Book cover for Materials Science and Engineering: An Introduction

Materials Science and Engineering: An Introduction

William D. Callister, Jr. David G. Rethwisch

ISBN #9780471736967

7th Edition

771 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter uses six diverse case studies to illustrate the principles and procedures of materials selection in engineering design. Key concepts include the derivation and use of performance indices—especially for torsional applications, the importance of materials selection charts, and economic as well as safety considerations. Additionally, the chapter covers detailed analyses of fatigue in valve springs, failure modes in automotive axles, and the specialized requirements for biomedical implants and microelectronic packaging. These examples emphasize that successful design depends on balancing mechanical properties, material cost, manufacturability, and application-specific challenges like biocompatibility and corrosion resistance.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

Ceramics Fabrication Techniques

The methods and processes used for shaping, forming, and consolidating ceramic materials into final products with desired properties.

Example Problems

Example 1

(a) Using the procedure as outlined in Section 22.2 ascertain which of the metal alloys listed in Appendix B have torsional strength performance indices greater than $10.0\left(\text { for } \tau_{f} \text { and } \rho \text { in units of } \mathrm{MPa} \text { and } \mathrm{g} / \mathrm{cm}^{3}\right.$ respectively $),$ and, in addition, shear strengths greater than $350 \mathrm{MPa}$. (b) Also using the cost database (Appendix C), conduct a cost analysis in the same manner as Section $22.2 .$ For those materials that satisfy the criteria noted in part a, and, on the basis of this cost analysis, which material would you select for a solid cylindrical shaft? Why?

Example 2

In a manner similar to the treatment of Section $22.2,$ perform a stiffness-to-mass performance analysis on a solid cylindrical shaft that is subjected to a torsional stress. Use the same engineering materials that are listed in Table 22.1 . In addition, conduct a material cost analysis. Rank these materials both on the basis of mass of material required and material cost. For glass and carbon fiber-reinforced composites, assume that the shear module are 8.6 and $9.2 \mathrm{GPa},$ respectively.

Example 3

(a) A cylindrical cantilever beam is subjected to a force $F$, as indicated in the figure below. Derive strength and stiffness performance index expressions analogous to Equations 22.9 and 22.11 for this beam. The stress imposed on the unfixed end $\sigma$ is $$\sigma=\frac{F L r}{I}$$,$L, r,$ and $I$ are, respectively, the length, radius, and moment of inertia of the beam. Furthermore, the beam-end deflection $\delta$ is $$\delta=\frac{F L^{3}}{3 E I}$$,(b) From the properties database presented in Appendix B, select those metal alloys with stiffness performance indices greater that 3.0 (for $E$ and $\rho$ in units of GPa and $\mathrm{g} / \mathrm{cm}^{3},$ respectively(c) Also using the cost database (Appendix C), conduct a cost analysis in the same manner as Section 22.2. Relative to this analysis and that in part b, which alloy would you select on a stiffness-per-mass basis?(d) Now select those metal alloys having strength performance indices greater than 14.0 (for $\sigma_{y}$ and $\rho$ in units of $\mathrm{MPa}$ and $\mathrm{g} / \mathrm{cm}^{3}$ respectively $),$ and rank them from highest to lowest $P$ (e) And, using the cost database, rank the materials in part d from least to most costly. Relative to this analysis and that in part d, which alloy would you select on a strength-per-mass basis? (f) Which material would you select if both stiffness and strength are to be considered relative to this application? Justify your choice.

Example 4

(a) Using the expression developed for stiffness performance index in Problem 22.D3(a) and data contained in Appen$\operatorname{dix} \mathrm{B},$ determine stiffness performance indices for the following polymeric materials: high-density polyethylene, polypropylene, poly(vinyl chloride), polystyrene, polycarbonate, poly(methyl methacrylate),poly(ethylene terephthalate), polytetrafluoroethylene, and nylon $6,6 .$ How do these values compare with those of the metallic materials? (Note: In Appendix B, where ranges of values are given, use average values.)(b) Now, using the cost database (Appen$\operatorname{dix} C),$ conduct a cost analysis in the same manner as Section $22.2 .$ Use cost data for the raw forms of these polymers. (c) Using the expression developed for strength performance index in Problem $22 . \mathrm{D} 3(\mathrm{a})$ and data contained in Appendix B, determine strength performance indices for these same polymeric materials.

Example 5

(a) A bar specimen having a square cross section of edge length $c$ is subjected to a uniaxial tensile force $F$, as shown in the following figure. Derive strength and stiffness performance index expressions analogous to Equations 22.9 and 22.11 for this bar. (b) From the properties database presented in Appendix B, select those metal alloys with stiffness performance indices greater than 26.0 (for $E$ and $\rho$ in units of Gpa and $\mathrm{g} / \mathrm{cm}^{3},$ respectively (c) Also using the cost database (Appendix $C$ ), conduct a cost analysis in the same manner as Section $22.2 .$ Relative to this analysis and that in part b, which alloy would you select on a stiffness-per-mass basis? (d) Now select those metal alloys having strength performance indices greater than 120 (for $\sigma_{y}$ and $\rho$ in units of $\mathrm{MPa}$ and $\mathrm{g} / \mathrm{cm}^{3}$ respectively $),$ and rank them from highest to lowest $P$ .(e) And, using the cost database, rank the materials in part d from least to most costly. Relative to this analysis and that in part d, which alloy would you select on a strength-per-mass basis? (f) Which material would you select if both stiffness and strength are to be considered relative to this application? Justify your choice.
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Step-by-Step Explanations

