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STRESS (σ) AND STRAIN (ε) :

|| EFFECTS OF LOADING ON A DEFORMABLE BODY ||

What are the effects of a force when applied on a rigid body? In general we know that the object would try to move along the direction of force. But, it is also seen that not every object would move as a result. It is generally seen that it is always difficult to move a massive body, when the magnitude of the applied force is small. It is due to friction, although.

Now, we shall ask a different question. Suppose we have a mechanical structure which consists of several components, and if we apply external forces on this structure, then what will be its effects on the components and as a whole to the structure?

Suppose we have a cubic block of mass (M), which is kept on a table. Now the gravity would exert a downward force (Mg), due to which, the block of mass would have a tendency to go downwards, but the table is there hence the block will exert a force on the table. The magnitude of this force would be equal to the force of gravity on the body i.e. (Mg). But, the table will try to push the cube upwards instead. This upward force given by the table arises obeying the Newton's Third law of motion that for every action there must be an equal and opposite reaction. Therefore, the force provided by the table will be exactly equal to the force acting on the table due to the block of mass (M).

Now at first we shall consider an assumed situation. Suppose we have a rigid rod of negligible mass and we shall apply a tensile load of F at the both side of the rod. The applied force F will try to increase the length L to L+dL. (dL) is called as "deformation", but as deformation takes place slowly a state of equilibrium would reach when there will be no more deformation, but again if we increase the magnitude of the applied force from F to F+dF, then further deformation occurs and again another equilibrium state will achieved. It indicates that after deformation occurs there another restoring force been induced inside the body of the rod that neutralizes the effect the force and thus deformation stops. At equilibrium condition the "Restoring Force" thus induced within the body is exactly equal and opposite to the external forces applied on the body. This external force is evenly distributed over the cross-sectional area of the body in this case. So we can measure the intensity of this internal "restoring force" by measuring the magnitude of this "restoring force" per unit cros-sectional area of the rod and we shall call it as "STRESS" developed within the body due to the application of external forces on the body & we shall denote it by the Greek alphabet "sigma" "σ". As the magnitude of the "Restoring Force" is exactly equal to the magnitude of the external force "F" applied on the cross-sectional area of the rod "A".

Then, by definition σ = (F/A)---- (i)

Where as deformation "dL" per unit length "L" is called as "Unit Deformation" or "STRAIN". It is denoted by the Greek alphabet "epsilon" "ε". It is a pure ratio hence dimensionless where as stress has the unit of pascal (Pa) or N/m2.

HOOKE'S LAW OF ELASTICITY:

As it is clear that deformation (dL) is somehow propotional to the applied force (F) and Restoring Force R is proportional to the deformation dL (as magnitude of F and R are equal to each other). It has been observed in the experiment that, if the Cross-sectional Area (A) and the undeformed length of the rod (L) remains constant, then
F is proportional to dL or F ∞ dL and for any specimen, when loading is within a certain range, Stress is proportional to the Strain or we can say,or σ ∞ ε.

HOOKE'S LAW: σ = E.ε

This relation is widely used for most of the elastic metal. The constant of proportionality, E is an important physical constant as its magnitude is the scale of measurement for Elasticity, a vital mechanical property of an Engineering Metal. Hence, the constant E is aptly named as "Modulus of Elasticity". The value of E is different for different metals and greater the magnitude of E indicates higher Elasticity of the metal.

The limit within which Hooke's Law is valid is called as "Proportional Limit".

What happens when we increase the applied load to induce stress beyond Proportional Limit?

We know from Hooke's Law that up to Proportional Limit, Stress induced in a body is directly proportional to the magnitude of Strain. When we still increase the magnitude of the applied load, the stress induced crosses the Proportional Limit, still Stress will be dependent on Strain as the material still retains some elasticity. The point of state up to which the body remains partially elastic is called as "elastic limit". Beyond elastic limit, the material loses its property of elasticity and a transition occurs when the material starts to exhibit plasticity in stead of elasticity. The point of state where plasticity starts to be the dominant property is known as Upper Yield Point. In this zone the process of Yielding or plastic deformation starts.

During yielding diformation propagates even if we actually decrease the load also. When the yielding zone comes to an end, deformation continues despite we lower the stress level. The end point of this yielding zone is known as Lower Yielding Point. After this material which is ductile may even show elastic behaviour before ultimately breaking. The highest magnitude of the stress just some time before the ultimate failure of the material is known as Ultimate Stress or Strength. The stress level at the time of breaking of the material is known as Breaking Stress.


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