Show that for a monatomic model solid the Young modulus E is given approximately by
E = 1/r0(d2U/dr2)x = x0
where U is the interaction potential energy of a pair of atoms distance r apart and r0 is their equilibrium separation. When a uniform beam is bent, the bending moment M is given by
M = EI/R,
where R is the radius of curvature of the arc and I is the second moment of area of the cross-section about the neutral axis. Use this relation to show that when a compressive stress is applied o the ends of a straight beam it will buckle when the stress reaches a critical value. Obtain an expression for this critical stress. A uniform round rod of length 1 m and radius 10 mm is subjected to a compressive stress. At what value of the compressive strain will the rod begin to buckle? (I for a uniform circular disc of radius R about its diameter is given by πR4/4.)