Young's Modulus

"Young's modulus ($Y$) is the ratio of stress to strain for a particular material.  young's modulus is a property of the material, and does not depend on the size or shape of an object.  The stiffer the material, the larger is Young's modulus."

The strain on a material is the fractional stretch, or the change in length divided by the original length.  This gives a ratio defined as $$strain = \frac{\Delta L}{L}$$The stress on a material is the tension force per unit of area.  This eliminates the dependency of thickness of the material by using both the tension force ($F_T$) and cross-sectional area of a wire ($A$).$$stress = \frac{F_T}{A}$$These ratios can be related down to the atomic level, the stress ($\frac{F_T}{A}$) being the force that each chain of atomic bonds must exert, and the strain ($\frac{\Delta L}{L}$) being the stretch of the interatomic bond.  The stiffer the material, the larger the modulus.  We write young's modulus as$$\frac{F_T}{A} = Y\frac{\Delta L}{L}$$$$Y = \frac{stress}{strain} = \frac{\frac{F_T}{A}}{\frac{\Delta L}{L}}$$This is dependent upon the fact that too large a stress will result in the material coming apart and breaking, a process known as "yielding."  The Young's modulus in terms of atomic quantities can be defined as:$$Y = \frac{\frac{k_{s,i}s}{d^2}}{\frac{s}{d}} = \frac{k_{s,i}}{d}$$$d$ represents the relaxed length of an interatomic bond, and the diameter of one atom.  The cross-sectional area of an atom is $d^2$, considering we are viewing the atom as occupying a cube of space in the crystal lattice versus its approximately spherical shape.
The stretch of the interatomic bond is $s$, or the $\Delta L$.
The interatomic force is $k_{s,i}$.

Tension Forces

Tension Forces are just forces that compensate for other forces.  For instance, when a ball is hanging motionless on a wire, the net force on the ball must be zero.  The gravity of the Earth is pulling it down, therefore there must be another force in the opposite direction compensating for this.

The force exerted by an object such as a wire or string is called a tension.$$\Delta \vec{p} = \vec{F}_{net}\Delta t$$$$0 = (\vec{F}_T - \vec{F}_{ext})\Delta t$$