields:
(12)
Physically, in this first version, which we call the material formulation, we consider how the material particles comprising the content of the body at a particular time are mapped into the Euclidean space E3 . On the other hand, we also can define a configuration as an embedding: ˆ : B → E3 .
(13)
In this version, which we call the body formulation, we are describing how the body points themselves are mapped into E3 . In the classical case, whereby is independent of time, this distinction is not necessary. But in the case of a growing body it provides us with a fruitful choice: we may operate on material particles or on body points. The notion of boundary growth can arise only in the first formulation. Boundary growth of the pair {B, } is said to take place whenever the set (∂B) depends on time. To establish the relation between both formulations, we start from the notion of local material density. Mathematically speaking, the mass density is a three-form. Thus, given the density 0 in (B), the density ˆ 0 in B is defined as the pullback of 0 by . More specifically, let (Cartesian) coordinate systems X I and Y ˛ be chosen in M and B, respectively, and let 0 (X, t) be the density at time t at the point X ∈ t (B). Denoting by H(t) the differential of t , the mass density at Y = −1 t (X) is given by: ˆ 0 (Y, t) = JH(Y,t) 0 (X, t),
(14)
where JA denotes the determinant of A. Notice that, in the context of our one-dimensional example, the Y ˛ -coordinates correspond to the single coordinate , while the map −1 corresponds to the t transformation given by Eq. (6). Consider next a scalar constitutive quantity, such as the free-energy density per unit volume of the material manifold (in a given chart X I ), and assume, for simplicity of the exposition, that it depends on the value of the local deformation gradient F = ∇ only. That is: =
(F; X, t).
(15)
The dependence on t (a dependence that may be mediated through internal variables) is intended to accommodate the possibility of phenomena of volumetric aging and remodeling. The particular case in which both the density and the constitutive equations are independent of time is of interest. Indeed, if this is the case and if the image (B) is also independent of time, there is no growth. If, on the other hand, (B) does depend on time, then we have a case of pure boundary growth. We consider now the second picture, whereby it is not the points in the material manifold M but those in the body manifold B that count. The energy per unit volume in the coordinate system Y is given by: ˆ (FH; Y, t) = JH
(F; (Y ), t).
(16)
Other constitutive equations can be handled in a similar way, with due attention to their vectorial or tensorial character. Notice that in this picture there is no apparent boundary growth, but there may be volumetric growth, remodeling and aging, for two reasons. The first one is the possible dependence on time of the constitutive law (15), and hence of (16). The second reason is that , and hence also H, may depend on t. To complete the passage from the material-particle representation to the body-point representation, we need to deal with the relation between the respective time-derivatives. A straightfor-
∂ ∂t
Y,t
∂ = ∂t
X,t
∂ + ∂X I
X,t
∂I ∂t
.
(17)
Y,t
Recall (see, e.g., Epstein and Maugin (2000)) that the mass balance equation in the absence of mass flow is given by:
∂0 ∂t
= ,
(18)
X,t
where is the mass production per unit volume in t (B). In the body formulation, it reads:
∂ ˆ 0 ∂t
= JH + ˆ 0 tr(Lˆ H ) + JH Y,t
∂0 ∂I , ∂X I ∂t
(19)
where we have made use of Eq. (14) and where:
Lˆ H = H−1
∂H ∂t
.
(20)
Y,t
In the case that we have called pure boundary growth, the first term on the right-hand side of Eq. (19) vanishes. If, in addition, the density 0 (X, t) is uniform (i.e., independent of X), then also the third term disappears and we are left with just the volumetric growth source deriving from the time-dependent transformation , namely:
∂ ˆ 0 ∂t
= ˆ 0 tr(Lˆ H ).
