International Journal of Solids and Structuresborja/pub/ijss2018(1).pdf · 2018. 11. 23. · K.C....

International Journal of Solids and Structures 143 (2018) 218–231

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International Journal of Solids and Structures

journal homepage: www.elsevier.com/locate/ijsolstr

Hyper-elastoplastic/damage modeling of rock with application to

porous limestone

K.C. Bennett a , b , R.I. Borja

a , ∗

a Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA b Fluid Dynamics and Solid Mechanics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

a r t i c l e i n f o

Article history:

Received 3 October 2017

Revised 31 January 2018

Available online 13 March 2018

Keywords:

Rock

Damage

Microfracture

Porosity

Eshelby stress

Drucker-Prager

a b s t r a c t

Relations between porosity, damage, and bulk plasticity are examined in the context of continuum dam-

age and hyper-elastoplasticity of porous rocks. Attention is given to a thermodynamically consistent

derivation of the damage evolution equations and their role in the constitutive equations, for which the

Eshelby stress is found to be important. The provided phenomenological framework allows for volumet-

ric damage associated with pore growth to be distinguished from the isochoric damage associated with

distributed microcracks, and a novel Drucker-Prager/cap type material model that includes damage evo-

lution is presented. The model is shown to capture well the hardening/softening behavior and pressure

dependence of the so-called brittle-ductile transition by comparison with confined triaxial compression

measurements from the literature. Non-linear finite element simulations are also provided of the predic-

tion of damage within porous limestone around a horizontal borehole wall.

© 2018 Elsevier Ltd. All rights reserved.

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1. Introduction

Porous rocks exhibit microfracture and changes in porosity un-

der inelastic deformation, which affect their strength and elastic

properties. Describing their inelastic deformation by means of clas-

sical plasticity theory typically requires attributing changes in the

relative void space (porosity) to bulk plasticity, representing di-

latant and compactive volume changes (e.g., Drucker, 1957; Borja

and Tamagnini, 1998; Fossum et al., 20 0 0; Borja and Choo, 2016 ,

among others). Such models often incorporate phenomenologi-

cal hardening/softening rules (e.g., Borja and Lee, 1990; Simo and

Meschke, 1993; Borja and Tamagnini, 1998; Tamagnini and Ciantia,

2016 ). Including the evolution of elastic properties with microfrac-

ture and changes in porosity typically requires introducing either

empirical dependence of elastic moduli (and hardening moduli) on

the extent of plastic strain (cf. Borja and Lee, 1990 , among others)

or, alternatively, a ductile damage law which relates the damaged

state of the material to the extent of plastic strain (cf. Hueckel,

1976; Ortiz, 1985; Ju, 1989 , among others).

Continuum damage mechanics (CDM), generally attributed to

being first described by Kachanov (1958) , provides a description

of elastic softening ascribed to material degradation by micro-

cracking and/or void growth. The coupling of CDM with elastoplas-

∗ Corresponding author.

E-mail addresses: [email protected] (K.C. Bennett), [email protected] (R.I. Borja).

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https://doi.org/10.1016/j.ijsolstr.2018.03.011

0020-7683/© 2018 Elsevier Ltd. All rights reserved.

icity, i.e., ductile damage theory, further provides a representation

f CDM as an energy-dissipative process, allowing for a thermody-

amically consistent representation of the material’s elastic con-

titutive response dependent on its damaged state (cf. Lemaitre,

985a, 1985b; Ju, 1989 ). In ductile damage theory, material dam-

ge affects both the elastic response and the material’s strength

y identifying the damage-effective stress, which provides a physi-

ally meaningful description of how a material becomes weakened

nder microcrack/void growth (cf. Lemaitre and Desmorat, 2005 ).

ombining hyper-elastoplasticity theory with a thermodynamically

onsistent description of damage for geomaterials poses longstand-

ng challenges, especially with respect to dilation and compaction

cf. Simo and Ju, 1987; Krajcinovic, 1996; Arson, 2014 ).

