Hindawi Publishing Corporation Journal of Nanomaterials Volume 2014, Article ID 836386, 7 pages http://dx.doi.org/10.1155/2014/836386

Research Article Elastic Properties of the Fabric Liner and Their Influence on the Wear Depth of the Spherical Plain

Xuejin Shen, Pandong Gao, Zhaolei Liu, and Xiaoyang Chen

Department of Mechanical Automation Engineering, Shanghai University, Shanghai 200072, China

Correspondence should be addressed to Xuejin Shen; [email protected]

Received 20 December 2013; Accepted 3 February 2014; Published 17 March 2014

Academic Editor: Yongsheng Zhang

Copyright © 2014 Xuejin Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The major failure mechanism of typical spherical plain bearings with self- is the wear of the woven fabric liner, which is an orthotropic composite of different elastic properties in different directions. The elastic properties of the liner are required for studying the tribological properties of the spherical plain bearings. This paper aims to develop an elastic property analysis model suitable for three commonly used fabric liners through a theoretical analysis of the elastic properties in order to obtain the parameter expression of the compliance matrix. The influence of the elastic properties on the wear depth of the spherical plain bearings is further investigated. Suggestions are made for the optimal design of the spherical plain bearings based on wear reduction.

1. Introduction and then to validate the analysis model in an experimental way and to investigate the influence of the elastic prop- Spherical with self-lubricating is a kind of erties on the wear depth of the spherical plain bearings. sliding bearings with low-speed multiaxial rotary oscillating Wewishtoproposesomesuggestionsforlinerstructural motion pattern. Widely used in mining and metallurgy, optimizing. aerospace, tank cannon gun system, and so forth [1, 2], the typical spherical plain bearing with self-lubricating consists of an inner ring with an outside spherical surface and an outer 2. Theoretical Analysis of Elastic ring which has an inner sphere surface with a woven fabric Properties of the Liner liner bonded to it, and its main failure form is the wear of the woven fabric liner. The current studies on woven fabric 2.1. Geometric Model of the Woven Fabric Liner. The woven liner of spherical plain bearing with self-lubricating largely fabric liner is mainly made up of matrix and reinforced fibers, concern the aspect of and wear experiments [3, 4], and the fiber distribution of the typical woven fabric liners while the elastic properties are obtained mainly by tensile is shown in Figure 1.AglobalCartesiancoordinatesystem (𝑋, 𝑌,𝑍) is set, where axis 𝑋,whoseincludedanglewith test experiments [5, 6] or numerical experiments [7–9]. Most 𝛽 of the experiments focus on the particular structure of the warp direction is , is along the horizontal direction, and 𝑌 𝑍 woven fabric liners [10–14]. Obtaining the elastic properties axes , arealongthedirectionsoffillandthicknessofthe of the liner is essential in studying the mechanical properties woven fabric liners, respectively. The angle between the warp 𝛾 ofsphericalplainbearings.Thestudyontheinfluenceofthe and fill directions is , the fibers in the warp direction are elastic properties of woven fabric liners on the wear depth of the mixture of (PTFE) and Nomex, the spherical plain bearings is almost blank. The purposes and only Nomex fibres are in fill direction. The Nomex fiber of this paper are to create an elastic properties analysis is used to improve the strength of the liner materials and model which is suitable for the three commonly used woven bonding properties, while the PTFE fiber is mainly used to fabric liners, to obtain the elastic properties of the plain and improve friction and wear properties of materials. The matrix stain liners using the method of the tensile test experiment, of the woven fabric liner is phenolic resin. 2 Journal of Nanomaterials

Fill

Fill Warp

Y Warp

𝛾 𝛽 X

(a) Plain liner (b) Twill liner

Fill

Warp

(c) Stain liner

Figure 1: Fiber distribution of the woven fabric liner.

