Mechanics of growing materials

Int. Z Mech. Sci. Vol. 20, pp. 493-504 © Pergamon Press Ltd., 1978. Printed in Great Britain

0020-7403F/810801--0493/$02.0010

MECHANICS OF GROWING MATERIALS J. L. NOWlNSKlt University of Delaware, Newark, DE 19711, U.S.A. (Received 26 October 1977; in revised form 21 February 1978) Summary--Assuming the constitutive equations in the form proposed by so-called linear hygrosteric materials, systematization and generalization of known for a class of three-dimensional problems are made. The behavior of an hollow cylinder subject to an internal uniform pressure, in the presence of acting in the material, is analysed in some detail.

a0, a,b a(t), t~(t), a0, b0 a~ d,, de, F0, Go d~j,d ~, dt~. g'J p p(t) p~, q, q(t),q0 r, 0, z t to t0~ tit, ~J ~j tf~ u~ v~= t~ x~ A, B, C, G, K A*, B* C,, C2 D/Dt E F G,, G2 Q,(r), Q2(r) a, ~8 8,, 8e 3, 3,,.2 x A,/~ 7r(t) p, po A, A*

Noll for the the solutions infinitely long mass sources

NOTATION material coefficients radii of circular tube coefficient, equation (2.28) coefficients, equation (2.14a) Euler deformation tensor metric tensor stimulation factor isotropic pressure coefficients, equation (1.1) strength of mass sources cylindrical coordinates time initial hydrostatic stress stress component, equation (3.6) Cauchy stress tensor extra stress stress flux spatial vector velocity cartesian rectangular system (also x, y, z) coefficients, equation (2.30a) coefficients, equation (4.1la) integration constants material derivative Young's modulus Newtonian viscosity coefficients, equation (2.34c) functions defined by (4.9a) preassigned constants exponents, equations (2.22) constant, equation (4.21) exponents, equations (2.15c) coefficients equations (2.11) Lam~ constants internal pressure mass density functions defined by (4.23).

I. INTRODUCTION T h e R a n d o m H o u s e D i c t i o n a r y defines the c o n c e p t of g r o w t h as " t h e i n c r e a s e b y n a t u r a l d e v e l o p m e n t , as a n y living o r g a n i s m , or part, b y a s s i m i l a t i o n of n u t r i m e n t " . A l t h o u g h r e a s o n a b l e in e s s e n c e , this d e f i n i t i o n s e e m s to b e t o o n a r r o w i n s c o p e , s i n c e the i d e a of g r o w t h m u s t n o t n e c e s s a r i l y be c o n f i n e d to o r g a n i c m a t t e r a l o n e . I n a b r o a d s e n s e of this w o r d , s u c h p r o c e s s e s i n v o l v i n g i n e r t m a t t e r as, f o r e x a m p l e , c r y s t a l g r o w t h , f o r m a t i o n of glaciers, m e l t i n g a n d a b l a t i o n , c o r r o s i o n , a n d r a d i o a c t i v e d e c a y , m a y also b e c a t e g o r i z e d as g r o w t h ( r e c k o n e d p o s i t i v e or n e g a t i v e as the c a s e tH. Fletcher Brown Professor Emeritus. 493

