Mechanical Properties of Metals Mechanical Engineering Notes | EduRev

Mechanical Engineering : Mechanical Properties of Metals Mechanical Engineering Notes | EduRev

 Page 1


Material Science 
 
Prof. Satish V. Kailas 
Associate Professor 
Dept. of Mechanical Engineering, 
Indian Institute of Science, 
Bangalore – 560012 
India 
Chapter 4. Mechanical Properties of Metals 
 
Most of the materials used in engineering are metallic in nature. The prime reason simply 
is the versatile nature of their properties those spread over a very broad range compared 
with other kinds of materials. Many engineering materials are subjected to forces both 
during processing/fabrication and in service. When a force is applied on a solid material, 
it may result in translation, rotation, or deformation of that material.  Aspects of material 
translation and rotation are dealt by engineering dynamics. We restrict ourselves here to 
the subject of material deformation under forces. Deformation constitutes both change in 
shape, distortion, and change in size/volume, dilatation. Solid material are defined such 
that change in their volume under applied forces in very small, thus deformation is used 
as synonymous to distortion. The ability of material to with stand the applied force 
without any deformation is expressed in two ways, i.e. strength and hardness. Strength is 
defined in many ways as per the design requirements, while the hardness may be defined 
as resistance to indentation of scratch.  
Material deformation can be permanent or temporary. Permanent deformation is 
irreversible i.e. stays even after removal of the applied forces, while the temporary 
deformation disappears after removal of the applied forces i.e. the deformation is 
recoverable. Both kinds of deformation can be function of time, or independent of time. 
Temporary deformation is called elastic deformation, while the permanent deformation is 
called plastic deformation. Time dependent recoverable deformation under load is called 
anelastic deformation, while the characteristic recovery of temporary deformation after 
removal of load as a function of time is called elastic aftereffect. Time dependent i.e. 
progressive permanent deformation under constant load/stress is called creep. For visco-
elastic materials, both recoverable and permanent deformations occur together which are 
time dependent. When a material is subjected to applied forces, first the material 
experiences elastic deformation followed by plastic deformation. Extent of elastic- and 
plastic- deformations will primarily depend on the kind of material, rate of load 
application, ambient temperature, among other factors. Change over from elastic state to 
plastic state is characterized by the yield strength ( s
0
) of the material. 
Forces applied act on a surface of the material, and thus the force intensity, force per unit 
area, is used in analysis. Analogous to this, deformation is characterized by percentage 
Page 2


Material Science 
 
Prof. Satish V. Kailas 
Associate Professor 
Dept. of Mechanical Engineering, 
Indian Institute of Science, 
Bangalore – 560012 
India 
Chapter 4. Mechanical Properties of Metals 
 
