Karihaloo 2001 Computers & Structures

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Homogenization-based multivariable element method for

pure torsion of composite shafts

B.L. Karihaloo a,*, Q.Z. Xiao a,b, C.C. Wu b

a Division of Civil Engineering, School of Engineering, Cardi University, Queen's Buildings, P.O. Box 686, Cardi CF24 3TB, UKb Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, China

Received 13 March 2000; accepted 28 June 2001

Abstract

Application of the hybrid stress element to torsion of composite shafts is restricted, if the volume fraction of re-

inforcement is very large. The homogenization method is the most suitable for such problems because it gives not only

the equivalent material properties but also detailed information of local elds with less computational cost. In this

paper, the extension of the homogenization method to the pure torsion of composite shafts with reinforcing bers

aligned along its axis and the application of high performance multivariable elements are studied. The incompatible

element based on a modied potential principle and the enhanced-strain element based on the HuWashizu principle

are appropriate for the analysis of the representative unit cell, whereas the hybrid stress element is appropriate for the

macro-homogenized problem. Numerical examples are provided and discussed. 2001 Elsevier Science Ltd. All rights

reserved.

Keywords: Composite shaft; Enhanced-strain element; Homogenization; Hybrid element; Incompatible element; Representative unitcell; Torsion

1. Introduction

In the mechanics of composite materials, it is of great

importance to determine the eective properties of the

composite from the distribution and basic properties of

constituents and the detailed distribution of elds on the

scale of microconstituents [14]. The hybrid stress ele-

ment introduced by the authors [5] is of course a usefultool in the analysis of composite shafts because it gives

accurate results for the warping displacement, the angle

of twist per unit length, as well as the shear stress by

using a relatively coarse mesh. However, if the volume

fraction of reinforcement is very large, it is not realistic to

obtain the microelds by this method because the degrees

of freedom needed to model the entire macrodomain

with a grid size comparable to that of the microscale

features are too many. In this case, the mathematical

homogenization method which has received considerable

attention in recent years [616] seems to be the most

suitable one. It is a kind of singular perturbation method

suitable for problems with boundary layers [17] that exist

at regions near the interfaces of dierent phases in a

heterogeneous medium. With the help of multiple scale

expansion, it gives not only the eective properties of thecomposite, but also detailed distribution of elds on the

scale of microconstituents with acceptable cost. In con-

trast to the most widely used methods in determining

the macro properties, i.e., the Eshelby method, the self-

consistent method, the MoriTanaka method, the dif-

ferential scheme and the bound theories [14], the

homogenization method takes into account the interac-

tion between phases naturally and avoids assumptions

other than the assumption of periodic distribution of

constituents. On the other hand, it accounts for micro-

structural eects on the macroscopic response without

explicitly representing the details of the microstructure in

the global analysis. The computational model at the

Computers and Structures 79 (2001) 16451660

www.elsevier.com/locate/compstruc

* Corresponding author. Tel.: +44-29-2087-4934; fax: +44-

29-2087-4597.E-mail address: [email protected] (B.L. Karihaloo).

0045-7949/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.

P II: S0 0 4 5 -7 9 4 9 (0 1 )0 0 0 9 1 -8


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lower scales is only needed if and when there is a necessity

to do so. In recent years it has been employed for the

solution of complex problems in conjunction with the

nite element (FE) method [810,1214]. Since accuracy

of the widely used isoparametric element method is not

satisfactory, high performance multivariable elementsare necessary to be introduced in the homogenization

method to improve the accuracy [14].

In this study, we will employ the homogenization

method for the solution of pure torsion of composite

shafts with bers aligned along its axis and study the

application of high performance multivariable elements.

In accordance with the mathematical homogenization

method, control dierential equations for the represen-

tative unit cell (RUC) are obtained in Section 2. The

corresponding variational principles are then deduced as

the basis of the FE method in Section 3. The formula-

tion and practical application of incompatible and en-hanced-strain elements to the analysis of RUC are

discussed in detail in Sections 4 and 5. A 4-noded in-

compatible element is also introduced. In Section 6, a

penalty function method is discussed to enforce the pe-

riodicity boundary condition of the RUC. In Section 7,

composite shafts of square as well as rectangular cross-

section reinforced with circular and elliptic bers are

analyzed as illustrative examples. For a square shaft

containing 16 bers, the problem is also solved directly

by the hybrid stress element. The shear modulus from

the homogenization method is compared with that ob-

tained by the hybrid stress element and by the Voigt

Reuss theory [4]. A comparison of the computed resultsshows common features of the local elds. Conclusions

and discussion follow in Section 8.