494

J.L. NOWlNSKI

may be). Actually, the so-called "law of growth" d N = --+AN dt, where N is a number of unspecified corpuscles and A a coefficient independent of N and the time t, is considered as applying in many fields of physics and chemistry. It seems to be an undeniable fact that processes of a predominantly physical character involving living matter should be governed by the same physical laws that govern processes involving ponderable matter. Am6ng these are the second law of thermodynamics, the principles of conservation of mass and energy, the balances of linear and angular momenta, and others. Probably the first, and rather elementary, attempt to apply the laws of physics to the growth of organic matter was made by D'Arcy T h o m p s o n in 1942. ~ H o w e v e r , prior to (and after) this data many experimental data on growth and its dependence on various stimulating and debilitating factors have been collected. To such factors belong light, heat, water or nutrient supply, the action of mechanical loads, and so on.? There has also been accumulated considerable literature on the mechanical properties of biological materials in vitro (see Ref. 5); literature of this kind, however, has little bearing on our understanding of the growth of living matter. Even more important, many of the known investigations were of a purely qualitative nature, those which were not were often too particular to provide sufficient material for general conclusions of a quantitative character. This remark concerns first of all the central question of any mechanical theory of real materials--the question of the constitutive equations. A significant development in the approach to the mechanics of growing materials took place just about the year 1970 when two remarkable papers appeared, written by Hsu, 2 and Strauss. 6 In the first of these the process of growth has been analysed quite extensively, and the influence of mechanical loads of various types investigated. The constitutive equations adopted by Hsu are of a differential character. A different line of approach was taken by Strauss; this author characterized the properties of growing materials by means of functional relations, and assumed that growing materials with m e m o r y lend themselves to description by means of bilinear functionals. In the present note, we adopt the standpoint of Hsu. To this purpose, we make use of a linearized version of the constitutive relations derived by Noll (Ref. 7, p. 31, and Ref. 8, p. 331) for the so-called fluent subclass of materials known as hygrosteric. A particular family of this subclass includes materials termed, by their inaugurator Truesdell, 9 hypoelastic materials of grade zero. Of course, the applicability of these or other constitutive equations for a characterization of the mechanical properties of the materials of the given class, may be decided only on the basis of experiments. For lack of the latter, it seems permissible to adopt a more liberal point of view, namely, to speculate what conclusions can be drawn from a set of constitutive equations which seem likely to describe at least some characteristic features of the given class. In the case investigated in the present paper, involving plant and bone tissues, the main mechanical feature seems to be the viscoelasticity of the materials. With regard to the osseous tissue, most researchers agree that the three-parameter model, known as the standard linear viscoelastic model, adequately describes the behavior of this material under load. I°'11 The botanic materials, on the other hand, are still a tabula rasa. From observations we know only that under the application of load they display a sudden deformation and, subsequently, an increase of the deformation with time if the load is maintained constant (the so-called creep phase). The simplest rheological model manifesting a similar behavior is the Maxwell model (Ref. 12, p. 6) analytically described by the constitutive equation dtr _ ql de dt Pl dt

1 tr, Pl

?For the literature compare Ref. (2). See also Refs. (3) and (4).

(1.1)

495

Mechanics of growing materials

w h e r e tr is t h e s t r e s s , e t h e s t r a i n , p l = F I E a n d q~ = F w i t h E as Y o u n g ' s m o d u l u s a n d F as t h e c o e f f i c i e n t o f N e w t o n i a n v i s c o s i t y . t A comparison of the preceeding equation with Noll's linearized constitutive e q u a t i o n (2.1) g i v e n b e l o w s h o w s t h a t b o t h t h e s e e q u a t i o n s b e l o n g to t h e s a m e c l a s s o f c o n s t i t u t i v e e q u a t i o n s c h a r a c t e r i z i n g t h e r a t e d e p e n d e n t m a t e r i a l s ; in p o i n t o f f a c t , the equations are almost identical. For want of something with a more rational f o u n d a t i o n , t h e s i m p l i c i t y o f N o l l ' s e q u a t i o n s u g g e s t s t h a t it m a y b e u s e d to s o m e a d v a n t a g e in t h e a n a l y s i s o f g r o w i n g m a t e r i a l s . Guided by this belief, we systematize, and somewhat generalize the solutions derived by Hsu for a number of particular three-dimensional problems. We also give a solution to the problem involving an infinite hollow cylinder made of a hygrosteric m a t e r i a l a n d s u b j e c t to a u n i f o r m i n t e r n a l p r e s s u r e . W h e t h e r b y c h a n c e o r n o t , t h e r e s u l t s o b t a i n e d f r o m t h e t h e o r e t i c a l f o r m u l a (4.23b), f o r c e r t a i n a r b i t r a r i l y s e l e c t e d values of the parameters a and/3 = 2a, do not deviate drastically from the practically e s t a b l i s h e d f a c t s v i a t h e f o r m u l a (4.23a); this h a p p e n s at l e a s t f o r t h e v a l u e s o f a < 0.3.

