Tensile characterization of unidirectional discontinuous bamboo
fibre/epoxy composites
D. Perremansa*, E. Trujilloa, L. Osorioa, A.W. Van Vuurea, J. Ivensa, I. Verpoesta
aDepartment
of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, Kasteelpark
Arenberg 44 bus 2450, 3001 Heverlee, Belgium
*Corresponding author. Tel. +3216321231; E-mail address: dieter.perremans@mtm.kuleuven.be
ABSTRACT
Bamboo culms consist of a nodal structure in which the fibres are as long as the internode length
(between 15 and 30 cm), limiting the length of the reinforcing fibres. The implementation of these
fibres to reinforce polymers in large-scale applications leads to the production of discontinuous
composites in unidirectional configuration. The influence of the average overlapping length between
adjacent fibre layers on the mechanical characteristics of aligned “short“ (5 cm) fibre bamboo-epoxy
composites is investigated using a tensile testing procedure. The overlapping patterns were
predefined at certain regular positions. Aligned discontinuous bamboo-epoxy composites with
different overlapping lengths are produced with a light RTM. Tensile testing results have indicated that
the overlapping patterns have a significant influence on the tensile strength of the composite samples
and low impact on the stiffness of the composite. A modified local load sharing model (LLS) was
applied to simulate the tensile strength of an aligned short fibre bamboo-epoxy composite with the
different overlapping patterns and also a random dispersion of fibre along the samples.
1. INTRODUCTION
1.1 General introduction and problem outline
Nowadays rational exploitation and use of sustainable natural resources are a necessity and they will
play a crucial role in the near future. In recent years there has been an increasing interest to
scientifically study the potential of bamboo fibre as reinforcing material for polymer matrix composites
[1-3]. Bamboo (Guadua angustifolia) fibres, referred in this study as technical fibres or fibre bundles
(composed of elementary fibres) and obtained after a standard extraction process, are becoming a
real alternative for glass fibres as reinforcement for composite materials.
Bamboo fibres, amongst other natural fibres, have one of the most favorable combinations of low
density (1.4g/cm³) and good mechanical properties that can compete with glass fibres in terms of
specific properties, as visible in figure 1 [4]. Furthermore, bamboo fibres are an inexhaustible
renewable bio-resource with high growth and CO2 fixation rate [5]. These advantages make that
bamboo fibres become a sustainable candidate to be exploited at larger industrial scale and that they
provide an alternative to reduce of the environment impact of composite structures by replacing less
environment-friendly fibres (e.g. glass fibre).
Figure 1: Representation of the specific mechanical characteristics of several natural fibres. Synthetic glass fibres are also
included to allow a comparison. Due to their low densities, the specific longitudinal stiffness and tensile strength of natural
fibres is similar to that of E-glass. Natural fibres can thus replace glass fibres in several applications. [4]
A large bottleneck that impeded the introduction of bamboo fibres as composite reinforcing materials
for many years has been the extraction of undamaged long fibres, but recently, a new environmental
friendly mechanical process was developed by KU Leuven to produce high quality long bamboo fibres
suitable to be used as a reinforcement in polymeric matrices [6]. In this process, the use of high
temperature, high pressure or chemicals is avoided, reducing both the damage introduced to the
fibres and the amount of energy required for the extraction of bamboo fibres, while at the same time
as well decreasing the environmental impact of the extraction method.
A second drawback towards the large-scale industrial application of bamboo fibre composites is the
discontinuous structure of the bamboo fibre culm, as visible in figure 2. The bamboo culm is divided in
internodes, in which the reinforcing fibres are well aligned, and nodes, where the reinforcing fibres
entangle and contribute as such to the buckling strength of the culm [7-8]. This structure, inherent to
the bamboo fibre plant, limits the extraction length of the reinforcing bamboo fibres to the internode
length. For a 48-month-old culm, the internode length of Guadua angustifolia fibres varies between 20
and 35 cm [7]. The fibre diameter ranges between 90 and 250 μm that in combination with its high
modulus gives a high bending stiffness of the fibres impeding the production of endless bamboo yarns
[7]. Therefore, an innovative approach towards the use of continuous bamboo fibre material is the
development of a preform of unidirectionally aligned bamboo fibres that will allow the use of existing
technology to produce high performance composite parts. The production of endless bamboo fibre
preform, will help to overcome the actual restriction of having discontinuous fibres.
