MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis CHAPTER 5 DESIGN AND ANALYSIS 5.1 INTRODUCTION The concept of designing a material to yield a desired set of properties has received impetus from the growing acceptance of composite materials.Inclusion of material design in the structural design process has had a significant effect on that process,particularly upon the preliminary design phase.In this pre- liminary design,a number of materials will be considered,including materials for which experimental ma- terials property data are not available.Thus,preliminary material selection may be based on analytically- predicted properties.The analytical methods are the result of studies of micromechanics,the study of the relationship between effective properties of composites and the properties of the composite constituents. The inhomogeneous composite is represented by a homogeneous anisotropic material with the effective properties of the composite. The purpose of this chapter is to provide an overview of techniques for analysis in the design of com- posite materials.Starting with the micromechanics of fiber and matrix in a lamina,analyses through sim- ple geometric constructions in laminates are considered. A summary is provided at the end of each section for the purpose of highlighting the most important concepts relative to the preceding subject matter.Their purpose is to reinforce the concepts,which can only fully be understood by reading the section. The analysis in this chapter deals primarily with symmetric laminates.It begins with a description of the micromechanics of basic lamina properties and leads into classical laminate analysis theory in an ar- bitrary coordinate system.It defines and compares various failure theories and discusses the response of laminate structures to more complex loads.It highlights considerations of translating individual lamina results into predicted laminate behavior.Furthermore,it covers loading situations and structural re- sponses such as buckling,creep,relaxation,fatigue,durability,and vibration. 5.2 BASIC LAMINA PROPERTIES AND MICROMECHANICS The strength of any given laminate under a prescribed set of loads is probably best determined by conducting a test.However,when many candidate laminates and different loading conditions are being considered,as in a preliminary design study,analysis methods for estimation of laminate strength be- come desirable.Because the stress distribution throughout the fiber and matrix regions of all the plies of a laminate is quite complex,precise analysis methods are not available.However,reasonable methods do exist which can be used to guide the preliminary design process. Strength analysis methods may be grouped into different classes,depending upon the degree of de- tail of the stresses utilized.The following classes are of practical interest: 1.Laminate level.Average values of the stress components in a laminate coordinate system are utilized. 2.Ply,or lamina,level.Average values of the stress components within each ply are utilized. 3.Constituent level.Average values of the stress components within each phase (fiber or matrix)of each ply are utilized. 4.Micro-level.Local stresses of each point within each phase are utilized. Micro-level stresses could be used in appropriate failure criteria for each constituent to determine the external loads at which local failure would initiate.However,the uncertainties,due to departures from the 5-1
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-1 CHAPTER 5 DESIGN AND ANALYSIS 5.1 INTRODUCTION The concept of designing a material to yield a desired set of properties has received impetus from the growing acceptance of composite materials. Inclusion of material design in the structural design process has had a significant effect on that process, particularly upon the preliminary design phase. In this preliminary design, a number of materials will be considered, including materials for which experimental materials property data are not available. Thus, preliminary material selection may be based on analyticallypredicted properties. The analytical methods are the result of studies of micromechanics, the study of the relationship between effective properties of composites and the properties of the composite constituents. The inhomogeneous composite is represented by a homogeneous anisotropic material with the effective properties of the composite. The purpose of this chapter is to provide an overview of techniques for analysis in the design of composite materials. Starting with the micromechanics of fiber and matrix in a lamina, analyses through simple geometric constructions in laminates are considered. A summary is provided at the end of each section for the purpose of highlighting the most important concepts relative to the preceding subject matter. Their purpose is to reinforce the concepts, which can only fully be understood by reading the section. The analysis in this chapter deals primarily with symmetric laminates. It begins with a description of the micromechanics of basic lamina properties and leads into classical laminate analysis theory in an arbitrary coordinate system. It defines and compares various failure theories and discusses the response of laminate structures to more complex loads. It highlights considerations of translating individual lamina results into predicted laminate behavior. Furthermore, it covers loading situations and structural responses such as buckling, creep, relaxation, fatigue, durability, and vibration. 5.2 BASIC LAMINA PROPERTIES AND MICROMECHANICS The strength of any given laminate under a prescribed set of loads is probably best determined by conducting a test. However, when many candidate laminates and different loading conditions are being considered, as in a preliminary design study, analysis methods for estimation of laminate strength become desirable. Because the stress distribution throughout the fiber and matrix regions of all the plies of a laminate is quite complex, precise analysis methods are not available. However, reasonable methods do exist which can be used to guide the preliminary design process. Strength analysis methods may be grouped into different classes, depending upon the degree of detail of the stresses utilized. The following classes are of practical interest: 1. Laminate level. Average values of the stress components in a laminate coordinate system are utilized. 2. Ply, or lamina, level. Average values of the stress components within each ply are utilized. 3. Constituent level. Average values of the stress components within each phase (fiber or matrix) of each ply are utilized. 4. Micro-level. Local stresses of each point within each phase are utilized. Micro-level stresses could be used in appropriate failure criteria for each constituent to determine the external loads at which local failure would initiate. However, the uncertainties, due to departures from the
