Questions and Answers of Principles Of Composite

Describe an experiment, and give the necessary equations for the measurement of the complex extensional (or longitudinal) modulus, $E_{x}^{*}$, of a symmetric laminated bar.
Describe an experiment, and give the necessary equations for the measurement of the complex through-thickness shear modulus, $G_{23}^{*}$, of a unidirectional, specially orthotropic, transversely
Describe an experiment, and give the necessary equations for measurement of the complex Young's modulus, $E_{\mathrm{m}}^{*}$, of an isotropic matrix material.
Describe an experiment, and give the necessary equations for the measurement of the complex longitudinal modulus, $E_{\mathrm{f} 1}^{*}$, of a reinforcing fiber.
Describe an experiment, and give the necessary equations for measurement of the through-thickness creep compliance, $S_{32}(t)$, of a specially orthotropic, transversely isotropic lamina. The plate
The specimen geometry and the frequency response curve for the second mode flexural vibration of a laminated composite cantilever beam specimen is shown in Figure 10.48. The specimen has length
Samples of unidirectional Kevlar 49/epoxy and S-glass/epoxy composites are subjected to elevated temperatures in an oven and the resulting thermal strains are measured by using strain gages oriented
For the linear part of the moisture absorption curve for a temperature of $77^{\circ} \mathrm{C}$ in Figure 5.12, and assuming a specimen thickness of $2.54 \mathrm{~mm}$, use the relevant
Repeat Example 5.1 for the cases where the hygrothermal conditions change from (a) RT dry to $300^{\circ} \mathrm{F} \mathrm{dry}$, and (b) from RT dry to RT wet, then compare with the results from
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
The dependence of the transverse (through-the-thickness) diffusivity of unidirectional AS/3501-5 graphite/epoxy composite on temperature is given in Figure 5.11. For a temperature of \(77^{\circ}
For the material described in Problems 5.2 and 5.3 above at a temperature of $77^{\circ} \mathrm{C}$, determine the time required for drying the material from $99.9 \%$ to $50 \%$ of its fully
Using only the linear part of the moisture absorption curve for a temperature of $77^{\circ} \mathrm{C}$ in Figure 5.12, and assuming a thickness of $2.54 \mathrm{~mm}$, estimate the diffusivity
For the composite properties and environmental conditions described in Examples 3.6, 4.7, and 5.3, determine the hygrothermally degraded values of the longitudinal and transverse tensile strengths.
The filament-wound E-glass/epoxy pressure vessel described in Example 4.4 is to be used in a hot-wet environment with temperature $T=100^{\circ} \mathrm{F}$ $\left(38^{\circ} \mathrm{C}\right)$
An orthotropic lamina forms one layer of a laminate which is initially at temperature $T_{0}$. Assuming that the lamina is initially stress free, that the adjacent laminae are rigid, that the
A carbon/epoxy lamina is clamped between rigid plates in a mold (Figure 5.17), while curing at a temperature of $125^{\circ} \mathrm{C}$. After curing, the lamina/mold assembly (still clamped
Samples of unidirectional Kevlar 49/epoxy and S-glass/epoxy composites are subjected to elevated temperatures in an oven and the resulting thermal strains are measured by using strain gages oriented
A unidirectional $45^{\circ}$ off-axis E-glass/epoxy composite lamina is supported on frictionless rollers between rigid walls as shown in Figure 5.18. The lamina is fixed against displacements in
An orthotropic lamina has thermal expansion coefficients $\alpha_{1}=-4.0$ $\left(10^{-6}\right) \mathrm{m} / \mathrm{m} / \mathrm{K}$ and \(\alpha_{2}=79\left(10^{-6}\right) \mathrm{m} /
A carbon/epoxy lamina having the properties listed in Problem 5.10 is clamped between two rigid plates as shown in Figure 5.17. If the lamina is heated from $20^{\circ} \mathrm{C}$ to \(120^{\circ}
A unidirectional E-glass/epoxy lamina is securely attached to rigid supports on both ends in two different ways as shown in Figure 5.19. In Figure 5.19a, the fiber direction is perpendicular to the
Derive Equation 5.35. 1 = Efiff +Emim1vm Ef1 Vf+ Em1Vm (5.35)
Develop an analytical model for determination of the CTE, $\alpha$, for a randomly oriented continuous fiber composite in terms of fiber and matrix properties and volume fractions. Assume that the
A unidirectional graphite/epoxy lamina having the properties described in Problem 5.10 is to be designed to have a CTE of zero along a particular axis. Determine the required lamina orientation for
