CHAPTER 2
Strain
Strain
When a force is applied to a body,
the body will be subjected to deformation
Chap. 2 : Strain
2-2
Normal Strain: The elongation or contraction of a line
segment per unit length
ε= (Δs’ – Δs) / Δs
Strain is dimensionless: m/m
Δs’ –Δs : elongation
Chap. 2 : Strain
2-3
Shear Strain: The change in angle between two line segments that
were originally perpendicular
γ= π/ 2 -θ’(in radians)
The shear strain is positive when the right angle decreases (θ’is
smaller than π / 2 )
Chap. 2 : Strain
2-4
Tensile testing
• Elongation δ= L L0 is measured
• δ is used to calculate the normal
strain in the specimen
Normal strain :
Normal stress :
A stress-strain diagram is obtained by plotting the various values
of the stress and corresponding strain in the specimen
Chap. 2 : Strain
2-5
Stress-strain diagram obtained from the standard tension test on a
structural steel specimen.
By plotting σ(ordinate) against ε(abscissa), we get a conventional
stress-strain diagram
• If the strain disappears when the stress is
removed, the material is said to behave
elastically.
• The largest stress for which this occurs is
called the elastic limit.
• When the strain does not return to zero after the
stress is removed, the material is said to behave
.
plastically
Chap. 2 : Strain
2-6
Stress-strain diagrams for various materials that fail
without significant yielding
Chap. 2 : Strain
2-7
HOOKE’S LAW
• Most engineering materials exhibit a linear
relationship between stress and strain with the elastic
region
• Discovered by Robert Hooke in 1676 using springs,
known as Hooke’s law
• E represents the constant of proportionality, also
called the modulus of elasticity or Young’s modulus
• E has units of stress, i.e., Pascals, MPa or Gpa
• If stress in material is greater than the proportional
limit, the stress-strain diagram ceases to be a straight
line and the equation is not valid.
• Modulus of elasticity E, can be used only if a material
has linear-elastic behavior.
Chap. 2 : Strain
2-8
Shearing Strain
• A cubic element subjected to a shear stress
will deform into a rhomboid. The
corresponding shear strain is quantified in
terms of the change in angle between the
sides,
 xy  f  xy
 
• A plot of shear stress vs. shear strain is
similar the previous plots of normal stress
vs. normal strain except that the strength
values are approximately half. For small
strains,
 xy  G  xy  yz  G  yz  zx  G  zx
where G is the modulus of rigidity or shear
modulus.
Chap. 2 : Strain
2-9
Poisson’s Ratio
• For a slender bar subjected to axial
loading:
x
x 
 y z  0
E
• The elongation in the x-direction is
accompanied by a contraction in the
other directions. Assuming that the
material is isotropic (no directional
dependence),
y  z  0
• Poisson’s ratio is defined as
z
lateralstrain  y

 
axialstrain
x
x
Chap. 2 : Strain
2 - 10
Chap. 2 : Strain
2 - 11
Chap. 2 : Strain
2 - 12
Chap. 2 : Strain
2 - 13
Relationships among the elastic constants
E
G
2(1   )
G is the modulus of rigidity
E is the modulus of Young

is the coefficient of Poisson
Chap. 2 : Strain
2 - 14
Chap. 2 : Strain
2 - 15
Chap. 2 : Strain
2 - 16
Chap. 2 : Strain
2 - 17
Chap. 2 : Strain
2 - 18