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Chapter 16: Problem 17

16.17 Compute the longitudinal tensile strength of an aligned glass fiber- epoxy matrix composite in which the average fiber diameter and length are \(0.010 \mathrm{~mm}\left(4 \times 10^{-4}\right.\) in.) and \(2.5\) \(\mathrm{mm}(0.10\) in.), respectively, and the volume fraction of fibers is \(0.40\). Assume that (1) the fiber-matrix bond strength is \(75 \mathrm{MPa}(10,900\) \(\mathrm{psi}\) ), (2) the fracture strength of the fibers is \(3500 \mathrm{MPa}(508,000 \mathrm{psi})\), and (3) the matrix stress at fiber failure is \(8.0 \mathrm{MPa}\) (1160 psi).

Short Answer

Expert verified
Answer: The longitudinal tensile strength of the aligned glass fiber-epoxy matrix composite is 1404.8 MPa.

Step by step solution

01

Identify the Critical Fiber Length

The critical fiber length is the minimum length of fibers required to achieve maximum tensile strength in a composite material. The critical fiber length is given by: \(l_{c}=\frac{\sigma_{b}D_{f}}{2\tau}\) where \(l_{c}\) is the critical fiber length, \(\sigma_{b}\) is the fiber-matrix bond strength, \(D_{f}\) is the average fiber diameter, and \(\tau\) is the matrix shear stress.
02

Calculate the Matrix Shear Stress

The matrix shear stress, \(\tau\), can be determined using the rule of mixtures for the fiber failure stress (\(\sigma_{f}\)) as follows: \(\tau = \frac{V_{f}\sigma_{f} - \sigma_{m}}{V_{m}}\) where \(V_{f}\) is the fiber volume fraction, \(V_{m}\) is the matrix volume fraction, and \(\sigma_{m}\) is the matrix stress at fiber failure.
03

Calculate the Critical Fiber Length

Using the equations obtained in Steps 1 and 2, calculate the critical fiber length (\(l_{c}\)).
04

Compare the Critical Fiber Length with the Given Fiber Length

Now, compare the calculated critical fiber length (\(l_{c}\)) with the given fiber length (\(l_{f}\)). If the given fiber length is greater than or equal to the critical fiber length, we can proceed to the next step. Otherwise, the problem needs to be reconsidered.
05

Calculate the Longitudinal Tensile Strength

Finally, we can calculate the longitudinal tensile strength of the composite material, \(\sigma_{c}\), using the following equation: \(\sigma_{c} = V_{f}\sigma_{f} + V_{m}\sigma_{m}\) Let's calculate the tensile strength based on the given data.
06

Calculation

\(\sigma_{b} = 75 \,\text{MPa}\) \(D_{f} = 0.010 \,\text{mm}\)
07

Calculation

\(V_{f} = 0.40\) \(\sigma_{f} = 3500 \,\text{MPa}\) \(\sigma_{m} = 8.0 \,\text{MPa}\) \(V_{m} = 1 - V_{f} = 1 - 0.40 = 0.60\) \(\tau = \frac{V_{f}\sigma_{f} - \sigma_{m}}{V_{m}} = \frac{0.40(3500) - 8.0}{0.60} = 2283.3\, \text{MPa}\)
08

Calculation

\(l_{c}=\frac{\sigma_{b}D_{f}}{2\tau}=\frac{75(0.010)}{2(2283.3)}=0.00164 \,\text{mm}\)
09

Comparison

\(l_{f} = 2.5\, \text{mm}\) (Given fiber length) Since \(l_{f} > l_{c}\), we can proceed to the next step.
10

Calculation

\(\sigma_{c} = V_{f}\sigma_{f} + V_{m}\sigma_{m} = 0.40(3500) + 0.60(8.0) = 1400 + 4.8 = 1404.8\, \text{MPa}\) Thus, the longitudinal tensile strength of the aligned glass fiber-epoxy matrix composite is \(1404.8 \,\text{MPa}\).

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Most popular questions from this chapter

\(16.16\) It is desired to produce an aligned carbon fiber-epoxy matrix composite having a longitudinal tensile strength of 750 MPa (109,000 psi). Calculate the volume fraction of fibers necessary if (1) the average fiber diameter and length are \(1.2 \times 10^{-2} \mathrm{~mm}\) (4.7 \(\times 10^{-4}\) in. \()\) and \(1 \mathrm{~mm}(0.04\) in. \()\), respectively; (2) the fiber fracture strength is \(5000 \mathrm{MPa}\) \((725,000 \mathrm{psi}) ;\) (3) the fiber-matrix bond strength is 25 MPa (3625 psi); and (4) the matrix stress at fiber failure is \(10 \mathrm{MPa}\) \((1450 \mathrm{psi})\)

16.29 Briefly describe laminar composites. What is the prime reason for fabricating these materials?

A large-particle composite consisting of tungsten particles within a copper matrix is to be prepared. If the volume fractions of tungsten and copper are \(0.60\) and \(0.40\), respectively, estimate the upper limit for the specific stiffness of this composite given the data that follow. \begin{tabular}{lcc} \hline & Specific Gravity & Modulus of Elasticity \((\boldsymbol{G P a})\) \\ \hline Copper & \(8.9\) & 110 \\ Tungsten & \(19.3\) & 407 \end{tabular}

The mechanical properties of aluminum may be improved by incorporating fine particles of aluminum oxide \(\left(\mathrm{Al}_{2} \mathrm{O}_{3}\right)\). Given that the moduli of elasticity of these materials are, respectively, \(69 \mathrm{GPa}\left(10 \times 10^{6} \mathrm{psi}\right)\) and 393 GPa ( \(\left.57 \times 10^{6} \mathrm{psi}\right)\), plot modulus of elasticity versus the volume percent of \(\mathrm{Al}_{2} \mathrm{O}_{3}\) in \(\mathrm{Al}\) from 0 to 100 vol\%, using both upper- and lower-bound expressions.

\(16.15\) Compute the longitudinal strength of an aligned carbon fiber-epoxy matrix composite having a \(0.25\) volume fraction of fibers, assuming the following: (1) an average fiber diameter of \(10 \times 10^{-3} \mathrm{~mm}\left(3.94 \times 10^{-4}\right.\) in. \()\), (2) an average fiber length of \(5 \mathrm{~mm}(0.20\) in.), (3) a fiber fracture strength of \(2.5 \mathrm{GPa}\) \(\left(3.625 \times 10^{5} \mathrm{psi}\right),(4)\) a fiber- matrix bond strength of \(80 \mathrm{MPa}(11,600 \mathrm{psi}),(5)\) a matrix stress at fiber failure of \(10.0 \mathrm{MPa}(1450\) psi), and (6) a matrix tensile strength of \(75 \mathrm{MPa}(11,000 \mathrm{psi})\)
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