Mechanical Properties of Solids: Stress-strain Curve and Hooke's Law

Mechanical properties of solids define the various characteristics of solids such as their resistance to deformation and their strength.

• Resistance to deformation is the resistance that is offered by an object to change its shape.
• Whereas strength elaborates on the ability of an object to resist the applied stress.
• When a solid is deformed, the atoms or molecules are displaced from their equilibrium position.
• This causes a change in interatomic distance.
• When the deforming force is removed, the intermolecular forces tend to drive them back to their original position.
• Thus the body regains its original shape and size.

Key Terms: Stress, Strain, Interatomic force, Strength, Elasticity, Plasticity, Ductility, Stress-strain curve, Hooke’s law, Young’s modulus of elasticity, Poisson's ratio, Elastic after effect, Elastic fatigue

Mechanical Properties of Solids

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Mechanical properties of solids focus on qualities such as deformation resistance and strength.

• Strength is the capacity of an object to endure applied stress, or how far it can withstand the force.
• The resistance to deformation of an object is its resistance to changing shape.
• If the object's resistance to deformation is low, it can quickly change shape and vice versa.

The following are the mechanical properties of solids

Elasticity 

Elasticity is defined as the property of an object by which the object regains its original shape and size after the removal of the applied force.

For example: If we stretch the rubber band to some extent and leave it then it will regain its original shape. 

Perfectly Elastic Body 

A Perfectly Elastic Body is defined as a body that recovers its original shape and size completely and immediately after the deforming force is removed.

For example: Phosphor bronze and Quartz fiber.

Plasticity

Plasticity is the property of an object in which the object changes its shape when deforming force is applied and never comes back to its original shape after the deforming force is removed.

It is the property of permanent deformation.

For example: All plastic objects.

Ductility 

Ductility is the property of an object in which it can be pulled in thin wires, plates, or sheets.

For example: Gold and Silver

Strength 

The ability to hold out against applied stress without any failure is known as strength. Compared to others, many objects have higher or more strength.

Resistance to Deformation

Solid materials have a definite size and shape.

• An external force is required to change the size and shape of the solid object.
• The shape of an object can be easily changed if the resistance to deformation is less. 
• Mechanical properties are those properties of solids that define their solidity.
• These properties include plasticity, elasticity, strength, and ductility. 

Stress 

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Under the action of an external force, if a body gets deformed then an internal force is framed at each and every section of the body that tends to restore the body to its original state.

• This internal force is known as Stress. 
• In simple words, stress is the internal restoring force per unit area.
• The magnitude of both internal and external forces is equal.
• The dimensional formula of stress is [ML^-1T^-2].
• Its SI unit is N/m² or Pascal.

If A is the cross-section area of the body and the force applied is F, then the stress will be

Stress = F/A

Types of Stress

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Stress is divided into three types

• Longitudinal Stress
• Shearing or Tangential Stress
• Bulk or Hydraulic Stress

Longitudinal Stress

When the deforming force of the cylindrical body is applied normally to the area of the cross-section, it is said to be Longitudinal Stress.

In longitudinal stress, a change takes place in the length of an object. 

Longitudinal Stress is divided into two types 

• Tensile Stress
• Compressional stress

Tensile stress

​Under the applied force effect, if there is an increase in the length of an object, then it is termed Tensile stress.

Shearing or Tangential Stress

Tangential stress is the restoring force per unit area when the force applied is parallel to the cross-sectional area of the body. The deforming force produces changes in the shape of the body when it is applied tangentially. 

Bulk or Hydraulic Stress

Hydraulic stress is the restoring force applied per unit area on the body or object by a fluid like water. 

Strain

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Strain is defined as a change in the size and shape of a body when a deforming force acts on the body. Simply, strain is the ratio of change in shape or size to the original shape or size. It is just a number and has no dimensions.

Types of Strain

There are three types of strain

• Shearing strain
• Longitudinal strain
• Volume strain

Shearing Strain

Shearing strain is the measurement of the relative displacement on the opposite faces of the body due to the shearing stress. It is represented by tanθ.

Longitudinal Strain

Due to the applied longitudinal stress, there is a change of length in the original length of the body. 

