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Constitutive modelling of the elastic–plastic, viscoplastic and damage behaviour of hard porous rocks within the unified theory of inelastic flow

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Abstract

This article presents a constitutive model for the elastic–plastic, viscoplastic and damage behaviour of hard porous rocks. The main hypotheses of the model are based on a large set of experimental data which are also presented in the paper. This constitutive model is of the over-stress type and is formulated within the unified theory of inelastic flow. An energy-based failure surface was considered to describe both short- and long-term behaviours within the same formulation. The inelastic yield surface is of static nature while the failure and damage surfaces are of dynamic nature. The kinetic law is written in terms of internal state variables that allow the description of how the frictional and the cohesive internal strengths of the material evolves. The reversible inelastic behaviour is also modelled using the “under-stress” concept, and considering that, it depends explicitly on the locked energy during the inelastic flow. In addition, this model is adapted to the porous nature of rocks such as iron ore that exhibit strong volumetric deformations and mean stress dependence. A large fraction of the volumetric straining is explained by damage mechanisms that also allow the accelerated creep to be modelled. Model parameters can easily be identified in the laboratory with commonly used mechanical tests. The constitutive model was implemented in a numerical code, and some qualitative simulations and comparisons with experimental curves showed the suitability of the model to reproduce both short- and long-term behaviours of porous rocks similar to iron ore.

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Abbreviations

H(x):

Heaviside function

<x> = ½ (x + |x|):

Macaulay brackets

t :

Time

x in :

Inelastic component

x vp :

Viscoplastic (time-dependent) component

\( \underline{\underline{{\dot{\varepsilon }}}} = \underline{\underline{{\dot{\varepsilon }}}}^{\text{e}} + \underline{\underline{{\dot{\varepsilon }}}}^{\text{in}} = \underline{\underline{{\dot{\varepsilon }}}}^{\text{e}} + \underline{\underline{{\dot{\varepsilon }}}}^{\text{p}} + \underline{\underline{{\dot{\varepsilon }}}}^{\text{vp}} \) :

Strain partition

\( \dot{\varepsilon }_{\text{v}}^{\text{in}} = \dot{\varepsilon }_{11}^{\text{in}} + \dot{\varepsilon }_{22}^{\text{in}} + \dot{\varepsilon }_{33}^{\text{in}} \) :

Inelastic volumetric strain

\( \beta = \dot{\varepsilon }_{\text{v}} /\dot{\gamma } \) :

Dilatancy coefficient

\( \sigma_{\text{eq}} = \sqrt {\frac{3}{2}\underline{\underline{s}} :\underline{\underline{s}} } \) :

Equivalent (Von Mises) deviatoric stress

\( \underline{\underline{s}} \) :

Deviatoric stress tensor

K :

Elastic bulk modulus

A :

Scaling parameter for kinetic law \( \dot{\gamma }^{\text{in}} \)

σ 0 :

Normalisation parameter (=1 MPa)

Y o :

Initial value of Y

Y r :

Slope of peak surface of plasticity

Y 2 :

Scaling parameter of Y

η o :

Initial value of η

η 1 :

Scaling parameter of η

w in,tot :

Total inelastic work

\( w_{{{\text{vol}}}}^{\text{in}} \) :

Volumetric inelastic work

w 1 :

Scaling parameter of \( w_{{{\text{d }}\hbox{max}}}^{\text{in}} \)

\( \underline{\underline{{\dot{\varepsilon }}}}^{\text{in*}} \) :

“Standard” inelastic* strain

\( \underline{\underline{{\dot{\varepsilon }}}}^{\text{b}} \) :

“Backward” (reverse) inelastic strain

\( \dot{\gamma }^{\text{f}} \) :

“Forward” inelastic distortion

\( Y^{*} \) :

“Standard” frictional strength

Y f :

“Forward” frictional strength

\( (\gamma^{{{\text{in}}*}} )_{{{ \hbox{max} }}} \) :

Maximal distortion

\( w_{{{\text{d}}\hbox{max}}}^{{{\text{in}}*}} \) :

Deviatoric “standard” inelastic* work at failure

\( w^{{{\text{in}}*,{\text{lock}}}} \) :

Locked “standard” inelastic* energy

\( w_{\hbox{max}}^{{{\text{b,lock}}}} \) :

Locked energy available for “backward” strain

w f,lock :

Locked “forward” energy

C :

Scaling parameter for kinetic law \( \dot{\gamma }^{\text{f}} \)

\( w^{{{\text{in}}*,{\text{diss}}}} \) :

Dissipated “standard” inelastic* energy

D :

Isotropic damage variable

\( w_{{{\text{d}}\,{\text{crit}}}}^{{{\text{in}}*}} \) :

Deviatoric inelastic energy at damage onset

ξ :