Relations between Eshelby’s stress tensor ( Eshelby, 1951; 1956;

975 ) and changes to a material’s free energy state during dam-

ge have long been recognized (e.g., Rice, 1968; Maugin, 1994;

rünig, 2004 ). Recently, a description of the role of the Eshelby

tress specifically in describing the hyper-elastoplasticity of materi-

ls that undergo large inelastic changes in volume (such as geoma-

erials) was provided by Bennett et al. (2016) . The implications of

ecognizing this are examined herein with respect to ductile dam-

ge theory of porous rocks, and a novel hyper-elastoplastic duc-

ile damage framework is provided. This work, contrary to that

f Bennett et al. (2016) , explores the relevance of the Eshelby

tress in formulating ductile damage constitutive models. Notably,

he constitutive theory unifies concepts of volumetric (also known

s “dilative” or “adhesive”) damage and dilation/compaction in a


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K.C. Bennett, R.I. Borja / International Journal of Solids and Structures 143 (2018) 218–231 219

Fig. 1. Micro-fracture and porosity changes evident in an SEM image of post-

indented Woodford shale in the indented region. Details of experiments are pro-

vided in Bennett et al. (2015) .

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hermodynamically consistent framework. The present work is mo-

ivated also by scanning electron microscope (SEM) observations

y the present authors of extensive micro-fracture and changes in

icro-void space induced by nanoindentation of a Woodford shale,

s described by Bennett et al. (2015) and shown in Fig. 1 .

Observations of microfracture in cohesive geomaterials are well

ocumented in the literature ( Cuss et al., 2003; Wong and Baud,

012; Bennett et al., 2015; Abousleiman et al., 2016; Hu et al.,

017; Semnani and Borja, 2017 ). Over the years, numerous contin-

um damage models have been proposed to model this type of

amage in rocks, both isotropic (e.g., Hajiabdolmajid et al., 2002;

icard and Bercovici, 2003; Hamiel et al., 2004; Liu et al., 2016;

ran-Manh et al., 2016 ) and anisotropic damage (e.g., Chen et al.,

010; Voyiadjis et al., 2012; Arson and Gatmiri, 2012 ), including

lso application to damage zones at faults and joints ( White, 2014;

ohri et al., 2014 ). The description of porosity changes as a dam-

ge and/or healing phenomena has, further, particularly been a

opic of recent interest ( Ju et al., 2012; Zhu and Arson, 2015; Ar-

on and Pereira, 2013; Arson and Vanorio, 2015; Le Pense et al.,

016 ). Micromechanical descriptions of damage have been pro-

ided for (non-cohesive) granular ( Zhu et al., 2010 ) and cohesive

Chazallon and Hicher, 1998; Zhu et al., 2008; Guéry et al., 2008;

hu et al., 2011; Shen et al., 2012 ) geomaterials, with microme-

hanical descriptions of porosity and damage provided in the con-

ext of Modified Cam-Clay material model ( Bignonnet et al., 2016 )

nd the degradation of cemented clays ( Ju et al., 2012; Nguyen

t al., 2014 ). The instability of granular assemblies ( Andrade et al.,

008; Chang and Meidani, 2013; Misra and Poorsolhjouy, 2015b;

ang et al., 2016 ) can also be described as a damage phenomenon

cf. Misra and Poorsolhjouy, 2015a ).

The equating of damage with pore growth appears to have been

rst proposed by Gurson (1977) , and has since found much utility

y others ( Salari et al., 2004; Arson and Pereira, 2013; Lebensohn

t al., 2013; Bronkhorst et al., 2016; Bennett et al., 2018 ). Contrar-

ly, some have ascribed damage purely to microfracture without in-

uced void space (e.g., Zhu et al., 2008; Chen et al., 2010; Buech-

er and Luscher, 2014 ). Another approach that effectually com-

ines some volumetric damage with deviatoric damage contribu-

ions was proposed by Ortiz (1985) for concrete, and has since

een widely used for describing damage of rocks (e.g., Chiarelli

t al., 2003; Parisio et al., 2015 , among others), where it is assumed

nly the positive principal stresses contribute to damage. Various

lternative approaches have been proposed to account for both vol-

metric and isochoric damage separately ( Lee and Fenves, 1998;

ažant et al., 20 0 0; Clayton, 20 06; Bakhtiary et al., 2014; White,

014; Shojaei et al., 2014; Clayton and Tonge, 2015; Grgic, 2016 ,

mong others).

