# Volumetric, Shear, Thermal Stress and Strain

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#### Volumetric stress:

• When the deforming forces are such that there is a change in the volume of the body, then the stress produced in the body is called volume stress.
• e.g. Solid sphere placed in a fluid under high pressure.
• Mathematically,

Volumetric Stress = Load / Area = Pressure Intensity = dP

•  S.I. Unit of stress is N m-2  or Pa (pascal) and its dimensions are [L-1M1T-2]. Units and dimensions of stress are the same as that of pressure.
• The internal restoring force per unit area developed in a body when the body is compressed uniformly from all sides is called hydrostatic stress or hydraulic stress.

#### Volumetric strain:

• When the deforming forces are such that there is a change in the volume of the body, then the strain produced in the body is called volume strain.
• Mathematically

Volumetric strain = – Change in volume (dV)/ Original Volume (V)

Negative sign indicates the decrease in the volume

• The volumetric strain has no unit and no dimensions.

#### Bulk Modulus of Elasticity:

• Within elastic limit, the ratio of volumetric stress to the corresponding volumetric strain in a body is always constant, which is called as Bulk modulus of elasticity.
• It is denoted by letter ‘K’. Its S.I. Unit of stress is N m-2  or Pa (pascal) and its dimensions are [L-1M1T-2].
• Mathematically, #### Compressibility:

• The reciprocal of bulk modulus of elasticity is called as compressibility.
• Mathematically

compressibility  = 1 / K

• Its S.I. unit is m2 N-1 or Pa-1 and its dimensions are  [L-1M-1T2].

### Shear Stress:

• When the deforming forces are such that there is a change in the shape of the body, then the stress produced is called shearing stress. Shear stress is also called as tangential stress. • Mathematically,

Shear stress = Shearing force (F) / Area under shear

• Its S.I. Unit of stress is N m-2  or Pa (pascal) and its dimensions are [L-1M1T-2].

#### Shear Strain:

• When the deforming forces are such that there is a change in the shape of the body, then the strain produced in the body is called shear strain.
• Shearing strain is defined as the ratio of relative displacement of any layer to its perpendicular distance from fixed layer.
• Mathematically,

tan θ = x/h

#### Modulus of Rigidity:

• Within elastic limit, the ratio of the shear stress to the corresponding shear strain in the body is always constant, which is called as modulus of rigidity.
• It is denoted by letter ‘η’. Its S.I. Unit of stress is N m-2  or Pa (pascal) and its dimensions are [L-1M1T-2].
• Consider a rigid body as shown in the figure which is fixed along the surface CD.  Let it be acted upon by tangential force F along surface AB as shown. Let lateral surface AD get deflected through angle θ as shown. The tangential force F per unit area of surface AB is called as shear stress. #### Characteristics of Moduli of Elasticity:

• Modulus of elasticity is the property of the material of a body and is independent of the stress and strain un the body.
• a material is said to be elastic if it has a greater value of modulus of elasticity.
• The modulus of elasticity for rigid bodies is infinity.
• Young’s modulus is the property of solids only. While bulk modulus exists for all the three states of matter.
• Gases posses two bulk moduli of elasticity. (i) isothermal bulk modulus Kiso = P and (ii) adiabatic bulk modulus Kadia = γP
• The elasticity of a substance decreases with the increase in the temperature.

#### Relation Between the Moduli of Elasticity: #### Concept of Thermal Stress:

• If a bar which is heated and prevented from expansion or heated rod is prevented from contraction as it cools, then stress is produced in the bar. This stress is called a thermal stress.
• If bar is prevented from expansion or contraction, the change in length is given by

Δl = l ∝ Δθ

Where,  Δl = change in length prevented.

l = original length of the rod

Δθ = Change in temperature.

∝ = coefficient of linear expansion of material of the rod. Where Y = Young’s modulus of elasticity.

If A is the area of cross-section of the bar then the force acting on the rod is

F = Thermal stress x area of cross-section

Force = Y α θ A

#### Elastic Hysteresis:

• This is a consequence of elastic after effect. The stress remains in the body, even if deforming force is removed. Thus the strain lags behind the stress. The lagging of strain behind stress when deforming force is removed is called elastic hysteresis. • The strain is found to be greater for the same value of the stress when it is being unloaded than when it is being loaded.
• The materials having low elastic hysteresis have also low elastic relaxation time. Hence quartz shows negligible elastic hysteresis.

#### Elastic Fatigue:

• Elastic fatigue is a property of an elastic body by virtue of which the behaviour becomes less elastic under the repeated application of deforming forces.
• Spring balance shows wrong readings after they have used or a long time is due to elastic fatigue.
• Bridges are declared unsafe after long use because their elastic strength decreases due to repeated strains.

#### Factor of Safety:

• In actual practice, no machine or structure part is subjected to very high stress. The maximum stress to which it is subjected is very much less and always within elastic limit. This stress is called working stress.
• The ratio of breaking stress to working stress is called the factor of safety.
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