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A constitutive equation for a dilute suspension of nearly straight, flexible fibres in an elastic medium
FibreScienceand Technology13(1980)303-315
A CONSTITUTIVE EQUATION FOR A DILUTE SUSPENSION OF NEARLY STRAIGHT, FLEXIBLE FIBRES IN AN ELASTIC MEDIUM N. PHAN-THIEN~
Department of Mechanical Engineering, University of Newcastle, New South Wales, 2308 (Australia)
SUMMARY
The slender-body approximation is applied to a dilute system of flexibly inextensible .fibres embedded in an elastic' medium. An explicit constitutive equation is derived/or an)' arbitral3' distribution of fibres in the undeJbrmed configuration relating the bulk stress in the composite and the homogeneous displacement field. Some typical deJbrmation fields are worked out in detail/or two cases of interest: the aligned-fibre system and the random distribution case. 'Bi-modulus" properties of such a composite are shown to arise due to the fact that the fibres cannot support compressive forces.
I.
INTRODUCTION
The calculation of the effective bulk properties of heterogeneous media is a subject that needs little introduction here. One dominant method of approaching this problem, taken by Hill I and admirably reviewed in a monograph by Hashin, 2 involves the application of variational theorems of the theory of elasticity to obtain upper and lower bounds and 'self-consistent' estimates for the effective moduli. The bounds are satisfactory at moderate concentrations of inclusions (see Hashin2); however, for rigid inclusions, they diverge, even at dilute concentration range. A second approach, valid at dilute concentrations of inclusions, makes use of the slender-body theory developed by Batchelor 3 and other authors (Cox, 4 Tillet 5) in Fluid Mechanics. This has been translated to Solid Mechanics by Phan-Thien, 6 and Russel and Acrivos. v The essence of this slender-body theory is to replace the actual slender inclusion by a line distribution of Kelvin-state. 8 By Kelvin-state we mean a singularity representing the effect of a concentrated force acting in an infinite elastic medium. With the right choice of the line density of the Kelvin-state distribution, the slip-stick condition on the body can be satisfied, and the force exerted on the t Permanent address: Department of Mechanical Engineering, University of Sydney, New South Wales, 2006 (Australia).
303 Fibre Science and Technology 0015-0568/80/0013-0303/$02.25 © Applied Science Publishers Ltd, England, 1980 Printed in Great Britain
304
N. PHAN-THIEN
inclusion by the matrix can be calculated. In reference 6, it is shown that, up to 0( 1/ln (21/Ro)), where 2/is the length and R o, a representative radius of the inclusion, this force is given by
8, e(lf = (~((3~+
3=4v
}
- 4 v ) ~ n ( Z l / R o ) ( 4 ( 1 - v) U~, U 2, U s
(1)
where E a n d v are the Young's modulus and Poisson's ratio of the elastic matrix and U is the homogeneous displacement field (at most linear in the position vector). In this paper, the deformation of a flexibly inextensible thread is examined in the light of the slender-body theory and the results obtained are used to derive a constitutive equation for a dilute suspension of nearly straight flexibly inextensible fibres. The assumption used here is that, in the undeformed configuration, the fibres are straight; consequently, in an infinitesimal deformation, the fibres remain nearly straight in the deformed configuration. Explicit results in a few typical deformation fields will be worked out, both for aligned and random distribution of fibres.
2.
THE KELVIN-STATE APPROXIMATION
The slender-body approximation s is essentially reproduced here for the completeness of the paper. Let us consider a slender body of length 2l immersed in an infinite elastic medium with Young's modulus E and Poisson's ratio v. R o is a representative radius of the body (refer to Fig. l).
[,,,% Fig. 1. Notation used. The basic idea is that the disturbance displacement field due to the presence of the body is approximately the same as that due to a suitably chosen line distribution of Kelvin-state. a By Kelvin-state we mean a singularity representing the effect of a concentrated force (Stokeslet in Fluid Mechanics, and hence in reference 4 this singularity is referred to as Kelvinlet). The displacement field at point x due to a force f applied at the origin is given by, 8
u(x) =8-~ff x 1 +
x,qf xaJ
Here, 1 is the identity tensor and x = (x. x) t/2 is the distance from the origin,
(1)
305
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
~-
E(1 - v) l+v
(2)
and = 3 - 4v
(3)
Thus if Kelvin-states are distributed over the interval [ - l, l] on the xl-axis then the resulting displacement at point x is ui(x) = ~ 1
t
"[(Xl
- s) 2 "1- r21lj2 +
.fi(s)ds
[(X1
(4)
Here, f(s) is the line density of the distribution, r 2 = x 2, + x~, s i = (s, O, 0) r and 6ij is the Kronecker delta. Cartesian tensor notation is used throughout the paper. Now, if the undisturbed displacement field is denoted by - U then, in order to satisfy the stick condition, the resulting displacement field u - U must be zero at the surface of the body. It is possible to choose f in such a way that this condition is satisfied everywhere near the axis of the body. In particular, the choice of a Kelvinstate distribution with 7rE f l = (1 + v)ln(2l/R o) gl 4~E f i = (1 + v)(3 - 4v)ln(21/Ro) g~
(j 4: 1)
(5)
where U may be a linear function of x (homogeneous deformation) satisfies the stick condition at the surface of the body up to 0(l/In (21/Ro). The force acting on the slender body is given by
f_ F =
8rtE(1- v)l (3-4v )r t fds = (1 + v)-~---4~l~(211Ro) \-4(i --- v) V~, Uz, U3 8nqt
( 3 - 4v ~- v) U,,
- ~ In (21/Ro)\4(i
)r U 2, U 3_
(6)
Note that the ratio of the force for displacement in any transverse direction to that for the longitudinal displacement of the body at the same amount is 4(1 - v)/(3 - 4v). In the next section eqn (6) will be used to study the deformation of a fibre.
