Scaling to zero of compressive modulus in disordered isostatic cubic networks

Area of Science:

• Physics
• Materials Science
• Network Theory

Background:

• Isostatic networks, with constraints equaling degrees of freedom, exhibit unique mechanical properties near collapse.
• Unlike hyperstatic networks, isostatic networks are highly sensitive to geometric disorder, affecting their elastic behavior.

Purpose of the Study:

• To investigate the influence of disorder and system size on the compressive load-bearing capacity of isostatic networks.
• To develop an analytical method for calculating the bulk compressive modulus (B) under varying conditions.

Main Methods:

• Considered square and cubic lattices with specified degrees of freedom and rotatable springs.
• Introduced 'hot-solid' disorder and fixed sites to model redundancies.
• Developed an analytic method to calculate the bulk compressive modulus (B).

Main Results:

• The bulk modulus (B) analytically and numerically decays as L⁻μ with system size (L).
• Decay exponents differ for small (μ_small) and large (μ_large) disorder.
• A scaling variable λ = L^(μ_large - μ_small)ε² characterizes disorder strength, with B(L,ε)L^μ_small = g(λ).
• Large disorder leads to a broader distribution of spring tensions, causing faster decay of B.

Conclusions:

• Disorder and system size critically affect the mechanical stability of isostatic networks under compression.
• The bulk modulus exhibits a discontinuous jump at zero disorder, indicating an unstable configuration in regular networks.