As by definition, coefficient of linear expansion \[\alpha =\frac{l}{L\Delta \theta }\]
\[\Rightarrow \] thermal strain \[\frac{l}{L}=\alpha \Delta \theta \]
So thermal stress \[=Y\alpha \Delta \theta \] [As Y = stress/strain]
And tensile or compressive force produced in the body \[=YA\alpha \Delta \theta \]
Note :
Increase in length of the composite rod (due to heating) will be equal to \[{{l}_{1}}+{{l}_{2}}=\]\[[{{L}_{1}}{{\alpha }_{1}}+{{L}_{2}}{{\alpha }_{2}}]\,T\] \[[\text{As}\,\,l=L\alpha \Delta \theta ]\]
and due to compressive force F from the walls due to elasticity, decrease in length of the composite rod will be equal to
\[\left[ \frac{{{L}_{1}}}{{{Y}_{1}}}+\frac{{{L}_{2}}}{{{Y}_{2}}} \right]\frac{F}{A}\] \[\left[ \text{As }l=\frac{FL}{AY} \right]\]
as the length of the composite rod remains unchanged the increase in length due to heating must be equal to decrease in length more... 
(1) It's value depends upon the nature of material of the body and the manner in which the body is deformed.
(2) It's value depends upon the temperature of the body.
(3) It's value is independent of the dimensions (length, volume etc.) of the body. There are three modulii of elasticity namely Young?s modulus (Y), Bulk modulus (K) and modulus of rigidity (h) corresponding to three types of the strain. 
(1) When the strain is small (< 2%) (i.e., in region OP) stress is proportional to strain. This is the region where the so called Hooke?s law is obeyed. The point P is called limit of proportionality and slope of line OP gives the Young?s modulus Y of the material of the wire. If \[\theta \] is the angle of OP from strain axis then \[Y=\tan \theta \] .
(2) If the strain is increased a little bit, i.e., in the region PE, the stress is not proportional to strain. However, the wire still regains its original length after the removal of stretching force. This behaviour is shown up to point E known as elastic limit or yield-point. The region OPE represents the elastic behaviour of the material of wire.
(3) If the wire is stretched beyond the elastic limit E, i.e., between EA, the strain increases much more rapidly and if the stretching force is removed the wire does not come back to its natural length. Some permanent increase in length takes place.
(4) If the stress is increased further, by a very small increase in it a very large increase in strain is produced (region AB) and after reaching point B, the strain increases even if the wire is unloaded and ruptures at C. In the region BC the wire literally flows. The maximum stress corresponding to B after which the wire begins to flow and breaks is called breaking or ultimate tensile strength. The region EABC represents the plastic behaviour of the material of wire.
(5) Stress-strain curve for different materials are as follows :
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The plastic region between E and C is small for brittle material and it will break soon after the elastic limit is crossed. Example : Glass, cast iron.
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The material of the wire more...
The ratio of change in configuration to the original configuration is called strain.
Being the ratio of two like quantities, it has no dimensions and units.
Strain are of three types :
(1) Linear strain : If the deforming force produces a change in length alone, the strain produced in the body is called linear strain or tensile strain.
\[\text{Linear strain}=\frac{\text{Change in length(}\Delta l\text{)}}{\text{Original length(}l\text{)}}\]
Linear strain in the direction of deforming force is called longitudinal strain and in a direction perpendicular to force is called lateral strain.
(2) Volumetric strain : If the deforming force produces a change in volume alone the strain produced in the body is called volumetric strain.
\[\text{Volumetric strain}=\frac{\text{Change in volume(}\Delta V\text{)}}{\text{Original volume(}V\text{)}}\]
(3) Shearing strain : If the deforming force produces a change in the shape of the body without changing its volume, strain produced is called shearing strain.
It is defined as angle in radians through which a plane perpendicular to the fixed surface of the cubical body gets turned under the effect of tangential force.
\[\varphi =\frac{x}{L}\]
Note :
When a force is applied on a body, there will be relative displacement of the particles and due to property of elasticity, an internal restoring force is developed which tends to restore the body to its original state.
The internal restoring force acting per unit area of cross section of the deformed body is called stress.
At equilibrium, restoring force is equal in magnitude to external force, stress can therefore also be defined as external force per unit area on a body that tends to cause it to deform.
If external force F is applied on the area A of a body then,
Stress \[=\frac{\text{Force }}{\text{Area}}=\frac{F}{A}\]
Unit : \[N/{{m}^{2}}\] (S.I.) , \[dyne/c{{m}^{2}}\] (C.G.S.)
Dimension : \[[M{{L}^{-1}}{{T}^{-2}}]\]
Stress developed in a body depends upon how the external forces are applied over it.
On this basis there are two types of stresses : Normal and Shear or tangential stress
(1) Normal stress : Here the force is applied normal to the surface.
It is again of two types : Longitudinal and Bulk or volume stress
(i) Longitudinal stress
(a) It occurs only in solids and comes in to picture when one of the three dimensions viz. length, breadth, height is much greater than other two.
(b) Deforming force is applied parallel to the length and causes increase in length.
(c) Area taken for calculation of stress is the area of cross section.
(d) Longitudinal stress produced due to increase in length of a body under a deforming force is called tensile stress.
(e) Longitudinal stress produced due to decrease in length of a body under a deforming force is called compressive stress.
(ii) Bulk or Volume stress
(a) It occurs in solids, liquids or gases.
(b) In case of fluids only bulk stress can be found.
(c) It produces change in volume and density, shape remaining same.
(d) Deforming force is applied normal to surface at all points.
(e) Area for calculation of stress is the complete surface area perpendicular to the applied forces.
(f) It is equal to change in pressure because change in pressure is responsible for change in volume.
(2) Shear or tangential stress : It comes into picture when successive layers of solid move on each other i.e. when there is a relative displacement between various layers of solid.
(i) Here deforming force is applied tangential to one of the faces.
(ii) Area for calculation is the area of the face on which force is applied.
(iii) It produces change in shape, volume remaining the same.
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In fact, the spring connecting the two molecules represents the inter-molecular force between them. On applying the deforming forces, the molecules either come closer or go far apart from each other and restoring forces are developed. When the deforming force is removed, these restoring forces bring the molecules of the solid to their respective equilibrium position \[(r={{r}_{0}})\] and hence the body regains its original form.
(6) Elastic limit : Elastic bodies show their property of elasticity upto a certain value of deforming force. If we go on increasing the deforming force then a stage is reached when on removing the force, the body will not return to its original state. The maximum deforming force upto which a body retains its property of elasticity is called elastic limit of the material of body. Elastic limit is the property of a body whereas elasticity is the property of material of the body.
(7) Elastic fatigue : The temporary loss of elastic properties because of the action of repeated alternating deforming force is called elastic fatigue. Due to elastic fatigue :
(i) Bridges are declared unsafe after a long time of their use.
(ii) Spring balances show wrong readings after they have been used for a long time.
(iii) We are able to break the wire by repeated bending.
(8) Elastic after effect : The time delay in which the substance regains its original condition after the removal of deforming force is called elastic after effect. It is the time for which restoring forces