A representative volume element (RVE) consisting of a cylindrical isotropic fiber embedded and perfectly bonded in a cylinder of isotropic matrix material is shown in Figure 5.23. If the ends of the
Derive Equations 5.26 and 5.30 . == Ef10f1Vf+ Em1m1Vm_ Ef1ff + Em1m1vm 1 Ef1Vf+ Em1Vm (5.26)
The interfacial shear stresses, $\tau$, and the fiber normal stresses, $\sigma_{f}$ acting on a differential element at a distance $x$ from the end of the fiber are shown in Figure 6.4, where
A short-fiber composite is to be modeled using the RVE in Figure 6.2b. Assuming that the matrix is rigid plastic in shear but that both the fiber and matrix are elastic in extension, develop an
Using the result from Problem 6.2, develop an expression for the longitudinal modulus of the RVE shown in Figure 6.2a that includes the effect of the matrix material at the fiber ends.Problem 6.2A
A carbon/epoxy single fiber test specimen is subjected to a uniaxial tensile stress that is increased until the fiber breaks up into pieces having a length of $0.625 \mathrm{~mm}$. If the fiber has
A linear elastic fiber of rectangular cross section is embedded in a linear elastic matrix material, and the composite is subjected to a uniaxial stress as shown in Figure 6.25a. The interfacial
The RVE for an aligned discontinuous fiber composite without matrix material at its ends is shown in Figure 6.4. Assume that when the RVE is loaded along the fiber direction, the interfacial shear
For the RVE in Figure 6.4, assume that the fiber length is greater than the ineffective length, and that the distribution of the fiber tensile normal stress is given by\[\begin{gathered}
In order to reduce material costs, a composite panel is to be made by placing fibers in the matrix material in an X-pattern of $\pm \alpha$ as shown in Figure 6.46, instead of randomly distributing
Determine the CTE for a randomly oriented fiber composite in terms of the longitudinal and transverse CTEs $\alpha_{1}$ and $\alpha_{2}$ of the corresponding unidirectional composite lamina.
Using micromechanics and the Tsai-Hill criterion, set up the equation for the averaged isotropic tensile strength for a randomly oriented short-fiber composite. The equation should be in terms of
Set up the equations for predicting the averaged isotropic shear modulus of a randomly oriented short-fiber composite. Your answer should be in terms of the fiber and matrix properties and volume
Using the Tsai-Hill criterion and the appropriate micromechanics equations, set up the equation for predicting the averaged isotropic shear strength for a randomly oriented short-fiber composite.
Verify the three predictions (i.e., Equations 3.27, 3.40, and 6.67) for the Young's modulus of the glass microsphere-reinforced polyester composite in Figure 6.42 for the specific case of a particle
Derive Equation 6.68. Veff 1+ Dact AR -(1.4) (6.68) R
A rectangular array of elliptical fibers is shown in Figure 3.5. Derive the relationship between the fiber volume fraction and the given geometrical parameters. What is the maximum possible fiber
A face-centered cubic array of round fibers is shown in Figure 3.6. Derive the relationship between the fiber volume fraction and the given geometrical parameters. What is the maximum possible fiber
A hybrid carbon-aramid/epoxy composite is made by randomly mixing continuous aligned fibers of the same diameter, so that there are two carbon fibers for each aramid fiber. The fibers are assumed to
Derive Equation 3.45. V12 = V12Vf+ VmVm (3.45)
Derive Equation 3.47. 1 Um +. (3.47) G12 Gf12 Gm
Using an elementary mechanics of materials approach, find the micromechanics equation for predicting the minor Poisson's ratio, $v_{21}$, for a unidirectional fiber composite in terms of the
A composite shaft is fabricated by bonding an isotropic solid shaft having shear modulus $G_{1}$, and outside radius $r_{1}$, inside a hollow isotropic shaft having shear modulus $G_{2}$ and
An RVE from a particle-reinforced composite is shown in Figure 3.12. The particle has a cross-sectional area $A_{\mathrm{p}}(x)$ that varies with the distance $x$, and the stresses and strains in
Using the result from Problem 3.9, determine the effective Young's modulus, $E_{x}$, for the RVE shown in Figure 3.13. In Figure 3.13, the reinforcing particle has a square cross section and is
A unidirectional composite is to be modeled by the RVE shown in Figure 3.14a, where the fiber and matrix materials are assumed to be isotropic and perfectly bonded together. Using a mechanics of