Figure 2: Guadua angustifolia culm showing its sectioning into nodes and internodes. In the nodes, the reinforcing fibres are
entangled, while in the internodes they are severely aligned [7-8].
To bring this to reality, is the aim of this current study; to perform a systematic characterization of the
effect of the overlapping length for a set of fibre bundles, but also to study the influence of a complete
randomized overlapping patterns of unidirectional single fibres into the continuous perform. This
characterization, benchmarked with the mechanical behaviour of a fully continuous UD fibre
composite, will show the feasibility to use highly oriented discontinuous bamboo fibres in continuous
preforms that can be applied in different existing manufacturing techniques.
1.2 Scientific modelling background
In this paper, the mechanical characteristics of aligned short-fibre bamboo-epoxy composites are
investigated using tensile test experiments. The experimental results are then compared with a series
of models that allow a prediction of the mechanical characteristics of these composites. The upcoming
paragraphs rally through the applied models to predict the longitudinal stiffness and tensile strength of
aligned unidirectional short-fibre composites and set forth their main assumptions.
1.2.1 Prediction of longitudinal stiffness of aligned UD short-fibre composites
The rule of mixtures is the simplest mechanical model to estimate the properties of a multiple
component system. It estimates the composite material properties by taking a volume weighed
average of the corresponding properties of the individual constituents. Concerning the prediction of
the longitudinal stiffness of a composite, this model assumes the presence of continuous aligned
fibres through-out the entire length of the composite. Furthermore, it adapts the isostrain assumption
that reflects the necessity that planar cross-sections remain planar during loading conditions. Applying
Hooke’s law and previous assumptions yields the following formula to estimate the longitudinal
stiffness of a composite:
(1)
The shear lag theory considers a single cylindrical linear elastic and isotropic fibre of finite length Lsf
and radius rsf that is encased in a concentric cylindrical shell of linear elastic, isotropic matrix with
radius Rm. A unidimensional stress-state situation is applied in which the matrix tensile strain
becomes equal to the applied strain (
) at a radial distance Rm from the fibre axis. Furthermore, it
assumes that the fibre axial stress vanishes at the fibre ends and that the shear force is constant at
concentric cylindrical surfaces around the fibre [9]. Full analysis of the model results in a length
depending modified rule of mixtures, as given in equations 2-4.
The Halpin-Tsai equations are a series of semi-empirical equations to determine the different moduli
of composites. Originally, they are developed to predict the entire set of moduli of continuous fibre
composites. However, by applying a number of restrictive assumptions, the results can be extended
towards short-fibre composites. The Halpin-Tsai equations are deduced from the work of Hermans
and Hill. They assume a composite in which the embedded phase consists of continuous and
perfectly aligned cylindrical fibres. The fibres are modelled as transversely isotropic and linearly
elastic. The matrix phase is considered to be linearly elastic and isotropic. Hermans analytically
establishes the strain field in the embedded system under an applied stress by assuming
displacement and stress continuity at the fibre radius. The stress and strain field are then volume
averaged to yield one of the stiffness tensor terms. By varying the applied stress, the entire stiffness
tensor can thus be deduced. Halpin and Tsai argue that these results (for the continuous fibre) can be
written in a more general form by the insertion of two variables, η and ζ, and two additional
assumptions. The first approximation yields that ζ is insensitive to the matrix Poisson coefficient. The
second, more inaccurate one, states that the composite contraction coefficient and longitudinal
stiffness must follow the rule of mixtures relations. The general form is shown in equations 5 and 6, in
which Z represents one of the composite moduli [10-12]. Equation 7 visualizes the necessary
assumptions.
