- Email: [email protected]
Spatial modelling of 3D woven variable thickness composite plate at the mesoscopic scale
Journal Pre-proofs Spatial modelling of 3D woven variable thickness composite plate at the mesoscopic scale Yu Zhou, Weidong Wen, Haitao Cui PII: DOI: Reference:
S0263-8223(19)31958-0 https://doi.org/10.1016/j.compstruct.2020.111946 COST 111946
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
24 May 2019 13 January 2020 17 January 2020
Please cite this article as: Zhou, Y., Wen, W., Cui, H., Spatial modelling of 3D woven variable thickness composite plate at the mesoscopic scale, Composite Structures (2020), doi: https://doi.org/10.1016/j.compstruct.2020.111946
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier Ltd.
Spatial modelling of 3D woven variable thickness composite plate at the mesoscopic scale Yu Zhou, Weidong WEN*, Haitao CUI College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China *Corresponding author. E-mail address: [email protected] Keyword: 3D woven; variable thickness plate; composite; geometric model
ABSTRACT The geometric details of variable thickness 3D woven composite plate, which features varying weft yarn cross section shapes and sizes at each layer and resulting diverse warp yarn path orientations, are quite different from the structure in constant thickness plate. A geometric model to handle these variances is presented in this paper which no doubt can also be used in constant thickness plate. Based on the idealized assumptions that the cross section shape of weft yarn is lenticular and warp yarn is rectangular, the inner structure details of variable thickness plate are illustrated in the efficient geometric model with weaving parameters and some measurements as input data. Two 3D woven sample materials are produced to confirm the accuracy of the proposed geometric model. Even though the idealizations in geometric model lead to some over predictions, the model still captures the majority of details in corresponding real sample architectures.
INTRODUCTION In pursuit of lighter weight and lower manufacturing cost of aircraft, the aerospace industry has always been looking for alternative materials that meet the mechanical performance requirements. Advanced composite material is a preferred choice. Most typically, the novel composite material with 3D woven reinforcement, widely known for the great flexibility in textile architecture and the ability to form net shape directly[1], receives arising attentions for its potential utilization in integrally formed engineering applications. However, to implement 3D woven composite to complex structure, the overall mechanical properties must be adequately established through experimental studies, analytical methods or simulation. One primary and essential step of the simulation is the geometric description of the reinforced textile. Other than the regular architecture of constant thickness plate, the reinforcement in variable thickness plate is characterized by intricate geometric details which vary at the location of different layers and columns. Thus it is difficult to describe the entire structure by simply repeating a typical segment. Over the past decades, much effort has been spent on the 3D woven textile meso-level geometric modeling. Observing that the regular 3D woven textile is featured with the continuous and repeated architecture, a unit cell, or named as RVE (representative volume element)[2] is selected to describe the details of the whole structure. There mainly exist two ways to build the model of RVE. One is adopting the idealized assumptions of the yarn cross section and paths to get a simplified geometric model (as presented in this paper). A number of studies based on this method have been published. Peirce [3] firstly used circle hypothesis to describe all the yarn cross section in the textile. With the development of micro observation technology, the idealized cross section shape has been improved to reflect the actual structure more accurately. Various yarn cross section assumptions has been presented from rectangle[4][5], racetrack, ellipse[6] to lens[7], while the yarn paths have been assumed to be straight line, broken line or different forms of curves[8][9]. Ali[10] took the distinctions of geometry details between warp and weft yarn into account, thus different assumptions were applied to the warp and weft yarn separately. The other way to build RVE model is based on optical measurement and image processing technology[11]. Desplentere[12] used X-ray micro-Computed Tomography (micro-CT) to explore micro structural
variation of four different 3D warp-interlaced fabrics. Besides, recent software packages like TexGen[13] enable users to generate a RVE model where yarn cross section shapes is no more constant. According to Fredrik Stig[14], variable yarn cross sections caused by the compaction of molding process were simulated in the form of combination of different curves while yarn paths were also consistent with the actual reinforcement. Those diverse modelling technique have been developed to describe the geometry of reinforcement in constant thickness plate as precisely as possible. For those complex 3D woven composite structures, two dry textile reinforcement forming methods were concluded. One is to form the shape of the textile directly which has high requirements on manufacturing equipment and processing techniques[15]. And the product size is restricted by the machines. The variable thickness plate to be discussed in this paper is one typical product of this way. The other is that the regular fabric deforms in the mold to gain a more complex structure[16][17]. In order to reach the shape, the fabric is subjected to complicated stress state where the yarns were compressed, bent or sheared. The local deformation can be large which modifies the mechanical properties to some extent. This method can be utilized to obtain the 3D woven T-beams and I-beams. S.Yan developed an analytical geometric model which is implemented in Texgen to effectively describe the features, warp yarn shifting together with cross-section bending and weft yarn flattening, in the noodle region of T-beams[18]. N. Vernet explored the compaction behavior at the preforming stage when specified fiber volume fraction are desired[19]. And P. Badel investigated the woven composite fabric deformation of double curvature shapes[20]. There is no existing publication to investigate the geometry and mechanical behavior of variable thickness composite plate. Thus the objective of this paper is to present a geometric model which can describe the details of the constant and variable thickness composite plate in general. The discussed variable thickness preform in this paper is directly formed by adopting weft yarns with various sizes in each layer. In this geometric model, weaving process parameters is obviously needed. In addition, a sample is also required to determine the yarn cross section shape. The proposed model is able to be used in subsequent mechanical analysis.
MATERIALS Two sample materials including the constant and variable thickness plate (thickness changes in the warp direction) were composited by Bismaleimide resin QY8911-IV and carbon fiber T300. All the plate was in 3D layer-to-layer style. The weaving parameters are listed below: the warp yarn space is set to be 1/10 centimeter while the weft yarn space is 1/3.5 centimeter. The constant plate consisted of 6 layers of weft and 5 layers of warp. The constant thickness preform is made of 3K carbon fiber for both weft and warp yarns. The variable thickness plate contained 3 regions: constant thickness region at both ends and variable thickness region in the middle. The preform used 3K\6K\12K tows as weft yarns and 3K tows as warp yarns. The warp yarn sizes in each layer of the variable thickness plate were shown in Figure 1.There were 10 layers of weft and 9 layers of warp in the whole region. In the constant region, each column of weft yarn kept the same yarn size. Variable thickness region includes 17 columns of weft yarn. In each column, there exists one layer of weft yarn that has different size from the adjacent columns leading to the thickness variation in the warp direction.
Figure 1. Different weft yarn size at each layer in the variable thickness composite plate
IMAGE ANALYSIS To characterize the intrinsic geometry features of the variable thickness plate, the optical microscope was used. The obtained images through Hirox KH-7700 were shown in Figure 2. According to the microscopy images in different view, we can observe that: (a) Due to the various weft yarn cross section in each layer, the warp yarn was no longer in the same path; (b) The cross section of the warp yarn is a relatively regular rectangle as shown in section A and section B while the weft yarns can be approximatively recognized as lens with different major and minor axis; (c) Between the two consecutive columns of weft yarn, each layer of warp yarn can be divided into three Parts: the contact Part with the weft yarns at both ends (Part 1 and Part 3) and the middle connecting Part (Part 2). In Part 1 and Part 3, the surface of the warp yarn is closely fitted with the weft yarn cross section, and in Part 2, the warp yarn is smooth and continuous.