Take the black box parts (in Figure 1)ofthelinersfor The radius, 𝑅𝑤, the inner angle, 𝜑𝑤,andthecross- theanalysisunitsandmakethefollowingassumptionsofthe sectional area, 𝐴𝑤, can be expressed as follows: microstructure of the fabric liners. 𝑤2 𝑑 2𝑤 𝑑 (1) The cross section of the yarn is lenticular shape [15]. 𝑅 = 𝑤 + 𝑤 ,𝜑=2 ( 𝑤 𝑤 ), 𝑤 𝑤 arcsin 2 2 Its shape and geometric dimensions are shown in Figure 2. 4𝑑𝑤 4 𝑑𝑤 +𝑤𝑤 𝑤 𝑑 (2) The width 𝑤 and the thickness 𝑤 of the warp yarn after 𝑑 𝑤 cutting along 𝑌 direction can be obtained as 𝐴 =𝑅2 𝜑 −𝑤 𝑅 + 𝑤 𝑤 . 𝑤 𝑤 𝑤 𝑤 𝑤 2 𝑤 𝑤𝑤 = ,𝑑𝑤 =𝑑, (1) sin 𝛾 (2)Warpandfillyarnshavethesamefiberdirections. The global coordinate system of the liner is (𝑋, 𝑌,𝑍),the where 𝑤 and 𝑑 represent the width and the thickness of coordinate system of the plane of the fiber yarn trace is local theyarninthecrosssection,respectively.Thesubscript𝑤 coordinate system (𝑢, V,𝑤),andthecoordinatesystemofthe indicates the warp yarn, and the subscript 𝑓 will be used in fiber yarn trace is local coordinate system (1, 2, 3). The local the fill yarn (an exception if it is labeled especially). The warp coordinate system (𝑢, V,𝑤)is obtained by 𝛽 angle rotation of yarns of the liner are taken as an example, to study here, and the global coordinate system (𝑋, 𝑌,𝑍) around 𝑍 axis, and the the fill yarns could be studied in the similar way. coordinate axes of (1, 2, 3) is parallel to the axes of (𝑢, V,𝑤) Journal of Nanomaterials 3

w yarn, the subscript 𝑤 refers to the warp direction, and the meaning of the subscript 𝑠/𝑤 below is the same as here. The volumes of the warp yarns in the basic study cell can be obtained as 𝑛 d 𝑉𝑠/𝑤 = ∑ 𝐴𝑤𝑖𝐿(𝑠/𝑤)𝑖,𝑛=2,3,4, (8) 𝑖=1 𝑛=2,3 𝜑 where , and 4 correspond to the plain, twill, and stain woven fabric liners, respectively. The meaning of the 𝑛 below R isthesameashere;𝑖=1represents the first warp yarn. The length of the basic study cell in the warp direction can Figure 2: Cross section of the fiber yarn. be written as

𝐿𝑤 =𝑛𝐿𝑠𝑤,𝑛=2,3,4. (9) Z, w The volume of the basic study cell is

𝑉=𝐿𝑤𝐿𝑓ℎ, (10) v 1 3 𝐿 𝜃 where 𝑓 indicates the length of the basic study cell in the fill direction, and ℎ represents the thickness of the study cell. Y ℎ=𝑡 u Since the thickness of the matrix is very thin, we assume . 𝛽 𝛾 The volume fraction of the warp yarns in the basic study 𝛽 cellcanbeobtainedas X 𝑉𝑠/𝑤 V = . (11) Figure 3: Coordinate systems of warp yarn. 𝑠/𝑤 𝑉 The fiber volume fraction of the warp yarns can be written as after rotating 𝜃 angle around V axis. The position relationship of the coordinate systems is shown in Figure 3. 𝑘𝐴 𝑓/𝑤 𝜓𝑤 = , (12) Fibers directions are divided into the straight parts and 𝐴𝑤 the curve parts shown in Figure 4. This model applies to plain, 𝑘 twill, and stain fabric liners. where indicates the number of the single fiber in the yarn, 𝐴𝑓/𝑤 represents the cross-sectional area of the single fiber, The thickness of two layers of fiber yarns can be written 𝑓 𝑤 𝑓/𝑤 as and subscripts and in refer to the single fiber and warp direction, respectively. The meaning of the subscript 𝑓/𝑤 𝑡=𝑑𝑤 +𝑑𝑓, (3) below is the same as here. Accordingly, the fiber volume fraction of the warp direc- where 𝑑𝑓 indicates the thickness of the fill yarn after cutting tion in the basic study cell can be calculated by along warp direction. 𝑛 𝑘𝐴 𝑓/𝑤 ∑𝑖=1 𝐿(𝑠/𝑤)𝑖 The bend radius of the yarn can be obtained as ]𝑓/𝑤 =𝜓𝑤]𝑠/𝑤 = , 𝑛=2,3,4. (13) 𝐿𝑤𝐿𝑓ℎ 𝑑𝑤 𝑟𝑤 =𝑅𝑓 + . (4) 2 2.2. Elastic Properties of the Woven Fabric Liner. The chamis model [16]isusedtocalculatetheelasticconstantsofthe Yarn-to-yarn distance is straight part of the yarn. The expressions are as follows:

2 2 𝑓 𝑚 𝑓 𝑚 𝐿𝑠𝑤 =2√𝑟𝑤 −𝑅𝑓. (5) 𝐸11 =𝑉𝑓𝐸11 +𝑉𝑚𝐸 , ]12 =𝑉𝑓]12 +𝑉𝑚] , 𝐸𝑚 The length of the curve part of the yarn is 𝐸22 = , 𝑚 𝑓 1−√𝑉𝑓 (1 − 𝐸 /𝐸22) 𝐿𝑐𝑤 =𝑟𝑤𝜑𝑓. (6) 𝐺𝑚 (14) 𝐺12 = , The length of the warp yarn can be calculated by 𝑚 𝑓 1−√𝑉𝑓 (1 − 𝐺 /𝐺12) 𝐿𝑠/𝑤 =𝑚𝐿𝑠𝑤 +2𝐿𝑐𝑤,𝑚=0,1,2, (7) 𝐺𝑚 𝐺23 = , where 𝑚 = 0, 1, 2 represents plain, twill, and stain woven 𝑚 𝑓 1−√𝑉𝑓 (1 − 𝐺 /𝐺 ) fabric liner, respectively. The subscript 𝑠 refers to the fiber 23 4 Journal of Nanomaterials

where 1 is radial direction along the direction of the fiber axis; and 2-3 plane shear modulus of the fiber, respectively; 𝑉𝑓 2, 3 are traverse directions lying in the plane perpendicular to indicates the fiber volume fraction of the yarn, namely, 𝜓𝑤 the fiber; 𝐸11 and 𝐸22 aretheelasticmodulioffiberyarnalong in the expression (12). the direction of 1 and 2, respectively; ]12 is Poisson’s ratio Thecurvepartoftheyarncanberegardedastheassem- ofthefiberyarnofthe1-2plane;𝐺12 and 𝐺23 are the shear blage of a number of infinitesimal straight yarn segments. The moduli of fiber yarn of the 1-2 and 2-3 planes, respectively; compliance matrix of the straight part of the warp yarn in the 𝑚 𝑚 𝑚 𝐸 , ] ,and𝐺 are the elastic modulus, Poisson’s ratio, local coordinate system (1, 2, 3) and the transition matrix [16] 𝐸𝑓 𝐸𝑓 ]𝑓 and shear modulus of the matrix, respectively; 11, 22, 12, between any two coordinate systems (𝑥1,𝑥2,𝑥3) and (𝑥,𝑦,𝑧) 𝑓 𝑓 𝐺12,and𝐺23 are the radial elastic modulus, traverse elastic can be written as modulus, in-plane Poisson’s ratio, in-plane shear modulus,

2 2 2 𝑙1 𝑙2 𝑙3 𝑙2𝑙3 𝑙3𝑙1 𝑙1𝑙2 𝑆11 𝑆12 𝑆12 000 [ ] [ ] [ 𝑚2 𝑚2 𝑚2 𝑚 𝑚 𝑚 𝑚 𝑚 𝑚 ] [𝑆12 𝑆22 𝑆23 000] [ 1 2 3 2 3 3 1 1 2 ] [𝑆 𝑆 𝑆 000] [ 2 2 2 ] 𝑠 [ 12 23 22 ] [ 𝑛1 𝑛2 𝑛3 𝑛2𝑛3 𝑛3𝑛1 𝑛1𝑛2 ] 𝑆𝑖𝑗 = [ ] ,𝑇=[ ] , (15) [ 000𝑆44 00] [ ] [ ] [2𝑚1𝑛1 2𝑚2𝑛2 2𝑚3𝑛3 𝑚2𝑛3 +𝑚3𝑛2 𝑚1𝑛3 +𝑚3𝑛1 𝑚1𝑛2 +𝑚2𝑛1] [ 0000𝑆55 0 ] [ ] [ 2𝑛1𝑙1 2𝑛2𝑙2 2𝑛3𝑙3 𝑙2𝑛3 +𝑙3𝑛2 𝑙1𝑛3 +𝑛1𝑙3 𝑙1𝑛2 +𝑙2𝑛1 ] [ 00000𝑆55] [2𝑙1𝑚1 2𝑙2𝑚2 2𝑙3𝑚3 𝑙2𝑚3 +𝑙3𝑚2 𝑙1𝑚3 +𝑙3𝑚1 𝑙1𝑚2 +𝑙2𝑚1 ]

