Physics 171.636: Modeling Across Multiple Length and Time Scales
Spring Semester, 2005

Homework 3: Due April 15

1) Consider a square lattice with equilibrium nearest-neighbor spacing d. Suppose nearest-neighbors are connected by springs with spring constant k and next-nearest-neighbors are connected by k’ springs.

a) Calculate the stiffness tensor K_ijkl from the formula given in class.

b) What happens in the limit k’®0?

2) Consider a one-dimensional chain with equilibrium spacing d between nearest-neighbors and spring constant k connecting them. A displacement of form u(x)=u₀ cos(qx) is present with equilibrium atoms at x=…, -2d, -d, 0, d, 2d, … . Consider the limit where u₀ is small and consider q corresponding to wavelengths of 2d, 4d, 8d, 16d, 32d and 64d.

a) What is the stiffness K (scalar) in elastic theory?

b) Compare the exact energy to the expression given by elastic theory for the above q. How does the error scale with q?

c) Consider a finite-element mesh with nodes at x = …, -8d, -4d, 0, 4d, 8d, … . What is the energy of the finite element model at the above q. How does the error scale with q.