2. Mathematical homogenization

Consider a uniform composite shaft of arbitrary

cross-section twisted by couples applied at the ends.

Without loss of generality, the origin of coordinates is

taken at the left end cross-section, with the x1- and x2-

axes as the principal axes of inertia, and the x3-axisalong the axis of the shaft and pointing to its other end.

According to St. Venant's theory of torsion, the dis-

placement components are [18]

u1 hx3x2

u2 hx3x1

u3 wx1;x2

1

where h represents the angle of twist per unit length

(clockwise about the x3-axis).

Assume the microstructure of the cross-section Xe to

be locally periodic with a period dened by a statistically

homogeneous volume element, denoted by the repre-

sentative volume element or unit cell Y, as shown in Fig.

1. In other words, the composite material is formed by a

spatial repetition of the unit cell. The shaft has two

length scales; a global length scale D that is of the order

of the section size, and a local length scale dthat is of the

order of the unit cell and proportional to the wavelength

of the variation of the microstructure. The size of the

unit cell is much larger than that of the constituents but

much smaller than that of the section. The relation be-tween the global coordinate system xi for the section and

the local system yi for the minimum repeated unit cell

can then be written as

yi xi

ei 1; 2 2

where e is a very small positive number representing the

scaling factor between the two length scales. The local

coordinate vector yi is regarded as a stretched coordinate

vector in the microscopic domain. For an actual hetero-

geneous body subjected to external forces, eld quan-

tities such as displacements, strains and stresses are

assumed to have slow variations from point to point

Fig. 1. Illustration of a problem with two length scales.

1646 B.L. Karihaloo et al. / Computers and Structures 79 (2001) 16451660


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with macroscopic (global) coordinate x as well as fast

variations with local microscopic coordinate y within a

small neighborhood of size e of a given point x. That is,

displacements, strains and stresses have two explicit

dependencies: one on the macroscopic level with coor-

dinates xi, and the other on the level of microconstitu-ents with coordinates yi

ue3 ue3x;y

ce3j ce3jx;y

se3j se3jx;y

3

where j 1, 2. Due to the periodicity of the micro-structure, functions ue3, c

e3j and s

e3j are assumed to be Y-

periodic, i.e., ue3x;y ue3x;y kY, c

e3jx;y c

e3jx;y

kY and se3jx;y se3jx;y kY, where Yyi is the size

of the unit cell, or the basic period of the stretched co-

ordinate system y and k is a non-zero integer.The unknown displacement ue3, the angle of twist per

unit length h, and the non-zero strain ce3j and stress se3j

can be solved from the following equations

Equilibrium :ose3j

oxj 0 in Xe 4

Kinematical :ce31ce32

o

ox1x2

o

ox2x1

" #ue3h

in Xe

5

Constitutive : se3i Ceijc

e3j in X

e 6

and

Mt

ZXe

s32x1 s31x2 dX 7

together with the traction free condition on the surface

of the shaft, and the traction and displacement condi-

tions at the interfaces between the microconstituents.

For the sake of simplicity and clarity, we assume that the

elds are continuous across the interfaces. The material

property tensor Ceij

is symmetric with respect to indices

i;j. Mt is the torque applied at the ends. The super-script e denotes Y-periodicity of the corresponding

function. The convention of summation over the re-

peated indices is used.

