Boltzmann superposition principle

Remarks

The idea is to approximate the strain history by a staircase of small steps $\Delta\varepsilon_i$ at times $s_i$. By time invariance each step produces the stress $G(t - s_i) \Delta\varepsilon_i$, and by linearity the stresses of all steps add up. Letting the steps become small turns the sum into the integral, a convolution of $G$ with $\dot\varepsilon$. Example: for a Maxwell material with $G(t) = E e^{-t/\tau}$ and a constant strain rate $\dot\varepsilon = r$ from $\varepsilon(0) = 0$, the formula gives $\sigma(t) = E r \tau (1 - e^{-t/\tau})$, which tends to the viscous value $\eta r$ with $\eta = E\tau$. The symmetric form $\varepsilon(t) = J(t) \sigma(0^+) + \int_0^t J(t - s) \dot\sigma(s) \, ds$ holds with the creep compliance $J$. In physics the principle is often taken as the definition of linear viscoelasticity; it fails for nonlinear materials and for aging materials such as young concrete.

Going deeper

For a student

The principle turns the constitutive law into a convolution, so that the Laplace transform changes it into the algebraic relation $\hat\sigma(s) = s \hat G(s) \hat\varepsilon(s)$. This is the basis of the correspondence principle and of the complex modulus used in oscillatory tests. Spring–dashpot models, whose relaxation moduli are Prony series, are special cases.