A few notes about viscoplastic rheologies

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.

💡 Research Summary

The paper presents a rigorous convex‑analysis framework for constructing a single dissipation potential that captures the combined effects of linear viscosity, perfect plasticity, and nonlinear power‑law viscosities. Starting from the simplest building blocks—a perfectly plastic element described by the 1‑homogeneous potential ζ₁(·)=σₐ|·| and a linear viscous element described by the 2‑homogeneous quadratic potential ζ₂(·)=½D|·|²—the authors examine two elementary configurations. In the parallel arrangement, the total stress belongs to the sum of sub‑gradients, σ∈∂ζ₁(ε)+∂ζ₂(ε), which is equivalent to σ∈∂(ζ₁+ζ₂)(ε). This yields the classic Bingham fluid law σ=σₐ ε/|ε|+Dε. In the series arrangement, the strain rate is split as ε=ε₁+ε₂ with σ=ζ₁′(ε₁)=ζ₂′(ε₂). Using the infimal convolution operator □, the combined dissipation potential becomes ζ_vp=ζ₁□ζ₂. For the linear‑viscous case this infimal convolution coincides with the Yosida approximation of ζ₁, producing a smooth “Huber‑type” potential that interpolates between a quadratic creep regime (|ε|≤σₐ/D) and a linear slip regime (|ε|>σₐ/D). The associated stress–strain relation is continuous and differentiable, overcoming the set‑valued sub‑gradient issue of the pure series model.

The authors then extend the analysis to three‑element configurations involving two viscous dampers and one plastic element, which are common in engineering practice. Two apparently different constructions (Fig. 4‑left and Fig. 4‑right) are shown to be mathematically equivalent. The left construction yields ζ_vp=(ζ₁□ζ₂)+ζ₃ with ζ₃=½D₃|·|², while the right construction first adds the plastic and the third viscous potential (eζ₁+eζ₃) and then takes the infimal convolution with the second viscous potential eζ₂. By appropriate parameter transformations (eσₐ=σₐ/(1+D₃/D₂), eD₂=D₂+D₃, eD₃=D₃/(1+D₃/D₂)), both expressions generate the same effective viscosity μ_eff(ε)=min{σₐ|ε|,D₂}+D₃. This formula captures a rate‑independent yield stress at low strain rates and a linear viscous response at high rates, providing a unified description of creep‑dominated and slip‑dominated regimes.

A significant portion of the paper is devoted to comparing the rigorous convex‑analysis results with the empirical harmonic‑mean approach widely used in geophysical mantle modeling. The harmonic mean μ_eff⁻¹=∑μ_i⁻¹ arises from assuming an additive split of the total strain rate into contributions from each element and then averaging the reciprocals of the individual viscosities. While this coincides with the convex‑analysis result for a collection of linear viscous elements (the effective viscosity is the harmonic mean multiplied by the number of elements), it diverges when any of the μ_i depend nonlinearly on their own strain rates. The authors illustrate this discrepancy with explicit examples, showing that the harmonic‑mean formula can either over‑ or under‑predict the true effective response, especially for shear‑thinning or shear‑thickening fluids.

Finally, the paper incorporates nonlinear power‑law (Norton‑Hoff) viscosities, characterized by the potential ζ(ε)=n/(n+1) D |ε|^{1+1/n}. The corresponding stress law σ=D |ε|^{1/n−1} ε yields an effective viscosity μ_eff(ε)=D |ε|^{1/n−1}. For n>1 the material exhibits shear‑thickening (common in magma flow), while n<1 describes shear‑thinning behavior (relevant for polymers or blood). By applying the infimal convolution to these power‑law potentials together with the plastic potential, the authors obtain a fully general, convex dissipation potential that subsumes all previously discussed linear and piecewise‑linear models. This unified framework ensures mathematical consistency (convexity, differentiability where appropriate) and provides clear guidelines for selecting model parameters in simulations of geophysical and engineering systems.

In conclusion, the work demonstrates that convex analysis offers a powerful, systematic method for synthesizing complex viscoplastic rheologies from elementary building blocks. It clarifies the conditions under which empirical averaging schemes are valid, highlights their limitations, and extends the theory to encompass realistic nonlinear viscosities. The resulting models are both mathematically robust and directly applicable to a wide range of problems, from mantle convection and fault slip to polymer processing and biomedical flow.