The displacement ue3x;y is expanded in powers ofthe small number e [614]

ue3x;y u03 x;y eu

13 x;y e

2u23 x;y 8

where u03 , u

13 , u

23 ,. . ., are Y-periodic functions with

respect to y. Substituting Eq. (8) into Eq. (5) gives the

expansion of the strain ce3j:

ce3jx;y e1c13j c

03j ec

13j 9

where

c13j

ou03

oyj

c03j c

0x3j c

0y3j

c0

x31 ou

03

ox1 hx2

c0

x32 ou

03

ox2 hx1

c0

y3j ou

13

oyj

c13j

ou13

oxjou

23

oyj

10

Substituting Eq. (9) into the constitutive relation (6)

gives the expansion of the stress se3j

se3jx;y e1s

13j s

03j es

13j 11

where

s13i Cijc

13j 12

s03i Cijc

03j 13

s13i Cijc

13j 14

Inserting the asymptotic expansion for the stress eld

(11) into the equilibrium equation (4) and collecting theterms of like powers in e gives equations

Oe2 :os

13j

oyj 0 15

Oe1 :os

13j

oxjos

03j

oyj 0 16

Oe :os

03j

oxjos

13j

oyj 0 17

We rst consider the Oe2

equilibrium equation (15) inY. Premultiplying it by u

03 , integrating over Y, followed

by integration by parts, yieldsZY

u03

os13j

oyjdY

IoY

u03 s

13j nj dC

ZY

ou03

oyjCji

ou03

oyidY 0 18

where oY denotes the boundary ofY. The boundary in-

tegral term in Eq. (18) vanishes due to the periodicity of

the boundary conditions in Y, because u03 and s

13j are

identical on the opposite sides of the unit cell, while the

corresponding normals nj are in opposite directions.

B.L. Karihaloo et al. / Computers and Structures 79 (2001) 16451660 1647


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Taking into account the positive deniteness of the

symmetric constitutive tensor Cij, we have

ou03

oyj 0 A u03 u

03 x 19

and

c13j x;y 0 s

13j x;y 0 20

Next, we proceed to the Oe1 equilibrium equation(16). From Eqs. (10) and (13) and taking into account

Eq. (20), it follows that

o

oyjCjic

0y3iu

13

oCji

oyjc

0x3i u

03 21

Based on the form of the right-hand side of Eq. (21)

which permits a separation of variables, u13 may be

expressed as

u13 x;y v

3j3 yc

0x3ju

03 22

where v3j3 y is a Y-periodic function dened in the unit

cell Y. Substituting Eq. (22) into Eq. (21), and taking

into account the arbitrariness of the macroscopic strain

eld, cx3ju03 within a unit cell, we have

o

oyjCjic

0y3iv

3k3 y

oCjk

oyj23

We now consider the Oe equilibrium equation (17).Substituting Eq. (22) into Eq. (10), followed by the latter

into Eq. (13), and nally the result into Eq. (17) yields

o

oxjCjidikh

v3k3;yi c0

x3ku03 i

os

13j

oyj 0 24

where dik is the Kronecker Delta. Integrating Eq. (24)

over the unit cell domain Y and taking into account the

periodicity of s13j yields

o

oxjCHjkc

0x3ku

03

h i 0 25

This is an equilibrium equation for a homogeneous

medium (cf. Eq. (4)) with constant material propertiesCHjk, which are usually termed as the homogenized or

eective material properties and are given by

CHjk 1

Y

ZY

Cjidik v3k3;yi

dY 26

where Y is the area of the unit cell.

3. Variational principles and nite elements

To solve the torsion of composite shafts by the ho-

mogenization method, together with numerical meth-

ods, e.g. the FE method adopted here, we will rst solve

for v3j3 y from Eq. (23) assuming it to be a Y-periodic

function dened in Y. The eective material properties

CHjk are given by Eq. (26). We then solve the homoge-

neous St Venant torsion problem by the hybrid stress

element [5] and obtain the macroscopic elds: warping

displacement u03 , the angle of twist per unit length h,strains c

0x3j and stresses (given by C

Hji c

0x3i ). If the distri-

bution of the microscopic elds in the neighborhood of

point x is of interest, we use Eq. (22) to calculate the

higher order displacement term, and then use Eqs. (10)

and (13) to calculate the higher order strain and stress

terms.

The key problem here is to develop powerful nite

element methods to solve Eq. (23).