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis assumed regular local geometry and the statistical variability of local strength make such a process im- practical. At the other extreme,laminate level stresses can be useful for translating measured strengths under single stress component tests into anticipated strength estimates for combined stress cases.However this procedure does not help in the evaluation of alternate laminates for which test data do not exist. Ply level stresses are the commonly used approach to laminate strength.The average stresses in a given ply are used to calculate first ply failure and then subsequent ply failure leading to laminate failure. The analysis of laminates by the use of a ply-by-ply model is presented in Section 5.3 and 5.4. Constituent level,or phase average stresses,eliminate some of the complexity of the micro-level stresses.They represent a useful approach to the strength of a unidirectional composite or ply.Micro- mechanics provides a method of analysis,presented in Section 5.2,for constituent level stresses. Micromechanics is the study of the relations between the properties of the constituents of a composite and the effective properties of the composite.Starting with the basic constituent properties,Sections 5.2 through 6.4 develop the micromechanical analysis of a lamina and the associated ply-by-ply analysis of a laminate. 5.2.1 Assumptions Several assumptions have been made for characterizing lamina properties. 5.2.1.1 Material homogeneity Composites,by definition,are heterogeneous materials.Mechanical analysis proceeds on the as- sumption that the material is homogeneous.This apparent conflict is resolved by considering homogene- ity on microscopic and macroscopic scales.Microscopically,composite materials are certainly heteroge- neous.However,on the macroscopic scale,they appear homogeneous and respond homogeneously when tested.The analysis of composite materials uses effective properties which are based on the aver- age stress and average strain. 5.2.1.2 Material orthotropy Orthotropy is the condition expressed by variation of mechanical properties as a function of orienta- tion.Lamina exhibit orthotropy as the large difference in properties between the 0 and 90 directions.If a material is orthotropic,it contains planes of symmetry and can be characterized by four independent elastic constants. 5.2.1.3 Material linearity Some composite material properties are nonlinear.The amount of nonlinearity depends on the prop- erty,type of specimen,and test environment.The stress-strain curves for composite materials are fre- quently assumed to be linear to simplify the analysis. 5.2.1.4 Residual stresses One consequence of the microscopic heterogeneity of a composite material is the thermal expansion mismatch between the fiber and the matrix.This mismatch causes residual strains in the lamina after curing.The corresponding residual stresses are often assumed not to affect the material's stiffness or its ability to strain uniformly. 5.2.2 Fiber composites:physical properties A unidirectional fiber composite(UDC)consists of aligned continuous fibers which are embedded in a matrix.The UDC physical properties are functions of fiber and matrix physical properties,of their volume 5-2
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-2 assumed regular local geometry and the statistical variability of local strength make such a process impractical. At the other extreme, laminate level stresses can be useful for translating measured strengths under single stress component tests into anticipated strength estimates for combined stress cases. However this procedure does not help in the evaluation of alternate laminates for which test data do not exist. Ply level stresses are the commonly used approach to laminate strength. The average stresses in a given ply are used to calculate first ply failure and then subsequent ply failure leading to laminate failure. The analysis of laminates by the use of a ply-by-ply model is presented in Section 5.3 and 5.4. Constituent level, or phase average stresses, eliminate some of the complexity of the micro-level stresses. They represent a useful approach to the strength of a unidirectional composite or ply. Micromechanics provides a method of analysis, presented in Section 5.2, for constituent level stresses. Micromechanics is the study of the relations between the properties of the constituents of a composite and the effective properties of the composite. Starting with the basic constituent properties, Sections 5.2 through 6.4 develop the micromechanical analysis of a lamina and the associated ply-by-ply analysis of a laminate. 5.2.1 Assumptions Several assumptions have been made for characterizing lamina properties. 5.2.1.1 Material homogeneity Composites, by definition, are heterogeneous materials. Mechanical analysis proceeds on the assumption that the material is homogeneous. This apparent conflict is resolved by considering homogeneity on microscopic and macroscopic scales. Microscopically, composite materials are certainly heterogeneous. However, on the macroscopic scale, they appear homogeneous and respond homogeneously when tested. The analysis of composite materials uses effective properties which are based on the average stress and average strain. 