BASIC PRINCIPLES OF FIBER COMPOSITE MATERIALS 25 requires data on the thermal expansion coefficients of the plies in the longitudinal and transverse directions,a1 and a2,respectively. Composite structural components used in aircraft are most often based on plies (sheets of unidirectional fibers or bi-directionally aligned woven fibers in a matrix) laminated together with the fibers at various orientations,as outlined in Chapter 1. Thus the properties required in the analysis are for a single ply of the composite,as described in Chapter 6.Although this analysis draws largely on data for the plies obtained from physical and mechanical testing of unidirectional composites,esti- mates of these properties provided by the micromechanical approach can provide useful approximate values of these properties when test data are unavailable. 2.3 Micromechanics As already mentioned,micromechanics utilizes microscopic models of composites,in which the fibers and the matrix are separately modelled.In most simple models,the fibers are assumed to be homogeneous,linearly elastic, isotropic,regularly spaced,perfectly aligned,and of uniform length.The matrix is assumed to be homogeneous,linearly elastic,and isotropic.The fiber/matrix interface is assumed to be perfect,with no voids or disbonds. More complex models,representing more realistic situations,may include voids,disbonds,flawed fibers(including statistical variations in flaw severity),wavy fibers,non-uniform fiber dispersions,fiber length variations,and residual stresses. Micromechanics2can,itself,be approached in three ways: (1)The mechanics of materials approach,which attempts to predict the behavior of simplified models of the composite material. (2)The theory of elasticity approach,which is often aimed at producing upper and lower bound exact analytical or numerical solutions. (3)The finite-element (F-E)approach based on t'wo-dimensional or three- dimensional models of varying degrees of sophistication. The most difficult aspect of the composite to model is the fiber/matrix interface, also known as the interphase,which can have a profound effect on strength and toughness.In view of this and other complexities,the F-E micromechanics approach offers by far the best prospect of success to predict strength behavior. Indeed,failure theories,described in Chapter 6,require local modelling at the micromechanical level for predicting the strength of actual components A common aim of both approaches is to determine the elastic constants and strengths of composites in terms of their constituent properties.As previously stated,the main elastic constants for unidirectional fiber composites are: EI=longitudinal modulus (i.e.,modulus in fiber direction) E2 transverse modulus
BASIC PRINCIPLES OF FIBER COMPOSITE MATERIALS 25 requires data on the thermal expansion coefficients of the plies in the longitudinal and transverse directions, al and a2, respectively. Composite structural components used in aircraft are most often based on phes (sheets of unidirectional fibers or bi-directionally aligned woven fibers in a matrix) laminated together with the fibers at various orientations, as outlined in Chapter 1. Thus the properties required in the analysis are for a single ply of the composite, as described in Chapter 6. Although this analysis draws largely on data for the plies obtained from physical and mechanical testing of unidirectional composites, estimates of these properties provided by the micromechanical approach can provide useful approximate values of these properties when test data are unavailable. 2.3 Micromechanics As already mentioned, micromechanics utilizes microscopic models of composites, in which the fibers and the matrix are separately modelled. In most simple models, the fibers are assumed to be homogeneous, linearly elastic, isotropic, regularly spaced, perfectly aligned, and of uniform length. The matrix is assumed to be homogeneous, linearly elastic, and isotropic. The fiber/matrix interface is assumed to be perfect, with no voids or disbonds. More complex models, representing more realistic situations, may include voids, disbonds, flawed fibers (including statistical variations in flaw severity), wavy fibers, non-uniform fiber dispersions, fiber length