Herein, a kinematic and thermodynamic description is provided

elating phenomenological damage parameters to both volumetric

nd isochoric deformation attributable to changes in porosity and

icro-fracture, respectively. The role of a spatial Eshelby-like stress

ensor in the damage-constitutive equations is described, which

s shown to be equivalent to the “Eshelby-zeta” stress tensor de-

cribed by Bennett et al. (2016) , but who made no connection with

amage mechanics. The coupling of critical state and ductile dam-

ge theory is provided in terms of a novel Drucker-Prager/Damage

DP-D) material model that consists of a smooth transition into a

ressure cap of the yield surface. Two distinct mechanisms of dam-

ge are considered: (1) volumetric damage attributed to changes

n void ratio (i.e., porosity), and (2) isochoric damage attributed

o the development of microcracks—with or without an associ-

ted void space opening (i.e., exclusive of their crack aperture).

his unifies two common but disparate descriptions of contin-

um damage, where the former description does not lend itself

ell to describing a statistical distribution of microcrack orienta-

ions, e.g., as necessary for describing anisotropic damage (c.f. Ken-

chi, 1984; Krajcinovic, 1996; Voyiadjis et al., 2012 ), and the latter

oes not account for crack openings along their length (apertures).

he two types of damage are developed with attention to ther-

odynamic restrictions, in particular the evolution of volumetric

amage is shown to arise directly from thermodynamic principles,

here damage is predicted to increase under dilation and reduce

n compression. Considering that this provides the connection be-

ween ductile damage and inelastic volumetric strain, the depen-

ence of the state of damage on isochoric plastic deformation is

hen introduced through a phenomenolgical isochoric (also called

eviatoric or shear) damage parameter, which has a clear physical

eaning as a measure of the shear induced microfracture. In or-

er to examine the ability of the proposed framework to represent

ardening/softening behavior through the evolution of damage, no

ther phenomenological hardening parameters are introduced, i.e.,

he proposed framework is not a mixed hardening/damage model

ut a purely ductile damage one.

The DP-D material model is developed for finite deforma-

ions, and the model is implemented with Abaqus non-linear fi-

ite element modeling software within a UMAT material sub-

outine. Calibration to the triaxial compression (TC) tests of

ajdova et al. (2004) on Tavel limestone are provided to show

ow the model is capable of capturing observed stress-strain and

ardening/softening behavior, in particular the observed pressure

ependence of the so-called brittle-ductile transition described in

hat work. Simulation of a horizontal wellbore in porous lime-

tone as described by Coelho et al. (2005) is also conducted, and

he model predictions of damage around the wellbore are exam-

ned. We note that issues of objectivity as they relate to mesh

ize dependence and localization of damage are not addressed

erein, as the present model is meant to address a (local) con-

inuum description of damage. The need to integrate global de-

criptions of damage and fracture (e.g., Clayton and Knap, 2015;

i et al., 2015; Zhang et al., 2016 ) with continuum damage the-

ry (cf. Chen et al., 20 0 0; Gao and Huang, 2003; Bazant, 2010 )

emains an issue of much present research effort (e.g., Verhoosel

t al., 2011; Li et al., 2015; Heyden et al., 2015; Weed et al.,

017; Tjioe and Borja, 2016 ), and is beyond the scope of this

resent work. We also note that the present work is restricted

o material isotropy, while extension to anisotropic ductile dam-

ge is part of ongoing research effort s. We refer to the works

f Nova (1980) ; Cazacu et al. (1998) ; Semnani et al. (2016) ;

iang et al. (2017) among others for a description of the

nisotropic plasticity of geomaterials and Al-Rub and Voyiad-

is (2003) ; Kondo et al. (2010) ; Chen et al. (2010) among others

or a description of anisotropic damage coupled with anisotropic

lasticity.