Corresponding to the equilibrium equation (23), the

virtual work principle states that

ZYdv

3k3

o

oyj Cji

ov3k3oyi dY ZY dv3k3 oCjkoyj dY 0 27a

where dv3k3 are arbitrary Y-periodic functions dened in

the unit cell Y. Integration of Eq. (27a) by parts yieldsIoY

dv3k3 Cjiov3k3oyi

nj ds

IoY

dv3k3 Cjknj ds

ZY

odv3k3oyj

Cji

ov3k3oyi

dY

ZY

odv3k3oyj

Cjk dY 0

The boundary integral terms in the above equation

vanish due to the Y-periodicity ofv3k3 and dv3k3 . Thus, we

haveZY

odv3k3oyj

Cjiov3k3oyi

dY

ZY

odv3k3oyj

Cjk dY 0 27b

Based on Eq. (27b), displacement elements can be es-

tablished in a standard manner.

It is easy to prove that Eqs. (27a) and (27b) is the rst

order variation of the following potential functional

PPv3k3

ZY

1

2

ov3k3oyj

Cjiov3k3oyi

dY

ZY

ov3k3oyj

Cjk dY 28

If we dene the strain

~ck3i ov3k3oyi

and the stress

~sk3j Cji ~ck3i

so that

~ck3i C1ij

~sk3j

which are Y-periodic functions in the unit cell, we have a

2-eld HellingerReissner functional



5/16

PHRv3k3 ; ~s

k3j

ZY

1

2~sk3iC

1ij ~s

k3j ~s

k3i

ov3k3oyi

oCjk

oyjv3k3

dY

29a

or equivalently

PHRv3k3 ; ~s

k3j

ZY

1

2~sk3iC

1ij

~sk3j ~sk3i

ov3k3oyi

Cjkov3k3oyj

dY

29b

By making use of the Lagrange multiplier method and

relaxing the compatibility condition in the potential

principle (28), or by employing Legendre transformation

on the HellingerReissner principle (29b), one arrives at

the 3-eld HuWashizu functional

PHWv3k3 ; ~c

k3j; ~s

k3j ZY

1

2~ck3iCij ~c

k3j ~s

k3i

~ck3i ov3k3oyi

Cjkov3k3oyj

dY 30

Based on the functionals (29) and (30), multivariable

nite elements can be established.

Although hybrid elements based on the Hellinger

Reissner principle or the HuWashizu principle can in

general improve the accuracy of the approximate dis-

placement and stress solutions, they will not be used

here as it is dicult to meet the Y-periodicity condition

of the stress on the boundary of the unit cell. The general

isoparametric elements are also not satisfactory because

of the gradients of v3k3 that appear in Eq. (26) in the

evaluation of the homogenized material properties. For

these reasons, in this paper we will instead introduce

displacement-incompatible elements based on the po-

tential (28) and enhanced-strain elements based on Eq.

(30).

4. Displacement-incompatible elements

Subdivide the unit cell domain Y into nite element

subdomains Ye, such that Ye Y, Ya Yb Y andoYa oYb Sab (a, b are arbitrary elements).

In each element, v3k3 is divided into a compatible part

v3k3q and an incompatible part v3k3k, so that the functional

(28) can be rewritten as

PPv3k3 v

3k3q v

3k3k

X

e

ZYe

1

2

ov3k3oyj

Cjiov3k3oyi

dY

ZYe

ov3k3q

oyjCjk dY

31

Taking the variation of the above functional, integrating

by parts and making use of the periodicity condition on

the outer boundary of the unit cell, yields

dPPv3k3

Xe

ZYe

dv3k3qo

oyjCji

ov3k3oyi

oCjk

oyj

dY

Xa;b ZSabdv3k3q Cji

ov3k3oyi

( Cjk

nj

a

Cjiov3k3oyi

Cjknj

b)ds

X

e

ZYe

odv3k3koyj

Cjiov3k3oyi

dY

The stationary condition of the functional (31) gives the

equilibrium equation (23) and the equilibrium of trac-

tion between the elements if the following condition is

met a priori

XeZYe odv

3k3k

oyj Cjiov3k3oyi dY 0

A convenient way to meet this condition is to satisfy the

following strong form (i.e. the sucient but not the

necessary condition) in each elementZYe

odv3k3koyj

Cjiov3k3oyi

dY 0

Since a constant stress state is recovered in each element

as its size is reduced to zero and since dv3k3k is arbitrary,

the above constraint reduces to the general patch test

condition (PTC) [19,20]ZYe

ov3k3koyj

dY 0 or equivalentlyIoYe

v3k3knj ds 0

32

The incompatible functions meeting the PTC can now

be easily formulated.

