Rock Mechanics and Rock Engineering

, Volume 50, Issue 10, pp 2677–2694 | Cite as

Validation of a New Elastoplastic Constitutive Model Dedicated to the Cyclic Behaviour of Brittle Rock Materials

  • B. Cerfontaine
  • R. Charlier
  • F. Collin
  • M. Taiebat
Original Paper


Old mines or caverns may be used as reservoirs for fuel/gas storage or in the context of large-scale energy storage. In the first case, oil or gas is stored on annual basis. In the second case pressure due to water or compressed air varies on a daily basis or even faster. In both cases a cyclic loading on the cavern’s/mine’s walls must be considered for the design. The complexity of rockwork geometries or coupling with water flow requires finite element modelling and then a suitable constitutive law for the rock behaviour modelling. This paper presents and validates the formulation of a new constitutive law able to represent the inherently cyclic behaviour of rocks at low confinement. The main features of the behaviour evidenced by experiments in the literature depict a progressive degradation and strain of the material with the number of cycles. A constitutive law based on a boundary surface concept is developed. It represents the brittle failure of the material as well as its progressive degradation. Kinematic hardening of the yield surface allows the modelling of cycles. Isotropic softening on the cohesion variable leads to the progressive degradation of the rock strength. A limit surface is introduced and has a lower opening than the bounding surface. This surface describes the peak strength of the material and allows the modelling of a brittle behaviour. In addition a fatigue limit is introduced such that no cohesion degradation occurs if the stress state lies inside this surface. The model is validated against three different rock materials and types of experiments. Parameters of the constitutive laws are calibrated against uniaxial tests on Lorano marble, triaxial test on a sandstone and damage-controlled test on Lac du Bonnet granite. The model is shown to reproduce correctly experimental results, especially the evolution of strain with number of cycles.


Fatigue Constitutive modelling Bounding surface model Cyclic loading Rock mechanics 

List of symbols

\(\alpha _0\)

Initial value of back-stress ratio of yield surface (–)


Back-stress ratio of yield surface (–)

\(\alpha ^\mathrm{b}\)

Back-stress ratio of boundary surface (–)

\(\alpha ^\mathrm{pc}\)

Back-stress ratio of fatigue surface (–)

\(\delta _\mathrm{l}\)

Compression/extension opening ratio (–)

\(\epsilon _1\)

Vertical strain (–)

\(\epsilon _3\)

Lateral strain (–)

\(\epsilon _\mathrm{q}\)

Deviatoric strain (–)

\(\epsilon _\mathrm{v}\)

Volumetric strain (–)

\(\epsilon _\mathrm{q}^\mathrm{p}\)

Plastic deviatoric strain (–)

\(\epsilon _\mathrm{v}^\mathrm{p}\)

Plastic volumetric strain (–)


Plastic multiplier (–)


Reduced deviatoric stress (–)


Poisson’s ratio (–)

\(\sigma _1\)

Vertical effective stress (MPa)

\(\sigma _3\)

Lateral effective stress (MPa)


Friction angle (–)


Hardening parameter (–)


Cohesion (MPa)


Young's modulus (MPa)

\({\mathbf {e}}\)

Strain vector (–)


Yield surface (–)


Limit surface (–)


Hardening direction of \(\alpha\) (–)


Hardening direction of \(p_\mathrm{c}\) (–)

\(n_{\alpha }\)

Parameter for hardening of \(\alpha\) (–)


Parameter for hardening of \(p_\mathrm{c}\) (–)


Mean stress (MPa)


Initial value of cohesion projection (MPa)


Cohesion projection (MPa)


Residual cohesion projection (MPa)


Deviatoric stress (MPa)

\({\mathbf {s}}\)

Stress vector (–)

\({\mathbf {v}}\)

Hardening vector (–)


Parameter for cohesion evolution (–)


Parameter for cohesion evolution (–)


Parameter for volumetric flow rule (–)


Parameter for deviatoric flow rule (–)

\({\mathbf {E}}\)

Elastic tensor (MPa)


Shear modulus (MPa)


Opening of the bounding surface (–)


Opening of the limit surface (–)


Opening of the fatigue surface (–)


Opening of the yield surface (–)


Number of cycles (–)



The authors gratefully acknowledge the financial support from Walloon Region (Belgium) and the SMARTWATER project. The first author would like to warmly thank the Pr. Derek Martin for his encouragement and data he provided.


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Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  • B. Cerfontaine
    • 1
  • R. Charlier
    • 1
  • F. Collin
    • 1
  • M. Taiebat
    • 2
  1. 1.Urban and Environmental EngineeringUniversity of LiegeLiegeBelgium
  2. 2.Department of Civil EngineeringUniversity of British ColumbiaVancouverCanada

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