Content uploaded by Zahur Ullah
Author content
All content in this area was uploaded by Zahur Ullah on Jul 20, 2015
Content may be subject to copyright.
Proceedings of the 22nd UK Conference of the
Association for Computational Mechanics in Engineering
2-4April2014,University ofExeter,Exeter
MULTISCALE MODELLING OF THE TEXTILE COMPOSITE
MATERIALS
*Zahur Ullah, Łukasz Kaczmarczyk, Michael Cortis and Chris J. Pearce
School of Engineering, Rankine Building, The University of Glasgow, Glasgow, UK, G12 8LT
*Zahur.Ullah@glasgow.ac.uk
ABSTRACT
This paper presents an initial computational multiscale modelling of the fibre-reinforced composite ma-
terials. This study will constitute an initial building block of the computational framework, developed
for the DURCOMP (providing confidence in durable composites)EPSRCproject,theultimategoalof
which is the use of advance composites in the construction industry, while concentrating on its major
limiting factor ”durability”. The use of multiscale modelling gives directly the macroscopic constitutive
behaviour of the structures based on its microscopically heterogeneous representative volume element
(RVE). The RVE is analysed using the University of Glasgow in-house parallel computational tool,
MoFEM (Mesh Oriented Finite Element Method), which is a C++ based finite-element code. A single
layered plain weave is used to model the textile geometry. ThegeometryoftheRVEmainlyconsistsof
two parts, the fibre bundles and matrix, and is modelled with CUBIT, which is a software package for
the creation of parameterised geometries and meshes. Elliptical cross sections and cubic splines are used
respectively to model the cross sections and paths of the fibrebundles,whicharethemaincomponents
of the yarn geometry. In this analysis, transversely isotropic material is introduced for the fibre bundles,
and elastic material is used for the matrix part. The directions of the fibre bundles are calculated using
apotentialflowanalysisacrossthefibrebundles,whicharethen used to define the principal direction
for the transversely isotropic material. The macroscopic strain field is applied using linear displacement
boundary conditions. Furthermore, appropriate interface conditions are used between the fibre bundles
and the matrix.
Key Words: multiscale modelling; composite material; Transverse isotropy; MoFEM; CUBIT
1. Introduction
Conventional materials, e.g. steel, aluminium and metallicalloyscannolongersatisfythedemandsfor
materials with exceptional mechanical properties and ultimately requires the design of new material [1].
These new materials are designed by changing their microconstituents at a scale, which is very small
as compared to the physical structures. Due to the complicated micro-structure of these materials, direct
macro-level modelling is not possible and requires a detailed modelling at the micro-level. Textile or fab-
ric composites is a class of these new materials which provides full flexibility of design and functionality
due to the mature textile manufacturing industry and is commonly used in many engineering applica-
tions, including ships, aircrafts, automobiles, civil structures and prosthetics [2]. Numerous analytical
and computational methods have been proposed to analyse textile composite materials, which includes
the calculation of the overall macro homogenised response and properties from the micro-heterogeneous
representative value element (RVE) [3] and is often referredasmicro-to-macrotransitionorhomogeni-
sation [4].
This paper presents the computational multiscale modellingofthetextilecomposites,usingtheUniver-
sity of Glasgow in-house computational tool MoFEM. The RVE inthiscaseconsistsoffibrebundles
and matrix, which is modelled and meshed in CUBIT using a Python parametrized script. CUBIT also
facilitates the insertion of interfaces between the fibres and matrix. Transversely isotropic material are
used for the fibres and isotropic martial are used for the matrix. Five material parameters are required
for the transversely isotropic material, i.e. Ep,νp,Ez,νpz and Gzp where Epand νpare Young’s mod-
ulus and Poisson’s ratio in the transverse direction respectively, while Ez,νpz and Gzp are the Young’s
modulus, Poisson’s ratio and shear modulus in the fibre directions respectively. For the matrix part, only
two material parameters are required, i.e. Young’s modulus Eand Poisson’s ratio ν.Although,periodic
boundary conditions [5, 6] gives better estimates of the homogenised response and properties as com-
pared to traction and linear displacement boundary conditions, linear displacement boundary conditions
are used in this paper due to its simple implementation. This will subsequently be extended to periodic
boundary conditions in future work. Fibre directions are calculated at each integration point by solving
apotentialflowproblem.