(21)
Y,t
Similarly, the momentum balance in the material formulation is given by:
∂vi 0 ∂t
= P i + T,IiI
(22)
X,t
where vi are the spatial components of the velocity field, P i are the components of the momentum source per unit volume in t (B), and T iI are the components of the material Piola stress. The boundary conditions are to be imposed on the possibly moving boundary (specifically, for the case of boundary growth). The motion of the boundary may be prescribed a priori or coupled by an evolution equation to the values of the field variables. In the body formulation, the velocity of the body points is obtained as:
∂xi (X I (Y ˛ , t)) vˆ = ∂t i
= vi + FIi Y,t
∂I . ∂t
(23)
The counterpart of Eq. (22) is: i
∂ˆv ˆ0 ∂t
i˛ = JH P i − i + Tˆ,˛
(24)
Y,t
The extra volumetric momentum-source term i is given explicitly as:
i = ˆ0
i
Fˆ ˛,ˇ
˛ i ∂2 ˛ ∂ ˛ ∂ ˇ i ∂
+ 2 vˆ ,˛ + Fˆ ˛ ∂t ∂t ∂t ∂t 2
(25)
where Fˆ is the deformation gradient of , ˆ as defined in Eq. (13), and
t is the inverse of the function t at fixed t. Example 3.1. Self-similar uniform boundary growth. Let the boundary of a body expand at a uniform rate b > 0 in a spherically symmetric manner. We set: X I = I (Y, t) = ıI˛ Y ˛ (1 + bt).
(26)
M. Epstein / Mechanics Research Communications 37 (2010) 453–457
A straightforward calculation yields: (LH )˛ ˇ =
∂ ˛
bY ˛
∂2 ˛
such that the field of inverse implant maps (i.e., the so-called intermediate configuration) is integrable.
2b2 Y ˛
b ı˛ , , = . =− 2 1 + bt ˇ ∂t 1 + bt ∂t 2 (1 + bt)
(27)
Assuming that for the original boundary growth problem the mass production and the momentum source P i vanish and the density 0 is uniform, the governing equations of the equivalent volumetric growth problem for the field variables ˆ 0 (Y, t) and xˆ i (Y, t) are: 3b ∂ ˆ0 , = ˆ0 1 + bt ∂t
(28)
and ˆ0
∂2 xˆ i i˛ = −i + Tˆ,˛ , ∂t 2
where i = ˆ0
∂2 xˆ i
(29)
b2 Y ˛ Y ˇ
∂Y ˛ ∂Y ˇ (1 + bt)2
−2
∂2 xˆ i bY ˛ ∂ˆxi 2b2 Y ˛ + ˛ ∂Y ∂t 1 + bt ∂Y ˛ (1 + bt)2
(30)
If the original constitutive law is given by: j
T iI = f iI (FJ ), we have:
(31)
2 iI Tˆ i˛ = (1 + bt) ı˛ I f
∂ˆxj
ˇ
ıJ
,
(32)
whose body-divergence is:
∂f iI 2 i˛ ˛ Tˆ,˛ = (1 + bt) ıI j ∂ZJ
A problem of boundary growth includes the specification of a time-dependent map t , whose physical meaning is the collection of material particles encompassed by the body at time t. After introducing the distinction between the material and the body representations, we have shown that the kinetics of a boundary growth process is equivalent to that of a purely volumetric growth counterpart represented by Eqs. (19) and (24). By equivalent we mean that the solutions of both problems yield the same distribution of spatial mass density at each instant of time. In the material representation all fields are referred to putative material particles, whose time of birth and/or death needs to be incorporated into the picture. In the body representation, on the other hand, no such accounting is necessary, since the body points persist throughout the entire deformation process. In a continuum theory the notion of material particle may thus play only an auxiliary role. Roughly speaking, the question may not so much be where a particle is at any given time but rather how many particles are to be found in any given spatial neighbourhood.
This work has been supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). References
∂2 xˆ j j J
4. Conclusion
Acknowledgement
∂Y ˇ 1 + bt
457
ˇ J
Z =(∂ˆxj /∂Y ˇ ) (ı /(1+bt))
ˇ ıJ
∂Y ˛ ∂Y ˇ 1 + bt
.
