2. Stress and Strain
Outline
1. Stress
2. Strain
3. Stress-strain curve
4. Important definitions
5. Hook’s law
6. Stress strain curve of different materials
4. Stress and Strain
What is stress?
Stress is the resistance force
generated per unit area.
Stress σ =
𝐹𝑜𝑟𝑐𝑒 (𝐹)
𝐶𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐴𝑟𝑒𝑎 (𝐴)
5. Stress and Strain
What is strain?
Strain or % Elongation is the ratio of change in length under the
application of force to the initial length.
Strain ε =
𝐶ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ (δ𝐿)
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ (𝐿)
6. Stress and Strain
Types of stress
1. Tensile Stress: Pulling force per unit area
2. Compressive stress: Pushing force per unit
area
3. Shear stress: Parallel force per unit area.
1. Single shear: Failure area is in one
place
2. Double shear: Failure area is in two
place
8. Stress and Strain
Important definitions:
1. Elastic limit(O to A): The elastic limit of a material is the maximum stress that can be
developed within it without causing permanent deformation. Beyond this limit, the material
will not come back to its original shape after force is removed.
2. Upper yield point (B): Material requires maximum stress to initiate plastic
deformation inside the material.
3. Lower yield point (C): Material requires minimum stress to initiate plastic
deformation inside the material.
4. Ultimate tensile stress (D): The maximum amount of stress a material can withstand
before starting to break.
5. Breaking stress (E): At this stress, the material breaks.
9. Stress and Strain
Strain Hardening
The elastic limit of a material is the maximum stress that can be developed within it
without causing permanent deformation
Necking
Necking is a type of plastic deformation observed in ductile materials subjected to tensile stress.
This deformation is characterized by a localized reduction in the cross-sectional area of the
material, giving it a "V" or "neck" shape. Necking begins after ultimate strength is reached. During
necking, the material can no longer withstand the maximum stress and the strain in the specimen
rapidly increases.
10. Stress and Strain
Hook’s law
When a material is loaded within its elastic limit, the stress is proportional to the strain
σ ∝ ε
Young Modulus/Modulus of elasticity
The Young's modulus (E) is a property of the material that tells us how easily it can stretch and
deform and is defined as the ratio of tensile stress (σ) to tensile strain (ε).
E =
𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑜𝑟 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑆𝑡𝑟𝑒𝑠𝑠
𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑜𝑟 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑆𝑡𝑟𝑎𝑖𝑛
=
σ
ε
11. Stress and Strain
Poisson’s ratio
Poisson's ratio is the ratio of lateral strain to
longitudinal strain in the direction of stretching
force.
Poisson’s ratio, μ =
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
= -
ϵt
ϵl
Lateral strain, ϵt = −
𝑑𝐵
𝐵
Longitudinal strain, ϵl =
𝑑𝐿
𝐿
12. Stress and Strain
Why do we need stress-strain curve?
By seeing the stress strain curve, we can select the proper
material for the manufacturing of a machine. From the stress-
strain curve, we can determine if our selected material will be
able to hold the load properly or not.
14. Stress and Strain Math
A 2.0-m-long steel rod has a cross-sectional area of 0.30cm2. The rod is a
part of a vertical support that holds a heavy 550-kg platform that hangs
attached to the rod’s lower end. Ignoring the weight of the rod, what is
the tensile stress in the rod and the elongation of the rod under the
stress? Young’s modulus for steel is Y=2.0×1011Pa.
16. Stress and Strain Math
A 5.0-m-long steel rod has a diameter of 0.1 m2. The rod is a part of a
vertical support that holds a heavy 30000 gm platform that hangs
attached to the rod’s lower end. Ignoring the weight of the rod, what is
the tensile stress in the rod and the elongation of the rod under the
stress? Young’s modulus for steel is Y=2.0×1011Pa.
Practice