𝑠 where subscripts 𝑖 and 𝑗 in 𝑆𝑖𝑗 refer to the row and column, and fill yarns have been obtained, their stiffness matrices can respectively. For the transition matrix 𝑇, also be obtained by inversing their compliance matrices. Then according to the volume fraction of the fibers in the warp 𝑙𝑖 = cos (𝑥𝑖,𝑥),𝑖 𝑚 = cos (𝑥𝑖,𝑦),𝑖 𝑛 = cos (𝑥𝑖,𝑧) and fill directions, the stiffness matrices of the woven fabric 𝑖 = 1, 2, 3. liners of spherical plain bearing with self-lubricating can be (16) calculated by 𝐶𝑐 = ] 𝐶𝑤 + ] 𝐶𝑓 +(1−] − ] )𝐶𝑚, The compliance matrix of the straight part of the warp 𝑖𝑗 𝑓/𝑤 𝑖𝑗 𝑓/𝑓 𝑖𝑗 𝑓/𝑤 𝑓/𝑓 𝑖𝑗 (19) yarn in the local coordinate system (1, 2, and 3) can be 𝐶𝑤 𝐶𝑓 𝐶𝑚 completely determined by the chamis model, based on the where 𝑖𝑗 , 𝑖𝑗 ,and 𝑖𝑗 indicate the stiffness matrices of the relationship between compliance matrix and elastic con- warp yarn, fill yarn, and matrix, respectively. ]𝑓/𝑤 and ]𝑓/𝑓 stants. The transition matrix 𝑇𝑤1 between the local coordinate refer to fiber volume fraction of the warp and fill direction, system (1, 2, and 3) and (𝑢, ],𝑤) and 𝑇𝑤2 between the local respectively. 𝑆𝑐 coordinate system (𝑢, ],𝑤)and the global coordinate system The compliance matrix 𝑖𝑗 of the woven fabric liner can be 𝐶𝑐 (𝑋, 𝑌,𝑍) can be obtained according to transition matrix 𝑇. obtained by inversing stiffness matrix 𝑖𝑗 , and the nine elastic 𝑐 Then the compliance matrix of the warp yarn in the global constants of the liner can be written as 𝐸11 =1/𝑆11, 𝐸22 = 𝑐 𝑐 𝑐 𝑐 𝑐 𝑐 coordinate system can be derived 1/𝑆22, 𝐸33 =1/𝑆33, ]12 =−𝑆12/𝑆11, ]23 =−𝑆23/𝑆22, ]13 = 𝑐 𝑐 𝑐 𝑐 𝑐 −𝑆13/𝑆11, 𝐺12 =1/𝑆66, 𝐺13 =1/𝑆55,and𝐺23 =1/𝑆44. 𝑠𝑤 𝑠 𝑇 𝑐𝐺 𝑠 𝑇 𝑇 𝑆𝑖𝑗 =𝑇𝑤2𝑆𝑖𝑗 𝑇𝑤2,𝑆𝑖𝑗 =𝑇𝑤2𝑇𝑤1𝑆𝑖𝑗 𝑇𝑤1𝑇𝑤2,