Most of the materials used in engineering are metallic in nature. The prime reason simply 
is the versatile nature of their properties those spread over a very broad range compared 
with other kinds of materials. Many engineering materials are subjected to forces both 
during processing/fabrication and in service. When a force is applied on a solid material, 
it may result in translation, rotation, or deformation of that material.  Aspects of material 
translation and rotation are dealt by engineering dynamics. We restrict ourselves here to 
the subject of material deformation under forces. Deformation constitutes both change in 
shape, distortion, and change in size/volume, dilatation. Solid material are defined such 
that change in their volume under applied forces in very small, thus deformation is used 
as synonymous to distortion. The ability of material to with stand the applied force 
without any deformation is expressed in two ways, i.e. strength and hardness. Strength is 
defined in many ways as per the design requirements, while the hardness may be defined 
as resistance to indentation of scratch.  
Material deformation can be permanent or temporary. Permanent deformation is 
irreversible i.e. stays even after removal of the applied forces, while the temporary 
deformation disappears after removal of the applied forces i.e. the deformation is 
recoverable. Both kinds of deformation can be function of time, or independent of time. 
Temporary deformation is called elastic deformation, while the permanent deformation is 
called plastic deformation. Time dependent recoverable deformation under load is called 
anelastic deformation, while the characteristic recovery of temporary deformation after 
removal of load as a function of time is called elastic aftereffect. Time dependent i.e. 
progressive permanent deformation under constant load/stress is called creep. For visco-
elastic materials, both recoverable and permanent deformations occur together which are 
time dependent. When a material is subjected to applied forces, first the material 
experiences elastic deformation followed by plastic deformation. Extent of elastic- and 
plastic- deformations will primarily depend on the kind of material, rate of load 
application, ambient temperature, among other factors. Change over from elastic state to 
plastic state is characterized by the yield strength ( s
0
) of the material. 
Forces applied act on a surface of the material, and thus the force intensity, force per unit 
area, is used in analysis. Analogous to this, deformation is characterized by percentage 
change in length per unit length in three distinct directions. Force intensity is also called 
engineering stress (or simply stress, s), is given by force divided by area on which the 
force is acting. Engineering strain (or simply strain, e) is given by change in length 
divided by original length. Engineering strain actually indicates an average change in 
length in a particular direction. According to definition, s and e are given as 
0
0
0
,
L
L L
e
A
P
s
-
= = 
where P is the load applied over area A, and as a consequence of it material attains the 
final length L from its original length of L
0
. 
Because material dimensions changes under application of the load continuously, 
engineering stress and strain values are not the true indication of material deformation 
characteristics. Thus the need for measures of stress and strain based on instantaneous 
dimensions arises. Ludwik first proposed the concept of, and defined the true strain or 
natural strain ( e) as follows:  
?
+
-
+
-
+
-
= ...
2
2 3
1
1 2
0
0 1
L
L L
L
L L
L
L L
e 
0
ln
0
L
L
L
dL
L
L
= =
?
e 
As material volume is expected to be constant i.e. A
0
L
0
=AL, and thus 
) 1 ln( ln ln
0
0
+ = = = e
A
A
L
L
e 
There are certain advantages of using true strain over conventional strain or engineering 
strain. These include (i) equivalent absolute numerical value for true strains in cases of 
tensile and compressive for same intuitive deformation and (ii) total true strain is equal to 
the sum of the incremental strains. As shown in figure-4.1, if L
1
=2 L
0
 and L
2
=1/2 L
1
=L
0
, 
absolute numerical value of engineering strain during tensile deformation (1.0) is 
different from that during compressive deformation (0.5). However, in both cases true 
strain values are equal (ln [2]). 
True stress ( s) is given as load divided by cross-sectional area over which it acts at an 
instant. 
) 1 (
0
0
+ = = = e s
A
A
A
P
A
P
s 
Page 3


Material Science 
 
Prof. Satish V. Kailas 
Associate Professor 
Dept. of Mechanical Engineering, 
Indian Institute of Science, 
Bangalore – 560012 
India 
Chapter 4. Mechanical Properties of Metals 
 