220 K.C. Bennett, R.I. Borja / International Journal of Solids and Structures 143 (2018) 218–231

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2. Theory describing damage, porosity, and micro-fracture

In the following, a description of the kinematics and adopted

notation is first given. The description is made in as con-

cise a manner as possible since the emphasis of the work is

the development of the damage-constitutive model. We refer to

Bennett et al. (2016) for further kinematic and thermodynamic

background relevant to the current work. Throughout, tensors are

written in boldface and their components are specified by sub-

script index notation. Summation convention is assumed for re-

peated indices. Matrix notation for the inner product is assumed,

whereas dyadic products are indicated by �. The double contrac-

tion of indices are indicated by double dots (: ). For example, the

second order identity tensor is represented as 1 and its compo-

nents by the Kronecker delta δij , such that the trace of a second

order tensor is given by tr [ (•) ] := (•) : 1 = δi j (·) i j . We make use

of similar standard notion for the natural log ln ( · ), the deviatoric

operator dev [ (•) ] := (•) − 1 / 3 tr [ (•) ] 1 , and the symmetric opera-

tor sym [(•)] := 1 / 2[(•) + (•) T ] , where the T superscript denotes

the transpose.

The concept of a representative elementary volume (REV) is as-

sumed, such that the material properties at a point describe the

microstructure, including herein microcracks and also void space.

Typically, fluids may fill the voids with time dependent dissipation

giving rise to effective stresses ( Biot, 1941; Terzaghi, 1944; Borja,

2006 ). All stresses described herein are always considered to be

in this regard effective stresses, and we note that the model pre-

sented here could be applied to coupled fluid/solid problems with

the addition of fluid phase diffusion laws (e.g., Choo and Borja,

2015; Wang and Sun, 2016 ).

We adopt standard mechanics sign convention, where, for ex-

ample, positive mean stress in compressive; however, when com-

paring to measurements where opposite (geomechanics) conven-

tion is used we sometimes report negative values in order to

match those figures and also adopt, for example, the convention

for positive “axial strain” to imply compression. Conventional no-

tation for the first stress invariant I (•) 1

:= tr [ (•) ] and the second

deviatoric stress invariant J (•) 2

:= 1 / 2 dev [ (•) ] : dev [ (•) ] is adopted,

where here ( •) denotes a stress measure. The mean stress (of any

particular stress measure) is then defined by p (•) := 1 / 3 I (•) 1

, and

the Mises stress as q (•) :=

√

3 J (•) 2

. Consistent with convention, un-

specified description of “mean stress” or “Mises stress” implies

with respect to Cauchy stress.

The deformation gradient is a function of position and time,

given by

F :=

ˆ F ( X , t) =

∂ ϕ ( X , t)

∂ X

, (1)

where x = ϕ ( X , t) describes a deformation from the reference to

current configurations of the material, i.e., ϕ t : B 0 → B t ⊂ R

3 . The

multiplicative split of the deformation gradient into elastic and

plastic parts is assumed, such that F = F e F p . The left Cauchy-Green

deformation tensor (Finger tensor) of B t is given by

b := F F T ; b i j := F iK F jK , (2)

and the logarithmic strain (Hencky strain) by

ε :=

1

2

ln [ b ] ; ε i j :=

1

2

ln [ b i j ] . (3)

The natural volume strain is then defined as e v := ε : 1 = ln [ J] =e e v + e

p v such that the natural deviatoric strain is ε := ε − 1 / 3 e v 1 ,

where J := det [ F ] = J e J p is the Jacobian determinant. This allows

for a representation of large strain elastoplasticity that follows

closely the small strain version, as has been described by Simo

(1992, 1998) and others. Note that intermediate configuration

quantities are represented with an over-bar, current configuration

n lower-case, and reference configuration in upper-case, whenever

ossible. For example infinitesimal volumes are mapped by the Ja-

obian determinant, J , by dv = J e d V and d V = J p dV, where J = J e J p .