2. GENERAL EQUATIONS Consider a viscoelastic material referred to a cartesian rectangular coordinate system x, i = l, 2, 3,. (xl = x, x2 = y, x3 = z) and defined by the linearized constitutive equations of Noli ([7], p. 31; [8], p. 331)~: ~j = (ao + a t , , + Ad~),$ii + 2btij + 2ttd~j,

(2.1)

where t o is the Cauchy stress tensor, ~i the objective stress rate (or stress flux)§, and where d~j = (vi.j + vi.i)/2

(2.2)

Dxi vi = ,~i =- D t

(2.3)

is the Euler deformation rate tensor with

as the velocity and Dui Dt

(2.4)

tgu~ + uljv~ at "

as the material derivative of a spatial vector u~. We note that an index preceded by a comma denotes differentiation with respect to the corresponding coordinate, and Einstein's summation convention over the repeated indices is used throughout; the coefficients a0, a, and b, as well as the Lam6 coefficients A and/x, are considered to be constants as long as the material is homogeneous and the conditions isothermal, hypotheses to be assumed from no on. Suppose now that the state of stress is defined by the matrix

tli =

t.(t)

0

0

0 0

t22(t ) 0

t33(t)

L

0

;

(2.5)

= t22(t)

this means that the state is homogeneous but changes with time. Suppose also that, in view of the fact that during any process of growth the motions are slow, the inertia effects can be disregarded and a quasi-static line of approach adopted. If this is so, the r.h.s, of the equations of motion, tlj.j = p -D-T'

(2.6)

may be set equal to zero. It should be clear that the stress system (2.5) satisfies the quasi-static equations of motion identically. Imagine now that the velocity field is defined by the equations vl = a x ,

v2 = B y ,

va = ~ z ,

(2.7)

where a and/3 are certain preassigned constants. tUndesirably, however, the creep deformation of a Maxwell model does not tend to a finite limiting value. ~:As introduced by Zaremba in 1903. §See Ref~ (8), p. 111. Below we employ the stress flux in the form proposed by Trusdell, Ref (9), equation (1.2). See equation (2.13).

496

J.L.

NOWINSKI

We first discuss the question of the mass density. We note that, since the body is supposed to be growing, the mass sources must be present within the body. Let the strength, q ( t ) , of the latter, per unit volume and per unit time, be independent of the position but remain a function of time. The equation of balance of mass then becomes (see Ref. 6, equation 2.5) ~t+PV~j

(2.8)

= q(t),

or, specifically, -t + (a + 2/3)0 = q ( t ) .

(2.9)

The solution of the preceding equation, if q ( t ) = qo is assumed to be independent of time, is P-- = e-(=+za"(1 - K ) + K,

(2.10)

p0

where p0 is the initial mass density, at the time t = 0, and K

q0

po(a + 2~)"

(2.11)

The mass being a non-negative quantity, remains so when t --*m. This gives the inequality a +2/3 > 0

(2.12)

if q0 > 0, as assumed above. It follows that, for the body to remain continuous, the velocity gradients, or the growth characteristics, a and AS, cannot be prescribed arbitrarily. Fig. 1 displays the variation of the mass density vs time for various values of the parameter K. It is seen that p decreases with time, and tends asymptotically to a limiting value which happens to be greater with increasing values of the coefficient r. If one decides to retain the nonlinear t e r m s t in the Truesdell stress flux (this does not lead to any complications in the calculations) [ii -- Oti~ + -- " ~ tij.ml)rn -- timl)i.m - tmil)Lm + ti~d,~,~,

(2.13)

then the constitutive equations (2.1) become at=, ~. d t t l , - 2atz2 = F0, at a t 2 2 + d2t:2 - a t N = Go, Ot

K

(2.14)

= 1.00

0.75

0.5,

.

.

.

.

0.2.'

ll.O

a2.0

t3.0

(u + 2~)t

FIG. 1. Variation of mass density vs time. t i t is, of course, conceivable, if one has a more formal turn of mind, to disregard these terms.