Most of the materials used in engineering are metallic in nature. The prime reason simply 
is the versatile nature of their properties those spread over a very broad range compared 
with other kinds of materials. Many engineering materials are subjected to forces both 
during processing/fabrication and in service. When a force is applied on a solid material, 
it may result in translation, rotation, or deformation of that material.  Aspects of material 
translation and rotation are dealt by engineering dynamics. We restrict ourselves here to 
the subject of material deformation under forces. Deformation constitutes both change in 
shape, distortion, and change in size/volume, dilatation. Solid material are defined such 
that change in their volume under applied forces in very small, thus deformation is used 
as synonymous to distortion. The ability of material to with stand the applied force 
without any deformation is expressed in two ways, i.e. strength and hardness. Strength is 
defined in many ways as per the design requirements, while the hardness may be defined 
as resistance to indentation of scratch.  
Material deformation can be permanent or temporary. Permanent deformation is 
irreversible i.e. stays even after removal of the applied forces, while the temporary 
deformation disappears after removal of the applied forces i.e. the deformation is 
recoverable. Both kinds of deformation can be function of time, or independent of time. 
Temporary deformation is called elastic deformation, while the permanent deformation is 
called plastic deformation. Time dependent recoverable deformation under load is called 
anelastic deformation, while the characteristic recovery of temporary deformation after 
removal of load as a function of time is called elastic aftereffect. Time dependent i.e. 
progressive permanent deformation under constant load/stress is called creep. For visco-
elastic materials, both recoverable and permanent deformations occur together which are 
time dependent. When a material is subjected to applied forces, first the material 
experiences elastic deformation followed by plastic deformation. Extent of elastic- and 
plastic- deformations will primarily depend on the kind of material, rate of load 
application, ambient temperature, among other factors. Change over from elastic state to 
plastic state is characterized by the yield strength ( s
0
) of the material. 
Forces applied act on a surface of the material, and thus the force intensity, force per unit 
area, is used in analysis. Analogous to this, deformation is characterized by percentage 
change in length per unit length in three distinct directions. Force intensity is also called 
engineering stress (or simply stress, s), is given by force divided by area on which the 
force is acting. Engineering strain (or simply strain, e) is given by change in length 
divided by original length. Engineering strain actually indicates an average change in 
length in a particular direction. According to definition, s and e are given as 
0
0
0
,
L
L L
e
A
P
s
-
= = 
where P is the load applied over area A, and as a consequence of it material attains the 
final length L from its original length of L
0
. 
Because material dimensions changes under application of the load continuously, 
engineering stress and strain values are not the true indication of material deformation 
characteristics. Thus the need for measures of stress and strain based on instantaneous 
dimensions arises. Ludwik first proposed the concept of, and defined the true strain or 
natural strain ( e) as follows:  
?
+
-
+
-
+
-
= ...
2
2 3
1
1 2
0
0 1
L
L L
L
L L
L
L L
e 
0
ln
0
L
L
L
dL
L
L
= =
?
e 
As material volume is expected to be constant i.e. A
0
L
0
=AL, and thus 
) 1 ln( ln ln
0
0
+ = = = e
A
A
L
L
e 
There are certain advantages of using true strain over conventional strain or engineering 
strain. These include (i) equivalent absolute numerical value for true strains in cases of 
tensile and compressive for same intuitive deformation and (ii) total true strain is equal to 
the sum of the incremental strains. As shown in figure-4.1, if L
1
=2 L
0
 and L
2
=1/2 L
1
=L
0
, 
absolute numerical value of engineering strain during tensile deformation (1.0) is 
different from that during compressive deformation (0.5). However, in both cases true 
strain values are equal (ln [2]). 
True stress ( s) is given as load divided by cross-sectional area over which it acts at an 
instant. 
) 1 (
0
0
+ = = = e s
A
A
A
P
A
P
s 
It is to be noted that engineering stress is equal to true stress up to the elastic limit of the 
material. The same applies to the strains. After the elastic limit i.e. once material starts 
deforming plastically, engineering values and true values of stresses and strains differ. 
The above equation relating engineering and true stress-strains are valid only up to the 
limit of uniform deformation i.e. up to the onset of necking in tension test. This is 
because the relations are developed by assuming both constancy of volume and 
homogeneous distribution of strain along the length of the tension specimen. Basics of 
both elastic and plastic deformations along with their characterization will be detailed in 
this chapter. 
 
