II. Yoffe E.H., 1951.The moving Griffith crack. Phil Mag 42, 739-750.

12. Freund L.B., 1990. Dynamic Fracture Mechanics. Cambridge University Press, New York.

13 . Maugin G.A., 1988. Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam.

14. Stroh A.N., 1962. Steady state problems in anisotropic elasticity. J Math Phys 41, 77-103.

15 . Lothe J., Barnett D.M., 1976. Integral formalism for surface waves in piezoelectric crystals. Existence considerations. J Appl Phys 47, 1799-1807.

16. . Suo Z., Kuo C. M., Barnett D. M. , Willis, J.R., 1992. Fracture mechanics for piezoelectric ceramics. J Mech Phys Solids 40, 739-765

Ting T.C.T., 1996. Anisotropic Elasticity: Theory and Applications. Oxford University Press, Oxford .

18 . Shindo Y., Watanabe K., Narita F., 2000. Electroelastic analysis of a piezoelectric ceramic with a central crack . Int J Engng Sci 38 , 1-19.

19 . Daros C.H., Antes H., 2000. On strong ellipticity conditions for piezoelectric materials of the crystal classes 6 mm and 622. Wave Motion 31,237-253.

20 . McHenry K.D., Koepke B.G. , 1983 . Electric fields effects on subcritical crack growth in PZT. In Fracture mechanics ofCeramics (Edited by R. C. Bradt, D. P. Hasselman and F. F. Lange). 5, 337-352.

21. Park S.B., Sun c.T. , \995 . Effect of electric fields on fracture of piezoelectric ceramics. Int J Fracture 70, 203-216.

22. Kumar S., Singh R. N., \996. Crack propagation in piezoelectric materials under combined mechanical and electrical loadings. Acta Mater 44. 173-200.

)

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~

~

THERMOELECTROELASTIC SOLUTIONS FOR INTERNAL BONE REMODELLING UNDER CONSTANT LOADS

Qing-Hua Qin Department ofMechanics, Tianjin University, Tianjin, 300072, Chian; Department ofAMMl University ofSydney, Sydney, NSW 2006, Australia

Abstract: A theoretical solution is presented for analysing thermoelectroelasti( problems of internal bone remodelling subjected to coupling tensile load external lateral pressure, electric load and thermal load. Numerical results an presented to show the effect of thermal and electric loaq on bone remodel lin! process, which is helpful for better understanding healing process of injurec bone

Key words : bone remodeling, piezoelectric, thermal field

1. INTRODUCTION

It is recognized that living bone is ~ontinually undergoing processes 0

growth, reinforcement and resorption te;?ned "remodelling" under interna orland external loads. These remodelling processes are the mechanisms b) which bone adapts its histological structure to changes in long tl rrn loading Generally, there are two kinds of bone remodelling: internal and surface (Frost, 1964). The distinction between them is as follows. Interna remodelling refers to the resorption or reinforcement of existing bone b) decreasing or increasing respectively, the bulk density of the bone withir fixed external boundaries. Surface remodelling refers to the resorption OJ

position of bone material on the external surface of the bone. The bone nodelling capacity has been investigated by many authors (Charnay & ~hantz, 1972; Cowin & Hegedus, 1976; Cowin & Buskirk, 1978, 1979; win & Firoozbakhsh, 1981; Hart et ai, 1984; Tsili, 2000). In recent years, ive research in the area of some tissues such as living bone and collagen ; shown these materials to be piezoelectric (Fukada & Yasuda, 57; 1964) and that the piezoelectric properties of bone play an important e in the development and growth of remodelling of the skeleton. Gjelsvik 173) presented a physical description of the remodelling of bone tissue, in ~s of very simplified form of the linear theory of piezoelectricity. lliams and Breger (1974) explored the applicability of stress gradient ory that could be applied for explaining the experimental data for a ltiiever bone beam subjected to constant end load and showed that the ,roximate gradient theory is in good agreement with the experimental a. Guzelsu (1978) presented a piezoelectric model for analysing ttilever dry bone beam subjected to a vertical end load. Johnson et al. 80) further addressed the problem of dry bone beam by presenting some oretical expressions for piezoelectric response to cantilever bending of beam. Demiray (1983) gave some theoretical descriptions on electro­

~hanical remodelling models of bones. In this work, an analytical solution for thermoelectroelastic problems of

:mal bone remodelling, based on the theory of adaptive elasticity(Cowin -Iegedus, 1976), is presented to study the effects of thermal and electric js on bone remodelling process. A numerical example is considered to w applicability of the proposed solution.

THERMOELECTROELASTIC THEORY OF \.

INTERNAL BONE .REMODELLING

Linear theory of thermoelectroelastic solid

Consider a hollow circular cylinder of linear thermopiezoeiectric bone ::rials subjected to axisymmetric loading. The constitutive equations for moelectroelastic field are (Mindlin, 1974)

~

cr 1'1' = CII Srr + C 12 Saa + C13Szz

cr as = C I2 S 1'1' + ci isaa + c13 Szz

a = zz cIJs rr + cl Jsaa + cJJ szz

- diEz - PIT

- tp, T ~-> - dl Ez

- d JEz - PJT (1) = - d4 Er' Dr =d4 szr + KIE ,.a zr C44 szr

Dz =d ,(S rI' + Saa ) + d Jszz + KJEz - P3T

kzHzhI' = krHr' hz =

where a ij , D;, S ij ' E;, represent the components of stress, electric

displacement, strain, and electric field intensity respectively, cij are elastic

stiffness, d; are piezoelectric constants, K; are dielectric permittivities, T

denotes the temperature change, P3 is the pyroelectric constant, P; are

stress-temperature coefficients, h are the heat flow, H; are heat intensity,

and k; are the heat conduction coefficients. The corresponding strains, electric fields, and heat intensity are respectively related to the displacements U;, electric potential q>, and temperature change T as

U r srr=urr , SSa= s zz =uzz szr=uzr+urz

. r ' " (2)

EI' = - <p ,I' Ez = -<p ,z, HI' = - T,r, H z = - T,z

For quasi-stationary behaviour in the absence of heat source, free electric charge and body forces , the above basic set of equations for thermopiezoelectric theory of bone is completed by adding the equations of equilibrium for heat flow, stress and electric displacements

8cr 1'1' 8cr Zl' a 1'1' - cr aa 0 8cr Zl' 8a zz cr zr - 0- - +--+ - -- ,- -+--+ = , az r8r 8z r (3)

8hr + 8h: + ~ = 08Dr + 8Dz + Dr = 0, 8r 8z r ' 8r 8z r

2.2 Equation for internal bone remodelling ,~

The equations of the theory of adaptive elasticity of Hegedus and

Cowin (1976) will be used in this study. The remodelling rate equation in cylindrical coordinates is

e= A' (e) + A,E (e)Er + Az£ (e)Ez (4)

+A;r(e)(srr +sBB)+A:z(e)szz +A:z(e)srz

where e is a change in the volume fraction of bone matrix material from its

reference value, say ~ 0 , A' (e), At (e) and A~ (e) are material

coefficients dependent upon the volume fraction e.

3. SOLUTION OF A HOLLOW CIRCULAR CYLINDER SUBJECTED TO EXTERNAL LOADS

We now consider a hollow circular cylinder of dry bone being subjected to an external temperature change To, a quasi-static axial load -pet), an external pressure p(t) and an electric load Do(t). The boundary conditions are

T = 0, a'T = a ril = a rz = Dr = 0, at r = a

T = To, a rr = -p, a ril =art = 0, Dr = Do, at r = b (5)

and

Is azzdS = -p (6)\

where a and b denote the inner and outer radii of the hollow cylinder, and S is the cross-sectional area. It is assumed that all the displacements, temperature and electrical potential except the axial displacement U z is independent of the z coordinate and that Uz may have linear dependence on z. Using (1) and (2), the differential equations (3) can be written as

(~+~~JT =0, C (~+ ~~-_IJu =P oT (6)or2 r or 1 1 or 2 r or r 2 I orI'

2

02 1 0 J (0 1 0 JC -+-- u +d -+-- fn=O (7)44 ( 2 z 4 2 't'• or r or or r or

d -+--0 2

1 0 J -K ( 0 2

1 0 J'.U -+-- (J>1t;'04 ( or 2 r or z I or2 r or '.'

