= Solution
The restoring force in ideal <rubber elasticity> is mainly entropic: stretching restricts the conformations of network strands. A <Gaussian chain> of $N_s$ statistical segments of length $b$ has extension <free energy> $3k_BTR^2/(2N_sb^2)$, so its small-extension spring constant is proportional to $T$. For an affine network with active-strand <number density> $\nu_s$, the shear modulus is approximately $G=\nu_sk_BT$. Increasing the <cross-link> density normally raises $G$ and shortens the strands; <polymer entanglement> can add effective constraints. Dangling ends and loops do not contribute like elastically active bridges, while junction fluctuations reduce the affine-network estimate.
For incompressible uniaxial stretching by ratio $\lambda$, the nominal tensile <stress> is $G(\lambda-\lambda^{-2})$ in the ideal Gaussian-network model. Finite contour length causes stronger stiffening as strands approach full extension; swelling dilutes the active strands and changes their reference conformations. A direct demonstration is to hang a load from a stretched rubber strip and warm it gently: an entropy-dominated network tends to contract under that fixed load. This <temperature> dependence contrasts with a model based solely on stretching bond energies. The simple formula assumes equilibrium, adequate cross-linking and temperatures away from glass transitions or degradation.
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