Scaling parameter of damage function

:

Fourth-order elasto-damaged stiffness tensor

ϕ d :

Damage porosity

ϕ 2 :

Scaling parameter for damage porosity

\( \tilde{H} \) :

Conjugate of the Heaviside function

\( \dot{x} \) :

Rate of x

x e :

Elastic (time-independent) component

x p :

Plastic (time-independent) component

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

Strain tensor

\( \dot{\gamma }^{\text{in}} = \sqrt {\frac{2}{3}\underline{\underline{{\dot{e}}}}^{\text{in}} :\underline{\underline{{\dot{e}}}}^{\text{in}} } \) :

Inelastic distortion

\( \underline{\underline{{\dot{e}}}}^{\text{in}} = \underline{\underline{{\dot{\varepsilon }}}}^{\text{in}} - \frac{1}{3}{\text{tr}}\,\underline{\underline{{\dot{\varepsilon }}}}^{\text{in}} \) :

Deviatoric strain tensor

\( \underline{\underline{\sigma }} \) :

Cauchy stress tensor

\( \sigma_{\text{m}} = \frac{1}{3}{\text{tr }}\underline{\underline{\sigma }} \) :

Mean stress

G :

Elastic shear modulus

F d :

Deviatoric yield function

n :

Scaling parameter for kinetic law \( \dot{\gamma }^{\text{in}} \)

Y :

Frictional strength (scalar state variable)

Y m :

Peak value of Y

Y 1 :

Scaling parameter of Y

η :

Cohesive strength (scalar state variable)

η m :

Maximal value of η

(γ in)max :

Maximal inelastic distortion

\( w_{\text{d}}^{\text{in}} \) :

Deviatoric inelastic work

\( w_{{{\text{d max}}}}^{\text{in}} \) :

Deviatoric inelastic work at failure

w 2 :

Scaling parameter of \( w_{{{\text{d }}\hbox{max}}}^{\text{in}} \)

\( \underline{\underline{{\dot{\varepsilon }}}}^{\text{f}} \) :

“Forward” inelastic strain

\( \dot{\gamma }^{{{\text{in}}*}} \) :

“Standard” inelastic* distortion

\( \dot{\gamma }^{\text{b}} \) :

“Backward” (reverse) inelastic distortion

\( \eta^{*} \) :

“Standard” cohesive strength

\( \eta^{\text{f}} \) :

“Forward” cohesive strength

\( w_{\text{d}}^{\text{in*}} \) :

Deviatoric “standard” inelastic* work

w in,lock :

Locked inelastic energy

x :

Recoverable fraction of \( w^{{{\text{in*,lock}}}} \)

w b,lock :

Locked “backward” energy

B :

Scaling parameter for kinetic law \( \dot{\gamma }^{\text{b}} \)

w in,diss :

Dissipated inelastic energy

\( w^{{{\text{in*,tot}}}} \) :

Total “standard” inelastic* energy

\( w_{\text{vol}}^{\text{in*}} \) :

Volumetric “standard” inelastic* work

F dam :

Energy-based damage function

\( \tilde{F}_{\text{d}} \) :

Deviatoric yield function with damage

:

Fourth-order elasticity stiffness tensor

ϕ 1 :

Scaling parameter for damage porosity

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Acknowledgments

This research was carried out thanks to subsidies from the Ministries for Industry and Research and the Lorraine Region within the GISOS (www.gisos.org) framework. The authors express their gratitude to these organisations.

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  1. GeoRessources Laboratory, University of Lorraine/CNRS/CREGU, BP 40, 54501, Vandoeuvre-lès-Nancy, France

    Dragan Grgic

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Appendix

Appendix

1.1 Procedure for the identification of the model parameters

The values of the model parameters can easily be obtained from classical laboratory tests: a set of triaxial compressive tests (i.e. with different confining pressures) at both high (0.004 MPa s−1 in our short-term experiments) and very low (5 × 10−6 MPa s−1 in our long-term experiments) loading rates and some multi-step (i.e. variable stress) creep tests.

1.2 Elastic parameters

The bulk modulus K is determined from an isotropic compressive test. The shear modulus G is determined from triaxial compressive tests at the first unloading–reloading cycle, i.e. at the beginning of the loading when the material is still undamaged.

1.3 Thresholds

The parameters Y o , η o , Y m and η m are the values of \( Y^{*} \) and \( \eta^{*} \) (Eq. 19) for initial and peak surfaces of inelasticity (Fig. 4), respectively. Within the coupled damage–plasticity scheme, η m (Eqs. 19, 20, 26) is the maximal internal cohesion of the undamaged material and has to be defined in such a way that (1 − D max)η m corresponds to the highest position of the damaged yield function (Eq. 38), with D max the maximal value of the damage variable D.