The Mori Tanaka model embroiders on the concept of an equivalent inclusion. In this case, an infinite
solid body with stiffness Cm that is initially in a stress-free state is considered. An inhomogeneous
inclusion with different material properties compared to the solid body is then introduced and the
entire system is subjected to a uniformly applied strain at infinity εA. The idea is to homogenize the
inhomogeneous inclusion by compensating for the material properties with the introduction of a virtual
transformation strain that leads to same stress state in the inclusion. In the Mori-Tanaka method the
influence of multiple inclusions on the strain field, compared to the one-inclusion system, is included
in an image strain. This image strain is approximated by its mean value, which is the same anywhere,
in all the inclusions and in the matrix. This mean field assumption implies that each inclusion only
feels the presence of the other inclusions indirectly through the total strain in the matrix. To obtain the
homogenized stiffness of a composite using the Mori-Tanaka method, the volume average of the
composite stresses is rewritten using Hooke’s Law. The strains in inclusions and matrix are then
linked to the externally applied strain using strain concentration tensors. The model assumes that the
total disturbance strain in the composite should be absent, which leads to equations 8-10, in which
represents the strain concentration tensor that links the total strain in the inclusion to the applied
strain. It can be stated in terms of the dilute strain concentration tensor
that relates the total strain
in the inclusion to the average matrix strain (which is the sum of the applied strain and the image
strain) [13].
It is remarked that this model additionally assumes that fibre and matrix are linearly elastic and
perfectly bonded through the entire deformation state. The matrix is usually considered isotropic,
whereas the fibres are modelled as transversely isotropic. The Mori-Tanaka algoritm does not depend
on the size of the inclusion, nor on their positional coordinates. It does depend on shape and
orientation of the inclusions, as well as their volume fraction [13].
1.2.2 Prediction of longitudinal tensile strength of aligned UD short-fibre composites
The rule of mixtures applies the iso-strain assumption, indicating that the component with the lowest
breakage strain will fail first. Since bamboo fibres, utilized in this research, are more brittle than the
epoxy matrix and since the produced composites contain high fibre volume fractions, they will break at
the breakage strain of the fibres, reducing the rule of mixtures to equation 11, in which
indicates
the longitudinal fibre tensile strength,
the longitudinal tensile stress in the matrix at fibre failure and
Vf and Vm the volume fractions of respectively fibre and matrix This explanation further assumes that
the stress concentration around one broken fibre initiates an avalanche of other fibre failures (at the
same strain) and that no mechanisms of crack propagation deceleration are present.
The Kelly-Tyson model is based on the shear lag theory and therefore puts forth the same
assumptions as previously described in paragraph 1.2.1. It embroiders on the interaction between
axial tensile stresses in the fibre and shear stresses at the interface of the fibre. The Kelly-Tyson
model further assumes interface failure to occur first and models the shear stress at the interface as a
constant in the debonded or yielded area [14]. By introducing the critical aspect ratio S c as the
smallest fibre aspect ratio at which the axial fibre tensile stress can just reach the fibre strength and
by assuming that fracture occurs when the axial fibre tensile stress reaches the fibre strength, the
composite strength is determined as stated in equation 12.
The global load sharing model (GLS), invented by Curtin, incorporates statistics in the estimation of
composite tensile strength. The model builds further on the pioneering work of Weibull, who deduced
a semi-empirical expression for the strength distribution of brittle materials based on a chain-of-links
system, and Rosen, who introduced the weakest-link theory in the prediction of the tensile strength of
fibre reinforced plastics. Curtin modifies Rosen’s chain-of-bundles idea by accounting for load
contributions of broken fibres. He argues that stress recovery occurs in the ineffective length due to
the presence of a presumed constant interfacial shear stress. In his model, the ineffective length is
calculated using the Kelly-Tyson approximations, leading to equation 13 for the longitudinal tensile
strength of composites [15-17].
