Introduction to simple stress and strain.

1. UNIT : I
STRESSES AND STRAIN

2. Concept of elasticity and plasticity
 Strength of Material: When an external force acts on a body, the body tend to
undergo some deformation. Due to the cohesion between the molecules, the
body resists deformation. The resistance by which material of the body
opposes the deformation is known as strength of materials.
 Elasticity: Property of material by which it returns to its original shape and
size after removing the applied load, is called elasticity. And the material
itself is said to be elastic.
 Plasticity: Characteristics of material by which it undergoes inelastic strains
(Permanent Deformation) beyond the elastic limit, known as plasticity. This
property is useful for pressing and forging.

4. Direct or Normal Stress
 When a force is transmitted through a body, the body tends to change its
shape or deform. The body is said to be strained
 Direct Stress =
Applied Force (F)
Cross Sectional Area (A)
 Units: Usually N/m2 (Pa), N/mm2 , MN/m2 , GN/m2 or N/cm2
 Note: 1 N/mm2 = 1 MN/m2 = 1 MPa

5. Unit of Stress
1
𝑁
𝑚2 = 1
𝑁
1000 𝑚𝑚 2 = 1
𝑁
106𝑚𝑚2 = 10−6 𝑁
𝑚𝑚2
1
𝑁
𝑚𝑚2 = 106 𝑁
𝑚2
{ Also, 1
𝑁
𝑚2 = Pascal = 1 Pa, and 106
is Mega (M)}
∴ 1
𝑁
𝑚𝑚2 = 1 MPa

6. Unit of Force
In MKS and SI units, the fundamental units are:
Metre – Length
Kilogram – mass
Second – time.
From the Newtons second law of motion,
Force ∝ rate of change of momentum.
∝ rate of change of (mass × velocity)
∝ mass × rate of change of velocity
∝ mass × acceleration.
Force = mass × acceleration.
 Unit – Kg 𝑚 𝑠2
= 9.81 N.
Difference is only when selecting the unit of force.
MKS - 𝑘𝑔 𝑚 𝑠2
or Kg - wt.
SI – Newton (N)

7. Direct or Normal Stress
 Direct stress may be tensile or compressive and result from forces acting
perpendicular to the plane of the cross-section

8. Direct or Normal Strain
 When loads are applied to a body, some deformation will occur resulting in a
change in dimension.
 Consider a bar, subjected to axial tensile loading force, F. If the bar extension
is dl and its original length (before loading) is l, then tensile strain is:
 Direct Strain ( ) =
Change in Length
Original Length
 i.e. = dl/l

9. Direct or Normal Strain
 As strain is a ratio of lengths, it is dimensionless.
 Similarly, for compression by amount, dl: Compressive strain = - dl/L
 Note: Strain is positive for an increase in dimension and negative for a
reduction in dimension.

10. Shear Stress and
Shear Strain
 Shear stresses are
produced by equal and
opposite parallel
forces not in line.
 The forces tend to
make one part of the
material slide over the
other part.
 Shear stress is
tangential to the area
over which it acts.

11. STRAIN
 It is defined as deformation per unit length
 it is the ratio of change in length to original length
 Tensile strain (ε) =
increase in length
Original length L
= δ / L (+ Ve)
 Compressive strain (ε) =
decrease in length
Original length L
= δ / L(- Ve)

12. Ultimate Strength
 The strength of a material is a measure of the stress that it can take when in
use. The ultimate strength is the measured stress at failure, but this is not
normally used for design because safety factors are required. The normal way
to define a safety factor is :
 safety factor =
stress at failure
stress when loaded
or
𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦
𝐷𝑒𝑚𝑎𝑛𝑑
=
Ultimate stress
Permissible stress
,
If
𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦
𝐷𝑒𝑚𝑎𝑛𝑑
≥ 1, Safe.

