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1)For the composite properties and environmental conditions described in Examples 3

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1)For the composite properties and environmental conditions described in Examples 3.6, 4.7, and 5.3, compare the reference and hygrothermally degraded values of the longitudinal compressive strength. Assume ν12 = 0.3. Compare and discuss the different effects that hygrothermal conditions have on longitudinal tensile and compressive strengths.

2)For the material described in Problems 5.2 and 5.3 above at a temperature of 77°C, determine the time required for drying the material from 99.9% to 50% of its fully saturated equilibrium moisture content.

3)Using Equation 5.6 for moisture diffusion, derive an equation for the time required for an initially dry material to reach 99.9% of its fully saturated equilibrium moisture content. The series in Equation 5.6 converges rapidly, so for the purposes of this problem, it is necessary only to consider the first term. The answer should be expressed in terms of the thickness, h, and the diffusivity, Dz

4)Repeat Example 5.1 for the cases where the hygrothermal conditions change from (a) RT dry to 300°F dry, and (b) from RT dry to RT wet, then compare with the results from Example 5.1. That is, compare the effect of combined elevated temperature and moisture (Example 5.1) on the moduli of the epoxy matrix and the transverse composite with the separate effects of elevated temperature (Problem 5)1(a)) and elevated moisture (Problem 5.1(b)) on these moduli.The composite described in Examples 3.2, 3.3, and 3.6 is to be used in a “hot–wet” environment with temperature T = 200°F (93°C) and resin moisture content Mr = 3%. If the glass transition temperature of the dry matrix resin is 350°F (177°C) and if the properties given in Examples 3.2, 3.3, and 3.6 are for a temperature of 70°F (21°C), determine the hygrothermally degraded values of the longitudinal and transverse moduli

6)An epoxy resin sample has a thickness h = 5 mm and a diffusivity D = 3 × 10−8 mm2 /s. Determine the moisture absorption of an initially dry sample after a time t = 100 days.

7)For a unidirectional IM-9/8551-7 carbon/epoxy composite under tensile loading, what is the critical fiber volume fraction, υfcrit, when the mode of failure is fiber failure as described in Figure 4.14a? How does this value of υfcrit compare with the typical fiber volume fraction, υf , for IM-9/8551-7 as  described in Table 4.1, and what is the physical meaning of this observation

8)Compare and discuss the estimated longitudinal compressive strengths of Scotchply 1002 E-glass/epoxy based on (a) fiber microbuckling and (b) transverse tensile rupture. Assume linear elastic behavior to failure. For the epoxy matrix, assume that the modulus of elasticity is Em = 3.79 GPa, the Poisson’s ratio is νm = 0.35, and the fiber volume fraction is υf = 0.45.

9)An element of an orthotropic lamina is subjected to an off-axis shear stress, τxy, as shown in Figure 4.7a. Using the Tsai–Hill criterion and assuming that the lamina strengths are the same in tension and compression, develop an equation relating the allowable value of τxy to the lamina strengths, sL, sT, and sLT, and the fiber orientation θ.

10)The Tsai–Wu interaction parameter F12 is determined from biaxial failure stress data. One way to generate a biaxial state of stress is by using a uniaxial 45° off-axis tension test. Derive the expression for F12 based on such a test, assuming that all the uniaxial and shear strengths are known.

11)An element of an orthotropic lamina having the properties given in Problem 4.3 is subjected to an off-axis tensile test, as shown in Figure 4.5. Using the maximum strain criterion, determine the values of σx at failure and the mode of failure for each of the following values of the angle θ: (a) 2°, (b) 30°, and (c) 75°.

12)An orthotropic lamina has the following properties: E s 1 = = 16 GPa M L Pa + 0 1800 ( ) Strength of a Continuous Fiber-Reinforced Lamina 165 E s 2 = = 1 GPa L MPa − 0 1400 ( ) ν12 = = 3 M T Pa + 0 4 . 0 ( ) s G s 12 = = 7 GPa T MPa ( ) − 230 sLT = 1 M 00 Pa Construct the failure surfaces in the σ1 − σ2 stress space for this material according to: (a) the maximum stress criterion, (b) the maximum strain criterion, and (c) the Tsai–Hill criterion.

13)If the applied stress σx is compressive in Problem 4.1, use the maximum stress criterion to determine (a) the value of the applied stress σx that would be required to produce longitudinal compressive failure, (b) the value of the applied stress σx that would be required to produce in-plane shear failure, (c) the value of the applied stress σx that would be required to produce transverse compressive failure, and (d) which of the failure modes described in parts (a), (b), and (c) would actually cause failure of the specimen, and why

14)Using the maximum strain criterion, determine the uniaxial failure stress, σx, for off-axis loading of the unidirectional lamina in Figure 4.5 if the material is AS/3501 carbon/epoxy and the angle θ = 30°.

15)The filament-wound pressure vessel described in Example 2.5 is fabricated from E-glass/epoxy having the lamina strengths listed in Table 4.1. Determine the internal pressure p, which would cause failure of the vessel according to (a) the maximum stress criterion and (b) the Tsai–Hill criterion.

16)A generally orthotropic lamina made from E-glass/470-36 vinylester composite material has the strength properties listed in Table 4.1 and the lamina orientation is θ = 30°. If the applied stresses in the off-axis coordinate system x,y are σx = 200 MPa, σy = −100 MPa, and τxy = 100 MPa, determine whether the lamina will fail or not according to the maximum stress criterion.

17)The fibers in a E-glass/epoxy composite are 0.0005 in. (0.0127 mm) in diameter before coating with an epoxy sizing 0.0001 in. (0.00254 mm) thick. After the sizing has been applied, the fibers are bonded together with more epoxy of the same type. What is the maximum fiber volume fraction that can be achieved? Using the fiber and matrix moduli given in Equation 3.33, determine the composite longitudinal modulus E1 and the composite transverse modulus E2 corresponding to the maximum fiber volume fraction.

18)Using the method of subregions, derive an equation for the transverse modulus, E2, for the RVE, which includes a fiber/matrix interphase region, as shown in Figure 3.27. Hint: The equation should reduce to Equation 3.54 when the fiber diameter is the same as the interphase diameter.

19)The Tsai–Hahn stress-partitioning factor for transverse loading, η2, defined in Equation 3.65 is used to derive Equation 3.66 for the transverse composite modulus, E2. Using a similar approach with a strainpartitioning parameter for longitudinal loading, η1, derive an equation for the longitudinal composite modulus, E1

20)Using an elementary mechanics of materials approach, find the micromechanics equation for predicting the minor Poisson’s ratio, ν21, for a unidirectional fiber composite in terms of the corresponding fiber and matrix properties and volume fractions. Assume that the fibers are orthotropic, the matrix is isotropic, and all materials are linear elastic. This derivation should be independent of the one in Problem 3.5.