3 PLY PROPERTIES It is of fundamental importance for the designer to understand and to know precisely the geometric and mechanical characteristics of the "fiber matrix"mixture which is the basic structure of the composite parts.The description of these charac- teristics is the object of this chapter. 3.1 ISOTROPY AND ANISOTROPY When one studies the mechanical behavior of elastic bodies under load (elasticity theory),one has to consider the following: An elastic body subjected to stresses deforms in a reversible manner. At each point within the body,one can identify the principal planes on which there are only normal stresses. The normal directions on these planes are called the principal stress directions. A small sphere of material surrounding a point of the body becomes an ellipsoid after loading. The spatial position of the ellipsoid relative to the principal stress directions enables us to characterize whether the material under study is isotropic or anisotropic.Figure 3.1 illustrates this phenomenon. Figure 3.2 illustrates the deformation of an isotropic sample and an anisotropic sample.In the latter case,the oblique lines represent the preferred directions along which one would place the fibers of reinforcement.One can consider that a longitudinal loading applied to an isotropic plate would create an extension in the longitudinal direction and a contraction in the transverse direction.The same loading applied to an anisotropic plate creates an angular distortion,in addition to the longitudinal extension and transversal contraction. In the simple case of plane stress,one can obtain the elastic constants using stress-strain relations. 2003 by CRC Press LLC
3 PLY PROPERTIES It is of fundamental importance for the designer to understand and to know precisely the geometric and mechanical characteristics of the “fiber + matrix” mixture which is the basic structure of the composite parts. The description of these characteristics is the object of this chapter. 3.1 ISOTROPY AND ANISOTROPY When one studies the mechanical behavior of elastic bodies under load (elasticity theory), one has to consider the following: An elastic body subjected to stresses deforms in a reversible manner. At each point within the body, one can identify the principal planes on which there are only normal stresses. The normal directions on these planes are called the principal stress directions. A small sphere of material surrounding a point of the body becomes an ellipsoid after loading. The spatial position of the ellipsoid relative to the principal stress directions enables us to characterize whether the material under study is isotropic or anisotropic. Figure 3.1 illustrates this phenomenon. Figure 3.2 illustrates the deformation of an isotropic sample and an anisotropic sample. In the latter case, the oblique lines represent the preferred directions along which one would place the fibers of reinforcement. One can consider that a longitudinal loading applied to an isotropic plate would create an extension in the longitudinal direction and a contraction in the transverse direction. The same loading applied to an anisotropic plate creates an angular distortion, in addition to the longitudinal extension and transversal contraction. In the simple case of plane stress, one can obtain the elastic constants using stress–strain relations. TX846_Frame_C03 Page 29 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
●M before stress application application of stress ⊙x y y Isotropic material:the axes of the Anisotropic material:the axes of ellipsoid coincide with the principal the ellipsoid are different from the stress axes principal stress axes Figure 3.1 Schematic of Deformation isotropic material anisotropic material Figure 3.2 Comparison between Deformation of an Isotropic and Anisotropic Plate 2003 by CRC Press LLC
Figure 3.1 Schematic of Deformation Figure 3.2 Comparison between Deformation of an Isotropic and Anisotropic Plate TX846_Frame_C03 Page 30 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
