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An Elasto-Viscoplastic Model for Soft Porous Rocks Within the Consistent Framework

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Abbreviations

F :

Bounding surface

f :

Loading surface

G e :

Elastic shear modulus

g :

Plastic potential

h :

Viscoplastic parameter on bounding surface

K e :

Elastic bulk modulus

k m :

Material parameter

m p :

Component of unit direction along the mean effective stress axis

m q :

Component of unit direction along the deviatoric stress axis

N :

Parameter controlling the shape of bounding surface

N ic :

Specific volume on isotropic compression line

p′:

Mean effective stress

\( p_{c}^{\prime } \) :

Size of loading surface

\( p_{0}^{\prime } \) :

Size of plastic potential

\( p_{\text{t}}^{\prime } \) :

Isotropic tensile strength

\( \bar{p}^{\prime } \) :

Mean effective stress on bounding surface

\( \bar{p}_{\text{c}}^{\prime } \) :

Size of bounding surface

\( \bar{p}^{\prime }_{{{\text{c}},{\text{i}}}} \) :

Plastic hardening parameter in initial condition

\( \bar{p}^{{\prime }}_{{{\text{c}},{\text{f}}}} \) :

Final value of hardening parameter

q :

Deviatoric stress

\( \bar{q} \) :

Deviatoric stress at bounding surface

R :

Shape parameter for bounding surface

t :

Time

χ :

Viscoplastic multiplier

ɛ q :

Deviatoric strain

ɛ v :

Volumetric strain

\( \varepsilon_{\text{v}}^{\text{vp}} \) :

Viscoplastic volumetric strain

ζ :

Plastic volumetric strain hardening modulus

\( \tilde{\zeta }\left( {{\text{d}}\varepsilon_{\text{v}}^{\text{vp}} } \right) \) :

Coupling hardening factor from plastic strain

ζ b :

Viscoplastic hardening parameter at bounding surface

ζ a :

Arbitrary strain modulus on loading surface

γ :

Plastic volumetric strain rate hardening modulus

\( \tilde{\gamma }\left( {{\text{d}}\dot{\varepsilon }_{\text{v}}^{\text{vp}} } \right) \) :

Coupling hardening factor from plastic strain rate

γ b :

Plastic strain rate hardening parameter at bounding surface

γ a :

Arbitrary strain rate modulus on loading surface

η p :

Slope of peak strength line in p′ ~ q plane

η :

Stress ratio

κ :

Slope of unloading–reloading lines in υ − ln p′ plane

λ :

Slope of isotropic compression line in υ − ln p′ plane

v :

Poisson’s ratio

υ :

Specific volume

\( \varvec{D}^{\text{e}} \) :

Elastic stiffness

m :

Unit direction of plastic flow

n :

Unit vector normal to the loading/bounding surface

\( \varvec{\delta} \) :

Identity vector

\( \varvec{\varepsilon} \) :

Strain vector in triaxial notation

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

Elastic strain vector

\( \varvec{\varepsilon}^{\text{vp}} \) :

Viscoplastic strain vector

\( \varvec{\sigma^{\prime}} \) :

Effective stress vector

\( \bar{\varvec{\sigma }}^{\prime } \) :

Effective stress on bounding surface

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Acknowledgements

This work is supported by National Natural Science Foundation of China (NSFC, No. 51508416) and Provincial Commonweal Science Foundation of Zhejiang (PCSFZ, No. 2017C33220). The financial support is gratefully acknowledged. Valuable suggestions from Professor Nasser Khalili (UNSW Australia) and anonymous reviewers are also acknowledged.

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  1. College of Architecture and Civil Engineering, Wenzhou University, Wenzhou, 325035, China

    Jianjun Ma

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  1. Jianjun Ma
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Ma, J. An Elasto-Viscoplastic Model for Soft Porous Rocks Within the Consistent Framework. Rock Mech Rock Eng 50, 3109–3114 (2017). https://doi.org/10.1007/s00603-017-1283-2

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