ALL 12 CHAPTERS COVERED
SOLUTIONS MANUAL
,Table of Contents
Chapter 1. Elastic Response of Solids
Chapter 2. Yielding and Plastic Flow
Chapter 3. Controlling Strength
Chapter 4. Time-Dependent Deformation
Chapter 5. Fracture: An Overview
Chapter 6. Elements of Fracture Mechanics
Chapter 7. Fracture Toughness
Chapter 8. Environment-Assisted Cracking
Chapter 9. Cyclic Stress and Strain Fatigue
Chapter 10. Fatigue Crack Propagation
Chapter 11. Analyses of Engineering Failures
Chapter 12. Consequences of Product Failure
,Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 1/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
CHAPTER 1
Review
1.1 In your own words, what are two differences between product testing and material
testing?
Possible answers include: (a) The goal of the two procedures is different. Whereas product
testing is design to determine the lifetime of a component under conditions that mimic real-
world use, material testing is intended to extract fundamental material properties that are
independent of the material’s use. (b) The specimen shape is different. Product testing must
use the material in the shape in which it will be used in the real product. Material testing uses
idealized specimen shapes designed to unambiguously determine one or more properties of
the material with the simplest analysis possible.
1.2 What are the distinguishing differences between elasticity, plasticity, and fracture?
Elasticity involves only deformation that is fully reversible when the applied load is removed
(even if it takes time to occur). Plasticity is permanent shape change without cracking, even
when no load exists. Fracture inherently involves breaking of bonds and the creation of new
surfaces. Often two or more of these processes take place simultaneously, but the contribution
of each can be separated from the others.
1.3 Write the definitions for engineering stress, true stress, engineering strain, and true
strain for loading along a single axis.
load P
eng = engineering stress = = (1-1a)
initial cross-sectional area A0
load P
true = true stress = = (1-2a)
instantaneous cross-sectional area Ai
change in length l f − l0
= engineering strain = = (1-1b)
eng
initial length l0
final length lf
true = true strain = ln = ln (1-2b)
initial length l0
1.4 Under what conditions is Eq. 1-4 valid? What makes it no longer useful if those
conditions are not met?
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional
purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of
this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the
copyright owner is unlawful.
, Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 1/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
P
= (l / l ) = (l / l ) = (1+ ) (1-4)
true i 0 eng i 0 eng eng
A0
This expression is true when volume is conserved. However, it is only useful if the cross-
sectional area is the same everyone on the test specimen. If this isn’t the case then the stress
and strain will vary from one part of the specimen to another.
1.5 Sketch Figure 1.3, curve ‘b’ (a ductile metal). Label it with the following terms,
indicating from which location on the curve each quantity can be identified or
extracted: elastic region, elastic-plastic region, proportional limit, tensile strength, onset
of necking, fracture stress.
onset of necking
tensile strength
fracture stress
proportional limit
elastic-plastic region
elastic region
stress
strain
1.6 On a single set of axes, sketch approximate atomic force vs. atom-separation curves like
the one shown in Fig. 1.4b for tungsten at temperatures of 200, 600, and 1000 K. Pay
close attention to the point x0 and the slope dF/dx for each of the curves you draw.
The key features of the plot are the increasing x0 spacing with increasing temperature (i.e.,
with thermal expansion) and the decreasing slope associated with decreased elastic modulus.
The plot is exaggerated but the trends are reasonable.
F
dF
dx
200 K
600 K
1000 K
x
x0 (1000 K)
x0 (600 K)
x0 (200 K)
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional
purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of
this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the
copyright owner is unlawful.
SOLUTIONS MANUAL
,Table of Contents
Chapter 1. Elastic Response of Solids
Chapter 2. Yielding and Plastic Flow
Chapter 3. Controlling Strength
Chapter 4. Time-Dependent Deformation
Chapter 5. Fracture: An Overview
Chapter 6. Elements of Fracture Mechanics
Chapter 7. Fracture Toughness
Chapter 8. Environment-Assisted Cracking
Chapter 9. Cyclic Stress and Strain Fatigue
Chapter 10. Fatigue Crack Propagation
Chapter 11. Analyses of Engineering Failures
Chapter 12. Consequences of Product Failure
,Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 1/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
CHAPTER 1
Review
1.1 In your own words, what are two differences between product testing and material
testing?
Possible answers include: (a) The goal of the two procedures is different. Whereas product
testing is design to determine the lifetime of a component under conditions that mimic real-
world use, material testing is intended to extract fundamental material properties that are
independent of the material’s use. (b) The specimen shape is different. Product testing must
use the material in the shape in which it will be used in the real product. Material testing uses
idealized specimen shapes designed to unambiguously determine one or more properties of
the material with the simplest analysis possible.
1.2 What are the distinguishing differences between elasticity, plasticity, and fracture?
Elasticity involves only deformation that is fully reversible when the applied load is removed
(even if it takes time to occur). Plasticity is permanent shape change without cracking, even
when no load exists. Fracture inherently involves breaking of bonds and the creation of new
surfaces. Often two or more of these processes take place simultaneously, but the contribution
of each can be separated from the others.
1.3 Write the definitions for engineering stress, true stress, engineering strain, and true
strain for loading along a single axis.
load P
eng = engineering stress = = (1-1a)
initial cross-sectional area A0
load P
true = true stress = = (1-2a)
instantaneous cross-sectional area Ai
change in length l f − l0
= engineering strain = = (1-1b)
eng
initial length l0
final length lf
true = true strain = ln = ln (1-2b)
initial length l0
1.4 Under what conditions is Eq. 1-4 valid? What makes it no longer useful if those
conditions are not met?
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional
purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of
this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the
copyright owner is unlawful.
, Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 1/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
P
= (l / l ) = (l / l ) = (1+ ) (1-4)
true i 0 eng i 0 eng eng
A0
This expression is true when volume is conserved. However, it is only useful if the cross-
sectional area is the same everyone on the test specimen. If this isn’t the case then the stress
and strain will vary from one part of the specimen to another.
1.5 Sketch Figure 1.3, curve ‘b’ (a ductile metal). Label it with the following terms,
indicating from which location on the curve each quantity can be identified or
extracted: elastic region, elastic-plastic region, proportional limit, tensile strength, onset
of necking, fracture stress.
onset of necking
tensile strength
fracture stress
proportional limit
elastic-plastic region
elastic region
stress
strain
1.6 On a single set of axes, sketch approximate atomic force vs. atom-separation curves like
the one shown in Fig. 1.4b for tungsten at temperatures of 200, 600, and 1000 K. Pay
close attention to the point x0 and the slope dF/dx for each of the curves you draw.
The key features of the plot are the increasing x0 spacing with increasing temperature (i.e.,
with thermal expansion) and the decreasing slope associated with decreased elastic modulus.
The plot is exaggerated but the trends are reasonable.
F
dF
dx
200 K
600 K
1000 K
x
x0 (1000 K)
x0 (600 K)
x0 (200 K)
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional
purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of
this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the
copyright owner is unlawful.
