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What is Stress?
Stress is magnitude of internal resisting forces and it can be defined as force per unit area
within materials that arises from external applied forces (pressure). By uneven heating, or
permanent deformation and that permits an accurate description and prediction of elastic,
plastic, and fluid behaviour. Stress is given by the following formula:
F
Stress (σ) = Force (F) /Area (A) σ
where, σ is the stress applied, F is the force applied A
and A is the area of the force application.
The unit of stress is N/m2.
Classifying Loads on Materials
• Normal Load (Axial load): Load is perpendicular to the supporting material.
• Tension Load: As the ends of material are pulled apart to make the material longer, the load
is called a tension load.
• Compression Load: As the ends of material are pushed into make the material smaller, the
load is called a compression load.
• Torsion Loads: Angular distortion on a component, such as a shaft, when a moment is
applied. (Twisting)
• Thermal Loads: Distortion caused be heating or cooling a material. A normal load is created
when the material is constrained in any direction in the plane that is constrained.
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Tension
Compression
Shear Load : Tangential load
Pulling apart
Strain
- Ratio of elongation of a material to the original
length
- unit deformation
e Lo e
ε
Lo
e : elongation (mm) L
Lo : unloaded(original) length of a material (mm)
ε : strain (mm/mm) - Unitless
Elongation:
e L Lo
L : loaded length of a material (mm)
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Longitudinal Strain
When a body is subjected to an axial tensile or compressive load, there is an
axial deformation in the length of the body. The ratio of axial deformation to
the original length of the body is known as longitudinal (or linear) strain. The
longitudinal strain is also defined as the deformation of the body per unit
length in the direction of the applied load.
Let L = Length of the body,
P = Tensile force acting on the body,
δL = Increase in the length of the body in the direction of P.
Then, longitudinal strain = δL /L
Lateral Strain
The strain at right angles to the direction of applied load is known as lateral strain. Let a rectangular bar
of length l, breadth b and depth d is subjected to an axial tensile load P as shown in Fig. The length of
the bar will increase while the breadth and depth will decrease.
Let δl = Increase in length,
δb = Decrease in breadth, and
δd = Decrease in depth.
lateral strain = δb/b or δd/d
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Poisson’s Ratio
The ratio of lateral strain to the longitudinal strain is a
constant for a given material, when the material is stressed
within the elastic limit. This ratio is called Poisson’s ratio and
it is generally denoted by μ. Hence mathematically,
Poisson’s ratio, µ = Lateral strain / Longitudinal strain
or Lateral strain = 𝜈 × longitudinal strain
For many metals and other alloys, the value of Poisson’s ratio
varies from 0.25 to 0.35. For rubber, its value ranges from 0.45 to
0.50.
• Theoretically, Poisson’s ratio for isotropic materials should be
1/4; furthermore, the maxi mum value for is 0.50.
• For isotropic materials, shear and elastic moduli are related to
each other and to Poisson’s ratio according to
E = 2G(1 + 𝜈)
In most metals, G is about 0.4E
• The body will regain its previous shape and size only
ELASTICITY AND ELASTIC
when the deformation caused by the external force,
LIMIT
• When an external force acts on a body, the body is within a certain limit. Thus, there is a limiting
tends to undergo some deformation. If the value of force up to and within which, the
external force is removed and the body comes deformation completely disappears on the removal
back to its original shape and size (which means of the force.
the deformation disappears completely), the • The value of stress corresponding to this limiting
body is known as elastic body. force is known as the elastic limit of the material.
• This property, by virtue of which certain • If the external force is so large that the stress
materials return back to their original position exceeds the elastic limit, the material loses to some
after the removal of the external force, is called extent its property of elasticity. If now the force is
elasticity. removed, the material will not return to its original
shape and size and there will be a residual
deformation in the material.
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ANELASTICITY
Time dependent elastic strain –
• In many engineering materials, an additional time-dependent component of elastic
strain exists.
• This means that elastic deformation continues even after stress is applied and does
not stop immediately.
• When the load is removed, the material takes some finite time to fully recover to its
original state.
This time-dependent elastic behavior is called anelasticity.
It occurs due to microscopic and atomistic processes that occur within the material
during deformation.
In metals, the anelastic component is usually small and is often neglected in engineering
applications.