3.
FIBRE TENSION
We consider a dilute suspension of finite inextensible fibres in an elastic medium. Arc length along the fibre is measured, by s, se[- 1, l]. The position of the centre line of the fibre is denoted by x(s). Undeformed variables are written in capital letters, thus
306
N. PHAN-THIEN
X(s) is the position of the centre line of the fibre in the undeformed configuration. Dashes are used to denote derivatives with respect to arc length. Since x' is a unit tangent to the fibre at x, x'. x' = 1
(7)
To preserve the arc length there must be a tension in the fibre to balance the force exerted by the matrix. If the force exerted on the fibre by the matrix is f(s) per unit length, then we have - f = (Tx')' = Tx" + T'x'
(8)
where T is the tension in the fibre. In a homogeneous deformation where the deformation gradient F is defined by t~x F = t~X
x = FX
(9)
we can use the result in Section 2 to model f. According to eqn (6) the frictional force/unit length for displacement parallel to the fibre is nrl/(l - v)In (21/Ro) and
4n~l/~ In (21/Ro) for transverse displacement. Thus we can write
f=~.ln(21/Ro)\
1 +~x'x'
FX-x)
= f~(1 - flx'x')(FX - x)
(10)
where _
4n~/ (ll)
~ In (21/Ro)
1
(12)
1+~ Equations (2)-(10) can be combined to give x = F X + ~ ITx,, +
1 T'x' f~(1 - / ~ )
(13)
In deriving eqn (13) we have used
(l--/~x'x')-I = 1 +
/~ x'x'
1--#
(14)
and x'. x" = 0
(15)
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
307
Also, by taking the derivative o f e q n (13) and noting that x ' . x' = x". x" we have the following differential equation for the tension T" - (1 - / a ) T x " . x" = Q(1 - g)(1 - x'. FX')
(16)
If the fibre is straight in the undeformed configuration then X' = P, where P is a unit vector denoting the undeformed configuration of the fibre. We assume that the fibre is straight initially throughout in which case eqn (16) becomes T" - (1 - g ) T x " . x " = f~(l - g)(1 - x ' . FP)
(17)
This second order differential equation is subjected to the boundary conditions that the tension vanishes at the ends of the fibre, i.e. T = 0 at s = + 1. With the tension given, the deformed configuration of the fibre can be found using eqn (13) given the undeformed configuration P. 3.1. Straight fibre A straight fibre has its position given by x = ps + a
(18)
where p is a unit vector and a = FA is the position of the centre of mass of the fibre. In the undeformed configuration, the position of the centre line is given by X = Ps + A
(19)
From eqns (13)-(17) the configuration of the deformed fibre can be shown to be p = FP/IFPI
where
Ixl
(20)
denotes the modulus of x, and the fibre tension is simply given by T = ½Q(I -/a)(IFPI - 1)(12 - s 2)
(21)
Note that if we write F = 1 + ? then y is the infinitesimal displacement gradient tensor (in the limit of small strain). In thiscase IFPI - 1 = 2p. ?P + 0(liE211) and so T is proportional to P . ?P. We will see that expression (21) is also valid when some deviation from the straight fibre is taken into account. 3.2. Nearly straight fibre If the deformed fibre is nearly straight we can write x = x o + ex I
(22)
where x o is given by eqn (18) and e is a perturbed quantity. Condition (7) requires that (up to 0(e2))x~ be orthogonal to p, x'l. p = 0
(23)
W e also express the fibre tension as
T = T O + eT x
(24)
308
N. PHAN-THIEN
S t r a i g h t f o r w a r d c a l c u l a t i o n s show t h a t p = FP/IFPI
(25)
T O = ½f~(l - / I ) ( I F P I - 1)(1 z - s z)
(26)
T~' = - - f ~ ( l - p ) x ' , . F P = 0
(27)
xl = ~(1 - p)(IFPI - 1)(/2 - s2)x~' - (IFPI - l)sx'l
(28)
F r o m eqn (27) a n d the b o u n d a r y c o n d i t i o n s , it is evident that T1 = 0. Thus, u p to first o r d e r t e r m s with respect to the c u r v a t u r e o f the fibre, we have
nE T(s) = 2(1 + v)ln(21/Ro) (IFPI - 1)(/2 - s 2)
(29)
T h e m a x i m u m tension occurs at the centre o f the fibre, i.e.
nEI 2 Tm~x = 2(1 + v ) l n ( 2 1 / R o ) ( [ F P I -
1)
(30)
F o r e x a m p l e , in a shear d e f o r m a t i o n where
[F]=
0
i
0
0
and P=
then IFPI - 1 = G ~
+ ½GZB 2
and
nEl 2 Tm~x = 2(1 + v)ln(21/Ro) Glt(a + ½Gfl) A n e a r l y straight fibre lying in the slip plane (fl = 0) will not experience a n y tension whatsoever. N o w , eqn (28) can be rewritten as
2s (/2 --
S2)X';
1 --/1 xl
2 (I --/~XIFPI - 1) x = 0
(31)
Since x 1 . p = 0, x 1 d o e s n o t have a n y c o m p o n e n t a l o n g p. I n t r o d u c e the o r t h o n o r m a l t r i a d p , q , r a n d express x I as x I =h+qq+
rr
(32)
W i t h o u t loss o f g e n e r a l i t y we put b = 0. q a n d r then o b e y
(P-s2)~r I
1-I,
-(1-/~)(IFPI-
1)
=
(33)
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
309
•e let re+l-
1
1 m - - 3 -4v
1 -p
(n - m ) ( n + m + 1 ) =
2 (1 - / ~ ) ( I F P I
- 1)
n in certain cases (where m and n are integers) the solution to eqn (33) is ( 1 - s2/I 2)- m/2. p~'(s/I)
ere p ~ ( x ) are the associated Legendre functions. Fhe only physical case corresponds to ~ < m < 1 (0 < v < ½). In general, eqn (33) no solution which is finite at both ends indicating that the perturbation scheme :d here is only valid away from the ends. Since we are only interested in the integral )perties, this perturbation scheme may be considered adequate for the purposes ?e.