 
4.1 Elastic deformation and Plastic deformation 
4.1.1 Elastic deformation 
Elastic deformation is reversible i.e. recoverable. Up to a certain limit of the applied 
stress, strain experienced by the material will be the kind of recoverable i.e. elastic in 
nature. This elastic strain is proportional to the stress applied. The proportional relation 
between the stress and the elastic strain is given by Hooke’s law, which can be written as 
follows: 
e s ? 
e s E = 
where the constant E is the modulus of elasticity or Young’s modulus,  
Though Hooke’s law is applicable to most of the engineering materials up to their elastic 
limit, defined by the critical value of stress beyond which plastic deformation occurs, 
some materials won’t obey the law. E.g.: Rubber, it has nonlinear stress-strain 
relationship and still satisfies the definition of an elastic material. For materials without 
linear elastic portion, either tangent modulus or secant modulus is used in design 
calculations. The tangent modulus is taken as the slope of stress-strain curve at some 
specified level, while secant module represents the slope of secant drawn from the origin 
to some given point of the s-e curve, as shown in figure-4.1. 
Page 4


Material Science 
 
Prof. Satish V. Kailas 
Associate Professor 
Dept. of Mechanical Engineering, 
Indian Institute of Science, 
Bangalore – 560012 
India 
Chapter 4. Mechanical Properties of Metals 
 
Most of the materials used in engineering are metallic in nature. The prime reason simply 
is the versatile nature of their properties those spread over a very broad range compared 
with other kinds of materials. Many engineering materials are subjected to forces both 
during processing/fabrication and in service. When a force is applied on a solid material, 
it may result in translation, rotation, or deformation of that material.  Aspects of material 
translation and rotation are dealt by engineering dynamics. We restrict ourselves here to 
the subject of material deformation under forces. Deformation constitutes both change in 
shape, distortion, and change in size/volume, dilatation. Solid material are defined such 
that change in their volume under applied forces in very small, thus deformation is used 
as synonymous to distortion. The ability of material to with stand the applied force 
without any deformation is expressed in two ways, i.e. strength and hardness. Strength is 
defined in many ways as per the design requirements, while the hardness may be defined 
as resistance to indentation of scratch.  
Material deformation can be permanent or temporary. Permanent deformation is 
irreversible i.e. stays even after removal of the applied forces, while the temporary 
deformation disappears after removal of the applied forces i.e. the deformation is 
recoverable. Both kinds of deformation can be function of time, or independent of time. 
Temporary deformation is called elastic deformation, while the permanent deformation is 
called plastic deformation. Time dependent recoverable deformation under load is called 
anelastic deformation, while the characteristic recovery of temporary deformation after 
removal of load as a function of time is called elastic aftereffect. Time dependent i.e. 
progressive permanent deformation under constant load/stress is called creep. For visco-
elastic materials, both recoverable and permanent deformations occur together which are 
time dependent. When a material is subjected to applied forces, first the material 
experiences elastic deformation followed by plastic deformation. Extent of elastic- and 
plastic- deformations will primarily depend on the kind of material, rate of load 
application, ambient temperature, among other factors. Change over from elastic state to 
plastic state is characterized by the yield strength ( s
0
) of the material. 
Forces applied act on a surface of the material, and thus the force intensity, force per unit 
area, is used in analysis. Analogous to this, deformation is characterized by percentage 
change in length per unit length in three distinct directions. Force intensity is also called 
engineering stress (or simply stress, s), is given by force divided by area on which the 
force is acting. Engineering strain (or simply strain, e) is given by change in length 
divided by original length. Engineering strain actually indicates an average change in 
length in a particular direction. According to definition, s and e are given as 
0
0
0
,
L
L L
e
A
P
s
-
= = 
where P is the load applied over area A, and as a consequence of it material attains the 
final length L from its original length of L
0
. 
Because material dimensions changes under application of the load continuously, 
engineering stress and strain values are not the true indication of material deformation 
characteristics. Thus the need for measures of stress and strain based on instantaneous 
dimensions arises. Ludwik first proposed the concept of, and defined the true strain or 
natural strain ( e) as follows:  
?
+
-
+
-
+
-
= ...
2
2 3
1
1 2
0
0 1
L
L L
L
L L
L
L L
e 
0
ln
0
L
L
L
dL
L
L
= =
?
e 
As material volume is expected to be constant i.e. A
0
L
0
=AL, and thus 
) 1 ln( ln ln
0
0
+ = = = e
A
A
L
L
e 
There are certain advantages of using true strain over conventional strain or engineering 
strain. These include (i) equivalent absolute numerical value for true strains in cases of 
tensile and compressive for same intuitive deformation and (ii) total true strain is equal to 
the sum of the incremental strains. As shown in figure-4.1, if L
1
=2 L
0
 and L
2
=1/2 L
1
=L
0
, 
absolute numerical value of engineering strain during tensile deformation (1.0) is 
different from that during compressive deformation (0.5). However, in both cases true 
strain values are equal (ln [2]). 
True stress ( s) is given as load divided by cross-sectional area over which it acts at an 
instant. 
) 1 (
0
0
+ = = = e s
A
A
A
P
A
P
s 
It is to be noted that engineering stress is equal to true stress up to the elastic limit of the 
material. The same applies to the strains. After the elastic limit i.e. once material starts 
deforming plastically, engineering values and true values of stresses and strains differ. 
The above equation relating engineering and true stress-strains are valid only up to the 
limit of uniform deformation i.e. up to the onset of necking in tension test. This is 
because the relations are developed by assuming both constancy of volume and 
homogeneous distribution of strain along the length of the tension specimen. Basics of 
both elastic and plastic deformations along with their characterization will be detailed in 
this chapter. 
 
