The stress due to the perpendicular or normal component of internal force is Normal stress.

`sigma_n = Lim_(Deltat ->0) (DeltaF_n)/(Delta A)`

Normal stress has two types:

1. Tensile stress
2. Compressive stress

If there is an increase in dimension or length of body in the direction of force applied, the stress is called Tensile stress.

Example A solid cylinder is stretched by two equal forces applied normal to its cross-section.

If there is decrease in dimension or length of a body due to force applied, the stress set-up is called compressive stress

Example A column supporting vertical load.

The stress set up due to force acting tangentially to the surface of a body is called shearing stress or tangential stress.

Shearing stress changes only shape of body by without changing its volume.

`t = F_t/A =` Tangential stress

If applied force is variable, then tangential stress is different at different points and at a given point is

`tau = Lim_(Delta t -> 0) (Delta F_t)/(Delta A)`

Example A shaft resists shear stress when it is twisted by external torque.

When a body is acted upon by forces such that

(i) The force at any point is normal to the surface.

(ii) The magnitude of force at any small area is proportional to the area.

The restoring force per unit area in the body is called volumetric stress.

`sigma_V = F/A`

Volumetric stress is same as pressure.


Example
If a block is kept completely immersed in water, hydrostatic pressure acts on a II sides of block.
The restoring force and hence the pressure set up in block is volumetric stress.

When a deforming force is applied on a body, there is change in shape, size of the body. The object is said to be strained or deformed.

Strain is defined as the ratio of change in dimension to the original dimension.

`text(Strain) = text(Change in dimension)/text(original dimension)`

Strain being the ratio of two like quantities has no units and hence it is a dimensionless number.

It is the ratio of change in length `(Delta l)` to the original length `(l)` .

`in_l = (Delta l)/l` = `text(change in length)/text(origin length)`

It is the ratio of change in volume `(Delta v)`to the original volume `(v)` of body.

`in_V = (Delta V)/V =` `text(change in volume)/text(original volume)`

It is the angle `(theta)` through which a face of a body originally perpendicular to fixed face is turned.

Shearing strain `(theta) = (Delta L)/L`

P = Proportional limit

E = Elastic limit

Y = Yield point

U T S =Ultimate Tensile strength

F = Fracture or Breaking point

OP `->` stress - strain graph is linear and it obeys Hook's law i.e.

stress `prop` strain

O E `->` Material remains elastic in this region but in PE it becomes non-linear i.e. Hook's law is not valid.

Y- slope of curve becomes zero. Material stars deforming under constant stress and behaves like a viscous liquid.

UTS is maximum stress material can withstand without failure.

F is point where material fractures or breaks.

Stress calculated is based on original cross-section but actually it is decreasing.

The delay in returning to the original configuration by an elastic body after removal of external load is called Elastic-after effect.

Elastic aftereffect is negligibly small for quartz fiber and phosphor bronze. Due to this reason the suspensions made from quartz and phosphor bronze are used in galvanometer and electrometers.

Glass has very large value of elastic aftereffect. It takes hours for glass fiber to return to its original state on removal of deforming force.

The loss of strength of material due to repeated loading and unloading cycle on the material is called Elastic Fatigue.

In stress - strain curve the region of plastic deformation lies on the right of elastic limit (E).


When body is loaded to stress beyond elastic limit and then unloaded, the unloading curve does not retraces the loading curve, and returns along different path, as shown in figure.

The area enclosed by OPEAA' is numerically equal to the energy lost per unit volume of body in the given cycle.

The loop `O E A A^'` is called the Hysteresis loop. `OA^'` is permanent deformation or the plastic strain or the residual strain.

On contrary, if the body is loaded and unloaded in the elastic region, it returns back by retracing the same original path. Hence no energy is lost in the loading unloading cycle. This indicates in elastic region, the internal forces acting between particles of the body are conservative forces.

When a metal rod or wire is elongated under a gradually increasing load, work is done by the load.

The work done by load is partially or completely stored as elastic potential energy of strain.

If the strain is within elastic limit, the work done is completely transferred into potential energy and can be recovered during gradual unloading.

Strain energy or potential energy stored in rod

Potential Energy stored in spring ` = U = 1/2 kx^2`

where , ` k = (YA)/l , x= Delta l =(Fl)/(A Y)`

`U = 1/2 ((YA)/l)((Fl)/(YA))^2 , U = 1/2 (F^2l)/(YA) = 1/2 F Delta l`


Rearranging the equation

`U= 1/2 (F/A)((Delta l)/l)(Al)`

`U= 1/2 text(stress)xxtext(strain)xxtext(volume)`

`U/text(Volume) = text(Elastic potential energy density)`

Energy density `(u) = 1/2 xx ` stress `xx` strain

(stress)/(strain) `= Y` or , stress `= Y xx` strain

:. Energy density `( u) = 1/2 xx Y xx (strai n)^2 ,` `u= 1/2 xx (stress^2)/Y`

`text(NOTE :)`

1 . The area under stress - strain curve gives the potential energy stored per unit volume of the specimen.

2. If stress applied to a specimen is not constant then, energy stored in elastic specimen is given by

`rho =` stress applied , `U =1/2 int rho^2 dv`

`dv =` Change in Volume

When there is increase or decrease in temperature, the body expands or contracts.

If deformation of body is prevented, the stresses areninduced in the body. These stresses are called Thermal stresses or Temperature stresses.

Consider rod AB fixed between two rigid supports.(fig)


Let I= length of rod A = Area of cross-section

Y = Young's modulus of rod

`alpha` = coefficient of thermal expansion of rod

Let temperature of rod increases by `Delta t` The increase in length of rod is given by

`Delta l = alpha l Delta T`

`in = alpha(Delta T)`

`sigma =Y in = Y alpha(Delta T)`

Where , `sigma = ` Thermal stress (Compressive)

Force on supports `= F = sigma A = Y A alpha (Delta T)`

Forces on Rod,(fig.)

When temperature of rod is increased, then compressive thermal stresses are induced in the rod and when temperature is decreased, then tensile thermal stresses are induced in the rod.