This document provides information on various mechanical properties of solids, including: - Solids have a definite shape and volume, with molecules held in fixed positions through electrical forces. - Solids can be modeled as springs connecting molecules, and exhibit elasticity by returning to their original shape/size when external forces are removed. - Crystalline solids have an ordered atomic structure, while amorphous solids have a random structure. - Stress, strain, elastic modulus, Young's modulus, bulk modulus, and shear modulus are defined and explained in relation to the deformation of solids under forces. - Different types of structures like arches are discussed in relation to their strength and load distribution properties.

1. Mechanical
properties of solids
chapter -9

2. Chapter 9
Solids and Fluids

3. States of Matter
● Solid
● Liquid
● Gas
● Plasma

4. Solids
● Has definite volume
● Has definite shape
● Molecules are held in
specific locations
○ by electrical forces
● vibrate about
equilibrium positions
● Can be modeled as
springs connecting
molecules

5. More About Solids
● External forces can be applied to the solid and compress the material
○ In the model, the springs would be compressed
● When the force is removed, the solid returns to its original shape and size
○ This property is called elasticity

6. Crystalline Solid
● Atoms have an
ordered structure
● This example is salt
○ Gray spheres
represent Na+ ions
○ Green spheres
represent Cl- ions

7. Amorphous Solid
● Atoms are arranged
almost randomly
● Examples include
glass

8. Liquid
● Has a definite volume
● No definite shape
● Exists at a higher temperature
than solids
● The molecules “wander”
through the liquid in a
random fashion
○ The intermolecular forces are
not strong enough to keep the
molecules in a fixed position

9. Gas
● Has no definite volume
● Has no definite shape
● Molecules are in constant random motion
● The molecules exert only weak forces on each other
● Average distance between molecules is large compared to the size of the
molecules

10. Plasma
● Matter heated to a very high temperature
● Many of the electrons are freed from the nucleus
● Result is a collection of free, electrically charged ions
● Plasmas exist inside stars

11. Deformation of Solids
● All objects are deformable
● It is possible to change the shape or size (or both)
of an object through the application of external
forces
● when the forces are removed, the object tends to
its original shape
○ This is a deformation that exhibits elastic behavior

12. Elastic Properties
● Stress is the force per unit area causing the
deformation
● Strain is a measure of the amount of
deformation
● The elastic modulus is the constant of
proportionality between stress and strain
○ For sufficiently small stresses, the stress is directly
proportional to the strain
○ The constant of proportionality depends on the
material being deformed and the nature of the
deformation

13. Elastic Modulus
● The elastic modulus can be thought of as the stiffness of the material
○ A material with a large elastic modulus is very stiff and difficult to deform
■ Analogous to the spring constant
○ stress = Elastic modulus x strain

14. Young’s Modulus:
Elasticity in Length
● Tensile stress is the ratio
of the external force to
the cross-sectional area
○ Tensile is because the
bar is under tension
● The elastic modulus is
called Young’s modulus

15. Young’s Modulus, cont.
● SI units of stress are Pascals, Pa
○ 1 Pa = 1 N/m2
● The tensile strain is the ratio of the change in length to the original length
○ Strain is dimensionless

16. Young’s Modulus, final
● Young’s modulus applies to
a stress of either tension or
compression
● It is possible to exceed the
elastic limit of the material
○ No longer directly
proportional
○ Ordinarily does not return to
its original length

17. Breaking
● If stress continues, it surpasses its ultimate
strength
○ The ultimate strength is the greatest stress the object
can withstand without breaking
● The breaking point
○ For a brittle material, the breaking point is just beyond
its ultimate strength
○ For a ductile material, after passing the ultimate
strength the material thins and stretches at a lower
stress level before breaking

18. Bulk Modulus, cont.
● Volume stress, ∆P, is
the ratio of the force
to the surface area
○ This is also the
Pressure
● The volume strain is
equal to the ratio of
the change in
volume to the
original volume

