Material science

1. Machakos Campus
Department of Orthopaedic
Trauma Medicine
Subject – Introduction to
Medical Mechanics
Topic – Stress and strain analysis
By
Mr. Oduor Wafulah
4th April, 2023.

2. Wafulah
Oduor
PRESENTATION OUTLINES
A. Stress
I. Terminologies related to stress
II. Types of stress
B. Strain
I. Terminologies related to strain
II. Types of strain
C. Relation Between Stress and Strain
D. Stress and strain Diagram

3. Wafulah
Oduor
STRESS :
Stresses are expressed as the ratio of the applied
force divided by the resisting area
Mathematically:
σ = Force /Area
Units:
N/m2 or Pascal.
1kPa = 1000Pa, 1 Mpa= 106 Pa

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Oduor
TERMINOLOGIES RELATED TO STRESS
Stressor:
A stressor is anything that has the effect of causing
stress.
Stress capacity:
While it is unclear precisely how much stress a
person can carry, since each person has some stress
in their lives, we say he/she has a capacity for stress.
Similarly in case of Rocks, how much capacity they
have to bear stress.
Stress-load:
Everyone, even children, must carry some amount of
stress in their daily lives. When we think of stress as
having an amount, or quantity, we refer to this as the
person’s stress-load. And here in case of rocks, we say
that how much an already existing stress is applied on a
rock.

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Oduor
TYPES OF STRESS
There are two types of stress
1) Normal Stress
I. Tensile stress
II. Compressive stress
2) Combine Stress
I. Shear stress
II. Tortional stress

6. Wafulah
Oduor
1)Normal Stress:
This is whereby the resisting area is perpendicular to the
applied force
1) Tensile Stress:
It is a stress induced in a body when it is subjected to
two equal and opposite pulls (Tensile force) as a result
of which there is tendency in increase in length.
It acts normal to the area and pulls on the area.
2) Compressive Stress:
Stress induced in a body, when subjected to two equal
and opposite pushes as a result of which there is a
tendency of decrease in length of the body.
It acts normal to the area and it pushes on the area.

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Oduor
2) Combined Stress:
 A condition of stress that cannot be represented by a single
resultant stress
1) Shear stress/Tangential stress:
Forces parallel to the area resisting the force cause shearing
stress.
 It differs to tensile and compressive stresses, which are
caused by forces perpendicular to the area on which they act.
2) Tortional stress:
 The stresses and deformations induced in a
circular shaft by a twisting moment.

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Oduor
Types of Stress:
Diagrams

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Oduor
STRAIN:
 When a body is subjected to some external
force, there is some change in the dimension
of the body.
 The ratio of change in dimension of body to
its original dimension is called as Strain.
 Strain is a dimensionless quantity.

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Oduor
Terminologies Related to Strain:
Longitudinal or Linear Strain
Strain that changes the length of a line without
changing its direction.
Can be either compressional or tensional.
Compression
Longitudinal strain that shortens an object.
Tension
Longitudinal strain that lengthens an object.

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Oduor
Shear
Strain that changes the angles of an object.
Shear causes lines to rotate.
Infinitesimal Strain
Strain that is tiny, a few percent or less.
Allows a number of useful mathematical
simplifications and approximations.
Finite Strain
Strain larger than a few percent.
Requires a more complicated mathematical
treatment than infinitesimal strain.

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Oduor
Homogeneous Strain
 Uniform strain.
 Straight lines in the original object remain straight.
 Parallel lines remain parallel.
 Circles deform to ellipses.
 Note that this definition rules out folding, since an
originally straight layer has to remain straight.
Inhomogeneous Strain
 How real geology behaves.
 Deformation varies from place to place.
 Lines may bend and do not necessarily remain parallel.

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Oduor
TYPES OF STRAIN
1. Tensile Strain
2. Compression Strain
3. Volumetric Strain
4. Shear Strain

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Oduor
1) Tensile Strain:
Ratio of increase in length to the original length of the
body when it is subjected to a pull force.
Tensile strain = Increase in length/ Original Length
= dL/L
2)Compressive Strain:
Ratio of decrease in Length to the original length of body
when it is subjected to push force.
Compressional Strain = Decrease in length/Original Length
= dL/L
3)Volumetric Strain:
Ratio of change of volume to the original volume.
Volumetric Strain= dV/V

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Oduor
Types of Strain
Diagrams

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Oduor
4) Shear Strain
Strain due to shearstresses.
Sign convection for direct strain
Tensile strains are considered positive in case
of producing increase in length.
Compressive strains are considered negative
in case of producing decrease in length.

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Oduor
Elasticity and Hooke’s Law
 All solid materials deform when they are
stressed, and as stress is increased, deformation
also increases
 Elasticity is the ability of a material returns to
its original size and shape on removal of load
causing deformation
 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

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Oduor
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

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Oduor
Relation between Stress and Strain
Hooke’s Law:
 It states that, the strain of the material is
proportional to the applied stress within the
elastic limit of that material
 Within Elastic limit the ratio of stress applied to
strain developed is constant.
 The constant is known as Modulus of elasticity
or Young’s Modulus or Elastic Modulus.

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Oduor
Relation between Stress and Strain
Mathematically:
 Hooke’s Law: Stress = Modulus (E) × Strain
𝐄 =
𝐒𝐭𝐫𝐞𝐬𝐬
𝐒𝐭𝐫𝐚𝐢𝐧
 Young’s Modulus E, is generally assumed to be same in
tension or Compression and for most of engineering
application has high Numerical value.
 Typically E = 210 X 109N/m2 for steel
 Modulus is a property of material under stress; the larger the
modulus the small deformation
 Stress (‘applied force’) – Units: Pascal= Pa=N/m2
 Strain (‘deformation’) – Units: Dimensionless
 Modulus: Same unit as stress

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Oduor
Stress and Strain Diagram

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Factor of Safety
 The load which any member of a machine
carries is called Working Load, and stress
produced by this load is the Working Stress
 The working stress must be less than the yield
stress, tensile strength or the ultimate stress
 This working stress is also called the
Permissible Stress or the Allowable Stress or
the Design Stress

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Oduor
Beams Theory
 In many engineering structures members are required to resist
forces that area applied laterally or transversely to their axes.
 These type of members are termed as beam
Definitions of a Beam
 A laterally loaded member, whose cross-sectional dimensions are
small as compared to its length
 A bar which is subjected to forces or couples that lie in a plane
containing the longitudinal axis of the bar.
 The forces are understood to act perpendicular to the longitudinal
axis of the bar
 A bar working under bending is generally termed as a beam

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Oduor
 The beams may be made from several usable
engineering materials such as;
a) Metal
b) Wood
c) Plastic
d) Concrete

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Oduor
Timoshenko Beam Theory
 The deflection of a beam is caused by bending
and shear when subjected to transverse
loading and can be obtained by solving the
equilibrium equations.
Torsion Theory
 Theory of pure torsion states that if a couple
is applied to a cylindrical rod in such a way
that the axis of the couple is coincident with
the axis of the rod.

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Oduor
Questions

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Oduor