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)