(33)
If the term i is negligible, it should be clear that the volumetric growth problem reduces to a case of ordinary anelastic evolution
Cowin, S.C., Hegedus, D.H., 1976. Bone remodeling. I. Theory of adaptive elasticity. J. Elasticity 6, 313–326. Epstein, M., Maugin, G.A., 2000. Thermomechanics of volumetric growth in uniform bodies. Int. J. Plasticity 16, 951–978. Segev, R., 1996. Growing bodies and the Eshelby tensor. Meccanica 31, 507–518. Skalak, R., Dasgupta, G., Moss, M., Otten, E., Dullemeijer, P., Vilmann, H., 1982. Analytical description of growth. J. Theor. Biology 94, 555–577.
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
a r t i c l e
i n f o
Article history: Received 23 December 2009 Received in revised form 14 March 2010 Available online 15 June 2010 Keywords: Surface growth Volumetric growth
a b s t r a c t 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 ramifications, has been drawn in the classical article by Skalak et al. (1982). In the case of volumetric growth, the full power of Continuum 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 legitimately 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 contradistinction 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 interior 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 gathered by further assuming that at each instant of time there exists a globally stress-free reference configuration. This reference configuration 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 configuration 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: [email protected]. 0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.06.004
is an elastic solid with a stress-free state defined uniquely up to a rotation and that the (stress-free, time-dependent) reference configuration 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 question 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
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M. Epstein / Mechanics Research Communications 37 (2010) 453–457
An example of such a transformation for the case depicted in Fig. 1 is the following: =
LX , L + kt
−∞ < X < ∞, t ≥ 0.
(7)
In the new coordinate plane , , the lines X = −L − kt and X = L + kt become, respectively, = −L and = +L, as can be verified by direct substitution. In this way, the problem is transformed into the standard form of the initial-boundary-value problem. In the new domain of definition, the mass density is given by: ˆ 0 (, ) = Fig. 1. Boundary growth of a supported bar.
that g(t) > f (t) for all t ≥ 0. The (elastic) constitutive equation is prescribed as:
T =E
∂x −1 ∂X
,
(1)
where T is the (relevant component of) the Piola stress, E is a (positive) modulus of elasticity and x is the longitudinal coordinate of points in the deformed bar. In the absence of body forces, the motion of the bar is governed by the law of momentum balance, whose local form is: ∂2 x ∂T − 0 2 = 0. ∂X ∂t
(2)
Taking into consideration the assumed constitutive law, the function x = x(X, t) is governed in this case by the one-dimensional wave equation: E
∂2 x ∂2 x − 0 2 = 0. ∂X 2 ∂t
(3)
The quantity c = E/0 represents the speed of sound in the bar. We assume that:
df (t) dg(t) dt , dt < c ∀t ≥ 0.
