The finite element (FE) method was used to calculate the axial and radial stress distributions as a function of axial distance, z, from the centre, and radius, r, in an elastic fibre surrounded by a plastic matrix. Plastic deformation of the matrix was considered to exert a uniform interfacial stress, τ , along half the length of the fibre. Axisymmetric models were created for uniform cylindrical, ellipsoidal, paraboloidal and conical fibres characterised by an axial ratio, q, and half length, L. Young&apos;s modulus for the material of the fibre and L were arbitrarily assigned values of unity, since they act as scaling factors; q also acts as a scaling factor but was assigned a value of 10 to create models with a fibrous appearance. For the cylindrical fibre, the axial stress increased linearly from the end towards the centre; the radial stress was more evenly distributed. At the other extreme, the conical fibre showed a uniform distribution of axial and radial stress. Results for ellipsoidal and paraboloidal fibres were intermediate between these two extremes. In general, the effect of taper is to lower peak stress at the fibre centre and make the stress distribution throughout the fibre more even. These results are in good agreement with recent analytical theories for the axial distribution of surface radial stress and axial stress along the fibre axis. However, FE models have the advantage of predicting full three-dimensional stress distributions. C 2000 Kluwer Academic Publisher

D W L Hukins

K J Mathias

K L Goh

R M Aspden

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JOURNAL OF MATERIALS SCIENCE35 (2000 )2493– 2497
Finite element analysis of the effect of fibre shape
on stresses in an elastic fibre surrounded
by a plastic matrix
K. L. GOH∗, K. J. MATHIAS∗, R. M. ASPDEN‡, D. W. L. HUKINS∗,§
∗Department of Bio-Medical Physics & Bio-Engineering, and ‡Department of Orthopaedic
Surgery, University of Aberdeen, Foresterhill, Aberdeen AB25 2ZD, UK
E-mail: d.hukins@biomed.abdn.ac.uk
The finite element (FE) method was used to calculate the axial and radial stress
distributions as a function of axial distance, z, from the centre, and radius, r, in an elastic
fibre surrounded by a plastic matrix. Plastic deformation of the matrix was considered to
exert a uniform interfacial stress, τ , along half the length of the fibre. Axisymmetric models
were created for uniform cylindrical, ellipsoidal, paraboloidal and conical fibres
characterised by an axial ratio, q, and half length, L. Young’s modulus for the material of
the fibre and L were arbitrarily assigned values of unity, since they act as scaling factors;
q also acts as a scaling factor but was assigned a value of 10 to create models with a
fibrous appearance. For the cylindrical fibre, the axial stress increased linearly from the end
towards the centre; the radial stress was more evenly distributed. At the other extreme, the
conical fibre showed a uniform distribution of axial and radial stress. Results for ellipsoidal
and paraboloidal fibres were intermediate between these two extremes. In general, the
effect of taper is to lower peak stress at the fibre centre and make the stress distribution
throughout the fibre more even. These results are in good agreement with recent analytical
theories for the axial distribution of surface radial stress and axial stress along the fibre
axis. However, FE models have the advantage of predicting full three-dimensional stress
distributions. C© 2000 Kluwer Academic Publishers
1. Introduction
This paper presents the results of a finite element (FE)
analysis of the stress distribution in a fibre, which need
not be a uniform cylinder, embedded in a plastic ma-
trix. In such a fibre composite system, deformation of
the matrix generates an interfacial shear stress,τ . For
an elastic matrixτ is not constant [1]. However, for a
plastic matrix the magnitude ofτ is constant along the
length of a fibre [2]. The plastic case is important for two
reasons. Firstly, it corresponds to the behaviour of the
matrix during failure. Secondly, a constant magnitude
for τ corresponds to the physically reasonable case of a
constant number of interactions per unit area between
fibre and matrix [3, 4]. There are well-established an-
alytical models which show thatτ generates an axial
tensile stress, but no compressive stress, in a uniform
cylindrical fibre [1, 5]. However, it has been found that
the fibres which reinforce some biological composites
are tapered [6–8].
Recently, a theory has been developed for the stress
