Strength prediction for bi-axial braided composites
by a multi-scale modelling approach
Chen Wang
1,2,3
, Yucheng Zhong
4
, P. F. Bernad Adaikalaraj
4
, Xianbai Ji
4
, Anish Roy
3
,
Vadim V. Silberschmidt
3
, and Zhong Chen
1,4,
*
1
School of Materials Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798,
Singapore
2
Institute for Sports Research, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
3
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK
4
Temasek Laboratories@NTU, 50 Nanyang Drive, Singapore 637553, Singapore
ABSTRACT
Braided textile-reinforced composites have become increasingly attractive as
protection materials thanks to their unique inter-weaving struct ures and
excellent energy-absorption capacity. However, development of adequate
models for simulation of failure processes in them remains a challenge. In this
study, tensile strength and progressive damage behaviour of braided textile
composites are predicted by a multi-scale modelling approach. First, a micro-
scale model with hexagonal arrays of fibres was built to compute effective
elastic constants and yarn strength under different loading conditions. Instead
of using cited values, the input data for this micro-scale model were obtained
experimentally. Subsequently, the results generated by this model were used as
input for a meso-scale model. At meso-scale, Hashin’s 3D with Stassi’s failure
criteria and a modified Murakami-type stiffness-degradation scheme was
employed in a user-defined subroutine developed in the general-purpose finite-
element software Abaqus/Standard. An overall stress–strain curve of a meso-
scale representative unit cell was verified with the experimental data. Numerical
studies show that bias yarns suffer continuous damage during an axial tension
test. The magnitudes of ultimate strengths and Young’s moduli of the studied
braided composites decreased with an increase in the braiding angle.
Introduction
Braided textile-reinforced composites have received
considerable attention in the recent years as protection
materials for various applications, including sports
products (e.g. helmets and shin guards) [1, 2]. Such
composites combine high structural stability with low
cost, excellent damage tolerance and energy absorption
thanks to yarn interlacing. The ease of incorporating
different types of yarns enables manufacture of
composites with a wide range of overall mechanical and
physical properties [3]. To aid product optimization at
the design stage, numerical models need to be devel-
oped, which, in turn, require a thorough understanding
of mechanical responses and energy-dissipation
mechanisms of braided composites.
To date, only limited research has been conducted on
prediction of mechanical properties and damage evo-
lution in braided composites [4]. Numerical tools such
as finite-element (FE) analysis were used to solve non-
linear dynamic problems associated with composite
failure [5–12]. Ivanov et al. [7] investigated failure of tri-
axial braided composites using the degradation
scheme of Murakami-Ohno and the damage evolution
law of Ladeveze. Xiao et al. [8] employed a sub-cell FE
representation of microstructure of textile composites to
predict their strength. Fang et al. [9]analysedarepre-
sentative volume cell (RVC) of braided composites with
a damage evolution model controlled by fracture
energy of constitutive materials. Prabhakar et al. [10]
considered kinking and splitting for fibre tows under
compressive load. Binienda et al. [11] studied an overall
stress–strain curve of 08/± 608 2D triaxially braided
composites with an advanced shell-element model. In
these studies, the meso-scale modelling approach was
widely used to obtain stress (and strain) distributions
throughout the braided structure. Song et al. [12]anal-
ysed an effect of a number of unit cells on compressive
failure of 2D tri-axial braided composites. However,
meso-scale modelling is rather challenging and should
be attempted based on three aspects [13]. First, the unit-
cell geometry should be realistic, since the dimensions
of yarn play an important role in deformation and
damage behaviour of the model. Second, effective
properties of yarn should be accurately determined.
Third, the modelling strategy implies that the details of
the physics below the yarn-level cannot be recovered.
A more advanced and adequate scheme, capturing
physics at the micro-level, is a multi-scale scheme that
can be used to link microscopic failure effects with
mesoscopic behaviour of the braided composites [14–18].
Consequently, homogenized mechanical properties of
yarns and ultimate strength of the composite can be
predicted more effectively using a multi-scale modelling
approach.Forinstance,Bednarcyketal.[19] utilized a
micro-mechanics theory known as Generalized Method of
Cells to represent non-linear behaviour of plain weave-
reinforced polymeric composites. Zhang et al. [20]
investigated a free-edge effect and progressive damage
of a single-layer braided composite, using simplified
Hashin’s 2D failure criteria. Zhang et al. [21, 22]pre-
sented a multi-scale computational model used to pre-
dict deformation, damage and failure responses of 3D
textile composites subjected to three-point bending. Xu
et al. [23] applied a Micro-mechanics of Failure theory to
predict tensile strength of braided structures and dam-
age initiation in yarns. It is clear that either classical
failure criteria or newly developed mechanical theory
were incorporated into multi-scale schemes for strength
prediction for braided composites. However, reliability
and accuracy of such schemes are still debatable [23–25].