If we refer to the 4-noded isoparametric element

shown in Fig. 2, the compatible displacement v3k3q is re-

lated to the nodal values eqk via the bilinear interpola-

tion functions

v3k3q Neqk 33

where

N N1 N2 N3 N4

and

Ni 141 nin1 gig

n; g represent the isoparametric coordinates, ni; gi arethe isoparametric coordinates of point i with the global

coordinates xi;yi, i 1, 2, 3, 4.The incompatible term v3k3k is related to the element

inner parameters ekk via the shape functions Nk

v3k3k Nkekk 34



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Here, two incompatible terms are employed in each ele-

ment as derived in Refs. [19,20]

Nk1 n2 D Nk2 g

2 D 35

D 2

3

J1

J0n

J2

J0g

where J0, J1 and J2 are related to the element Jacobian asfollows

jJj J0 J1n J2g

a1b3 a3b1 a1b2 a2b1n a2b3 a3b2g

36

and coecients ai and bi i 1; 2; 3 are dependent onthe element nodal coordinates

a1 b1

a2 b2

a3 b3

264

375

1

4

1 1 1 11 1 1 1

1 1 1 1

24

35

x1 y1x2 y2x3 y3x4 y4

2664

3775 37

With the above assumed displacements (33) and (34), we

have

ov3k3

oy1ov3k

3

oy2

8


7/16

The stationary condition of the functional (41) gives the

equilibrium equation (23), the stressstrain relations and

the equilibrium of traction between the elements if the

following condition is met a priori

XeZ

Yed~sk3i ~ckk3idY 0

Following the procedure employed in Section 4, the

above constraint can be simplied to the PTC [20,23]ZYe

~ckk3idY 0 42

It is evident that Eq. (42) is an alternative formulation of

the PTC (32), if the enhanced strain ~ckk3i corresponds to

the incompatible displacement v3k3k.

The nite element based on the stationary condition

of functional (41) requires an independent approxima-

tion of three elds: v3k3 , ~ckk3j and ~s

k3j. In the enhanced-

strain element, however, the independent stress eld is

eliminated by selecting it to be orthogonal to the en-

hanced strain eld, i.e.ZYe

~sk3i ~ckk3idY 0 43

Thus, the two independent elds for the enhanced-strain

formulation are the displacement v3k3 and the enhanced

assumed strains ~ckk3j. The formulation here is the same as

in Section 4 above, provided ~ckk3j are interpolated from

the element inner parameters as follows

~ckk31~ckk32

( ) Bkk

k 44

Moreover, if the assumed strains ~ckk3j in Eq. (44) corre-

spond to the incompatible displacement v3k3k in Eq. (34),

the enhanced-strain formulation will be equivalent to

the incompatible-displacement formulation discussed in

Section 4. Therefore, only NQ6 introduced in Section 4

will be employed in the computations to follow. Note

however that the stress in the enhanced-strain formula-

tion can be recovered with the help of the orthogonal-

ization condition (43), as suggested in Ref. [21].

6. Enforcing the periodicity boundary condition in the

analysis of the RUC

Assembling the discretized equations of equilibrium

of all elements, yields the following system of equilib-

rium equations

Kqk fk 45

Two dierent loading cases need to be analyzed in order

to determine the characteristic deformations of the unit

cell. The periodicity condition of the boundary dis-

placement can conveniently be enforced by a penalty

function technique [24]. Eq. (45) is the EulerLagrange

equation of the following functional

Pqk 12

qkTKqk qkTfk 46

The periodicity condition yields the following constraint

Rqk 0

If a couple of nodes, i and j, on the boundary have the

same displacement because of the periodicity condition,

i.e.

