3 Micromechanical Analysis of a Lamina Chapter Objectives Develop concepts of volume and weight fraction(mass fraction)of fiber and matrix,density,and void fraction in composites. Find the nine mechanical and four hygrothermal constants:four elastic moduli,five strength parameters,two coefficients of thermal expansion,and two coefficients of moisture expansion of a unidirec- tional lamina from the individual properties of the fiber and the matrix,fiber volume fraction,and fiber packing. Discuss the experimental characterization of the nine mechanical and four hygrothermal constants. 3.1 Introduction In Chapter 2,the stress-strain relationships,engineering constants,and fail- ure theories for an angle lamina were developed using four elastic moduli, five strength parameters,two coefficients of thermal expansion(CTE),and two coefficients of moisture expansion(CME)for a unidirectional lamina. These 13 parameters can be found experimentally by conducting several tension,compression,shear,and hygrothermal tests on unidirectional lamina (laminates).However,unlike in isotropic materials,experimental evaluation of these parameters is quite costly and time consuming because they are functions of several variables:the individual constituents of the composite material,fiber volume fraction,packing geometry,processing,etc.Thus,the need and motivation for developing analytical models to find these param- eters are very important.In this chapter,we will develop simple relationships for the these parameters in terms of the stiffnesses,strengths,coefficients of thermal and moisture expansion of the individual constituents of a compos- ite,fiber volume fraction,packing geometry,etc.An understanding of this 203 2006 by Taylor Francis Group,LLC
203 3 Micromechanical Analysis of a Lamina Chapter Objectives • Develop concepts of volume and weight fraction (mass fraction) of fiber and matrix, density, and void fraction in composites. • Find the nine mechanical and four hygrothermal constants: four elastic moduli, five strength parameters, two coefficients of thermal expansion, and two coefficients of moisture expansion of a unidirectional lamina from the individual properties of the fiber and the matrix, fiber volume fraction, and fiber packing. • Discuss the experimental characterization of the nine mechanical and four hygrothermal constants. 3.1 Introduction In Chapter 2, the stress–strain relationships, engineering constants, and failure theories for an angle lamina were developed using four elastic moduli, five strength parameters, two coefficients of thermal expansion (CTE), and two coefficients of moisture expansion (CME) for a unidirectional lamina. These 13 parameters can be found experimentally by conducting several tension, compression, shear, and hygrothermal tests on unidirectional lamina (laminates). However, unlike in isotropic materials, experimental evaluation of these parameters is quite costly and time consuming because they are functions of several variables: the individual constituents of the composite material, fiber volume fraction, packing geometry, processing, etc. Thus, the need and motivation for developing analytical models to find these parameters are very important. In this chapter, we will develop simple relationships for the these parameters in terms of the stiffnesses, strengths, coefficients of thermal and moisture expansion of the individual constituents of a composite, fiber volume fraction, packing geometry, etc. An understanding of this 1343_book.fm Page 203 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
204 Mechanics of Composite Materials,Second Edition 10000 Nonhomogeneous lamina Homogeneous lamina FIGURE 3.1 A nonhomogeneous lamina with fibers and matrix approximated as a homogeneous lamina. relationship,called micromechanics of lamina,helps the designer to select the constituents of a composite material for use in a laminated structure. Because this text is for a first course in composite materials,details will be explained only for the simple models based on the mechanics of materials approach and the semi-empirical approach.Results from other methods based on advanced topics such as elasticity are also explained for completeness. As mentioned in Chapter 2,a unidirectional lamina is not homogeneous. However,one can assume the lamina to be homogeneous by focusing on the average response of the lamina to mechanical and hygrothermal loads(Figure 3.1).The lamina is simply looked at as a material whose properties are different in various directions,but not different from one location to another. Also,the chapter focuses on a unidirectional continuous fiber-reinforced lamina.This is because it forms the basic building block of a composite structure,which is generally made of several unidirectional laminae placed at various angles.The modeling in the evaluation of the parameters is dis- cussed first.This is followed by examples and experimental methods for finding these parameters. 