The damage evolution law of Ladeveze is attractive for
UD composites because of its simplicity, but it needs
modification when applying to braided composites. No
criterion is universally accepted by designers as ade-
quate under general loading conditions, since some of
the adopted classical criteria are not capable to capture
initiation of damage of braided composites effectively.
Moreover, some failure modes such as fibre splitting,
matrix cracking or interface failure are not presented
adequately [26].
The aim of this study is to attempt a multi-scale
modelling framework accounting for the underlying
physical mechanisms that drive deformation and
damage in the composite under static tensile loading
states. In this study, a micro-scale model was first
built with hexagonal arrays of fibres to obtain effec-
tive elastic constants and strengths of yarns under
different loading conditions. The input data for the
micro-scale model were exp erimentally measured in
our previous work [27, 28]. The results of micro-scale
modelling were used as input for material properties
of the meso-scale mod el. Hashin’s 3D and Stassi’s
failure criteria were presented with a stiffness-
degradation model in a user-defined subroutine for
the FE software Abaqus/Standa rd. The overall
stress–strain curve obtained with the meso-scale
model was correlated with experimental data.
Finally, the predictive capability of the developed
model was illustrated with some case studies.
Multi-scale modelling approach
Geometric representative unit cells
Micro-scale modelling aims at obtaining effective
properties of the yarn of a braided composite.
Microstructure of the yarn is similar to that of a
lamina in terms of fibre, matrix and interface. In our
micro-scale model, fibres were arranged hexagonally
considering high-fibre volume fractions in fibre
bundles of braided composites. Previous studies [29,
30] demonstr ated that predictions of elastic moduli
and strengths based on hexagonal and random
arrangements were very similar Specifically, this
volume fraction was assumed as 0.8 herein with a
representative unit cell (RUC) as depicted in Fig. 1,in
which 2pr
2
/ab = 0.8. Here, a and b are the length and
width of the RUC, respectively, while r is the radius
of the carbon fibre.
A bi-axial braided textile preform consists of
interlaced ?h and -h bias yarns [23, 31] (Fig. 2). In
creating unit cells, these components were modelled
separately using SolidWorks
TM
. Bias yarns were
created by sweeping a cross-sect ion with an elliptical
shape along a predefined undulating path. From a
careful observation of a complex microstr ucture of a
braided textile, a repeating unit can be identified as
shown in Fig. 2a, b. The geometric parameters
marked in Fig. 2 include the braiding angle h, width
and thickness of braiding yarns w and t, respectively,
the distance between neighbouring yarns e and the
gap bet ween the interlacing yarns. In this work, all
the dimensions were measured for the real braided
architecture. The width and thickness of ya rns were 3
and 0.314 mm, resp ectively. A cross-section of the
yarn was modelled as elliptical shape, with the value
of e and the gap between positive and negative bias
tows set as 0.2 and 0.05 mm. The global fibre volume
fractions (V
f
) of the unit cell were set to be 50 % for all
the braiding angles; based on it, dimensions of the
matrix block were chosen.
To facilitate a subsequent FE analysis, the diamond
braided textile unit cell was further merged with a
matrix block as a composite volume element—a
r
b
a
Figure 1 Geometry of
hexagonal micro-unit cell.
w
t
ε
2θ
gap
(b)
(a)
(c)
(d)
+θ
bias yarn
+θ
-θ
-θ bias yarn
Figure 2 Architecture of bi-axial braided textiles (a); meso-scale
model representation (b); the RUC of composite (c); and its side
view of RUC (d).
meso-scale RUC, as shown in Fig. 2c and d. Such
RUC represents a repeating part of microstructure of
the modelled specimen. Since bias yarns do not ter-
minate at the edges parallel to the longitudinal
direction, they interlock with each other to transmit
an applied load.
Mesh generation and boundary conditions
Four-node tetrahedron eleme nts (C3D4) were used to
mesh the micro-scale unit cell, including both fibre
and matrix (Fig. 1). Zero-thickness cohesive elements
(COH3D8) were located at the fibre/epoxy interfaces.
As a unit cell is a small RUC of the yarn, the use of
periodic boundary conditions (PBCs) devised by Xia
et al. [32] is essential. In terms of the unit cell studied
here, the PBCs and minimization of mesh mis-
matches were achieved through increasing the
number of unit cells analysed in a single simulation,
while merging mismatched nodes on contacting
faces. According to our previous work [31], seven
independent boundary conditions (BCs) in the form
of uniform displacements were specified to obtain the
material properties of fibre tows, as shown in Fig. 3.