Figure 2. Microscopy images of variable thickness composite plate in weft and warp direction view(Section A and Section B) Based on the structural characteristics of 3D woven variable thickness plate, the following assumptions are accepted in this paper for the geometric model: (a) The twist of yarn cross section in the plate caused by the shear stress is ignored; (b) The resin area is supposed to be filled evenly without pores and inclusions; (c) Within the yarn, the fiber volume fraction is assumed to be equal everywhere; (d) The weft yarn path is straight and the cross section is lenticular which is determined by the yarn size; (e) The shape of warp yarn cross section is assumed to be rectangular. The warp yarn path is divided into three Parts, as shown in Figure 2. The curve in Part 1 and Part 3 is influenced by the contacted weft yarn while the curve in Part 2, the middle connecting Part, is in the form of cubic polynomial; (f).For the simplicity in calculation, the vertical misalignment of weft yarns has not been taken into consideration in this idealized geometric model.
GEOMETRICAL MODELLING (1)The selection of a typical segment Different from the constant plate, yarns of variable thickness plate is disparate in each layer. A RVE just including one layer of yarns is unable to describe the whole geometric details. In fact, due to the size changes of weft yarn at each column and row, there will no more be a Repeated Volume Element. However, to directly describe the whole variable thickness plate demands a large number of parameters and also will cause meshing problems in further analysis of mechanical behaviors. Similar to the concept of RVE, thus a typical segment (TS for short) that consists of two consecutive weft yarn columns and all the layers through thickness, as shown in Figure 3, is presented in this paper where k is the weft layer while i means the weft yarn column. The position of a weft yarn can be determined by these two subscripts.
Here we locate the Cartesian coordinate in a TS. The x axis is along the warp direction and the y axis is along the weft direction while z axis is along the thickness direction. The origin point is set in the middle of the weft yarn cross section center line in column i.
Figure 3. Representation of the 3D variable thickness plate and definition of the TS, marked with black dash lines. For the definition of the idealized geometry, as shown in Figure 4, geometry parameters as well as basic TS dimensions are required. (a) length of TS :
Lx (mm) Lx
10 Cx Mw
(1)
Where Mw is the number of weft yarns in per centimeter which determines the weft yarn spacing. Cx is a modified coefficient to reflect the compaction influences, it is obtained through micro images measurements. (b) width of TS: Ly (mm) Ly
10 Mj
(2)
Where M j is the number of warp yarns in per centimeter which determines the warp yarn space. (c) Two weft column thickness of TS:
Liz and Liz1 (mm) m
L （m+1） H j Hw(i,k ) i z
(3)
k 1
m
Liz1 （m+1） H j Hw(i1,k ) k 1
(4)
Where m is the weft layer ， H j is the height of the warp yarn,
Hw(i,k ) and Hw(i1,k ) are the height of weft yarn in
column i, layer k and layer k+1 respectively, weft yarn (i,k) and weft yarn (i,k+1) for short. In addition, W w( i , k ) is the width of weft yarn (i,k).
Figure 4. TS with the parameters needed for its definition (2)Weft yarn cross section Here a power ellipse[21] was introduced to describe the various cross section of weft yarn with different size generally. It is defined as: 2
2
x zn 1 a b
(5)
With the major and minor ellipse axes, a and b, the realistic weft yarn cross section can be generated with the different value of the exponent n. As shown in Figure 5, when n>1, the shape is lenticular. When n=1, it is elliptical and when n<1, it is a rectangle with rounded edge.
Figure 5 Example of power elliptical shapes with different exponents n[21] And the area of the cross section S can be calculated with the help of gamma function[21], which is defined as:
1 ( (n 2)) 2 S 2ab 1 ( (n 3)) 2 Eq.5 can be changed to get the expression of weft yarn’s boundary curve:
(6)
n
x x0 2 2 z f ( x ) 1 b z 0 ， x [ x0 , x0 a ] a
(7)
Where ( x0 , z0 ) is the center point position of warp yarn cross section; f ( x ) is the function to describe the 1/4 weft yarn’s boundary curve. The whole curve line can be obtained by symmetry and angular symmetry transformation. Replacing a, b in this function with the geometry parameters W w( i , k ) ,
Hw(i,k ) , with the subscript（i，k）, the 1/4 curve of
weft yarn in any position can be described as:
2 x 2 x (i ,k) 2 0 f (i ,k ) ( x) 1 (i ,k ) W w
n( i ,k ) 2
H w(i ,k ) H (i ,k ) z0(i ,k) ， x ( x0(i ,k) , x0(i ,k) w ) (8) 2 2
Where ( x0( i ,k) , z 0( i ,k) ) is the center point position of weft yarn (i,k) cross section. Similarly, Eq.6 can be expressed as:
Aw(i ,k )
Where
Aw(i,k ) and n(i,k)
H
(i ,k ) w
(i ,k ) w
W 2
( (
n(i ,k ) 2 2
n(i ,k ) 3 2
) (9)
) (i , k )
is the area and exponent of weft yarn (i,k) respectively. W w( i , k ) , Hw
required through the observation of the micro images, then
n(i,k)
(i , k )
and Aw
are
can be calculated through Eq.9. Thus the weft yarn cross
section curve can be obtained. (3)Warp yarn cross section For simplicity, the shape of warp yarn is assumed to be rectangular. The parameters of warp yarn cross section, W j and A j which are width and area of warp yarn cross section respectively, are obtained:
Wj Hj
10 Mj Aj Wj
(10)
(11)
Figure 6． The path curve of warp yarn: the contact part with the weft yarns at both ends (Part 1 and Part 3) and the middle connecting part (Part 2) (4)Warp yarn path The path curve of warp yarn is continuous and smooth and it can be divided into 3 Parts as introduced above. Here we introduce the scheme to obtain the expression of warp yarn path curve. As shown in Figure 6, the Part 1 lower surface AB of warp in layer k is closely contacted with the weft yarn (i,k) while The Part 3 upper surface FE is closely contacted with the weft yarn(i,k+1). The upper warp yarn surface is parallel to the lower surface. Thus we can obtain that: k f AB ( x) f(i,k) ( x) k f HG ( x) f(i,k) ( x) H j
f FEk ( x) f (i+1,k+1) ( x) k fCD ( x) f (i+1,k+1) ( x) H j
Ww(i ,k ) ) 2
(12)
Ww(i+1,k+1) (i+1,k+1) , x0 ) 2
(13)
x ( x0(i ,k) , x0(i ,k )
x( x
(i+1,k+1) 0
k k k k There f AB ( x ) , f HG ( x ) , f FE ( x ) , f CD ( x ) are the expression of the warp yarn curve AB, HG, FE, CD in layer k
respectively. In this paper, cubic polynomial functions are used to describe the Part 2 of warp yarn, the middle connecting section, which is specifically defined as:
gk (x) c1k x3 c2k x2 c3k x c4k
(14)
k k k k The coefficients c1 , c2 , c3 , c4 are determined by solve the equation set below. Taking the upper curve GF in Part
2 as an example, the curve HG and curve GF are connected at point G where these two curves are both continuous. In the same way, the curve GF and curve FE are connected and continuous at point F. so we can get that:
k g k ( xG ) c1k xG 3 c2k xG 2 c3k xG c4k f HG ( xG ) k ( xG ) f HG g k ( xG ) 3c1k xG 2 2c2k xG c3k 3 2 k k k k k g k ( xF ) c1 xF c2 xF c3 xF c4 f FE ( xF ) 2 k k k k ( xF ) f FE g k ( xF ) 3c1 xF 2c2 xF c3
(15)
Once the equation set is solved, the expression of curve GF can be obtained, so is the curve BC. Thus we can gain the expression of the whole warp yarn. (5)Fiber volume fraction (FVF)
p j is the warp yarn FVF, implicitly assumed to be equal among all the warp yarn path, while p w( i , k ) is the weft yarn (i ,k ) (i,k) FVF. p j and p w can be acquired by :
pj
pw(i,k )
Tt S T 1000 Aj 1000 Aj
T 1000 A
(i , k ) w
Tt S 1000 Aw(i,k )
(16)
(17)
Where T is the yarn linear density (g/1000m), S is the yarn size (K), Tt is the yarn linear density when S equals to 1K, ρ is density (g/cm3). The volume V of the TS, total warp yarn volume Vj and volume of weft yarn (i,k) are calculated below:
V Lx L y
( Liz Liz1 ) 2
(18)
Vj = Aj Ly
(19)
Vw(i,k ) Aw(i,k ) Ly
(20)
The overall fiber volume fraction P then can be obtained:
P
V j p j (Vw(i ,k ) pw(i ,k ) ) V
p j Aj ( Aw(i ,k ) pw(i ,k ) ) Lx
( Liz Liz1 ) 2
(21)
(6)Measurements of the sample material As shown in Table 1, all the parameters for the geometric model are summarized. Apart from the weaving parameters, measurements of the sample material from the micro images are also required. The measurements of the warp yarn cross section with different size and warp yarn in the sample material are listed in Table 2 where the mean value is given. In addition, the calculated FVF and exponent n are also shown in Table 2.