M e c h a n i c s of growing materials

497

where dl = - a + 2l] - a - 2b, de = a - 2(a + b), Fo = a o + A ( a + 2/3) + 2/~a,

Go = ao + A (a + 2/]) + 2/#3.

(2.14a)

Solution of the preceding equations is straightforward and gives . a ( d 2 F o + 2aG0) d l d 2 - 2a 2 '

t . = 1 [Cl(vl + d2) e v'' + C2(y2 + d2) e wt] * t2e = Ci e ,~t + C2 e

~t

, aFo +

-r ~

,

diGo

(2.15a) (2.15b)

where yl.2 = ~1 { - ( d 1+ de) -+ V'[(dl - de) e + 8ae]},

(2.15c)

d t d e - 2a e = ytye,

(2.15d)

and C~ and Ce are integration c o n s t a n t s to be determined later. It is evident that the roots y~, l = l, 2, are always real, but they m a y be either positive or negative. For w a n t of a n y data concerning the material coefficients a and di, i = 1, 2, we v e n t u r e the following crude analysis..Suppose for a m o m e n t that one is interested in a uniaxial case. F r o m equations (2.1), and for tt~ = (OtH/Ot)~ t m there follows ill = ao + (a + 2 b ) t , + (A + 2/~)dll.

(2.16)

A c o m p a r i s o n of the foregoing equation with Maxwell's equation (1.1) suggests that the following c o r r e s p o n d e n c e exists: (A + 2/~) ~

>q~=Et, pl

(2.17a)

-(a +2b) ~

1 E ~ ~-~= ~ ;

(2.17b)

thus, a +2b <0.

(2.17c)

W e o b s e r v e that the last inequality does not contradict Noll's inequalities (Ref. 7, equations 2.4) b#0,

3a + 2 b # 0 .

(2.18)

In view of the fact ([12], p. 84) that for p~ ~ 0 the Maxwell material t r a n s f o r m s into a viscous fluid (the ratio ql/Pz = p c e, with c as the wave velocity, remaining finite), it s e e m s permissible to e x p e c t that llp~ is large and, c o n s e q u e n t l y , that la + 2 h i ~ a , ~

(2.19)

for the slow p r o c e s s e s u n d e r study. F r o m the first two equations (2.14a), then dt ~- - ( a "+ 2b),

de ~ - 2 ( a + b),

(2.20)

and, by (2.15a),

O n a c c o u n t of the inequality (2.17c), therefore, there is r e a s o n to conjecture that both y~ and y, are negative. W e thus set, for clarity, 81 = -'y~,

8e = - y 2 ,

w h e r e 8~, 8e > 0. t N o t e that for P o i s s o n ' s ratio v = 0 we have, in fact, A + 2p. = E.

(2.22)

498

J.L.

NOWINSK!

Let us now concentrate on what m a y be called the natural growth. By this term we decide to understand a growth that is not affected by the action of an external load, i.e. is a growth free from stress: tij -= 0.

(2.23)

(ao + Adrr)~ij + 2l~d,~ = O,

(2.24)

The constitutive equations for the natural growth,

are obtained direct from equation (2.1); they yield the relations between the gradients of the growth velocities. A s s u m e that, in accordance with the situation existing in the problem under discussion, d~=0

for

i#j.

(2.25)

We then easily convince ourselves that, after disregarding the integration functions, we have v~ = - a ~x,

vy = - a ~y,

v: = - a ~z,

(2.26)

and* dli = - a n .

i = I. 2.3.

(2.27)

where a~

ao 2/x + 3A"

(2.28)

It follows that, in the case considered, the natural growth is isotropic. Since, by its very nature, the concept of growth involves the idea of expansion, and since the coefficients A, t~ > 0, then it is natural to conjecture that the coefficient a0 is negative. We note that, in the natural growth, the coefficients a and /3 and not independent but are related. Namely, a =/3 = - a * ,

(2.29)

as one could anticipate. U n d e r the stress- and deformation-rate s y s t e m s a s s u m e d before (equations 2.5 and 2.7), the growth ceases, of course, to remain natural and acquires the properties of what m a y be called transversely isotropic growth (with the plane yz as the plane of isotropy). Following Hsu, 2 it is possible to introduce the concept of the stimulation factor, p, defined as the ratio of the e x p a n s i o n d. = 3A + (3B + G)t, + (3C + K)t,.