19. Bulk Modulus, final
● A material with a large bulk modulus is difficult to
compress
● The negative sign is included since an increase in
pressure will produce a decrease in volume
○ B is always positive
● The compressibility is the reciprocal of the bulk
modulus

20. Notes on Moduli
● Solids have Young’s, Bulk, and Shear moduli
● Liquids have only bulk moduli, they will not undergo a shearing or tensile
stress
○ The liquid would flow instead

21. Ultimate Strength of Materials
● The ultimate strength of a material is the maximum force per unit area the
material can withstand before it breaks or factures
● Some materials are stronger in compression than in tension

22. Post and Beam Arches
● A horizontal beam is
supported by two
columns
● Used in Greek temples
● Columns are closely
spaced
○ Limited length of
available stones
○ Low ultimate tensile
strength of sagging stone
beams

23. Semicircular Arch
● Developed by the
Romans
● Allows a wide roof span
on narrow supporting
columns
● Stability depends upon
the compression of the
wedge-shaped stones

24. Gothic Arch
● First used in Europe
in the 12th century
● Extremely high
● The flying buttresses
are needed to prevent
the spreading of the
arch supported by
the tall, narrow
columns

25. Density
● The density of a substance of uniform composition is defined as its mass per
unit volume:
● Units are kg/m3 (SI) or g/cm3 (cgs)
● 1 g/cm3 = 1000 kg/m3

26. Density, cont.
● The densities of most liquids and solids vary slightly with changes in
temperature and pressure
● Densities of gases vary greatly with changes in temperature and pressure

27. Shear Modulus:
Elasticity of Shape
● Forces may be parallel
to one of the object’s
faces
● The stress is called a
shear stress
● The shear strain is the
ratio of the horizontal
displacement and the
height of the object
● The shear modulus is S

28. Shear Modulus, final
●
● S is the shear modulus
● A material having a
large shear modulus is
difficult to bend

29. Bulk Modulus:
Volume Elasticity
● Bulk modulus characterizes the response of an object to uniform squeezing
○ Suppose the forces are perpendicular to, and act on, all the surfaces
■ Example: when an object is immersed in a fluid
● The object undergoes a change in volume without a change in shape

30. Specific Gravity
● The specific gravity of a substance is the ratio of its density to the density of
water at 4° C
○ The density of water at 4° C is 1000 kg/m3
● Specific gravity is a unitless ratio

31. Strength Of Materials :
• Strength of materials, also called Mechanics of
materials, is a subject which deals with the behavior
of solid objects subject to stresses and strains.
• The study of strength of materials often refers to
various methods of calculating the stresses and strains
in structural members, such as beams, columns, and
shafts.

32. Elasticity
Introduction
When a force is applied to a body, there are two possibilities
1)
either state of the body changes
i.e. body moves form its initial position
F

33. Elasticity
2)
or shape and/or size changes
If shape and/or size changes then that change is called as Deformation
and the corresponding force is called as deforming force .
F

34. Elasticity
Bodies
Rigid
Elastic
Non-rigid
Plastic
By the use of Deformation all bodies can be
classified as -

35. Rigid Body:
• A rigid body is defined as a body on which
the distance between two points never
changes whatever be the force applied on it.
• Practically, there is no rigid body.
Rigid body is a body whose shape and/or size does not
change even if a very high force is applied.

36. Deformable body:
A deformable body is defined as a body on which the distance between
two points changes under action of some forces when applied on it.
Non rigid body is a body whose shape and/or size
changes if a very large force is applied.

37. Elasticity
Property possessed by material body by virtue of which a material body
opposes to change its shape or size or both when subjected to deforming
force and regains its original shape and size when deforming forces are
removed
Elasticity
Elastic Body : An elastic body is a body which regains its original shape and/or size
when deforming forces are removed.
e.g. : Rubber

38. Elasticity
Plasticity :
Property possessed by material bodies by virtue of which it do not
regain its original shape or size or both after removal of external
deforming force
Plastic Body :
A plastic body is a body which does not regain its original
shape and/or size when deforming forces are removed.
e.g. : clay, dough etc.