QUESTION

How can we derive the mass of material required for a cylindrical shaft under a given twisting moment, and how does that lead to a performance index?\nStep-by-step Answer:\nStep 1: Begin with the shear stress equation for a solid circular cylinder under torsion, relating applied twisting moment, polar moment of inertia, and shaft geometry.\nStep 2: Introduce a safety factor by replacing the shear stress with the allowable shear strength (stress divided by the safety factor).\nStep 3: Express the mass of the shaft in terms of its volume (using density and cylindrical volume formulas).\nStep 4: Substitute the geometric expression for the radius (or cross-sectional area) derived from the mass equation into the stress equation, isolating mass as a function of material properties (density and strength) and geometric parameters (shaft length, twisting moment).\nStep 5: Rearrange the resulting expression to define a performance index, P, which is the reciprocal of the density-to-strength ratio, indicating that a higher P corresponds to a light yet strong material ideal for the design.\n\n- Topic: Fatigue Analysis in Valve Spring Design \nQuestion: How is the expected fatigue life of a valve spring determined and what role does shot peening play?\nStep-by-step Answer:\nStep 1: Determine the shear stresses produced in the helical spring when a compressive force is applied, using the geometry of the spring and equations for twisting moment and deflection.\nStep 2: Identify the fatigue limit of the spring material (often a steel alloy) and compare it to the computed stress amplitude from cyclic loading to ensure sustainable operation.\nStep 3: Recognize that repeated loading, especially under high strain rates, may reduce the fatigue life; therefore, incorporate design measures (such as increased coil or wire diameter) if needed.\nStep 4: Explain that shot peening is used to introduce compressive residual stresses at the surface, substantially increasing the fatigue limit by mitigating the initiation of cracks.\nStep 5: Conclude by comparing the computed stress amplitude to the enhanced fatigue limit (post shot peening) to verify that the design is marginal or satisfactory for long-term cyclic loading.\n\n"

STEP-BY-STEP ANSWER:

Step 1: Begin with the shear stress equation for a solid circular cylinder under torsion, relating applied twisting moment, polar moment of inertia, and shaft geometry.\nStep 2: Introduce a safety factor by replacing the shear stress with the allowable shear strength (stress divided by the safety factor).\nStep 3: Express the mass of the shaft in terms of its volume (using density and cylindrical volume formulas).\nStep 4: Substitute the geometric expression for the radius (or cross-sectional area) derived from the mass equation into the stress equation, isolating mass as a function of material properties (density and strength) and geometric parameters (shaft length, twisting moment).\nStep 5: Rearrange the resulting expression to define a performance index, P, which is the reciprocal of the density-to-strength ratio, indicating that a higher P corresponds to a light yet strong material ideal for the design.\n\n- Topic: Fatigue Analysis in Valve Spring Design \nQuestion: How is the expected fatigue life of a valve spring determined and what role does shot peening play?\nStep-by-step Answer:\nStep 1: Determine the shear stresses produced in the helical spring when a compressive force is applied, using the geometry of the spring and equations for twisting moment and deflection.\nStep 2: Identify the fatigue limit of the spring material (often a steel alloy) and compare it to the computed stress amplitude from cyclic loading to ensure sustainable operation.\nStep 3: Recognize that repeated loading, especially under high strain rates, may reduce the fatigue life; therefore, incorporate design measures (such as increased coil or wire diameter) if needed.\nStep 4: Explain that shot peening is used to introduce compressive residual stresses at the surface, substantially increasing the fatigue limit by mitigating the initiation of cracks.\nStep 5: Conclude by comparing the computed stress amplitude to the enhanced fatigue limit (post shot peening) to verify that the design is marginal or satisfactory for long-term cyclic loading.\n\n"
Final Answer:

"- Topic: Strength Performance Index for a Torsionally Stressed Shaft \nQuestion: How can we derive the mass of material required for a cylindrical shaft under a given twisting moment, and how does that lead to a performance index?\nStep-by-step Answer:\nStep 1: Begin with the shear stress equation for a solid circular cylinder under torsion, relating applied twisting moment, polar moment of inertia, and shaft geometry.\nStep 2: Introduce a safety factor by replacing the shear stress with the allowable shear strength (stress divided by the safety factor).\nStep 3: Express the mass of the shaft in terms of its volume (using density and cylindrical volume formulas).\nStep 4: Substitute the geometric expression for the radius (or cross-sectional area) derived from the mass equation into the stress equation, isolating mass as a function of material properties (density and strength) and geometric parameters (shaft length, twisting moment).\nStep 5: Rearrange the resulting expression to define a performance index, P, which is the reciprocal of the density-to-strength ratio, indicating that a higher P corresponds to a light yet strong material ideal for the design.\n\n- Topic: Fatigue Analysis in Valve Spring Design \nQuestion: How is the expected fatigue life of a valve spring determined and what role does shot peening play?\nStep-by-step Answer:\nStep 1: Determine the shear stresses produced in the helical spring when a compressive force is applied, using the geometry of the spring and equations for twisting moment and deflection.\nStep 2: Identify the fatigue limit of the spring material (often a steel alloy) and compare it to the computed stress amplitude from cyclic loading to ensure sustainable operation.\nStep 3: Recognize that repeated loading, especially under high strain rates, may reduce the fatigue life; therefore, incorporate design measures (such as increased coil or wire diameter) if needed.\nStep 4: Explain that shot peening is used to introduce compressive residual stresses at the surface, substantially increasing the fatigue limit by mitigating the initiation of cracks.\nStep 5: Conclude by comparing the computed stress amplitude to the enhanced fatigue limit (post shot peening) to verify that the design is marginal or satisfactory for long-term cyclic loading.\n\n"

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Common Mistakes

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