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Granular solid dynamics with eutaraxy and hysteresis

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Abstract

The dynamics of grain fabrics is captured by means of a hidden state variable , named eutaraxy, which quantifies the propensity for a heat-like micro-seismicity due to disturbing actions. Its increase by driving and decrease by halting are captured by an evolution equation with switch functions. The specific elastic energy depends on the elastic strain and the void ratio so that a Coulomb condition and a maximal void ratio denote critical points. The rate-independent evolution equation of the elastic strain tensor implies a micro-seismically activated relaxation. Monotonous deformations lead to a hypoplastic response, and thereafter isochoric ones tend to steady states . The hysteresis of repeated reversals, which requires rate-independence and stability, is captured without further parameters, and likewise the accumulation of anelastic strain by cycles with small amplitudes. The parameters are physically defined and can be calibrated by means of triaxial tests. Limitations and possible extensions are outlined.

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Abbreviations

\(D_r\equiv (e_{\max }-e)/(e_{\max }-e_{\min })\) :

Relative density via void ratio e

\(\sigma '_{ij}\) :

Fabric or effective stress tensor

\(p'\equiv \frac{1}{3}\sigma '_{ii}\) :

Mean fabric pressure

\(\epsilon _{ij}\) :

Strain tensor with amount \(\epsilon \equiv \sqrt{\epsilon _{ij}\epsilon _{ij}}\ll 1\) and volumetric strain \(\epsilon _v\equiv \epsilon _{ii}\)

\(\epsilon ^{e}_{ij}\) :

Elastic strain tensor with volumetric and deviatoric invariants \(\epsilon ^{e}_v\) and \(J^{e}_2\)

\(w_e(\epsilon ^{e}_v,J^{e}_2, D_r) \) :

Specific elastic energy stiffness and friction factors \(B_d\) and b

\(\sigma ^{e}_{ij}\equiv \partial w_e/\partial \epsilon ^{e}_{ij}\) :

Elastic stress tensor with first and second invariants \(p^{e}\) and \(\bar{\tau }^{e}\)

\(\alpha , \alpha _h\) :

Transfer factor and its hypoplastic upper bound

\(\chi , \chi _h\) :

Eutaraxy and its hypoplastic upper bound

\(c_{\chi }, \chi _a, \epsilon _r\) :

Factor and auxiliary variables for the evolution of \(\chi \)

\(\lambda _v/\lambda _d\) :

Ratio of volumetric and deviatoric relaxation factors, factor \(c_{\lambda }\) for it.

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Acknowledgements

I owe valuable hints to Roberto Cudmani (Munich), Gerhard Huber (Karlsruhe), Demetrios Kolymbas (Innsbruck), Mario Liu (Tübingen), Andrzej Niemunis (Kalsruhe) and Torsten Wichtmann (Weimar). This is gratefully acknowledged.

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  1. Emeritus, Institute of Soil and Rock Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Gerd Gudehus

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Gudehus, G. Granular solid dynamics with eutaraxy and hysteresis. Acta Geotech. 15, 1173–1187 (2020). https://doi.org/10.1007/s11440-019-00820-y

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