(2.30)

occurring under the action of an external load. to the expansion 3A, taking place in the a b s e n c e of load; there 3B+G-

3a+2b 3A + 2p.'

3C+K=

1 3A + 2/z'

3A

3a0 3A + 2 ~ '

(2.30a)

so that, finally, the stimulation factor b e c o m e s

P =

(3a + 2 b ) t , - i , + 3ao 3ao '

(2.31)

or, by virtue of the constitutive equations (2.14),

P

(a + 2g)(3A + 2p.) 3ao

(2.32)

Bearing in mind the inequality (2.12), and our former conjecture regarding the sign of the coefficient a0 (i.e. a0 < 0), we arrive at the conclusion that, at least in the case considered, the stimulation factor is positive. Let us now return to the constitutive equations (2.15a,b) and a s s u m e that at the initial time, that is, at t = 0, we have tll(0) = t01,

t22(0) = t , ( 0 ) = to.,,

(2.33)

where to, and to: are certain prescribed constants. T h e preceding initial stress s y s t e m is an admissible s y s t e m due to the fact that the quasi-static equations of motion are satisfied, and the b o u n d a r y conditions are not specified. t A n underscore s u s p e n d s the s u m m a t i o n .

M e c h a n i c s of growing materials

499

A longer, but more trivial, calculation now gives: 1 ~(to2 - Gt)(8,. - d2) - Gz + atol (to,. - GO(dz - 8,) + G2 - atol } IH = a ( ~'2--"~ (d2 - 8 ~) e -~'' 4 6,. - 6~ (d., - 62) e -s2, + G2 ,

(2.34a)

t22

(2.34b)

(to2- Gi)(828,.-_d2)si - G,. + atol e-S,t + (toz - G~)(d26z-_6~)61+ G,. - atol e_~,.t + GI,

where aFo + diGo GI = d l d , _ 2a z, G2 = a(dzFo + 2aGo) did2 - 2a 2

(2.34c)

We observe that, as time proceeds, the stresses in the body tend asymptotically to limiting values equal to t , = G,_/a and t22 = G~, respectively. 3. P A R T I C U L A R C A S E S 1. H2vpoelastic material o / g r a d e zero In this particular case (Ref. 9, e q u a t i o n s 4.2), a o = a = b = 0;

(3.1)

thus, dt=2~-a,

d2=a,

"yl.~ = { --(/3 (/] + - aa) )'

(3.2)

From equations (2.4), we then find dtH F(2/] - a ) t . dt ~t~2

= X ( a + 2 / ] ) + 2/~a

+ tzt:: = A(a + 2/3) + 2/~/],

(3.3)

and this, apart f r o m the notation, agrees with the results obtained by G r e e n (Ref. 13, equation 2.6), if the f u n c t i o n s P and Q of that author are a s s u m e d to be independent of position. For the details of the further analysis we refer the reader to the paper of Green. 2. Uniaxial tension in the z-direction In this particular case we set

t,.z = t3a -- 0,

(3.4)

and find easily that GI = O,

G~, = atoz.

(3.5)

Equation (2.34a) then yields t , = G--22 a = t0~ at all times t.

(3.6)

The coefficients of growth, a and/3, are n o w determined from equations (3.5) which read (A + 2#~ + toOa + 2(A - to0/3 = - a o - (a + 2b)t0h Xa + 2(a + tx )13 = - a o - atot,

. (3.7)

This result agrees with that obtained in Ref. (14, p. 20), if one replaces the stress flux by the simple time derivative and modifies the notation appropriately. From equations (3.7) we now find immediately that 2 a = ~ {p,ao + [ao + ~,a + 2(A + / z ) b ]tot + at2o~}, 1

/] = ~ {2/~ao + [ao - 2Ab + 2t~a ]tol + atgl}, where A = 2[(3A + 2g.)p. + (2A + g.)tol].