The local load sharing (LLS) model deroutes from the intention to predict the composite tensile
strength in an analytical way. It incorporates matrix shear loading to re-distribute the applied load
between adjacent fibres and therefore can account more accurately for fibre breakages. Usually, a
spring element model is considered in which the fibre is represented by axial springs in the
longitudinal direction and the matrix by shear springs in the transverse direction. Each of the spring
elements is assigned a stiffness matrix. Strength variation is included through a Weibull distribution.
At each strain increment, the stresses in the springs are compared to the fibre strength. If the fibre
strength is exceeded, the spring element stiffness is removed and a stress re-distribution is
performed. At each step the normalized composite stress is also calculated. This iterative scheme is
repeated until the relative difference in two subsequent composite stress values is larger than a preset value. Monte Carlo simulation is then performed and the composite strength is taken as the
average of the simulated values [18].
2. MATERIALS AND METHODS
Bamboo culms (Guadua angustifolia) are extracted from a typical bamboo plantation in Colombia,
specifically from the Coffee Region, at 1.300 meters above sea level. Technical fibres are extracted
from the bamboo culms using a mechanical extraction process that neither uses chemicals nor high
temperature. The maximum length of the extracted fibres is the internode length, which for 48 month
culms is reported to be between 20 and 35 cm. The fibre diameter ranges between 90 and 280 µm.
The bamboo fibres are layer-wise positioned in a unidirectional way in a channel of predefined
dimensions (28x2cm). In a first stage, discontinuities in the fibre preform are introduced by the
application of a repetitive series of slits over half the width of each layer. The length between two halfwidth slits originating at the same side within one layer is referred to as the slit length and fixed at
5cm. Subsequently, a series of slits of which the initial position is shifted over a certain length,
referred to as the overlapping length (LO), are applied within the same layer, but toward the other side.
The overlapping length is varied as a function of the slit length and takes on the values of 0.5, 1.5 and
2.5cm respectively. For each stacking layer, the initial position of all the slits is shifted by half a slit
length (LS) in order to reduce stress concentrations. No slits are inserted in the clamp area. The entire
pattern construction is visible in figure 3.
Figure 3: The explanation of the inversion symmetry that exists between adjacent fibre assemblies. The left-hand side of the
figure shows the upper view of the alternating fibre assemblies. The cuts that are made in one of the assemblies are also made
in the neighbouring assemblies, but from the other edge on. The right part reveals the lateral view of the composite. Solid
yellow lines indicate that the fibre cut is made to the viewer’s side. Dotted yellow lines represent fibre cuts to the back of the
composite.
In a second stage, the effect of the slits is reduced by reducing the cutting length to one fifth of the
composite width and by randomizing the overlapping length between different fibre bundles. To
establish such a configuration, a random number generator (RNG) is asked to produce a collection of
20 numbers in between 0 and 250. The end value of the RNG expresses the length dimension of the
fibre in mm, as given by the ASTM standard. The applied random number generator makes use of the
random variation in atmospherical noise to bring forth the desired set of values. The values are linked
to the fibre bundles in the order they are generated, starting from the bottom wall of the mould
channel and continuing until a fibre layer is completely filled. This numbering sequence is visualized
with red indices for the bottom fibre assembly in figure 4. The collection of RNG values expresses the
distance between the cuts and the left side of the fibre bundles. To avoid breakage of the composite
samples in the clamped area during tensile testing, values that are located between the interval [0,50]
are shifted 50mm to the right and values within the interval [200,250] are shifted 100mm to the left.
Figure 4: Schematic clarification of the linking procedure to relate the RNG-values to the fibre cuts in each assembly. The
sequence is shown for the first 3 RNG-values. To highligth the cuts, the remaining side fibre bundles are not depicted.
In a third stage, the short fibre bundles are replaced by cut individual technical bamboo fibres. These
short bamboo fibres are randomly placed in the channel according to the randomization sequence of
figure 4.