With the measurements and the material parameters, finally we can build the geometric model. Table 1 Required parameters for the geometric model Weaving parameters
Mw , M j , m
Material parameters
S, Tt ,
Measurements
Ww(i,k ) , Hw(i,k ) , Aw(i,k ) , A j
Calculated value
pj pw(i,k ) n(i,k) , ,
Table 2 Measurements of the weft yarn and warp yarn cross section sizes in the sample material Area（μm2）
Width（μm） Height（μm）
FVF
Exponent n
Weft(3K)
166337
1454
186
67.73%
1.722
Weft(6K)
325999
1781
285
69.02%
1.450
Weft(12K)
634272
2265
436
70.95%
1.452
Warp(3K)
161173
-
-
69.80%
-
VALIDATION and DISCUSSION To validate the proposed modeling geometry that can be applied to describe the details of the constant and variable thickness plate generally, here two cases are studied. Case1: Geometric model of the constant thickness composite plate. The constant thickness composite plate was prepared as mentioned above. The material properties of the fiber and resin together with the weaving parameters are shown in Table 3. With the given value as well as the data from Table 2, the final result of the geometric modelling scheme is illustrated in Figure 7, where a comparison between the simulated structure and the constant 3D woven sample is also presented. As can be seen there is good agreement between the simulated and real sample architectures where position and cross section of interior layer weft yarn as well as warp path match well, even though the shape of surface weft yarn differs slightly from the actual structure. Meanwhile, a more detailed comparison of the cross section and yarn path was shown in Table 4, where the good agreement is more visible. The error of thickness prediction potentially caused by the compaction effect is still in a reasonable range. Table 3 Input data to the geometric modelling Symbols
Value
Mw , M j , m
10, 3.5, 6
S, Tt , , Cx
3, 66, 1.76,1
Figure 7. Comparison between generated geometric model and the observed structure of constant thickness plate Table 4 The predicted value and measurements of geometry details in the constant thickness plate Wrap
Weft
Overall
Thickness
FVF(%)
(μm)
204
42.46%
2217.6
1454
186
39.60%
2244.4
9.41%
8.82%
6.74%
1.21%
area
width
height
angle
area
width
height
( mm 2 )
(μm)
(μm)
(°)
( mm 2 )
(μm)
(μm)
Measurements
0.160
1019
176.5
18.05
0.167
1329
Predicted Value
0.161
1000
161.2
18.26
0.166
RE(Relative Error)
0.63%
1.86%
8.67%
1.15%
0.60%
Case2: Geometric model of the variable thickness composite plate Here the predicted geometric model of variable thickness plate is investigated. The sample material is produced as mentioned in Materials. Similar to the input data from Table 3, the layer m is set to be 10 meanwhile Cx is set to be 1.041. And the weft yarn size is given according to Figure 1.The micro images of 12 weft yarn columns (from 1 to 12) from sample 1 and sample 2 are displayed to validate the geometry methodology in Figure 8. Even though the sample architecture is more complicated than the idealized structure, the final obtained geometric model somehow captures a lot of the complexity obviously in terms of inner weft cross section size and shape changes as well as resulting warp yarn path diversities. Taking the segment between column 7 and column 8 of sample 1 as an example, a further comparison is performed at a relatively high level of detail. As can be seen in Figure 9 (a), there is satisfying agreement of interior weft yarn between the simulated model and the real sample architecture in terms of shape and size as well as position, but the surface weft yarn differs from the observed structure since the real surface weft yarns slide and are flatten under compaction during the processing. In the view of section A from Figure 9(a), a good correlation of interior warp yarn in position and cross section sizes is visible between the sample (Figure 9 (c)) and the model (Figure 9 (d)). Results from Table5 show that the geometric model precisely describes the warp yarn cross section. Meanwhile the predicted weft yarn cross section area and height comply well with the measurements from the sample except that the model over-predicts the width of 3K and 12K weft yarn at the error of 12.28% and 11.85% respectively which are still in a reasonable range. The angles of wrap yarn path are compared between the model and the observation in Figure 9 (b) and Table 6. There is excellent correlation in interior warp yarn layer while the predicted warp yarn paths in the exterior layer have lower slope over the real warp yarn. A plausible reason for this discrepancy is still the ignorance of the surface compaction. The surface weft yarns are further curved due to the slides which result in the higher slope of warp yarn over them.
It is of interest to look at the thickness changes at the position of each weft yarn column. As shown in Figure 10, the model over-predicts the thickness compared to the measurements from both sample 1 and sample 2. The max error occurs in the column 3 of sample 1 at the value of 15.08%. However, the predicted thickness is much closer to the observed thickness from column 8 to column 12 where the min error is 1.94% at column 12 in sample 2. This is partially explained that the interior weft sizes in the relatively thicker part, from column 1 to column 6, are almost 3K while the surface weft sizes are constantly 6K. Thus the compaction on surface 6K weft yarn has a relatively greater influence on the overall thickness from column1 to column 6. However once the interior weft size is larger than the exterior, the compaction effect on thickness of the plate is gradually reduced. The locations in warp direction (x axis) of weft cross section center at each row and column have also been investigated in Figure 11 where the vertical misalignment of weft yarns is quantified. Results show that there are differences between the samples in weft yarn shift. After adopting the modified coefficient Cx in the calculation of Lx (the distance between weft columns), the predicted weft yarn cross section center position in x axis lies in the middle of those measured from sample 1 and sample 2. Thus even though the proposed geometric model ignores the variances in x axis, it still reasonably reflects the real weft yarn cross section center position at each column in warp direction. Table 5 Comparison between the measured and predicted cross section sizes of yarn in column 7 and column8 from sample1 Area(μm 2)
Yarn
Width(μm)
Height(μm)
Measured
Predicted
RE
Measured
Predicted
RE
Measured
Predicted
RE
Weft(3K)
160827
166337
3.43%
1295
1454
12.28%
183
186
1.64%
Weft(6K)
346540
325999
5.93%
1969
1781
9.55%
305
285
6.55%
Weft(12K)
632811
634272
0.23%
2025
2265
11.85%
430
436
1.40%
Warp(3K)
163593
161173
1.48%
1100
1000
9.10%
172
161
6.40%
Table 6 Comparisons between the predicted and measured angle of warp yarns through column 7 and column 8 wefts from sample1 AN1
AN2
AN3
AN4
AN5
AN6
AN7
AN8
AN9
Measured
19.78
20.19
18.71
26.06
36.25
41.42
25.41
14.04
22.07
Predicted
15.33
16.04
18.43
27.18
39.73
40.27
27.33
13.88
18.43
Relative Error
22.50%
20.55%
1.50%
4.30%
9.60%
2.78%
7.56%
1.14%
16.49%
Figure 8.Comparison between modelled geometry and the observed microscopy image of sample 1 and sample 2through column1 to column 12 in variable thickness plate
Figure 9.A detailed comparison between the model and the real architecture in both weft view ((a) and (b)) and warp view ((c) and (d)) of a segment through column 7 and column 8 from sample 1 6.5 6.0 5.5 5.0
