UDK 66.017:519.61/.64:620.17
Original scientific article/Izvirni znanstveni članek
ISSN 1580-2949 MTAEC9, 46(2)97(2012)
LINEAR TWO-SCALE MODEL FOR DETERMINING THE MECHANICAL PROPERTIES OF A TEXTILE COMPOSITE MATERIAL
LINEARNI DVOSTOPENJSKI MODEL ZA DOLOČITEV MEHANSKIH LASTNOSTI TEKSTILNEGA KOMPOZITA
Tomaš Kroupa1, Petr Janda2, Robert Zemčik3
1University of West Bohemia in Pilsen, Department of Mechanics, Univerzitni 22, 306 14, Plzen, Czech Republic 2University of West Bohemia in Pilsen, Department of Machine design, Univerzitni 22, 306 14 Plzen, Czech Republic 3University of West Bohemia in Pilsen, Department of Mechanics, Univerzitni 22, 306 14, Plzen, Czech Republic
kroupa@kme.zcu.cz
Prejem rokopisa - received: 2011-02-01; sprejem za objavo - accepted for publication: 2011-10-10
The engineering mechanical constants for a description of mechanical macro-scale models of carbon and aramid textile composite materials are calculated using finite-element analyses. Two sub-scale models of representative volumes are used. The micro-scale model represents a periodically repeated volume consisting of fibers and a matrix within each interweaved yarn. The meso-scale model represents a unit cell of four interweaved yarns, which is repeated within the whole composite with the properties obtained from a micro-scale model and matrix. The finite-element models are built with the commercial packages Siemens NX 7.5 and MSC.Marc 2008r1 using subroutines.
Keywords: composite, textile, linear, carbon, aramid, epoxy, tensile, finite-element analysis
Z uporabo metode končnih elementov so izračunane inženirske konstante za opis mehanskega makrodimenzionalnega ogljik-aramidnega modela kompozita. Uporabljena sta dva poddimenzionalna modela za manjše ustrezne prostornine. Mikrodimenzionalni model je periodično ponavljanje prostornine, ki se ponavlja za ves kompozit za matico in vpleteno vlakno. Mezodimenzionalni model je spletna celica iz štirih vpletenih vlaken in se ponavlja v vsem kompozitu z lastnostmi mikromodela in matice. Modeli končnih elementov so vgrajeni v paketa Siemens NX 7.5 in MSC.Marc 2008r1 z uporabo podrutin.
Ključne besede: kompozit, tekstil, linearen, ogljik, aramid, epoksi, natezne lastnosti, končni elementi
1 INTRODUCTION
A knowledge of the precise values of the mechanical properties of materials is crucial for the capability of models to predict the behavior of analyzed structures. This is also the case for the modeling of composite materials. Several material properties of the composites can be calculated directly from experimental results (Young's moduli from tensile tests, etc.). The presented paper is aimed at a determination of the elasticity constants of textile composites using sub-scale models to determine the properties that cannot be measured and calculated directly from the experiment (shear modulus, etc.). The models were used for the prediction of the elasticity constants of two materials with a simple plain weave. This type of material was chosen because of the possibility to measure directly the Young's moduli in the principal material directions using tensile tests. Nevertheless, the shear modulus was fitted on the linear part of the measured curves using a gradient-optimization algorithm and the Poisson's ratios of the whole textile composites were identified using a digital image correlation1. The material data of the constituents were given by the manufacturer and the dimensions of the periodically repeated volume (unit-cell element - UCE) of the textiles were measured using a digital camera.
2 EXPERIMENT
The effective material parameters were determined using simple tensile tests performed on a Zwick/ Roell Z050 test machine on thin strips with the dimensions given in Table 1.