Longitudinal strain = change in length /original length

Volume strain

Volume strain is a strain that is produced by hydraulic pressure. It is defined as the ratio of change in volume to the original volume.

Hooke's Law

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According to Hooke's Law, strain and stress are corresponding to each other within the elastic limit. 

Thus,

Stress ∝ Strain

⇒ Stress/Strain = K

Here, K is the proportionality constant known as the modulus of elasticity.

• Hooke's law is said to be empirical law and is valid for most materials.
• Some materials like human muscle, rubber, etc do not obey Hooke's law.

Stress-Strain Curve

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A curve drawn between stress and strain is called the stress-strain curve.

• When stress and stress are drawn along the y-axis and x-axis respectively, a linear graph is formed in the ideal situation of Hooke’s law.
• However, when actual experiments are drawn, a curve is formed known as the stress-strain curve.
• To understand the tensile strength of a material, a stress-strain curve is very useful.
• This curve varies from material to material.

The stress-strain curve is shown in the below diagram. 

• The curve OA is a straight line. It indicates that stress is proportional to strain.
• Hooke’s law is obeyed up to point A. Point A is the proportional limit.
• After point A, stress is not proportional to strain. AB curved portion is obtained.
• Once the load is removed, the body recovers its original dimension.
• Point B is the elastic limit or yield point.
• The corresponding stress to point B is said to be the yield strength.
• Beyond B, the curve shows plastic deformation.
• Here, the strain increases more quickly than stress.
• Point D on the curve indicates the tensile strength of a material.
• The length of the wire increases without any additional load in the region between B and D.
• This region is said to be the plastic region.

Elastic Moduli

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The modulus of elasticity or Elastic Moduli is the ratio of strain and stress. It is one of the basic characteristics of the material.

Elastic Moduli can be of three types

• Young’s modulus
• Shear modulus
• Bulk modulus.

Young's Modulus of Rigidity

Young's Modulus of Rigidity is defined as the ratio of longitudinal stress to the longitudinal strain within the elastic limit.

• The symbol of Young's Modulus is Y.
• In comparison to other materials, the Young Modulus of metals has high values. 

Y= Normal Stress / Longitudinal Strain

Bulk Modulus of Rigidity

The ratio of normal stress to volumetric strain within the elastic limit is known as the Bulk Modulus of Rigidity. It is represented by B.

B = Normal Stress / Volumetric Strain

Bulk Moduli of Materials

Modulus of Rigidity or Shear Modulus

The ratio of shearing stress to shearing strain is known as the Modulus of Rigidity or Shear Modulus. It is represented by G.

G = Shearing Stress / Shearing Strain

Shear Moduli of a Few Common Materials

Poisson's Ratio

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Poisson's Ratio is defined as the ratio of lateral or transverse strain to longitudinal or axial strain.

• It is the ratio of two strains.
• It has no dimension or unit.
• The theoretical range of Poisson's Ratio is between – 1 to 0.5
• Its practical range is between 0 to 0.5.
• It is represented by σ.

σ = Lateral strain / Longitudinal strain

Relation Between Young’s Modulus, Bulk Modulus, Shear Modulus, and Poisson’s Ratio

• Y = 3B (1-2σ)
• Y = 2G (1+σ)
• σ = 3D-2G2G+6B
• 9Y = 1B + 3G

Elastic Fatigue

Under the action of repeating alternate deforming force, the property of an elastic body in which its behavior becomes less elastic is known as Elastic Fatigue.

• It takes place when a body comes under repeated deforming forces or strains, even within the elastic limit.
• Under such circumstances, the body loses its elasticity property.

Things to Remember

• In a solid, atoms or molecules are bound together by interatomic or intermolecular forces.
• The force that produces a change in the normal positions of the molecules is known as the deforming force.
• The property due to which a body regains its original configuration is known as elasticity.
• The property due to which a body does not regain its original configuration is known as elasticity.
• The restoring force developed per unit area in a body is called stress.
• The strain is the relative change in the dimensions of a body resulting from external forces.
• According to Hooke’s law, stress is directly proportional to strain.