variations, and residual stresses. Micromechanics 2 can, itself, be approached in three ways: (1) The mechanics of materials approach, which attempts to predict the behavior of simplified models of the composite material. (2) The theory of elasticity approach, which is often aimed at producing upper and lower bound exact analytical or numerical solutions. (3) The finite-element (F-E) approach based on t'wo-dimensional or threedimensional models of varying degrees of sophistication. The most difficult aspect of the composite to model is the fiber/matrix interface, also known as the interphase, which can have a profound effect on strength and toughness. In view of this and other complexities, the F-E micromechanics approach offers by far the best prospect of success to predict strength behavior. Indeed, failure theories, described in Chapter 6, require local modelling at the micromechanical level for predicting the strength of actual components. A common aim of both approaches is to determine the elastic constants and strengths of composites in terms of their constituent properties. As previously stated, the main elastic constants for unidirectional fiber composites are: E1 = longitudinal modulus (i.e., modulus in fiber direction) E2 = transverse modulus
26 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES v12=major Poisson's ratio (i.e.,ratio of contraction in the transverse direction consequent on an extension in the fiber direction) G12 in-plane shear modulus a1 longitudinal thermal expansion coefficient a2=transverse expansion coefficient The main strength values required are: of=longitudinal strength (both tensile and compressive) a2=transverse strength (both tensile and compressive) 72=shear strength where the superscript u refers to ultimate strength. 2.4 Elastic Constants 2.4.1 Mechanics of Materials Approach The simple model used in the following analyses is a single,unidirectional ply,or lamina,as depicted in Figure 2.2.Note that the representative volume element shown,is the full thickness of the single ply and that the simplified"two- dimensional"element is used in the following analyses.The key assumptions used in connection with this model are indicated in Figure 2.3. 2.4.1.1 E1 Longitudinal Modulus.The representative volume element under an applied stress is shown in Figure 2.3a.The resultant strain E is assumed to be common to both the fiber and matrix.The stresses felt by the fiber,matrix,and composite are,respectively,of om,and 1.Taking Erand Em as the fiber and matrix moduli,respectively,then: Of=Er81,Om Em81,1=E181 (2.1) The applied stress acts over a cross-sectional area A consisting of Af,the fiber cross- section,andAm,the matrix cross-section.Because the fibers and matrix are acting in parallel to carry the load: 1A =OfAf+omAm or o1=ofVf+omVm (2.2) where V=Af/A fiber volume fraction and Vm Am/A =1-V=matrix volume fraction. Substituting equation(2.1)into equation(2.2)gives: E1 EfVf EmVm (2.3) Equation (2.3)is a "rule-of-mixtures"type of relationship that relates the composite property to the weighted sum of the constituent properties
26 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES "1)12 = major Poisson's ratio (i.e., ratio of contraction in the transverse direction consequent on an extension in the fiber direction) G12 = in-plane shear modulus al = longitudinal thermal expansion coefficient o¢ 2 = transverse expansion coefficient The main strength values required are: 0~1 = longitudinal strength (both tensile and compressive) 0-u22 = transverse strength (both tensile and compressive) ~12 = shear strength where the superscript u refers to ultimate strength. 2.4 Elastic Constants 2.4.1 Mechanics of Materials Approach The simple model used in the following analyses is a single, unidirectional ply, or lamina, as depicted in Figure 2.2. Note that the representative volume element shown, is the full thickness of the single ply and that the simplified "twodimensional" element is used in the following analyses. The key assumptions used in connection with this model are indicated in Figure 2.3. 