3.2 Volume and Mass Fractions,Density,and Void Content Before modeling the 13 parameters of a unidirectional composite,we intro- duce the concept of relative fraction of fibers by volume.This concept is critical because theoretical formulas for finding the stiffness,strength,and hygrothermal properties of a unidirectional lamina are a function of fiber volume fraction.Measurements of the constituents are generally based on their mass,so fiber mass fractions must also be defined.Moreover,defining the density of a composite also becomes necessary because its value is used in the experimental determination of fiber volume and void fractions of a composite.Also,the value of density is used in the definition of specific modulus and specific strength in Chapter 1. 3.2.1 Volume Fractions Consider a composite consisting of fiber and matrix.Take the following symbol notations: 2006 by Taylor Francis Group,LLC
204 Mechanics of Composite Materials, Second Edition relationship, called micromechanics of lamina, helps the designer to select the constituents of a composite material for use in a laminated structure. Because this text is for a first course in composite materials, details will be explained only for the simple models based on the mechanics of materials approach and the semi-empirical approach. Results from other methods based on advanced topics such as elasticity are also explained for completeness. As mentioned in Chapter 2, a unidirectional lamina is not homogeneous. However, one can assume the lamina to be homogeneous by focusing on the average response of the lamina to mechanical and hygrothermal loads (Figure 3.1). The lamina is simply looked at as a material whose properties are different in various directions, but not different from one location to another. Also, the chapter focuses on a unidirectional continuous fiber-reinforced lamina. This is because it forms the basic building block of a composite structure, which is generally made of several unidirectional laminae placed at various angles. The modeling in the evaluation of the parameters is discussed first. This is followed by examples and experimental methods for finding these parameters. 3.2 Volume and Mass Fractions, Density, and Void Content Before modeling the 13 parameters of a unidirectional composite, we introduce the concept of relative fraction of fibers by volume. This concept is critical because theoretical formulas for finding the stiffness, strength, and hygrothermal properties of a unidirectional lamina are a function of fiber volume fraction. Measurements of the constituents are generally based on their mass, so fiber mass fractions must also be defined. Moreover, defining the density of a composite also becomes necessary because its value is used in the experimental determination of fiber volume and void fractions of a composite. Also, the value of density is used in the definition of specific modulus and specific strength in Chapter 1. 3.2.1 Volume Fractions Consider a composite consisting of fiber and matrix. Take the following symbol notations: FIGURE 3.1 A nonhomogeneous lamina with fibers and matrix approximated as a homogeneous lamina. Nonhomogeneous lamina Homogeneous lamina 1343_book.fm Page 204 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Micromechanical Analysis of a Lamina 205 volume of composite,fiber,and matrix,respectively Pm=density of composite,fiber,and matrix,respectively. Now define the fiber volume fraction V and the matrix volume fraction V as and Vin =Va Ve (3.1a,b) Note that the sum of volume fractions is V,+Vm=1, from Equation (3.1)as Uf+Um=Vc. 3.2.2 Mass Fractions Consider a composite consisting of fiber and matrix and take the following symbol notation:m=mass of composite,fiber,and matrix,respectively. The mass fraction (weight fraction)of the fibers (W)and the matrix(W) are defined as w,and W,= 0哑 W= (3.2a,b) Note that the sum of mass fractions is Wr+Wm =1, 2006 by Taylor Francis Group,LLC
Micromechanical Analysis of a Lamina 205 vc,f,m = volume of composite, fiber, and matrix, respectively ρc,f,m = density of composite, fiber, and matrix, respectively. Now define the fiber volume fraction Vf and the matrix volume fraction Vm as and (3.1a, b) Note that the sum of volume fractions is , from Equation (3.1) as 3.2.2 Mass Fractions Consider a composite consisting of fiber and matrix and take the following symbol notation: wc,f,m = mass of composite, fiber, and matrix, respectively. The mass fraction (weight fraction) of the fibers (Wf ) and the matrix (Wm) are defined as (3.2a, b) Note that the sum of mass fractions is , V v v f f c = , V v v m m c = . V V f m + = 1 v v f m + = vc . W w w f f c = , and W w w m m c = . W W f m + = 1 1343_book.fm Page 205 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