The solution to the heat conduction equation (6)1 satisfying bound2

conditions (5) can be written as

T = In(r / a) To In(b / a)

It is easy to prove that equations (6)-(8) will be satisfied if we assume

B(t) liJr[ln(r / a) - 1) ur = A(t)r+--+ (1

r CII

U = zC(t) + D(t)r[ln(r / a) -1], <p = F(t)r[In(r / a) -1) (1 z

where A, B, C, D and F are unknown variables to be determined bound;

conditions, and liJ = PITo . Substituting «(10) and (11) into (2), 21n(b / a)

later into (1), we obtain

B(t) . O'rr =A(t)(c II +CI2)--2-(Cll-CI2)

r

C12 r r]+C\3C(t)+w -(In--I)-In­[ c" a a

_ A B(t) [C12 r r aee - (t)(CII +CI2 )+-2-(C" -CI2 ) + c\3C(t) +liJ -In--In-­

r c" a a (

a =2A(t)CI3 +C33C(t)+liJ:13 [21nCr/a)-I]-P3To ~n(~/.a? (zz "

a zr = [c D(t) + d F(t)] In(r / a), Dr = [d4 D(t) - KI F(t)) In(r / a)44 4

CII

d In(r I a)A(t)d, + C(t)d3+ UJ - I [21n(r I a) -I] - P3TO ------'-_--'- ­ (16) CII In(b l a)

undary conditions (5) and (6) on stress and electric displacement hat

'; 44 D(t) + d4 F( t) = 0, [d4D(t) - KI F(t)] In(b I a) = Do (1 7)

B(t) C _nA(t)(CII +CI2 )--2-(CIJ -CI2 )+C C(t)--UJ-OJ3 (18)a C

ll

B(t)A(t)(c'l +CI2 )--2-(C -C )

II I2b (19)

C 12+cIJC(t)+w -(In--b I) -. In -bJ =-P[

CII a a

2 2 '.n(b - a )[2A(t)CI3 + C(t)C33 - F; To] + F2 To =-P (20)

F;' = 1 (C1313 1 J In(b l a) CII -133

't

'. F2' = nb 2 (~131 -13 3J (2l)

CII

town functions AU), B(t), CC/), D(/) and F(t) are readily found from as

I ( (J' (J' C33 C I2 F2'To + P(t) , J:;;- C33 I [ 2 TO+ p(t)] + W"-- + 2 2 CJ3 - F; T cl3 3 CI I n(b - a ) o

(22)

B(t) = a213;[I3;To + pet)] .. ~..•.. (23)

CII -C12

C(t) = J.([F;·T - F2'To + P(t)]F3 0 n(b2-a2) (C II +C12 )

(24)

- 2cJ3 (J: [(J;To +p(t)] - 2C13 CI2W"J

d 4DO(t) (25)

D(t) = (d 2 + C K) In(b I a)4 44 I

F(t) = - C44 DO(t) (26)(d; + C44 K I ) In(b I a)

where

b2 • 2

F3 = C33(CII + C12) - 2c13 , (27)13;=(a2 _b2)' ~> ~ (::: ~ IJ

Using expressions (22)-(26), the displacements Un Uz and electrical potential cp are given by

r ( •• C33 CI2 F2'To+ pet) • J U r = -. c 33 l3! [13 2To + p(t)] + UJ-- + 2 2 C13 - F; TOcl3

F3 CII n(b - a )

2a 13;[I3;To + p(t)] UJr[ln(r l a)-I]+ + - -=----'------'---=­r(c l ! -C I2 ) Cli

(28)

z ([. F2'To + P(t)] · • • U z ·=---. F;TO~ 2 2 (CII+CI2)-2cl3(JI[(J2TO+p(t)]

F3 n(b -a )

_ 2c13 cI2 W"J+ d 4 Do(t)r[ln(rla)-I]

(d; + C44 K I ) In(b I a)cII

(29)

""

_ c44 Do(t)r[ln(r / a) -1]<p - - --;;-~-------'­ (30)

(d; + C44 K, ) In(b / a)

ress and electric displacement may be found by introducing (22)-(26) 2)-( 16). Substituting (28)-(30) into (2), and later into (4), we have

