When a material fractures energy is required to stretch and

When a material fractures, energy is required to stretch and break bonds. If an extremely brittle ceramic is used as an example, we can ignore the energy dissipated in dislocation motion because the ceramic will crack before it plastically deforms. If every bond requires 10 eV of energy to break, how much energy (in Joules) is necessary increase the size of a crack by 1 square inch? Assume the material forms a simple cubic structure with a = 1 nm. How much total energy is absorbed when a 3 inch diameter cylinder cracks all the way through?

Solution

Given: a = 1*10^-9 m, A = 1 in² = 0.00064516 m², E_bond = 10 eV = 1.6022×10¹⁹ J

a) Now,

E_bond = 2*e*(a*a)*A

here, e = energy necessary to increase the size of crack by 1 in²

substituting and solving, we get

e = 1241.7075 J/m²

b) d = 3 in = 0.0762 m

area = 3.14*0.25*0.0762² = 4.5604*10^-3 m²

So total energy absorbed in cylinder = 1241.7075*4.5604*10^-3 = 5.66268 J