𝜃 (17) 3. Elastic Properties Test and Validation of 𝑐𝑤 1 𝑐𝐺 𝑆𝑖𝑗 = ∫ 𝑆𝑖𝑗 𝑑𝜃, the Analysis Model 𝜃 0 Experiment specimens include two types of woven fabric 𝑆𝑠𝑤 𝑆𝑐𝐺 𝑆𝑐𝑤 where 𝑖𝑗 , 𝑖𝑗 ,and 𝑖𝑗 signify the compliance matrices of the liners: one is the plain woven fabric liner with PTFE fibers in straight part, curve part yarn segment, and curve part yarn in both the warp and fill directions; the other is the stain woven the global coordinate system (𝑋, 𝑌,𝑍),respectively. fabric liner with PTFE and Nomex fibers in the warp direc- The compliance matrix of the warp yarn can be written as tion and Nomex fiber in the fill direction. The experiment is implemented by Zwick/Roell (BZ2.5/TS1S) test machine. 𝑆𝑤 =𝜆 𝑆𝑠𝑤 +𝜆 𝑆𝑐𝑤, 𝑖𝑗 𝑠𝑤 𝑖𝑗 𝑐𝑤 𝑖𝑗 (18) During the experiment, the displacement control mode is 𝜆 =2𝐿 /𝐿 𝜆 =1−𝜆 used, and the loading speed is 0.2 mm/min. A CCD camera where 𝑐𝑤 𝐶𝑤 𝑠/𝑤 and 𝑠𝑤 𝑐𝑤 indicate is placed at the normal of the specimen surface to record the length fractions of the curve part and the straight part, images while loading. Precision for strain measurement is respectively. 50 𝜇𝜀. The sequential images are subjected to DIC (digital 𝑆𝑓 The compliance matrix of the fill yarn 𝑖𝑗 can be obtained image correlation) analysis, and then we can acquire the in a similar way. Since the compliance matrices of the warp elastic properties of the liners through the corresponding Journal of Nanomaterials 5

w

Ls Lcw w

3 1

𝜃 t h

Warp Fill

𝜑f Curve u r R Straight part part w f

Figure 4: Geometric model of fibers directions.

Table 1: Geometric parameters of woven fabric composites.

∘ Thickness of yarn (mm)𝑑 ( ) Width of yarn (mm) (𝑤) Lamina thickness Inner angle ( ) Fiber volume fraction of Liner types 𝜑 𝜓 Warp Fill Warp Fill (mm) (h) ( 𝑤) the yarn ( 𝑤) Plain/stain liners 0.19 0.51 0.38 90 0.70

Table 2: Material elastic properties. 180

Material 𝐸11 (GPa) 𝐸22 (GPa) 𝐺12 (GPa) 𝐺13 (GPa) ]12 160 0.75 0.75 0.28 0.28 0.30 PTFE fiber m) 𝜇 Nomex fiber 6.70 6.70 2.69 2.69 0.23 140 Phenolic resin 2.88 2.88 1.01 1.01 0.42 matrix 120

100 Maximum wear depth ( depth wear Maximum stress-strain curves. The structural and material parameters 80 are shown in Tables 1 and 2 [17]. The elastic properties of plain and stain liners can be calculated by applying the 60 structural and material parameters to Sections 2.1 and 2.2, 0 100 200 300 400 500 respectively. The comparison between the computational Elastic modulus E1 (GPa) and experimental results is listed in Table 3.Thecoordinate Figure 5: Influencing of elastic modulus 𝐸1 on the maximum wear system (𝑋, 𝑌,𝑍) is replaced by (1, 2, and 3) to align with the depth. conventional means of expression. Table 3 shows that the calculation results of plain woven fabric liner are in good agreement with the experimental data. Relative error of the warp elastic modulus of the stain woven Figure 5 to Figure 8 are the curves of influencing of fabric liner is a little bit bigger, but within the scope of the the elastic properties on the maximum wear depth of the engineering allowable error. spherical plain bearing after 25000 cycles of oscillating. Figures 5 and 6 show that the elastic moduli of the liner have a tremendous impact on the maximum wear depth of 4. Influence of Elastic Properties of the Liner the spherical plain bearing. The elastic moduli 𝐸1 (𝐸2)and𝐸3 on Wear Depth of Spherical Plain Bearing have an opposite influence on the maximal wear depth when they vary from 0 GPa to 65 GPa. With 𝐸1 (𝐸2)increasing,the Wear simulation was realized by commercial finite element maximal wear depth is firstly increased by 96.34% and then software ABAQUS. Wear simulation program is designed decreased by 30.61%. With the increase of 𝐸3, the maximal using Python language and the elastic properties of the liner wear depth is firstly decreased by 49.31% and then increased ischangedtoderivethevariationtrendofthemaximum by 28.33%. When the elastic moduli are over 65 GPa, the wear depth of spherical plain bearing after 25000 cycles of maximal wear depth increases firstly and then decreases and oscillating. increases with the increase of 𝐸1 (𝐸2)and𝐸3,respectively. 6 Journal of Nanomaterials

Table 3: Comparison between the computational and experimental results.

