Rubber Band (2D): Gent Stress-Strain Curve, Mass Bounce and Gough-Joule Heating

Presets

Live readouts

Graphs

Nominal stress against stretch: Gent model (solid), neo-Hookean (dashed), Hooke tangent (dotted) and the measured path (MPa)

Extension of the band against time (cm)

Temperature rise of the band (kelvin above room temperature)

Energy (J): kinetic, elastic, gravity, total including dissipation (mass mode); work and dissipation (tensile mode)

Bounce frequency of a hanging mass against its mass (Hz): the band is a nonlinear spring

Numerical checks

running...

How to use

Press Release / restart to drop the mass from the chosen stretch, or choose the Tensile test preset. Try: (1) Bounce a mass, then raise the mass and watch the equilibrium marker and the bounce frequency (last graph); (2) Fast stretch (adiabatic) and see the band warm by several kelvin, then cool back; (3) Stiff EPDM cord to hit the strain-stiffening limit; (4) turn Thermal model off to see the pure mechanical response; (5) compare the Gent and neo-Hookean curves near lambda = 4. Space pauses, R restarts.

The science

Rubber is an entropic spring. Stretching a rubber band uncoils tangled polymer chains. The restoring force comes mostly from the loss of entropy, not from stretching bonds, so the stiffness is proportional to absolute temperature: G = n k T. That is why a stretched band that is warmed pulls harder (Gough-Joule effect), and why stretching it quickly makes it warm: for an ideal rubber the internal energy does not change, so the work you do is released as heat.

Hyperelastic law. For an incompressible strip stretched by lambda = L / L0 the Gent model gives the strain energy density and the nominal stress (force per unstretched cross-section) W = -(G Jm / 2) ln(1 - (I1 - 3)/Jm), I1 = lambda^2 + 2/lambda, sigma = dW/dlambda = G (lambda - lambda^-2) / (1 - (I1 - 3)/Jm) For Jm -> infinity this is the neo-Hookean law sigma = G (lambda - lambda^-2). At small strain sigma ~ 3 G (lambda - 1), so the Young modulus is E = 3 G (rubber is incompressible, Poisson ratio 1/2). The finite limit Jm is the chain-extensibility limit: the stress diverges as I1 - 3 -> Jm (strain stiffening), which is why a band suddenly gets very hard to stretch near its limit.

Dynamics. The loop has two strands, so the load-bearing area is A0 = 2 w t. With the hanging mass m and viscous damping c: m x'' = m g - A0 sigma(lambda) - c x', lambda = 1 + x / L0 Equilibrium satisfies A0 sigma(lambda_eq) = m g. Small bounces about it have omega^2 = k / m with k = A0 sigma'(lambda_eq) / L0; because the curve stiffens, a heavier mass bounces at a different frequency than a linear spring would (last graph). If the band goes slack (lambda < 1) it carries no load and the mass moves ballistically.

Heat. With heat capacity rho c V0 and Newtonian cooling time constant tau_c: rho c V0 dT/dt = F v - (T - T_room) rho c V0 / tau_c F v is the mechanical power into the band (negative on release, so the band cools below room temperature when it contracts). G is scaled by T / 293 K.

Tensile mode. A crosshead stretches the loop as lambda(t) = 1 + (lambda_max - 1)(1 - cos 2 pi f t)/2. The viscous term c x' makes the loading and unloading paths differ: the area of the force-extension loop is the energy dissipated per cycle.

Honest simplifications. (1) Real rubber also shows the Mullins effect, permanent set, rate-dependent (viscoelastic) modulus and crystallisation at large strain; here damping is a single viscous dashpot. (2) The temperature is uniform through the band and the cooling is a single time constant. (3) Thermal expansion (the thermoelastic inversion at small strain) is neglected. (4) The two strands share the load equally and the hook is frictionless. (5) The band is a 1D element; the visual widths are drawn from the contraction 1/sqrt(lambda).

Checks run in the page. (a) The damped mass settles at the root of A0 sigma = m g found by bisection; (b) the measured small-bounce period against 2 pi sqrt(m/k); (c) mechanical energy conservation over 5 s of undamped bouncing; (d) the temperature rise in an adiabatic stretch against W(lambda) / (rho c); (e) the small-strain force against 3 G A0 (lambda - 1); (f) the area of the force-extension loop against the integral of c v^2 dt.