The resin system used in this master thesis consists of an Epikote 828 LVEL epoxy resin and a
DYTEK DCH-99 diaminocyclohexane hardener in a mass ratio of 500/76. This resin system requires a
curing temperature of 70°C for an hour and an additional post-curing step at 150°C for an hour. The
mass quantities of each constituent are measured using a Mettler PJ4000 with an accuracy of 1g. To
homogenize the resin system, a 70W Kika Labortechnik RW 20.n mixing element with an operational
velocity of 400rotations/min is applied during 10minutes. As this mixing element introduces air
bubbles in the resin system, the resin system is put in a 1.6kW Heraeus VT6060M degassing
machine for a period of time of 25minutes. Samples are producing using vacuum infusion with a top
plate to control the thickness.
Tensile test samples are prepared according to ASTM D3039. Measurements are carried out using an
Instron 4467 with a load cell of 30kN. The crosshead speed is 2mm/min. The dimensions of the
specimens are 250x25x2mm. The samples are mechanically clamped. Sand paper is inserted
between the clamps and the samples to prevent slippage. The gauge length is set at 150mm. An
Instron type 2620-604 extensometer with a gauge length of 25mm is used to accurately measure
displacement data. Data is gathered at a sampling rate of 10points/second. For every test
configuration, 6 samples are tested.
3. RESULTS AND DISCUSSION
3.1 UD continuous fibre composites
The results of the tensile tests performed on the continuous fibre composite samples are depicted in
table 2. This table reveals that the experimental values respectively reach 92% and 79% of the
longitudinal tensile stiffness and strength as obtained using the rule of mixtures, which assumes a
perfect composite.
Table 2: Results of the tensile test of the unidirectional continuous fibre composites. The efficiency factor is calculated as the
ratio between the mechanical characteristic of the sample and the same characteristic as calculated using the rule of mixtures
at the same volume fraction. The results are given with a representative volume fraction of 40%.
Mechanical characteristic
Longitudinal stiffness (GPa)
Longitudinal tensile strength (MPa)
Strain at breakage (/)
Efficiency
factor (%)
92.3 ± 6.4
78.9 ± 4.6
/
Average value
17.6 ± 1.4
222.5 ± 13.0
1.4 ± 0.2
The fracture plane is examined using SEM at secondary electron mode with an applied voltage of 2kV
to avoid charging of the sample. The result is depicted in figure 5. The left-hand side of figure 5
indicates a brittle fracture. It reveals that cracks mostly propagate in the same cross-section, cutting
the fibres along its way. This points out that a strong fibre-matrix interface is present and that the resin
impregnates the fibres very well, which is as well attested by the resin filled lumen on the right-hand
side of figure 5. It is also visible from figure 5 that there is a good dispersion of the fibres. The layerwise construction of the composite is not reflected in the final configuration.
Figure 5: SEM- observations of the fracture plane of continuous fibre composites after tensile testing. The left-hand side gives
an overview of the brittle fracture. The right-hand side reveals the presence of parenchyma cells and provides evidence of the
prematurity of the bamboo fibres through the presence of an unusually large, resin-filled lumen channel.
3.2 UD discontinuous fibre bundle composites with fixed overlapping length
The results of the tensile tests, performed on UD-discontinuous composites in which the overlapping
length is fixed, are given in table 3.
Table 3: Results of the tensile tests of the discontinuous fibre bundle composites with fixed overlapping length. The efficiency
factor is calculated as the ratio between the mechanical characteristic of the sample and the same characteristic as calculated
using the rule of mixtures at the same volume fraction. * The results are given with a representative volume fraction of 40%.
