A.3 ALTERNATIVE EQUATIONS FOR THE ELASTIC AND THERMAL PROPERTIES OF A LAMINA Property Chamis [1] Tsai-Hahn [2] Eu Same as Equation 3.36 Same as Equation 3.36 E22 ErEm (Vr+m2vm)ErEm Er-VVi(Er -Em) (EmnVr +722VmEr) GGm G12 (Vr+m2Vm)GrGm Gr-VVE(Gr-Gm) (GmV:+712VmGr) 12 Same as Equation 3.37 Same as Equation 3.37 11 Same as Equation 3.58 a2 an Vi (1-V)(1+Eul) E K陆 Krve+Km(1-vr) Kz ViKr Km (1-)Km+K(K:-Ka) Note:In Tsai-Hahn equations for E22 and Gi2,n22 and n2 are called stress-partitioning parameters.They can be determined by fitting these equations to respective experimental data. Typical values of nz2 and m2 for epoxy matrix composites are: Fiber Type Carbon Glass Kevlar-49 722 0.5 0.516 0.516 12 0.4 0.316 0.4 Thermal conductivity. REFERENCES 1.C.C.Chamis,Simplified composite micromechanics equations for hygral,thermal and mechanical properties,SAMPE Quarterly.15:14(1984). 2.S.M.Tsai and H.T.Hahn,Introduction to Composite Materials,Technomic Publish- ing Co.,Lancaster,PA (1980). 2007 by Taylor Francis Group,LLC
REFERENCES 1. C.C. Chamis, Simplified composite micromechanics equations for hygral, thermal and mechanical properties, SAMPE Quarterly, 15:14 (1984). 2. S.M. Tsai and H.T. Hahn, Introduction to Composite Materials, Technomic Publishing Co., Lancaster, PA (1980). A.3 ALTERNATIVE EQUATIONS FOR THE ELASTIC AND THERMAL PROPERTIES OF A LAMINA Property Chamis [1] Tsai–Hahn [2] E11 Same as Equation 3.36 Same as Equation 3.36 E22 EfEm Ef ffiffiffiffi vf p (Ef Em) (vf þ h22vm)EfEm (Emvf þ h22vmEf) G12 GfGm Gf ffiffiffiffi vf p (Gf Gm) (vf þ h12vm)GfGm (Gmvf þ h12vmGf) n12 Same as Equation 3.37 Same as Equation 3.37 a11 Same as Equation 3.58 a22 afT ffiffiffiffi vf p þ(1 ffiffiffiffi vf p ) 1 þ vfnm Ef E11 am K* 11 Kfvf þ Km(1 vf) K22 (1 ffiffiffiffi vf p )Km þ ffiffiffiffi vf p KfKm Kf ffiffiffiffi vf p (Kf Km) Note: In Tsai–Hahn equations for E22 and G12, h22 and h12 are called stress-partitioning parameters. They can be determined by fitting these equations to respective experimental data. Typical values of h22 and h12 for epoxy matrix composites are: Fiber Type Carbon Glass Kevlar-49 h22 0.5 0.516 0.516 h12 0.4 0.316 0.4 * Thermal conductivity. 2007 by Taylor & Francis Group, LLC.
A.4 HALPIN-TSAI EQUATIONS The Halpin-Tsai equations are simple approximate forms of the generalized self-consistent micromechanics solutions developed by Hill.The modulus val- ues based on these equations agree reasonably well with the experimental values for a variety of reinforcement geometries,including fibers,flakes,and ribbons.A review of their developments is given in Ref.[1]. In the general form,the Halpin-Tsai equations for oriented reinforcements are expressed as p1+5nvr a=1- with n=/pm)-1 (pr/pm)+5 where p =composite property,such as E11,E22,G12,G23,and v23 Pr =reinforcement property,such as Er,Gr,and v Pm=matrix property,such as Em,Gm,and vm =a measure of reinforcement geometry,packing geometry,and loading conditions Vr =reinforcement volume fraction Reliable estimates for the factor are obtained by comparing the Halpin-Tsai equations with the numerical solutions of the micromechanics equations [2-4]. For example, =2+40v10 for Eu. g=2”+40v40 forE22, -(月m +40v10 for G12, where /w,and t are the reinforcement length,width,and thickness,respect- ively.For a circular fiber,/=le and t=w=de,and for a spherical reinforce- ment,/=t=w.The term containing v in the expressions for is relatively small up to v=0.7 and therefore can be neglected.Note that for oriented continuous fiber-reinforced composites,-oo,and substitution of n into the Halpin-Tsai equation for Eu gives the same result as obtained by the rule of mixture. Nielsen [5]proposed the following modification for the Halpin-Tsai equa- tion to include the maximum packing fraction,v*: 2007 by Taylor&Francis Group.LLC
A.4 HALPIN–TSAI EQUATIONS The Halpin–Tsai equations are simple approximate forms of the generalized self-consistent micromechanics solutions developed by Hill. The modulus values based on these equations agree reasonably well with the experimental values for a variety of reinforcement geometries, including fibers, flakes, and ribbons. A review of their developments is given in Ref. [1]. In the general form, the Halpin–Tsai equations for oriented reinforcements are expressed as p pm ¼ 1 þ zhvr 1 hvr with h ¼ (pr=pm) 1 (pr=pm) þ z where p ¼ composite property, such as E11, E22, G12, G23, and n23 pr ¼ reinforcement property, such as Er, Gr, and nr pm ¼ matrix property, such as Em, Gm, and nm z ¼ a measure of reinforcement geometry, packing geometry, and loading conditions vr ¼ reinforcement volume fraction Reliable estimates for the z factor are obtained by comparing the Halpin–Tsai equations with the numerical solutions of the micromechanics equations [2–4]. For example, z ¼ 2 l t þ 40v10 r for E11, z ¼ 2 w t þ 40v10 r for E22, z ¼ w t 1:732 þ 40v10 r for G12, where l, w, and t are the reinforcement length, width, and thickness, respectively. For a circular fiber, l ¼ lf and t ¼ w ¼ df, and for a spherical reinforcement, l ¼ t ¼ w. The term containing vr in the expressions for z is relatively small up to vr ¼ 0.7 and therefore can be neglected. Note that for oriented continuous fiber-reinforced composites, z ! 1, and substitution of h into the Halpin–Tsai equation for E11 gives the same result as obtained by the rule of mixture. Nielsen [5] proposed the following modification for the Halpin–Tsai equation to include the maximum packing fraction, v r : 2007 by Taylor & Francis Group, LLC.