Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 16: Problem 16

\(16.16\) 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 \(10 \mathrm{MPa}\) \((1450 \mathrm{psi})\)

Short Answer

Expert verified
Answer: The volume fraction of fibers needed is approximately 0.158 or 15.8%.

Step by step solution

01

Identify the Given Data

The given data include: - Desired longitudinal tensile strength, \(σ_L = 750\,\text{MPa}\) - Fiber diameter, \(d_f = 1.2 × 10^{-2}\,\text{mm}\) - Fiber length, \(l_f = 1\,\text{mm}\) - Fiber fracture strength, \(σ_f = 5000\,\text{MPa}\) - Fiber-matrix bond strength, \(τ = 25\,\text{MPa}\) - Matrix stress at fiber failure, \(σ_m = 10\,\text{MPa}\)
02

Use the Kelly-Tyson Equation

To determine the volume fraction of fibers necessary to achieve the desired longitudinal tensile strength, we will use the Kelly-Tyson equation: \(σ_L = V_f σ_f [η_L η_0 + (1 - η_L) \rho]\) where: - \(σ_L\) is the desired longitudinal tensile strength - \(V_f\) is the volume fraction of fibers - \(σ_f\) is the fiber fracture strength - \(η_L\) is the fiber length efficiency factor - \(η_0\) is the fiber orientation efficiency factor - \(ρ\) is the fiber reinforcement efficiency factor: \(\rho = \frac{τ} {(σ_m\frac{l_f}{2d_f}). \frac{σ_f}{σ_m}}\)
03

Calculate the Fiber Length Efficiency Factor, \(η_L\)

The fiber length efficiency factor, \(η_L\), is given by: \(η_L =\frac{l_f}{l_c + l_f}\) where \(l_c\) is the critical fiber length, which can be calculated using: \(l_c = \frac{d_f τ}{σ_f}\) Insert the given values and calculate \(l_c\): \(l_c = \frac{(1.2 × 10^{-2}\,\text{mm})(25\,\text{MPa})}{(5000\,\text{MPa})} = 6 × 10^{-3}\,\text{mm}\) Now, calculate \(η_L\): \(η_L = \frac{1\,\text{mm}}{6 × 10^{-3}\,\text{mm} + 1\,\text{mm}} \approx 0.994\)
04

Calculate the Fiber Orientation Efficiency Factor, \(η_0\)

Since the fibers are aligned, the fiber orientation efficiency factor, \(η_0\), is equal to 1.
05

Calculate the Fiber Reinforcement Efficiency Factor, \(ρ\)

Using the given data, calculate the fiber reinforcement efficiency factor, \(ρ\): \(ρ = \frac{25\,\text{MPa}}{(10\,\text{MPa}\frac{1\,\text{mm}}{2(1.2 × 10^{-2}\,\text{mm})}). \frac{5000\,\text{MPa}}{10\,\text{MPa}}} = 0.0185\)
06

Calculate the Volume Fraction of Fibers, \(V_f\)

Using the Kelly-Tyson equation, calculate the volume fraction of fibers, \(V_f\): \(750\,\text{MPa} = V_f (5000\,\text{MPa})\left[ (0.994)(1) + (1 - 0.994)(0.0185)\right]\) Solving for \(V_f\), we get: \(V_f ≈ 0.158\) The volume fraction of fibers necessary to achieve the desired longitudinal tensile strength is approximately 0.158 or 15.8%.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(16.14\) A continuous and aligned fiber-reinforced composite having a cross- sectional area of \(1130 \mathrm{~mm}^{2}\left(1.75 \mathrm{in} .^{2}\right)\) is subjected to an \(\mathrm{ex}\) ternal tensile load. If the stresses sustained by the fiber and matrix phases are \(156 \mathrm{MPa}\) \((22,600 \mathrm{psi})\) and \(2.75 \mathrm{MPa}(400 \mathrm{psi})\), respectively; the force sustained by the fiber phase is \(74,000 \mathrm{~N}\left(16,600 \mathrm{lb}_{6}\right)\); and the total longitudinal strain is \(1.25 \times 10^{-3}\), determine (a) the force sustained by the matrix phase, (b) the modulus of elasticity of the composite material in the longitudinal direction, and (c) the moduli of elasticity for fiber and matrix phases.

(a) What is the distinction between matrix and dispersed phases in a composite material? (b) Contrast the mechanical characteristics of matrix and dispersed phases for fiberreinforced composites.

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).

(a) List four reasons why glass fibers are most commonly used for reinforcement. (b) Why is the surface perfection of glass fibers so important? (c) What measures are taken to protect the surface of glass fibers?

16.7 (a) For a fiber-reinforced composite, the efficiency of reinforcement \(\eta\) is dependent on fiber length \(l\) according to $$ \eta=\frac{l-2 x}{l} $$ where \(x\) represents the length of the fiber at each end that does not contribute to the load transfer. Make a plot of \(\eta\) versus \(l\) to \(l=40\) \(\mathrm{mm}\) (1.6 in.), assuming that \(x=0.75 \mathrm{~mm}\) \((0.03 \mathrm{in} .)\) (b) What length is required for a \(0.80\) efficiency of reinforcement?
See all solutions

Recommended explanations on Physics Textbooks

Nuclear Physics

Read Explanation

Kinematics Physics

Read Explanation

Famous Physicists

Read Explanation

Quantum Physics

Read Explanation

Fluids

Read Explanation

Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free