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Stress strain relationship of Elastic and Plastic material.

Stress is the force applied to an object. If we imagine a vertical column of material, along any imaginary horizontal plane within this column, the material above the plane because of its weight pushes downward on the material below the plane. Similarly the part of column below the plane pushes upward with an equal force on the material above the plane. This mutual action and reaction along the surface constitutes the Stress. Here the imaginary plane may be horizontal, vertical, or inclined.

Strain may be defined as the deformation caused by the stress. Strain may be dilation which is a change in the volume or distortion which is a change in the form or both.

Stress strain relationship of Elastic and Plastic material:

When a body is subjected to directed forces for a short period of time then the body usually passes through three stages of deformation. At first, the deformation is elastic; that is, if the stress is withdrawn then the body returns to its original shape and size. If the stress exceeds the elastic limit than the deformation is plastic; that is, the specimen only partially returns to its original shape even if the stress is removed. When there is a continued increase in the stress, one or more fractures develop and the specimen eventually fails by rupture.

ELASTIC DEFORMATION

At the room temperature and pressure and under stress, the most brittle rocks behave elastically until they fail by rupture. For such rocks the elastic limit or yield point is the stress at rupture.

If a solid cylinder of rock is subjected to stress parallel to its long axis, it will lengthen under tension and shorten under compression. The ratio of the stress to the deformation is a measure of the property of the rock to resist deformation.

E= σ / ɛ   …………………………………………………………………………………… (1)

Where E is the Young’s Modulus (also called modulus of elasticity), σ is stress and ɛ   is strain.

ɛ   =Δ l / lo   …………………………………………………………………………………. (2)

Δ l is the change in length, lo is the original length.

Under tension the diameter of a cylinder subjected to tension parallel to the axis   becomes

Smaller; under compression parallel to the axis the diameter becomes greater. Poisson’s ratio

is the ratio of transverse strain to the axial strain.

υ = Δ d / do  / Δ l / lo   ……………………………………………………………………….(4)

where υ is Poisson’s ratio, Δ d is change in diameter, do is the original diameter.

If the diameter of the cylinder is decresed by 0.00025 cms, poisson’s ratio is 0.25; this is a

good average of rocks. Rigidity measures the resistance to change in shape i.e. ratio of shear stress to shear strain.

……………………………………………..……………………………… (5)

 

Where G is rigidity modulus

…..……………………………………..………………………………(6)

 

Stress strain

Figure 1 Rigidity Square acdf is deformed into parallelogram bcef.

The bulk modulus or incompresisibility is

K = Δ h    Δ V / V

Where K is the bulk modulus, Δ h is the change in hydrostatic pressure, Δ V is the change in

volume, V is the original volume.

Elastic deformation is primarily of importance in analyzing the tidal deformation of solid

earth  and in investigating the transmission of  seismic waves through the earth. It is of even

more direct significance to structural geology in studying the elastic rebound associated with

Earthquakes in the fracturing of rocks to produce joints and faults.

Plastic Deformation:

Most rocks at room temperature and pressure fail by rupture before attempting the stage of

plastic deformation, most rocks at significantly high temperature and confining pressure

deform plastically . This plastic deformation is not recoverable. That is, if the stress is

removed the material does not return it its original shape.

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