(4)
Under these conditions, the problem represented by Eq. (3) with the boundary conditions: x(f (t), t) = f (t),
x(g(t), t) = g(t)
∀t ≥ 0,
(5)
and with (consistent) initial conditions of position and velocity on the segment [−L, L] at t = 0, is a well-posed hyperbolic problem. It represents a rod supported at both ends, where the supports occupy at each instant of time the positions of the ends of the growing beam. A similar formulation can be conceived for free ends, in which case the X-derivative is made to vanish by the boundary conditions. Fig. 1 shows the particular case f (t) = −L − kt and g(t) = L + kt where k is a positive constant smaller than c. A few characteristic lines suggest schematically how this problem can be solved by the method of characteristics for the case of fixed supports. There is, on the other hand, an alternative approach to solve this problem, namely, by “straightening” its domain so as to convert it into the rectangular domain of a standard wave-equation formulation. To achieve this end, it is sufficient to provide any smooth (at least C 2 ) and smoothly invertible transformation, (X, t) → (, ), that renders f (t) and g(t) constant (−L and L, respectively). In particular, we consider transformations of the form: = (X, t) =t
(6)
∂X 0 . ∂
(8)
Clearly, this density varies with time. We may then say that, as far as the new and “nicer” configuration (nicer because it is independent of time) is concerned, there is a phenomenon of volumetric growth, not experienced by the original (boundary growing) configuration. Similarly, the constitutive Eq. (1) is transformed into:
Tˆ = E
∂ˆx/∂ −1 ∂X/∂
,
(9)
where we have denoted by x = xˆ (, ) the spatial placements in terms of the new reference configuration. Notice that in the onedimensional case under consideration the transformed Piola stress Tˆ does not need any corrective multipliers in terms of T. The physical meaning of Eq. (9) is that, as far as the new reference configuration is concerned, in addition to mass growth, the body undergoes also a process of remodeling. There is still an additional price to be paid for the new domain: a concomitant modification of the equation of motion and of the boundary conditions. The details of the transformed equation of motion can be obtained by direct substitution. For our purposes, however, what is of interest is that this equation can be cast in the form: ∂Tˆ ∂2 xˆ − ˆ 0 2 = . ∂ ∂
(10)
The term appearing on the right-hand side of this equation is a sort of reactive body force absent in the original formulation. Its exact expression is not of interest at this point. We note, however, that this is a “follower force”, in the sense that it is a function of the deformation gradient ∂ˆx/∂, the velocity gradient ∂2 xˆ /∂∂ and the second gradient of the deformation ∂2 xˆ /∂ 2 . This sort of material Coriolis force (which vanishes if is independent of t) is the result of having adopted a peculiar reference configuration, whose legitimacy, however, cannot be disputed. Recall that in the field equations of volumetric growth one usually accounts for additional terms that represent momentum sources other than the canonical source represented by the passively incoming mass (see, e.g., Cowin and Hegedus (1976), or Epstein and Maugin (2000)). Recall, moreover, that such additional terms are not necessarily subject to the principle of material frame indifference, since they are not necessarily constitutive in nature. They may represent, for example, the reaction of a colony of bacteria to the local slope or curvature of the continuum substrate. For an observer intent on keeping a fixed reference configuration, the phenomenon that we originally described as boundary growth is nothing of that sort. No particles are created or destroyed anywhere. Instead, this observer reckons a process of volumetric growth and remodeling with volumetrically distributed momentum sources. To be sure, in biological applications, the speed of the boundary growth is several orders of magnitude smaller than the speed of sound. As a result, the magnitude of the body forces is negligible. A case of particular interest because of its illuminating simplicity is given by the growth functions: f (t) = −L + kt and g(t) = L + kt. In this case, the two bounding lines of the original domain are
M. Epstein / Mechanics Research Communications 37 (2010) 453–457
455
Fig. 2. Creeping boundary growth.
parallel. In physical terms, what is lost on one end is matched by an equal growth on the other end. The transformation = X − kt clearly does the job of straightening the domain. In this particular case, therefore, the left-hand side of the equation of motion (2) remains formally unchanged, and the only price to be paid for the steady material snake-like creeping of the bar to the right, is given by the appearance of the reactive body force:
= ˆ0
k2
∂2 xˆ ∂2 xˆ − 2k 2 ∂∂ ∂
,
prescribed explicitly. More realistically, this motion may be coupled with the solution by means of an evolution equation (specifying, for example, the speed of the bar end as a function of the reaction at the corresponding support). In closing this section we notice that, in the one-dimensional case, a global stress-free configuration always exists. We can, therefore, envisage also the inverse transformation from volumetric to boundary growth of the very special kind just described.
