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An Elastoplastic Model for Partially Saturated Collapsible Rocks

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Abstract

A unified elastoplastic model for describing the stress–strain behavior of partially saturated collapsible rocks is proposed. The elastic–plastic response due to loading and unloading is captured using bounding surface plasticity. The coupling effect of hydraulic and mechanical responses is addressed by applying the effective stress concept. Special attention is paid to the rock–fluid characteristic curve (RFCC), effective stress parameter, and suction hardening. A wide range of saturation degree is considered. The characteristics of mechanical behavior in partially saturated collapsible rocks are captured for all cases considered.

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Abbreviations

\(A\) :

Sign for loading direction

\(F\) :

Bounding surface

\(f\) :

Loading surface

\(G^{\text{e}}\) :

Elastic shear modulus

\(g\) :

Plastic potential

\(h\) :

Hardening modulus

\(h_{\text{b}}\) :

Hardening modulus at bounding surface

\(h_{\text{f}}\) :

Additive hardening modulus

\(J_{2}\) :

Second invariant of deviatoric stress

\(K_{\text{bulk}}^{\text{e}}\) :

Elastic bulk modulus

\(k_{\text{m}}\) :

Material parameter

\(M\) :

Total mass of porous rock

\(M_{\alpha }\) :

Mass of phase \(\alpha\) (\(\alpha = \text{s},\;\text{w},\;\text{a}\) represents solids, water, and air, respectively)

\(M_{\text{cs}}\) :

Slope of critical state line (CSL)

\(M_{\text{cs}}^{ - }\) :

Slope of CSL for triaxial extension

\(M_{\text{cs}}^{ + }\) :

Slope of CSL for triaxial compression

\(m_{p}\) :

Component of unit direction along the mean effective stress axis

\(N\) :

Parameter controlling shape of bounding surface

\(N\left( s \right)\) :

Specific volume on isotropic compression line at mean effective stress of 1 kPa or 1 MPa

\(p^{\prime}\) :

Mean effective stress

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

Size of loading surface

\(p^{\prime}_{\text{co}}\) :

Maximum pressure experienced in the past

\(p^{\prime}_{\text{cs}}\) :

Mean effective stress at critical state

\(p^{\prime}_{0}\) :

Size of plastic potential

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

Mean effective stress on bounding surface

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

Size of bounding surface

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

Initial value of hardening parameter

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

Final value of hardening parameter

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

Air pressure

\(\dot{P}_{\text{a}}\) :

Increment of air pressure

\(P_{\text{w}}\) :

Water pressure

\(\dot{P}_{\text{w}}\) :

Increment of water pressure

\(q\) :

Deviatoric stress

\(\bar{q}\) :

Deviatoric stress at bounding surface

\(R\) :

Material parameter in bounding surface

s :

Suction

\(\dot{s}\) :

Increment of suction

S i :

Initial suction value

\(s_{\text{ae}}\) :

Air entry value

\(s_{\text{e}}\) :

Suction value at saturation state changing point

\(s_{\text{ex}}\) :

Air expulsion value

\(s_{\text{rd}}\) :

Suction reversal point on drying path

\(s_{\text{rw}}\) :

Suction reversal point on wetting path

\(S_{\text{eff}}\) :

Effective degree of saturation

\(S_{\text{r}}\) :

Degree of saturation

\(S_{\text{re}}\) :

Residual degree of saturation

\(t\) :

Sign for loading direction

\(V\) :

Total volume of partially saturated rock

\(V_{\alpha }\) :

Volume of phase \(\alpha\) (\(\alpha = \text{s},\;\text{w},\;\text{a}\))

\(\alpha\) :

Material number (\(\alpha = \text{s},\;\text{w},\;\text{a}\))

\(\chi\) :

Effective stress parameter

\(\varepsilon_{q}\) :

Deviatoric strain

\(\varepsilon_{p}\) :

Volumetric strain

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

Plastic volumetric strain

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

Plastic volumetric strain rate

\(\Delta \varepsilon_{p}^{\text{p}}\) :

Plastic volumetric strain increment

\(\varphi^{\prime}\) :

Drained friction angle

\(\varGamma \left( s \right)\) :

Specific volume on critical state line at mean effective stress of 1 kPa or 1 MPa

\(\gamma \left( s \right)\) :

Suction hardening function

\(\eta\) :

Stress ratio

\(\eta_{\text{p}}\) :

Slope of peak strength line in \(p^{\prime}-q\) plane

\(\kappa\) :

Slope of elastic unloading–reloading line in \(\upsilon-\ln p^{\prime}\) plane

\(\lambda \left( s \right)\) :

Slope of isotropic compression line in \(\upsilon-\ln p^{\prime}\) plane

\(\lambda_{\text{p}}\) :

Index for pore size distribution

\(v\) :

Poisson ratio

\(\theta\) :

Lode angle

\(\upsilon\) :

Specific volume

\(\upsilon_{\text{cs}}\) :

Specific volume at critical state line

\(\upsilon_{\text{i}}\) :

Initial value of a specific state

\(\upsilon_{\text{IC}}\) :

Specific volume at isotropic compression line

\(\zeta\) :

Slope of drying–wetting transition line in \(\ln S_{\text{r}}-\ln s\) plane

\(\varsigma\) :

Slope of drying–wetting transition line in \(\ln \chi -\ln s\) plane

\(\psi\) :

Incremental effective stress parameter

\(\varOmega\) :

Material parameter

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

Elastic stiffness

\(\varvec{m}\) :

Unit direction of plastic flow

\(\varvec{n}\) :

Unit vector normal to loading/bounding surface

\(\varvec{\delta}\) :

Identity vector

\(\varvec{\varepsilon}\) :

Strain vector

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

Incremental form of strain

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

Incremental form of elastic strain

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

Incremental form of plastic strain

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

Deviatoric strain vector

\(\varvec{S}\) :

Deviatoric stress vector

\(\varvec{\sigma}\) :

Total stress vector

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

Effective stress vector

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

Effective stress on bounding surface

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

Incremental form of effective stress

\(\varvec{\sigma}_{\text{net}}\) :

Net stress vector

\(\dot{\varvec{\sigma }}_{\text{net}}\) :

Incremental form of net stress

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Acknowledgments

The author thanks Professor Nasser Khalili and Dr. Gaofeng Zhao (University of New South Wales, Australia) for their constructive comments and review of this manuscript. This work is supported by Key Scientific and Technological Innovation Team of Zhejiang Province (no. 2011R50020), Key Scientific and Technological Innovation Team of Wenzhou (no. C20120006). This financial support is gratefully acknowledged.

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

    Jianjun Ma

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Ma, J. An Elastoplastic Model for Partially Saturated Collapsible Rocks. Rock Mech Rock Eng 49, 455–465 (2016). https://doi.org/10.1007/s00603-015-0751-9

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