Thickness Lz(mm)
4.5 4.0 3.5 3.0 2.5 2.0
Measurements from Sample 1 Measurements from Sample 2 Predicted Value
1.5 1.0 0.5 0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Weft Column
Figure 10. Predicted and measured thickness of the variable thickness composite plate
15 14
Sample 1 Predicted Sample 2
13 12 11
Weft Row
10
Col 1
Col 2
0.0
2.5
Col 3
Col 4
Col 5
Col 6
Col 7
Col 8
Col 9
Col 10
Col 11 Col 12
9 8 7 6 5 4 3 2 1 0
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
Weft Yarn Centre Location in X axis (mm) Figure 11 . Measured and predicted location of weft yarn cross section center in x axis at different row and column
CONCLUSIONS An accurate description of the textile geometry is the basis for further computer analysis. The geometric model proposed in this paper is aimed at the variable thickness composite plate, which also can be used in constant composite plate of course. The modelling requires weaving processing parameters together with measurements from sample material as input data. The comparisons with two sample composite plates confirm that the presented geometric model can be successfully utilized in constant and variable thickness 3D woven composite plate generally. The results from the model containing weft yarn cross section shape and size as well as warp yarn paths agree well with the correlated real structure. The mean predicted error is about 4.34% when applying the model to constant plate. To predict the geometric details of variable thickness plate, the average discrepancy of surface weft yarn cross section sizes is 7.34% while average discrepancy of the interior is 5.14%. When predicting the warp yarn sizes, the mean error is about 5.66%. The agreement between the predicted and the sample of inner layer warp yarn path angle is striking where mean error is 6.77% while the prediction of the surface warp yarn path angle is not so satisfying where the error is 22.50% and 16.49% respectively. The predicted thickness at each weft yarn column is reasonable with the mean error at 8.50%. According to the plausible explanation that the main factor causing these discrepancies may be the compaction behavior during manufacturing process, the geometric model can be further improved by taking the surface compact effect into consideration. Even though the idealizations of the proposed geometric model in this paper lead to some over predictions such in the thickness of the plate as well as 3K and 6K weft cross section width, it still captures the majority of the details overall in real sample architectures. To conclude, a relatively efficient approach to generate a geometric model of variable thickness 3D composite plate has been presented. It could handle the various weft yarn cross sections at each layer and the resulted different warp yarn paths in the variable thickness 3D woven composite plate. After exporting the modelled geometry to a mesh-generating software, thus the geometric model could be utilized for the further mechanical analysis.
Acknowledgements This work has been supported by National Science and Technology Major Project (2017-IV-0007-0044).The authors would also like to express their gratitude to editors and reviewers for their comments and suggestions.
REFERENCES [1]
Stig F, Hallström S. A modelling framework for composites containing 3D reinforcement. Compos Struct
2012;94:2895–901. doi:10.1016/j.compstruct.2012.03.009. [2]
Ivanov DS, Lomov S V. Modelling the structure and behaviour of 2D and 3D woven composites used in aerospace
applications. Elsevier Ltd; 2014. doi:10.1016/B978-0-85709-523-7.00002-5. [3]
Peirce FT. The geometry of cloth structure. J Text Inst Trans 1937. doi:10.1080/19447023708658809.
[4]
Tan P, Tong L, Steven GP. Modelling for predicting the mechanical properties of textile composites - A review.
Compos Part A Appl Sci Manuf 1997. doi:10.1016/S1359-835X(97)00069-9. [5]
Ivanov I, Tabiei A. Three-dimensional computational micro-mechanical model for woven fabric composites. Compos
Struct 2001. doi:10.1016/S0263-8223(01)00121-0. [6]
Buchanan S, Grigorash A, Archer E, McIlhagger A, Quinn J, Stewart G. Analytical elastic stiffness model for 3D
woven orthogonal interlock composites. Compos Sci Technol 2010. doi:10.1016/j.compscitech.2010.05.019. [7]
Quinn J, McIlhagger R, McIlhagger AT. A modified system for design and analysis of 3D woven preforms. Compos.
Part A Appl. Sci. Manuf., 2003. doi:10.1016/S1359-835X(03)00084-8. [8]
Kuhn JL, Charalambides PG. Modeling of Plain Weave Fabric Composite Geometry. J Compos Mater 1999.
doi:10.1177/002199839903300301. [9]
Sheng SZ, Van Hoa S. Modeling of 3D angle interlock woven fabric composites. J Thermoplast Compos Mater 2003.
doi:10.1177/0892705703016001206. [10] Hallal A, Younes R, Fardoun F, Nehme S. Improved analytical model to predict the effective elastic properties of 2.5D interlock woven fabrics composite. Compos Struct 2012. doi:10.1016/j.compstruct.2012.03.019. [11] Salvo L, Cloetens P, Maire E, Zabler S, Blandin JJ, Buffière JY, et al. X-ray micro-tomography an attractive characterisation technique in materials science. Nucl. Instruments Methods Phys. Res. Sect. B Beam Interact. with Mater. Atoms, 2003. doi:10.1016/S0168-583X(02)01689-0. [12] Desplentere F, Lomov S V., Woerdeman DL, Verpoest I, Wevers M, Bogdanovich A. Micro-CT characterization of variability in 3D textile architecture. Compos Sci Technol 2005. doi:10.1016/j.compscitech.2005.04.008. [13] Lin H, Brown LP, Long AC. Modelling and Simulating Textile Structures Using TexGen. Adv Mater Res 2011. doi:10.4028/www.scientific.net/amr.331.44. [14] Stig
F,
Hallström
S.
Spatial
modelling
of
3D-woven
textiles.
Compos
Struct
2012;94:1495–502.
doi:10.1016/j.compstruct.2011.12.003. [15] Mountasir A, Loser M, Hoffmann G, Cherif C, Grossmann K. 3D Woven Near-Net-Shape Preforms for Composite Structures. Adv Eng Mater 2016;18:391–6. doi:10.1002/adem.201500441.
[16] Vernet N, Ruiz E, Advani S, Alms JB, Aubert M, Barburski M, et al. Experimental determination of the permeability of engineering textiles: Benchmark II. Compos Part A Appl Sci Manuf 2014. doi:10.1016/j.compositesa.2014.02.010. [17] Ouagne P, Ouahbi T, Park CH, Bréard J, Saouab A. Continuous measurement of fiber reinforcement permeability in the
thickness
direction:
Experimental
technique
and
validation.
Compos
Part
B
Eng
2013.
doi:10.1016/j.compositesb.2012.06.007. [18] Yan S, Zeng X, Long A. Meso-scale modelling of 3D woven composite T-joints with weave variations. Compos Sci Technol 2019;171:171–9. doi:10.1016/j.compscitech.2018.12.024. [19] Vernet N, Trochu F. Analysis and modeling of 3D Interlock fabric compaction behavior. Compos Part A Appl Sci Manuf 2016;80:182–93. doi:10.1016/j.compositesa.2015.10.024. [20] Badel P, Sallé EV, Maire E, Boisse P. Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale. Int J Mater Form 2009;2:189–92. doi:10.1007/s12289-009-0572-z. [21] Gommer F, Brown LP, Wedgwood KCA. Analytical method using gamma functions for determining areas of power elliptical
shapes
for
use
in
geometrical
doi:10.1016/j.compositesa.2015.11.031.
textile
models.
Compos
Part
A
Appl
Sci
Manuf
2016;81:222–4.
Yu Zhou：Methodology, Software, Investigation, Writing Original Draft, Visualization. Weidong WEN：Conceptualization, Resources, Data Curation, Supervision. Haitao CUI：Validation, Formal analysis, Writing - Review & Editing.
