All questions are related to a material that has a simple cu

All questions are related to a material that has a simple cubic structure with an inter-atomic energy in the form of U_m = -3.2/r + 5.2.0 times 10^-6/r^8 (eV or 1.6 times 10^-19 J)(The unit for r is nm) What is the lattice constant a and the smallest equilibrium distance r_o between the atoms? What is the bonding energy E_b in J? What is the elastic constant, E, in Pa? What is the surface energy, gamma, in J/m^2 for (100) face. Calculate the force that drives Atom A slides on the array of Atoms B, C, and D when Atom A is at the middle point between Atoms C and D as shown in the diagram. Use the force to calculate the critical shear stress for yielding. tau_crss. If this material experiences a stress of 20 MPa in [100] direction, what is critical crack length (in millimeter) for brittle fracture?

Solution

The hard core radius r0 is obtained by finding the derivative of the potential and setting it to zerohis gives r0

= (130*10^-7)^1/7 = 2.004 Angstroms

Assuming a cubic structure, from elementary Materials Theory, the lattice constant is a = 2*r since 1/8th of each atom fills the the corners, unit cells has one atomvolume

Lattice constant = 2*r0 = 4.008 Angstroms

The bonding energy is obtained by substituting ro in the expression for U

The Youngs Modulus is obtained froma general expression Y = (m-n) nA/(r0 ^(n+3)) for a cubic

here m= 8, n=1

A =3.2*7/16 in the units given, convert to Pa