Closed-form evaluation of two-dimensional static lattice sums

Area of Science:

• Condensed matter physics
• Materials science
• Computational physics

Background:

• Static lattice sums (S2 for conductivity, T2 for elasticity) are crucial for modeling composite materials.
• Calculating these sums for 2D lattices is often complex due to conditional convergence.

Purpose of the Study:

• To derive closed-form analytical formulas for 2D static lattice sums S2 and T2.
• To establish exact relationships between S2 and T2 for various lattice structures.
• To obtain asymptotic analytical formulas for effective tensors of 2D composites with circular inclusions.

Main Methods:

• Utilizing complete elliptic integrals of the first and second kind.
• Developing closed-form expressions for S2 and T2.
• Applying these formulas to derive effective tensor properties up to the third order in concentration.

Main Results:

• Closed-form formulas for S2 and T2 derived using elliptic integrals.
• Asymptotic analytical formulas for effective tensors of 2D composites with circular inclusions obtained.
• Exact relations between S2 and T2 established for different lattices.
• Specific values S2=π for square/hexagonal lattices and T2=π/2 for hexagonal lattices confirmed/deduced.

Conclusions:

• The derived formulas provide an efficient method for calculating static lattice sums and effective properties.
• The results are applicable to the analysis of 2D composite materials with circular inclusions.
• The established relations and specific values offer valuable insights into lattice sum behavior.