5.2.1.2 Material orthotropy Orthotropy is the condition expressed by variation of mechanical properties as a function of orientation. Lamina exhibit orthotropy as the large difference in properties between the 0° and 90° directions. If a material is orthotropic, it contains planes of symmetry and can be characterized by four independent elastic constants. 5.2.1.3 Material linearity Some composite material properties are nonlinear. The amount of nonlinearity depends on the property, type of specimen, and test environment. The stress-strain curves for composite materials are frequently assumed to be linear to simplify the analysis. 5.2.1.4 Residual stresses One consequence of the microscopic heterogeneity of a composite material is the thermal expansion mismatch between the fiber and the matrix. This mismatch causes residual strains in the lamina after curing. The corresponding residual stresses are often assumed not to affect the material's stiffness or its ability to strain uniformly. 5.2.2 Fiber composites: physical properties A unidirectional fiber composite (UDC) consists of aligned continuous fibers which are embedded in a matrix. The UDC physical properties are functions of fiber and matrix physical properties, of their volume
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis fractions,and perhaps also of statistical parameters associated with fiber distribution.The fibers have,in general,circular cross-sections with little variability in diameter.A UDC is clearly anisotropic since proper- ties in the fiber direction are very different from properties transverse to the fibers. Properties of interest for evaluating stresses and strains are: Elastic properties Viscoelastic properties-static and dynamic Thermal expansion coefficients Moisture swelling coefficients Thermal conductivity Moisture diffusivity A variety of analytical procedures may be used to determine the various properties of a UDC from volume fractions and fiber and matrix properties.The derivations of these procedures may be found in Refer- ences 5.2.2(a)and (b). 5.2.2.1 Elastic properties The elastic properties of a material are a measure of its stiffness.This information is necessary to determine the deformations which are produced by loads.In a UDC,the stiffness is provided by the fi- bers;the role of the matrix is to prevent lateral deflections of the fibers.For engineering purposes,it is necessary to determine such properties as Young's modulus in the fiber direction,Young's modulus trans- verse to the fibers,shear modulus along the fibers and shear modulus in the plane transverse to the fi- bers,as well as various Poisson's ratios.These properties can be determined in terms of simple analyti- cal expressions. The effective elastic stress-strain relations of a typical transverse section of a UDC,based on aver- age stress and average strain,have the form: G11=ne1+0E22+0e33 62=0e1+(k*+G2)E22+(k*-G2)E3时 5.2.2.1(a) 33=CG11+(k-G)e22+(k*+G)e33 612=2Ge12 023=2G2e23 5.2.2.1(b) G13=2G1813 with inverse 1- i2- i2- Ef11E02 Ei033 1 -二-专可11十E支022“E533 5.2.2.1(c) 3.啦、 V23- 1 EOI :Eon+厨0g where an asterisk(*)denotes effective values.Figure 5.2.2.1 illustrates the loadings which are associ- ated with these properties. The effective modulus k'is obtained by subjecting a specimen to the average state of stress E22=33 with all other strains vanishing in which case it follows from Equations 5.2.2.1(a)that (G22+33)=2k*(E22+e33) 5.2.2.1(d) 5-3
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-3 fractions, and perhaps also of statistical parameters associated with fiber distribution. The fibers have, in general, circular cross-sections with little variability in diameter. A UDC is clearly anisotropic since properties in the fiber direction are very different from properties transverse to the fibers. Properties of interest for evaluating stresses and strains are: Elastic properties Viscoelastic properties - static and dynamic Thermal expansion coefficients Moisture swelling coefficients Thermal conductivity Moisture diffusivity A variety of analytical procedures may be used to determine the various properties of a UDC from volume fractions and fiber and matrix properties. The derivations of these procedures may be found in References 5.2.2(a) and (b). 5.2.2.1 Elastic properties The elastic properties of a material are a measure of its stiffness. This information is necessary to determine the deformations which are produced by loads. In a UDC, the stiffness is provided by the fibers; the role of the matrix is to prevent lateral deflections of the fibers. For engineering purposes, it is necessary to determine such properties as Young's modulus in the fiber direction, Young's modulus transverse to the fibers, shear modulus along the fibers and shear modulus in the plane transverse to the fibers, as well as various Poisson's ratios. These properties can be determined in terms of simple analytical expressions. The effective elastic stress-strain relations of a typical transverse section of a UDC, based on average stress and average strain, have the form: 11 * 11 * 22 * 33 22 * 11 * 2 * 22 * 2 * 33 33 * 11 * 2 * 22 * 2 * 33 = n + + = + (k + G ) + (k - G ) = + (k - G ) + (k + G ) σε ε ε σε ε ε σε ε ε A A A A 5.2.2.1(a) 12 1 * 12 23 2 * 23 13 1 * 13 = 2G = 2G = 2G σ ε σ ε σ ε 5.2.2.1(b) with inverse 11 1 * 11 12* 1 * 22 12* 1 * 33 22 12* 1 * 11 2 * 22 23* 2 * 33 33 12* 1 * 11 23* 2 * 22 2 * 33 = 1 E - E - E = - E + 1 E - E = - E - E + 1 E ε σ ν σ ν σ ε ν σ σ ν σ ε ν σ ν σ σ 5.2.2.1(c) where an asterisk (*) denotes effective values. Figure 5.2.2.1 illustrates the loadings which are associated with these properties. The effective modulus k* is obtained by subjecting a specimen to the average state of stress 22 33 ε ε = with all other strains vanishing in which case it follows from Equations 5.2.2.1(a) that ( + ) = 2 k (+) 22 33 * σσ εε 22 33 5.2.2.1(d)