Liner types Elastic properties Computational results Test results Error

𝐸11 (GPa) 1.95 1.98 1.35%

Plain liner 𝐸22 (GPa) 1.95 1.98 1.35%

𝐺12 (GPa) 0.66 0.63 4.39%

𝐸11 (GPa) 3.29 2.96 11.15%

Stain liner 𝐸22 (GPa) 3.36 3.23 4.02%

𝐺12 (GPa) 1.19 1.09 9.17%

140 135

130 130 125 m) 120 m) 𝜇 𝜇 120 110 115 100 110 90 105

80 100 Maximum wear depth ( depth wear Maximum Maximum wear depth ( depth wear Maximum 95 70 90 60 0 10 20 30 40 50 60 70 80 90 85 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Elastic modulus E3 (GPa) Poisson’s ratios

Figure 6: Influencing of elastic modulus 𝐸3 on the maximum wear v12 depth. v13, v23

Figure 8: Influencing of Poisson’s ratios on the maximum wear 135 depth. 130 125

m) 120 𝐺 𝜇 modulus 12 ranges from 35 GPa to 60 GPa. The maximal 115 wear depth of the bearing almost keeps the same and is under 110 the minimum state when 𝐺13 and 𝐺23 change from 20 GPa to 105 80 GPa. 100 Figure 8 suggests that the maximum wear depth will 95 increase and decrease when the Poisson’s ratio ]12 increases 90 in the area of 0.1 to 0.15 or 0.3 to 0.35 and 0.15 to 0.3 or 0.35 Maximum wear depth ( depth wear Maximum 85 to 0.4, respectively. The maximal wear depth will be minimal 80 when Poisson’sratio ]12 is 0.3. Poisson’s ratios ]13 and ]23 have 75 no effect on the maximal wear depth. 0 10 20 30 40 50 60 70 80 90 The wear depth of the spherical plain bearing at prede- Shear moduli (GPa) termined conditions cannot be greater than allowed limit