Hooke’s Law and Young’s modulus or Modulus of elasticity,
• Hooke’s Law states that when a material is loaded within elastic limit, the stress is
proportional to the strain produced by the stress.
• This means the ratio of the stress to the corresponding strain is a constant within the
elastic limit. This constant is known as Modulus of Elasticity or Modulus of Rigidity or
Modulus of Elasticity.
𝑵𝒐𝒓𝒎𝒂𝒍 𝑺𝒕𝒓𝒆𝒔𝒔 𝝈
= 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝑜𝑟
𝑪𝒐𝒓𝒓𝒆𝒔𝒑𝒐𝒏𝒅𝒊𝒏𝒈 𝑺𝒕𝒓𝒂𝒊𝒏 𝜺
where σ = Normal stress, 𝜀 = Strain and E = Young’s modulus
The SI unit for the modulus of elasticity is gigapascal (GPa), where 1 GPa = 109 N/m2 = 103 MPa.
Room-Temperature Elastic and Shear
Moduli and Poisson’s Ratio for Various
Metal Alloys
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Modulus of Elasticity (Young’s Modulus of Rigidity or Shear
Modulus Modulus.
The ratio of tensile stress or compressive The ratio of shear stress to the
stress to the corresponding strain is a corresponding shear strain within the
constant. elastic limit, is known as Modulus of
This ratio is known as Young’s Modulus or Rigidity or Shear Modulus. This is
Modulus of Elasticity and is denoted by E. denoted by C or G.
𝑺𝒉𝒆𝒂𝒓 𝑺𝒕𝒓𝒆𝒔𝒔 𝝉
𝑮= = Or 𝝉 = 𝑮𝜸
𝑻𝒆𝒏𝒔𝒊𝒍𝒆 𝒔𝒕𝒓𝒆𝒔𝒔 𝑪𝒐𝒎𝒑𝒓𝒆𝒔𝒔𝒊𝒗𝒆 𝑺𝒕𝒓𝒆𝒔𝒔 𝑺𝒉𝒆𝒂𝒓 𝑺𝒕𝒓𝒂𝒊𝒏 𝜸
𝑬= 𝒐𝒓
𝑻𝒆𝒏𝒔𝒊𝒍𝒆 𝒔𝒕𝒓𝒂𝒊𝒏 𝑪𝒐𝒎𝒑𝒓𝒆𝒔𝒔𝒊𝒗𝒆 𝒔𝒕𝒓𝒂𝒊𝒏
𝝈
𝑬=
𝜺
Elastic Deformation
Deformation in which stress and strain are proportional is
called elastic deformation;
• a plot of stress (ordinate) versus strain (abscissa) results in a
linear relationship, as shown in Figure.
• The slope of this linear segment corresponds to the modulus
of elasticity E.
• This modulus may be thought of as stiffness, or a material’s
resistance to elastic deformation.
• The greater the modulus, the stiffer the material, or the
smaller the elastic strain that results from the application of
a given stress.
• Elastic deformation is nonpermanent, which means that
when the applied load is released, the piece returns to its
Figure: Schematic stress–strain
diagram showing linear elastic
original shape.
deformation for loading and
unloading cycles.
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Elastic Deformation
• There are some materials (i.e., gray cast
iron, concrete, and many polymers) for
which this elastic portion of the stress–
strain curve is not linear as shown in figure.
• For this nonlinear behavior, either the
tangent or secant modulus is normally used.
• The tangent modulus is taken as the slope
of the stress–strain curve at some specified
level of stress, whereas the secant modulus
represents the slope of a secant drawn from Figure: Schematic stress–strain diagram showing
nonlinear elastic behavior and how secant and
the origin to some given point of the stress-
tangent moduli are determined.
strain curve.
Plastic Deformation
Figure(a) represent the tensile stress–strain behavior into
the plastic region for a typical metal.
Plastic Deformation:
• Begins beyond 0.005 strain.
• Stress-strain proportionality breaks (Nonlinear
behavior).
• Permanent, nonrecoverable deformation occurs.
Figure shows:
• Gradual transition from elastic to plastic regions.
• Curvature at the onset of plastic deformation.
• Stress increases more rapidly with strain in the plastic
zone.
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Plastic Deformation
Plastic Deformation at the Atomic scale: From an atomic perspective,
• Plastic deformation occurs when atomic bonds break with original atomic neighbors and reform
bonds with new neighbors.