dimensions 1×1 Ex G Figure 3.3 Stress-Strain Behavior in an Isotropic Material 3.1.1 Isotropic Materials The following relations are valid for a material that is elastic and isotropic. One can write the stress-strain relation (see Figure 3.3)in matrix form as' 1 0 E Ox E 0 o, Yxr 0 0 G in these equations,are also the small strains that are obtained in a classical manner from the displacements us and uy as:Ex=du ldx;=du,ldy;Ysy=duldy duldx. 2003 by CRC Press LLC
3.1.1 Isotropic Materials The following relations are valid for a material that is elastic and isotropic. One can write the stress–strain relation (see Figure 3.3) in matrix form as1 Figure 3.3 Stress–Strain Behavior in an Isotropic Material 1 In these equations, ex,ey,gxy are also the small strains that are obtained in a classical manner from the displacements ux and uy as: ex = ∂ux /∂x; ey = ∂uy /∂y; gxy = ∂ux /∂y + ∂uy /∂x. ex e y Ó g xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ 1 E -- n E –-- 0 n E –-- 1 E -- 0 0 0 1 G --- sx sy Ó txy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = TX846_Frame_C03 Page 31 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
dimensions 1×1 Ex=Ex -Vyx Ey Ex叶 ey=Ey -Vxy Ex Gx Figure 3.4 Deformation in an Anisotropic Material There are three elastic constants:E,v,G.There exists a relation among them as: E G=21+V The above relation shows that a material that is isotropic and elastic can be characterized by two independent elastic constants:E and v. 3.1.2 Anisotropic Material The matrix equation for anisotropic material (see Figure 3.4)is 1 0 Ey 知 0 1马0 0 2003 by CRC Press LLC
There are three elastic constants: E, n, G. There exists a relation among them as: The above relation shows that a material that is isotropic and elastic can be characterized by two independent elastic constants: E and n. 3.1.2 Anisotropic Material The matrix equation for anisotropic material (see Figure 3.4) is Figure 3.4 Deformation in an Anisotropic Material G E 2 1( ) + n = -------------------- ex e y Ó g xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ 1 Ex ----- nyx Ey –------- 0 nxy Ex –------- 1 Ey ---- 0 0 0 1 Gxy -------- sx sy Ó txy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = TX846_Frame_C03 Page 32 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
Note that the stress-strain matrix above is symmetric.?The number of distinct elastic constants is five: Two moduli of elasticity:Ex and E Two Poisson coefficients:Vax and v,and ■One shear modulus:Gg In fact there are only four independent elastic constants:EE Go and v(or v).The fifth elastic constant can be obtained from the others using the symmetry relation: Ex Vay Vy E 3.2 CHARACTERISTICS OF THE REINFORCEMENT-MATRIX MIXTURE We denote as ply the semi-product "reinforcement resin"in a quasi-bidimen- sional form.'This can be A tape of unidirectional fiber matrix, ■A fabric+matrix,or ■Amat+matrix. These are examined more below. 3.2.1 Fiber Mass Fraction Fiber mass fraction is defined as Mass of fibers M= Total mass In consequence,the mass of matrix is Mass of matrix Total mass with Mm 1-M To know more about the development on this point,refer to Section 9.2 and Exercise 18.1.2. 3Refer to Section 13.2. Such condition exists in the commercial products.These are called preimpregnated or SMC (sheet molding compound).One can also find non-preformed mixtures of short fibers and resin.These are called premix or BMC (bulk molding compound). 2003 by CRC Press LLC
Note that the stress–strain matrix above is symmetric.2 The number of distinct elastic constants is five: Two moduli of elasticity: Ex and Ey, Two Poisson coefficients: nyx and nxy, and One shear modulus: Gxy. In fact there are only four independent elastic constants:3 Ex, Ey, Gxy, and nyx (or nxy). The fifth elastic constant can be obtained from the others using the symmetry relation: 3.2 CHARACTERISTICS OF THE REINFORCEMENT–MATRIX MIXTURE We denote as ply the semi-product “reinforcement + resin” in a quasi-bidimensional form.4 This can be A tape of unidirectional fiber + matrix, A fabric + matrix, or A mat + matrix. These are examined more below. 3.2.1 Fiber Mass Fraction Fiber mass fraction is defined as In consequence, the mass of matrix is with 2 To know more about the development on this point, refer to Section 9.2 and Exercise 18.1.2. 3 Refer to Section 13.2. 4 Such condition exists in the commercial products. These are called preimpregnated or SMC (sheet molding compound). One can also find non-preformed mixtures of short fibers and resin. These are called premix or BMC (bulk molding compound). nxy nyx Ex Ey = ----- Mf Mass of fibers Total mass = ----------------------------------- Mm Mass of matrix Total mass = ------------------------------------- Mm = 1 – Mf TX846_Frame_C03 Page 33 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