.() A;C44 DO(t) In(r / a) A:rd4Do(t) In(r / a) e + 2 + 2 +

(d4 +c44 K,)In(b/a) (d4 +c44

K,)In(b/a)

4;~ ( " C33C'2 F;To + pet) ., JT c33 /3,[/32 TO+P(t)]+1ll--+ 2 2 c - F;1 C

133 c" !r(b - a ) 13 0

A,j~1ll[21n(r / a) - 1] A:~ ([F'r, _F;To + P(t)]( ) ------ -+. '0 2 2 c,' + C' 2

ell F3 !r(b -a )

, • • 2C' 3C' 2111 J.c13 /3, [/32 To + p(t)] - (31) c"

Ice we do not know the exact values of the material functions £ , .

Ai (e), Ai; (e) , Cij, 0, ~j, Kj and P3 we use approxImate forms of

Ir small value of e. Following the way of Cowin and Buskirk(1978),

C C C2e2 AE() A £o A E' AS() A'o AS' ; 0 + ,e + , i e == i + e i' Ij e == ii + e i; (32)

\ o e , 0 o e , 0ci; +~(C!i -cy ), d i (e) == di + - (d

i - d ),

So i

o e , 0 o e , 0~ i + ~(Pi - Pi)' K,(e) = Ki + ~(K i - Ki ),

o e , 0 P3 + SO (P3 - P3 ) (33)

A SO A slC C A EO A EI 0 I dO d' /30 /31 0 , 0 , " 2, i' i ' ii ' y,CU'CI)' i' i ' i' i ,Ki ,K"p)

.re material constants. Using these approximations the remodelling ion (31) can be simplified as the form

e=a(e2 -2~e+y) (34) ~.

by neglect of terms of e3 , where a , ~ and yare constants . The solution to

(34) is straightforward and has been discussed by Hegedus and Cowin(l976) . For the reader's benefit , the solution process is briefly

2described below. Let e, and e2 denote solutions to e - 2~e + y = 0, i.e .

e'.2 = ~ ± (~ 2 _ y)"2 (35)

When 13 2 < y , el and e 2 are a pair of complex conjugate, the solution of (34) is

e(t) =~ + ~(y - ~2) tan(at~(y - ~2 ) + arctan ~(y - ~2) ] (36) ~ - eo

where e=eo is the initial condition. When 13 2 = y, the solution is

e, - eo (37)e(t) = e, -1 + aCe, - eo)t

Finally, when ~ 2 > y, we have

e(t) == e,(eo - e2 ) + e2(e, - eo)exp(a(e l - e2)t) (38)

(eo -e2 )+(el -eo)exp(a(e, -e2 )t)

Since it has been proved that both the solutions (36) and (37) are physically unlikely [Cowin and Buskirk(l978)] , we will use the solution (38) in our numerical analysis.

4. NUMERICAL EXAMPLE "

As numerical illustration of the proposed analytical solution, we consider a femur with a=20mm and b=30mm. The material prowerties assumed for the bone are

C" = 15(1+e)GPa, CI2 =6.6(1+e)GPa, C33 = 12(1 +e)GPa,=c13

C44 = 4.4(1 +e)GPa, ~I =0.621(1+e) x 10 5 NK - l m-2 ,

~ 3 =0.551(1+e)x10 5 NK-'m-2 , P3 = 0.0133(1+e)CK- l m -2

,

....-..

).435(l+e)C/m 2, d 3 =1.75(1+e)C/m2 ,

14(1+e)C/m2 , K J =111.5(l+e)K ,o

~6(1 + e)Ko , Ko == 8.85 xl 0-12 C 2INm 2 = pennitivity of free space

remodelling rate coefficients are assumed to be

l=3.09 x I0-9sec- , C1 =2 x I0 -7sec- l, C2 = 10-6sec - I,

As' = ASO A s1 = ASO = A SI -lASO

= = = 10-5sec" " n n n n ,

A F.O = A £J = 10-15 V -1m! sec = 10-15 N ·J CI secr r

tial inner and outer radii are assumed to be

aO = 10 mm, = 15 mmbo

is assumed. In the calculation, u,. (t) « ao has been assumed for

of simplicity, i.e., aCt) and bet) may be approximated by ao and boo ficients A(t)-D(t), F(t) are determined from expressions (22)-(26) to