2. Theoretical background
Computational multiscale modelling is used in this paper to analyse the textile composite ma-
terials, in which a heterogeneous RVE is associated with eachintegrationpointofthemacro-
homogenous structure as shown in Figure 1, in which B⊂R3and B⊂R3are macro and mi-
cro domains respectively. The calculation of the RVE boundary conditions from the macro-strain
integration point
Strain
Stress
microstructure
macrostructure
Wwarp
hgap warp
HRVE
WRVE
LRVE
WweftWweft
Hweft
hgap weft
Vgap
Figure 1: Transition from macro-to-micro and micro-to-macro
ε=!ε11 ε22 ε33 2ε12 2ε23 2ε31 "Tat macroscopic integration point x=!x1x2x3"T
is known as macro-to-micro transition, while subsequent calculation of the homogenised stress σ=
!σ11 σ22 σ33 σ12 σ23 σ31 "Tand tangent moduli is known as micro-to-macro transition. The
macro-strain is applied as linear displacement boundary conditions, which leads to satisfaction of Hill-
Mandel principle [7], i.e.
ε:σ=1
V#V
ε:σdV,(1)
where Vis the volume of the RVE, while σand εare stresses and strains associated with a point y=
!y1y2y3"Tof the RVE. The micro displacement field u=!u1u2u3"T,iswrittenas
u=u∗+$
u,(2)
where u∗is known as Taylor displacements and $
uis the unknown displacement fluctuations. The Taylor
component is written as
u∗
i=DT
iε,i=1,2,···n,(3)
where nis the number of nodes and Diis the coordinate matrix and is given as [4]
Di=1
2
2y100
02y20
002y3
y2y10
0y3y2
y30y1
i
.(4)
Finally, the homogenised stress is calculated as
σ=1
V
nb
+
i=1
Difext
i,(5)
where nbis the number of nodes on the boundary ∂Bof the RVE, and fext
iis the external nodal force
vector.
3. Numerical example
AsampleRVE,whichwasusedin[2],isusedherewiththesamegeometrical and material parameters,
as shown in Figure 1, where the subscripts warp and weft represent the corresponding directions of
fibre bundles. The geometrical and material parameters are defined in Table 1. This RVE is referred
Parameters Values Parameters Values
Wwar p 0.3 Wwe f t 0.3
Hwar p 0.1514 Hwe f t 0.0757
hgap war p 0.09 hgap we f t 1.2
LRVE 3.0 vgap 0.012
WRVE 0.8
HRVE 0.3
Fibres properties Matrix Properties
EpEzνpνzGpz Eν
40 270 0.26 0.26 24 35 0.35
Table 1: RE V g e o m e t r i c al and mate r i a l p r o p e r ties (a l l d i m e n sions in mm while Eand Gare in GPa)
(a) Textile RVE (b) 0/90 non-crimp RVE
oo(c) Fibres directions
x
y
z
Figure 2: Crimp and non-crimp RVEs and sample fibre directions
as unbalanced, where the dimensions of fibre bundles are different in warp and weft directions. The
manufacturing processes and crossing of the warp and weft yarn will lead to non-circular cross sections
of the fibre bundles; therefore, elliptical cross sections are used in this paper, which are then sweeped
over the cubic spline fibres’ path to generate the fibres. Four-node tetrahedral elements are used for both
the fibre bundles and the matrix, while six-nodes prism elements are used as an interface between fibres
and matrix.
The textile RVE is analysed using two different meshes with 41,193 and 106,011 DOFs and is subjected
to 1 % strain in xdirection, i.e. εxx.ThefinestmeshandcoordinatessystemareshowninFigure2(a),
where xand zare warp and weft directions respectively, while sample fibredirectionsvectorareshown
in Figure 2(c). The resulting homogenised stress σxx versus applied strain εxx for the two meshes and a
reference value from [2] are shown in Table 2, in which Mesh-2 with 106,011 DOFs provides satisfactory
results. The small difference between current and reference results may be due to theuseoflenticular
cross-sections for the fibres, use of 8-node 3D linear brick element and 4- node linear tetrahedron ele-
ment for fibres and matrix respectively and the use of perfect bonding between fibres and matrix in [2].