LO=50%
Mechanical
characteristic
Longitudinal
tensile stiffness
(GPa)
Longitudinal
tensile strength
(MPa)
Strain at
breakage (/)
LO=30%
Efficiency
Average
factor
value*
(%)
Efficiency
factor (%)
Average
value*
91.1±5.1
16.5±2.2
89.8±7.7
28.9±4.9
81.6±13.2
/
0.5±0.1
LO=10%
Efficiency
factor (%)
Average
value*
16.9±1.5
86.6±7.8
16.3±1.5
28.3±4.2
79.9±11.9
30.0±6.5
84.6±18.2
/
0.5±0.1
/
0.5±0.1
As expected, the introduction of slits seriously reduces the mechanical characteristics of the
composite samples. The strength is almost bisected with the insertion of these weak points and the
efficiency factor drops to approximately 30%. The strain falls back from approximately 1.2% to only
0.5%. It is argued that the introduction of these slits results in the presence of a matrix enriched area
in the composite, as visible in figure 6. The application of a tensile force produces intensive stress
concentrations around the discontinuous fibre bundles. These stress concentrations eventually result
in shear yielding of the matrix and trigger crack initiation. The initiation of multiple tiny cracks in the
different resin rich zones is proposed. In one of the matrix enriched areas, several tiny voids
coalescence to form a critical crack. The stress singularities at the tip of this critical crack are initially
reduced due to plastic yielding of the matrix. However, the limited ductility of the epoxy resin facilitates
crack propagation.
Figure 6: The left-hand side shows the presence of resin-rich zones in the fibre slit of a UD-discontinuous composite with
LO=30%, observed by SEM. The layer-wise construction seems visible. It is noted that below the thick resin zone in the middle,
the fibre bundles are still present. The right-hand side indicates the brittle fracture, accelerated by the slit presence, of a typical
discontinuous UD bamboo fibre composites.
3.3 UD discontinuous fibre bundle composites with random overlapping length
The results of the tensile tests, performed on UD-discontinuous composites with random overlapping
length, are given in table 4. As compared to the composites with fixed overlapping length, the
reduction of the slit width and the randomization of the overlap allow an increase in efficiency factor
by ±5% for the longitudinal tensile strength. SEM observation however points out a similar fracture
phenomenon as given in the previous paragraph. This indicates that the slit width still acts as a crack
initiator leading to premature failure of the samples.
Table 4: Results of the tensile tests of the discontinuous fibre bundle composites with random overlapping length. The
efficiency factor is calculated as the ratio between the mechanical characteristic of the sample and the same characteristic as
calculated using the rule of mixtures at the same volume fraction. * The results are given with a representative volume fraction
of 40%.
Mechanical characteristic
Longitudinal stiffness (GPa)
Longitudinal tensile
strength (MPa)
Strain at breakage (/)
Efficiency factor
(%)
89.3 ± 5.2
35.5 ± 6.3
Average value*
/
0.7 ± 0.1
16.8 ± 1.0
100.1 ± 17.8
3.4 UD discontinuous fibre composites with random overlapping length
The latter set of experiments is designed to fully remove the crack initiator effect of the slits. The
results of the tensile tests, performed on UD-discontinuous composites with random overlapping
length, are given in table 5.
Table 5: Results of the tensile tests of the discontinuous fibre composites with random overlapping length. The efficiency factor
is calculated as the ratio between the mechanical characteristic of the sample and the same characteristic as calculated using
the rule of mixtures at the same volume fraction. * The results are given with a representative volume fraction of 40%.
Mechanical characteristic
Longitudinal stiffness (GPa)
Longitudinal tensile
strength (MPa)
Strain at breakage (/)
Efficiency factor
(%)
90.1 ± 4.6
63.7 ± 3.8
Average value*
/
0.83 ± 0.03
17.2 ± 0.8
190.9 ± 20.8
These results clearly postulate that randomization of the discontinuities in bamboo fibre composites is
adamant to utilize the good mechanical properties of the fibres. Introduction of randomized
discontinuities leads to a preservation of 85% of the longitudinal tensile strength of a continuous
unidirectional bamboo fibre epoxy composite. By varying the fibre ends over the length of the sample,
the overall stress fields surrounding the discontinuities are expected to be significantly reduced,
decelerating as such the initiation of cracks. The longitudinal tensile testing specimen still exhibit a
brittle fracture, but the cross-sectional plane of fracture is scattered more randomly along the length of
the samples, indicating that the contribution of the additive stress concentrations, induced by the
application of discontinuous ends, indeed is rather small.