Table 3.1 Common Fiber Volume Fractions in Different Processes Molding Process Fiber Volume Fraction Contact Molding 30% Compression Molding 40% Filament Winding 60%-85% Vacuum Molding 50%-80% 3.2.2 Fiber Volume Fraction Fiber volume fraction is defined as = Volume of fiber Total volume As a result,the volume of matrix is given as Vm= Volume of matrix Total volume withs Vm=1- Note that one can convert from mass fraction to volume fraction and vice versa.If p,and p are the specific mass of the fiber and matrix,respectively,we have =M+匹 Pr ViPL My=Vrpr+Vupm Py P Depending on the method of fabrication,the common fiber volume fractions are as shown in Table 3.1. 3.2.3 Mass Density of a Ply The mass density of a ply can be calculated as p=total mass total volume 5In reality,the mixture of fiber/matrix also includes a small volume of voids,characterized by the porosity of the composite.One has then V+V+Ve=1,in which V denotes the percentage of"volume of void/total volume."V is usually much less than 1 (See Exercise 18.1.11). 2003 by CRC Press LLC
3.2.2 Fiber Volume Fraction Fiber volume fraction is defined as As a result, the volume of matrix is given as with5 Note that one can convert from mass fraction to volume fraction and vice versa. If rf and rm are the specific mass of the fiber and matrix, respectively, we have Depending on the method of fabrication, the common fiber volume fractions are as shown in Table 3.1. 3.2.3 Mass Density of a Ply The mass density of a ply can be calculated as Table 3.1 Common Fiber Volume Fractions in Different Processes Molding Process Fiber Volume Fraction Contact Molding 30% Compression Molding 40% Filament Winding 60%–85% Vacuum Molding 50%–80% 5 In reality, the mixture of fiber/matrix also includes a small volume of voids, characterized by the porosity of the composite. One has then Vm + Vf + Vp = 1, in which Vp denotes the percentage of “volume of void/total volume.” Vp is usually much less than 1 (See Exercise 18.1.11). Vf Volume of fiber Total volume = ---------------------------------------- Vm Volume of matrix Total volume = -------------------------------------------- Vm = 1 – Vf Vf Mf rf ---- Mf rf ---- Mm r m + ----- = ----------------- Mf Vfrf Vfrf + Vmrm = ------------------------------ r total mass total volume = ------------------------------- TX846_Frame_C03 Page 34 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
Table 3.2 Ply Thicknesses of Some Common Composites M H E glass 34% 0.125mm R glass 68% 0.175mm Kevlar 65% 0.13mm H.R.Carbon 68% 0.13mm The above equation can also be expanded as p=mass of fiber mass of matrix total volume total volume volume of fiber erpvolume of matrix Pm total volume total volume or P=prVr+puVm 3.2.4 Ply Thickness The ply thickness is defined as the number of grams of mass of fiber mor per m of area.The ply thickness,denoted as b,is such that: b×l(m)=total volume=total volume× mof fiber volume x p or b= mof Vres One can also express the thickness in terms of mass fraction of fibers rather than in terms of volume fraction. =m[】 Table 3.2 shows a few examples of ply thicknesses 3.3 UNIDIRECTIONAL PLY 3.3.1 Elastic Modulus The mechanical characteristics of the fiber/matrix mixture can be obtained based on the characteristics of each of the constituents.In the literature,there are theoretical as well as semi-empirical relations.As such,the results from these relations may not always agree with experimental values.One of the reasons is 2003 by CRC Press LLC
