 # Elasticity and Plasticity

By now, you probably know about the concept of elasticity. In layman terms, it means that some substances get back to their original shape after being stretched. You have played with a slingshot. Haven’t you? That is an elastic material. Let us get into the concepts of elasticity and plasticity and learn more about these two properties of matter.

### Suggested Videos        ## Elasticity and Plasticity

Elasticity is the property of a body to recover its original configuration (shape and size) when you remove the deforming forces. Plastic bodies do not show a tendency to recover to their original configuration when you remove the deforming forces. Plasticity is the property of a body to lose its property of elasticity and acquire a permanent deformation on the removal of deforming force.

## Stress

The restoring force (F) per unit area (A) is called stress. The unit of stress in  S.I system is N/m2 and in C.G.S-dyne/cm2. The dimension of stress = [M1L-1T-2]. Stress is given by,

Stress = F/A ## Types of Stress

Stress could be of the following types:

• Normal stress:- Normal stress has the restoring force acting at right angles to the surface.
• Compressional stress:- This stress produces a decrease in length per volume of the body.
• Tensile stress:- This stress results in an increase in length per volume of the body.
• Tangential stress:- Stress is said to be tangential if it acts in a direction parallel to the surface.

### Strain

The strain is the relative change in configuration due to the application of deforming forces. It has no unit or dimensions. The strain could be of the following types:

• Longitudinal Strain: It is the ratio between the change in length (l) to its original length (L). Longitudinal strain = l/L
• Lateral Strain: The lateral strain is the ratio between the change in diameter to its original diameter when the cylinder is subjected to a force along its axis. Lateral strain = change in diameter /original diameter
• Volumetric Strain: It is the ratio between the change in volume (v) to its original volume (V). Volume strain = v/V.

## Hooke’s Law It states that within elastic limits, stress is proportional to strain. Within elastic limits, tension is proportional to the extension. So, Stress ∝ Strain or F/A∝l/L. Therefore, we have for:

• Stretching: Stress = Y×strain or Y=FstretchL/A(l)
• Shear: Stress = η×strain or η=FshearL/A(l)
• Volume Elasticity: Stress = B×strain  or B = – P/(v/V)

### Proportionality Coefficients

• The coefficient of elasticity: It is basically the ratio between stress and strain.
• Young’s modulus of elasticity (Y): It is the ratio between normal stress to the longitudinal strain. Y = normal stress/longitudinal strain  = (F/A)/(l/L) = (Mg×L)/(πr2×L)
• Bulk modulus of elasticity (B): It is the ratio between normal stress to the volumetric strain. B = normal stress/volumetric strain  = (F/A)/(v/V) = pV/v

## Other Important Terms

### Compressibility

The compressibility of a material is the reciprocal of its bulk modulus of elasticity. Compressibility = 1/B.

### Workdone in Stretching

• Workdone, W = ½ ×(stress)×(strain)×(volume) = ½ Y(strain)2×volume = ½ [(stress)2/Y]×volume
• Potential energy stored, U = W = ½ ×(stress)×(strain)×(volume)
• Potential energy stored per unit volume, U = ½ ×(stress)×(strain)

### Workdone During Extension (Energy Density)

W =½ F×l = ½ tension ×extension

### Elastomer

Elastometer produces a large strain with a small stress.

### Elastic Fatigue

The phenomenon by virtue of which a substance exhibits a delay in recovering its original configuration if it had been subjected to a stress for a longer time, is called elastic fatigue.

### Poisson’s Ratio (σ)

Poisson’s ratio of the material of a wire is the ratio between lateral strains per unit stress to the longitudinal strain per unit stress. σ = lateral strain/longitudinal strain = β/α = (ΔD/D)/(ΔL/L). Values of σ lie between -1 and 0.5.

### Relations Among Elastic Constants

• B= Y/[3(1-2σ)]
• η = Y/[2(1+ σ)]
• 9/Y = 3/η + 1/B
• σ = [3B-2η]/[6B+2η]

## Solved Examples For You

Q: Which of the following is/are true about deformation of a material?

1. Deformation capacity of the plastic hinge and resilience of the connections are essential for good plastic behaviour.
2. Deformation capacity equations considering yield stress and gradient of the moment.
3. Different materials have different deformation capacity.
4. All of the above.

Solution: D) In a well-designed steel frame structure, inelastic deformation under severe seismic loading is confirmed in beam plastic hinges located near the beam-to-column connections. Thus, deformation capacity of the plastic hinge and resilience of the connections are essential for good plastic behaviour at the hinge is strongly influenced by the difference of material properties.

Generally, the material properties are specified in terms of yield stress and/or ultimate strength. However, the characteristics of the materials are not defined by only these properties. Thus, the characteristics of various materials aren’t reflected in present building codes, particularly on deformation capacity classification.

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