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Polymer Chain Mechanics

Describes the stress-strain relationship in polymer chains, often utilizing Hooke's Law or more complex models incorporating chain flexibility and entanglement.
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The statement of the theorem

Let F:Body→Body′\mathbf{F}: \text{Body} \to \text{Body}' be the deformation gradient mapping the reference configuration to the current configuration. The Cauchy stress tensor σ\boldsymbol{\sigma} is derived from the strain energy density function W(F)W(\mathbf{F}) via the relationship: \nσ=∂W∂FFT\boldsymbol{\sigma} = \frac{\partial W}{\partial \boldsymbol{\mathbf{F}}} \mathbf{F}^T \nFor an incompressible, neo-Hookean approximation, the strain energy density is defined as W(F)=12μ(tr(FTF)−3)+κ(det(F)−1)W(\mathbf{F}) = \frac{1}{2} \mu (\text{tr}(\mathbf{F}^T \mathbf{F}) - 3) + \kappa (\text{det}(\mathbf{F}) - 1), where μ\mu is the shear modulus and κ\kappa is the bulk modulus.
Source: Wikipedia