We have no conflicts of interest to declare.
S0263-8223(19)31958-0 https://doi.org/10.1016/j.compstruct.2020.111946 COST 111946
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
24 May 2019 13 January 2020 17 January 2020
Please cite this article as: Zhou, Y., Wen, W., Cui, H., Spatial modelling of 3D woven variable thickness composite plate at the mesoscopic scale, Composite Structures (2020), doi: https://doi.org/10.1016/j.compstruct.2020.111946
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier Ltd.
Spatial modelling of 3D woven variable thickness composite plate at the mesoscopic scale Yu Zhou, Weidong WEN*, Haitao CUI College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China *Corresponding author. E-mail address: [email protected] Keyword: 3D woven; variable thickness plate; composite; geometric model
ABSTRACT The geometric details of variable thickness 3D woven composite plate, which features varying weft yarn cross section shapes and sizes at each layer and resulting diverse warp yarn path orientations, are quite different from the structure in constant thickness plate. A geometric model to handle these variances is presented in this paper which no doubt can also be used in constant thickness plate. Based on the idealized assumptions that the cross section shape of weft yarn is lenticular and warp yarn is rectangular, the inner structure details of variable thickness plate are illustrated in the efficient geometric model with weaving parameters and some measurements as input data. Two 3D woven sample materials are produced to confirm the accuracy of the proposed geometric model. Even though the idealizations in geometric model lead to some over predictions, the model still captures the majority of details in corresponding real sample architectures.
INTRODUCTION In pursuit of lighter weight and lower manufacturing cost of aircraft, the aerospace industry has always been looking for alternative materials that meet the mechanical performance requirements. Advanced composite material is a preferred choice. Most typically, the novel composite material with 3D woven reinforcement, widely known for the great flexibility in textile architecture and the ability to form net shape directly[1], receives arising attentions for its potential utilization in integrally formed engineering applications. However, to implement 3D woven composite to complex structure, the overall mechanical properties must be adequately established through experimental studies, analytical methods or simulation. One primary and essential step of the simulation is the geometric description of the reinforced textile. Other than the regular architecture of constant thickness plate, the reinforcement in variable thickness plate is characterized by intricate geometric details which vary at the location of different layers and columns. Thus it is difficult to describe the entire structure by simply repeating a typical segment. Over the past decades, much effort has been spent on the 3D woven textile meso-level geometric modeling. Observing that the regular 3D woven textile is featured with the continuous and repeated architecture, a unit cell, or named as RVE (representative volume element)[2] is selected to describe the details of the whole structure. There mainly exist two ways to build the model of RVE. One is adopting the idealized assumptions of the yarn cross section and paths to get a simplified geometric model (as presented in this paper). A number of studies based on this method have been published. Peirce [3] firstly used circle hypothesis to describe all the yarn cross section in the textile. With the development of micro observation technology, the idealized cross section shape has been improved to reflect the actual structure more accurately. Various yarn cross section assumptions has been presented from rectangle[4][5], racetrack, ellipse[6] to lens[7], while the yarn paths have been assumed to be straight line, broken line or different forms of curves[8][9]. Ali[10] took the distinctions of geometry details between warp and weft yarn into account, thus different assumptions were applied to the warp and weft yarn separately. The other way to build RVE model is based on optical measurement and image processing technology[11]. Desplentere[12] used X-ray micro-Computed Tomography (micro-CT) to explore micro structural
variation of four different 3D warp-interlaced fabrics. Besides, recent software packages like TexGen[13] enable users to generate a RVE model where yarn cross section shapes is no more constant. According to Fredrik Stig[14], variable yarn cross sections caused by the compaction of molding process were simulated in the form of combination of different curves while yarn paths were also consistent with the actual reinforcement. Those diverse modelling technique have been developed to describe the geometry of reinforcement in constant thickness plate as precisely as possible. For those complex 3D woven composite structures, two dry textile reinforcement forming methods were concluded. One is to form the shape of the textile directly which has high requirements on manufacturing equipment and processing techniques[15]. And the product size is restricted by the machines. The variable thickness plate to be discussed in this paper is one typical product of this way. The other is that the regular fabric deforms in the mold to gain a more complex structure[16][17]. In order to reach the shape, the fabric is subjected to complicated stress state where the yarns were compressed, bent or sheared. The local deformation can be large which modifies the mechanical properties to some extent. This method can be utilized to obtain the 3D woven T-beams and I-beams. S.Yan developed an analytical geometric model which is implemented in Texgen to effectively describe the features, warp yarn shifting together with cross-section bending and weft yarn flattening, in the noodle region of T-beams[18]. N. Vernet explored the compaction behavior at the preforming stage when specified fiber volume fraction are desired[19]. And P. Badel investigated the woven composite fabric deformation of double curvature shapes[20]. There is no existing publication to investigate the geometry and mechanical behavior of variable thickness composite plate. Thus the objective of this paper is to present a geometric model which can describe the details of the constant and variable thickness composite plate in general. The discussed variable thickness preform in this paper is directly formed by adopting weft yarns with various sizes in each layer. In this geometric model, weaving process parameters is obviously needed. In addition, a sample is also required to determine the yarn cross section shape. The proposed model is able to be used in subsequent mechanical analysis.
MATERIALS Two sample materials including the constant and variable thickness plate (thickness changes in the warp direction) were composited by Bismaleimide resin QY8911-IV and carbon fiber T300. All the plate was in 3D layer-to-layer style. The weaving parameters are listed below: the warp yarn space is set to be 1/10 centimeter while the weft yarn space is 1/3.5 centimeter. The constant plate consisted of 6 layers of weft and 5 layers of warp. The constant thickness preform is made of 3K carbon fiber for both weft and warp yarns. The variable thickness plate contained 3 regions: constant thickness region at both ends and variable thickness region in the middle. The preform used 3K\6K\12K tows as weft yarns and 3K tows as warp yarns. The warp yarn sizes in each layer of the variable thickness plate were shown in Figure 1.There were 10 layers of weft and 9 layers of warp in the whole region. In the constant region, each column of weft yarn kept the same yarn size. Variable thickness region includes 17 columns of weft yarn. In each column, there exists one layer of weft yarn that has different size from the adjacent columns leading to the thickness variation in the warp direction.
Figure 1. Different weft yarn size at each layer in the variable thickness composite plate
IMAGE ANALYSIS To characterize the intrinsic geometry features of the variable thickness plate, the optical microscope was used. The obtained images through Hirox KH-7700 were shown in Figure 2. According to the microscopy images in different view, we can observe that: (a) Due to the various weft yarn cross section in each layer, the warp yarn was no longer in the same path; (b) The cross section of the warp yarn is a relatively regular rectangle as shown in section A and section B while the weft yarns can be approximatively recognized as lens with different major and minor axis; (c) Between the two consecutive columns of weft yarn, each layer of warp yarn can be divided into three Parts: the contact Part with the weft yarns at both ends (Part 1 and Part 3) and the middle connecting Part (Part 2). In Part 1 and Part 3, the surface of the warp yarn is closely fitted with the weft yarn cross section, and in Part 2, the warp yarn is smooth and continuous.
Figure 2. Microscopy images of variable thickness composite plate in weft and warp direction view(Section A and Section B) Based on the structural characteristics of 3D woven variable thickness plate, the following assumptions are accepted in this paper for the geometric model: (a) The twist of yarn cross section in the plate caused by the shear stress is ignored; (b) The resin area is supposed to be filled evenly without pores and inclusions; (c) Within the yarn, the fiber volume fraction is assumed to be equal everywhere; (d) The weft yarn path is straight and the cross section is lenticular which is determined by the yarn size; (e) The shape of warp yarn cross section is assumed to be rectangular. The warp yarn path is divided into three Parts, as shown in Figure 2. The curve in Part 1 and Part 3 is influenced by the contacted weft yarn while the curve in Part 2, the middle connecting Part, is in the form of cubic polynomial; (f).For the simplicity in calculation, the vertical misalignment of weft yarns has not been taken into consideration in this idealized geometric model.
