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Static equilibrium and elasticity

Statics is about studying the forces and torques when an object is in static equilibrium. Static equilibrium refers to the state of an object where there is no translational or rotational motion of the object. Statics is widely used when building bridges and structures. It is used to analyze the stability and load capacity of structures.

Conditions for static equilibrium

An object in static equilibrium has no translational or rotational motion. Therefore, there is no net force or net torque acting on the object. So, we have two conditions for static equilibrium,

(i) Net force on the object is zero: $F_{net}=0$ and

(ii)Net torque is zero: $\tau_{net} = 0$

Since force is a vector, we use the components for the first condition:

$F_x=0$;$F_y=0$ and $F_z=0$

Elasticity

Elasticity in the broader sense is about studying the effect of forces on changing the size (length or volume) or shape of objects.The term elasticity also refers to an object's material property by which the object to come back to its original state when a force that deform the object is removed.

When you apply a force on an object that deform the object to some degree. In most cases the deformation is so small to be visible. But when you remove that force, if the object can comeback to its original state, we say the applied force is within a limit called elastic limit.

Stress in an object

A force applied on an object creates stress within the object. Stress is defined as force per unit area:

$stress=\dfrac{F}{A}$

where $A$ is the area over which a force $F$ is applied on the object. If you stretch a wire or compress a column by applying a force, then $A$ is the area of cross section of the wire or the column.
compressing a column

Types of stress

There are three types of stress depends on what the applied force does on the object. If the force stretches the object, then the stress is called tensile stress. If the force compresses the object, the stress is compressive stress and if the force shear the object, it is shear stress.

Linear strain or strain

When you apply a force to stretch a wire or compress a column, the length of the object changes. The ratio of the change in length to the original length of the object is called linear strain or just strain.

$strain=\dfrac{\Delta l}{l_0}$

where $l_0$ is the original length and $\Delta l$ is the change in length of the object due to the application of the force.
stretching a wire by a force

Young's modulus

When the applied force is within the elastic limit, the ratio of stress to the linear strain is constant for a given material of the object. The ratio is called the Young's modulus of the material of the object:

$E=\dfrac{stress}{linear\:strain}$

i.e., $E=\dfrac{F.l_0}{A \Delta l}$
Young's modulus is a material property. Different materials have different values of Young's modulus. It is a constant and is independent of the object's length.

Shear stress and shear strain

A shearing force applied on a side of an object while the opposite side is fixed, changes the shape of the object. In the following figure, a shearing force is applied on the top side of the object, while the bottom side is fixed.
shearing an object
Shear strain is
$shear \: strain=\dfrac{\Delta l}{l_0}$
and
$shear \: stress=\dfrac{F}{A}$

where $A$ is the area of the top side.

Shear modulus

It is the ratio of the shear stress to the shear strain:

$shear\:modulus=\dfrac{shear\:stress}{shear\:strain}$
$G=\dfrac{F.l_0}{A \Delta l}$

Volume strain

When an object is submerged in a fluid, the fluid pressure exerts forces on the surface of the object. As a result, the object is compressed and its volume changes.

The ratio of the change in volume to the original volume is called, volume strain:

$Volume \:strain=\dfrac{\Delta V}{V_0}$
where $V_0$ is the original volume of the object and $\Delta V$ is the change in volume due to the pressure change.

Bulk modulus of elasticity

The ratio of the change in pressure to the volume strain is constant for a given material of the object. This ratio is called bulk modulus of the material of the object. Bulk modulus is

$B=\dfrac{change\:in\:pressure}{volume \: strain}$

(or)

$B=-\dfrac{V_0.\Delta P}{\Delta V}$

where $\Delta P$ is the pressure change the object undergoes.

A minus sign is included in the above equation to make $B$ positive as when the pressure increases (positive $\Delta P$) volume decreases ($\Delta V$ is negative).

Note that all the three moduli of elasticity: $E$, $G$ and $B$ are the properties of materials. They are independent of the size or shape of the objects.

Fracture and strength of materials

An object can handle only a certain amount of stress. Too much stress results in an object's fracture. Maximum stress an object can handle is called the strength of the material of the object. Since there are three types of stresses, there are three types of strengths: tensile strength, compressive strength and shear strength.