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Experimental and Modelling Investigations of the Coupled Elastoplastic Damage of a Quasi-brittle Rock

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Abstract

Triaxial compression tests are conducted on a quasi-brittle rock, limestone. The analyses show that elastoplastic deformation is coupled with damage. Based on the experimental investigation, a coupled elastoplastic damage model is developed within the framework of irreversible thermodynamics. The coupling effects between the plastic and damage dissipations are described by introducing an isotropic damage variable into the elastic stiffness and yield criterion. The novelty of the model is in the description of the thermodynamic force associated with damage, which is formulated as a state function of both elastic and plastic strain energies. The latter gives a full consideration on the comprehensive effects of plastic strain and stress changing processes in rock material on the development of damage. The damage criterion and potential are constructed to determine the onset and evolution of damage variable. The return mapping algorithms of the coupled model are deduced for three different inelastic corrections. Comparisons between test data and numerical simulations show that the coupled elastoplastic damage model is capable of describing the main mechanical behaviours of the quasi-brittle rock.

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Abbreviations

σ 1 :

Major principal stress

σ 3 :

Minor principal stress, confining pressure

σ cc, σ ci, σ cd, σ p and σ r :

Characteristic stresses of crack closure, initiation, damage, peak strength and residual strength, respectively

ɛ p1 :

Major principal plastic strain

ɛ p3 :

Minor principal plastic strain

E :

Damaged elastic modulus

ω :

Damage variable

\( {\varvec{\upsigma}} \) :

Stress tensor

\( {\mathbf{S}} \) :

Deviatoric stress tensor

\( {\kern 1pt} p \) :

Mean stress

\( {\kern 1pt} q \) :

Deviatoric shear stress

θ :

Lode’s angle

J 3 :

The third invariant of the deviatoric stress tensor

\( {\varvec{\upvarepsilon}} \) :

Strain tensor

\( {\varvec{\upvarepsilon}}^{e} \) :

Elastic strain tensor

\( {\varvec{\upvarepsilon}}^{p} \) :

Plastic strain tensor

ϕ :

Thermodynamic potential

ϕ e :

Elastic part of the thermodynamic potential

ϕ p :

Plastic part of the thermodynamic potential

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

Internal plastic variable or equivalent plastic shear strain

R :

Thermodynamic force associated with plasticity

Y :

Thermodynamic force associated with damage

Y e :

Elastic part of the thermodynamic force associated with damage

Y p :

Plastic part of the thermodynamic force associated with damage

λ p (dλ p):

Plastic multiplier (increment)

λ ω (dλ ω):

Damage multiplier (increment)

\( {\mathbf{e}}^{{{\kern 1pt} p}} \) (\( {\text{d}}{\mathbf{e}}^{{{\kern 1pt} p}} \)):

Plastic deviatoric strain tensor (increment)

\( {\varvec{\updelta}} \) :

Kronecker tensor

ɛ pv (dɛ pv ):

Plastic volumetric strain (increment)

ɛ p m (dɛ p m ):

Mean plastic volumetric strain (increment)

F p :

Plastic criterion

G p :

Plastic potential

F ω :

Damage criterion

G ω :

Damage potential

\( {\mathbf{D}} \) :

Elastic damage stiffness tensor

\( {\mathbf{D}}_{0} \) :

Initial elastic stiffness tensor

\( {\mathbf{D}}^{ep} \) :

Elastoplastic stiffness tensor

\( {\mathbf{D}}^{e\omega } \) :

Elastodamage stiffness tensor

\( {\mathbf{D}}^{ep\omega } \) :

Coupled elastoplastic damage stiffness tensor

\( {\mathbf{D}}^{p\omega } \) and \( {\mathbf{D}}^{\omega p} \) :

Coupled plastic–damage stiffness tensor

\( {\kern 1pt} \chi^{p\omega } \) :

A comprehensive function reflecting plastic hardening and damage softening effects on the evolution of yielding surface

H p :

Plastic modulus

H ω :

Damage modulus

H pω and H ωp :

Coupled plastic–damage moduli

e r :

Tiny value set for numerical computation

E 0 and ν 0 :

Elastic parameters: representing initial elastic modulus and Poisson’s ratio, respectively

α 1 and \( {\kern 1pt} \kappa \) :

Drucker–Prager strength parameters: representing internal friction and cohesion, respectively

α 2 :

Parameter: representing shear–volumetric dilation

h 0 and h 1 :

Plastic hardening parameters: representing initial and final yielding thresholds, respectively

b :

Parameter: representing plastic hardening rate

ω c :

Parameter: representing asymptotic damage at the residual deformation stage

β :

Parameter: representing damage evolution rate

r :

Weight coefficient

a :

Parameter: representing damage softening

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Acknowledgements

The work presented in this paper was financially supported by the National Natural Science Foundation of China (Grant No. 51609070).

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  1. Department of Civil Engineering, Yunnan Minzu University, Kunming, 650504, China

    Jiu-Chang Zhang

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Zhang, JC. Experimental and Modelling Investigations of the Coupled Elastoplastic Damage of a Quasi-brittle Rock. Rock Mech Rock Eng 51, 465–478 (2018). https://doi.org/10.1007/s00603-017-1322-z

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