(3.8a)

500

J.L.

NOWINSKI

3. l s o t r o p i c state o f stress

In this case, a = ~-=a0,

(3.9a)

t . = t22 = t33 ~ tic(t),

(3.9b)

and d, = a o - a - 2b,

d2 = a o - 2a - 2b,

F0 = Go = a0 + (3A + 2p.)a0, 82 = a 0 - 2 b ,

81=ao-(3a+2b), G2 = a G t , Gi

(3.10)

ao+ (3A + 2p.)a0

a0 - (3a + 2b) " Finally, G~t)-2-~= ( - ~ - 1 )

e-~'' + 1,

(3.11)

where to is the initial hydrostatic stress. It is seen that the (positive or negative) process of relaxation d e p e n d s on the relations (to/GO ~ 1 (Fig. 2). For (to/GO > I, the relative stress (tidGO decreases with time and for t--*co tends to the limiting value ( t J G O = 1. On the contrary, for (to/GO < 1, the relative stress increases with time to its limiting value ( t J G O = 1. For to = G~, the stress remains c o n s t a n t (see Ref. 15, p. 24). T h e p h e n o m e n a just described are g e r m a n e to the Maxwell model, and should c o m e as no surprise if one bears in mind our earlier r e m a r k s on the similarity between equations (l.l) and (2.1).

tis

G1 2.0"

to

1.5"

t

1.0

to Gi

0.5"

0

--:O.5

t

I

t

t.O

2.0

3.0

~ ~t

FIG. 2. Relaxation of isotropic stress for various values of to/G~. 4. C I R C U L A R T U B E U N D E R I N T E R N A L P R E S S U R E A n infinitely long circular tube of inner and outer radii t~0 and bo, respectively, is subject to an internal pressure w(t) varying in time. It is natural to refer the tube to a cylindrical coordinate s y s t e m x ~--- r, x 2E 0 and x 3 ~ z. T h e latter being a curvilinear s y s t e m , it is required to take into account the levels of indices (whether subscripts or superscripts), and to replace the ordinary differentiation by covariant differentiation.? It is easily s h o w n that for a vector v ~ - ( v m, v 2, v 3) referred to a cylindrical frame, a s s u m i n g axial s y m m e t r y and independence of the coordinate z, we have

;1= - ~ - ,

v :2

Or

r"

The equation of balance of m a s s (2.8) now b e c o m e s D E + pv :~= q.

~See Ref. (16, Section 1.12). A covariant derivative will be denoted by a semicolon.

(4.2)

Mechanics of growing materials

501

From the definition of the incompressibility of the material, assumed from now on, it follows that the mass density does not depend on time, Dp/Dt = 0. From the definition of the homogeneity of the material, on the other hand, it follows that the mass density does not depend on position. Thus, finally, p = P0, a constant. If we agree, moreover, that q = q0, a constant, then equation (4.2) becomes

Ovi+v.r__~J=p0' qo

i

di~" v :t -ffi~

(4.3)

where •

1

~-

i

d/ = ~ (v j + v :0-

(4.3a)

The solution of the equation (4.3) is v'

r + 2--~o r, = C--

(4.4)

where C is a constant provided that the motion is steady. In this case, the velocity of growth, v ~, does not explicitly depend on time. Let us, for simplicity, disregard the nonlinear terms in the stress flux (2.13). Then, Noll's constitutive equations (2.1) take the form

a?" a---t-= (ao + a?k ~ + Adkk)g ~ + 2b? 'j + 2p,d °,

(4.5)

where the fundamental metric tensor g'~ becomes g " = g33 = 1, g~2 = (l/r2), and gii = 0 for i # j. Let us now introduce, for the incompressible material considered here, the symbol

?,'J = t u + pgii

(4.5a)

for the so-called extra stress. The latter constitutes the sum of the symmetrical stress tensor, t ij, and the isotropic pressure, p(r, t), to be determined later from additional conditions. The equation of motion in the quasi-static form now becomes

?~-g'Jpj = 0 ,

(4.6)

or explicitly, ?" - r2F z

a?"