39. Elasticity
Distinguish between
Plastic Body
Elastic Body
Elastic bodies regain original
dimensions after removal of external
forces
Plastic bodies do not regain original
dimensions after removal of external
forces
In elastic bodies intermolecular
forces are strong
In Plastic bodies intermolecular
forces are weak
Magnitude of external force required
to change dimension is larger
Magnitude of external force required
to change dimension is small
E.G. Steel ,Iron , Quartz etc… E.G. Mud , clay , Pleistocene etc…
https://www.youtube.com/watch?v=YKpvYF0hVDE

40. Elasticity
elastic forces or restoring forces
such force are called as elastic forces or restoring forces
The direction of internal restoring force is exactly
opposite to applied force
and magnitude is same as that of applied force.
When external force is applied on a body , it gets deformed
during deformation of body , internal forces are produced in body
which opposes the change in dimensions of body
Deforming force
Internal restoring
force

41. • Dimensional formula of the stress =
𝑀𝐿𝑇 −2
𝐿2
∴ = 𝑀𝐿−1𝑇 −2
• The SI unit of stress is =𝑁𝑚−2 = 𝑃𝑎𝑠𝑐𝑎𝑙
Stress (𝝈):
Stress is the applied force or system of forces
that tends to deform a body.

42. Strain (𝜺)
• The change in the size or shape of a body due to
the deforming force.
• The ratio of change in configuration to the
original configuration is called strain.
• i.e.
𝑆𝑡𝑟𝑎𝑖𝑛, =
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛
• Strain is Dimensionless hence no unit

43. Elasticity
Definition of Stress and strain
Stress
Stress is defined as the internal
restoring force per unit area.
The internal restoring force is
equal and opposite to the deforming
force. i.e. the applied force.
Hence Stress is also defined as
the applied force per unit area.
Strain is defined as the ratio of
change in dimension to original
dimension.
Since strain is the ratio of identical
quantities therefore, it has no units and
dimensions.
Strain
Strain =
Change in Dimensions
Original Dimensions
Units :
Dimensions : [M1 L-1 T-2]
C.G.S. Unit : dyne/cm2
S.I.. Unit : N/m2
https://www.youtube.com/watch?v=AnwrItEvqh4

44. END OF DAY-1

45. Elasticity
Types of Stress and strain
Depending on nature of deformation stress and strain is of three types
1 ) Tensile Stress
If a force F = mg is applied to a wire of radius
r having area of cross section A = πr2
When force is applied at right angle to cross
sectional area of wire then wire gets
elongated and gives to stress which is
perpendicular to cross sectional area. This
stress is called as tensile stress
L
l
Mg
Tensile stress =
Applied Force
Area
1 ) Tensile Strain
Is define as change in length per
unit original length
Tensile Strain = l /L

46. Elasticity
2 ) Volume
Stress
Types of Stress and strain
Stress produced in body due to change in volume
is called as volume stress
When a body is subjected to increase in pressure
then body is compressed .
Thus the volume of body changes.
Volume stress =
F
A
= dP
= Change in Pressure
2 ) Volume
Strain
V-dv
Change in volume per unit original
volume is called as volume strain
Volume strain =
dv
V
https://www.youtube.com/watch?v=OeuZ8y0_w6A

47. Elasticity
Types of Stress and strain
3 ) Shearing Stress
When a tangential force is applied
to one face of body with other
face of body as fixed then the
shape changes and stress is
produced in the body .
The stress produced due to change in shape of body is called as shearing stress
A B
C
D
P Q
R
S
A B
C
D
P Q
R
S
F
A′ B′
Q′
P′
θ θ
Shearing stress
https://www.youtube.com/watch?v=x-UiYHyAUaM

48. Elasticity
Types of Stress and strain
3 ) Shearing Strain
Ration of lateral displacement of layer to its distance
from fixed layer is called as shearing strain
F
P P′ Q Q′
x
h
S R
θ
Shearing strain =
Lateral displacement of any layer
Its perpendicular distance from the fixed layer
=
PP'
PS
=
x
h
tan θ ≈ θ [... x <<< h
hence tan θ ≈ θ ]
Shearing
Strain
=

49. Elastic Limit
• Is the maximum stress from which an elastic
body will recover its original state after the
removal of the deforming force.
• It differs widely for different materials.
• It is very high for a substance like steel and low
for a substance like lead.

