@LECTSOLUTIONSSTUVIA
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 stress
tru (1-2a)
instantaneous cross-sectional areaAi
e
change in length lf l0
engineering strain
(1-1b)
eng
initial length l0
final lf
true true strain ln ln (1-2b)
length l0
initial
length
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.
,1.4 Under what conditions is Eq. 1-4 valid? What makes it no longer useful if
those conditions are not met?
,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)
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional
x0 (600 K)
purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of
x0 (200that
this work beyond K) permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the
copyright owner is unlawful.
,1.7 State the critical difference in the processing behavior of thermoplastics vs. thermosets.
Thermoplastics can be melted and resolidified multiple times, so processing often involves
several heating, forming, and cooling steps. Thermosets harden by a one-time chemical
reaction so there cannot be any additional forming operations after the cross-linking operation
takes place.
1.8 What happens to the stiffness of a polymer as the temperature Tg is exceeded?
For what group of polymers is this change the greatest? The smallest?
The stiffness of a polymer decreases above the glass transition temperature, sometimes
dramatically. The effect is the largest for amorphous, uncross-linked polymers. It is the
smallest for highly cross-linked polymers (such as certain epoxies).
1.9 Write typical values of E for diamond, steel, aluminum, silicate glass,
polystyrene, and silicone rubber subjected to small strains (note that the latter
value is not included in this chapter, but is widely available). Clearly indicate
the units for each value.
The following values are not intended to represent any particular processing method or alloy
composition; they are rounded average values for certain material families.
Diamond ~ 1000 GPa
Steel ~ 200 GPa
Aluminum ~ 70 GPa
Silicate glass ~ 70 GPa
Polystyrene ~ 3 GPa
Silicone rubber ~ 10 MPa (0.010 GPa)
1.10 What is the purpose of a plasticizer, and what specific effect on room
temperature behavior is likely when a plasticizer is added?
A plasticizer is added to a polymer to break up the molecular interactions, allowing more
chain mobility than would otherwise be possible for that particular polymer at the
temperature of interest. At room temperature, therefore, the polymer is more likely to have a
low elastic modulus (i.e., a ordinarily-hard polymer may become flexible).
1.11 Identify a minimum of two structural characteristics and two mechanical
characteristics that set elastomers apart from other classes of materials (including
other polymers).
Elastomers are amorphous and moderately cross-linked. They tend to display significant
changes in stiffness as their use temperate exceeds Tg, but they do not melt at even higher
temperature.
1.12 Define what is meant by uniaxial, biaxial and triaxial loading.
Uniaxial loading occurs along a single direction, biaxial along two directions, and triaxial
,Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 1/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
along three. Note that there may be multiaxial strains even when the loading is restricted to
one or two directions.
1.13 State one advantage and disadvantage of compression testing.
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.
,An advantage may be to avoid failure due to tensile cracking at low loads (as in the case for
ceramics and glasses), and therefore to allow exploration of degrees of plasticity impossible to
achieve under tensile loading. One disadvantage would be the difficulty in achieving ideal
friction-free conditions between the specimen and the loading platen.
1.14 Is buckling failure initiated by an elastic, plastic, or cracking process? Explain.
Buckling failure is initially an elastic process in which the member deflects in a direction
perpendicular to the loading axis. This failure may then be followed by plasticity or fracture,
but these processes are not inherent in buckling.
1.15 What is the difference between the resilience and the strain energy density of a
material under load? Illustrate your answer by reproducing Figure 1.3, curve
‘b’ (a ductile metal), and annotating it appropriately.
Resilience is a measure of the maximum elastic strain energy stored in the material before the
onset of plasticity. The strain energy density is a more general term that is a measure of the
stored elastic energy at any point during a mechanical test. It may be greater or less than the
resilience, depending on the hardening or softening behavior that takes place after plastic
deformation begins.
stress
strain energy density at the
point of necking
strain
resilience
1.16 Sketch Figure 1.3, curve ‘b’ (a ductile metal) and show on the figure the difference
between the proportional limit and the offset yield strength.
,Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 1/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
stress
offset yield strength
proportional limit
strain
0.2%
1.17 Describe when and why bend testing (flexural testing) is most advantageous.
Bend testing may be used for any class of materials. It can be used to assess elastic or plastic
properties. It is particularly useful when the material is only available in the shape of a
rectangular prism, or when the material would be likely to fail prematurely due to extreme
flaw sensitivity (as is usually the case for brittle ceramic and glass materials).
1.18 Where can the maximum stress be found for a rectangular bar undergoing
3-point bending? 4-point bending?
The maximum stress in 3-point bending is found in two locations: at the top and bottom
surfaces directly aligned with the central load point. In 4-point bending, the maximum stress
is also found on the top and bottom surfaces, but it exists at a constant level between the inner
(closer) load points.
1.19 Write the basic isotropic form of Hooke’s law relating stress and strain for uniaxial
tension/compression loading and shear loading. Define all quantities.
Linear elastic uniaxial tension/compression is described by E , where is stress, E is
Young’s modulus, and is strain. An analogous form exists for shear loading, for which
= G . In this expression, is the shear stress, G is the shear modulus, and is the
shear strain.
1.20 Why do we define engineering and true stresses for tension/compression
loading but not for shear loading?
In tension and compression the cross-sectional area bearing the load changes during
deformation, so it is often necessary to account for this change. In shear loading there is
distortion of the material but the area over which the force is distributed does not change, so
there is no need for a true stress definition.
1.21 Sketch a pair of pliers squeezing an object and use it to show why the hinge pin
is under shear loading.
When the clamping force is applied to an object, a reaction force must exist at the pin. The two
jaw faces experience equal and opposite clamping forces, so the pin must experience equal and
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.
, opposite reaction forces at its two ends. These create shear stress within the pin.