qki qk

j

the above condition is equivalent to

Ri; i 1 Ri;j 1 Ri; l T i;j 0

In order to satisfy the above periodicity constraint by a

penalty function technique the functional (46) is modi-

ed as

ePqk 12

qkT

Kqk qkT

fk a

2qk

T

RTRqk 47

where a is a large positive number and taken to be 104 in

our computations. Thus, instead of Eq. (45), we will

solve the following equations

K aRTRqk fk 48

7. Numerical examples

As the performance of the element using incompati-

ble functions dened in Eq. (35) and of the hybrid stress

element to be used here for torsion of shafts has been

extensively studied in Refs. [5,19,20], no standard per-

formance tests for both elements are included in thispaper. However, to illustrate the method described

above, we solve the torsion of a composite shaft with

square cross-section (length of side 80), as shown inFig. 3(a). Assume that the microstructure of the cross-

section is locally periodic with a period dened by a

RUC shown in Fig. 3(b), i.e. it consists of an isotro-

pic circular ber of diameter 2a embedded in an iso-

tropic square matrix with side 4a. a 5 is adapted inthis study. The problem is solved in two stages. First,

we solve the RUC by using the incompatible element

NQ6 introduced in Section 4, with the periodicity

boundary condition enforced by the penalty function

approach discussed in Section 6. We obtain the eld v3k3



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and its derivatives o=oyjv3k3 and calculate the homog-enized moduli from Eq. (26). Second, we solve the tor-

sion of the square shaft shown in Fig. 3(a) with the

homogenized moduli obtained at step one above, by

using the hybrid stress element introduced in Ref. [5]. In

this way, we calculate the warping displacement, tor-

sional rigidity and the angle of twist per unit length, aswell as the shear stresses and strains. With the results so

obtained, we can calculate the rst order warping dis-

placement from Eq. (22) and the local strain and stress

elds from Eqs. (10) and (13), respectively. For the

present illustrative purpose, we choose e 0:25. Thecomplete shaft section from which the RUC has been

extracted is shown in Fig. 3(c). In the gures to follow,

lled triangles represent computed data. In all the gures

that illustrate the stress distribution, a line segment

represents the distribution within an element. In Figs. 5,

6(b) and (c), the solid line represents the polynomial t

of the corresponding computed data that is not satis-

factorily smooth.

The RUC shown in Fig. 3(b) is discretized into 896

quadrilateral elements and 929 nodes, as shown in Fig.

4(a). According to the denition of the RUC, its size

should be enlarged four times as e 0:25. However,numerical results show that the results are unaected by

whether or not the RUC size is enlarged, allowing us to

use the original RUC size. Care must be taken in en-forcing the periodicity boundary condition at corner

nodes. For the four corner nodes, i, j, k and l, shown in

Fig. 3(b), the periodicity condition yields

qki qk

j qkk q

kl

The above condition can be rewritten as

qki qk

j

qkj qkk

qkk qkl

Fig. 3. Geometry of a composite shaft of square prole: (a) square prole, (b) RUC, (c) square shaft with 16 bres.



9/16

and treated conveniently by the procedure discussed in

Section 6. The ber and the matrix are considered to be

isotropic with the shear moduli, Gf 10 and Gm 1,respectively. The computed homogenized shear moduli

are

C11 C12Sym C22

1:38271 0:00138

Sym 1:38467 49

Thus the macroscopic behavior of the composite shaft is

also isotropic. The numerical results for the character-

istic displacements v3k3 and their derivatives o=oyjv3k3

are saved for later use.

The isotropic shaft of square cross-section shown in

Fig. 3(a) is now analyzed with the homogenized shear

moduli (49) obtained above. Only a quarter of the cross-

section, the shaded part shown in Fig. 3(a), is dicretized

because of symmetry. The warping displacements are

xed on the axes of symmetry. The employed FE mesh

with 400 quadrilateral elements and 441 nodes is shown

in Fig. 4(b). One unit of torque is applied on the quarter

section with its units being consistent with those of the

shear moduli. The computed result for the torsional ri-

gidity 4 1:9927 106 is very close to the accurate value7:9856 106 obtained from the formula [16]

Torsional rigidity 0:141G2b4 50

where the shear modulus G 1:38271, and the length ofside of the square cross-section 2b 80 in the presentexample. The numerical results for the local elds near

or along the interface between the ber and the matrix

adjacent to the point with global co-ordinates x1 30,x2 30 are shown in Figs. 5 and 6. Fig. 5(a)(c) showthe results along the line 36y16 7, y2 0 near the pointP in Fig. 3(b). Fig. 5(a) shows the distribution of

warping displacement. Fig. 5(b) shows the polynomial

tting of the computed shear stress sxz, on the scale of

the gure results given by the upper and lower ele-

ments adjacent to the line cannot be distinguished. Fig.