Figure 3.15 shows an RVE for an elementary mechanics of materials model of the same type as shown in Figure 3.7, but with transverse deformation along the 2 direction prevented by rigid supports
Using the method of subregions, derive an equation for the transverse modulus, $E_{2}$, for the RVE, which includes a fiber/matrix interphase region, as shown in Figure 3.27. Matrix Fiber Sed d
Derive Equation 3.59. me me + ax ay =0 (3.59)
For a unidirectional composite with a rectangular fiber array (Figure 3.18), use the equations of elasticity to set up the displacement boundary value problem for the determination of the transverse
For the quarter domain of an RVE in Figure 3.23, a uniform transverse normal stress $\bar{\sigma}_{x}$ is applied on the plane $x=D / 2$. Set up the equations describing the boundary conditions
The fibers in a E-glass/epoxy composite are 0.0005 in. $(0.0127 \mathrm{~mm})$ in diameter before coating with an epoxy sizing $0.0001 \mathrm{in} .(0.00254 \mathrm{~mm})$ thick. After the sizing
Derive Equation 3.66. Uf 2vm + (3.66) E2 Vi+n2vm Et Em
Show that a value of $\xi=0$ reduces the Halpin-Tsai equation (Equation 3.63) to the inverse rule of mixtures Equation 3.40, whereas a value $\xi=\infty$ reduces it to the rule of mixtures
An orthotropic lamina has the following properties:\[E_{1}=160 \mathrm{GPa} \quad s_{\mathrm{L}}^{(+)}=1800 \mathrm{MPa}\]\[\begin{aligned} & E_{2}=10 \mathrm{GPa} \quad s_{\mathrm{L}}^{(-)}=1400
Using the material properties from Problem 4.3 and assuming that the stiffnesses are the same in tension and compression, determine the allowable off-axis shear stress, $\tau_{x y}$ at
An element of a balanced orthotropic lamina is under the state of stress shown in Figure 4.11. The properties of the lamina are:Using the maximum strain criterion, determine whether or not failure
If some of the compliances and strengths of an orthotropic lamina satisfy certain conditions, the maximum strain criterion failure surface will intercept the horizontal axis at a point like
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
Repeat Problem 4.7 for an off-axis compression test.Problem 4.7An element of an orthotropic lamina having the properties given in Problem 4.3 is subjected to an off-axis tensile test, as shown in
A material having the properties given in Problem 4.3 is subjected to a biaxial tension test, and the biaxial failure stress is found to be $\sigma_{1}=\sigma_{2}=35 \mathrm{MPa}$. Determine the
The Tsai-Wu interaction parameter $F_{12}$ is determined from biaxial failure stress data. One way to generate a biaxial state of stress is by using a uniaxial $45^{\circ}$ off-axis tension test.
An element of an orthotropic lamina is subjected to an off-axis shear stress, $\tau_{x y}$, as shown in Figure 4.7a. Using the Tsai-Hill criterion and assuming that the lamina strengths are the
A uniaxial off-axis tensile test is conducted as shown in Figure 4.5. Using the Tsai-Hill criterion and assuming that the lamina strengths are the same in tension and compression, develop an equation
An orthotropic AS/3501 carbon/epoxy lamina (see Tables 2.2 and 4.1) is subjected to the plane stress condition $\sigma_{x}=1000 \mathrm{MPa}, \sigma_{y}=50 \mathrm{MPa}$, and \(\tau_{x y}=50
Determine the longitudinal tensile strength of the hybrid carbon/aramid/ epoxy composite described in Problem 3.4 and Figure 3.10 of Chapter 3 if the fiber packing array is square with the closest
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
Using the maximum strain criterion and micromechanics, set up the equations for predicting the averaged isotropic strength of a randomly oriented continuous fiber composite. Your answer should be
Assuming that the failure mode for longitudinal compression of unidirectional E-glass/epoxy with fiber volume fraction $v_{\mathrm{f}}=0.6$ is a transverse tensile rupture due to Poisson strains,
For the IM-9/8551-7 carbon/epoxy composite rod design of Problem 3.3 in Chapter 3, what would be the increase in the longitudinal tensile strength compared with that of the original 6061-T6 aluminum
For a cylindrical particle, derive the relationship between the ratio of surface area to volume, A/V, and the particle aspect ratio, l/d, and verify the shape of the curve shown in Figure 1.3. FIGURE
Compare the total surface area of a group of N small-diameter spherical particles with that of a single large-diameter spherical particle having the same volume.