13. Strain
 We must also define strain. In engineering this is not a measure of force but is
a measure of the deformation produced by the influence of stress. For tensile
and compressive loads:
 Strain is dimensionless, i.e. it is not measured in metres, kilograms etc. strain
(ε) =
increase in length x
original length L
 For shear loads the strain is defined as the angle γ This is measured in radians
 shear strain (γ)=
shear displacement x
width L

14. Shear Stress and
Shear Strain
 Shear strain is the distortion
produced by shear stress on an
element or rectangular block as
above. The shear strain,
(gamma) is given as
 𝛾 =
𝑇𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑎𝑙 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐴𝐷
 𝛾 =
𝐷𝐷′
ℎ
=
𝑑𝑙
ℎ
= 𝑡𝑎𝑛𝜑 A B
C
D C’
D’
P
L
h
𝜑 𝜑
𝑑𝑙
Resistance (R)
𝑺𝒉𝒆𝒂𝒓 𝑭𝒐𝒓𝒄𝒆, 𝝉 =
𝑺𝒉𝒆𝒂𝒓 𝑹𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 (𝑹)
𝑺𝒉𝒆𝒂𝒓 𝑨𝒓𝒆𝒂 (𝑨)
=
𝑹
𝑨
=
𝑷
𝑳×𝟏

15. Shear Stress and Shear Strain
 For small φ, γ = φ
 Shear strain then becomes the change in the right angle.
 It is dimensionless and is measured in radians.

16. Elastic and Plastic deformation

17. Modulus of Elasticity
 If the strain is "elastic" Hooke's law may be used to define
 Youngs Modulus E =
Stress
= Strain
=
W
x
L
A
 Young's modulus is also called the modulus of elasticity or stiffness and is a
measure of how much strain occurs due to a given stress. Because strain is
dimensionless Young's modulus has the units of stress or pressure

18. Volumetric Strain
 Hydrostatic stress refers to tensile or compressive stress in all dimensions
within or external to a body
 Hydrostatic stress results in change in volume of the material.
 Consider a cube with sides x, y, z. Let dx, dy, and dz represent increase in
length in all directions.
 i.e. new volume = (x + dx) (y + dy) (z + dz)

19. Volumetric Strain
 Neglecting products of small quantities:
 New volume = x y z + z y dx + x z dy + x y dz
 Original volume = x y z
 = z y dx + x z dy + x y dz
 Volumetric strain=
z y dx + x z dy + x y dz
x y z
 ε v = dx/x + dy/y + dz/z
 ε v =εx +ε y +ε z

20. Elasticity and Hooke’s Law
 All solid materials deform when they are stressed, and as stress is increased,
deformation also increases.
 If a material returns to its original size and shape on removal of load causing
deformation, it is said to be elastic.
 If the stress is steadily increased, a point is reached when, after the removal
of load, not all the induced strain is removed.
 This is called the elastic limit.

21. Hooke’s Law
 Stress and strain has linear
relationship between O to A.
 i.e., 𝜎 𝛼 𝜀
 Hook's law applies only between
O and A, which is the
proportionality limit.
 Not proportional beyond point
A.
 Slope of the line OA represents
Youngs modulus (E)
 Higher slope means stiffer and
stronger material.
 It is a measure of the stiffness
of a material.
𝐸 =
𝜎
𝜀
Slope
O

22.  O – origin
 A- Proportionality limit
 B – Elastic limit
 C – Upper yield point
 D – Lower yield point
 E – Ultimate tensile strength
 F – Fracture point
O

23. Hooke’s Law
 States that providing the limit of proportionality of a material is not
exceeded, the stress is directly proportional to the strain produced.
 If a graph of stress and strain is plotted as load is gradually applied, the first
portion of the graph will be a straight line.
 The slope of this line is the constant of proportionality called modulus of
Elasticity, E or Young’s Modulus.
 It is a measure of the stiffness of a material.
 Modulus of Elasticity,=
Direct stress
Direct strain
 E=
σ
ε

24. Stress strain relation in 2 D Cases.
In (1D) stress system, 𝜎 ∝ 𝜀 and
𝜎
𝜀
= E.
𝜎 = Normal stress, 𝜀 = Strain and E = Youngs modulus.
2D stress system 
Longitudinal strain =
𝛿𝐿
𝐿
Lateral strain =
𝛿𝑏
𝑏
Note: If longitudinal strain is tensile, lateral stain will be
compressive. If longitudinal stain is compressive, lateral strain
will be tensile.