5(c) shows the computed shear stress syz, data linked by

solid and broken lines represent respectively the results

Fig. 4. Discretised meshes used in the computation: (a) mesh of the RUC shown in Fig. 3(b), (b) mesh of a quarter of the cross-section

shown in Fig. 3(a), (c) mesh of a quarter of the cross-section shown in Fig. 3(c).



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obtained from the upper and lower elements adjacent to

the line in question. From the results it is seen that the

gradient of the warping displacement changes rapidly

across the interface y1 5 and that the distribution of

sxz but not of syz is continuous across the interface. The

distribution of warping displacement, and of normal

and tangential shear stresses along the interface, which

are given by

Fig. 5. Numerical results on the line 36y1 6 7, y2 0, from the homogenisation method: (a) distribution of warping displacement, (b)distribution of sxz, (c) distribution of syz.



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sn sxz cosu syz sinu

st sxz sinu syz cosu51

where u is the angle from the axis y1 as shown in Fig.

3(b), are plotted in Fig. 6 (a)(c). In Fig. 6(b) and (c),

data linked by broken lines represent the results ob-

tained from the matrix side, the continuous solid line

represents the polynomial t of the results obtained

from the ber side of the interface. These results show

that the warping displacement and normal shear stress

Fig. 6. Numerical results along the interface from the homogenisation method: (a) distribution of warping displacement, (b) distri-

bution of the normal shear stress sn, (c) distribution of the tangential shear stress st.



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sn vary continuously across the interface, whereas the

tangential shear stress st has a signicant discontinuity.

Although it will not be possible to compare the re-

sults with those obtained by the present homogenization

method, we will still solve directly the torsion of the

composite shaft shown in Fig. 3(c) by the hybrid stresselement introduced in Ref. [5] to illustrate some typical

features of local elds adjacent to the interface. Again,

only a quarter of the cross-section is needed to be

dicretized because of symmetry. The warping displace-

ments are xed on the axes of symmetry. The FE mesh

with 3584 quadrilateral elements and 3649 nodes is

shown in Fig. 4(c). One unit of torque is applied on the

quarter section with its units being consistent with those

of the shear modulus. The computed result for torsional

rigidity is 4 1:9456356 106, which according to theformula (50) corresponds to an isotropic shaft with

shear modulus 1.34754. The result is reasonably close tothat obtained by the homogenization method (49). The

latter predicts larger values of moduli because the em-

ployment of the periodic boundary condition makes the

system stier. The result given by the homogenization

method is also within the lower bound 1.215 and the

upper bound 2.767 as per the VoigtReuss theory [4].

Zhao and Weng [25] have derived the nine eective

elastic constants of an orthotropic composite reinforced

with monotonically aligned, uniformly dispersed elliptic

cylinders using the EshelbyMoriTanaka method. The

problem studied above is the special case that the rein-

forcements are bers with circular cross-section. The

two shear moduli relevant to torsion given by Zhao andWeng [25] are

C11

Gm 1

cfcma1a

GmGfGm

;C22

Gm 1

cfcm

1a Gm

GfGm

52

where cf and cm are volume fractions of ber and matrix,

respectively, and a is the cross-sectional aspect ratio of

the reinforced ber. In our case, cf p=4, cm 1 p=4and a b=a 1, and hence the eective shear moduli

C11 4:595947 C22 given by Eq. (52) are unreason-ably higher than the results by the direct FE analysis, as

well as the results (49) by the homogenization method

mentioned above. They are also above the upper bound

of the VoigtReuss theory. The EshelbyMoriTanaka

method cannot give good results, especially for high

volume fraction of reinforcements, because Eshelby's

tensor is based on the inclusion in an innite matrix,

which takes into account of the interaction between re-

inforcements in a very weak sense. On the other hand, it

is evident that the homogenization method has the ad-

vantage of taking the interaction between phases into

account naturally and of not having to make assump-

tions such as isotropy of material.