 
4.1 Elastic deformation and Plastic deformation 
4.1.1 Elastic deformation 
Elastic deformation is reversible i.e. recoverable. Up to a certain limit of the applied 
stress, strain experienced by the material will be the kind of recoverable i.e. elastic in 
nature. This elastic strain is proportional to the stress applied. The proportional relation 
between the stress and the elastic strain is given by Hooke’s law, which can be written as 
follows: 
e s ? 
e s E = 
where the constant E is the modulus of elasticity or Young’s modulus,  
Though Hooke’s law is applicable to most of the engineering materials up to their elastic 
limit, defined by the critical value of stress beyond which plastic deformation occurs, 
some materials won’t obey the law. E.g.: Rubber, it has nonlinear stress-strain 
relationship and still satisfies the definition of an elastic material. For materials without 
linear elastic portion, either tangent modulus or secant modulus is used in design 
calculations. The tangent modulus is taken as the slope of stress-strain curve at some 
specified level, while secant module represents the slope of secant drawn from the origin 
to some given point of the s-e curve, as shown in figure-4.1. 
 
Figure-4.1: Tangent and Secant moduli for non-linear stress-strain relation. 
If one dimension of the material changed, other dimensions of the material need to be 
changed to keep the volume constant. This lateral/transverse strain is related to the 
applied longitudinal strain by empirical means, and the ratio of transverse strain to 
longitudinal strain is known as Poisson’s ratio ( ?). Transverse strain can be expected to 
be opposite in nature to longitudinal strain, and both longitudinal and transverse strains 
are linear strains. For most metals the values of ? are close to 0.33, for polymers it is 
between 0.4 – 0.5, and for ionic solids it is around 0.2. 
Stresses applied on a material can be of two kinds – normal stresses, and shear stresses. 
Normal stresses cause linear strains, while the shear stresses cause shear strains. If the 
material is subjected to torsion, it results in torsional strain. Different stresses and 
corresponding strains are shown in figure-4.2.  
 