Or

+ - -

r

ap

Or

=

O.

(4.6a)

To avoid too many technicalities, which would obscure rather than clarify the matter, let us specialize the class of materials to be examined by assuming that a =0.

(4.7)

This assumption does not violate Noll's inequalities (2.18) as long as one agrees that b # 0. Bearing in mind the rules of juggling the indices (see Ref. 16, Section 1.10), we find

_vt_C

l qo

(4.8)

Consequently it is possible to cast the constitutive equations (4.5) into the form

0? I' at

2b?" = Ql(r),

OF ~ O---i--2bTn = Q2(r),

(4.9)

where

Q,(r)=ao+A_~oo+2tt [qo

~-

C~

rW'

O2(,)=(.0+~ ~)lmr + 2t~ /q0+C~ 1 ~-opo 7 / r m'

(4.9a)

The solution of the preceding equations is elementary and yields

~,,=D,e~,_l[A

2bC'~ I

~, -r-7-/

r2?22 = D2 e TM - 2~ [ A + 2bC~

7-/'

where Di, i = 1, 2, are integration constants and

(4. lO)

502

J . L . NOWINSK! A = a0+ (~t +A) q0. p0

(4.10a)

It is to be noted that, in view of the inequality (2.17c) and our assumption (4.7), the coefficient b appears to be negative. Suppose now that the initial stresses are those that occur in an elastic solid. This assumption assures us that the preassigned initial stresses satisfy the required compatibility conditions. Evidently, the elastic solution to the problem under examination is the well-known classical Lain6 solution (see Ref. 17, p. 59), ~.~1= _ B ~ + A , ' r-

,r22 = B* + A* 7

(4.1 l) '

where A*

402 *r(0), bo2 - 402

B *=

a°25°2 ~r(0), 502 - 402

(4.11a)

and *r(0) is the value of the internal pressure, 7r(t), at the time t = 0. With these in mind, we impose the condition that, at time t = 0 , the physical stress components t-" and r2t "22 a r e equal to z " and T 22, respectively. This gives D, = D, = A* + A zo "

C = - bB* I.t

(4.12)

so that, finally,

Equation of motion (4.6a) shows that the isotropic pressure is not a function of position, that is, p = p(t).

(4.14)

Since the outer surface of the tube is free from stress we arrive at the boundary condition t )l[5(t), t] = t-tt[b(t), t] - p ( t ) = 0.

(4.15)

This equation immediately determines the value of p ( t ) . It is evident that, in the velocity field defined by equation (4.4), the motion of a generic particle of the tube is described by the differential equation (4.16)

d r = ( C + - ~ o p o r ) dt.

Hence, the position, ,~(t), at time t of a particle originally (i.e. at t = 0) at the place r = 4o is determined by

a2(t) = (402 + 2C po% e,qotm,,_ 2C po. qo/

qo

(4.17a)

Similarly, for a particle originally at r = bo we have bZ(t) = (502 + 2C Po~ e,q¢o0,, _ 2C 0o. qoo/ qo

(4.17b)

At a fixed time, then, and for qo # 0, 5"(t) - ~ 5(t ) = e (q0/~),(5o2_ 402).

(4. i 8)

If q 0 = 0 , that is if mass sources are absent or inactive, then the preceding equation reduces to the well-known condition of incompressibility of the material (see Ref. 16, equation 3.4.6). In the presence of mass sources, however, the area of the cross section of the tube increases unboundedly with time [since (qot/p) is essentially positive). We can express this fact by stating that the body "grows". Equations (4.17a, b) give the inner and the outer radii of the tube in its deformed state. The second boundary condition, so far left aside, t"[~(t), t] = t-"[~(t), t] - p ( t ) = -*r(t)

(4.19)

specifies the value of the internal pressure It(t), acting on the tube. This result completes the solution of the problem under investigation, since a list of explicit formulae for