The distribution of warping displacement and shear

stresses along the line corresponding to Fig. 5 and the

interface corresponding to Fig. 6 are plotted in Figs. 7

and 8. (51) has been used to obtain the normal and

tangential shear stresses in Fig. 8(b) and (c). A com-

parison of Figs. 5 and 6 with Figs. 7 and 8, respectively,shows the obvious dierences of the results obtained by

the homogenization method and the direct hybrid stress

element. The dierences are to be expected in view of the

limited number of bers that can be economically han-

dled by the hybrid stress element. The homogenization

method is suitable for problems involving a large num-

ber of periodically distributed reinforcements so that the

RUC occupies only a ``point'' in the physical domain

[12]. The computed stress elds by the hybrid stress

element are smoother than those obtained by the ho-

mogenization method and smoothing techniques are

unnecessary for the former since dierentiations areavoided in the computations. Notwithstanding these

dierences, the results by the two methods reveal the

common features of the local elds: a signicant dis-

continuity exists in the tangential shear stress, while

other elds are continuous adjacent to the interface.

Having gained condence in the accuracy of the in-

compatible element NQ6 developed from the homoge-

nization theory in predicting the eective shear moduli,

we study below the eect of the cross-sectional shape of

the reinforcing bers. The RUC used is illustrated in

Fig. 9(a), i.e. an elliptic cylindrical ber is embedded in

the matrix of rectangular shape; the pattern of discreti-

zation is similar to Fig. 4(b). The material properties ofthe ber as well as the matrix and their volume fractions

are selected as above. The variation of the computed C11and C22 with the aspect ratio of the ber is plotted in

Fig. 9(b). Again, the results from Eq. (52), i.e., the

EshelbyMoriTanaka method, are unreasonably high-

er. However, the predicted trend is the same with an

increase in b=a, C11 decreases, and C22 increases.

8. Conclusions and discussion

The homogenization method is most suitable for

problems involving a large number of periodically dis-

tributed reinforcements so that the RUC can be re-

garded as a ``point'' in the physical domain. It gives not

only the equivalent material properties but also detailed

information of local elds with much lower computa-

tional cost. Such detailed information of the elds on

the scale of microconstituents is almost impossible to

obtain by using the hybrid stress element, because of the

enormous degrees of freedom needed to model the en-

tire macrodomain with a grid size comparable to that of

the microscale features. When the number of the rein-

forcement is not very large, numerical results by the



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homogenization method without the terms of order

higher than one are usually quantitatively dierent from

those obtained by the direct hybrid stress element. The

inclusion of higher terms may improve numerical ac-

curacy, but it inevitably complicates the procedure. In

the determination of the equivalent properties and in the

evaluation of the local elds, the derivatives of the

characteristic displacement v3k3 are needed. Therefore,

the widely used isoparametric elements are not suitable

for the analysis of the RUC. The hybrid stress element is

also limited because it is dicult to enforce the period-

icity condition on the assumed stresses. From a practical

point of view, the incompatible element based on the

modied potential principle and the enhanced-strain

Fig. 7. Numerical results on the line 36y16 7, y2 0, from the hybrid stress element method: (a) distribution of warping displace-ment, (b) distribution of sxz, (c) distribution of syz.



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element based on the 3-eld HuWashizu principle are

the most appropriate for the analysis of RUC, whereas

the hybrid stress element is appropriate for the macro-

homogenized problem.

For torsion of ber reinforced composite shafts, a

signicant discontinuity exists in the tangential shear

stress, while other elds are continuous along the in-

terface between the ber and the matrix.

Fig. 8. Numerical results along the interface from the hybrid stress element method: (a) distribution of warping displacement,

(b) distribution of the normal shear stress sn, (c) distribution of the tangential shear stress st.



15/16

Acknowledgements

CC Wu and QZ Xiao acknowledge the nancial

support from the National Natural Science Foundation

of China under grant number 19772051.

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