(11) 3. The general case
which, as already pointed out, is negligible in most applications, since usually k c. The boundary conditions are: xˆ (−L, ) = −L + k and xˆ (L, ) = L + k. Fig. 2 shows the history of a material particle in the boundary-growth representation. In this case, the observer presumes to know what a “real” particle is, but pays the price of birth and mortality for each such particle. In the straightened domain of Fig. 3, on the other hand, particles are immortal and their world lines correspondingly infinite. It may be claimed that the first representation is better because it has a simpler constitutive law. But this fact cannot be known a priori in all cases. In real phenomena, involving three-dimensional bodies, there may not exist a stress-free reference configuration and the decision as to which portion of the growth is happening at the boundary and which portion occurs in the interior cannot be made canonically. Any contemplated motion of the material boundary (that is, in the material body-time manifold), whether directly or constitutively prescribed, can be transformed away by the technique just shown. The very notion of material particle is thus literally in the eyes of the beholder. If what matters in the end is the solution of the initial-boundary value problem in the sense that it delivers the mass density as a function of physical space and time, then which “particles” are responsible for the presence of such density may not be an answerable, or even relevant, question. Remark 2.1. One may reasonably claim that a particle can be identified as the carrier of a mark, such as an ink dot drawn on the instantaneous boundary of the body. If this dot disappears as it is covered by material, it is reasonable to claim that surface growth has taken place. These ideas, however, do not fall within the realm of a mechanical theory. We have considered the case in which the motion of the boundary of the time-dependent stress-free reference configuration is
The preceding example suggests that a similar point of view may be adopted for the general three-dimensional case. This generalization, however, is not a trivial one as it requires a more sophisticated conceptual framework due, in part, to the fact that, whereas the boundary of a line consists of just two points, the boundary of a body is a two-dimensional manifold. We start by assuming the existence of a material universe, namely, a three-dimensional manifold M of material particles with no a-priori mechanical properties. In this context, a material body is defined as a connected trivial three-dimensional body manifold B together with an embedding : B → M. Without this embedding, the manifold B serves mainly the purpose of establishing, once and for all, the topology of the body. Accordingly, the body manifold B is a time-independent entity consisting of points without any fixed a-priori material content, just as the line segment −L ≤ ≤ L in our one-dimensional example. The embedding can be considered as serving the function of capturing material particles, just like a cookie cutter applied to a sheet of dough. In the terminology of Segev (1996), who was the first to use this idea in the context of growth, the embedding is called a content. In the conventional treatment of Continuum Mechanics, this embedding is established ab initio and, therefore, can be incorporated into the very definition of the body, without any explicit mention. In the case of growth and remodeling, however, we may allow the capturing embedding to be a (smooth) function of the time variable t (measured with respect to some defining event). When this fact needs to be emphasized, we will use the notation t . Proceeding to the notion of configuration, we immediately see that (in a manner somewhat analogous to the Lagrangian and Eulerian formulations of the equations of balance) we have two possibilities to define a configuration. On the one hand, we may
Fig. 3. Volumetric-growth representation of the creeping bar.
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M. Epstein / Mechanics Research Communications 37 (2010) 453–457
define a configuration as an embedding: : (B) → E3 .
ward application of the chain rule yields:
(12)
Physically, in this first version, which we call the material formulation, we consider how the material particles comprising the content of the body at a particular time are mapped into the Euclidean space E3 . On the other hand, we also can define a configuration as an embedding: ˆ : B → E3 .