GEOMETRICAL MODELLING (1)The selection of a typical segment Different from the constant plate, yarns of variable thickness plate is disparate in each layer. A RVE just including one layer of yarns is unable to describe the whole geometric details. In fact, due to the size changes of weft yarn at each column and row, there will no more be a Repeated Volume Element. However, to directly describe the whole variable thickness plate demands a large number of parameters and also will cause meshing problems in further analysis of mechanical behaviors. Similar to the concept of RVE, thus a typical segment (TS for short) that consists of two consecutive weft yarn columns and all the layers through thickness, as shown in Figure 3, is presented in this paper where k is the weft layer while i means the weft yarn column. The position of a weft yarn can be determined by these two subscripts.
Here we locate the Cartesian coordinate in a TS. The x axis is along the warp direction and the y axis is along the weft direction while z axis is along the thickness direction. The origin point is set in the middle of the weft yarn cross section center line in column i.
Figure 3. Representation of the 3D variable thickness plate and definition of the TS, marked with black dash lines. For the definition of the idealized geometry, as shown in Figure 4, geometry parameters as well as basic TS dimensions are required. (a) length of TS :
Lx (mm) Lx
10 Cx Mw
(1)
Where Mw is the number of weft yarns in per centimeter which determines the weft yarn spacing. Cx is a modified coefficient to reflect the compaction influences, it is obtained through micro images measurements. (b) width of TS: Ly (mm) Ly
10 Mj
(2)
Where M j is the number of warp yarns in per centimeter which determines the warp yarn space. (c) Two weft column thickness of TS:
Liz and Liz1 (mm) m
L （m+1） H j Hw(i,k ) i z
(3)
k 1
m
Liz1 （m+1） H j Hw(i1,k ) k 1
(4)
Where m is the weft layer ， H j is the height of the warp yarn,
Hw(i,k ) and Hw(i1,k ) are the height of weft yarn in
column i, layer k and layer k+1 respectively, weft yarn (i,k) and weft yarn (i,k+1) for short. In addition, W w( i , k ) is the width of weft yarn (i,k).
Figure 4. TS with the parameters needed for its definition (2)Weft yarn cross section Here a power ellipse[21] was introduced to describe the various cross section of weft yarn with different size generally. It is defined as: 2
2
x zn 1 a b
(5)
With the major and minor ellipse axes, a and b, the realistic weft yarn cross section can be generated with the different value of the exponent n. As shown in Figure 5, when n>1, the shape is lenticular. When n=1, it is elliptical and when n<1, it is a rectangle with rounded edge.
Figure 5 Example of power elliptical shapes with different exponents n[21] And the area of the cross section S can be calculated with the help of gamma function[21], which is defined as:
1 ( (n 2)) 2 S 2ab 1 ( (n 3)) 2 Eq.5 can be changed to get the expression of weft yarn’s boundary curve:
(6)
n
x x0 2 2 z f ( x ) 1 b z 0 ， x [ x0 , x0 a ] a
(7)
Where ( x0 , z0 ) is the center point position of warp yarn cross section; f ( x ) is the function to describe the 1/4 weft yarn’s boundary curve. The whole curve line can be obtained by symmetry and angular symmetry transformation. Replacing a, b in this function with the geometry parameters W w( i , k ) ,
Hw(i,k ) , with the subscript（i，k）, the 1/4 curve of
weft yarn in any position can be described as:
2 x 2 x (i ,k) 2 0 f (i ,k ) ( x) 1 (i ,k ) W w
n( i ,k ) 2
H w(i ,k ) H (i ,k ) z0(i ,k) ， x ( x0(i ,k) , x0(i ,k) w ) (8) 2 2
Where ( x0( i ,k) , z 0( i ,k) ) is the center point position of weft yarn (i,k) cross section. Similarly, Eq.6 can be expressed as:
Aw(i ,k )
Where
Aw(i,k ) and n(i,k)
H
(i ,k ) w
(i ,k ) w
W 2
( (
n(i ,k ) 2 2
n(i ,k ) 3 2
) (9)
) (i , k )
is the area and exponent of weft yarn (i,k) respectively. W w( i , k ) , Hw
required through the observation of the micro images, then
n(i,k)
(i , k )
and Aw
are
can be calculated through Eq.9. Thus the weft yarn cross
section curve can be obtained. (3)Warp yarn cross section For simplicity, the shape of warp yarn is assumed to be rectangular. The parameters of warp yarn cross section, W j and A j which are width and area of warp yarn cross section respectively, are obtained:
Wj Hj
10 Mj Aj Wj
(10)
(11)
Figure 6． The path curve of warp yarn: the contact part with the weft yarns at both ends (Part 1 and Part 3) and the middle connecting part (Part 2) (4)Warp yarn path The path curve of warp yarn is continuous and smooth and it can be divided into 3 Parts as introduced above. Here we introduce the scheme to obtain the expression of warp yarn path curve. As shown in Figure 6, the Part 1 lower surface AB of warp in layer k is closely contacted with the weft yarn (i,k) while The Part 3 upper surface FE is closely contacted with the weft yarn(i,k+1). The upper warp yarn surface is parallel to the lower surface. Thus we can obtain that: k f AB ( x) f(i,k) ( x) k f HG ( x) f(i,k) ( x) H j
f FEk ( x) f (i+1,k+1) ( x) k fCD ( x) f (i+1,k+1) ( x) H j
Ww(i ,k ) ) 2
(12)
Ww(i+1,k+1) (i+1,k+1) , x0 ) 2
(13)
x ( x0(i ,k) , x0(i ,k )
x( x
(i+1,k+1) 0
k k k k There f AB ( x ) , f HG ( x ) , f FE ( x ) , f CD ( x ) are the expression of the warp yarn curve AB, HG, FE, CD in layer k
respectively. In this paper, cubic polynomial functions are used to describe the Part 2 of warp yarn, the middle connecting section, which is specifically defined as:
gk (x) c1k x3 c2k x2 c3k x c4k
(14)
k k k k The coefficients c1 , c2 , c3 , c4 are determined by solve the equation set below. Taking the upper curve GF in Part
2 as an example, the curve HG and curve GF are connected at point G where these two curves are both continuous. In the same way, the curve GF and curve FE are connected and continuous at point F. so we can get that:
k g k ( xG ) c1k xG 3 c2k xG 2 c3k xG c4k f HG ( xG ) k ( xG ) f HG g k ( xG ) 3c1k xG 2 2c2k xG c3k 3 2 k k k k k g k ( xF ) c1 xF c2 xF c3 xF c4 f FE ( xF ) 2 k k k k ( xF ) f FE g k ( xF ) 3c1 xF 2c2 xF c3
(15)
Once the equation set is solved, the expression of curve GF can be obtained, so is the curve BC. Thus we can gain the expression of the whole warp yarn. (5)Fiber volume fraction (FVF)
p j is the warp yarn FVF, implicitly assumed to be equal among all the warp yarn path, while p w( i , k ) is the weft yarn (i ,k ) (i,k) FVF. p j and p w can be acquired by :
pj
pw(i,k )
Tt S T 1000 Aj 1000 Aj
T 1000 A
(i , k ) w
Tt S 1000 Aw(i,k )
(16)
(17)
Where T is the yarn linear density (g/1000m), S is the yarn size (K), Tt is the yarn linear density when S equals to 1K, ρ is density (g/cm3). The volume V of the TS, total warp yarn volume Vj and volume of weft yarn (i,k) are calculated below:
V Lx L y
( Liz Liz1 ) 2
(18)
Vj = Aj Ly
(19)
Vw(i,k ) Aw(i,k ) Ly
(20)
The overall fiber volume fraction P then can be obtained:
P
V j p j (Vw(i ,k ) pw(i ,k ) ) V
p j Aj ( Aw(i ,k ) pw(i ,k ) ) Lx
( Liz Liz1 ) 2
(21)
(6)Measurements of the sample material As shown in Table 1, all the parameters for the geometric model are summarized. Apart from the weaving parameters, measurements of the sample material from the micro images are also required. The measurements of the warp yarn cross section with different size and warp yarn in the sample material are listed in Table 2 where the mean value is given. In addition, the calculated FVF and exponent n are also shown in Table 2.
