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 $k_{s,i}$.