(13)
In this version, which we call the body formulation, we are describing how the body points themselves are mapped into E3 . In the classical case, whereby is independent of time, this distinction is not necessary. But in the case of a growing body it provides us with a fruitful choice: we may operate on material particles or on body points. The notion of boundary growth can arise only in the first formulation. Boundary growth of the pair {B, } is said to take place whenever the set (∂B) depends on time. To establish the relation between both formulations, we start from the notion of local material density. Mathematically speaking, the mass density is a three-form. Thus, given the density 0 in (B), the density ˆ 0 in B is defined as the pullback of 0 by . More specifically, let (Cartesian) coordinate systems X I and Y ˛ be chosen in M and B, respectively, and let 0 (X, t) be the density at time t at the point X ∈ t (B). Denoting by H(t) the differential of t , the mass density at Y = −1 t (X) is given by: ˆ 0 (Y, t) = JH(Y,t) 0 (X, t),
(14)
where JA denotes the determinant of A. Notice that, in the context of our one-dimensional example, the Y ˛ -coordinates correspond to the single coordinate , while the map −1 corresponds to the t transformation given by Eq. (6). Consider next a scalar constitutive quantity, such as the free-energy density per unit volume of the material manifold (in a given chart X I ), and assume, for simplicity of the exposition, that it depends on the value of the local deformation gradient F = ∇ only. That is: =
(F; X, t).
(15)
The dependence on t (a dependence that may be mediated through internal variables) is intended to accommodate the possibility of phenomena of volumetric aging and remodeling. The particular case in which both the density and the constitutive equations are independent of time is of interest. Indeed, if this is the case and if the image (B) is also independent of time, there is no growth. If, on the other hand, (B) does depend on time, then we have a case of pure boundary growth. We consider now the second picture, whereby it is not the points in the material manifold M but those in the body manifold B that count. The energy per unit volume in the coordinate system Y is given by: ˆ (FH; Y, t) = JH
(F; (Y ), t).
(16)
Other constitutive equations can be handled in a similar way, with due attention to their vectorial or tensorial character. Notice that in this picture there is no apparent boundary growth, but there may be volumetric growth, remodeling and aging, for two reasons. The first one is the possible dependence on time of the constitutive law (15), and hence of (16). The second reason is that , and hence also H, may depend on t. To complete the passage from the material-particle representation to the body-point representation, we need to deal with the relation between the respective time-derivatives. A straightfor-
∂ ∂t
Y,t
∂ = ∂t
X,t
∂ + ∂X I
X,t
∂I ∂t
.
(17)
Y,t
Recall (see, e.g., Epstein and Maugin (2000)) that the mass balance equation in the absence of mass flow is given by:
∂0 ∂t
= ,
(18)
X,t
where is the mass production per unit volume in t (B). In the body formulation, it reads:
∂ ˆ 0 ∂t
= JH + ˆ 0 tr(Lˆ H ) + JH Y,t
∂0 ∂I , ∂X I ∂t
(19)
where we have made use of Eq. (14) and where:
Lˆ H = H−1
∂H ∂t
.
(20)
Y,t
In the case that we have called pure boundary growth, the first term on the right-hand side of Eq. (19) vanishes. If, in addition, the density 0 (X, t) is uniform (i.e., independent of X), then also the third term disappears and we are left with just the volumetric growth source deriving from the time-dependent transformation , namely:
∂ ˆ 0 ∂t
= ˆ 0 tr(Lˆ H ).
(21)
Y,t
Similarly, the momentum balance in the material formulation is given by:
∂vi 0 ∂t
= P i + T,IiI
(22)
X,t
where vi are the spatial components of the velocity field, P i are the components of the momentum source per unit volume in t (B), and T iI are the components of the material Piola stress. The boundary conditions are to be imposed on the possibly moving boundary (specifically, for the case of boundary growth). The motion of the boundary may be prescribed a priori or coupled by an evolution equation to the values of the field variables. In the body formulation, the velocity of the body points is obtained as:
∂xi (X I (Y ˛ , t)) vˆ = ∂t i
= vi + FIi Y,t
∂I . ∂t
(23)
The counterpart of Eq. (22) is: i
∂ˆv ˆ0 ∂t
i˛ = JH P i − i + Tˆ,˛
(24)
Y,t
The extra volumetric momentum-source term i is given explicitly as:
i = ˆ0
i
Fˆ ˛,ˇ
˛ i ∂2 ˛ ∂ ˛ ∂ ˇ i ∂
+ 2 vˆ ,˛ + Fˆ ˛ ∂t ∂t ∂t ∂t 2
(25)
where Fˆ is the deformation gradient of , ˆ as defined in Eq. (13), and
t is the inverse of the function t at fixed t. Example 3.1. Self-similar uniform boundary growth. Let the boundary of a body expand at a uniform rate b > 0 in a spherically symmetric manner. We set: X I = I (Y, t) = ıI˛ Y ˛ (1 + bt).
