All questions are related to a material that has a simple cubic structure with a

Question

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?

Explanation / Answer

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

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