With the measurements and the material parameters, finally we can build the geometric model. Table 1 Required parameters for the geometric model Weaving parameters
Mw , M j , m
Material parameters
S, Tt ,
Measurements
Ww(i,k ) , Hw(i,k ) , Aw(i,k ) , A j
Calculated value
pj pw(i,k ) n(i,k) , ,
Table 2 Measurements of the weft yarn and warp yarn cross section sizes in the sample material Area（μm2）
Width（μm） Height（μm）
FVF
Exponent n
Weft(3K)
166337
1454
186
67.73%
1.722
Weft(6K)
325999
1781
285
69.02%
1.450
Weft(12K)
634272
2265
436
70.95%
1.452
Warp(3K)
161173
-
-
69.80%
-
VALIDATION and DISCUSSION To validate the proposed modeling geometry that can be applied to describe the details of the constant and variable thickness plate generally, here two cases are studied. Case1: Geometric model of the constant thickness composite plate. The constant thickness composite plate was prepared as mentioned above. The material properties of the fiber and resin together with the weaving parameters are shown in Table 3. With the given value as well as the data from Table 2, the final result of the geometric modelling scheme is illustrated in Figure 7, where a comparison between the simulated structure and the constant 3D woven sample is also presented. As can be seen there is good agreement between the simulated and real sample architectures where position and cross section of interior layer weft yarn as well as warp path match well, even though the shape of surface weft yarn differs slightly from the actual structure. Meanwhile, a more detailed comparison of the cross section and yarn path was shown in Table 4, where the good agreement is more visible. The error of thickness prediction potentially caused by the compaction effect is still in a reasonable range. Table 3 Input data to the geometric modelling Symbols
Value
Mw , M j , m
10, 3.5, 6
S, Tt , , Cx
3, 66, 1.76,1
Figure 7. Comparison between generated geometric model and the observed structure of constant thickness plate Table 4 The predicted value and measurements of geometry details in the constant thickness plate Wrap
Weft
Overall
Thickness
FVF(%)
(μm)
204
42.46%
2217.6
1454
186
39.60%
2244.4
9.41%
8.82%
6.74%
1.21%
area
width
height
angle
area
width
height
( mm 2 )
(μm)
(μm)
(°)
( mm 2 )
(μm)
(μm)
Measurements
0.160
1019
176.5
18.05
0.167
1329
Predicted Value
0.161
1000
161.2
18.26
0.166
RE(Relative Error)
0.63%
1.86%
8.67%
1.15%
0.60%
Case2: Geometric model of the variable thickness composite plate Here the predicted geometric model of variable thickness plate is investigated. The sample material is produced as mentioned in Materials. Similar to the input data from Table 3, the layer m is set to be 10 meanwhile Cx is set to be 1.041. And the weft yarn size is given according to Figure 1.The micro images of 12 weft yarn columns (from 1 to 12) from sample 1 and sample 2 are displayed to validate the geometry methodology in Figure 8. Even though the sample architecture is more complicated than the idealized structure, the final obtained geometric model somehow captures a lot of the complexity obviously in terms of inner weft cross section size and shape changes as well as resulting warp yarn path diversities. Taking the segment between column 7 and column 8 of sample 1 as an example, a further comparison is performed at a relatively high level of detail. As can be seen in Figure 9 (a), there is satisfying agreement of interior weft yarn between the simulated model and the real sample architecture in terms of shape and size as well as position, but the surface weft yarn differs from the observed structure since the real surface weft yarns slide and are flatten under compaction during the processing. In the view of section A from Figure 9(a), a good correlation of interior warp yarn in position and cross section sizes is visible between the sample (Figure 9 (c)) and the model (Figure 9 (d)). Results from Table5 show that the geometric model precisely describes the warp yarn cross section. Meanwhile the predicted weft yarn cross section area and height comply well with the measurements from the sample except that the model over-predicts the width of 3K and 12K weft yarn at the error of 12.28% and 11.85% respectively which are still in a reasonable range. The angles of wrap yarn path are compared between the model and the observation in Figure 9 (b) and Table 6. There is excellent correlation in interior warp yarn layer while the predicted warp yarn paths in the exterior layer have lower slope over the real warp yarn. A plausible reason for this discrepancy is still the ignorance of the surface compaction. The surface weft yarns are further curved due to the slides which result in the higher slope of warp yarn over them.
It is of interest to look at the thickness changes at the position of each weft yarn column. As shown in Figure 10, the model over-predicts the thickness compared to the measurements from both sample 1 and sample 2. The max error occurs in the column 3 of sample 1 at the value of 15.08%. However, the predicted thickness is much closer to the observed thickness from column 8 to column 12 where the min error is 1.94% at column 12 in sample 2. This is partially explained that the interior weft sizes in the relatively thicker part, from column 1 to column 6, are almost 3K while the surface weft sizes are constantly 6K. Thus the compaction on surface 6K weft yarn has a relatively greater influence on the overall thickness from column1 to column 6. However once the interior weft size is larger than the exterior, the compaction effect on thickness of the plate is gradually reduced. The locations in warp direction (x axis) of weft cross section center at each row and column have also been investigated in Figure 11 where the vertical misalignment of weft yarns is quantified. Results show that there are differences between the samples in weft yarn shift. After adopting the modified coefficient Cx in the calculation of Lx (the distance between weft columns), the predicted weft yarn cross section center position in x axis lies in the middle of those measured from sample 1 and sample 2. Thus even though the proposed geometric model ignores the variances in x axis, it still reasonably reflects the real weft yarn cross section center position at each column in warp direction. Table 5 Comparison between the measured and predicted cross section sizes of yarn in column 7 and column8 from sample1 Area(μm 2)
Yarn
Width(μm)
Height(μm)
Measured
Predicted
RE
Measured
Predicted
RE
Measured
Predicted
RE
Weft(3K)
160827
166337
3.43%
1295
1454
12.28%
183
186
1.64%
Weft(6K)
346540
325999
5.93%
1969
1781
9.55%
305
285
6.55%
Weft(12K)
632811
634272
0.23%
2025
2265
11.85%
430
436
1.40%
Warp(3K)
163593
161173
1.48%
1100
1000
9.10%
172
161
6.40%
Table 6 Comparisons between the predicted and measured angle of warp yarns through column 7 and column 8 wefts from sample1 AN1
AN2
AN3
AN4
AN5
AN6
AN7
AN8
AN9
Measured
19.78
20.19
18.71
26.06
36.25
41.42
25.41
14.04
22.07
Predicted
15.33
16.04
18.43
27.18
39.73
40.27
27.33
13.88
18.43
Relative Error
22.50%
20.55%
1.50%
4.30%
9.60%
2.78%
7.56%
1.14%
16.49%
Figure 8.Comparison between modelled geometry and the observed microscopy image of sample 1 and sample 2through column1 to column 12 in variable thickness plate
Figure 9.A detailed comparison between the model and the real architecture in both weft view ((a) and (b)) and warp view ((c) and (d)) of a segment through column 7 and column 8 from sample 1 6.5 6.0 5.5 5.0