(26)
M. Epstein / Mechanics Research Communications 37 (2010) 453–457
A straightforward calculation yields: (LH )˛ ˇ =
∂ ˛
bY ˛
∂2 ˛
such that the field of inverse implant maps (i.e., the so-called intermediate configuration) is integrable.
2b2 Y ˛
b ı˛ , , = . =− 2 1 + bt ˇ ∂t 1 + bt ∂t 2 (1 + bt)
(27)
Assuming that for the original boundary growth problem the mass production and the momentum source P i vanish and the density 0 is uniform, the governing equations of the equivalent volumetric growth problem for the field variables ˆ 0 (Y, t) and xˆ i (Y, t) are: 3b ∂ ˆ0 , = ˆ0 1 + bt ∂t
(28)
and ˆ0
∂2 xˆ i i˛ = −i + Tˆ,˛ , ∂t 2
where i = ˆ0
∂2 xˆ i
(29)
b2 Y ˛ Y ˇ
∂Y ˛ ∂Y ˇ (1 + bt)2
−2
∂2 xˆ i bY ˛ ∂ˆxi 2b2 Y ˛ + ˛ ∂Y ∂t 1 + bt ∂Y ˛ (1 + bt)2
(30)
If the original constitutive law is given by: j
T iI = f iI (FJ ), we have:
(31)
2 iI Tˆ i˛ = (1 + bt) ı˛ I f
∂ˆxj
ˇ
ıJ
,
(32)
whose body-divergence is:
∂f iI 2 i˛ ˛ Tˆ,˛ = (1 + bt) ıI j ∂ZJ
A problem of boundary growth includes the specification of a time-dependent map t , whose physical meaning is the collection of material particles encompassed by the body at time t. After introducing the distinction between the material and the body representations, we have shown that the kinetics of a boundary growth process is equivalent to that of a purely volumetric growth counterpart represented by Eqs. (19) and (24). By equivalent we mean that the solutions of both problems yield the same distribution of spatial mass density at each instant of time. In the material representation all fields are referred to putative material particles, whose time of birth and/or death needs to be incorporated into the picture. In the body representation, on the other hand, no such accounting is necessary, since the body points persist throughout the entire deformation process. In a continuum theory the notion of material particle may thus play only an auxiliary role. Roughly speaking, the question may not so much be where a particle is at any given time but rather how many particles are to be found in any given spatial neighbourhood.
This work has been supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). References
∂2 xˆ j j J
4. Conclusion
Acknowledgement
∂Y ˇ 1 + bt
457
ˇ J
Z =(∂ˆxj /∂Y ˇ ) (ı /(1+bt))
ˇ ıJ
∂Y ˛ ∂Y ˇ 1 + bt
.
(33)
If the term i is negligible, it should be clear that the volumetric growth problem reduces to a case of ordinary anelastic evolution
Cowin, S.C., Hegedus, D.H., 1976. Bone remodeling. I. Theory of adaptive elasticity. J. Elasticity 6, 313–326. Epstein, M., Maugin, G.A., 2000. Thermomechanics of volumetric growth in uniform bodies. Int. J. Plasticity 16, 951–978. Segev, R., 1996. Growing bodies and the Eshelby tensor. Meccanica 31, 507–518. Skalak, R., Dasgupta, G., Moss, M., Otten, E., Dullemeijer, P., Vilmann, H., 1982. Analytical description of growth. J. Theor. Biology 94, 555–577.