\( (\gamma^{in*} )_{{{ \hbox{max} }}} \), the maximal inelastic distortion the material can support, should be determined from uniaxial or triaxial compressive tests with very slow loading rate, as illustrated in Fig. 23a.

Fig. 23
figure 23

a Identification procedure for the parameter \( (\gamma^{{{\text{in}}*}} )_{{{ \hbox{max} }}} \) from a triaxial compressive test on iron ore. b Identification procedure for the parameters Y 1 and Y 2 from a set of triaxial compressive tests on iron ore

Full size image

The evolution law of \( w_{{{\text{d max}}}}^{\text{in*}} \) (Eq. 19), the maximal deviatoric energy, is obtained from a set of triaxial compressive tests (i.e. with different confining pressures). It is necessary to plot a graph showing the maximal deviatoric energy of each test versus the stress triaxiality ratio at failure. w 1 is the slope of this curve and w 2 is the maximal deviatoric energy at failure for a uniaxial stress path.

\( w_{{{\text{d crit}}}}^{\text{in*}} \), the deviatoric inelastic energy at the damage onset (Eq. 37), is supposed to be equal to zero. Indeed, according to our experimental results (Fig. 7), damage (i.e. micro-cracking) appears at the very beginning of inelastic flow.

1.4 Viscoplasticity parameters

As supposed in Sect. 3, the volumetric strains are principally due to damage mechanisms. In order to respect this assumption, the inelastic dilatancy coefficient β in (Eqs. 10–11) has to be equal to 0, thus inducing isovolumetric inelasticity.

The scaling parameters Y 1, Y 2 and η 1 of the frictional and cohesive strengths, Y and η (Eq. 19), respectively, have to be determined from a set of triaxial compressive tests (i.e. with different confining pressures) performed with a very slow loading rate in such a way that the maximal inelastic distortion \( (\gamma^{{{\text{in}}*}} )_{{{ \hbox{max} }}} \) is mobilised. A graph needs to be plotted showing the slope and the value at origin of the yield surface for different values of the inelastic distortion. Hence, the parameters Y 1, Y 2 and η 1 can be determined from nonlinear adjusting methods. Figure 23b illustrates the identification procedure for the parameters Y 1 and Y 2 from a set of triaxial compressive tests on iron ore.

The scaling parameters A and n of the kinetic law (Eq. 25) are generally identified from multi-step creep tests, using nonlinear adjusting methods (e.g. numerical nonlinear least squares fitting method). These parameters should be determined for low stress levels, i.e. when damage and therefore volumetric strains are not of great importance. After a complete unloading (i.e. zero stress step) and stabilization of the reverse creep, it is also possible to identify the fraction of the recoverable inelastic energy x (Eq. 21) and the scaling parameter B (Eq. 22). We found almost the same value for the scaling parameters A (Eq. 25) and C (Eq. 24). These parameters were determined from the multi-step creep test on iron ore represented in Figs. 6 and 19.

1.5 Damage-related parameters

For the determination of ξ (Eq. 35), it is necessary to plot the damage variable D (0 < D < 1) versus the ratio \( w_{\text{d}}^{\text{in*}} /w_{{{\text{d max}}}}^{\text{in*}} \). During triaxial compressive tests with constant strain rate, the former is calculated from the decrease of elastic properties (Eqs. 39–40) obtained using unloading–reloading cycles for example, whereas the latter is calculated by measuring the inelastic deformations. A linear regression on the experimental points is then necessary to obtain D max, which corresponds to ξ, for \( w_{\text{d}}^{\text{in*}} /w_{{{\text{d max}}}}^{\text{in*}} { = 1} \). Figure 24a illustrates the identification procedure for the parameter ξ from a triaxial compressive test on iron ore.

Fig. 24
figure 24

a Identification procedure for the parameter ξ = D max from a triaxial compressive test on iron ore. b Identification procedure for the parameters ϕ 1 and ϕ 2 from a triaxial compressive test on iron ore

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The parameters ϕ 1 and ϕ 2 of the damage porosity ϕ d (Eq. 41) can be identified from triaxial compressive tests with constant strain rate. Since we considered that the volumetric strains are principally due to damage mechanisms (i.e. β in = 0), ϕ d corresponds to the volumetric inelastic deformation. It is necessary to plot the volumetric inelastic deformation versus the damage variable D and make a nonlinear regression to determine ϕ 1 and ϕ 2. This is illustrated in Fig. 24b.

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Grgic, D. Constitutive modelling of the elastic–plastic, viscoplastic and damage behaviour of hard porous rocks within the unified theory of inelastic flow. Acta Geotech. 11, 95–126 (2016). https://doi.org/10.1007/s11440-014-0356-6

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