About Spring Network Simulation
This simulation places 50 nodes randomly in a 2D canvas and connects pairs of nodes with probability p. The resulting network is a random spring lattice — each bond acts as a linear spring with rest length equal to the initial node separation. The mean degree z = p(N-1) controls the mechanical behavior of the network.
James Clerk Maxwell showed in 1864 that a pin-jointed frame in d dimensions requires at least 2dN bonds to be rigid. For large random networks in 2D, this translates to a critical mean degree z_c = 4. Below z_c the network is floppy, possessing f = 2N - B - 3 zero-energy deformation modes (mechanisms). Above z_c the network is stressed-rigid. Adjust the connectivity slider to watch the network cross the rigidity transition.
When a load is applied (shear, compression, or tension), boundary nodes are displaced affinely and interior nodes relax via gradient descent on the elastic energy E = Σ½k(|r_ij| - L0_ij)². Springs are colored by stress: blue for compression (shorter than rest length), red for tension (longer), and gray for neutral. Line width scales with stress magnitude.
Frequently Asked Questions
What is the Maxwell rigidity criterion?
James Clerk Maxwell showed in 1864 that a pin-jointed frame in d dimensions with N nodes and B bonds is rigid when B ≥ dN - d(d+1)/2. For large 2D networks this simplifies to mean degree z_c = 2d = 4: below this threshold the network has floppy (zero-energy) deformation modes; above it the network is stressed-rigid.
What is a floppy mode?
A floppy mode is a collective deformation of the network that costs zero energy — a mechanism. Below the Maxwell threshold z_c, the network has f = 2N - B - 3 independent floppy modes. These are soft modes that allow large deformation at negligible energy cost, making the material mechanically weak.
How do spring networks model amorphous solids?
Amorphous solids (glasses, gels, fiber networks) lack crystalline order but have mechanical rigidity. Random spring networks capture their essential physics: the degree distribution controls the moduli, and the rigidity transition at z_c = 2d is analogous to a geometric phase transition seen in granular matter, colloidal gels, and biopolymer networks.
What is affine vs non-affine displacement?
Under an applied strain, an affine displacement is the simple, homogeneous deformation you would see in a continuum solid (every point moves in proportion to its position). Non-affine displacements are the extra, heterogeneous rearrangements that occur in disordered networks to accommodate stress — they are largest near the rigidity threshold and are measured by the non-affinity parameter Γ.
How do elastic moduli arise from spring networks?
The bulk modulus K and shear modulus G of a random spring network depend on the connectivity p and spring stiffness k. Near the rigidity threshold, both moduli vanish as power laws: G ~ (z-z_c)^f with critical exponent f≈1.4 in 2D. This power-law vanishing is analogous to percolation critical behavior.
What is the rigidity percolation transition?
As connectivity p increases from 0, the network transitions from floppy to rigid at the critical connectivity p_c corresponding to z_c = 4. This rigidity percolation transition is distinct from connectivity percolation (where a path first spans the system) and occurs at higher p. Near p_c, the network shows critical fluctuations and diverging length scales.
Where do random spring networks appear in biology?
The cytoskeleton of living cells is a biopolymer network of actin filaments and microtubules cross-linked by proteins. The connectivity of these networks determines cell stiffness and mechanosensing. The extracellular matrix, collagen gels, and fibrin clots also behave as random spring networks with rigidity transitions tuned by cross-link density.
What is the difference between stretching and bending rigidity?
Pure spring (central-force) networks have rigidity only above z_c = 4 in 2D. Networks with angular (bending) springs between connected bonds are rigid at all z > 2, because bending stiffness stabilizes mechanisms that central-force springs cannot resist. Real materials often have both, with bending dominating at low connectivity.
How is network rigidity related to granular materials?
Granular packings (sand, gravel) near jamming behave as random spring networks. At the jamming point, each grain touches exactly z_c = 2d neighbors on average — the minimal number for marginal rigidity. The shear modulus vanishes at jamming and grows as G ~ (z - z_c)^1 above it, in agreement with spring-network theory.
Why do fiber networks show non-linear stiffening?
Biopolymer fiber networks (collagen, actin) stiffen dramatically under large strain — the modulus can increase 100-fold. This strain-stiffening occurs because initially floppy networks are recruited toward rigidity as the strain aligns bonds and exhausts floppy modes. The onset of stiffening corresponds to a strain-induced rigidity transition.