G12 0.114 mm based on the standard MIL-B-81820. Thus the G13,G23 elastic moduli 𝐸1 and 𝐸2 should change from 0 GPa to 40 GPa or 56 GPa to 113 GPa, 𝐸3 should be greater than 12 GPa, the Figure7:Influencingofshearmodulionthemaximumweardepth. shear modulus 𝐺12 should change from 0 GPa to 46 GPa or 54 GPa to 80 GPa, and Poisson’sratios ]12 should change from 0.16 to 0.34 or 0.36 to 0.45. Because the elastic constants 𝐺13, 𝐺 ] ] The main movement pattern of the spherical plain bear- 23, 13,and 23, have a little or no influence on the maximal ing is multiaxial rotary oscillating motion in a lower speed, so wear depth, they can be ignored. thereisanimpactofthein-planeshearperformanceonwear depth of the spherical plain bearing, which cannot be ignored. 5. Conclusions As shown in Figure 7,withtheshearmodulus𝐺12 increasing, the maximal wear depth firstly decreases, then increases, and An elastic property analysis model, which is suitable for then decreases again. And it varies greatly when the shear three commonly used woven fabric liners, is described in Journal of Nanomaterials 7 this paper. The influence of the elastic properties on the wear References depth of the spherical plain bearings is further investigated. The following is a summary. [1]B.C.Kim,D.C.Park,H.S.Kim,andD.G.Lee,“Development of composite spherical bearing,” Composite Structures,vol.75, no. 1–4, pp. 231–240, 2006. (1) A general elastic properties analysis model adjusting [2] M. F. Fleszar, “Thermal behaviour of teflon/phenolic liners in to the plain, twill, and stain liners was built. self-lubricating bearings,” Journal of Thermal Analysis,vol.49, no. 1, pp. 219–226, 1997. (2) For the plain woven fabric liner, the computational [3] D. Xiang, Z. Yao, and J. 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For the stain woven fabric liner, [4]Q.Guo,Y.-F.Song,H.-B.Qiao,andW.-L.Luo,“Thefriction the relative error of the warp elastic modulus is 11.15%, and wear properties of the spherical plain bearings with self- which is a little larger but within the scope of the lubricating composite linear in oscillatory movement,” Journal engineering allowable error, and the relative errors of of Wuhan University of Technology,vol.19,pp.86–91,2004. fill elastic and in-plane shear modulus are 4.02% and [5]B.j.Pang,S.Y.Du,J.C.Han,X.D.He,andY.Yan,“Experimen- 9.17%, respectively. tal investigation of three-dimensional four-directional braided carbon/epoxy composites,” Acta Materiae Compositae Sinica, (3) The influence of the liner elastic properties onthe vol. 16, no. 4, pp. 136–141, 1999. wear of the spherical plain bearing was analyzed [6] C.-K. Yang, “Research of mechanical properties of 3D braided basedonthefiniteelementmethod.Theresultsshow composite materials,” Journal of Materials Engineering,no.7,pp. that the liner elastic properties, which are the elastic 33–36, 2002. [7]Z.Xia,C.Zhou,Q.Yong,andX.Wang,“Onselectionof moduli 𝐸1 (𝐸2)and𝐸3,theshearmodulus𝐺12,and repeated unit cell model and application of unified periodic Poisson’s ratio ]12, have the greater impacts on the boundary conditions in micro-mechanical analysis of compos- maximal wear depth of the woven fabric liner of the ites,” International Journal of Solids and Structures,vol.43,no. spherical plain bearing. The results also show that 2,pp.266–278,2006. the shear moduli 𝐺13 (𝐺23) have a small effect and ] V [8] K. Xu and X. Xu, “Prediction of elastic constants and simulation Poisson’s ratios 13 ( 23)havenoeffectontheliner’s of stress field of 3D braided composites based on the finite wear. element method,” Acta Materiae Compositae Sinica,vol.24,no. 3,pp.178–185,2007. (4) For the woven fabric liner of the spherical plain [9] W.-Y. Zhang, Z.-H. Yao, X.-F. Yao, and Y.-P. Cao, “Numerical bearing based on the standard MIL-B-81820, the model of woven fabric composites,” Engineering Mechanics,vol. elastic moduli 𝐸1, 𝐸2 should change from 0 GPa to 21, no. 3, pp. 55–60, 2004. 𝐸 40 GPa or 56 GPa to 113 GPa, 3 should be greater [10] P. Vandeurzen, J. Ivens, and I. Verpoest, “A three-dimensional than 12 GPa, the shear modulus 𝐺12 should change micromechanical analysis of woven-fabric composites: I. Geo- from 0GPa to 46GPa or 54GPa to 80GPa, and metric analysis,” Composites Science and Technology,vol.56,no. Poisson’s ratios V12 should change from 0.16 to 0.34 11, pp. 1303–1315, 1996. or0.36to0.45,soastomeetthewearrequirement [11] J. L. Kuhn and P. G. Charalambides, “Modeling of plain weave ofthebearing.Theelasticconstants𝐺13, 𝐺23, ]13, fabric composite geometry,” Journal of Composite Materials,vol. and ]23 havealittleornoinfluenceonthemaximal 33,no.3,pp.188–220,1999. wear depth; therefore they can be ignored. Above [12] T. C. Lim, “Elastic stiffness of three-phase composites by the suggestions could be used for the optimal design of generalized mechanics-of-materials (GMM) approach,” Journal the spherical plain bearings based on wear reduction, of Thermoplastic Composite Materials,vol.15,no.2,pp.155–168, soastoselectthesuitablegeometricandmaterial 2002. parameters of the woven fabric composite. [13] R. Wang, J.-K. Wang, and L. Wu, “Prediction for elastic properties of plain weave fabric composites,” Acta Materiae Compositae Sinica,vol.19,no.1,pp.90–94,2002. Conflict of Interests [14] L. Wang and Y. Yan, “Micro analysis and experimental study of the elastic properties of braided composites structure,” Acta The authors declare that there is no conflict of interests Materiae Compositae Sinica,vol.21,no.4,pp.152–156,2004. regarding the publication of this paper. [15] S.-K. Lee, J.-H. Byun, and S. H. Hong, “Effect of fiber geometry on the elastic constants of the plain woven fabric reinforced alu- Acknowledgments minum matrix composites,” Materials Science and Engineering A,vol.347,no.1-2,pp.346–358,2003. This work is supported by the COSTIND of China (D.50- [16] Z. M. Huang, An Introduction to Micromechanics of Composites, 0109-11-002) and High and New Engineering Program of The Press of Science, 2004. Shanghai (D.51-0109-09-001). The authors wish to thank [17] L. Deters, F. Mueller, and M. Berger, “Self-lubricating dry ProfessorQ.JaneWangatNorthwesternUniversityofUSA rubbing bearings-fundamentals and methods of calculation,” for valuable discussions. 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