distribution in tapered fibres embedded in a plastic
matrix [4]. The fibres were modelled as a cone, a
paraboloid and an ellipsoid with an axial ratioq (see
Section 2.1 for further details). The theory predicts that
§ Author to whom all correspondence should be addressed.
there will be a radial compressive stress,σr , as well as an
axial tensile stress,σz, in these tapered fibres given by
σr = −(τ/q) fr (Z) and σz = τq fz(Z) (1)
where the form of fr (Z) and fz(Z) depend on the
fibre shape and are both unity for a cone. HereZ
represents a fractional distance between the centre of
a fibre and its end (see Section 2.2 for further details).
Thusτ/q acts as a scaling factor for the radial stress
andτq acts as a scaling factor for the axial stress [4].
Strictly,σr is the radial stress at the surface of the fibre
andσz is the axial stress along the fibre axis. The FE
odel was developed for two reasons. Firstly, it was
developed to check the predictions of the theory. The
second reason was to investigate the radial distribution
of the radial and axial stresses. These distributions
need not be constant and cannot be predicted by the
analytical approach. In this paper the FE model is used
to investigate stress distributions in a uniform cylinder,
a cone, a paraboloid and an ellipsoid when subjected
to an interfacial stress of constant magnitude.
0022–2461 C© 2000 Kluwer Academic Publishers 2493
(a)
(b)
(c)
(d)
Figure 1 Models for (a) uniform cylindrical, (b) ellipsoidal, (c) parab-
oloidal and (d) conical fibres. Each model is developed from a curve in the
upper right-hand quadrant which is reflected and rotated (see Section 2.1)
to generate an axisymmetric fibre (see Section 2.5). For each shape the
radius,ro, at the centre of the fibre and the half length,L, of a fibre are
marked.
2. Methods
2.1. Fibre models
Fig. 1 shows upper right-hand quadrants of plane sec-
tions through fibres with the following shapes: a uni-
form cylinder, an ellipsoid, a paraboloid and a cone. A
cylindrical polar coordinate system (r , φ, z) was de-
fined to have its origin at the fibre centre, O, wherer
denotes the radius of the fibre (which may vary along
its length),φ is an azimuthal angle andz lies along
the fibre axis. The axial ratio was defined byq= L/ro,
wherero is the radius of the fibre at the origin andL is
the half length of the fibre. The complete fibre model
can be generated by reflecting about OA (in Fig. 1) and
rotating about OC (in Fig. 1a) or OB (in Fig. 1b–d).
For the purposes of the FE analysis, the shape of the
fibre surface was defined by the equations which de-
scribe the appropriate cross-section for 0≤ z≤ L. For
a uniform cylinder this is given by
r = ro. (2)
An ellipsoidal fibre is defined by
r = ro
√
1−
(
z
L
)2
, (3)
a paraboloidal fibre by
r = ro
√
1−
(
z
L
)
, (4)
and a conical fibre by
r = ro
[
1−
(
z
L
)]
. (5)
Equations 2–5 were used to generate the coordinates
that described the profiles of the two-dimensional mod-
els for each of the four fibres. Axisymmetric com-
puter models were created from these coordinates using
ANSYS (version 5.4; ANSYS, Inc., Houston, PA).
2.2. Fibre properties
The Young’s modulus of the fibre material was assigned
a value of unity since it acts as a scaling factor. Poisson’s
ratio was assigned a single value (0.3) since this paper
is concerned solely with determining stresses which are
independent of its value (unlike strains). Sinceq acts as
a scaling factor (see Section 1), it was assigned a value
of 10 which yields models with a fibrous appearance
(Fig. 1) but avoids difficulties in meshing models with
large values ofq. SinceZ= z/L, in Section 1, it also
follows that L acts as a scaling factor when stress is
plotted as a function ofz; therefore, it was assigned a
value of unity.
2.3. Meshing and optimisation
Each model was initially meshed using the default
mesh generator in ANSYS. Element shape and ele-
ment size [9] were varied to produce stress distributions
which were smooth curves and optimised the level of
agreement with the analytical models. Six-node trian-
gular elements were found to be better, according to
these criteria, than eight-node quadrilateral elements.
The number of elements used for each shape ranged
from approximately 13000 (cone) to 15000 (cylinder).
The computational time was not affected appreciably
by the number of elements. Therefore, the maximum
number of elements allowed by ANSYS was used.
2.4. Boundary conditions
To ensure that the model had the symmetry described
in Section 2.1, the nodes lying along OA, in Fig. 1,