• Unlike elastic deformation, atoms do not return to their original positions after stress removal.
The mechanism of this deformation is different for crystalline and amorphous materials.
• For crystalline solids, deformation is accomplished by means of a process called slip, which involves
the motion of dislocations.
• Plastic deformation in non-crystalline solids (as well as liquids) occurs by a viscous flow mechanism
Figure : Atomic rearrangements that accompany the motion of an edge dislocation as it moves in response to an applied shear stress. (a) The extra half-plane of atoms is labeled A.
(b) The dislocation moves one atomic distance to the right as A links up to the lower portion of plane B; in the process, the upper portion of B becomes the extra half-plane. (c) A
step forms on the surface of the crystal as the extra half-plane exits. (Adapted from A. G. Guy, Essentials of Materials Science, McGraw-Hill Book Company, New York, 1976, p. 153.)
Stress and Strain Diagram
For Ductile and Brittle
Materials
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(a) Stress-Strain Curves for Ductile Materials
If a mild steel bar of uniform cross-
sectional area is subjected to gradually
increasing axial tensile force (generally
is done in Universal Testing Machine)
till failure of the bar occurs, and if we
plot the graph for stress and strain, the
following curve (Fig. 1.1) may be
obtained:
mild steel bar for Tensile test.jpeg
Fig.1.1 : Stress – strain Diagram for Ductile Material
Necking formation.jpeg
The curve may be divided into following parts:
Portion OA: This portion is absolutely straight, where the stress is proportional to strain and
the material obeys Hooke’s law. The value of stress at point A is called proportional limit.
Portion AB: In this portion, Hook’s law is not obeyed, although the material may still be
elastic. The point B indicates the elastic limit.
Portion BC: In this portion, the metal shows an appreciable strain even without further
increase in stress and the strain is not fully recoverable when load is removed.
Portion CC': Yielding commences in this portion and there is a drop of stress at the point C'
immediately after yielding commences at C. The point C' is termed as lower yield point and
C is called upper yield point.
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Portion C'D: After yielding has taken place at C', further straining takes place at this
portion by increasing the stress and the stress–strain curve continues to rise up to the
points D. Strain in this portion is about 100 times that of portion O to C. At the point D,
the bar begins to form a local neck. The point D is termed as ultimate tensile stress point.
Ultimate stress is calculated at this point.
Portion DE: In this portion, the load falling off from the maximum until fracture at E
takes place. The point E is termed as fracture or breaking point and the corresponding
stress is called breaking stress.
(b) Stress Strain Curves for Brittle
Materials
Materials which show very small elongation
before
they fracture are called brittle materials. The
shape of curve for high carbon steel is shown
in fig. and is typical of many brittle materials
such as G.I., concrete and high strength light
alloys. For most brittle materials the
permanent elongation (i.e. increase in length)
is less than 10%. Fig. : Stress – strain Diagram for Brittle Material
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Principal Mechanical Properties
Those characteristics of the materials which describe their behaviour under external loads are
known as Mechanical Properties. The most important and useful mechanical properties are:
Strength
• It is the resistance offered by a material when subjected to external loading. So, stronger the
material the greater the load it can withstand. Depending upon the type of load applied the
strength can be tensile, compressive, shear or torsional.
• The maximum stress that any material will withstand before destruction is called its ultimate
strength (Point D as shown in Fig. 1.1).
Elasticity
• Elasticity of a material is its power of coming back to its original position after deformation
when the stress or load is removed. Elasticity is a tensile property of its material.
• The greatest stress that a material can endure without taking up some permanent set is called
elastic limit (Point B as shown in Fig. 1.1.).
Principal Mechanical Properties
Stiffness (Rigidity)
• The resistance of a material to deflection is called stiffness or rigidity. Steel is stiffer or more
rigid than aluminium.
• Stiffness is measured by Young’s modulus E. The higher the value of the Young’s modulus,
the stiffer the material. E is the ratio of stress over strain and is given by the slope of line O–A.
Plasticity
• The plasticity of a material is its ability to undergo some degree of permanent deformation
without failure. Plastic deformation will take place only after the elastic range has been
exceeded, beyond point b.
• Plasticity is an important property and widely used in several mechanical processes like
forming, shaping, extruding and many other hot and cold working processes.
• In general, plasticity increases with increasing temperature and is a favorable property of
material for secondary forming processes.