2.4.1.1 E1 Longitudinal Modulus. The representative volume element under an applied stress is shown in Figure 2.3a. The resultant strain E is assumed to be common to both the fiber and matrix. The stresses felt by the fiber, matrix, and composite are, respectively, try, trm, and trl. Taking E/and Em as the fiber and matrix moduli, respectively, then: Off = Efel, Orm = Emel, o'1 = Elel (2.1) The applied stress acts over a cross-sectional areaA consisting of Af, the fiber crosssection, andAm, the matrix cross-section. Because the fibers and matrix are acting in parallel to carry the load: oq A = o'fAy -~- O'mA m or O" 1 = o'f Vf -Jr- o'm V m (2.2) where VT = AT/A = fiber volume fraction and Vm = Am/A = 1 - VU = matrix volume fraction. Substituting equation (2.1) into equation (2.2) gives: E1 = EfVf + EmVm (2.3) Equation (2.3) is a "rule-of-mixtures" type of relationship that relates the composite property to the weighted sum of the constituent properties
BASIC PRINCIPLES OF FIBER COMPOSITE MATERIALS 27 thickness t Model Volume element thickness t Simplified 2.D element Matrix Fiber Matrix Fig.2.2 Model and representative volume element of a unidirectional ply. Experimental verification of equation(2.3)has been obtained for many fiber/ resin systems;examples of the variation of Ei with V for two glass/polyester resin systems are shown in Figure 2.4. 2.4.1.2 E2 Transverse Modulus.As shown in Figure 2.3b,the fiber and matrix are assumed to act in series,both carrying the same applied stress o2.The transverse strains for the fiber,matrix,and composite are thus,respectively: >ef=1 02 02 02 em=2= (2.4) where E is the effective transverse modulus of the fiber
BASIC PRINCIPLES OF FIBER COMPOSITE MATERIALS 27 Model T , _L ,...j/T thickness t Volume element Simplified 2-D element Fig. 2.2 1 T ~ ! 2 ...... Matrix F///~ .~o~ ~'//~ Matrix J_ thickness t Model and representative volume element of a unidirectional ply. Experimental verification of equation (2.3) has been obtained for many fiber/ resin systems; examples of the variation of El with Vf for two glass/polyester resin systems are shown in Figure 2.4. 2.4.1.2 E2 Transverse Modulus. As shown in Figure 2.3b, the fiber and matrix are assumed to act in series, both carrying the same applied stress o.2. The transverse strains for the fiber, matrix, and composite are thus, respectively: o.2 o-2 o.2 = ~ = -- (2.4) > 8f ' 8m ~ gm' 82 E2 where E} is the effective transverse modulus of the fiber
28 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES Undeformed element 1 Matrix Fbr7Z7☑ Matrix L Applied stresses Deformations Relationship E,E V,+Em (1-V,) 3. Determination of E 作。v,E,+1v,1Em b.Determination of E △w12 2"V,+vm1-V, c.Determination of v2 1/G2=V,/6,+(1-,)/Gm △m12 d.Determination of G Fig.2.3 Models for the determination of elastic constants by the "mechanics of materials'”approach
28 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES b. Applied stresses a. Determination of E 1 lllltll ill 111~ o, Determination of E= Determination of ul= - T , IJJJJJJJA, l Determination of G~z Undeformed element T~/~ Matrix | .Fi.ber ///~1 _LI Matrix I I_ --I i L -I Deformations =I HaLl-- F ..... -]. -Law b///////~ s Ii il Z~WIZ /t.f~ A AmlZ Relationship E= =EfVf+E m (1-Vf) l/E= = vf/Ef+(1-Vf)/E m ul==hVf+Um (1-Vf) 1/G1= = Vf/Gf + (1 - Vf) / G m Fig. 2.3 Models for the determination of elastic constants by the "mechanics of materials" approach
BASIC PRINCIPLES OF FIBER COMPOSITE MATERIALS 29 60 50 40 (84) 30 20 8 10 0 00.10.20.30.40.50.60.70.8 Vr Fig.2.4 E1 versus fiber volume fraction V for two glass/polyester systems. Deformations are additive over the width W,so that: △W=△W+△Wm or 82W =ef(VW)+8m(VmW) (2.5) Substitution of equation (2.4)into equation(2.5)yields: 1 Vr Vm 每+ (2.6) Experimental results are in reasonable agreement with equation (2.6)as shown, for example,in Figure 2.5,for a glass/polyester composite
BASIC PRINCIPLES OF FIBER COMPOSITE MATERIALS 29 O.. t.~ 60 50 40 30 20 10 ~d 0 0 0 ¢1 ¢ 0 I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Vf Fig. 2.4 E1 versus fiber volume fraction Vf for two glass/polyester systems. Deformations are additive over the width W, so that: or aW = aW~ + aWm 82W = ~,f(Wf W) "]- ,~m(Vm W) (2.5) Substitution of equation (2.4) into equation (2.5) yields: 1 _ Vf ~ Vm (2.6) E2 E~ Em Experimental results are in reasonable agreement with equation (2.6) as shown, for example, in Figure 2.5, for a glass/polyester composite