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis Unlike the other properties listed above,k'is of little engineering significance but is of considerable ana- lytical importance. E V12 V2 E2 G FIGURE 5.2.2.1 Basic loading to define effective elastic properties. Only five of the properties in Equations 5.2.2.1(a-c)are independent.The most important interrela- tions of properties are: n=E+4k*3 5.2.2.1(e) 0=2k*vi2 5.2.2.1(⑤ 4 14超 5.2.2.1(g) 2 k =1+ 1 5.2.2.1(h) 1+4k* G E E2 G= 5.2.2.10 2(1+w23) Computation of effective elastic moduli is a very difficult problem in elasticity theory and only a few simple models permit exact analysis.One type of model consists of periodic arrays of identical circular fibers,e.g.,square periodic arrays or hexagonal periodic arrays (References 5.2.2.1(a)-(c)).These models are analyzed by numerical finite difference or finite element procedures.Note that the square ar- ray is not a suitable model for the majority of UDCs since it is not transversely isotropic. The composite cylinder assemblage(CCA)permits exact analytical determination of effective elastic moduli (Reference 5.2.2.1(d)).Consider a collection of composite cylinders,each with a circular fiber core and a concentric matrix shell.The size of the cylinders may vary but the ratio of core radius to shell radius is held constant.Therefore,the matrix and fiber volume fractions are the same in each composite 5-4
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-4 Unlike the other properties listed above, k* is of little engineering significance but is of considerable analytical importance. FIGURE 5.2.2.1 Basic loading to define effective elastic properties. Only five of the properties in Equations 5.2.2.1(a-c) are independent. The most important interrelations of properties are: * 1 * * 12 *2 n = E + 4 k ν 5.2.2.1(e) * * 12* A = 2 k ν 5.2.2.1(f) 4 E = 1 G + 4 2 E * 2 * 12 *2 1 * ν 5.2.2.1(g) 2 1- = 1+ k 1+4 k E G 23* * * 23 *2 1 * 2 * ν F ν H G I K J 5.2.2.1(h) 2 * 2 * 23* G = E 2(1+ ) ν 5.2.2.1(i) Computation of effective elastic moduli is a very difficult problem in elasticity theory and only a few simple models permit exact analysis. One type of model consists of periodic arrays of identical circular fibers, e.g., square periodic arrays or hexagonal periodic arrays (References 5.2.2.1(a) - (c)). These models are analyzed by numerical finite difference or finite element procedures. Note that the square array is not a suitable model for the majority of UDCs since it is not transversely isotropic. The composite cylinder assemblage (CCA) permits exact analytical determination of effective elastic moduli (Reference 5.2.2.1(d)). Consider a collection of composite cylinders, each with a circular fiber core and a concentric matrix shell. The size of the cylinders may vary but the ratio of core radius to shell radius is held constant. Therefore, the matrix and fiber volume fractions are the same in each composite
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis cylinder.One strength of this model is the randomness of the fiber placement,while an undesirable fea- ture is the large variation of fiber sizes.It can be shown that the latter is not a serious concern. The analysis of the CCAgives closed form results for the effective properties.k,and Gi and closed bounds for the properties G2,E2,and v23.Such results will now be listed for isotropic fibers with the necessary modifications for transversely isotropic fibers (References 5.2.2(a)and 5.2.2.1(e)- k'=km(kf+Gm)Vm+kf(km+Gm)vf (kf+Gm)Vm+(km+Gm)vf 5.2.2.10 =km+1 Vf Vm (kf-km)(km+Gm) E时=Emvm+Ervr+灯VmPVmvt Vm+vf+ kf km G 5.2.2.1(k) ≌Emvm+Efvf The last is an excellent approximation for all UDC. (Vi-Vm k Vm Vf vi2 Vm Vm+VfVf+- n kf 5.2.2.10 Vm+Vf+ 1 kf km Gm Gj=Gm GmYm+Gr(1+vr) Gm(1+vf)+GfVm Vf 5.2.2.1(m) =Gm+一 1 +Vm (Gf-Gm)2Gm As indicated earlier in the CCA analysis for G does not yield a result but only a pair of bounds which are in general quite close (References 5.2.2(a),5.2.2.1(d,e)).A preferred alternative is to use a method of approximation which has been called the Generalized Self Consistent Scheme(GSCS).According to this method,the stress and strain in any fiber is approximated by embedding a composite cylinder in the effective fiber composite material.The volume fractions of fiber and matrix in the composite cylinder are those of the entire composite.Such an analysis has been given in Reference 5.2.2(b)and results in a quadratic equation for G2.Thus, G2 +2B G2+C=0 5.2.2.1(n) Gm Gm where A =3vvi(r-1X(r+n) 5.2.2.1(o) t[rnm+nm-(Ynm-n)villvr nm(Y-1)-(rnm+1)] B =-3vv(-1+)++(-1)vr+I](m-1(y+-2(Y7m-7m)v 5.2.2.1(p) +'(nm+1y-1)[y++(ynm-n)] 2 C=3vv (Y-1)(r+n)+[rnm+(r-1)v+l++(rnm-n)vl 5.2.2.1(q) 5-5
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-5 cylinder. One strength of this model is the randomness of the fiber placement, while an undesirable feature is the large variation of fiber sizes. It can be shown that the latter is not a serious concern. The analysis of the CCA gives closed form results for the effective properties, kE n * * * * * ,, ,, 1 ν 12 A and 1 * G and closed bounds for the properties 2 * G , 2 * E , and 23* ν . Such results will now be listed for isotropic fibers with the necessary modifications for transversely isotropic fibers (References 5.2.2(a) and 5.2.2.1(e)). * mf mm f m mf f mm m mf m f f m m m m k = k (k + G ) v + k (k + G )v (k + G ) v + (k + G )v = k + v 1 (k - k ) + v (k + G ) 5.2.2.1(j) 1 * mm ff f m 2 m f f m m m f f E = E v + E v + 4( - ) v v + v k + 1 G ~ E v + E v ν ν v k m f − 5.2.2.1(k) The last is an excellent approximation for all UDC. 12* mm ff f m m f m f m f f m m = v + v + (- ) 1 k - 1 k v v v k + v k + 1 G νν ν ν ν F H G I K J 5.2.2.1(l) 1 * m mm f f m f fm m f f m m m G = G G v + G (1+ v ) G (1+ v ) + G v = G + v 1 (G - G ) + v 2G 5.2.2.1(m) As indicated earlier in the CCA analysis for 2 * G does not yield a result but only a pair of bounds which are in general quite close (References 5.2.2(a), 5.2.2.1(d,e)). A preferred alternative is to use a method of approximation which has been called the Generalized Self Consistent Scheme (GSCS). According to this method, the stress and strain in any fiber is approximated by embedding a composite cylinder in the effective fiber composite material. The volume fractions of fiber and matrix in the composite cylinder are those of the entire composite. Such an analysis has been given in Reference 5.2.2(b) and results in a quadratic equation for 2 * G . Thus, A G G +2B G G + C = 0 2 2 * m 2 * m F H G I K J F H G I K J 5.2.2.1(n) where A = 3 ( -1)( + ) +[ + - ( - ) ][ ( -1) -( +1)] f m2 f m f m m f f 3 f m m ν ν γ γ η γ η ηη γ η η ν ν η γ γ η 5.2.2.1(o) B = - 3 ( -1)( + ) + 1 2 [ + ( -1) +1][( -1)( + ) - 2( - ) ] + 2 ( +1)( -1)[ + + ( - ) ] f m2 f m f m f mm f 3 f m f mf f 3 ν ν γ γ η γ η γ ν η γ η γ η η ν ν η γ γ η γ η η ν 5.2.2.1(p) C = 3 ( -1)( + ) +[ + ( -1) +1][ + + ( - ) ] f m2 f m f f mf f 3 ν ν γ γ η γ η γ ν γ η γ η η ν 5.2.2.1(q)