Table 1: Dimensions of the strips Tabela 1: Dimenzije traka
		Carbon	Aramid
Length	mm	100.00	100.00
Width	mm	10.00	10.00
Thickness	mm	0.30	0.35
Three types of specimens were used for each material. One of two principal directions of the textile fabric form the angles 0°, 45° and 90° with the direction of the loading force. Once the force-displacement diagrams (Figures 1 and 2) were measured, the Young's moduli in directions 1 and 2 and the shear modulus were fitted on the linear parts of the curves using a combination of a plane-stress finite-element (FE) model and the gradient-optimization algorithm implemented in OptiSLang software (the methodology is described in2,3). A digital image correlation1 was used for the calculation
		Carbon/Epoxy	Aramid/Epoxy
El	GPa	31.05	15.85
E2	GPa	29.73	15.66
Gl2	GPa	1.83	1.24
Vl2	-	0.19	0.31
2	J" 1
Figure 3: Material axes: yarn (left), textile (right) Slika 3: Osi materiala: vlakno (levo), tekstil (desno)
Figure 1: Force-displacement diagram (gray) for Carbon/Epoxy, with fitted parts (black)
Slika 1: Odvisnost sila - premik (sivo) za ogljik/epoksi s približki (črno)

Figure 2: Force-displacement diagrams (gray) for Aramid/Epoxy, with fitted parts (black)
Slika 2: Odvisnost sila - premik (sivo) za aramid/epoksi s približki (črno)
of the Poisson's ratios. The elasticity parameters of the textiles are shown in Table 2.
Table 2: Elasticity parameters of textile composites Tabela 2: Parametri elastičnosti za tekstilne kompozite
Figure 4: UCE geometry of yarn with Vf = 0.6 (fibers - black, matrix
- gray)
Slika 4: UCE-geometrija vlakna z Vf = 0.6 (vlakna - črno, matica -sivo)
Table 3: Dimensions of the unit cell of the yarn Tabela 3: Dimenzije spletne celice
l1	[-1	1
l2	[-1	4
l3	[-1	4
3 UNIT-CELL ELEMENTS
Axes orientations in UCE (Figure 4) are shown in Figure 3. Dimensionless geometry of UCE of yarns is shown in Table 3. The ideal distribution of fibers in the yarns, the perfect saturation of the yarns by the matrix and the volume fiber fractions Vf = 0.6 and Vf = 0.7 are considered in the calculations.
Figure 5: Photograph of Carbon/Epoxy specimen Slika 5: Posnetek vzorca ogljik/epoksi
B [wb .WB ]
Figure 6: Photograph of Aramid/Epoxy specimen Slika 6: Posnetek vzorca aramid/epoksi
The dimensions of the UCE of both materials, which were measured using detailed photographs provided by a Canon EOS 400D digital camera (Figures 5 and 6) are shown in Table 4.
Table 4: Dimensions of textile unit cells Tabela 4: Dimenzije spletne celice tekstila
mm
mm
mm
Carbon/Epoxy
.5.00
5.00
0.30
Aramid/Epoxy
3.00
3.00
0.3.5
4 BOUNDARY CONDITIONS
In the FE model of the UCE (Figure 7) it is necessary to invoke pure tensile conditions or pure shear conditions to determine the elasticity parameters. Furthermore, the UCE has to be fixed in space to eliminate rigid body modes. Finally, the periodic boundary conditions have to be satisfied. The effect of the periodic boundary condition on the UCE with two periodically tied faces is sketched in Figure 8.
Figure 8: Scheme of the effect of the periodic boundary conditions Slika 8: Shema učinka periodičnosti mejnih pogojev
Each corresponding pair of nodes on opposite faces of the FE model must fulfill the conditions 4-6
-u A = du, V B - V A = dv = di for i=\...N
Ub-UA wl -w
(P)
where i is the index of the corresponding constrained faces; N is the total number of the periodically constrained faces; u, v and w are the displacements in the 1, 2 and 3 direction; and d'u, dV, dWw are displacements of the appropriate retained nodes in which the loading is applied (Figure 9). The UCE of the yarns is periodically tied in all three directions and the UCE of the textiles is tied in direction 1 and 2 (Figure 9).
For the determination of the Young's modulus E1 and the Poisson's ratio Vi2 of the textile composite the model is loaded with Oi ^ 0 and the other loadings are equal to zero. Normal strains in direction 1 and 2 are calculated as
d; d22 = , £2 =	(e12)
h '2
and the Young's modulus and Poisson's ratio are
E„
E, =
(E1v12)
Figure 7: UCE geometry of Carbon/Epoxy textile composite (yarns -black, matrix - gray)
Slika 7: UCE-geometrija kompozita ogljik/epoksi (trakovi - črno, matica - sivo)
Figure 9: Boundary conditions and links used for periodic boundary conditions for UCE of Aramid/Epoxy textile
Slika 9: Mejni pogoji in povezave, uporabljene za periodične mejne pogoje za UCE aramid/epoksi tekstil
1
For the determination of the shear modulus G12 the model is loaded by 712 ^ 0. The other loadings are equal to zero. The shear strain in plane 12 is calculated as
y 12 =
and the shear modulus is
d 2
I1 I2
G12 =
y 1
(g12)
(G12)
The same scheme is used for the determination of the elasticity constants in the other directions or planes.