Page 5


Material Science 
 
Prof. Satish V. Kailas 
Associate Professor 
Dept. of Mechanical Engineering, 
Indian Institute of Science, 
Bangalore – 560012 
India 
Chapter 4. Mechanical Properties of Metals 
 
Most of the materials used in engineering are metallic in nature. The prime reason simply 
is the versatile nature of their properties those spread over a very broad range compared 
with other kinds of materials. Many engineering materials are subjected to forces both 
during processing/fabrication and in service. When a force is applied on a solid material, 
it may result in translation, rotation, or deformation of that material.  Aspects of material 
translation and rotation are dealt by engineering dynamics. We restrict ourselves here to 
the subject of material deformation under forces. Deformation constitutes both change in 
shape, distortion, and change in size/volume, dilatation. Solid material are defined such 
that change in their volume under applied forces in very small, thus deformation is used 
as synonymous to distortion. The ability of material to with stand the applied force 
without any deformation is expressed in two ways, i.e. strength and hardness. Strength is 
defined in many ways as per the design requirements, while the hardness may be defined 
as resistance to indentation of scratch.  
Material deformation can be permanent or temporary. Permanent deformation is 
irreversible i.e. stays even after removal of the applied forces, while the temporary 
deformation disappears after removal of the applied forces i.e. the deformation is 
recoverable. Both kinds of deformation can be function of time, or independent of time. 
Temporary deformation is called elastic deformation, while the permanent deformation is 
called plastic deformation. Time dependent recoverable deformation under load is called 
anelastic deformation, while the characteristic recovery of temporary deformation after 
removal of load as a function of time is called elastic aftereffect. Time dependent i.e. 
progressive permanent deformation under constant load/stress is called creep. For visco-
elastic materials, both recoverable and permanent deformations occur together which are 
time dependent. When a material is subjected to applied forces, first the material 
experiences elastic deformation followed by plastic deformation. Extent of elastic- and 
plastic- deformations will primarily depend on the kind of material, rate of load 
application, ambient temperature, among other factors. Change over from elastic state to 
plastic state is characterized by the yield strength ( s
0
) of the material. 
Forces applied act on a surface of the material, and thus the force intensity, force per unit 
area, is used in analysis. Analogous to this, deformation is characterized by percentage 
change in length per unit length in three distinct directions. Force intensity is also called 
engineering stress (or simply stress, s), is given by force divided by area on which the 
force is acting. Engineering strain (or simply strain, e) is given by change in length 
divided by original length. Engineering strain actually indicates an average change in 
length in a particular direction. According to definition, s and e are given as 
0
0
0
,
L
L L
e
A
P
s
-
= = 
where P is the load applied over area A, and as a consequence of it material attains the 
final length L from its original length of L
0
. 
Because material dimensions changes under application of the load continuously, 
engineering stress and strain values are not the true indication of material deformation 
characteristics. Thus the need for measures of stress and strain based on instantaneous 
dimensions arises. Ludwik first proposed the concept of, and defined the true strain or 
natural strain ( e) as follows:  
?
+
-
+
-
+
-
= ...
2
2 3
1
1 2
0
0 1
L
L L
L
L L
L
L L
e 
0
ln
0
L
L
L
dL
L
L
= =
?
e 
As material volume is expected to be constant i.e. A
0
L
0
=AL, and thus 
) 1 ln( ln ln
0
0
+ = = = e
A
A
L
L
e 
There are certain advantages of using true strain over conventional strain or engineering 
strain. These include (i) equivalent absolute numerical value for true strains in cases of 
tensile and compressive for same intuitive deformation and (ii) total true strain is equal to 
the sum of the incremental strains. As shown in figure-4.1, if L
1
=2 L
0
 and L
2
=1/2 L
1
=L
0
, 
absolute numerical value of engineering strain during tensile deformation (1.0) is 
different from that during compressive deformation (0.5). However, in both cases true 
strain values are equal (ln [2]). 
True stress ( s) is given as load divided by cross-sectional area over which it acts at an 
instant. 
) 1 (
0
0
+ = = = e s
A
A
A
P
A
P
s 
It is to be noted that engineering stress is equal to true stress up to the elastic limit of the 
material. The same applies to the strains. After the elastic limit i.e. once material starts 
deforming plastically, engineering values and true values of stresses and strains differ. 
The above equation relating engineering and true stress-strains are valid only up to the 
limit of uniform deformation i.e. up to the onset of necking in tension test. This is 
because the relations are developed by assuming both constancy of volume and 
homogeneous distribution of strain along the length of the tension specimen. Basics of 
both elastic and plastic deformations along with their characterization will be detailed in 
this chapter. 
 