25. Stress strain relation in 2 D Cases.
 Poisson’s Ratio (𝜇) =
𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑎𝑟𝑖𝑛
𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
(or) Lateral strain = - 𝜇 × 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛. (As
lateral strain is opposite in sign to longitudinal
strain.)
 Determines how a body responds to a applied
stresses.

26. Stress strain relation in 2 D Cases.
Auxetic material

27. Stress strain relation in 2 D Cases.
 Let 𝜎 1 = Normal stress in x- direction and
𝜎 2 = Normal stress in y – direction.
Strain produced by 𝜎 1
Longitudinal strain produced by 𝜎 1 will be equal
𝜎 1
𝐸
. (x axis)
Lateral strain produced by 𝜎 1 will be equal to −𝜇
𝜎 1
𝐸
. (y axis)
Strain produced by 𝜎 2
Longitudinal strain produced by 𝜎 2 will be equal
𝜎 2
𝐸
. (y axis)
Lateral strain produced by 𝜎 2 will be equal to −𝜇
𝜎 2
𝐸
. (x axis)
Total Strain
Let, 𝑒1the total stain in x direction =
𝜎 1
𝐸
−𝜇
𝜎 2
𝐸
𝑒2the total stain in y direction =
𝜎 2
𝐸
−𝜇
𝜎 1
𝐸
A B
C
D
𝜎 1
𝜎 1
𝜎 2
𝜎 2

28. Stress strain relation in 3 D Cases.
 Strain produced by 𝜎 1
The stress 𝜎 1 will produce strain in the direction of x and also in the
direction y and z.
Longitudinal Strain in the direction of x =
𝜎 1
𝐸
(x axis)
Lateral strain in y direction= −𝜇
𝜎 1
𝐸
. (y axis)
Lateral strain in z direction = −𝜇
𝜎 1
𝐸
. (z axis)
Strain produced by 𝜎 2
Longitudinal Strain in the direction of y =
𝜎 2
𝐸
(y axis)
Lateral strain in x direction = −𝜇
𝜎 2
𝐸
(x axis)
Lateral strain in z direction = −𝜇
𝜎 2
𝐸
(z axis)
x
y
z
𝜎 1
𝜎 2
𝜎 3

29. Stress strain relation in 3 D Cases.
Strain produced by 𝜎 3
Longitudinal strain in the direction of z =
𝜎 3
𝐸
(z axis)
Lateral strain in x direction = −𝜇
𝜎 3
𝐸
(x axis)
Lateral strain in y direction = −𝜇
𝜎 3
𝐸
(y axis)
Total Strain (let 𝑒1, 𝑒2, and 𝑒3 be the total strain in x, y, and z direction respectively)
𝑒1 =
𝜎 1
𝐸
−𝜇
𝜎 2
𝐸
−𝜇
𝜎 3
𝐸
(𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 𝑖𝑛 𝑥 𝑎𝑥𝑖𝑠).
𝑒2 =
𝜎 2
𝐸
−𝜇
𝜎 1
𝐸
−𝜇
𝜎 3
𝐸
(𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 𝑖𝑛 𝑦 𝑎𝑥𝑖𝑠).
𝑒3 =
𝜎 3
𝐸
−𝜇
𝜎 1
𝐸
−𝜇
𝜎 2
𝐸
(𝑡𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 𝑖𝑛 𝑧 𝑎𝑥𝑖𝑠).
The above 3 equations give stress and strain relationship for 3 orthogonal normal stress system.