5 INPUT PARAMETERS
Carbon (Toray T600) and Aramid (Twaron K1055) fibers are transversely isotropic materials. Their elasticity parameters are given in Table 5.
Table 5: Elasticity parameters for fibers Tabela 5: Parametri elasti~nosti za vlakna
		Carbon	Aramid
E1	GPa	230.00	104.00
E2	GPa	7.05	5.40
E3	GPa	7.05	5.40
V12	-	0.30	0.40
V23	-	0.30	0.40
V31	-	0.02	0.02
G12	GPa	50.00	12.00
G23	GPa	50.00	12.00
G31	GPa	50.00	12.00
Figure 11: Deformed FE model of the UCE of the textile under shear loading in plane 12 (shown values of shear stress r12) Slika 11: Deformiran FE-model UCE za tekstil pri strižni obremenitvi v ravnini 12 (prikazane vrednosti strižne napetosti r12)
Table 6: Elasticity parameters for the matrix Tabela 6: Parametri elasti~nosti za matico
		Epoxy
E	GPa	3.00
v	-	0.30
The matrix is manufactured from epoxy resin (MGS® L 285) and hardener (MGS® 285). It is considered to be a linear isotropic material (Table 6).
6 RESULTS
The effect of the periodic boundary conditions is shown in Figures 10 and 11. Opposite faces of the UCE are deformed in the same shape. The elastic properties of the yarns are shown in Table 7. The results are shown for both fiber volume fractions (Vf). Similarly, the results for the textiles are shown for both Vf (Table 8).
Table 7: Calculated elasticity parameters of the yarns Tabela 7: Izra~unani parametri elasti~nosti za spleta
Figure 10: Deformed FE model of the UCE of yarn under shear loading in plane 23 (shown values of shear stress r23) Slika 10: Deformiran FE-model za UCE-spleta pri strižni obremenitvi v ravnini 23 (prikazane vrednosti strižne napetosti r23)
		Carbon/Epoxy		Aramid/Epoxy	
Vf	-	0.60	0.70	0.60	0.70
E1	GPa	138.87	161.54	63.46	73.55
E2	GPa	7.05	8.33	4.36	4.60
E3	GPa	7.05	8.33	4.36	4.60
v12	-	0.30	0.30	0.36	0.37
v23	-	0.36	0.34	0.40	0.40
v31	-	0.02	0.02	0.02	0.02
G12	GPa	4.26	5.90	3.42	4.34
G23	GPa	3.88	5.45	3.15	4.01
G31	GPa	4.26	5.90	3.42	4.34
Table 8: Calculated elasticity parameters for textiles					
Tabela 8: Izra~unani parametri elasti~nosti za				tekstil	
		Carbon/Epoxy		Aramid/Epoxy	
Vf	-	0.60	0.70	0.60	0.70
E1	GPa	27.75	31.89	14.08	15.72
E2	GPa	27.75	31.89	14.08	15.72
G12	GPa	2.60	3.30	2.21	2.60
v12	-	0.33	0.33	0.28	0.29
2
12
7 CONCLUSION
Multi-scale models for the prediction of the elasticity constants of textile composites with a simple plain weave were developed and presented. Good agreement of the Young's moduli was achieved between the calculated and experimental values. However, the shear moduli are slightly over-predicted. The calculated Poisson's ratios were calculated with acceptable accuracy only for the Aramid/Epoxy textile. Future research will be aimed at the non-linear, plastic and damage behavior of the matrix, the damage behavior of the fibers and an investigation of the imperfections and the unit-cell element dimensions of the textiles.
Acknowledgement
The work has been supported by the projects GA P101/11/0288 and European project NTIS - New
Technologies for Information Society No. CZ.1.05/ 1.1.00/02.0090.
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