stress and strain

 The force applied to the material per unit area is called
Stress.
 The maximum stress a material can stand before it breaks is
called the breaking stress.
 Mathematically:
Stress =
𝐹𝑜𝑟𝑐𝑒
𝐴𝑟𝑒𝑎
𝜎 =
𝐹
𝐴
where σ is the stress
F is the force
A is the surface area.
 Units : Nm-2 or Pa
Prepared by : Syed Abuzar 1

 The internal force generated within the material to
oppose the Mechanical deformation or relative change
in shape or size of a material is known as Strain.
 Mathematically:
Strain =
𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛
𝑙𝑒𝑛𝑔𝑡ℎ
∈=
∆𝐿
𝐿
where ∆L = extension measured in meter.
L = original length measured in meter.
 Strain is dimensionless quantity and has no units as it is a
ratio of two lengths measured in meter.
 When a material is stretched, the change in length and the
strain are positive. When it is compressed, the change in
length and strain are negative.

 Strains which involve no length changes but which
do change angles is called Shear Strain.

 The distance moved by any point of material in the
(x,y,z) directions are denoted respectively by (u,v,w).
so
∈ 𝑥=
𝜕𝑢
𝜕𝑥
∈ 𝑦=
𝜕𝑣
𝜕𝑦
∈ 𝑧=
𝜕𝑤
𝜕𝑧
Where ∈ is strain in x,y and z direction.

 The action of stresses which will cause change in volume of
the material is known as volumetric strain
Volumetric strain = e =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒
𝑜𝑟𝑖𝑔𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
Consider the rectangular solid illustrated. The original shape is
on the left and the deformed shape on the right.

 The original volume is given by 𝑉𝑜 = ∆x∆y∆z. For the
deformed body on the right, the volume is given by:
V = ∆x(1+∈ 𝑥).∆y(1+∈ 𝑦). ∆z(1+ ∈ 𝑧 )
V = ∆x.∆y. ∆z(1+∈ 𝑥 +∈ 𝑦+∈ 𝑧+∈ 𝑥∈ 𝑦+∈ 𝑦∈ 𝑧+∈ 𝑧∈ 𝑥+∈ 𝑥∈ 𝑦∈ 𝑧)
 As strains are small, so that we can neglect all the high
order terms so
V ≈ ∆x.∆y. ∆z(1+∈ 𝑥 +∈ 𝑦+∈ 𝑧)
V = 𝑉𝑜(1+∈ 𝑥 +∈ 𝑦 + ∈ 𝑧)
 Now, using the definition of volumetric strain
e =
𝑉−𝑉𝑜
𝑉𝑜
= ∈ 𝑥 +∈ 𝑦 + ∈ 𝑧

 According to Hook’s law
𝑆𝑡𝑟𝑒𝑠𝑠
𝑆𝑡𝑟𝑎𝑖𝑛
= constant
Where constant is known as an elastic modulus or simply a modulus.
 The relation of Stress and Strain within the elastic limits :
𝐸 =
𝜎 𝑎
∈ 𝑎
𝑘𝑔/𝑚2
(1)
Where
E is Young’s modulus
𝜎 𝑎 is the axial stress, 𝑘𝑔/𝑚2
∈𝑎 is axial strain, m/m
 Materials undergo strain when they are subject to stress.
 The relationship between stress and strain is different for different
materials, and can be appreciated by plotting stress against strain.

 Suppose the stress is tensile, and the specimen of material is
stretched along x. Then, it will get thinner across the direction
of stretching(y and z directions).
 The strains will be negative along y and z. They are related to
the strain along x by a positive constant v.

 Stress along x produces transverse strains:
∈ 𝑦= ∈ 𝑧= -v∈ 𝑥
Where v is known as Poisson’s ratio.
From (1)
∈ 𝑦= ∈ 𝑧= -v
𝜎 𝑥
𝐸
This is true when we only have stress along x.
 In general, we have stress along y and z also. These
other stresses will cause transverse strains also. The
transverse strains along the same axis arising from
different stresses simply added together.

 When three stresses act, the general result is:
∈ 𝑥 =
1
𝐸
(𝜎𝑥 − 𝑣(𝜎 𝑦+𝜎𝑧))
∈ 𝑦 =
1
𝐸
(𝜎 𝑦 − 𝑣(𝜎 𝑥+𝜎𝑧))
∈ 𝑧 =
1
𝐸
(𝜎𝑧 − 𝑣(𝜎 𝑥+𝜎 𝑦))
 In 2D, 𝜎𝑧 = 0 so:
∈ 𝑥 =
1
𝐸
(𝜎 𝑥 − 𝑣𝜎 𝑦)
∈ 𝑦 =
1
𝐸
(𝜎 𝑦 − 𝑣𝜎 𝑥)
∈ 𝑧 =
−𝑣
𝐸
(𝜎 𝑥+𝜎 𝑦)

 A strain gauge is a sensor whose resistance varies with
applied force. It converts force, pressure, tension, weight,
etc into a change in electrical resistance which can then be
measured.
 Unit: Microstrain
“change in length in 𝜇𝑚 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ”.

