190 3-D textile reinforcements in composite materials three parameters,i.e.wale density,W,course density,C,and the yarn diam- eter,d. In the fabric plane,we set the global rectangular axis Ox to be parallel to the wale direction and Oy to the course direction.Suppose that the OQ portion of the loop has a center at C with a total angle o,i.e.OCQ=o.'ad' denotes the radius of projection of the loop,i.e.the length between O and C,where a is a constant.Q is the point at which the central axis of this loop joins the central axis of the loop with a center F.H and J are the points at which the yarns of adjacent loops (loops with centers at C and B)cross over.The angles OCB=w and HCB=0.Let P be any point on the central axis of the loop and the angle of the projection of the loop portion from O to P be 0,OCP =0.The co-ordinates of P are given by x=ad1-cosθ) y=adsin [6.1] 、 -叫】 where h is a constant used for representing maximum height hd(at Q)of 20 the central axis above the plane of the fabric.The parameters a,h and o in (6.1)are determined from the following formulae: 1 4= 4Wdsino [6.2] :ssauppy dl p=π+sin cwu-coP C'd -tan- [wu-cd) [6.3] -s)sin [6.4 (2a 2a-1 sin [6.5] [6.6可 The yarn diameter d can be expressed in terms of the linear density(D,) of the yarn and packing fraction(K)of fibers in the yarn as 2D, d=3110xpK ×10-2(cm) [6.7] where pr is the density of fiber (g/cm)
three parameters, i.e. wale density, W, course density, C, and the yarn diameter, d. In the fabric plane, we set the global rectangular axis Ox to be parallel to the wale direction and Oy to the course direction. Suppose that the OQ portion of the loop has a center at C with a total angle j, i.e. OCQ = j. ‘ad’ denotes the radius of projection of the loop, i.e. the length between O and C, where a is a constant. Q is the point at which the central axis of this loop joins the central axis of the loop with a center F. H and J are the points at which the yarns of adjacent loops (loops with centers at C and B) cross over. The angles OCB = y and HCB = f. Let P be any point on the central axis of the loop and the angle of the projection of the loop portion from O to P be q, OCP = q. The co-ordinates of P are given by [6.1] where h is a constant used for representing maximum height hd (at Q) of the central axis above the plane of the fabric. The parameters a, h and j in (6.1) are determined from the following formulae: [6.2] [6.3] [6.4] [6.5] [6.6] The yarn diameter d can be expressed in terms of the linear density (Dy) of the yarn and packing fraction (K) of fibers in the yarn as [6.7] where rf is the density of fiber (g/cm3 ). d D K y = ¥ ( ) - 2 3 10 10 2 prf cm f = Ê - Ë ˆ ¯ - cos 1 2 1 2 a a y j = - Ê Ë ˆ ¯ - sin sin 1 2 2 1 a a h = Ê Ë ˆ ¯ Ê Ë ˆ ¯ È Î Í ˘ ˚ ˙ - sin sin p y j p f j 1 j p = + [ ] + - ( ) Ê Ë Á Á ˆ ¯ ˜ ˜ - ( ) - È Î Í ˘ ˚ ˙ - - sin tan 1 2 2 2 22 2 1 2 1 2 2 1 1 C d C W Cd C W Cd a Wd = 1 4 sinj z hd = - Ê Ë ˆ ¯ È Î Í ˘ ˚ ˙ 2 1 cos p q j y ad = sinq x ad = - ( ) 1 cosq 190 3-D textile reinforcements in composite materials RIC6 7/10/99 8:12 PM Page 190 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:57 AM IP Address: 158.132.122.9
Knitted fabric composites 191 From Fig.6.7,it is clear that the orientation of the yarn in a knit loop (MNOQP)can be determined from knowing the orientation of the yarn in the portion OQ.We may assume that the OQ portion is an assemblage of a series of straight segments.Let (x-,y-1,n-)and (xn.y,m)be the co- ordinates of start and end points of the (n-1)th yarn segment(see Fig. 6.8).The orientation of the segment in 3-D co-ordinates can be specified using two angles,0,and 0.,where 0,is the angle between the z-axis and the yarn segment and 0,the angle between the x-axis and the projected straight line of the segment on the x-y plane.These two angles are important in our geometric analysis.They are determined as [6.8] [6.9] WV LS:Og:ZI I IOZ 'ZZ Anur 'Aupines Z 3 (X y) Y (Xn-1 yn-1 Z-D e. X 6.8 Representation of a segment of yarn
From Fig. 6.7, it is clear that the orientation of the yarn in a knit loop (MNOQP) can be determined from knowing the orientation of the yarn in the portion OQ. We may assume that the OQ portion is an assemblage of a series of straight segments. Let (xn-1, yn-1, zn-1) and (xn,yn,zn) be the coordinates of start and end points of the (n - 1)th yarn segment (see Fig. 6.8). The orientation of the segment in 3-D co-ordinates can be specified using two angles, qx and qz, where qz is the angle between the z-axis and the yarn segment and qx the angle between the x-axis and the projected straight line of the segment on the x–y plane. These two angles are important in our geometric analysis. They are determined as [6.8] qz [6.9] nn nn n n tg xx yy z z = ( ) - + - ( ) - È Î Í Í ˘ ˚ ˙ ˙ - - - - 1 1 2 1 2 1 qx n n n n tg y y x x = - - Ê Ë ˆ ¯ - - - 1 1 1 Knitted fabric composites 191 6.8 Representation of a segment of yarn. RIC6 7/10/99 8:12 PM Page 191 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:57 AM IP Address: 158.132.122.9