3.5 COMPARISON OF THE TENSILE TEST RESULTS WITH MODELS
3.5.1 Stiffness comparison
In this paragraph, the longitudinal stiffness of the composites with a fixed overlapping length is
compared with the predictions of several models. The chosen models are the rule of mixtures, the
shear lag theory, the Mori-Tanaka model and the Halpin-tsai equations. The results of the
comparative scheme are given in table 6. As no statistical difference is detected between the
longitudinal stiffness of the discontinuous composites, the results of the different overlapping
distances are combined to yield an approximated average.
Table 6: Comparison of the longitudinal stiffness of composites with a fixed overlapping length with the predictions of several
models. As there are no significant differences between the longitudinal stiffness of the discontinuous composites, their results
are combined to yield an approximate average. The table indicates that the Halpin-Tsai equations give the best estimation.
Mechanical
characteristic
Experimental
average
Rule of
Mixtures
Shear lag
theory
MoriTanaka
model
HalpinTsai
equation
Longitudinal
tensile
stiffness (GPa)
16.7 ± 1.7
18.8
18.6
18.9
17.2
Table 6 reveals that the semi-empirical Halpin-Tsai equation gives the best estimation of the
longitudinal tensile stiffness of the discontinuous bamboo fibre reinforced epoxy composites. The
parameter that reflects the geometrical packing of the reinforcing fibres (ζ) is set at two times the
aspect ratio of the fibres. This yields a value of 76.9. The remaining misfit is attributed to an induced
misalignment in the composite preparation step.
The rule of mixtures gives a relatively large over-prediction of the longitudinal stiffness of the
discontinuous composites. This is allocated to the large simplification, set forth in this model, by
assuming the fibres to be infinitely long and applying the iso-strain condition.
The shear lag theory overthrows the fibre continuity necessity and accounts for a stress recovery
zone at the edges of the short fibre that reduces the average axial stress (and stiffness). This results
in a slightly better approximation of the longitudinal stiffness. It assumes however that the fibres can
be modelled to fit the cylindrical shape. Bamboo fibres do not satisfy this requirement. The hypothesis
that the shear force is constant at concentric surfaces around the fibre does not only depend on radial
coordinates, but also on the angular position. The entire derivation therefore needs an adaptation to fit
the geometrical shape of the bamboo fibre.
The application of the Mori-Tanaka model requires knowledge of the different engineering constants
that set up the stiffness or compliance matrix. Considering the bamboo fibres as transversely isotropic
and the epoxy matrix as isotropic, a total of 8 engineering constants are required. However, the
necessary transverse stiffness and Poisson ratio’s for bamboo fibres are unknown. To surpass this
obstacle, the bamboo fibres are also assumed to be isotropic with a fixed Poisson ratio of 0.3. The
result of table 6 reveals that the negligence of the anisotropic behaviour is unjustified.
3.5.2 Strength comparison
3.5.2.1 Adaptation of the local load sharing model
The local load sharing model implements interactions between adjacent fibres that lead to a stress redistribution through the application of matrix shear spring elements. The original model only allows the
prediction of the tensile strength of continuous fibre composites. With each strain increment, the
resulting tensile stresses in the axial fibre spring elements are compared to the fibre strength. If the
fibre strength is exceeded, the fibre spring element is removed from the system, simulating a broken
fibre. However, it is remarked that, in previous considerations, the continuous fibre composite actually
behaves as a discontinuous one once multiple axial fibre spring elements are removed from the
system at incremented strains. This knowledge thrives the reformulation of the original concept in
order to implement the discontinuities as broken axial fibre spring elements at an initial strain
increment. To avoid numerical instabilities, the strength of each of the slit elements is reduced to
9*10-9MPa. The value of the matrix shear spring element stiffness is determined by fitting the
simulated average tensile strength of the continuous composite with the experimental values. The
simulations are repeated ten times to allow statistical variations. The list of introduced parameters in
the local load sharing model is given by table 7.