Explain qualitatively why sandwich structures (Figure 1.5) have such high flexural stiffness-to-weight ratios. Describe the key parameters affecting the flexural stiffness-to-weight ratio of a
A support cable in a structure must be 5 m long and must withstand a tensile load of 5 kN with a safety factor of 2.0 against tensile failure. Assuming a solid cylindrical cross section for the cable
A flywheel for energy storage is modeled as a rotating thin-walled cylindrical ring (t ≪ r) as shown in Figure 1.7. Find the equation for the tensile stress in the ring as a function of the mean
Describe some practical physical limitations on the use of the RRIM processin molding of chopped FRP matrix composites.
Describe some practical physical limitations on the use of the RTM process in fabrication of composite sandwich structures which consist of composite face sheets and foam or honeycomb core.
Describe a possible sequence of fabrication processes that might be used to manufacture the helicopter rotor blade in Figure 1.12. Note that several different materials and fiber lay-ups are used.
A thin-walled filament-wound composite pressure vessel has fibers wound at a helical angle θ, as shown in Figure 1.45. Ignore the resin matrix material and assume that the fibers carry the entire
A filament-wound E-glass/epoxy pressure vessel has a diameter of 50 in. (127 cm), a wall thickness of 0.25 in. (6.35 mm), and a helical wrap angle θ = 54.74°. Using a netting analysis and a safety
The 2000 mm long composite bar shown in Figure 1.46 consists of an aluminum bar having a modulus of elasticity EAl = 70 GPa and length LAl = 500 mm, which is securely fastened to a steel bar having
The concrete composite post in Figure 1.47 is 1.2 m long with a 0.3 m × 0.3 m square cross section. The post is reinforced by four vertical steel rods of the same length having a cross-sectional
The composite bar system in Figure 1.48 consists of a steel bar and a bronze bar that are both securely attached to a rigid block and rigid supports. The system is loaded with a total load P at the
The composite post in Figure 1.49 has the same properties and dimensions as in Problem 1.13, except that there is a gap Δ = 0.1 mm between the top of the cover plate on the post and the upper
For the original concrete composite post design of Problem 1.13, assume that the steel rods are made of 4340 steel, and that the rods are to be replaced by IM9 carbon fiber bundles of the same
For the original 4340 steel-reinforced concrete post design of Problem 1.13 and the new IM9 carbon fiber-reinforced concrete post design of Problem 1.16, compare the tensile stress-to-tensile
Using an example of static equilibrium of an element in pure two-dimensional (2D) shear stress, prove that the shear stresses are symmetric (i.e., prove that $\sigma_{i j}=\sigma_{j i}$ when \(i eq
A representative section from a composite lamina is shown in Figure 2.7 along with the transverse stress and strain distributions across the fiber and matrix materials in the section. The composite
The stiffness matrix $[C]$ for a specially orthotropic material associated with the principal material axes $(1,2,3)$ is given in Equation 2.16. Prove that, when the material is specially
A balanced orthotropic, or square symmetric lamina, is made up of $0^{\circ}$ and $90^{\circ}$ fibers woven into a fabric and bonded together, as shown in Figure 2.10.a. Describe the
Derive Equation 2.27. = S22 E S11S22 S2 1-V12V21 - S12 V12E2 SS22 S2 1-V12V21 S11 - S11S22-S2 1 = == S66 G12 = E2 1-V12V21 Q11 = Q12= Q22 Q66 = Q21 (2.27)
Find all components of the stiffness and compliance matrices for a specially orthotropic lamina made of AS/3501 carbon/epoxy.
For a specially orthotropic, transversely isotropic material the "plane strain bulk modulus," $K_{23}$, is an engineering constant that is defined by the stress conditions
Describe the measurements that would have to be taken and the equations that would have to be used to determine $G_{23}, v_{32}$, and $E_{2}$ for a specially orthotropic, transversely isotropic
Derive the first of Equation 2.40 for the off-axis modulus, $E_{x}$. 2V12 E-[1 + (1-2)+1] Ex = Ey E E 1 2V12 + 1 1 G12 E2 1 E2 2V12 - E 2G12 1 1 sc E E1 E2 E2 G12 Gxy = 1 G12 Vxy = Ex ($4 V12 +c

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