4.
CONSTITUTIVE EQUATION
rxt, we consider a dilute suspension of flexibly inextensible fibres in an elastic ,,dium which is under a homogeneous deformation characterised by F defined in a (9). The fibres are assumed to be straight in the undeformed configuration and nder so that their tensions are given by T(s) = Tmax 1 - ~-
(34)
l'he ensemble-average stress in a statistically homogeneous suspension of fibres is = ~
adV
(35)
tere ~fibre stress if r is inside a fibre, a(r) = (matrix stress if otherwise. ' the use of the identity (tr~jXk).~ = trek, with the comma denoting covariant rivative, it can be shown that 9 = ~
adV +
n .ardA
m
(36)
• a
Lere Vm is the matrix volume in V, A~, the bounding surface of fibre e and the rnmation sign is taken over all fibres in V.
310
N. PHAN-THIEN
Consistent with the approximation adopted here, the fibre-contributed stress is given by V
T
1
which is 2 "~q" Tma,pplnRo2
where c~ is the volume fraction of the fibres; 1 - cf = Cmis the volume fraction of the matrix. If we denote by a ~m~the matrix stress and let ~b(p) be the probability density of p, then the ensemble-average stress is given by
1
= CmO'tm~ + ~ C f ' ( 1
E(21/Ro)2 f v)ln(2l/Ro )
+
dlFPI
(IFPI - 1)ppO(p)d3p
(37)
I >0"
Note that only fibres with positive tension, i.e. (IFP[ - 1) > 0, contribute to the stress. For infinitesimal deformation we write F = 1 + ~, where ~ + ?r) is the classical strain tensor. Now (IFPI - l)pp = (P. ?PXPP + yPP + pp),r _ 2P. ),PPP + 0(Ih,ll2)) and thus, up to terms of order
1
(38)
(11~'11) the constitutive equation is given by
(21/Ro)2 trEy: fr PPPP~k(P) d3p (21/Ro) :PP>0
"¢ ---~CmO'(m) "]- 12(1 + v)ln
(39)
It is emphasised again, that only fibres with positive tension, i . e . P . y P > 0, contribute to the stress. Hence, if '(,~m~and c(~¢~denote the fourth-order elasticity tensors of the matrix and the composite, respectively, then c61¢1is given by
(21/R°)2 v)ln(21/Ro) cfE f~
c~tcl _____CmC~m~ q_ 12(1 +
PPPP¢(p)d3p
(40)
:PP>0
Some of the composite properties will now be calculated for two cases of interest, the aligned-fibre case and the random distribution case.
5.
ALIGNED FIBRES
In this case, ~(R) = 6(R - P) where P is the unit vector specifying the configuration of the fibre in the undeformed configuration. From now on, the superscript 'c' will
311
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
denote the composite property. Also, we let the components of P in a rectangular Cartesian co-ordinate system be P = (1,0,0) r
(41)
i.e. we consider a dilute suspension of fibres aligned in the x-direction. Extension to an arbitrary orientation of P is straightforward.
5.1. Longitudinal Young's modulus and Poisson's ratio For this mode of deformation, the strain is given by
[~] =
0
- vl°y
0
0
°01
(42)
- v~°~
with v~c~ being the composite Poisson's ratio and the stresses are given by Zll = Etc~Y 322 = 333 = 0
(43)
Now, since ?: P P = 7, all fibres contribute to the stress if 1' > 0. Consequently for y >0, 322 = 333 = Cm I - " - ~ V 3'
-- }V
VIcl
= 0
that is, vICe= v
(44)
Furthermore,
(21/Ro) ~ v)ln(21/Ro) qE''
311 = CmE'}' "4" 12(1 +
and hence the composite Young's modulus is given by
(21/Ro) 2 E~C~ = cmE + 12(1 + v)ln (21/Ro) q E
(45)
On the other hand, in an uniaxial compression characterised by eqn (19) where Y < 0, the fibres will not contribute anything to the bulk stress, and we obtain
E~C~ o m p =. CmE
(46)
In general, the uniaxial Young's moduli in compression and extension are different for a more complicated case where P = (a, b,c)r; a z + b 2 + c z = 1. 5.2. Rigidity modulus Jbr transrerse shear The strain ~, is given by
312
N. PHAN-THIEN
[~] =
0
0
0
0
(47)
Now if P is given by eqn (18), y: P P = 0 and thus the fibres do not contribute to the stress; consequently, the composite rigidity modulus is given by G ~c) =
CmE (48) 2(1 + v) However, for P = (a, b, c) r, then those fibres with 7ba > 0 will contribute to the stress and the composite rigidity modulus can be shown to be, (for 7ba > 0) c r a g --
(21/Ro): 1 -, 2 2 G ~c) = cmG Jr 12(1 +- -v)ln(21/Ro) ,,c, L o a
6.
(49)
RANDOM DISTRIBUTION OF FIBRES
Next, we consider the case of random distribution of fibres where ¢(P) = 4~
(50)
c~(IPI - 1)
Extension to an arbitrary distribution ~(P) is straightforward.