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis 7=Gf/Gm 5.2.2.1(0 7m=3-4vm 5.2.2.1(s) f=3-4y 5.2.2.1() To compute the resulting E2 and v23,use Equations 5.2.2.1(g-h).It is of interest to note that when the GSCS approximation is applied to those properties for which CCA results are available(see above Equa- tions 5.2.2.1(j-m)),the CCA results are retrieved. For transversely isotropic fibers,the following modifications are necessary (References 5.2.2(a)and 5.2.2.1(e): For k* kr is the fiber transverse bulk modulus For E1.V12 Er=EIf V=VIf kr as above For Gi Gr=Gir For G2 Gt=G2f nr=1+2G2/k Numerical analysis of the effective elastic properties of the hexagonal array model reveals that the values are extremely close to those predicted by the CCA/GSCS models as given by the above equa- tions.The results are generally in good to excellent agreement with experimental data. The simple analytical results given here predict effective elastic properties with sufficient engineering accuracy.They are of considerable practical importance for two reasons.First,they permit easy deter- mination of effective properties for a variety of matrix properties,fiber properties,volume fractions,and environmental conditions.Secondly,they provide the only approach known today for experimental determination of carbon fiber properties. For purposes of laminate analysis,it is important to consider the plane stress version of the effective stress-strain relations.Let x3 be the normal to the plane of a thin unidirectionally-reinforced lamina.The plane stress condition is defined by 033=013=623=0 5.2.2.1(u Then from Equations 5.2.2.1(b-c) 1 e11= Ef O1-E 022 V12- 1 E1+2 E22= 5.2.2.1() 2E2= G The inversion of Equation 5.2.2.1(v)gives 11=Ci1E11+Ci2E22 o22=Ci2E1+C22E22 5.2.2.1(w 612=2G2e12 where 5-6
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-6 γ = Gf m / G 5.2.2.1(r) η m = 3- 4ν m 5.2.2.1(s) f = 3 f η - 4ν 5.2.2.1(t) To compute the resulting 2 * E and 23* ν , use Equations 5.2.2.1(g-h). It is of interest to note that when the GSCS approximation is applied to those properties for which CCA results are available (see above Equations 5.2.2.1(j-m)), the CCA results are retrieved. For transversely isotropic fibers, the following modifications are necessary (References 5.2.2(a) and 5.2.2.1(e)): For k* kf is the fiber transverse bulk modulus For E1 12 * * ,ν Ef = E1f νf = ν1f kf as above For G1 * Gf = G1f For G2 * Gf = G2f ηf = 1 + 2G2f/kf Numerical analysis of the effective elastic properties of the hexagonal array model reveals that the values are extremely close to those predicted by the CCA/GSCS models as given by the above equations. The results are generally in good to excellent agreement with experimental data. The simple analytical results given here predict effective elastic properties with sufficient engineering accuracy. They are of considerable practical importance for two reasons. First, they permit easy determination of effective properties for a variety of matrix properties, fiber properties, volume fractions, and environmental conditions. Secondly, they provide the only approach known today for experimental determination of carbon fiber properties. For purposes of laminate analysis, it is important to consider the plane stress version of the effective stress-strain relations. Let x3 be the normal to the plane of a thin unidirectionally-reinforced lamina. The plane stress condition is defined by 33 13 23 σσσ = = =0 5.2.2.1(u) Then from Equations 5.2.2.1(b-c) 11 1 * 11 12* 2 * 22 22 12* 1 * 11 2 * 22 12 12 1 * = 1 E - E = E + 1 E 2 = G ε σ ν σ ε ν σ σ ε σ 5.2.2.1(v) The inversion of Equation 5.2.2.1(v) gives 11 11* 11 12* 22 22 12* 11 22* 22 12 2 * 12 = C + C = C +C = 2G σεε σ εε σ ε 5.2.2.1(w) where
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis E Cil= 1-yiE吃/E1 2吃 Ci=1-v吃E 5.2.2.1(X) E陵 C1-v吃EEl For polymer matrix composites,at the usual 60%fiber volume fraction,the square of vi2 is close enough to zero to be neglected and the ratio of E2/Ei is approximately 0.1-0.2.Consequently,the following approximations are often useful. Cil~E1 Ci2≈vi2E吃 C22≈E2 5.2.2.1y) 5.2.2.2 Viscoelastic properties The simplest description of time-dependence is linear viscoelasticity.Viscoelastic behavior of poly- mers manifests itself primarily in shear and is negligible for isotropic stress and strain.This implies that the elastic stress-strain relation O11+022+33=3K(E1+e22+e33) 5.2.2.2(a) where K is the three-dimensional bulk modulus,remains valid for polymers.When a polymeric specimen is subjected to shear strain gi2 which does not vary with time,the stress needed to maintain this shear strain is given by o12()=2G()Ei2 5.2.2.2(b) and G(t)is defined as the shear relaxation modulus.When a specimen is subjected to shear stress,oi2. constant in time,the resulting shear strain is given by c2(0=280oi2 5.2.2.2(c) and g(t)is defined as the shear creep compliance. Typical variations of relaxation modulus G(t)and creep compliance g(t)with time are shown in Figure 5.2.2.2.These material properties change significantly with temperature.The relaxation modulus de- creases with increasing temperature and the creep compliance increases with increasing temperature, which implies that the stiffness decreases as the temperature increases.The initial value of these proper- ties at"time-zero"are denoted Go and go and are the elastic properties of the matrix.If the applied shear strain is an arbitrary function of time,commencing at time-zero,Equation 5.2.2.2(b)is replaced by w02G2 5.2.2.2(d) Similarly,for an applied shear stress which is a function of time,Equation 5.2.2.2(c)is replaced by 920=280om20+56 -)dad 5.2.2.2(e) dt' The viscoelastic counterpart of Young's modulus is obtained by subjecting a cylindrical specimen to axial strain gil constant in space and time.Then 11()=E)i1 5.2.2.2(⑤ and E(t)is the Young's relaxation modulus.If the specimen is subjected to axial stress,o11,constant is space and time,then e11(t)=e()oi1 5.2.2.2(g) and e(t)is Young's creep compliance.Obviously E(t)is related to K and G(t),and e(t)is related to k and g(t).