were constrained to prevent any displacement in the
z-direction.
The constant shear stress,τ , tangential to the surface
of the fibre was assigned a value of unity as it acts as
a scaling factor (see Section 1). This interfacial stress
gives rise to an interfacial force. The interfacial force,
Fi , acting on thei th node, along AB, is given by
Fi = τπ l i (ri + ri+1) (6)
wherel i is the distance between thei th and (i + 1)th
nodes;ri andri+1 are the radii of these nodes and, hence,
of the toroidal surface element. Equation 6 relatesτ to
Fi by multiplying it by the outer surface area of this
element which is the surface of the frustum of a cone.
For input into the FE model, each value ofFi was re-
solved into components parallel with and perpendicular
to thez-axis of the cylindrical polar coordinate system
(Section 2.1).
2494
2.5. Solving and postprocessing
The FE models were solved for the stresses using the
default solver provided by ANSYS. Equation 1 shows
that τ/q acts as a multiplier for the radial stress,σr .
Sinceτ andq were assigned arbitrary values,σr q/τ
was plotted as a function ofz andr using Matlab (ver-
sion 5.0; The MathWorks, Inc., Massachusetts, USA).
Similarly, Equation 1 shows thatτq acts as a multiplier
for the axial stress,σz. Sinceτ andq were assigned ar-
bitrary values,σz/τq was plotted as a function ofz and
r . These plots enable values ofσr andσz to be calcu-
lated for a fibre with a known value ofq when subjected
to a known value ofτ .
3. Results
The axial stress distributions for the different shapes
of fibre are shown in Fig. 2. For the uniform cylindri-
cal fibre, the stress rose linearly, in the axial direction,
from zero at its end to a maximum value;σz/τq had its
greatest value of 2 at the centre of the fibre (Fig. 2a).
For the ellipsoidal (Fig. 2b) and paraboloidal (Fig. 2c)
fibres, the stresses also increased from zero at the end
but were distributed more evenly than for the uniform
cylinder. In the ellipsoidal fibre,σz/τq reached a maxi-
mum value of about 1.5 at the centre; in the paraboloidal
fibre it reached a value of about 1.3. The stress was con-
stant along the whole length of the conical fibre with
σz/τq= 1 (Fig. 2d). Note that the uniform cylinder,
ellipsoid, paraboloid and cone are progressively more
tapered in shape. Tapering the fibre has the effect of
lowering the peak stress at its centre and making the dis-
tribution more uniform over its length. Variation across
the fibre radius for a uniform cylindrical fibre was usu-
ally small (less than 0.5%) with the highest values along
the fibre axis; an exception occurred at the surface, near
the plane of symmetry, but is likely to be an artefact
of the modelling procedure. There was no apparent de-
pendence of stress on radius for a conical fibre and,
as before, the ellipsoidal and paraboloidal fibres had
distributions intermediate between the extremes of the
uniform cylinder and the cone.
The radial stress distributions at the fibre surface are
defined by the radial component of the applied shear
stress, which is compressive; the distribution of radial
stress values is shown in Fig. 3 for all four fibre shapes.
In practice,σr q/τ was plotted, instead of the stress,σr ,
for the reason discussed in Section 2.5. For the cylindri-
cal fibre, the radial stress is zero across the whole radius
over most of the length of the fibre; however, the stress
becomes increasingly compressive at the fibre end and
there is a region of tensile stress near the fibre centre
(Fig. 3a). The conical fibre shows no variation of radial
stress (Fig. 3d). There is a constant compressive stress
over the whole cross-section of the fibre, corresponding
to the radial component of the applied shear stress. Once
again the ellipsoidal and paraboloidal fibres show be-
haviour intermediate to these extremes, with the stress
distribution in the less tapered ellipsoidal fibre (Fig. 3b)
being similar to that of the cylinder and that of the more
tapered paraboloidal fibre showing a distribution closer
to that of the cone.
(a)
(b)
(c)
(d)
Figure 2 Distribution of σz/τq plotted against axial displacement,z,
and radius,r , for (a) uniform cylindrical, (b) ellipsoidal, (c) paraboloidal
and (d) conical fibres. Hereσz is the axial stress induced in a fibre of
axial ratioq (see Section 2.1) as a result of an interfacial stressτ (see
Section 1).
4. Discussion
The axial distribution of axial stress,σz, determined
from the FE models is in excellent agreement with the-
oretical predictions. Fig. 4a compares the FE results