50. Limits of proportionality
•If a tensile force applied to a uniform bar
of mild steel is gradually increased and the
corresponding extension of the bar is
measured, then provided the applied force
is not too large, a graph depicting these
results is likely to be as shown in Figure.
•Since the graph is a straight line,
extension is directly proportional to the
applied force. (Hooke’s Law)
The point on the graph where extension is no longer proportional
to the applied force is known as the limit of proportionality.

53. Limits of elasticity
• As mentioned, limits of proportionality …
Just beyond this point the material can
behave in a non-linear elastic manner, until
the elastic limit is reached.
• If the applied force is large, it is found that
the material becomes plastic and no longer
returns to its original length when the force
is removed.
In short, The value of force up to and within which, the
deformation entirely disappears on removal of the force is
called limit of elasticity

57. https://www.youtube.com/watch?v=zc3b6LdDFtY

58. END OF DAY-2

59. Elasticity
Hook’s Law
Within elastic limits stress is directly proportional to strain
. . . Stress ∝ Strain
i.e.
Strain
= Constant = E
Stress
E is a constant of proportionality, or
modulus of elasticity OR coefficient of elasticity
Robert Hook
Units : S.I. Unit : N/m2
C.G.S. Unit : dyne /cm2
Dimensions : [M1 L-1 T-2]
Hooke's Law is obeyed only if the deformation is within
the elastic limit.
beyond the elastic limit,
stress is not proportional to the strain and Hooke's
law is not obeyed.
https://www.youtube.com/watch?v=yAIb3T9DPyE

60. Elasticity
Elastic Constants
1) Young’s Modulus( Y)
The ration of tensile stress to tensile strain is called as
Young’s Modulus
. . .
. . .
. . .
. . .
Units : S.I. Unit : N/m2
C.G.S. Unit : dyne /cm2
Dimensions : [M1 L-1 T-2]
Only Solids have definite length.
Therefore,
Young’s modulus is the property of
solids only

64. Question 9. 2. Figure shows the strain-stress curve for a given material. What
are (a) Young’s modulus and (b) approximate yield strength for this material?

66. HOMEWORK EXERCISE - 9.4 AND 9.5, 9.11,

67. END OF DAY 3

68. Elasticity
Elastic Constants
2) Bulk Modulus( K)
The ratio of volume stress to volume strain is called as
Bulk Modulus
. . .
. . .
. . .
Units : S.I. Unit : N/m2
C.G.S. Unit : dyne /cm2
Dimensions : [M1 L-1 T-2]
Solid , liquid and gases all have Bulk modulus
Reciprocal of Bulk modulus is called as
compressibility

72. HOMEWORK - 9.13

73. Elasticity
Elastic Constants
3) Modulus of rigidity ( η)
The ratio of shearing stress to shearing strain is called as
Modulus of Rigidity
. . .
. . .
If θ is small then tanθ = 0
. . .
Units : S.I. Unit : N/m2
C.G.S. Unit : dyne /cm2
Dimensions : [M1 L-1 T-2]
Particular shape.
Therefore,
Only solids possess modulus of rigidity

76. Question 9. 7. Four identical hollow cylindrical columns of mild steel support a big
structure of mass 50,000 kg. The inner and outer radii of each column are 30 cm and 60
cm respectively. Assuming the load distribution to be uniform, calculate the
compressional strain of each column. Young’s modulus, Y = 2.0 x 10^11 Pa

77. HOMEWORK - 9.8 AND 9.9

78. https://www.youtube.com/watch?v=50VYcOryCXo&feature=emb_logo

80. https://www.youtube.com/watch?v=YI9ke-cy_1g&feature=emb_logo