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Similarities between GSH, hypoplasticity and KCR

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Abstract

Accounting for elasto-plastic motion in granular media, hypoplasticity is a state-of-the-art constitutive model derived from data accumulated over many decades. In contrast, GSH, a hydrodynamic theory, is derived from general principles of physics, with comparatively few inputs from experiments, yet sporting an applicability ranging from static stress distribution via elasto-plastic motion to fast dense flow, including non-uniform ones such as a shear band. Comparing both theories, we find great similarities for uniform, slow, elasto-plastic motion. We also find that proportional paths and the Goldscheider rule used to construct barodesy, another, more recent constitutive model, are natural results of GSH’s equations. This is useful as it gives these constitutive relations a solid foundation in physics and, in reverse, GSH a robust connection to reality. The same symbiotic relation exists between GSH and KCR, or Kamrin’s non-local constitutive relation, a model that was successfully employed to account for a wide shear band in split-bottom cells.

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Abbreviations

\(v_{ij}\) :

\(\equiv \frac{1}{2}(\nabla _iv_j+\nabla _jv_i)\), the strain rate,

\(u_{ij}\) :

\(\equiv \varepsilon ^{elast}_{ij}\), the elastic strain,

\(\pi _{ij}\) :

the elastic stress,

\(\sigma _{ij}\) :

the Cauchy stress,

\(x^*_{ij}\) :

the traceless part of \(x_{ij}\),

\(\Delta\) :

\(\equiv -u_{\ell \ell }\),

\(P_\Delta\) :

\(\equiv \pi _{\ell \ell }/3\),

P :

\(\equiv \sigma _{\ell \ell }/3\),

\(v_s\) :

\(\equiv \sqrt{v^*_{ij}v^*_{ij}} \equiv ||v^*_{ij}||\),

\(u_s\) :

\(\equiv \sqrt{ u^*_{ij}u^*_{ij}}\),

\(\pi _s\) :

\(\equiv \sqrt{ \pi ^*_{ij}\pi ^*_{ij}}\),

\(\sigma _s\) :

\(\equiv \sqrt{ \sigma ^*_{ij}\sigma ^*_{ij}}\).

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Authors and Affiliations

  1. Central South University, Changsha, 410083, China

    Yimin Jiang

  2. Theoretische Physik, Universitt Tübingen, 72076, Tübingen, Germany

    Mario Liu

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Correspondence to Mario Liu.

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Jiang, Y., Liu, M. Similarities between GSH, hypoplasticity and KCR. Acta Geotech. 11, 519–537 (2016). https://doi.org/10.1007/s11440-016-0461-9

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