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