with the theoretical axial distribution ofσz/τq for a
2495
(a)
(b)
(c)
(d)
Figure 3 Distribution of σr q/τ plotted against axial displacement,z,
and radius,r , for (a) uniform cylindrical, (b) ellipsoidal, (c) paraboloidal
and (d) conical fibres. Hereσr is the radial stress induced in a fibre of
axial ratioq (see Section 2.1) as a result of an interfacial stressτ (see
Section 1).
uniform cylinder. The theoretical expression was cal-
culated using the generally accepted theory for a plastic
matrix [1, 5] which is a special case of the general the-
ory for fibres of any shape [4]. The result for a uniform
cylinder has also been obtained previously [10] by ap-
plying Filon’s analysis [11]. Similarly good agreement
between FE results and the theoretical axial distribu-
tion of σz/τq was obtained for ellipsoidal (Fig. 4b),
paraboloidal (Fig. 4c) and conical fibres (Fig. 4d). In
these cases, the theory for tapered fibres [4] was used
to calculate the theoretical stress distributions.
(a)
(b)
(c)
(d)
Figure 4 FE results (filled circles) compared with theoretical results
(continuous line) forσz/τq plotted against axial distance,z, for (a) uni-
form cylindrical, (b) ellipsoidal, (c) paraboloidal and (d) conical fibres.
Hereσz is the axial stress induced in a fibre of axial ratioq (see Section
2.1) as a result of an interfacial stressτ (see Section 1). Formulae for
calculating theoretical results are published elsewhere [4].
The axial distribution of surface radial stress,σr , de-
termined from the FE models is also in excellent agree-
ment with theoretical predictions (Fig. 5). The same
theories were used to calculate the theoretical distri-
butions of radial stress for the uniform cylindrical [1],
ellipsoidal, paraboloidal and conical [4] fibres as were
used for axial stress.
Thus the FE models agree with the predictions of the-
ory, if τ is considered to be constant along the length
of a fibre. However, FE models have a further appli-
cation. Current theories are restricted to predicting the
axial distribution of radial stress, at the fibre surface
only, and of axial stress, along the fibre axis only. In
contrast, FE models can provide information on the full
three-dimensional stress distribution throughout a fibre.
Previously the FE method has been used to investigate
the interfacial stress distribution for a uniform cylindri-
cal elastic fibre surrounded by an elastic matrix [12].
Now it has been used to investigate stress distributions
2496
(a)
(b)
(c)
(d)
Figure 5 FE results (filled circles) compared with theoretical results
(continuous line) forσr q/τ plotted against axial distance,z, for (a) uni-
form cylindrical, (b) ellipsoidal, (c) paraboloidal and (d) conical fibres.
Hereσr is the radial stress induced in a fibre of axial ratioq (see Sec-
tion 2.1) as a result of an interfacial stressτ (see Section 1). Formulae
for calculating theoretical results are published elsewhere [4].
in fibres, which need not be uniform cylinders, when
embedded in a plastic matrix.
The FE results enable us to draw two important con-
clusions on the effect of taper on the stress distribution
within an elastic fibre embedded in a plastic matrix.
Firstly, comparison of Figs 2 and 3 shows that radial
stresses are much less than axial stresses. In order to
understand this conclusion note thatτ is a constant
and that, for real fibres,q will be very large; thusσz
will be much greater thanσr . Secondly, increasing the
taper lowers the peak stress at the fibre centre and dis-
tributes the axial stress more evenly (see Section 3). In
the extreme case of a conical fibre, the axial and radial
stresses are both evenly distributed throughout the fi-
bre. Since a cone has only one-third the volume of a
uniform cylinder of the same length and radius, coni-
cal fibres then represent a more efficient use of a given
volume of reinforcing material than uniform cylindrical
fibres.
Acknowledgements
We thank Dr. J. R. Meakin for comments on the paper,
the University of Aberdeen for an award of a university
postgraduate studentship (to KLG) and the Medical Re-
search Council for an award of a senior fellowship to
(RMA).
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Received 17 March
and accepted 22 November 1999
2497


Finite element analysis of the effect of fibre shape on stresses in an elastic fibre surrounded

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Finite element analysis of the effect of fibre shape on stresses in an elastic fibre surrounded

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