192 3-D textile reinforcements in composite materials Equations 6.8 and 6.9 imply that only relative co-ordinates of the yarn are important.Therefore,we can replace the unit cell shown in Fig.6.7 with the unit cell in Fig.6.9(a).The unit cell in Fig.6.9(a)can be further divided into four identical sub-cells.Each sub-cell consists of two impregnated yarns which cross over each other.This sub-cell is known as the crossover model [16]and is represented in Fig.6.9(b).Using the crossover model,a unit cell can be constructed.Repeating the unit cell in the fabric plane will obviously reproduce the complete plain knitted fabric structure.We thus only need to investigate the crossover model which is taken as a representative volume.The co-ordinates of the first yarn in the model are given by Equa- tion 6.1 with 0s0s o.To determine the co-ordinates of the second yarn easily,we choose its starting point to be nearer to the end point of the first yarn.The co-ordinates of the points on the second yarn are thus given by x2m=2ad-2wIg(w) 1 2nd 1 wo'ssaudmaupaypoow/:dny 2W and= Xn2nd =X2nd-Xnlst ynand=yi2nd-yalst znm2md=zm1n≥2,3，. 三3ngA 6.4.3 Estimation of fiber volume fraction Based on the above-mentioned geometric model,the fiber volume fraction of the knitted fabric composite is given by [11]: V =D,LCWx10 [6.10] 9p:At where ng is the number of plies of the fabric in the composite,t is the thick- ness of the composite measured in centimeters,A is the planar area over which W and C are measured,and Ls is the length of yarn in one loop of the unit cell which can be represented approximately by Ls =4(ad) [6.11] Let us apply Equation 6.10 to the knitted glass fiber fabric reinforced epoxy composites described in Section 6.3.Knitted fabrics with W=2 loops/cm and C=2.5 loops/cm,are made using 1600 denier (D)glass fiber yarns (fiber density pr=2.54g/cm).Composites with single and four plies of
Equations 6.8 and 6.9 imply that only relative co-ordinates of the yarn are important. Therefore, we can replace the unit cell shown in Fig. 6.7 with the unit cell in Fig. 6.9(a). The unit cell in Fig. 6.9(a) can be further divided into four identical sub-cells. Each sub-cell consists of two impregnated yarns which cross over each other. This sub-cell is known as the crossover model [16] and is represented in Fig. 6.9(b). Using the crossover model, a unit cell can be constructed. Repeating the unit cell in the fabric plane will obviously reproduce the complete plain knitted fabric structure. We thus only need to investigate the crossover model which is taken as a representative volume. The co-ordinates of the first yarn in the model are given by Equation 6.1 with 0 £q£j. To determine the co-ordinates of the second yarn easily, we choose its starting point to be nearer to the end point of the first yarn. The co-ordinates of the points on the second yarn are thus given by z1 2nd = z1 1st xn 2nd = x1 2nd - xn 1st yn 2nd = y1 2nd - yn 1st zn 2nd = zn 1st n ≥ 2, 3, . . . 6.4.3 Estimation of fiber volume fraction Based on the above-mentioned geometric model, the fiber volume fraction of the knitted fabric composite is given by [11]: [6.10] where nk is the number of plies of the fabric in the composite, t is the thickness of the composite measured in centimeters, A is the planar area over which W and C are measured, and Ls is the length of yarn in one loop of the unit cell which can be represented approximately by [6.11] Let us apply Equation 6.10 to the knitted glass fiber fabric reinforced epoxy composites described in Section 6.3. Knitted fabrics with W = 2 loops/cm and C = 2.5 loops/cm, are made using 1600 denier (Dy) glass fiber yarns (fiber density rf = 2.54 g/cm3 ). Composites with single and four plies of L ad s ª 4( )j V n D L CW At k y f s f = ¥ - 9 10 5 r y W 1 2 1 2 nd = x ad Wtg 1 2 1 2 2nd = - ( ) y 192 3-D textile reinforcements in composite materials RIC6 7/10/99 8:12 PM Page 192 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:57 AM IP Address: 158.132.122.9