206 Mechanics of Composite Materials,Second Edition from Equation (3.2)as 0r+0m=0c· From the definition of the density of a single material, Wc=rVcr ws=ror,and (3.3a-c) Wm=YmUm Substituting Equation(3.3)in Equation(3.2),the mass fractions and vol- ume fractions are related as vand W= Wm= P碰Vw (3.4a,b) Pe in terms of the fiber and matrix volume fractions.In terms of individual constituent properties,the mass fractions and volume fractions are related by P WI= Pm一Vf' PLV+Vm Pm Wn= 1 Vm (3.5a,b) L(1-Vm)+Vm P One should always state the basis of calculating the fiber content of a composite.It is given in terms of mass or volume.Based on Equation(3.4), it is evident that volume and mass fractions are not equal and that the mismatch between the mass and volume fractions increases as the ratio between the density of fiber and matrix differs from one. 2006 by Taylor Francis Group,LLC
206 Mechanics of Composite Materials, Second Edition from Equation (3.2) as . From the definition of the density of a single material, (3.3a–c) Substituting Equation (3.3) in Equation (3.2), the mass fractions and volume fractions are related as (3.4a, b) in terms of the fiber and matrix volume fractions. In terms of individual constituent properties, the mass fractions and volume fractions are related by . (3.5a, b) One should always state the basis of calculating the fiber content of a composite. It is given in terms of mass or volume. Based on Equation (3.4), it is evident that volume and mass fractions are not equal and that the mismatch between the mass and volume fractions increases as the ratio between the density of fiber and matrix differs from one. w f m + w = wc w r v w r v w r v c c c f f f m m m = = = , , . and f f c W = V f , ρ ρ and m m c W = V m, ρ ρ f f m f m f m W = f V +V V , ρ ρ ρ ρ W V V m V f m m m = m − + 1 1 ρ ρ ( ) 1343_book.fm Page 206 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Micromechanical Analysis of a Lamina 207 3.2.3 Density The derivation of the density of the composite in terms of volume fractions is found as follows.The mass of composite w.is the sum of the mass of the fibers w,and the mass of the matrix w as We=Wf+wm. (3.6) Substituting Equation(3.3)in Equation(3.6)yields Pevc=PrUr+PmUmr and UfPm vc Pe=PyVe (3.7) Using the definitions of fiber and matrix volume fractions from Equation (3.1), Pe=PVr+puVm (3.8) Now,consider that the volume of a composite v is the sum of the volumes of the fiber v and matrix () 0c=0r十0m· (3.9) The density of the composite in terms of mass fractions can be found as 工_W+W血 (3.10) Pe Pr Pm Example 3.1 A glass/epoxy lamina consists of a 70%fiber volume fraction.Use proper- ties of glass and epoxy from Table 3.1*and Table 3.2,respectively,to deter- mine the +Table 3.1 and Table 3.2 give the typical properties of common fibers and matrices in the SI sys- tem of units,respectively.Note that fibers such as graphite and aramids are transversely isotro- pic,but matrices are generally isotropic.The typical properties of common fibers and matrices are again given in Table 3.3 and Table 3.4,respectively,in the USCS system of units. 2006 by Taylor Francis Group,LLC
Micromechanical Analysis of a Lamina 207 3.2.3 Density The derivation of the density of the composite in terms of volume fractions is found as follows. The mass of composite wc is the sum of the mass of the fibers wf and the mass of the matrix wm as (3.6) Substituting Equation (3.3) in Equation (3.6) yields and . (3.7) Using the definitions of fiber and matrix volume fractions from Equation (3.1), (3.8) Now, consider that the volume of a composite vc is the sum of the volumes of the fiber vf and matrix (vm): . (3.9) The density of the composite in terms of mass fractions can be found as (3.10) Example 3.1 A glass/epoxy lamina consists of a 70% fiber volume fraction. Use properties of glass and epoxy from Table 3.1* and Table 3.2, respectively, to determine the * Table 3.1 and Table 3.2 give the typical properties of common fibers and matrices in the SI system of units, respectively. Note that fibers such as graphite and aramids are transversely isotropic, but matrices are generally isotropic. The typical properties of common fibers and matrices are again given in Table 3.3 and Table 3.4, respectively, in the USCS system of units. w w c f = + wm. ρ ρ c c v v = + f f ρmvm , ρ ρ c f ρ f c m m c v v v v = + c f ρ ρ = V f + ρmV m. v v c f = + vm 1 = W + W . c f f m m ρ ρ ρ 1343_book.fm Page 207 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC