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Elastic limit is the limit up to which the object will undergo elastic deformation. When a force is applied to an object, it undergoes changes in its dimension. For some materials, this change is not seen with the naked eyes, but it does occur at molecular levels.
As soon as this external force is removed the object regains its original shape. Alternatively, if the applied force is large the object will undergo permanent deformation. This dimensional change will be lasting and the object will not get back to its original form.
Elastic and Plastic Deformation
An object is said to undergo elastic deformation when it reverts to its actual shape when the external force is removed. However, when the external force is large, the elastic deformation transforms into plastic deformation. Unlike elastic deformation, plastic deformation causes a permanent change in the dimensions of the object.
Elastic Limit
As discussed previously, an elastic deformation causes a temporary change in the dimension of an object. The limit up to which the object will undergo elastic deformation is known as the elastic limit. Within the elastic limit, the object will always have reversible dimensional changes. Once the object is subjected to an external force that exceeds the elastic limit, permanent plastic deformation will take place.
Stress and Strain
When an external deforming force is applied to an object, the object develops an equal and opposite restorative force within itself. This force divided per unit area is called stress.
Stress = Force/Area
SI unit of stress is newton per meter square (N/m2) or pascal (Pa)
Dimensional Formula is ML-1T-2
A stressed object undergoes dimensional changes. The changes in dimension when divided by the original dimension give the measurement of Strain. Depending upon the type of change, the strain is classified as longitudinal strain, shearing strain, and volumetric strain.
- Longitudinal Strain → Change in length / Original length
- Shearing Strain → Change in displacement / Original length
- Volume Strain → Change in volume / Original length
Strain has no units and dimensional formula
Stress α Strain
or
\(\frac {Stress} {Strain} = E\)
StressStrain = E
Where E is a constant known as modulus of elasticity.
Hooke's law is an Empirical law. It is followed by most of the materials with few exceptions.
Stress-Strain Relationship
The Stress and Strain relationship can be described using a graph. The below graph shows the typical representation of the stress-strain relationship for a metal. The stress-strain graph will vary from material to material. From this graph several observations can be drawn:
- The region O to A in the graph is linear. Here the Hooke's law is obeyed.
- From point A to point B, the graph is non-linear which means that the stress is no longer proportional to strain. However, the metal at this point returns to its original shape.
- Point B is called the elastic limit or the yield point. The corresponding stress at this point is called the Yield Strength of the material.
- The region from B to D shows that the strain in the metal increases rapidly even though there is a small change in stress.
- The region of the graph where Hooke's law is observed is the region of elastic deformation
- If the stress is removed at point C, the metal still undergoes deformation which is plastic deformation.
- Point D on the graph represents the ultimate tensile strength of the material. Beyond this point, even if low force is applied the metal fractures at point E.
- When the distance between points D and E is small the material is said to be brittle. Conversely, when the distance between these two points is large, the material is said to be ductile.
Limit of Proportionality
The limit of proportionality is the highest possible applied stress at which the stress and strain are directly proportional. Within the proportional limit, the stress-strain graph is a straight line and Hooke's law is applicable.
Yield Point
When a material is stressed beyond its elastic limit, it undergoes plastic deformation. The point at which the transition from elastic to plastic deformation begins is known as the Yield point of the material.
Elastic Limit vs Proportionality Limit vs Yield Point
| Elastic Limit | Proportionality Limit | Yield Point |
|---|---|---|
| Limit within which the material under stress gets back to its original shape and size once the stress is removed | Limit within which the Hooke's law is obeyed and stress-strain graph is linear | The point at which object transitions from elastic to plastic deformation. |
Things to Remember based on Elastic Limit
- Elastic limit is the limit up to which the object will undergo elastic deformation.
- Stress is the force acting on a unit area.
- The changes in dimension when divided by the original dimension give the measurement of Strain.
- SI unit of stress is Newton per meter square (N/m²) or pascal (Pa).
- There is no unit for strain.
- Strain is classified as longitudinal strain, shearing strain, and volumetric strain.
- Stress- Strain relationship is established by Hooke’s law.
Sample Questions based on Elastic Limit
Ques. What is elasticity in a material? (2 marks)
Ans. Elasticity is described as the property by which material goes back to its previous dimensions when stress is removed.
Ques. Materials that stretch more are said to be more elastic. Is this true or false? (2 marks)
Ans. Contrary to the popular belief, a material that stretches more is not more elastic than the material that stretches less. In fact, material that stretches to a lesser extent for a given load is considered to be more elastic.
Ques. What is the point at which a material undergoes plastic deformation from an elastic deformation? (2 marks)
Ans.The point at which a material changes from elastic to plastic deformation is known as Yield.
Ques. What are elastomers? Give some examples. (2 marks)
Ans. Elastomers are materials that have a high elastic limit that is they can be stretched to cause large strains. The aorta present in the human heart is an elastomer.
Ques. What would the stress/strain ratio be if stress is doubled? (2 marks)
Ans. The stress-strain ratio is a constant known as the modulus of elasticity. So, if the stress is doubled the strain decreases by half and the stress-strain ratio remains constant.
Ques. What kind of material has zero moduli of elasticity? (2 marks)
Ans. Materials exhibiting plastic behavior have zero moduli of elasticity. The stress-strain graph for such materials is a horizontal line.
Ques. What law is known as the elasticity law? (2 marks)
Ans. Hooke's law is also known as the elasticity law. It states that the stress is proportional to strain within the elastic limit of an object.
Ques. Figure shows the strain-stress curve for a given material. What are (a) Young’s modulus and (b) approximate yield strength for this material? (3 marks)
Ans. 
(a) Young’s modulus of the material (Y) is given by
Y =Stress/Strain
=150 x 106/0.002
150 x 106/2 x 10-3
=75 x 109 Nm-2
=75 x 1010 Nm-2
(b)Yield strength of a material is defined as the maximum stress it can sustain. From graph, the approximate yield strength of the given material
= 300 x 106 Nm-2
= 3 x 108 Nm-2 .
Ques.The stress-strain graphs for materials A and B are shown in figure. (3 marks)
The graphs are drawn to the same scale.
(a) Which of the materials has the greater Young’s modulus?
(b) Which of the two is the stronger material?
Ans. (a) From the two graphs we note that for a given strain, stress for A is more than that of B. Hence Young’s modulus =(Stress /Strain) is greater for A than that of B.
(b) Strength of a material is determined by the amount of stress required to cause fracture. This stress corresponds to the point of fracture. The stress corresponding to the point of fracture in A is more than for B. So, material A is stronger than material B.
Ques. Read the ‘allowing two statements below carefully and state, with reasons, if it is true or false.
(a) The Young’s modulus of rubber is greater than that of steel;
(b) The stretching of a coil is determined by its shear modulus. (2 marks)
Ans. (a) False. The-Young’s modulus is defined as the ratio of stress to the strain within the elastic limit. For a given stretching force elongation is more in rubber and quite less in steel. Hence, rubber is less elastic than steel.
(b) True. Stretching of a coil is determined by its shear modulus. When equal and opposite forces are applied at opposite ends of a coil, the distance, as well as shape of helicals of the coil change and it involves shear modulus.
Ques. A steel cable with a radius of 1.5 cm supports a chairlift at a ski area. If the maximum stress is not to exceed 108 Nm-2 what is the maximum load the cable can support ? (2 marks)
Ans. 
Ques. Define Young's Modulus of elasticity. (2 marks)
Ans. For a solid, in the form of a wire or a thin rod, Young’s modulus of elasticity within elastic limit is defined as the ratio of longitudinal stress to longitudinal strain. It is given as:
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