6.1. Hydrostatic deformation In this mode of deformation, the strain is given by
[~1 =
[! ° °0j ~
0
(51)
= ~,l
7
It is evident that v : P P = 7. Furthermore, for a random distribution,
(52)
~ P P ~ ( P ) d3p = ~1
Thus, if ~ > 0, the stress is given by
f CmE
(21/R°)2
1
= ( I - 2v + 36(1 + v)ln(21/Ro) cfE, 71
(53)
and the bulk modulus of the composite can be obtained as
K~)
_
cmE
3(1 - 2 v )
+
(21/Ro)2 108(1 + v)ln(21/Ro) cfE
(54)
313
FLEXIBLE FIBRES IN A N ELASTIC M E D I U M
On the other hand, if ? < 0 (hydrostatic compression), the fibres do not contribute to the bulk stress in which case we have I~ ~) -
cmE
3(1 - 2v)
- cmK
(55)
Note that, due to the random distribution of fibres, we have effectively a homogeneous solid on a macroscopic scale. 6.2. Plane strain bulk modulus To calculate the plane strain bulk modulus, let us consider the following strain field
[~,]
(56)
=
0 In a spherical co-ordinate system where P = (sin 0 c o s 4~, sin 0 sin q~, cos 0),
(57)
~,:PP = ?sin z 0 and thus all fibres contribute to the stress if, and only if, ? > 0.
Assume that this is the case, then the stresses are given by
(2l/Ro) 2
CmE'):
(58)
Tll = (1 + v)(1 - 2 v ) + 45(1 + v)ln(2l/Ro)cfE?
(59)
"[22 = "Cll
Cmgv~ T33 =
2(1 + v ~ - - - 2v)
+
(21/Ro)2
(60)
90(1 + v)ln(21/Ro) cfE?
All other stresses are zero. The composite plane strain bulk modulus is k (c) =
f
CmE
(21/Ro)2
2(1 + v)(1 - 2v) + 90(1 + v)ln(21/Ro) cfE
cmE k.~+
v)(1--2v) = cmk
y>0 (61)
Y <0
6.3. Rigidity modulus for transverse shear Again, in this deformation field the strain tensor y is given by eqn (24). In a spherical co-ordinate system we have ~,:PP = ~ sin 2 0 sin ~bcos 4~
(62)
314
N. PHAN-THIEN
So ify > 0, only those fibres whose configuration with 0 < ~ < ~/2 contribute to the stress. The stresses are then given by
cmE7
[000]
(21/Ro)27n- 1 cfE ; d O + 48(1 + v)ln(21/Ro)
[ z ] - 2(1 + v-----~ 0 0 000
x sina 0 sin ~bcos $
2
dq~
sin2 0 cos 2 q~ sin 2 0 sin ~bcos tk sin 0 cos 0 cos 4~] ,, sin2 0 sin2 ~b sin 0 cos 0 sin 4~/ cos ~ 0
J
(63) Carrying out the integration, we find,
(21/Ro)2n- 1 zll = 180(1 + v)ln(21/Ro) crE~ = r22
(21/Ro)2nt33
----"
1
360(1 + v) In (21/Ro) cfE7
cmE~ z12 - 2 (v1 ~+
(64) (65)
(21/Ro) 2 + 720(1 + v)ln(21/Ro)cf E~
(66)
Thus, the material can support a normal stress in shear deformation and hence it is non-Hookean in this sense. The shear rigidity is
(21/Ro) 2 G~C~= cmG~m~+ 720(1 + v)ln(21/Ro) cfE
(67)
For ~, < 0, it can be shown that the above formulae are valid. Consequently the shear rigidity is given by eqn (67). 6.4. Uniaxial displacement modulus Lastly we look at a uniaxial displacement field where the strain is given by
[r]=
0
0
0
0
(68)
Evidently, ~:PP = ~p2 and hence all fibres contribute if ~ > 0. In this case,
zij = Cm "(1 + v)(1 - 2 v) ~i~ + ~
?ij
(2l/Ro) 2 + 150(1 + v)ln(2l/Ro) cfE(~, 6ii + 2?ij)
(69)
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
315
from which the uniaxial displacement modulus is given by E(C~ _ z , , = CmE(l - v) - 7 (1 + v)(1 - 2 v )
7.
(21/Ro) 2
+ 60(1 + v ) l n ( 2 1 / R o ) c f E
(70)
FINALREMARKS
We have seen how the slender-body theory can be adapted to the mechanics of a dilute suspension of flexibly inextensible fibres in an elastic matrix. Due to the fact that the fibres are incapable of supporting compressive forces, m a n y 'bi-modulus' properties are revealed. Loosely speaking there are two extensional moduli, one for tension and the other compression. The present formulation also allows for an arbitrary distribution of fibres (such a distribution is assumed known from the outset) and this is a clear advantage over the alternative method of calculating the elasticity tensor. Further work has been done on the dilute suspension of nearly rigid rods with Young's modulus Efemploying the sort of theory presented in this paper. This will be a later publication, t°
ACKNOWLEDGEMENT I wish to thank Dr John D. Atkinson of Sydney University for his helpful comments on this paper. The support of the Internal Research Assessment Committee is also gratefully acknowledged.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8.
R. HILL,J. Mech. Phys. Solids, II (1963) p. 357. Z. HssmN, NASA Contractor Report No. 1974 (1972). G. K. BATeHELOR,J. Fluid Mech., 41 (1970) p.419. R. G. Cox, J. Fluid Mech., 44 (1970) p. 791; 45 (1971) p.625. J. P. K. TILLEr,J. FluM Mech., 44 (1970) p.401. N. PUAN-Tm~, Fibre Sci. and Tech., 12 (1979) p. 235. W. B. Rus,~L and A. Acmvos, ZAMP, 23 0972) p.4M. M. E. GuirIN, Handbuch Der Physik, Band VI a/2 (Ed. C. Truesdell) Springer-Verlag,New York 1972, p. 174. 9. L. D. LANDAUand E. M. LWSHITZ,Theory of Elasticily, Pergamon, London 1959, p.29. 10. N. PMAN-Tm~,R. R. HUmOOLand Y. W. MAI, Fibre Sci. and Tech., in press.