The above equation can also be expanded as or 3.2.4 Ply Thickness The ply thickness is defined as the number of grams of mass of fiber mof per m2 of area. The ply thickness, denoted as h, is such that: or One can also express the thickness in terms of mass fraction of fibers rather than in terms of volume fraction. Table 3.2 shows a few examples of ply thicknesses. 3.3 UNIDIRECTIONAL PLY 3.3.1 Elastic Modulus The mechanical characteristics of the fiber/matrix mixture can be obtained based on the characteristics of each of the constituents. In the literature, there are theoretical as well as semi-empirical relations. As such, the results from these relations may not always agree with experimental values. One of the reasons is Table 3.2 Ply Thicknesses of Some Common Composites Mf H E glass 34% 0.125 mm R glass 68% 0.175 mm Kevlar 65% 0.13 mm H.R. Carbon 68% 0.13 mm r mass of fiber total volume -------------------------------- mass of matrix total volume = + ------------------------------------- volume of fiber total volume ---------------------------------------rf volume of matrix total volume = + -------------------------------------------- rm r r = fVf + rmVm h ¥ 1 m2 ( ) total volume total volume mof fiber volume ¥ rf = = ¥ ------------------------------------------- h mof Vfrf = --------- h mof 1 rf ---- 1 rm ------ 1 – Mf Mf -------------- Ë ¯ Ê ˆ = + TX846_Frame_C03 Page 35 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
Table 3.3 Fiber Elastic Modulus Glass Carbon Carbon E Kevlar H.R. H.M. fiber longitudinal modulus in e direction Efe (MPa) 74.000 130,000230,000 390.000 fiber transverse modulus in t direction Ef (MPa) 74,000 5400 15,000 6000 fiber shear modulus Gfa (Mpa) 30,000 12,000 50,000 20,000 fiber Poisson ratio via 0.25 0.4 0.3 0.35 Isotropic Anisotropic because the fibers themselves exhibit some degree of anisotropy.In Table 3.3, one can see small values of the elastic modulus in the transverse direction for Kevlar and carbon fibers,whereas glass fiber is isotropic. With the definitions in the previous paragraph,one can use the following relations to characterize the unidirectional ply: Modulus of elasticity along the direction of the fiber E is given by? E ErVt+EmVm or Ee=E'+Em(1-月 In practice,this modulus depends essentially on the longitudinal mod- ulus of the fiber,E because E<<E (as Em resin/Eglss6%). Modulus of elasticity in the transverse direction to the fiber axis,E: In the following equation,E represents the modulus of elasticity of the 6This is due to the drawing of the carbon and Kevlar fibers during fabrication.This orients the chain of the molecules. Chapter 10 gives details for the approximate calculation of the moduli E.E Ge and v which lead to these expressions. 2003 by CRC Press LLC
because the fibers themselves exhibit some degree of anisotropy. In Table 3.3, one can see small values of the elastic modulus in the transverse direction for Kevlar and carbon fibers, whereas glass fiber is isotropic. 6 With the definitions in the previous paragraph, one can use the following relations to characterize the unidirectional ply: Modulus of elasticity along the direction of the fiber El is given by 7 E = Ef Vf + EmVm or In practice, this modulus depends essentially on the longitudinal modulus of the fiber, Ef , because Em << Ef (as Em resin/Ef glass # 6%). Modulus of elasticity in the transverse direction to the fiber axis, Et: In the following equation, Eft represents the modulus of elasticity of the Table 3.3 Fiber Elastic Modulus Glass Carbon Carbon E Kevlar H.R. H.M. fiber longitudinal modulus in direction Ef (MPa) 74,000 130,000 230,000 390,000 fiber transverse modulus in t direction Eft (MPa) 74,000 5400 15,000 6000 fiber shear modulus Gft (Mpa) 30,000 12,000 50,000 20,000 fiber Poisson ratio vft 0.25 0.4 0.3 0.35 Isotropic 6 This is due to the drawing of the carbon and Kevlar fibers during fabrication. This orients the chain of the molecules. 7 Chapter 10 gives details for the approximate calculation of the moduli E, Et , Gt and ut which lead to these expressions. Anisotropic Ï Ô Ô Ô Ô Ô Ì Ô Ô Ô Ô Ô Ó E EfVf Em 1 – Vf = + ( ) TX846_Frame_C03 Page 36 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
