Short Notes: Elastic Constants | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Elastic Constants
Elastic Constants: Stress produces a strain, but how much strain is produced 
depends on the solid itself. The solid is then characterised by an elastic modulus 
that relates strain to stress.
Stress c r
-------------------- — Elastic Modulus = —
Strain £
Different types of stresses and their corresponding strains within elastic limit are 
related which are referred to as elastic constants. The three types of elastic 
constants (moduli) are:
• Modulus of elasticity or Young’s modulus (E),
• Bulk modulus (K) and
• Modulus of rigidity or shear modulus (M, C or G).
• Young’s modulus
Page 2


Elastic Constants
Elastic Constants: Stress produces a strain, but how much strain is produced 
depends on the solid itself. The solid is then characterised by an elastic modulus 
that relates strain to stress.
Stress c r
-------------------- — Elastic Modulus = —
Strain £
Different types of stresses and their corresponding strains within elastic limit are 
related which are referred to as elastic constants. The three types of elastic 
constants (moduli) are:
• Modulus of elasticity or Young’s modulus (E),
• Bulk modulus (K) and
• Modulus of rigidity or shear modulus (M, C or G).
• Young’s modulus
Rigidity modulus
f
\
shear stress 
shear strain
= hearModuhis
G = -t- = -5 — ^- =
£. St i f A» 7
G is modulus of rigidity, Ft is shear stress which is also designated as y is shear 
strain
• Bulk modulus
Normal inward forces 
compress the solid
bulk stress 
bulk strain
Bulk Modulus
K =
PV
~sv
Poisson's Ratio:
• The three stresses and strains do not operate independently.
• Stresses produce strains in lateral directions as the solid tries to retain its 
original volume.
• When an axial force is applied along the longitudinal axis of a bar, the length 
of a bar will increase but at the same time its lateral dimension (width) will be 
decreased so, it is called as Poisson' ratio.
Page 3


Elastic Constants
Elastic Constants: Stress produces a strain, but how much strain is produced 
depends on the solid itself. The solid is then characterised by an elastic modulus 
that relates strain to stress.
Stress c r
-------------------- — Elastic Modulus = —
Strain £
Different types of stresses and their corresponding strains within elastic limit are 
related which are referred to as elastic constants. The three types of elastic 
constants (moduli) are:
• Modulus of elasticity or Young’s modulus (E),
• Bulk modulus (K) and
• Modulus of rigidity or shear modulus (M, C or G).
• Young’s modulus
Rigidity modulus
f
\
shear stress 
shear strain
= hearModuhis
G = -t- = -5 — ^- =
£. St i f A» 7
G is modulus of rigidity, Ft is shear stress which is also designated as y is shear 
strain
• Bulk modulus
Normal inward forces 
compress the solid
bulk stress 
bulk strain
Bulk Modulus
K =
PV
~sv
Poisson's Ratio:
• The three stresses and strains do not operate independently.
• Stresses produce strains in lateral directions as the solid tries to retain its 
original volume.
• When an axial force is applied along the longitudinal axis of a bar, the length 
of a bar will increase but at the same time its lateral dimension (width) will be 
decreased so, it is called as Poisson' ratio.
V
Ad
~ d
Aw
— w
A l / l A l / l
• Value of Poisson's ratio is same in tension and compression 
Under uniaxial loading
• OS p s 0.5
• p = 0 for cork
• p = 0.5 For perfectly plastic body (Rubber)
• p = 0.25 to 0.42 for elastic metals
• p = 0.1 to 0.2 for concrete
• p = 0.286 mild steel
• p is greater for ductile metals than for brittle metals.
Volumetric Strain (ev): It is defined as the ratio of change in volume to the initial 
volume.
=
Change involunte _ AV 
Initial volume V
Volumetric Strain Due to Single Direct Stress: The ratio of change in volume to 
original volume is called volumetric strain.
ev = e7 + e2 + 63
AV AL AB AT 
V ~ L B T
Volumetric strain:
P
e. = — u
P P
e, = — u.e, = - u — 
E' ' E
er = ^ , e y = A ( l- 2 //)
For the circular bar of diameter d:
V = - d zL
4
Page 4


