@LECTSOLUTIONSSTUVIA
ALL 12 CHAPTERS
COVERED
SOLUTIONS MANUAL
, @LECTSOLUTIONSSTUVIA
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
eng engineering stress (1-1a)
P
initial cross-sectional area A0
true (1-2a)
load
true stress
P
instantaneous cross-sectional area Ai (1-1b)
change in length lf
engineering strain
l0
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?
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. 2/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
P
(l / l ) (l / l ) (1 ) (1-4)
true
A0 i 0 eng i 0 eng eng
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
Excerpts from this K)
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. 3/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
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
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.
,Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 4/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
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.
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. 5/162
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 opposite reaction forces at its two ends. These create shear stress within the pin.
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. 6/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
top jaw
applied force bottom jaw
clamping force
reaction force
1.22 Write out the most general expression for tension or compression strain along a single
axis resulting from all possible applied stresses, assuming that the material is elastically
isotropic.
1
xx v( yy zz )
xx xx yy zz
E E
E E
1.23 Write out the most general expression for shear strain along a single axis resulting from
all possible applied stresses, assuming that the material is elastically isotropic.
xy
xy
G
Sketch and name the stress state present in the skin of a cylindrical thin-walled pressure
vessel. Repeat for the strain state.
biaxial stress triaxialstrain
1.24 What is the name of the matrix, Sij?
The Compliance Matrix.
1.25 Why can the compliance and stiffness tensors for cubic and orthotropic materials be
greatly simplified from the general case?
In cubic and orthotropic materials there are several directions that are structurally identical,
and therefore have identical elastic properties. Furthermore, the high degree of symmetry
reduces the degree of coupling between applied stresses and induced strains (e.g., the
absence of XY, YZ, or XZ shear strains generated by XX, YY, and ZZ stresses).
1.26 Describe the geometric criteria that differentiate orthotropic and cubic symmetry.
Orthotropic materials have three distinct a, b, and c axes that are each separated by interior
angles of = = =90°. Cubic materials also have three axes separated by = = =90°,
but the three axes are identical.
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.
, 1.27 Define hydrostatic stress state.
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.
ALL 12 CHAPTERS
COVERED
SOLUTIONS MANUAL
, @LECTSOLUTIONSSTUVIA
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
eng engineering stress (1-1a)
P
initial cross-sectional area A0
true (1-2a)
load
true stress
P
instantaneous cross-sectional area Ai (1-1b)
change in length lf
engineering strain
l0
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?
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. 2/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
P
(l / l ) (l / l ) (1 ) (1-4)
true
A0 i 0 eng i 0 eng eng
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
Excerpts from this K)
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. 3/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
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
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.
,Deformation and Fracture Mechanics of Engineering Materials, 5th ed. Problem Solutions p. 4/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
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.
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. 5/162
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 opposite reaction forces at its two ends. These create shear stress within the pin.
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. 6/162
Draft document, Copyright R. Hertzberg, R. Vinci, J. Hertzberg 2009
top jaw
applied force bottom jaw
clamping force
reaction force
1.22 Write out the most general expression for tension or compression strain along a single
axis resulting from all possible applied stresses, assuming that the material is elastically
isotropic.
1
xx v( yy zz )
xx xx yy zz
E E
E E
1.23 Write out the most general expression for shear strain along a single axis resulting from
all possible applied stresses, assuming that the material is elastically isotropic.
xy
xy
G
Sketch and name the stress state present in the skin of a cylindrical thin-walled pressure
vessel. Repeat for the strain state.
biaxial stress triaxialstrain
1.24 What is the name of the matrix, Sij?
The Compliance Matrix.
1.25 Why can the compliance and stiffness tensors for cubic and orthotropic materials be
greatly simplified from the general case?
In cubic and orthotropic materials there are several directions that are structurally identical,
and therefore have identical elastic properties. Furthermore, the high degree of symmetry
reduces the degree of coupling between applied stresses and induced strains (e.g., the
absence of XY, YZ, or XZ shear strains generated by XX, YY, and ZZ stresses).
1.26 Describe the geometric criteria that differentiate orthotropic and cubic symmetry.
Orthotropic materials have three distinct a, b, and c axes that are each separated by interior
angles of = = =90°. Cubic materials also have three axes separated by = = =90°,
but the three axes are identical.
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.
, 1.27 Define hydrostatic stress state.
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.