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 stress
tru (1-2a)
instantaneous cross-sectional areaAi
e
change in length lf l0
engineering strain
(1-1b)
eng
initial length l0
final lf
true true strain ln ln (1-2b)
length l0
initial
length
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.
,1.4 Under what conditions is Eq. 1-4 valid? What makes it no longer useful if
those conditions are not met?
,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)
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional
x0 (600 K)
purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of
x0 (200that
this work beyond K) permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the
copyright owner is unlawful.
,1.7 State the critical difference in the processing behavior of thermoplastics vs. thermosets.
Thermoplastics can be melted and resolidified multiple times, so processing often involves
several heating, forming, and cooling steps. Thermosets harden by a one-time chemical
reaction so there cannot be any additional forming operations after the cross-linking operation
takes place.
1.8 What happens to the stiffness of a polymer as the temperature Tg is exceeded?
For what group of polymers is this change the greatest? The smallest?
The stiffness of a polymer decreases above the glass transition temperature, sometimes
dramatically. The effect is the largest for amorphous, uncross-linked polymers. It is the
smallest for highly cross-linked polymers (such as certain epoxies).
1.9 Write typical values of E for diamond, steel, aluminum, silicate glass,
polystyrene, and silicone rubber subjected to small strains (note that the latter
value is not included in this chapter, but is widely available). Clearly indicate
the units for each value.
The following values are not intended to represent any particular processing method or alloy
composition; they are rounded average values for certain material families.
Diamond ~ 1000 GPa
Steel ~ 200 GPa
Aluminum ~ 70 GPa
Silicate glass ~ 70 GPa
Polystyrene ~ 3 GPa
Silicone rubber ~ 10 MPa (0.010 GPa)
1.10 What is the purpose of a plasticizer, and what specific effect on room
temperature behavior is likely when a plasticizer is added?
A plasticizer is added to a polymer to break up the molecular interactions, allowing more
chain mobility than would otherwise be possible for that particular polymer at the
temperature of interest. At room temperature, therefore, the polymer is more likely to have a
low elastic modulus (i.e., a ordinarily-hard polymer may become flexible).
1.11 Identify a minimum of two structural characteristics and two mechanical
characteristics that set elastomers apart from other classes of materials (including
other polymers).
Elastomers are amorphous and moderately cross-linked. They tend to display significant
changes in stiffness as their use temperate exceeds Tg, but they do not melt at even higher
temperature.
1.12 Define what is meant by uniaxial, biaxial and triaxial loading.
Uniaxial loading occurs along a single direction, biaxial along two directions, and triaxial
,Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 1/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
along three. Note that there may be multiaxial strains even when the loading is restricted to
one or two directions.
1.13 State one advantage and disadvantage of compression testing.
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.
,An advantage may be to avoid failure due to tensile cracking at low loads (as in the case for
ceramics and glasses), and therefore to allow exploration of degrees of plasticity impossible to
achieve under tensile loading. One disadvantage would be the difficulty in achieving ideal
friction-free conditions between the specimen and the loading platen.
1.14 Is buckling failure initiated by an elastic, plastic, or cracking process? Explain.
Buckling failure is initially an elastic process in which the member deflects in a direction
perpendicular to the loading axis. This failure may then be followed by plasticity or fracture,
but these processes are not inherent in buckling.
1.15 What is the difference between the resilience and the strain energy density of a
material under load? Illustrate your answer by reproducing Figure 1.3, curve
‘b’ (a ductile metal), and annotating it appropriately.
Resilience is a measure of the maximum elastic strain energy stored in the material before the
onset of plasticity. The strain energy density is a more general term that is a measure of the
stored elastic energy at any point during a mechanical test. It may be greater or less than the
resilience, depending on the hardening or softening behavior that takes place after plastic
deformation begins.
stress
strain energy density at the
point of necking
strain
resilience
1.16 Sketch Figure 1.3, curve ‘b’ (a ductile metal) and show on the figure the difference
between the proportional limit and the offset yield strength.
,Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 1/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
stress
offset yield strength
proportional limit
strain
0.2%
1.17 Describe when and why bend testing (flexural testing) is most advantageous.
Bend testing may be used for any class of materials. It can be used to assess elastic or plastic
properties. It is particularly useful when the material is only available in the shape of a
rectangular prism, or when the material would be likely to fail prematurely due to extreme
flaw sensitivity (as is usually the case for brittle ceramic and glass materials).
1.18 Where can the maximum stress be found for a rectangular bar undergoing
3-point bending? 4-point bending?
The maximum stress in 3-point bending is found in two locations: at the top and bottom
surfaces directly aligned with the central load point. In 4-point bending, the maximum stress
is also found on the top and bottom surfaces, but it exists at a constant level between the inner
(closer) load points.
1.19 Write the basic isotropic form of Hooke’s law relating stress and strain for uniaxial
tension/compression loading and shear loading. Define all quantities.
Linear elastic uniaxial tension/compression is described by E , where is stress, E is
Young’s modulus, and is strain. An analogous form exists for shear loading, for which
= G . In this expression, is the shear stress, G is the shear modulus, and is the
shear strain.
1.20 Why do we define engineering and true stresses for tension/compression
loading but not for shear loading?
In tension and compression the cross-sectional area bearing the load changes during
deformation, so it is often necessary to account for this change. In shear loading there is
distortion of the material but the area over which the force is distributed does not change, so
there is no need for a true stress definition.
1.21 Sketch a pair of pliers squeezing an object and use it to show why the hinge pin
is under shear loading.
When the clamping force is applied to an object, a reaction force must exist at the pin. The two
jaw faces experience equal and opposite clamping forces, so the pin must experience equal and
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.
, opposite reaction forces at its two ends. These create shear stress within the pin.
