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A Constitutive Model for Strain-Controlled Strength Degradation of Rockmasses (SDR)

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Abstract

The paper describes a continuum, rate-independent, incremental plasticity constitutive model applicable in weak rocks and heavily fractured rockmasses, where mechanical behaviour is controlled by rockmass strength rather than structural features (discontinuities). The model describes rockmass structure by a generalised Hoek–Brown Structure Envelope (SE) in the stress space. Stress paths inside the SE are nonlinear and irreversible to better simulate behaviour at strains up to peak strength and under stress reversals. Stress paths on the SE have user-controlled volume dilatancy (gradually reducing to zero at large shear strains) and can model post-peak strain softening of brittle rockmasses via a structure degradation (damage) mechanism triggered by accumulated plastic shear strains. As the SE may strain harden with plastic strains, ductile behaviour can also be modelled. The model was implemented in the Finite Element Code Simulia ABAQUS and was applied in plane strain (2D) excavation of a cylindrical cavity (tunnel) to predict convergence-confinement curves. It is shown that small-strain nonlinearity, variable volume dilatancy and post-peak hardening/softening strongly affect the predicted curves, resulting in corresponding differences of lining pressures in real tunnel excavations.

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Abbreviations

SDR:

Constitutive model for Strength Degradation of Rockmasses

SE:

Structure Envelope

SPR:

Stress Path Reversal

SSR:

Stress Sign Reversal

a, σ c, m b, s :

Genearlised Hoek–Brown parameters

b :

Rotation of the SE axis o with respect to the isotropic (σ) axis

\({\mathbf{C}}^{e}\) :

Tangent non-plastic stiffness matrix

c :

Primary strength anisotropy parameter varied along each of the five shearing directions (values c 1 , c 2 , … c 5 along deviatoric axes S 1 , S 2 , … S 5)

d :

State variable controlling the tensile strength of the rockmass, d = (s/m b ) σ c

d in, d fin :

Initial and final values of structure variable d

H :

Plastic hardening modulus

I :

Unit isotopic tensor

K, G :

Nonlinear bulk and shear moduli

m b,in, m b,fin :

Initial and final values of structure variable m b

n :

Material constant controlling the non-plastic stiffness mean stress dependency

P :

Plastic potential tensor

P, P′:

Volumetric plastic potential and plastic potential deviator

p atm :

Atmospheric pressure (p atm ≅ 101.3 kPa)

Q :

Gradient of the SE

Q, Q′:

Isotropic and deviatoric components of gradient Q

q :

Scalar stress deviator (shear stress)

r :

Radial distance along the springline measuring from the tunnel centre (r ≥ R, where R is the tunnel diameter)

\(\varvec{s}\) :

Stress deviator

u R :

Wall tunnel convergence

u r :

Radial convergence along the springline

\(\dot{W}\) :

Second-order total work

\(\dot{W}^{p}\) :

Second-order plastic work

γ :

Material constant controlling the rate of stiffness reduction

\(\dot{\varvec{\varepsilon }}\) :

Strain increment

\(\dot{\varepsilon }\,,\;\dot{\varvec{e}}\) :

Volumetric strain and strain deviator increments

\(\dot{\varvec{\varepsilon }}^{e}\) :

Non-plastic strain increment

\(\dot{\varepsilon }^{e} \,,\;\dot{\varvec{e}}^{e}\) :

Non-plastic volumetric strain and strain deviator increments

\(\dot{\varvec{\varepsilon }}^{p}\) :

Plastic strain increment

\(\dot{\varepsilon }^{p} \,,\;\dot{\varvec{e}}^{p}\) :

Plastic volumetric strain and strain deviator increments

\(\varepsilon_{\text{a}}\) :

Axial strain

\(\varepsilon_{q}^{p}\) :

Scalar deviatoric plastic strain

\(\dot{\varepsilon }_{q}^{p}\) :

Scalar deviatoric plastic strain increment

\(\varepsilon_{{q,f_{1} }}^{p}\) :

Material constant equal to the accumulated plastic shear strain when variable m b reaches the final value (m b,fin)

\(\varepsilon_{{q,f_{2} }}^{p}\) :

Material constant equal to the accumulated plastic shear strain when variable d reaches the final value (d fin)

\(\varepsilon_{\text{vol}}\) :

Volumetric strain

ζ, n pq :

Material constants controlling plastic volume dilatancy

θ in, θ fin :

Initial and final values of variable θ

\(\dot{\varLambda }\) :

Scalar plastic multiplier

ν :

Poisson’s ratio

ξ, θ :

State variables controlling the volumetric and deviatoric stiffness nonlinearity and irreversibility along non-plastic paths

ξ in, ξ fin :

Initial and final values of variable ξ

ξ 0, θ 0 :

Values of variable ξ and θ at the SE axis

ξ SSR, θ SSR :

Values of variable ξ and θ at SSR

\(\varvec{\sigma}\) :

Stress tensor

\(\sigma\), p :

Mean stress

\(\dot{\varvec{\sigma }}\) :

Stress increment

\(\dot{\sigma }\,,\;\dot{\varvec{s}}\) :

Mean stress and stress deviator increments

σ 1, σ 2, σ 3 :

Major, intermediate and minor principal stresses

σ rev :

Stress at SPR

σ SSR :

Stress at SSR

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Acknowledgements

The present research has received partial funding from the European (FP7) project New Technologies for Tunnelling and Underground works (NeTTUN) under Grant Agreement 280712.

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Authors and Affiliations

  1. Geotechnical Department, School of Civil Engineering, National Technical University of Athens (NTUA), 9, Iroon Polytechniou Street, 157 80, Zografou, Greece

    A. Kalos & M. Kavvadas

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  1. A. Kalos
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  2. M. Kavvadas
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Correspondence to A. Kalos.

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Kalos, A., Kavvadas, M. A Constitutive Model for Strain-Controlled Strength Degradation of Rockmasses (SDR). Rock Mech Rock Eng 50, 2973–2984 (2017). https://doi.org/10.1007/s00603-017-1288-x

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  • DOI: https://doi.org/10.1007/s00603-017-1288-x

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