Thickness Lz(mm)
4.5 4.0 3.5 3.0 2.5 2.0
Measurements from Sample 1 Measurements from Sample 2 Predicted Value
1.5 1.0 0.5 0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Weft Column
Figure 10. Predicted and measured thickness of the variable thickness composite plate
15 14
Sample 1 Predicted Sample 2
13 12 11
Weft Row
10
Col 1
Col 2
0.0
2.5
Col 3
Col 4
Col 5
Col 6
Col 7
Col 8
Col 9
Col 10
Col 11 Col 12
9 8 7 6 5 4 3 2 1 0
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
32.5
35.0
Weft Yarn Centre Location in X axis (mm) Figure 11 . Measured and predicted location of weft yarn cross section center in x axis at different row and column
CONCLUSIONS An accurate description of the textile geometry is the basis for further computer analysis. The geometric model proposed in this paper is aimed at the variable thickness composite plate, which also can be used in constant composite plate of course. The modelling requires weaving processing parameters together with measurements from sample material as input data. The comparisons with two sample composite plates confirm that the presented geometric model can be successfully utilized in constant and variable thickness 3D woven composite plate generally. The results from the model containing weft yarn cross section shape and size as well as warp yarn paths agree well with the correlated real structure. The mean predicted error is about 4.34% when applying the model to constant plate. To predict the geometric details of variable thickness plate, the average discrepancy of surface weft yarn cross section sizes is 7.34% while average discrepancy of the interior is 5.14%. When predicting the warp yarn sizes, the mean error is about 5.66%. The agreement between the predicted and the sample of inner layer warp yarn path angle is striking where mean error is 6.77% while the prediction of the surface warp yarn path angle is not so satisfying where the error is 22.50% and 16.49% respectively. The predicted thickness at each weft yarn column is reasonable with the mean error at 8.50%. According to the plausible explanation that the main factor causing these discrepancies may be the compaction behavior during manufacturing process, the geometric model can be further improved by taking the surface compact effect into consideration. Even though the idealizations of the proposed geometric model in this paper lead to some over predictions such in the thickness of the plate as well as 3K and 6K weft cross section width, it still captures the majority of the details overall in real sample architectures. To conclude, a relatively efficient approach to generate a geometric model of variable thickness 3D composite plate has been presented. It could handle the various weft yarn cross sections at each layer and the resulted different warp yarn paths in the variable thickness 3D woven composite plate. After exporting the modelled geometry to a mesh-generating software, thus the geometric model could be utilized for the further mechanical analysis.
Acknowledgements This work has been supported by National Science and Technology Major Project (2017-IV-0007-0044).The authors would also like to express their gratitude to editors and reviewers for their comments and suggestions.
REFERENCES [1]
Stig F, Hallström S. A modelling framework for composites containing 3D reinforcement. Compos Struct
2012;94:2895–901. doi:10.1016/j.compstruct.2012.03.009. [2]
Ivanov DS, Lomov S V. Modelling the structure and behaviour of 2D and 3D woven composites used in aerospace
applications. Elsevier Ltd; 2014. doi:10.1016/B978-0-85709-523-7.00002-5. [3]
Peirce FT. The geometry of cloth structure. J Text Inst Trans 1937. doi:10.1080/19447023708658809.
[4]
Tan P, Tong L, Steven GP. Modelling for predicting the mechanical properties of textile composites - A review.
Compos Part A Appl Sci Manuf 1997. doi:10.1016/S1359-835X(97)00069-9. [5]
Ivanov I, Tabiei A. Three-dimensional computational micro-mechanical model for woven fabric composites. Compos
Struct 2001. doi:10.1016/S0263-8223(01)00121-0. [6]
Buchanan S, Grigorash A, Archer E, McIlhagger A, Quinn J, Stewart G. Analytical elastic stiffness model for 3D
woven orthogonal interlock composites. Compos Sci Technol 2010. doi:10.1016/j.compscitech.2010.05.019. [7]
Quinn J, McIlhagger R, McIlhagger AT. A modified system for design and analysis of 3D woven preforms. Compos.
Part A Appl. Sci. Manuf., 2003. doi:10.1016/S1359-835X(03)00084-8. [8]
Kuhn JL, Charalambides PG. Modeling of Plain Weave Fabric Composite Geometry. J Compos Mater 1999.
doi:10.1177/002199839903300301. [9]
Sheng SZ, Van Hoa S. Modeling of 3D angle interlock woven fabric composites. J Thermoplast Compos Mater 2003.
doi:10.1177/0892705703016001206. [10] Hallal A, Younes R, Fardoun F, Nehme S. Improved analytical model to predict the effective elastic properties of 2.5D interlock woven fabrics composite. Compos Struct 2012. doi:10.1016/j.compstruct.2012.03.019. [11] Salvo L, Cloetens P, Maire E, Zabler S, Blandin JJ, Buffière JY, et al. X-ray micro-tomography an attractive characterisation technique in materials science. Nucl. Instruments Methods Phys. Res. Sect. B Beam Interact. with Mater. Atoms, 2003. doi:10.1016/S0168-583X(02)01689-0. [12] Desplentere F, Lomov S V., Woerdeman DL, Verpoest I, Wevers M, Bogdanovich A. Micro-CT characterization of variability in 3D textile architecture. Compos Sci Technol 2005. doi:10.1016/j.compscitech.2005.04.008. [13] Lin H, Brown LP, Long AC. Modelling and Simulating Textile Structures Using TexGen. Adv Mater Res 2011. doi:10.4028/www.scientific.net/amr.331.44. [14] Stig
F,
Hallström
S.
Spatial
modelling
of
3D-woven
textiles.
Compos
Struct
2012;94:1495–502.
doi:10.1016/j.compstruct.2011.12.003. [15] Mountasir A, Loser M, Hoffmann G, Cherif C, Grossmann K. 3D Woven Near-Net-Shape Preforms for Composite Structures. Adv Eng Mater 2016;18:391–6. doi:10.1002/adem.201500441.
[16] Vernet N, Ruiz E, Advani S, Alms JB, Aubert M, Barburski M, et al. Experimental determination of the permeability of engineering textiles: Benchmark II. Compos Part A Appl Sci Manuf 2014. doi:10.1016/j.compositesa.2014.02.010. [17] Ouagne P, Ouahbi T, Park CH, Bréard J, Saouab A. Continuous measurement of fiber reinforcement permeability in the
thickness
direction:
Experimental
technique
and
validation.
Compos
Part
B
Eng
2013.
doi:10.1016/j.compositesb.2012.06.007. [18] Yan S, Zeng X, Long A. Meso-scale modelling of 3D woven composite T-joints with weave variations. Compos Sci Technol 2019;171:171–9. doi:10.1016/j.compscitech.2018.12.024. [19] Vernet N, Trochu F. Analysis and modeling of 3D Interlock fabric compaction behavior. Compos Part A Appl Sci Manuf 2016;80:182–93. doi:10.1016/j.compositesa.2015.10.024. [20] Badel P, Sallé EV, Maire E, Boisse P. Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale. Int J Mater Form 2009;2:189–92. doi:10.1007/s12289-009-0572-z. [21] Gommer F, Brown LP, Wedgwood KCA. Analytical method using gamma functions for determining areas of power elliptical
shapes
for
use
in
geometrical
doi:10.1016/j.compositesa.2015.11.031.
textile
models.
Compos
Part
A
Appl
Sci
Manuf
2016;81:222–4.
Yu Zhou：Methodology, Software, Investigation, Writing Original Draft, Visualization. Weidong WEN：Conceptualization, Resources, Data Curation, Supervision. Haitao CUI：Validation, Formal analysis, Writing - Review & Editing.
We have no conflicts of interest to declare.