🔄 Rubber Network Elasticity

Neo-Hookean rubber: stress σ = G(λ - λ⁻²) where G = nkT is the shear modulus and n is cross-link density. See stress-stretch curve, Mooney-Rivlin fit, and the Gent model limiting stretch.

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Stress-stretch curves · Blue = Neo-Hookean · Green = Mooney-Rivlin · Orange = Gent · Drag slider to update

How it Works

Rubber elasticity arises from the entropic elasticity of cross-linked polymer chains. At rest, chains adopt random-coil configurations with maximum entropy. Stretching reduces chain entropy, and the Gaussian statistics of random walks gives a linear restoring force per chain. Summing over all network strands gives the macroscopic stress-stretch relation.

Three constitutive models are shown: Neo-Hookean (simplest, one parameter G), Mooney-Rivlin (two parameters C₁, C₂, better for moderate strains), and Gent (accounts for finite extensibility through Jₘ, the maximum value of I₁ - 3, causing stress to diverge at limiting stretch).

Neo-Hookean: σ = G·(λ − λ⁻²), G = nRT
Mooney-Rivlin: σ = 2(C₁ + C₂·λ⁻¹)·(λ − λ⁻²)
Gent: σ = G·(λ − λ⁻²)·Jₘ / [Jₘ − (λ² + 2λ⁻¹ − 3)]
I₁ = λ² + 2λ⁻¹ (first invariant, uniaxial)

Frequently Asked Questions

What is rubber elasticity?

Rubber elasticity is the large, reversible deformation behavior of cross-linked polymer networks. Unlike metals, rubber elasticity is entropic in origin: stretching reduces the conformational entropy of polymer chains, and the restoring force is an entropic spring.

What is the neo-Hookean model?

The neo-Hookean model gives stress as σ = G(λ - λ⁻²) for uniaxial tension, where G = nkT is the shear modulus, n is the cross-link density, k is Boltzmann's constant, T is temperature, and λ is the stretch ratio.

What is the stretch ratio λ?

The stretch ratio λ = L/L₀ is the ratio of deformed length to original length. For incompressible rubber, uniaxial stretching (λ > 1) causes lateral contraction with λ_y = λ_z = λ⁻¹/². Engineering strain ε = λ − 1.

What is the Mooney-Rivlin model?

Mooney-Rivlin extends neo-Hookean by adding a second invariant term: σ = 2(C₁ + C₂/λ)(λ - λ⁻²). The Mooney plot of σ/2(λ - λ⁻²) vs 1/λ gives a straight line with slope C₂ and intercept C₁. It better fits real rubber at moderate stretches.

What is the Gent model?

The Gent model accounts for the finite extensibility of polymer chains via parameter Jₘ. As λ approaches the limiting stretch, stress diverges, mimicking strain stiffening observed in real rubber networks.

Why does G = nkT?

From statistical mechanics of Gaussian chains, each network strand stores entropic energy proportional to kT. Summing over n network strands per unit volume gives shear modulus G = nkT. Higher cross-link density leads to higher G.

What is strain-induced crystallization?

At high stretch ratios (λ > 4–6 for natural rubber), polymer chains become aligned and can crystallize, causing an upturn in the stress-strain curve. This greatly increases tear resistance and is why natural rubber outperforms synthetic rubber in many applications.

How does temperature affect rubber modulus?

For ideal rubber elasticity, G = nkT, so modulus increases linearly with absolute temperature. This thermo-elastic inversion (rubber stiffens on heating) contrasts with metals that soften. At very low T, rubber undergoes glass transition and becomes rigid.

What is the Mullins effect?

The Mullins effect is stress-softening observed in filled rubbers after the first large deformation. Reloading follows a lower stress path than the first loading. It is attributed to rupture of filler-polymer bonds and is partially recovered upon heating.

What are common cross-linking methods for rubber?

Vulcanization with sulfur (natural rubber) creates polysulfide bridges. Peroxide cross-linking creates carbon-carbon bonds for better heat resistance. Radiation cross-linking is used for silicone and polyethylene. Cross-link density controls hardness and modulus.

About this simulation

This sandbox plots the true stress vs. stretch ratio for a cross-linked rubber network under three constitutive models at once. Drag the cross-link density and temperature sliders and watch the shear modulus G = nkT recompute live, then compare how the Mooney-Rivlin C₂ term bends the curve away from ideal neo-Hookean behavior and how the Gent model's finite-extensibility limit Jₘ makes stress shoot up as the chains run out of slack.

🔬 What it shows

Three overlaid stress-stretch curves — blue neo-Hookean σ = G(λ − λ⁻²), dashed green Mooney-Rivlin with an extra C₂ term, and orange Gent — sharing the same shear modulus G but diverging as stretch λ grows. A dashed vertical marker shows λ_max, the point where the Gent model's denominator Jₘ − (I₁ − 3) hits zero and stress theoretically becomes infinite.

🎮 How to use

Move the cross-link density n and temperature T sliders to change G = nRT (scaled to kPa); adjust the Mooney C₂ factor to blend in the second Mooney-Rivlin term; drag Gent Jₘ to tighten or loosen the finite-extensibility ceiling. Use the Model dropdown to isolate one curve, or press Reset / hit the R key to redraw.

💡 Did you know?

Rubber elasticity is almost entirely entropic — stretching a rubber band doesn't stretch chemical bonds much, it just uncoils random-walk polymer chains, which is why, unusually, rubber gets stiffer (not softer) as you heat it, since G is directly proportional to T.

Frequently asked questions

Why does the stress curve suddenly shoot upward at high stretch in the Gent model?

The Gent model divides by Jₘ − (I₁ − 3), where I₁ = λ² + 2/λ is the first strain invariant. As λ increases, I₁ − 3 grows toward Jₘ, shrinking the denominator toward zero, so stress diverges. This models the physical limit where polymer chains between cross-links become fully extended and can stretch no further — the λ_max marker on the chart shows exactly where this happens.

What does moving the cross-link density slider actually change?

Cross-link density n sets how many polymer strands exist per unit volume between vulcanization points. Since shear modulus G = nRT, raising n directly raises G, stiffening every curve at once — you'll see the "Shear modulus G" stat and all three curves scale up together while their basic shapes stay the same.

Why does rubber get stiffer when heated, unlike most solids?

Because G = nkT (here approximated with the gas constant R per mole), the shear modulus is directly proportional to absolute temperature. This is the opposite of metals, which soften with heat. Physically, higher T means more vigorous thermal motion driving chains back toward disordered, high-entropy coils, so the entropic restoring force — and hence stiffness — increases.

What's the difference between the Mooney-Rivlin and neo-Hookean curves?

Neo-Hookean assumes a single-parameter strain energy function, giving σ = G(λ − λ⁻²). Mooney-Rivlin adds a second term controlled by the C₂ factor slider, giving σ = 2(C₁ + C₂/λ)(λ − λ⁻²). Setting the Mooney C₂ factor to 0 collapses the green dashed curve exactly onto the blue neo-Hookean curve; raising it bends the curve to better match real rubber's behavior at moderate stretches.

What is λ_max and why does the plot stop there?

λ_max is the stretch ratio at which the Gent model's denominator reaches zero, found numerically in the code by scanning λ until λ² + 2/λ − 3 ≥ Jₘ. Beyond this point the Gent stress is undefined (infinite), so the simulation caps the plotted stretch range just below λ_max and marks it with a dashed orange vertical line labeled "λ_max".