A CONSTITUTIVE EQUATION FOR A DILUTE SUSPENSION OF NEARLY STRAIGHT, FLEXIBLE FIBRES IN AN ELASTIC MEDIUM N. PHAN-THIEN~
Department of Mechanical Engineering, University of Newcastle, New South Wales, 2308 (Australia)
SUMMARY
The slender-body approximation is applied to a dilute system of flexibly inextensible .fibres embedded in an elastic' medium. An explicit constitutive equation is derived/or an)' arbitral3' distribution of fibres in the undeJbrmed configuration relating the bulk stress in the composite and the homogeneous displacement field. Some typical deJbrmation fields are worked out in detail/or two cases of interest: the aligned-fibre system and the random distribution case. 'Bi-modulus" properties of such a composite are shown to arise due to the fact that the fibres cannot support compressive forces.
I.
INTRODUCTION
The calculation of the effective bulk properties of heterogeneous media is a subject that needs little introduction here. One dominant method of approaching this problem, taken by Hill I and admirably reviewed in a monograph by Hashin, 2 involves the application of variational theorems of the theory of elasticity to obtain upper and lower bounds and 'self-consistent' estimates for the effective moduli. The bounds are satisfactory at moderate concentrations of inclusions (see Hashin2); however, for rigid inclusions, they diverge, even at dilute concentration range. A second approach, valid at dilute concentrations of inclusions, makes use of the slender-body theory developed by Batchelor 3 and other authors (Cox, 4 Tillet 5) in Fluid Mechanics. This has been translated to Solid Mechanics by Phan-Thien, 6 and Russel and Acrivos. v The essence of this slender-body theory is to replace the actual slender inclusion by a line distribution of Kelvin-state. 8 By Kelvin-state we mean a singularity representing the effect of a concentrated force acting in an infinite elastic medium. With the right choice of the line density of the Kelvin-state distribution, the slip-stick condition on the body can be satisfied, and the force exerted on the t Permanent address: Department of Mechanical Engineering, University of Sydney, New South Wales, 2006 (Australia).
303 Fibre Science and Technology 0015-0568/80/0013-0303/$02.25 © Applied Science Publishers Ltd, England, 1980 Printed in Great Britain
304
N. PHAN-THIEN
inclusion by the matrix can be calculated. In reference 6, it is shown that, up to 0( 1/ln (21/Ro)), where 2/is the length and R o, a representative radius of the inclusion, this force is given by
8, e(lf = (~((3~+
3=4v
}
- 4 v ) ~ n ( Z l / R o ) ( 4 ( 1 - v) U~, U 2, U s
(1)
where E a n d v are the Young's modulus and Poisson's ratio of the elastic matrix and U is the homogeneous displacement field (at most linear in the position vector). In this paper, the deformation of a flexibly inextensible thread is examined in the light of the slender-body theory and the results obtained are used to derive a constitutive equation for a dilute suspension of nearly straight flexibly inextensible fibres. The assumption used here is that, in the undeformed configuration, the fibres are straight; consequently, in an infinitesimal deformation, the fibres remain nearly straight in the deformed configuration. Explicit results in a few typical deformation fields will be worked out, both for aligned and random distribution of fibres.
2.
THE KELVIN-STATE APPROXIMATION
The slender-body approximation s is essentially reproduced here for the completeness of the paper. Let us consider a slender body of length 2l immersed in an infinite elastic medium with Young's modulus E and Poisson's ratio v. R o is a representative radius of the body (refer to Fig. l).
[,,,% Fig. 1. Notation used. The basic idea is that the disturbance displacement field due to the presence of the body is approximately the same as that due to a suitably chosen line distribution of Kelvin-state. a By Kelvin-state we mean a singularity representing the effect of a concentrated force (Stokeslet in Fluid Mechanics, and hence in reference 4 this singularity is referred to as Kelvinlet). The displacement field at point x due to a force f applied at the origin is given by, 8
u(x) =8-~ff x 1 +
x,qf xaJ
Here, 1 is the identity tensor and x = (x. x) t/2 is the distance from the origin,
(1)
305
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
~-
E(1 - v) l+v
(2)
and = 3 - 4v
(3)
Thus if Kelvin-states are distributed over the interval [ - l, l] on the xl-axis then the resulting displacement at point x is ui(x) = ~ 1
t
"[(Xl
- s) 2 "1- r21lj2 +
.fi(s)ds
[(X1
(4)
Here, f(s) is the line density of the distribution, r 2 = x 2, + x~, s i = (s, O, 0) r and 6ij is the Kronecker delta. Cartesian tensor notation is used throughout the paper. Now, if the undisturbed displacement field is denoted by - U then, in order to satisfy the stick condition, the resulting displacement field u - U must be zero at the surface of the body. It is possible to choose f in such a way that this condition is satisfied everywhere near the axis of the body. In particular, the choice of a Kelvinstate distribution with 7rE f l = (1 + v)ln(2l/R o) gl 4~E f i = (1 + v)(3 - 4v)ln(21/Ro) g~
(j 4: 1)
(5)
where U may be a linear function of x (homogeneous deformation) satisfies the stick condition at the surface of the body up to 0(l/In (21/Ro). The force acting on the slender body is given by
f_ F =
8rtE(1- v)l (3-4v )r t fds = (1 + v)-~---4~l~(211Ro) \-4(i --- v) V~, Uz, U3 8nqt
( 3 - 4v ~- v) U,,
- ~ In (21/Ro)\4(i
)r U 2, U 3_
(6)
Note that the ratio of the force for displacement in any transverse direction to that for the longitudinal displacement of the body at the same amount is 4(1 - v)/(3 - 4v). In the next section eqn (6) will be used to study the deformation of a fibre.
3.
