The Boltzmann superposition principle states that the strain response of a viscoelastic material is a superposition of the responses to all previous stress histories. Assuming no stress before time $t=0$, the constitutive equation can be written as: $\begin{equation} \gamma(t)=\int_0^t J(t-t^\prime)\dot{\sigma}(t^\prime)\mathrm{d}t^\prime \end{equation}$ Applying the Laplace transform to this equation yields: $\begin{align} \hat{\gamma}(s)&=\hat{J}(s)\hat{\dot{\sigma}}(s)\\ &=\hat{J}(s)\left(s\hat{\sigma}(s)-\sigma(0^{-})\right) \\ &=\hat{J}(s)\left(s\hat{G}(s)\hat{\dot{\gamma}}(s)-\sigma(0^{-})\right)\\ &=\hat{J}(s)\left(s\hat{G}(s)(s\hat{\gamma}(s)-\gamma(0^-))-\sigma(0^{-})\right)\\ […]

Persistence length, $L_p$, is a fundamental mechanical property that quantifies the bending stiffness of a polymer. It’s defined as the characteristic length scale over which the correlation of bond angles decays. This correlation is expressed as the average cosine of the angle, $\theta$, between bonds separated by a distance $s$ along the chain: $\langle\cos(\theta(s))\rangle = […]

A Gaussian chain, when averaged over all conformations and orientations, exhibits isotropic behavior. This means it can be treated as a sphere with a radius equal to its radius of gyration, $R_g$. However, an exercise in the second chapter of Rubinstein’s Polymer Physics demonstrates that $R_g^2$ becomes asymmetric when the coordinate frame is aligned with […]

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