• Due to this properties various metal can be transformed into different products of required
shape and size. This conversion into desired shape and size is effected either by the
application of pressure, heat or both.
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Principal Mechanical Properties
Ductility
• Ductility of a material enables it to draw out into thin wire on application of the load. Mild
steel is a ductile material.
• The wires of gold, silver, copper, aluminium, etc. are drawn by extrusion or by pulling
through a hole in a die due to the ductile property.
• The ductility decreases with increase of temperature.
• The per cent elongation and the reduction in area in tension is often used as empirical
measures of ductility.
Malleability
• Malleability of a material is its ability to be flattened into thin sheets without cracking by hot
or cold working.
• Aluminium, copper, tin, lead, steel, etc. are malleable metals.
• Lead can be readily rolled and hammered into thin sheets but can not be drawn into wire.
• Ductility is a tensile property, whereas malleability is a compressive property. Malleability
increases with increase of temperature.
Principal Mechanical Properties
Brittleness
• The brittleness of a material is the property of breaking without much permanent distortion.
There are many materials, which break or fail before much deformation take place. Such
materials are brittle e.g., glass, cast iron.
• Therefore, a non-ductile material is said to be a brittle material.
• Usually the tensile strength of brittle materials is only a fraction of their compressive strength.
• A brittle material should not be considered as lacking in strength.
• It only shows the lack of plasticity. On stress-strain diagram, these materials don’t have yield
point and value of E is small.
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Principal Mechanical Properties
Toughness
The toughness of a material is its ability to withstand both plastic and elastic deformations. It is a
highly desirable quality for structural and machine parts to withstand shock and vibration.
Manganese steel, wrought iron, mild steels are tough materials.
For Ex: If a load is suddenly applied to a piece of mild steel and then to a piece of glass the mild
steel will absorb much more energy before failure occurs. Thus, mild steel is said to be much
tougher than a glass.
• Toughness is a measure of the amount of energy a material can absorb before actual fracture
or failure takes place. “The work or energy a material absorbs is called modulus of toughness”
Toughness is also resistance to shock loading. It is measured by a special test on Impact
Testing Machine.
Principal Mechanical Properties
Hardness
• Hardness is closely related to strength. It is the ability of a material to resist scratching,
abrasion, indentation, or penetration.
• It is directly proportional to tensile strength and is measured on special hardness testing
machines by measuring the resistance of the material against penetration of an indentor of
special shape and material under a given load.
Hardenability
• Hardenability is the degree of hardness that can be imparted to metal by process of
hardening.
• A metal capable of being hardened throughout its structure is said to have high hardenability.
• The material is heated above a certain temperature and then suddenly quenched in a cold oil
or water bath.
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Principal Mechanical Properties
Resilience
• Resilience is the capacity of a material to absorb energy when it is
deformed elastically and then, upon unloading, to have this energy
recovered.
• The associated property is the modulus of resilience, Ur, which is
the strain energy per unit volume required to stress a material from
an unloaded state up to the point of yielding.
𝜺
𝑼𝒓 = ∫𝟎 𝒚 𝝈𝒅𝜺 … … … … … … … … (𝒊)
Assuming a linear elastic region, we have
𝟏
𝑼𝒓 = 𝝈𝒚 𝜺𝒚 …………………(ii)
𝟐
where 𝜀 is the strain at yielding.
Figure: Schematic representation
𝟏 𝟏 𝝈𝒚 𝝈𝟐𝒚
𝑼𝒓 = 𝝈𝒚 𝜺𝒚 = 𝝈𝒚 = ……….(iii) showing how modulus of resilience
𝟐 𝟐 𝑬 𝟐𝑬 (corresponding to the shaded area)
For SI units, this is joules per cubic meter (J/m3, equivalent to Pa). is deter mined from the tensile
*The maximum energy which can be stored in a body up to elastic stress–strain behavior of a material.
limit is called the proof resilience.
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Que: When a brass rod of diameter 6 mm is subjected to a tension of 5 × 103 N, the
diameter changes by 3.6 × 10-4 cm. Calculate the longitudinal strain and Poisson’s ratio for
brass given that E for the brass is 9 × 1010 N/m²
Que. A metal wire of length 1.5 m is loaded and an elongation of 2 mm is
produced. If the diameter of the wire is 1 mm, find the change in the diameter of
the wire when elongated. Poisson’s ratio v = 0.24
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