Two scalar variables of damage, d

c and d

s , are distinguished, as-

ociated with volumetric and isochoric damage, respectively. The

olumetric part d

c , which we call also the comprehensive dam-

ge parameter, is isotropic because it is directly related to porosity,

hich cannot be given a statistically preferential direction or align-

ent according to standard REV theory (cf. Hill, 1963; Borja, 2006 ).

n advantage of the present model is that because it separates the

olumetric from the isochoric contributions to the overall damage,

t allows for future work to consider the isochoric damage to be

nisotropic. This is significant because an anisotropic damage ten-

or can then be assembled from the statistical distribution of mi-

rocracks when the cracks are considered to have no opening by

he method first proposed by Lubarda and Krajcinovic (1993) (see

lso Voyiadjis et al., 2012 ), which is analogous to the development

f the micro-mechanical fabric tensor used to represent the distri-

ution of particle contact planes in granular assemblies (e.g., as de-

cribed in Chang and Bennett, 2015 ). We emphasize, however, that

he present work considers only an isotropic distribution of micro-

racks, allowing for the isochoric damage to be represented by a

calar variable. Subsequently, it will be shown that d

s is taken as a

on-linear function of its conjugate energy release rate y s , similar

o the general form proposed by Lemaitre and Chaboche (1994) ,

s =

ˆ d

s (y s ) . The terms “deviatoric damage” and “shear damage”

re variably used in the literature for what we call here “iso-

horic damage,” and we use these terms interchangeably as

ell.

The porosity ( n ) is the ratio of the volume of the voids to the

otal volume in the current configuration, i.e, n := dv v / dv , where

he total volume is the sum of the solid and void volumes, i.e.,

v = dv s + dv v . Changes in porosity result from both plastic and

lastic deformation; however, it is convenient to define the poros-

ty relative to the plastically-deformed/elastically-unloaded state,

.e., the intermediate configuration B . In this case, the definition

f porosity becomes n := d V v /d V . This is the definition of poros-

ty used in this work, which is the assumption that change in

orosity is attributed to plastic volumetric deformation, and plas-

ic volumetric deformation is likewise attributed solely to changes

n porosity. This assumption is analogous to the assumption that

e ≈ 1, which is reasonable if the elastic volume change is negligi-

le in relation to inelastic volume change, for which case n ≈ n .

e note that this assumption is commonly justifiable for geo-

aterials. For example, consider triaxial compression tests where

nitial and final specimen porosities (before loading and after

nloading) can differ significantly relative to the elastic volume

train exhibited during the test. A further justification for this as-

umption is that the solid grains are expected to exhibit elas-

ic volumetric deformation but not plastic. Hence, all of the plas-

ic deformation is reasonably attributed to changes in porosity,

ut the separation of bulk elastic deformation into that of the

olid and that of the void fractions is not obvious without mak-

ng some other assumptions (or perhaps higher resolution model-

ng, e.g., micromechanical modeling). With this definition of poros-

ty, the plastic Jacobian determinant, J p , is related to porosity

y

p =

ρ0

ρ=

d V

dV

=

d V

s + d V

v

d V

s + d V

v =

d V − d V

v + d V

v

dV

= 1 − n 0 +

d V

v

dV

,

(4)

here because of near-incompressibility of the solid constituent,

v s = d V s = d V s , and the subscript 0 is used throughout to de-

ote initial (or reference) values. Similarly, the definition of poros-

ty provides


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T

d

d

:=

dv v

dv ≈ d V

v

d V

=

d V − dV

s

d V

=

d V − dV + dV

v

d V

=

d V − dV + f 0 dV

d V

= 1 − 1 − n 0

J p . (5)

ince we should expect that damage is at least in part attributable

o changes in porosity (see references in the introduction), we

dentify that there is a volumetric damage parameter proportional

o porosity by α, i.e., d

c = αn . It is noted that this relationship

ould in general be nonlinear, where, for example, α could depend

n the plastic strain history. However, we do not need to specify αxplicitly here nor make any assumptions about how d

c is related

o the total damage, only recognize the proportionality for consid-

ration in Section 3 ; the precise form will be determined by the

hoice of constitutive equations in Section 4 , where an expression

or d

c as a function of volumetric plastic deformation and model

arameters is obtained. The expression for the proportionality be-

ween volumetric damage and J p can thus be written as

c = αn = α(

1 − 1 − n 0

J p

), (6)