Furthermore, The effect of fibres dimensions and crimp pattern are analysed, for which a new 0o/90o
non-crimp RVE with 103,095 DOFs (shown in Figure 2(b)) is generated and is subjected to the same
strain state. Comparison of the homogenised stress σxx for both crimp and non-crimp RVEs are given
in Table 3, where relatively lower value of homogenised stresses σxx in the crimp RVE is due to the
waviness of the fibre bundles. Furthermore, both crimp and non-crimp RVEs are subjected to 1 % strain
in zdirection, i.e. εzz and comparison of their homogenised stress in the zdirection, i.e. σzz are shown in
Tab l e 3, where aga in σzz is lower for the crimp RVE. Due to the small size and higher waviness of the
weft fibre bundles, the values of σzz are relatively smaller than the corresponding values of σxx.
σxx (MPa)
εxx (%) Mesh-1 Mesh-2 Reference
1749.82 508.771 541.278
Table 2 : σxx versus εxx for different mesh levels
σxx (MPa) σzz (MPa)
εxx (%) Crimp Non-Crimp εzz (%) Crimp Non-Crimp
1508.771 751.507 183.9317 125.065
Table 3: Co m p a r i s o n o f σxx versus εxx and σzz versus εzz for crimp and non-crimp RVE
4. Conclusions
This paper described an initial computational modelling framework for the DURACOMP project. Tex-
tile composite RVE geometry, which consists of two parts, i.e. fibre bundles and matrix is modelled and
meshed using CUBIT, where fibres are modelled using cubic spline with elliptical cross sections. The
University of Glasgow in-house computational tool MoFEM is used to analyse the RVE using trans-
versely isotropic material for the fibre bundles and isotropic material for the matrix. Linear displacement
boundary conditions and elastic interfaces between fibre bundles and matrix are used in this paper. Direc-
tion of the fibre bundles are calculated using a potential flow analysis. Two different level of meshes are
used to solve the RVE, and it is found that the homogenised stress calculated in the case of Mesh-2 are in
agoodagreementwiththereferencesolution.Itisalsofoundthathomogenisedstressinthecaseofthe
crimp RVE is lower than the corresponding non-crimp RVE. Furthermore, it is also observed that due to
the relatively smaller dimensions and more waviness patternfortheweftfibrebundles,thehomogenised
stress σzz is lower than the corresponding stress σxx in the warp direction.
Acknowledgements
The authors gratefully acknowledge the support of the UK Engineering and Physical Sciences Research
Council through the Providing Confidence in Durable Composites (DURACOMP) project (Grant Ref.:
EP/K026925/1).
References
[1] D. Peri´c, E.A. de Souza Neto, A.J. Carneiro Molina, and M.Partovi. Computational Methods in
Applied Sciences,volume7,pages165–185. SpringerNetherlands,2007.
[2] T. W. Chua. Multi–scale modeling of textile composites. Master’s thesis, Department of Mechanical
Engineering, Technische Universiteit Eindhoven, January 2011.
[3] V.G. Kouznetsova. Computational homogenization for the multi-scale analysisofmulti-phasemate-
rials.PhDthesis,TechnischeUniversiteitEindhoven,TheNetherlands, 2002.
[4] C. Miehe and A. Koch. Computational micro-to-macro transitions of discretized microstructures
undergoing small strains. Archive of Applied Mechanics,72(4-5):300–317,2002.
[5] Ł.Kaczmarczyk,C.J.Pearce,andN.Bi´cani´c. Scaletransition and enforcement of RVE boundary
conditions in second-order computational homogenization.International Journal for Numerical
Methods in Engineering,74(3):506–522,2008.
[6] G. Soni, R. Singh, M. Mitra, and B. G. Falzon. Modelling matrix damage and fibre-matrix interfa-
cial decohesion in composite laminates via a multi-fibre multi-layer representative volume element
(M2RVE). International Journal of Solids and Structures,51(2):449–461,2014.
[7] R. Hill. On constitutive macro-variables for heterogeneous solids at finite strain. In Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences,volume326,pages
131–147. The Royal Society, 1972.