(See Reference 5.2.2.2(a).) The basic problem is the evaluation of the effective viscoelastic properties of a UDC in terms of matrix viscoelastic properties and the elastic properties of the fibers.(It is assumed that the fibers themselves do not exhibit any time-dependent properties.)This problem has been resolved in general fashion in Ref- 5-7
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-7 11* 1 * 12 * 2 * 1 12* 12* 2 * 12 * 2 * 1 22* 2 * 12 * 2 * 1 C = E 1- E / E C = E 1- E / E C = E 1- E / E 2 2 2 ν ν ν ν 5.2.2.1(x) For polymer matrix composites, at the usual 60% fiber volume fraction, the square of 12* ν is close enough to zero to be neglected and the ratio of 2 * 1 * E / E is approximately 0.1 - 0.2. Consequently, the following approximations are often useful. 11* * 12* 12* 2 * 22* 2 * C ≈≈ ≈ E1 C E ν C E 5.2.2.1(y) 5.2.2.2 Viscoelastic properties The simplest description of time-dependence is linear viscoelasticity. Viscoelastic behavior of polymers manifests itself primarily in shear and is negligible for isotropic stress and strain. This implies that the elastic stress-strain relation σ11 22 33 11 22 33 +σ +σ = 3K(ε + ε + ε ) 5.2.2.2(a) where K is the three-dimensional bulk modulus, remains valid for polymers. When a polymeric specimen is subjected to shear strain 12° ε which does not vary with time, the stress needed to maintain this shear strain is given by σ ε 12 (t) = 2 G(t) 12° 5.2.2.2(b) and G(t) is defined as the shear relaxation modulus. When a specimen is subjected to shear stress, 12° σ , constant in time, the resulting shear strain is given by 12 12 (t) = 1 2 ε σ g(t) ° 5.2.2.2(c) and g(t) is defined as the shear creep compliance. Typical variations of relaxation modulus G(t) and creep compliance g(t) with time are shown in Figure 5.2.2.2. These material properties change significantly with temperature. The relaxation modulus decreases with increasing temperature and the creep compliance increases with increasing temperature, which implies that the stiffness decreases as the temperature increases. The initial value of these properties at "time-zero" are denoted Go and go and are the elastic properties of the matrix. If the applied shear strain is an arbitrary function of time, commencing at time-zero, Equation 5.2.2.2(b) is replaced by 12 12 o t 12 (t) = 2 G(t) (0) + 2 G(t- t ) d dt’ σ ε dt ε z ′ ′ 5.2.2.2(d) Similarly, for an applied shear stress which is a function of time, Equation 5.2.2.2(c) is replaced by 12 12 o t 12 (t) = 1 2 g(t) (0) + 1 2 g(t- t ) d dt’ ε σ dt σ z ′ ′ 5.2.2.2(e) The viscoelastic counterpart of Young's modulus is obtained by subjecting a cylindrical specimen to axial strain 11° ε constant in space and time. Then σ ε 11(t) = E(t) 11° 5.2.2.2(f) and E(t) is the Young's relaxation modulus. If the specimen is subjected to axial stress, 11° σ , constant is space and time, then ε σ 11(t) = e(t) 11° 5.2.2.2(g) and e(t) is Young's creep compliance. Obviously E(t) is related to K and G(t), and e(t) is related to k and g(t). (See Reference 5.2.2.2(a).) The basic problem is the evaluation of the effective viscoelastic properties of a UDC in terms of matrix viscoelastic properties and the elastic properties of the fibers. (It is assumed that the fibers themselves do not exhibit any time-dependent properties.) This problem has been resolved in general fashion in Ref-
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis erences 5.2.2.2(b)and (c).Detailed analysis shows that the viscoelastic effect in a UDC is significant only for axial shear,transverse shear,and transverse uniaxial stress. For any of average strains E22,23,and 12 constant in time,the time-dependent stress response will be G22()=E2()E22 G23(0=2G2(0e23 5.2.2.2(h) 612)=2Gi(082 =CONST a CONST. c(0)÷a/Ee e(t) CREEP c■CONST, o(0)=E6- 11 a(t) c=CONST. E=CONST. o(t) RELAXATION FIGURE 5.2.2.2 Typical viscoelastic behavior. For any of stresseso22,023,and 12 constant in time,the time-dependent strain response will be E220=e2(0o2 8230=2820o23 5.2.2.20 Tag where material properties in Equations 5.2.2.2(h)are effective relaxation moduli and the properties in Equations 5.2.2.2(i)are effective creep functions.All other effective properties may be considered elastic. This implies in particular that if a fiber composite is subjected to stresso(t)in the fiber direction,then 可11)=Ei810 5.2.2.20) E22(0=E33()=i2E11() 5-8
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-8 erences 5.2.2.2(b) and (c). Detailed analysis shows that the viscoelastic effect in a UDC is significant only for axial shear, transverse shear, and transverse uniaxial stress. For any of average strains 22 23 12 εε ε , ,and constant in time, the time-dependent stress response will be 22 2 * 22 23 2 * 23 12 1 * 12 (t) = E (t) (t) = 2G (t) (t) = 2G (t) σ ε σ ε σ ε 5.2.2.2(h) FIGURE 5.2.2.2 Typical viscoelastic behavior. For any of stresses 22 23 σ σ, , and σ12 constant in time, the time-dependent strain response will be 22 2 * 22 23 2 * 23 12 1 * 12 (t) = e (t) (t) = 1 2 g (t) (t) = 1 2 g (t) ε σ ε σ ε σ 5.2.2.2(i) where material properties in Equations 5.2.2.2(h) are effective relaxation moduli and the properties in Equations 5.2.2.2(i) are effective creep functions. All other effective properties may be considered elastic. This implies in particular that if a fiber composite is subjected to stress 11 σ (t) in the fiber direction, then 11 1 * 11 22 33 12* 11 (t) E (t) (t) = (t) (t) σ ε ε ε ν ε ≈ ≈ 5.2.2.2(j)