warp fibers matrix transverse direction longitudinal direction Unidirectional ply Unidirectional fabric Figure 3.5 Orientations in Composite Layers fiber in the direction that is transverse to the fiber axis as indicated in Table 3.3. 1 Er=Em 1-+y Shear modulus Ge:An order of magnitude of this modulus is given in the following expression,in which G represents the shear modulus of the fiber (as shown in Table 3.3). a-叨+ Poisson coefficient v:The Poisson coefficient represents the contraction in the transverse direction t when a ply is subjected to tensile loading in the longitudinal direction (see Figure 3.5). Ver vvr+VmVm Modulus along any direction:The modulus along a certain direction in plane et,other than along the fiber and transverse to the fiber,"is given in the expression below,where c cose and s sine.Note that this modulus decreases rapidly as one is moving away from the fiber direction (see Figure 3.6). Ex g++2c2 E 2Gu E, The calculation of these moduli is shown in details in Chapter 11. 2003 by CRC Press LLC
fiber in the direction that is transverse to the fiber axis as indicated in Table 3.3. Shear modulus Gt: An order of magnitude of this modulus is given in the following expression, in which Gflt represents the shear modulus of the fiber (as shown in Table 3.3). Poisson coefficient nt: The Poisson coefficient represents the contraction in the transverse direction t when a ply is subjected to tensile loading in the longitudinal direction (see Figure 3.5). Modulus along any direction: The modulus along a certain direction in plane t, other than along the fiber and transverse to the fiber, 8 is given in the expression below, where c = cosq and s = sinq. Note that this modulus decreases rapidly as one is moving away from the fiber direction (see Figure 3.6). Figure 3.5 Orientations in Composite Layers 8 The calculation of these moduli is shown in details in Chapter 11. Et Em 1 1 – Vf ( ) Em Eft + ----Vf = ----------------------------------- Gt Gm 1 1 – Vf ( ) Gm Gft + ------Vf = ------------------------------------- nt = nfVf + nmVm Ex 1 c 4 E --- s 4 Et --- 2c 2 s 2 1 2Gt -------- nt Et – ---- Ë ¯ Ê ˆ + + = ----------------------------------------------------------- TX846_Frame_C03 Page 37 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC
Ex 0°0 90° Figure 3.6 Off-axis Modulus load load rupture rupture METAL UNIDIRECTIONAL elongation elongation Figure 3.7 Loading Curves of Metal and Unidirectional Composite 3.3.2 Ultimate Strength of a Ply The curves in Figure 3.7 show the important difference in the behavior between classical metallic materials and the unidirectional plies.These differences can be summarized in a few points as There is lack of plastic deformation in the unidirectional ply.(This is a disadvantage.) Ultimate strength of the unidirectional ply is higher.(This is an advantage.) There is important elastic deformation for the unidirectional ply.(This can be an advantage or a disadvantage,depending on the application;for example,this is an advantage for springs,arcs,or poles.) When the fibers break before the matrix during loading along the fiber direction,one can obtain the following for the composite: &n[+1-吟 2003 by CRC Press LLC
3.3.2 Ultimate Strength of a Ply The curves in Figure 3.7 show the important difference in the behavior between classical metallic materials and the unidirectional plies. These differences can be summarized in a few points as There is lack of plastic deformation in the unidirectional ply. (This is a disadvantage.) Ultimate strength of the unidirectional ply is higher. (This is an advantage.) There is important elastic deformation for the unidirectional ply. (This can be an advantage or a disadvantage, depending on the application; for example, this is an advantage for springs, arcs, or poles.) When the fibers break before the matrix during loading along the fiber direction, one can obtain the following for the composite: Figure 3.6 Off-axis Modulus Figure 3.7 Loading Curves of Metal and Unidirectional Composite s rupture sf rupture Vf 1 – Vf ( ) Em Ef = + ------ TX846_Frame_C03 Page 38 Monday, November 18, 2002 12:05 PM © 2003 by CRC Press LLC