A few notes about viscoplastic rheologies

TL;DR

This work develops a convex-analysis framework to unify viscoplastic rheologies by constructing a single dissipation potential $\zeta_{\rm vp}$ from elemental viscous and plastic components. It shows how parallel and serial combinations, via $\zeta_{\rm vp}=\zeta_1+\zeta_2$ or $\zeta_{\rm vp}=\zeta_1\Box\zeta_2$, yield rate-dependent responses characterized by $\bm\sigma=\zeta_{\rm vp}'(\bm\varepsilon)$ and an effective viscosity $\mu_{\rm eff}$, with explicit forms in Bingham and Norton–Hoff examples. The paper extends to three-element configurations and nonlinear viscosities (power-law, Herschel–Bulkley), deriving both exact and regularized expressions (e.g., Yosida approximations) and highlighting when harmonic-mean models align with but also diverge from the rigorous convex-convolution constructions. The approach facilitates robust, thermodynamically consistent large-strain modeling and can incorporate elasticity and internal variables, temperature, volume, and other state dependencies. Overall, it provides a principled pathway to derive and compare viscoplastic models used in geophysics and materials science.

Abstract

The rigorous tools of convex analysis are used to examine various serial and parallel combinations of linear viscosity and perfect plasticity. Nonlinear viscosities are also considered. The general aim is to synthesize a single convex ``viscoplastic'' dissipation potential from the potentials of particular viscous or plastic elements. Rigorous serial-viscosity models are then compared with empirical models based on harmonic means, which are commonly used for various geomaterials.

Figures (7)

• Figure 1: Two options in combination of a viscous damper with a perfect-plasticity element with the activation threshold $\sigma_\text{\!\sc a}^{}$.
• Figure 2: Schematic illustration of the convex nonsmooth dissipation potential $\zeta_{\rm vp}$ from (\ref{['GSM-Jeffreys-plast']}) with both $D_2>0$ and $\sigma_\text{\!\sc a}^{}\!>0$, its subdifferential $\partial\zeta_{\rm vp}$ which is set-valued at 0, and its (smooth) conjugate $\zeta_{\rm vp}^*$ and its derivative (= single-valued inverse of $\partial\zeta_{\rm vp}$) used to model (rate-dependent) plasticity; cf. also JirBaz02IAS.
• Figure 3: Schematic illustration of the convex smooth dissipation potential $\zeta_{\rm vp}$ from (\ref{['GSM-Jeffreys-plast+']}) with both $D_2>0$ and $\sigma_\text{\!\sc a}^{}\!>0$, its continuous differential $\zeta_{\rm vp}'$, and its (nonsmooth) conjugate $\zeta_{\rm vp}^*$ and its derivative (= set-valued inverse of $\zeta_{\rm vp}'$) used to model plasticity combined with a creep.
• Figure 4: Two (mutually equivalent) variants of a combination of two viscous dampers with a perfect plasticity with the activation thresholds $\sigma_\text{\!\sc a}^{}$ and $\widetilde{\sigma}_\text{\!\sc a}^{}$ leading to a continuous ${\zeta_{\rm vp}^*}\!\!'$.
• Figure 5: Schematic illustration of the bi-visco-plastic model from from Figure \ref{['fig-visco-plastic-1']}-left leading to a continuously differentiable ${\zeta_{\rm vp}^*}\!\!'$ with ${\zeta_{\rm vp}^*}$ continuously differentiable.

Theorems & Definitions (5)

• Remark 1: Serial linear viscosities.
• Remark 2: The harmonic mean general.
• Remark 3: Alternative phenomenological combination of visco-plasticity.
• Example 1: Serial combination of the diffusive and the dislocation viscosities.
• Remark 4: Generalized Maxwell rheologies.