r in other words,

p =

1 − n 0

1 − d

c /α. (7)

he relation between J p and d

c in Eq. (7) is remarkably similar to

he relation often used in modeling spall damage of metals at large

trains, which appears to be first obtained by Davison et al. (1977) ,

he difference being in only the scaling factor α and the initial

orosity 1 The relation between changes in damage and porosity is

hen found to be given by

d

c = α�n = α(n − n 0 ) = α(1 − n 0 ) (

1 − 1

J p

). (8)

q. (8) provides an explicit expression for the plastic Jacobian de-

erminant in terms of the damage increment when volumetric

amage is related to volumetric plasticity,

p =

1

1 − �d c

α(1 −n 0 )

. (9)

e emphasize that this relationship is based only on the classi-

al assumption that there is some proportionality between damage

nd porosity and that we do not require α to be a fixed material

arameter in the derivation of our material model in what follows.

t will be shown in Section 5 that the proposed model can be used

o solve for α in Eq. (8) by providing { J p , d

c }.

. Thermodynamic consistency

The second law of thermodynamics, neglecting temperature ef-

ects, can be expressed in the form of the Clausius-Planck inequal-

ty (cf. Truesdell and Noll, 2004; Simo, 1998 ),

B t

(d : σ − ρ ˙

)dv ≥ 0 , (10)

here d = sym [ F F −1 ] is the deformation rate, σ is the Cauchy

tress, ρ is density, and is the specific Helmholtz free-energy.

ennett et al. (2016) have shown that for a material undergoing

1 Equivalence to the spall damage expression of Davison et al. (1977) is obtained

y setting α = 1 in the absence of initial porosity.

w

l

d

ulk plasticity this implies that the hyperelastic constitutive equa-

ion for the stress is given for the current configuration represen-

ation of elastoplasticty in terms of the so-called “zeta” stress, ζ,

y

:= J e σ = 2

∂( ρ)

∂ b

e b

e . (11)

n order for a material model to be thermodynamically consistent,

t was shown in Bennett et al. (2016) that the plastic dissipation

D) must satisfy

= ξ : d

p − q : ˙ z ≥ 0 , (12)

here d p is the plastic part of the deformation rate d = d p + d e , z

nd q are respectively vectors of internal state variables (ISV’s) and

heir conjugate thermodynamic forces, and

:= J e σ − ( ρ) 1 = ζ − ( ρ) 1 (13)

s an Eshelby-like stress measure we call the Eshelby-zeta stress.

In this work, the free-energy is assumed to consist of only

he elastic stored energy and internal state variables associated

ith damage (e.g., there is no additional hardening potential). It

s taken to be of the functional form,

=

ˆ ( b

e , d

c , d

s ) , (14)

here b e is the elastic Finger tensor, and d

c and d

s are ISV’s de-

cribing the damaged state of the material. The dissipation inequal-

ty hence takes the form

= ξ : d

p − y c ˙ d

c − y s ˙ d

s ≥ 0 , (15)

here y c = ∂ d c ( ρ) and y s = ∂ d s ( ρ) are the energy release rates

ssociated with comprehensive (i.e., reduction of the total free-

nergy) and purely deviatoric damage (i.e., microfractures), respec-

ively.

Under the supposition of non-associated flow, the dissipation

otential G and yield function F are taken to be of the forms, re-

pectively,

:=

ˆ G ( ξ, y c , y s ; d

c , d

s ) ; F :=

ˆ F ( ξ, y c , y s ; d

c , d

s ) , (16)

ith the dissipation potential taken to be of the additive form

: = G p ( ξ, y c ; d

c , d

s ) + G d (y c , y s ; d

c , d

s )