 
4.1 Elastic deformation and Plastic deformation 
4.1.1 Elastic deformation 
Elastic deformation is reversible i.e. recoverable. Up to a certain limit of the applied 
stress, strain experienced by the material will be the kind of recoverable i.e. elastic in 
nature. This elastic strain is proportional to the stress applied. The proportional relation 
between the stress and the elastic strain is given by Hooke’s law, which can be written as 
follows: 
e s ? 
e s E = 
where the constant E is the modulus of elasticity or Young’s modulus,  
Though Hooke’s law is applicable to most of the engineering materials up to their elastic 
limit, defined by the critical value of stress beyond which plastic deformation occurs, 
some materials won’t obey the law. E.g.: Rubber, it has nonlinear stress-strain 
relationship and still satisfies the definition of an elastic material. For materials without 
linear elastic portion, either tangent modulus or secant modulus is used in design 
calculations. The tangent modulus is taken as the slope of stress-strain curve at some 
specified level, while secant module represents the slope of secant drawn from the origin 
to some given point of the s-e curve, as shown in figure-4.1. 
 
Figure-4.1: Tangent and Secant moduli for non-linear stress-strain relation. 
If one dimension of the material changed, other dimensions of the material need to be 
changed to keep the volume constant. This lateral/transverse strain is related to the 
applied longitudinal strain by empirical means, and the ratio of transverse strain to 
longitudinal strain is known as Poisson’s ratio ( ?). Transverse strain can be expected to 
be opposite in nature to longitudinal strain, and both longitudinal and transverse strains 
are linear strains. For most metals the values of ? are close to 0.33, for polymers it is 
between 0.4 – 0.5, and for ionic solids it is around 0.2. 
Stresses applied on a material can be of two kinds – normal stresses, and shear stresses. 
Normal stresses cause linear strains, while the shear stresses cause shear strains. If the 
material is subjected to torsion, it results in torsional strain. Different stresses and 
corresponding strains are shown in figure-4.2.  
 
Figure 4.2: Schematic description of different kinds of deformations/strains. 
Analogous to the relation between normal stress and linear strain defined earlier, shear 
stress ( t) and shear strain ( ?) in elastic range are related as follows: 
? t G = 
where G is known as Shear modulus of the material. It is also known as modulus of 
elasticity in shear. It is related with Young’s modulus, E, through Poisson’s ratio, ?, as  
) 1 ( 2 ? +
=
E
G 
Similarly, the Bulk modulus or volumetric modulus of elasticity K, of a material is defined 
as the ratio of hydrostatic or mean stress ( s
m
) to the volumetric strain ( ?). The relation 
between E and K is given by 
) 2 1 ( 3 ?
s
-
=
?
=
E
K
m
 
Let s
x
, s
y
 and s
z
 are linear stresses and e
x
, e
y
 and e
z
 are corresponding strains in X-, Y- 
and Z- directions, then 
3
z y x
m
s s s
s
+ +
= 
Volumetric strain or cubical dilatation is defined as the change in volume per unit 
volume.  
z y x z y x
e e e e e e + + ˜ - + + + = ? 1 ) 1 )( 1 )( 1 ( , 
m
e 3 = ? 
where e
m
 is mean strain or hydrostatic (spherical) strain defined as  
3
z y x
m
e e e
e
+ +
= 
An engineering material is usually subjected to stresses in multiple directions than in just 
one direction. If a cubic element of a material is subjected to normal stresses s
x
, s
y
, and 
s
z
, strains in corresponding directions are given by 
[] ) (
1
z y x x
E
s s ? s e + - = , [ ] ) (
1
x x y y
E
s s ? s e + - = , and [ ] ) (
1
y x z z
E
s s ? s e + - = 
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