FIBRE TENSION
We consider a dilute suspension of finite inextensible fibres in an elastic medium. Arc length along the fibre is measured, by s, se[- 1, l]. The position of the centre line of the fibre is denoted by x(s). Undeformed variables are written in capital letters, thus
306
N. PHAN-THIEN
X(s) is the position of the centre line of the fibre in the undeformed configuration. Dashes are used to denote derivatives with respect to arc length. Since x' is a unit tangent to the fibre at x, x'. x' = 1
(7)
To preserve the arc length there must be a tension in the fibre to balance the force exerted by the matrix. If the force exerted on the fibre by the matrix is f(s) per unit length, then we have - f = (Tx')' = Tx" + T'x'
(8)
where T is the tension in the fibre. In a homogeneous deformation where the deformation gradient F is defined by t~x F = t~X
x = FX
(9)
we can use the result in Section 2 to model f. According to eqn (6) the frictional force/unit length for displacement parallel to the fibre is nrl/(l - v)In (21/Ro) and
4n~l/~ In (21/Ro) for transverse displacement. Thus we can write
f=~.ln(21/Ro)\
1 +~x'x'
FX-x)
= f~(1 - flx'x')(FX - x)
(10)
where _
4n~/ (ll)
~ In (21/Ro)
1
(12)
1+~ Equations (2)-(10) can be combined to give x = F X + ~ ITx,, +
1 T'x' f~(1 - / ~ )
(13)
In deriving eqn (13) we have used
(l--/~x'x')-I = 1 +
/~ x'x'
1--#
(14)
and x'. x" = 0
(15)
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
307
Also, by taking the derivative o f e q n (13) and noting that x ' . x' = x". x" we have the following differential equation for the tension T" - (1 - / a ) T x " . x" = Q(1 - g)(1 - x'. FX')
(16)
If the fibre is straight in the undeformed configuration then X' = P, where P is a unit vector denoting the undeformed configuration of the fibre. We assume that the fibre is straight initially throughout in which case eqn (16) becomes T" - (1 - g ) T x " . x " = f~(l - g)(1 - x ' . FP)
(17)
This second order differential equation is subjected to the boundary conditions that the tension vanishes at the ends of the fibre, i.e. T = 0 at s = + 1. With the tension given, the deformed configuration of the fibre can be found using eqn (13) given the undeformed configuration P. 3.1. Straight fibre A straight fibre has its position given by x = ps + a
(18)
where p is a unit vector and a = FA is the position of the centre of mass of the fibre. In the undeformed configuration, the position of the centre line is given by X = Ps + A
(19)
From eqns (13)-(17) the configuration of the deformed fibre can be shown to be p = FP/IFPI
where
Ixl
(20)
denotes the modulus of x, and the fibre tension is simply given by T = ½Q(I -/a)(IFPI - 1)(12 - s 2)
(21)
Note that if we write F = 1 + ? then y is the infinitesimal displacement gradient tensor (in the limit of small strain). In thiscase IFPI - 1 = 2p. ?P + 0(liE211) and so T is proportional to P . ?P. We will see that expression (21) is also valid when some deviation from the straight fibre is taken into account. 3.2. Nearly straight fibre If the deformed fibre is nearly straight we can write x = x o + ex I
(22)
where x o is given by eqn (18) and e is a perturbed quantity. Condition (7) requires that (up to 0(e2))x~ be orthogonal to p, x'l. p = 0
(23)
W e also express the fibre tension as
T = T O + eT x
(24)
308
N. PHAN-THIEN
S t r a i g h t f o r w a r d c a l c u l a t i o n s show t h a t p = FP/IFPI
(25)
T O = ½f~(l - / I ) ( I F P I - 1)(1 z - s z)
(26)
T~' = - - f ~ ( l - p ) x ' , . F P = 0
(27)
xl = ~(1 - p)(IFPI - 1)(/2 - s2)x~' - (IFPI - l)sx'l
(28)
F r o m eqn (27) a n d the b o u n d a r y c o n d i t i o n s , it is evident that T1 = 0. Thus, u p to first o r d e r t e r m s with respect to the c u r v a t u r e o f the fibre, we have
nE T(s) = 2(1 + v)ln(21/Ro) (IFPI - 1)(/2 - s 2)
(29)
T h e m a x i m u m tension occurs at the centre o f the fibre, i.e.
nEI 2 Tm~x = 2(1 + v ) l n ( 2 1 / R o ) ( [ F P I -
1)
(30)
F o r e x a m p l e , in a shear d e f o r m a t i o n where
[F]=
0
i
0
0
and P=
then IFPI - 1 = G ~
+ ½GZB 2
and
nEl 2 Tm~x = 2(1 + v)ln(21/Ro) Glt(a + ½Gfl) A n e a r l y straight fibre lying in the slip plane (fl = 0) will not experience a n y tension whatsoever. N o w , eqn (28) can be rewritten as
2s (/2 --
S2)X';
1 --/1 xl
2 (I --/~XIFPI - 1) x = 0
(31)
Since x 1 . p = 0, x 1 d o e s n o t have a n y c o m p o n e n t a l o n g p. I n t r o d u c e the o r t h o n o r m a l t r i a d p , q , r a n d express x I as x I =h+qq+
rr
(32)
W i t h o u t loss o f g e n e r a l i t y we put b = 0. q a n d r then o b e y
(P-s2)~r I
1-I,
-(1-/~)(IFPI-
1)
=
(33)
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
309
•e let re+l-
1
1 m - - 3 -4v
1 -p
(n - m ) ( n + m + 1 ) =
2 (1 - / ~ ) ( I F P I
- 1)
n in certain cases (where m and n are integers) the solution to eqn (33) is ( 1 - s2/I 2)- m/2. p~'(s/I)
ere p ~ ( x ) are the associated Legendre functions. Fhe only physical case corresponds to ~ < m < 1 (0 < v < ½). In general, eqn (33) no solution which is finite at both ends indicating that the perturbation scheme :d here is only valid away from the ends. Since we are only interested in the integral )perties, this perturbation scheme may be considered adequate for the purposes ?e.