lo5.242+3.351e)IO -6 T + 1.776xlO-7 P-1.255 x l0 - p

o (39)l+e

B(t) = 3.726 X 10-10 T _ 2.143 X 10-14

o 1 p (40)+e \

(7.309 + 2.585e)10-6 T, 1.381 x I0- 'O p-3 .196 x l0-7P o + (41)

l+e

D(t) = 0.487 Do (t) (42)l+e

9 1.878 x 10 Do(t)F(t) = (43)

l+e

e=(3.09 x l0-3 +0.2e+e 2) x l0 -6 +6.744xl0-6 1n(rla)Do(t)

+(l+e)[4.117e+IO.211n(r l a)-1.930] x l0 -f,·Ta . ,

+ 0.356 X 10-12 pet) -1.129 X 10-15 pet) (44)

We distinguish following four loading cases:

0.025 .

D =2D' OO2 r D' = IO-JCm·2 o

0.015 Do=D' e

0.01

o,oos D = -D'

0 t~.~~~··-· · · -- - ··~-· -··-~ · · -·-~ - - ·· -~· · ·P... .. .. . .o.OOS Do =-2D'

-0,01 0 500 ,000 1.000,000 1,500,000 2,000 ,000 2.500.000

t (sec )

Fig. I Variation of e with time I Cr., == p == P =0)

(1) Do(t) = n x lO -3 Cm·2 (n=-2, -1, 1, 2), and others are zero.

Substituting it to the remodelling equation( 44) with r = bo ' we have

0.00240(1- e-2227 XIO'7()e(t) = . 7 for n=-2;

0.0114 + 0.211e-2227 x'o' (

0.000343(1- e-L965x IO ·7 () eCt) =----.....:.....----.....!...7 for n=·I;

0.00173 - 0.198e - 1 965x IO · (

1291 Xe(t) = 0.00584(1 - e - !O'7() 7

for n=I ; 0.0355 - 0.165e -129Ix IO · ( f 000858(1 e-7,528X l0" ()

e(t) =' - for n=2. 0.0624 - 0.138e-7528xl0" (

shows the variation of e(t) with time t for each of the above cases. e seen from Fig. 1 that the rate of bone remodelling process will along with increase of external electric displacement Do, i.e., the

'ill be more porous at any given time for a larger value of Do. It is ng that the porosity of the femur will be reduced when Do is less

critical value DrO ' In this example the critical value D is rO

nately - 1.15 x 10-3 Cm·2•

0 .014

Tu = 100' C 0.0 12

e 0.01 To = 50' C

0 .008

0.006

0 .004 .- To =-50'C

0.002 To = -IOO'C

a a 500.000 1.000.000 1.500.000 2.000.000 2 ,500,000

t (sec)

Fig. 2 Variation ore with time t (Do = P =p = 0)

=n x 50· C (n=-2, -1, 1, 2), and others are zero. In this case we

1 Eq.(44) withr = b :o

0.000884(1- e -1.877x\O-7/) '­e(t)= for n=-2 ;\ 0.00458 - 0.193e -1877X,0-7(

0.00199(1- e-l773xI0 -7 /)eel) = for 12=-1;

0 .0106 - O,188e- J.773x,0 -7(

0.00419(1- e-1544xI0-7 ()e(t)= for n=l ;

0 .0236 - 0.178e-1544x'0-7 (

0.00528(1- e-1415xI0-7/)eel) = for n=2.

0 .0308 - 0.172e -1.415xI0-7(

Figure 2 shows the effects of temperature change on bone remodelling rate when Do=p(t)=P(t)=O. It is found from Fig. 2 t~t the temperature change always increase the porosity of the femur. It is evident that the bone remodelling rate will increase along with the increase of temperature.