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis where Ei and vi2 are the elastic results of Equations 5.2.2.2(k)with matrix properties taken as initial (elastic)matrix properties.Similar considerations apply to the relaxation modulus k. The simplest case of the viscoelastic properties entering into Equations 5.2.2.2(h-i)is the relaxation modulus Gi(t)and its associated creep compliance gi(t).A very simple result has been obtained for fi- bers which are infinitely more rigid than the matrix (Reference 5.2.2(a)).For a viscoelastic matrix,the results reduce to Gi()=Gm(1+vt 1-vf 5.2.2.2(k) g1()=gm(0) 1-vf 1+vf This results in an acceptable approximation for glass fibers in a polymeric matrix and an excellent ap- proximation for boron fibers in a polymeric matrix.However,the result is not applicable to the case of carbon or graphite fibers in a polymeric matrix since the axial shear modulus of these fibers is not large enough relative to the matrix shear modulus.In this case,it is necessary to use the correspondence prin- ciple mentioned above (References 5.2.2(a)and 5.2.2.2(b)).The situation for transverse shear is more complicated and involves complex Laplace transform inversion.(Reference 5.2.2.2(c)). All polymeric matrix viscoelastic properties such as creep and relaxation functions are significantly temperature dependent.If the temperature is known,all of the results from this section can be obtained for a constant temperature by using the matrix properties at that temperature.At elevated temperatures, the viscoelastic behavior of the matrix may become nonlinear.In this event,the UDC will also be nonlinearly viscoelastic and all of the results given here are not valid.The problem of analytical determi- nation of nonlinear properties is,of course,much more difficult than the linear problem(See Reference 5.2.2.2(d). 5.2.2.3 Thermal expansion and moisture swelling The elastic behavior of composite materials discussed in Section 5.2.2.1 is concerned with externally applied loads and deformations.Deformations are also produced by temperature changes and by ab- sorption of moisture in two similar phenomena.A change of temperature in a free body produces thermal strains while moisture absorption produces swelling strains.The relevant physical parameters to quantify these phenomena are thermal expansion coefficients and swelling coefficients. Fibers have significantly smaller thermal expansion coefficients than do polymeric matrices.The ex- pansion coefficient of glass fibers is 2.8 x 10 in/in/F(5.0 x 10m/m/C)while a typical epoxy value is 30 x 10 in/in/F(54 x 10 m/m/C).Carbon and graphite fibers are anisotropic in thermal expansion.The expansion coefficients in the fiber direction are extremely small,either positive or negative of the order of 0.5 x 10 in/in/F(0.9 x 10 m/m/C).To compute these stresses,it is necessary to know the thermal expansion coefficients of the layers.Procedures to determine these coefficients in terms of the elastic properties and expansion coefficients of component fibers and matrix are discussed in this section. When a laminate absorbs moisture,there occurs the same phenomenon as in the case of heating. Again,the swelling coefficient of the fibers is much smaller than that of the matrix.Free swelling of the layers cannot take place and consequently internal stresses develop.These stresses can be calculated if the UDC swelling coefficients are known. Consider a free cylindrical specimen of UDC under uniform temperature change AT.Neglecting tran- sient thermal effects,the stress-strain relations(Equation 5.2.2.1(c))assume the form 5-9
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-9 where 1 * E and 12* ν are the elastic results of Equations 5.2.2.2(k) with matrix properties taken as initial (elastic) matrix properties. Similar considerations apply to the relaxation modulus k* . The simplest case of the viscoelastic properties entering into Equations 5.2.2.2(h-i) is the relaxation modulus 1 * G (t) and its associated creep compliance 1 *g (t). A very simple result has been obtained for fibers which are infinitely more rigid than the matrix (Reference 5.2.2(a)). For a viscoelastic matrix, the results reduce to 1 * m f f 1 * m f f G (t) = G (t)1+ v 1- v g (t) = g (t) 1- v 1+ v 5.2.2.2(k) This results in an acceptable approximation for glass fibers in a polymeric matrix and an excellent approximation for boron fibers in a polymeric matrix. However, the result is not applicable to the case of carbon or graphite fibers in a polymeric matrix since the axial shear modulus of these fibers is not large enough relative to the matrix shear modulus. In this case, it is necessary to use the correspondence principle mentioned above (References 5.2.2(a) and 5.2.2.2(b)). The situation for transverse shear is more complicated and involves complex Laplace transform inversion. (Reference 5.2.2.2(c)). All polymeric matrix viscoelastic properties such as creep and relaxation functions are significantly temperature dependent. If the temperature is known, all of the results from this section can be obtained for a constant temperature by using the matrix properties at that temperature. At elevated temperatures, the viscoelastic behavior of the matrix may become nonlinear. In this event, the UDC will also be nonlinearly viscoelastic and all of the results given here are not valid. The problem of analytical determination of nonlinear properties is, of course, much more difficult than the linear problem (See Reference 5.2.2.2(d)). 