4.
CONSTITUTIVE EQUATION
rxt, we consider a dilute suspension of flexibly inextensible fibres in an elastic ,,dium which is under a homogeneous deformation characterised by F defined in a (9). The fibres are assumed to be straight in the undeformed configuration and nder so that their tensions are given by T(s) = Tmax 1 - ~-
(34)
l'he ensemble-average stress in a statistically homogeneous suspension of fibres is = ~
adV
(35)
tere ~fibre stress if r is inside a fibre, a(r) = (matrix stress if otherwise. ' the use of the identity (tr~jXk).~ = trek, with the comma denoting covariant rivative, it can be shown that 9 = ~
adV +
n .ardA
m
(36)
• a
Lere Vm is the matrix volume in V, A~, the bounding surface of fibre e and the rnmation sign is taken over all fibres in V.
310
N. PHAN-THIEN
Consistent with the approximation adopted here, the fibre-contributed stress is given by V
T
1
which is 2 "~q" Tma,pplnRo2
where c~ is the volume fraction of the fibres; 1 - cf = Cmis the volume fraction of the matrix. If we denote by a ~m~the matrix stress and let ~b(p) be the probability density of p, then the ensemble-average stress is given by
1
= CmO'tm~ + ~ C f ' ( 1
E(21/Ro)2 f v)ln(2l/Ro )
+
dlFPI
(IFPI - 1)ppO(p)d3p
(37)
I >0"
Note that only fibres with positive tension, i.e. (IFP[ - 1) > 0, contribute to the stress. For infinitesimal deformation we write F = 1 + ~, where ~ + ?r) is the classical strain tensor. Now (IFPI - l)pp = (P. ?PXPP + yPP + pp),r _ 2P. ),PPP + 0(Ih,ll2)) and thus, up to terms of order
1
(38)
(11~'11) the constitutive equation is given by
(21/Ro)2 trEy: fr PPPP~k(P) d3p (21/Ro) :PP>0
"¢ ---~CmO'(m) "]- 12(1 + v)ln
(39)
It is emphasised again, that only fibres with positive tension, i . e . P . y P > 0, contribute to the stress. Hence, if '(,~m~and c(~¢~denote the fourth-order elasticity tensors of the matrix and the composite, respectively, then c61¢1is given by
(21/R°)2 v)ln(21/Ro) cfE f~
c~tcl _____CmC~m~ q_ 12(1 +
PPPP¢(p)d3p
(40)
:PP>0
Some of the composite properties will now be calculated for two cases of interest, the aligned-fibre case and the random distribution case.
5.
ALIGNED FIBRES
In this case, ~(R) = 6(R - P) where P is the unit vector specifying the configuration of the fibre in the undeformed configuration. From now on, the superscript 'c' will
311
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
denote the composite property. Also, we let the components of P in a rectangular Cartesian co-ordinate system be P = (1,0,0) r
(41)
i.e. we consider a dilute suspension of fibres aligned in the x-direction. Extension to an arbitrary orientation of P is straightforward.
5.1. Longitudinal Young's modulus and Poisson's ratio For this mode of deformation, the strain is given by
[~] =
0
- vl°y
0
0
°01
(42)
- v~°~
with v~c~ being the composite Poisson's ratio and the stresses are given by Zll = Etc~Y 322 = 333 = 0
(43)
Now, since ?: P P = 7, all fibres contribute to the stress if 1' > 0. Consequently for y >0, 322 = 333 = Cm I - " - ~ V 3'
-- }V
VIcl
= 0
that is, vICe= v
(44)
Furthermore,
(21/Ro) ~ v)ln(21/Ro) qE''
311 = CmE'}' "4" 12(1 +
and hence the composite Young's modulus is given by
(21/Ro) 2 E~C~ = cmE + 12(1 + v)ln (21/Ro) q E
(45)
On the other hand, in an uniaxial compression characterised by eqn (19) where Y < 0, the fibres will not contribute anything to the bulk stress, and we obtain
E~C~ o m p =. CmE
(46)
In general, the uniaxial Young's moduli in compression and extension are different for a more complicated case where P = (a, b,c)r; a z + b 2 + c z = 1. 5.2. Rigidity modulus Jbr transrerse shear The strain ~, is given by
312
N. PHAN-THIEN
[~] =
0
0
0
0
(47)
Now if P is given by eqn (18), y: P P = 0 and thus the fibres do not contribute to the stress; consequently, the composite rigidity modulus is given by G ~c) =
CmE (48) 2(1 + v) However, for P = (a, b, c) r, then those fibres with 7ba > 0 will contribute to the stress and the composite rigidity modulus can be shown to be, (for 7ba > 0) c r a g --
(21/Ro): 1 -, 2 2 G ~c) = cmG Jr 12(1 +- -v)ln(21/Ro) ,,c, L o a
6.
(49)
RANDOM DISTRIBUTION OF FIBRES
Next, we consider the case of random distribution of fibres where ¢(P) = 4~
(50)
c~(IPI - 1)
Extension to an arbitrary distribution ~(P) is straightforward.