0.005,

.- --------- ---- ---~ - p=4l\.1Pa

p=6l\.1Pa

p=21\.1Pa

e

~ .Ol

p=8l\.1Pa -0.015 LI ---~-~----'~----'-----'--~--'

a 500.000 1.000.000 1.500.000 2.000,000 2.500,00

t (sec)

Fig. 3 Variation of e with time t (Do = To = 0 and P = 1500N)

(3) p(t) =nx 2MPa(n=1, 2, 3 and 4), P(t)= 1500N, and others are zero. In this case we have from Eq.(44) :

0.00137(1_e-1858X'0 -7( ) eel) = for n=l ;

0.00708 - 0.193e-1.858X'0-7(

0.000892(1 - e-2087 xI0 -7 () e(t)= for n=2;

0.00436 + 0.204e-2087x I0-7 (

e(t) = 0 . 00315~_e-2293X '0 -7 () for n=3;

0.0147 + 0.215e-2393x,0-7 (

0.00541(1_e -2483 XW7{) eel) = for n=4.

0.0241 + 0.224e-2483xw 7/ .,

The results for this loading case are shown in Fig . 3 to study the study the effect of external pressure on bone remodelling process. It is evident that

",<.

exists a critical value PrO above which the porosity of the femur will

duced. The critical value PrO in this problem is approximately [Pa.

3 2oCt) = nx 1O- Cm- (n=l, 2, 3 and 4), pet) =8MPa, P(t)= 1500N, =0. In this case we have from Eq.(44) :

0.0109(1- e-289IXIO-7 () eU)= · ~ for n=-2;0.0446 + 0.245e-2.89IxIO /

eel) =__ e_-2_6_95_XI_O-_7/~)0._0_0_8_15...,:('-1___ . for n=-l;0.0347 + 0.235e-2695xW-7/

e(t) = .~ ___e_-_2.2_51_ 7/!....)0._0~02_6_6~(l XI_O-_(or n=l;0.0125 + 0.213e-225Ixw7/

0.0000852(1- e-1.991 xIO-7/)eel) =---_-.C.___-!.. for n=2.0.000427 - O. I9ge-199lxIQ"7/

we loading case was considered to study coupling effect of electric chanical loads on bone remodelling rate. Figure 4 shows the al results of volume fraction change for different values of electric ment Do when To=O, P(t)= 1500N and p(t)=8MPa. It can be seen g. 4 that the bone remodelling rate will decrease along with the

of elastic displacement Do provided Do c:; 2 x 10-3 Cm-2 . . \

....

0005 1'- --_______________~

° l~ -~()_ :=~_I?:_ O.OOS

Do = D' -0,01

0.015

-0.02 D' = 10-lCm-2 Do =-D'

.025

-0.03 L , Do =-2D', o 500,000 1,000,000 1,500.000 2,000,000 2,500,000

t (sec)

Fig. 4 Variation of e with time t (7;,= p =0 and p = 8MPa)

5. CONCLUSION

The problem of thennopiezoelectric bone remodelling has been addressed within the framework of adaptive elastic theory. An analytical therrnoelectroelastic solution for bone materials has been derived through use of adaptive elastic theory. Numerical study indicates that both electric and thennal load can affect bone remodelling process. There exist some critical points for external pressure and electric displacement at which the bone remodefling rate reach its lowest values. This property can be utilized in controlling healing process of injured bone.

.ACKNOWLEDGEMENTS

The financial support from the Tianjin University and the Australian Research Council is acknowledged.

REFERENCES

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Cowin S.c., Van Buskirk W.c. Internal bone remodelling induced by a medullary pin.

Journal of Biomechanics 1978; 11 :269-75

Cowin S.c., Van Buskirk W .c. Surface bone remodelling induced by a medullary pin.

Journal of Biomechanics 1979; 12:269-76

Cowin S.c., Firoozbakhsh K. Bone remodelling-·of diaphysial surfaces under constant load:

'"Theoretical predictions. Journal of Biomechanics 1981; 14:471-84

CowinS.C., Hegedus D.M. Bone remodelling I: Theory of adaptive elasticity. Journal of

Elasticity 1976: 6:313-26

Demiray H. Electro-Mechanical remodelling of bones. Int J Eng Sci 1983; 21: 1 117-26

Frost H.M. "Dynamics of bone remodelling." In Bone biodynamics, H.M . Frost, ed. Boston:

Little & Brown, 1964.

Fukada E., Yasuda I. On the piezoelectric effect of bone. J Phys Soc Japan 1957; 12: 1158-62

"'... -­