Since carbon fibres are a transversely isotropic
material, subscript 1 denotes fibre direction and 2 and
3 denote transverse directions. A global coordinate
system was employed for the whole model.
For yarns and the pure matrix block in the meso-
scale unit cell, four-node tetrahedron elements
(C3D4) were used to discretise the complex yarn
architecture inside the RUC (Fig. 4a). The mesh
density should be sufficient for an adequate intro-
duction of geometry of the undulated tow. Since the
pure matrix region between the yarns was very thin
(*0.02–0.05 mm), a number of elements required to
attain acceptable mesh quality was relative ly high
compared to that for yarns. A mesh-convergence
study was carefully carried out to avo id any mesh-
dependent results. Unlike a micro-scale mod el, a
simple non-periodic boundary condition was used in
meso-scale RUCs to predict ultimate strengths of the
braided composite as shown in Fig. 4b. To apply
PBCs, opposite sides of the mo del must have identi-
cal nodal coordinates and a constraint equation
should be used to tie each node pair. However, this
becomes difficult to impose as node pairs are not
always placed symmetrically on either side because
of an irregular mesh used to discretise the model.
Instead, in our modelling, the lateral sides of the unit
cell were left free to move, while a displacement
boundary condition was applied at the top surface of
the unit cell and the bottom surface was constrained
with a pin boundary condition (Fig. 4b). A detailed
comparison studies [33, 34] of PBCs and non-periodic
boundary conditions for braided composites show
that the difference was minimal in case of uniaxial
loading conditions. This justifies the chosen mod-
elling approach.
In the meso-scale model, the matrix material was
assumed to be isotropic and braiding yarns were
transversely isotropic. Assigning mater ial orientation
Figure 3 Boundary
conditions of micro-scale unit
cell for longitudinal properties
(a), transverse properties (b),
in-plane shear (c), out-of-plane
shear (d), and Poisson ratio
(e).
to yarns is one of the important steps because of
yarn’s undulations inside the unit cell. In this work,
orientation of yarns was assigned discretely, defining
a normal surface and principal axis (fibre direction).
With this method, undulations and tilt regions were
assigned with precise material orien tation at all
locations of the mesh in comparison with global
coordinate system, as shown in Fig. 5.
Failure criteria and st
iffness-degradation
model
In the micro-scale model, a maximum stress failure
criterion was deemed appropriate in describing
damage initiation of carbon fibres, as
r
f
X
fT
or jr
f
jX
fC
; ð1Þ
where X
T
and X
C
are the tensile and compressive
strengths, respectively, the subscript f denotes carbon
fibre, and r
f
is the normal stress component along the
longitudinal fibre direction. At fibre failure (Eq. 1),
the Young’s modulus was reduced to zero instanta-
neously [35]. It should be noted that failure mecha-
nisms of fibre tows under longitudinal compression
are quite complicated [36]. In order to obtain consis-
tent values of the longitudinal modulus under ten-
sion and compression, that strong bonding between
fibres and matrix was assumed leading to fibre rup-
ture rather than buckling or kinking.
A mo dified von Mises criterio n (the Stassi’s crite-
rion), which accounts for two strength parameters,
was employed to capture damage initiation in pure
matrix both for micro- and meso-scale models.
Although the matrix in the unit cell was considered
isotropic, tensile failure strength of epoxy matrix is
generally lower than the compressive one. This is due
to the influence of hydrostatic pressure (a first
invariant of the stress tensor) besides deviatoric stress
components on the tensile strength. Christensen [37]
modified the Stassi’s criterion for materials with dif-
ferent strengths in compression and tension as
1
X
mT
1
X
mC
3P þ
1
X
mT
X
mC
r
2
vm
1; ð2Þ
where P and r
vm
are the hydrostatic pressure and von
Mises stress, respectively. Subscript m represents
epoxy matrix in this paper.
Fibre Tow
Pure Matrix
C3D4-Tetrahedron Element Boundary Condition for RUC
Displacement Control
(a) (b)
2
1
3
1
2
3
Figure 4 Meshing unit cell of
bi-axial braided composite
(a) and displacement-
controlled boundary condition
(b).
1
3
1
1
22
2
2
3
3
3
3
z
3
3
3
y
Figure 5 Segmentation of individual bias yarns and local coordinate systems (Blue arrows indicate local direction of ‘‘1’’; yellow arrows
indicate local direction of ‘‘2’’; and red arrows indicate local direction of ‘‘3’’).
