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