 1856: Lord Kelvin first reported on a relationship
between strain and the resistance of wire conductors.
 1930’s : Charles Kearns made the first notable use of
bonded resistance strain gauges to measure the
vibratory strains in high performance propeller
blades.
 1937/8 : Arthur Ruge discovered that small diameter
wires made of electrical resistance alloys could be
bonded to a structure to measure surface strain.
 1952: At that time printing circuits were emerging,
and Saunders-Roe developed the idea of making a
strain gauge by etching the pattern for the gauge
from a thin foil.

 Based on Principles of working :
1. Mechanical
2. Electrical
3. Piezoelectric
 Based on Construction :
1. Wire-wound Type.
2. Foil type.
3. Semiconductor type.

 Wire-wound type gauges are further classified into two types:
1. Bonded Strain Gauge
A fine resistance wire diameter 0.025mm which is bent
again and again as shown in diagram. This is done to increase
the length of the wire so that it permits a uniform distribution of
stress. This resistance wire is placed between the two carrier
bases (paper, Bakelite or Teflon) which are cemented to each
other.
The carrier base protects the
gauge from damages. Leads
are provided for electrically
connecting the strain gauge
to a measuring instrument
(Wheatstone bridge).

 They are reasonably inexpensive.
 They can pull off overall accuracy of better than +/-0.10%.
 They are available in a short gauge length and have small
physical size.
 These strain gauges are only moderately affected by
temperature changes.
 They are extremely sensitive and have low mass.
 Bonded resistance strain gages can be employed to
measure both static and dynamic strain.
 These types of strain gauges are appropriate for a wide
variety of environmental conditions. They can measure
strain in jet engine turbines operating at very high
temperatures and in cryogenic fluid applications at
temperatures as low as -452℉ (-269℃).

2. Unbonded Strain Gauge:
It consists up of wire stretched between two points
in an insulating medium such as air. One end of the wire
is fixed and the other end is attached to a movable
element.
 It has preloaded resistance
wires connected to a
initially balanced
Wheatstone bridge.

 A small motion of movable base increases tension in
two wires while decreasing it in two others (the wires
are initially stretched enough so that they can never go
slack) due to which bridge become unbalance because
resistance of bridge changes.
 The output voltage is proportional to input
displacement which can be calibrated in terms of
Strain.

 The arrangement of an unbonded strain gauges consists of the
following. Two frames P and Q carrying rigidly fixed insulated
pins as shown in diagram. These two frames can move relative
with respect to each other and they are held together by a
spring loaded mechanism.
 A fine wire resistance strain
gauge is stretched around the
insulated pins.
 The strain gauge is connected
to a wheat stone bridge.

 When a force is applied on the structure under study
(frames P & Q), frames P moves relative to frame Q,
and due to this strain gauge will change in length and
cross section. That is, the strain gauge is strained. This
strain changes the resistance of the strain gauge and
this change in resistance of the strain gauge is
measured using a wheat stone bridge. This change in
resistance when calibrated becomes a measure of the
applied force and change in dimensions of the
structure.

 Used in force, pressure and acceleration measurement.
 Used in places where the gauge is to be detached and
used again and again.
 The range of this gauge is +/- 0.15% strain.
 This gauge has a very high accuracy.
Disadvantage:
 It occupies more space.

 The foil strain gage has metal foil on the electric
insulator of the thin resin, and gage leads attached.

 The strain gage is bonded to the measuring object with
a dedicated adhesive.
 Strain occurring on the measuring site is transferred to
the strain sensing element via adhesive and the resin
base.
 For accurate measurement, the strain gage and adhesive
should be compatible with the measuring material and
operating conditions such as temperature, etc.

 Semiconductor strain gauges are used where a very
high gauge factor and a small envelope are required.
 They depend for their action upon piezo-resistive
effect(the change in the value of the resistance due to
change in resistivity).
Materials:
 Semi-conducting materials such as silicon and
germanium are used as resistive materials.

 A typical semiconductor strain gauge is formed by the
semiconductor technology.
 Semiconducting filaments of length from 2mm to 10mm and
thickness of 0.05mm are bonded on suitable insulating
substrates.
 The gold leads are usually employed for making electrical
contacts.
 The electrodes are formed by
vapour deposition.
 The assembly is placed in a
protective box.

 When the strain is applied to the semiconductor
element a large change in resistance occur which can
be measured with the help of a wheatstone bridge.
 The strain can be measured with high degree of
accuracy due to relatively high change in resistance.
 A temperature compensated semiconductor strain
gauge can be used to measure small strains of the order
of micro-strain.

 The gauge factor of semiconductor strain gauge is very
high, about ±130.
 They are useful in measurement of very small strains of
the order of 0.01 micro-strains due to their high gauge
factor.
 The semiconductor strain gauge has much higher and
stable output.
 It has long life and durable.
 It possesses a high frequency response of 1012 Hz.
 They can be manufactured in very small sizes, their
lengths ranging from 0.7 to 7.0 mm.

 They are very sensitive to changes in temperature.
 Linearity of the semi-conductor strain gauge is poor
due to the higher output.
 Semi-conductor strain gauges are more expensive.
 Difficult to attach to the object under study.
 Limited strain range typically 3000 to 10,000
microstrain.

stress and strain

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stress and strain