6.1. Hydrostatic deformation In this mode of deformation, the strain is given by
[~1 =
[! ° °0j ~
0
(51)
= ~,l
7
It is evident that v : P P = 7. Furthermore, for a random distribution,
(52)
~ P P ~ ( P ) d3p = ~1
Thus, if ~ > 0, the stress is given by
f CmE
(21/R°)2
1
= ( I - 2v + 36(1 + v)ln(21/Ro) cfE, 71
(53)
and the bulk modulus of the composite can be obtained as
K~)
_
cmE
3(1 - 2 v )
+
(21/Ro)2 108(1 + v)ln(21/Ro) cfE
(54)
313
FLEXIBLE FIBRES IN A N ELASTIC M E D I U M
On the other hand, if ? < 0 (hydrostatic compression), the fibres do not contribute to the bulk stress in which case we have I~ ~) -
cmE
3(1 - 2v)
- cmK
(55)
Note that, due to the random distribution of fibres, we have effectively a homogeneous solid on a macroscopic scale. 6.2. Plane strain bulk modulus To calculate the plane strain bulk modulus, let us consider the following strain field
[~,]
(56)
=
0 In a spherical co-ordinate system where P = (sin 0 c o s 4~, sin 0 sin q~, cos 0),
(57)
~,:PP = ?sin z 0 and thus all fibres contribute to the stress if, and only if, ? > 0.
Assume that this is the case, then the stresses are given by
(2l/Ro) 2
CmE'):
(58)
Tll = (1 + v)(1 - 2 v ) + 45(1 + v)ln(2l/Ro)cfE?
(59)
"[22 = "Cll
Cmgv~ T33 =
2(1 + v ~ - - - 2v)
+
(21/Ro)2
(60)
90(1 + v)ln(21/Ro) cfE?
All other stresses are zero. The composite plane strain bulk modulus is k (c) =
f
CmE
(21/Ro)2
2(1 + v)(1 - 2v) + 90(1 + v)ln(21/Ro) cfE
cmE k.~+
v)(1--2v) = cmk
y>0 (61)
Y <0
6.3. Rigidity modulus for transverse shear Again, in this deformation field the strain tensor y is given by eqn (24). In a spherical co-ordinate system we have ~,:PP = ~ sin 2 0 sin ~bcos 4~
(62)
314
N. PHAN-THIEN
So ify > 0, only those fibres whose configuration with 0 < ~ < ~/2 contribute to the stress. The stresses are then given by
cmE7
[000]
(21/Ro)27n- 1 cfE ; d O + 48(1 + v)ln(21/Ro)
[ z ] - 2(1 + v-----~ 0 0 000
x sina 0 sin ~bcos $
2
dq~
sin2 0 cos 2 q~ sin 2 0 sin ~bcos tk sin 0 cos 0 cos 4~] ,, sin2 0 sin2 ~b sin 0 cos 0 sin 4~/ cos ~ 0
J
(63) Carrying out the integration, we find,
(21/Ro)2n- 1 zll = 180(1 + v)ln(21/Ro) crE~ = r22
(21/Ro)2nt33
----"
1
360(1 + v) In (21/Ro) cfE7
cmE~ z12 - 2 (v1 ~+
(64) (65)
(21/Ro) 2 + 720(1 + v)ln(21/Ro)cf E~
(66)
Thus, the material can support a normal stress in shear deformation and hence it is non-Hookean in this sense. The shear rigidity is
(21/Ro) 2 G~C~= cmG~m~+ 720(1 + v)ln(21/Ro) cfE
(67)
For ~, < 0, it can be shown that the above formulae are valid. Consequently the shear rigidity is given by eqn (67). 6.4. Uniaxial displacement modulus Lastly we look at a uniaxial displacement field where the strain is given by
[r]=
0
0
0
0
(68)
Evidently, ~:PP = ~p2 and hence all fibres contribute if ~ > 0. In this case,
zij = Cm "(1 + v)(1 - 2 v) ~i~ + ~
?ij
(2l/Ro) 2 + 150(1 + v)ln(2l/Ro) cfE(~, 6ii + 2?ij)
(69)
FLEXIBLE FIBRES IN AN ELASTIC MEDIUM
315
from which the uniaxial displacement modulus is given by E(C~ _ z , , = CmE(l - v) - 7 (1 + v)(1 - 2 v )
7.
(21/Ro) 2
+ 60(1 + v ) l n ( 2 1 / R o ) c f E
(70)
FINALREMARKS
We have seen how the slender-body theory can be adapted to the mechanics of a dilute suspension of flexibly inextensible fibres in an elastic matrix. Due to the fact that the fibres are incapable of supporting compressive forces, m a n y 'bi-modulus' properties are revealed. Loosely speaking there are two extensional moduli, one for tension and the other compression. The present formulation also allows for an arbitrary distribution of fibres (such a distribution is assumed known from the outset) and this is a clear advantage over the alternative method of calculating the elasticity tensor. Further work has been done on the dilute suspension of nearly rigid rods with Young's modulus Efemploying the sort of theory presented in this paper. This will be a later publication, t°
ACKNOWLEDGEMENT I wish to thank Dr John D. Atkinson of Sydney University for his helpful comments on this paper. The support of the Internal Research Assessment Committee is also gratefully acknowledged.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8.
R. HILL,J. Mech. Phys. Solids, II (1963) p. 357. Z. HssmN, NASA Contractor Report No. 1974 (1972). G. K. BATeHELOR,J. Fluid Mech., 41 (1970) p.419. R. G. Cox, J. Fluid Mech., 44 (1970) p. 791; 45 (1971) p.625. J. P. K. TILLEr,J. FluM Mech., 44 (1970) p.401. N. PUAN-Tm~, Fibre Sci. and Tech., 12 (1979) p. 235. W. B. Rus,~L and A. Acmvos, ZAMP, 23 0972) p.4M. M. E. GuirIN, Handbuch Der Physik, Band VI a/2 (Ed. C. Truesdell) Springer-Verlag,New York 1972, p. 174. 9. L. D. LANDAUand E. M. LWSHITZ,Theory of Elasticily, Pergamon, London 1959, p.29. 10. N. PMAN-Tm~,R. R. HUmOOLand Y. W. MAI, Fibre Sci. and Tech., in press.