503

Mechanics of growing materials

t " , t n, and p(t) does not seem to be of special interest. We note, however, that, as the time goes by, the components of the extra stress (4.13) tend asymptotically to some non-zero limiting values (recall that b <0). Considering the fact that there exists an enormous variety of living materials--starting, say, from plants and ending with mammals--it is illusory to expect that any particular constitutive equations are able to adequately describe all classes of known materials. This conclusion remains true, of course, with regard to the specific constitutive equations (2.1) proposed by Noll. There seems to be nothlng basically wrong, however, if one gives the rein to one's imagination and speculates as follows. Suppose that we confine our theorization to the growth of the trunk of a tree--and denote by 6 the relation qo/po. Let the hole in the cylinder representing the core of the trunk be very small, so that ~o, ti ,~ bo, b.

(4.20)

If we now assume that the rate of increase of the outer radius of the cylinder, that is the rate of growth of the tree-rings, is constantt db

~-

= 3"b0,

(4.21)

where 3' is a constant, then /~ = (1 + yt)/~o,

(4.22)

A* =--/~o~ b: = (1 + 3'02.

(4.23a)

and

This may be compared with A

•

6"

~ o ~" es'

(4.23b)

obtained from equation (4.18). To arrive at something more tangible than a qualitative evaluation of our results, let us normalize the two preceding equations by the requirement that, for t tending to zero, the equations differ by quantities of the second and higher orders in time. Upon observing that the equation (4.23a) has a form reminiscent of the sum of the first three terms in the power series expansion of the function e ~t, we conclude that the exponent 8 may be set equal to 23'. With this value in mind, we calculate the values of A* and A versus time for 3' = 0.05, 0.1, 0.2 and 0.3. We ":observe that, at least for the values of 3' < 0-3, the corresponding curves for A* and A shown in Fig. 3 do not deviate drastically from each other.

A,A

Ai / 8

=o.6

I0 .

A

_A_

\/ /

5-

=

~~ ~ ~ . r . . ~

~ , ~ . ~ Y =o.2 8 =0.2 7" =0. I 8 =0.1 y:o.o5

I

0

i I

J 2

FIG. 3. Graphs of A and A* vs time

I 3 for

various values of X and 8

i 4 =

t

2%

tSuch an assumption is suggested by an examination of the actual distribution of the tree-rings repeatedly displayed, say, in Ref. (3). A striking example is provided by the cross section of a sequoia tree, about 2000 years old, exhibited in Longwood Gardens, Kennett Square, Pennsylvania.

504

I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

J . L . NOWlNSKI REFERENCES W. D'ARCY THOMPSON, On Growth and Form. Cambridge University Press, London (1942). FENG-HSIANGHSU, J. Biomech. l, 303 (1968). W. J. GLOCK, Principles and Methods of Tree-Ring Analysis. Carnegie Institution, Washington 0937). F. W. WEber and K. V. TmMANN, Phytohormones. Macmillan, New York 0937). H'. YAMADA,Strength of Biological Materials. Williams & Wilkins, Baltimore 0970). A. M. STRAUSS,Publ. Inst. Mathem., Nouv. Ser. 12(26), 131 (1971). W. NOLL, J. Rat. Mech. Anal. 4, 3 0955). A. C. EeaNGEN, Mechanics of Continua. Wiley, New York 0967). C. TRUESDELL,J. Rat. Mech. Anal. 4, 83 (1955). E. SEDLIN, Acta Orthop. Scand. Suppl. 83 (1965). V. H. FRANKELand A. H. BURSTEIN,Orthopaedic Biomechanics. Lea & Febiger, Philadelphia (1970). W. FLUOGE, Viscoelasticity. Blaisdeil, Waltham, Massachusetts (1967). A. E. GREEN, .7. Rat. Mech. Anal. 5, 637 (1956). FENG HSIANGHSU, Ph.D. Thesis, Rice University (1965). I. I. GOLBERO, Mechanical Behavior of Polymers. Chimya, Moscow 0970) (In Russian). A. E. GREEN and W. ZERNA, Theoretical Elasticity. Clarendon Press, Oxford (1954). S. P. TIMOSHENKOand J. N. GOODIER, Theory of Elasticity. McGraw-Hill, New York (1951).