5.2.2.3 Thermal expansion and moisture swelling The elastic behavior of composite materials discussed in Section 5.2.2.1 is concerned with externally applied loads and deformations. Deformations are also produced by temperature changes and by absorption of moisture in two similar phenomena. A change of temperature in a free body produces thermal strains while moisture absorption produces swelling strains. The relevant physical parameters to quantify these phenomena are thermal expansion coefficients and swelling coefficients. Fibers have significantly smaller thermal expansion coefficients than do polymeric matrices. The expansion coefficient of glass fibers is 2.8 x 10-6 in/in/F° (5.0 x 10-6 m/m/C°) while a typical epoxy value is 30 x 10-6 in/in/F° (54 x 10-6 m/m/C°). Carbon and graphite fibers are anisotropic in thermal expansion. The expansion coefficients in the fiber direction are extremely small, either positive or negative of the order of 0.5 x 10-6 in/in/F° (0.9 x 10-6 m/m/C°). To compute these stresses, it is necessary to know the thermal expansion coefficients of the layers. Procedures to determine these coefficients in terms of the elastic properties and expansion coefficients of component fibers and matrix are discussed in this section. When a laminate absorbs moisture, there occurs the same phenomenon as in the case of heating. Again, the swelling coefficient of the fibers is much smaller than that of the matrix. Free swelling of the layers cannot take place and consequently internal stresses develop. These stresses can be calculated if the UDC swelling coefficients are known. Consider a free cylindrical specimen of UDC under uniform temperature change ∆T. Neglecting transient thermal effects, the stress-strain relations (Equation 5.2.2.1(c)) assume the form
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis 1 E时o1 1 Ej 022- Ei033+aiAT 22. 1 EO11+ V23 臣o2Eo3+ai△T 5.2.2.3(a） i2-V23- 1 E33= Eio1Eo2 E陵o33+吱4T where a-effective axial expansion coefficient a2-effective transverse expansion coefficient It has been shown by Levin (Reference 5.2.2.3(a))that there is a unique mathematical relationship between the effective thermal expansion coefficients and the effective elastic properties of a two-phase composite.When the matrix and fibers are isotropic ai=am+ af-am 3(1-2y12) 1 11 E Km] Kf Km 5.2.2.3(b) antag [331-2i2)1 2k* E Km」 Kf Km where am,af matrix,fiber isotropic expansion coefficients Km,Kf matrix,fiber three-dimensional bulk modulus Ei,vi2.k" effective axial Young's modulus,axial Poisson's ratio, and transverse bulk modulus These equations are suitable for glass/epoxy and boron/epoxy.They have also been derived in Refer- ences 5.2.2.3(b)and(c).For carbon and graphite fibers,it is necessary to consider the case of trans- versely isotropic fibers.This complicates the results considerably as shown in Reference 5.2.2.1(c)and (e). Frequently thermal expansion coefficients of the fibers and matrix are functions of temperature.It is not difficult to show that Equations 5.2.2.3(b)remain valid for temperature-dependent properties if the elastic properties are taken at the final temperature and the expansion coefficients are taken as secant at that temperature. To evaluate the thermal expansion coefficients from Equation 5.2.2.3(b)or (c),the effective elastic properties,k',El,and vi2 must be known.These may be taken as the values predicted by Equations 5.2.2.1(j-I)with the appropriate modification when the fibers are transversely isotropic.Figures 5.2.2.3(a) and(b)shows typical plots of the effective thermal expansion coefficients of graphite/epoxy. When a composite with polymeric matrix is placed in a wet environment,the matrix will begin to ab- sorb moisture.The moisture absorption of most fibers used in practice is negligible;however,aramid fi- bers alone absorb significant amounts of moisture when exposed to high humidity.The total moisture absorbed by an aramid/epoxy composite,however,may not be substantially greater than other epoxy composites. 5-10
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-10 11 1 * 11 12* 1 * 22 12* 1 * 33 1 * 22 12* 1 * 11 2 * 22 23* 2 * 33 2* 33 12* 1 * 11 23* 2 * 22 2 * 33 2 * = 1 E - E - E + T = - E + 1 E - E + T = - E - E + 1 E + T ε σ ν σ ν σ α ε ν σ σ ν σ α ε ν σ ν σ σ α ∆ ∆ ∆ 5.2.2.3(a) where 1 * α - effective axial expansion coefficient 2 * α - effective transverse expansion coefficient It has been shown by Levin (Reference 5.2.2.3(a)) that there is a unique mathematical relationship between the effective thermal expansion coefficients and the effective elastic properties of a two-phase composite. When the matrix and fibers are isotropic 1 * m f m f m 12* 1 * m 2 * m f m f m * 12* 1 * m = + - 1 K - 1 K 3(1- 2 ) E - 1 K = + - 1 K - 1 K 3 2 k - 3(1- 2 ) E - 1 K α α αα ν α α αα ν L N M O Q P L N M O Q P 5.2.2.3(b) where m f α ,α - matrix, fiber isotropic expansion coefficients Km f ,K - matrix, fiber three-dimensional bulk modulus 1 * 12* * E , , ν k - effective axial Young's modulus, axial Poisson's ratio, and transverse bulk modulus These equations are suitable for glass/epoxy and boron/epoxy. They have also been derived in References 5.2.2.3(b) and (c). For carbon and graphite fibers, it is necessary to consider the case of transversely isotropic fibers. This complicates the results considerably as shown in Reference 5.2.2.1(c) and (e). Frequently thermal expansion coefficients of the fibers and matrix are functions of temperature. It is not difficult to show that Equations 5.2.2.3(b) remain valid for temperature-dependent properties if the elastic properties are taken at the final temperature and the expansion coefficients are taken as secant at that temperature. To evaluate the thermal expansion coefficients from Equation 5.2.2.3(b) or (c), the effective elastic properties, * 1 * k ,E , and 12* ν must be known. These may be taken as the values predicted by Equations 5.2.2.1(j-l) with the appropriate modification when the fibers are transversely isotropic. Figures 5.2.2.3(a) and (b) shows typical plots of the effective thermal expansion coefficients of graphite/epoxy. When a composite with polymeric matrix is placed in a wet environment, the matrix will begin to absorb moisture. The moisture absorption of most fibers used in practice is negligible; however, aramid fibers alone absorb significant amounts of moisture when exposed to high humidity. The total moisture absorbed by an aramid/epoxy composite, however, may not be substantially greater than other epoxy composites