Table 7: Overview of the different parameters that are initially set in the calculation of the tensile strength using the local load
sharing model.
.
Parameter
Fibre Young’s modulus (GPa)
Fibre radius (μm)
Weibull scaling parameter (MPa)
Weibull modulus (/)
Length of Weibull parameters
(mm)
Value
43
80
4952
2.2
Parameter
Matrix Young’s modulus (GPa)
Volume fraction (%)
Composite length (mm)
Effective matrix shear yield stress (GPa)
Value
2.73
40
150
32
100
Length of fibre element (mm)
1
3.5.2.2 Discussion of the different modelling results
In this section, several of the discussed models are administered to predict the tensile strength of the
bamboo-epoxy short-fibre composite. The results of these models are visualized in table 8. This table
distinguishes between the models that are fit to incorporate the proposed overlapping pattern. The
results of the continuous fibre composites are given as a reference.
Table 8: Overview of the different modelling attempts to predict the tensile strength of bamboo-epoxy unidirectional short-fibre
composites. The experimental results are also included as a comparison. It is remarked that the good correspondence between
the tensile strength of the continuous fibre composite, calculated with LLS simulations, and the experimental results is due to
the fitting of the stiffness of the matrix shear spring elements.
Applied
model
(
)
UD
continuous
composites
Experimental
222.5 ±
13.0
RoM
Kelly-Tyson
GLS
248.6
LLS
219.3 ±
14.5
UD discontinuous fibre bundle composites
LO=10%
84.6 ±
18.2
LO=30%
79.9 ±
11.9
96.3 ±
11.7
98.1 ±
14.5
LO=50% LO=random
81.6 ±
100.1 ±
13.2
17.8
262.9
250.8
131.8
95.7 ±
125.7 ±
13.8
19.6
UD
discontinuous
fibre
composites
random
190.9 ± 20.8
213 ± 15.8
Although most of the analytical models manage to predict the longitudinal tensile strength of
continuous bamboo fibre epoxy composite within a reasonable degree of certainty, they all fail to
accurately predict the longitudinal tensile strength of discontinuous samples. There are several
sources of inaccuracy for that, nevertheless, the main reason is the impossibility to include in the
mentioned models certain parameters, such as: the statistical variation of the fibre stiffness, the
incorrect assumption of the ineffective length in which stress recovery takes place, the introduction of
accurate Weibull parameters and the impossibility to incorporate stress redistribution from the broken
fibre to the adjacent fibres. All these parameters seem to have an important contribution on the
prediction of the longitudinal tensile strength of the composites.
A numerical method appears to be necessary to predict the tensile strength of discontinuous
composites with sufficient accuracy. The local load sharing model allows the distribution of the applied
load between adjacent fibres by the insertion of matrix shear spring elements. The implementation of
the different patterns in numerical code has been addressed in the previous section. The modified
local load sharing model is able to predict the composite tensile strength within an error margin of
15%. These differences are accounted to a reduced shear yield stress upon increased loading. The
load transfer to adjacent fibres is incorporated by the presence of matrix shear spring elements. The
local load sharing model assigns a constant stiffness value to these spring elements, while
experimental results show a decrease of the shear yield stress upon increased loading. Additional
errors may also be explained by the insertion of incorrect Weibull parameters.
CONCLUSION
Because of high longitudinal tensile strength and low weight, bamboo fibre reinforced epoxy
composites show high potential to industrially substitute glass fibre composites. The industrial scaleup of large bamboo fibre composites requires the invention of a processing technique to diminish the
effect of the discontinuous character of bamboo fibres, inherent to the bamboo culm composition.
Through an optimized production method, the insertion of slits that lower the longitudinal tensile
strength by more than 60% is avoided. The full randomization of discontinuous fibres is achievable,
reducing the longitudinal tensile strength by only 15%. Furthermore, a local load sharing model is
developed that incorporates the effect of discontinuities and is able to semi-empirically predict the
longitudinal tensile strength of discontinuous bamboo fibre epoxy composites within an error margin of
15%.
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