Elastic Constants
Elastic Constants: Stress produces a strain, but how much strain is produced 
depends on the solid itself. The solid is then characterised by an elastic modulus 
that relates strain to stress.
Stress c r
-------------------- — Elastic Modulus = —
Strain £
Different types of stresses and their corresponding strains within elastic limit are 
related which are referred to as elastic constants. The three types of elastic 
constants (moduli) are:
• Modulus of elasticity or Young’s modulus (E),
• Bulk modulus (K) and
• Modulus of rigidity or shear modulus (M, C or G).
• Young’s modulus
Rigidity modulus
f
\
shear stress 
shear strain
= hearModuhis
G = -t- = -5 — ^- =
£. St i f A» 7
G is modulus of rigidity, Ft is shear stress which is also designated as y is shear 
strain
• Bulk modulus
Normal inward forces 
compress the solid
bulk stress 
bulk strain
Bulk Modulus
K =
PV
~sv
Poisson's Ratio:
• The three stresses and strains do not operate independently.
• Stresses produce strains in lateral directions as the solid tries to retain its 
original volume.
• When an axial force is applied along the longitudinal axis of a bar, the length 
of a bar will increase but at the same time its lateral dimension (width) will be 
decreased so, it is called as Poisson' ratio.
V
Ad
~ d
Aw
— w
A l / l A l / l
• Value of Poisson's ratio is same in tension and compression 
Under uniaxial loading
• OS p s 0.5
• p = 0 for cork
• p = 0.5 For perfectly plastic body (Rubber)
• p = 0.25 to 0.42 for elastic metals
• p = 0.1 to 0.2 for concrete
• p = 0.286 mild steel
• p is greater for ductile metals than for brittle metals.
Volumetric Strain (ev): It is defined as the ratio of change in volume to the initial 
volume.
=
Change involunte _ AV 
Initial volume V
Volumetric Strain Due to Single Direct Stress: The ratio of change in volume to 
original volume is called volumetric strain.
ev = e7 + e2 + 63
AV AL AB AT 
V ~ L B T
Volumetric strain:
P
e. = — u
P P
e, = — u.e, = - u — 
E' ' E
er = ^ , e y = A ( l- 2 //)
For the circular bar of diameter d:
V = - d zL
4
V L d " L ^ E
Ad P
— r = e ; = -fi —
e i = 70 - 2p) e
E
Volumetric Strain due to Three Mutually Perpendicular Stress System: When a 
body is subjected to identical pressure in three mutually perpendicular direction, 
then the body undergoes uniform changes in three directions without undergoing 
distortion of shape.
A B
Three stress system
P , P 2 +P2
= — -JU— ------
^ E E
R R + R
e, = — - u—-----L
' E E
R R + R
e. = — - u — ---- -
• ’ E ‘ E
e ^ . = ev +e2 +e.
'R + R + R '| 
E i
^ = ( 1 - 2 ^ )
or
Shear Modulus or Modulus of Rigidity (G)
shear stress r
(j = — ------------- = —
shear strain < j >
• At principal planes, shear stress is always zero.
• Planes of maximum shear stress also contain normal stress.
Relationship of E , G , K and p :
Modulus of rigidity:
G =
E
2(1 + /u)
Bulk modulus:
Page 5


Elastic Constants
Elastic Constants: Stress produces a strain, but how much strain is produced 
depends on the solid itself. The solid is then characterised by an elastic modulus 
that relates strain to stress.
Stress c r
-------------------- — Elastic Modulus = —
Strain £
Different types of stresses and their corresponding strains within elastic limit are 
related which are referred to as elastic constants. The three types of elastic 
constants (moduli) are:
• Modulus of elasticity or Young’s modulus (E),
• Bulk modulus (K) and
• Modulus of rigidity or shear modulus (M, C or G).
• Young’s modulus
Rigidity modulus
f
\
shear stress 
shear strain
= hearModuhis
G = -t- = -5 — ^- =
£. St i f A» 7
G is modulus of rigidity, Ft is shear stress which is also designated as y is shear 
strain
• Bulk modulus
Normal inward forces 
compress the solid
bulk stress 
bulk strain
Bulk Modulus
K =
PV
~sv
Poisson's Ratio:
• The three stresses and strains do not operate independently.
• Stresses produce strains in lateral directions as the solid tries to retain its 
original volume.
• When an axial force is applied along the longitudinal axis of a bar, the length 
of a bar will increase but at the same time its lateral dimension (width) will be 
decreased so, it is called as Poisson' ratio.
V
Ad
~ d
Aw
— w
A l / l A l / l
• Value of Poisson's ratio is same in tension and compression 
Under uniaxial loading
• OS p s 0.5
• p = 0 for cork
• p = 0.5 For perfectly plastic body (Rubber)
• p = 0.25 to 0.42 for elastic metals
• p = 0.1 to 0.2 for concrete
• p = 0.286 mild steel
• p is greater for ductile metals than for brittle metals.
Volumetric Strain (ev): It is defined as the ratio of change in volume to the initial 
volume.
=
Change involunte _ AV 
Initial volume V
Volumetric Strain Due to Single Direct Stress: The ratio of change in volume to 
original volume is called volumetric strain.
ev = e7 + e2 + 63
AV AL AB AT 
V ~ L B T
Volumetric strain:
P
e. = — u
P P
e, = — u.e, = - u — 
E' ' E
er = ^ , e y = A ( l- 2 //)
For the circular bar of diameter d:
V = - d zL
4
V L d " L ^ E
Ad P
— r = e ; = -fi —
e i = 70 - 2p) e
E
Volumetric Strain due to Three Mutually Perpendicular Stress System: When a 
body is subjected to identical pressure in three mutually perpendicular direction, 
then the body undergoes uniform changes in three directions without undergoing 
distortion of shape.
A B
Three stress system
P , P 2 +P2
= — -JU— ------
^ E E
R R + R
e, = — - u—-----L
' E E
R R + R
e. = — - u — ---- -
• ’ E ‘ E
e ^ . = ev +e2 +e.
'R + R + R '| 
E i
^ = ( 1 - 2 ^ )
or
Shear Modulus or Modulus of Rigidity (G)
shear stress r
(j = — ------------- = —
shear strain < j >
• At principal planes, shear stress is always zero.
• Planes of maximum shear stress also contain normal stress.
Relationship of E , G , K and p :
Modulus of rigidity:
G =
E
2(1 + /u)
Bulk modulus:
K =
E
3(1-2M)
or
9 KG 
1K + G
3 K - 2 G 
6 K + 2G
Material Number of Independent 
elastic constant
Homogeneous & Isotropic 2
Orthotropic fWoodl 9
Anisotropic 21