= G p ( ξ, y c ; d

c , d

s ) + G c (y c ; d

c ) + G s (y s ; d

s ) , (17)

here G p is the (non-associative) plastic potential and the G d =

c + G s is the damage potential, assumed to be additively decom-

osed into comprehensive and deviatoric parts, G c and G s , respec-

ively. Although additive decomposition into plastic and damage

arts is typical (cf. Lemaitre and Desmorat, 2005 ), the assumption

hat the damage potential can be further additively decomposed

s justified by recognizing the functional dependence of the plas-

ic potential on y c due to the Eshelby-zeta stress (but not y s ). This

otivates the concept adopted here that G c can be somehow re-

ated to G p . In anticipation of providing a damage evolution equa-

ion consistent with Eq. (6) , we take the form of this relation to be

c := c αG p . (18)

his provides a convenient canonical form of the comprehensive

amage evolution equation in terms of the plastic potential,

˙

c = − ˙ λ∂G c ∂y c

= − ˙ λc α∂G p ∂y c

, (19)

here ˙ λ is the plastic multiplier. Similarly, the shear damage evo-

ution equation is given by

˙

s = − ˙ λ∂G s ∂y s

, (20)


Table 1

List of damage state variables and associated symbols.

Type Symbol Definition

Damage internal variables d c volumetric damage variable

d s shear damage variable

Damage functions �c volumetric damage function

�s shear damage function

ω

c := �c −1 inverse of volumetric damage function

ω

s := �s −1 inverse of shear function

Energy release rates y c volumetric energy release rate

y s shear energy release rate

Damage potentials G d = G c + G s total damage potential

G c volumetric damage potential

G s shear damage potential

T

T

o

ζ

w

ζ

N

s

ξ

T

t

y

w

p

t

a

c

E

ξ

s

ξ

w

ζ

W

4

f

F

T

e

s

t

d

t

e

and the flow rule by

d

p =

˙ λ∂G p ∂ ξ

. (21)

Along with the standard so called KKT and consistency conditions

(cf. Borja, 2013 ), this provides the complete canonical form of the

(hyper-elastoplastic/damage) constitutive and evolution equations.

4. Modeling choices and implementation

The material model presented in this section can be called

a Drucker-Prager/Damage (DP-D) model. It is loosely based on

the Drucker-Prager type model first presented by Regueiro and

Ebrahimi (2010) and further modified by Bennett et al. (2016) . This

model incorporates a two-invariant Drucker-Prager yield surface

with a smooth transition to a pressure cap (DP-C), and has been

further developed herein (incorporating damage) especially for the

purpose of modeling the constitutive behavior of rocks. It is noted,

however, that Drucker-Prager type models inclusive of a pressure

cap have found broad application in the modeling of porous ma-

terials in general, and the model as presented here may be sim-

ilarly extendable (with proper selection of model parameters). It

is emphasized that the relationship that is described in the fol-

lowing between the damage-energy release rate and the Eshelby-

zeta stress of Eq. (13) provides a useful connection between en-

forcing thermodynamic consistency and providing a description of

the hardening/softening behavior of a porous material.

4.1. Helmholtz free energy

The free-energy function is taken to depend on the state

of damage described by the scalar damage variables d

c and d

s

through the damage functions �c =

ˆ �c (d

c ) and �s =

ˆ �s (d

s ) , re-

spectively. It is considered to be additively composed of volumet-

ric and deviatoric elastic stored energy potentials in the functional

form

=

ˆ (e e v , ε

e , d

c , d

s )

= �c (d

c ) ·(

˜ v (e e v ) + �s (d

s ) · ˜ s ( ε

e ) ). (22)

The damage functions 0 < �c ≤ 1 and 0 < �s ≤ 1 range between

1 for the undamaged state and approach 0 for the completely

damaged state of the material. Consistent with the principle

of strain equivalence ( Lemaitre, 1985a, 1985b ), damage-effective

stress and the corresponding damage-effective free-energy are

identified. Damage-effective quantities are represented with tilde

notation, such that the per unit volume damage-effective volumet-

ric and deviatoric elastic stored energy potentials are given respec-

tively by

ρ ˜ v (e e v ) :=

K 0

2

e e 2

v ; ρ ˜ s ( ε

e ) := μ0 ε

e : ε

e , (23)

where K 0 and μ0 are the damage-effective bulk and shear moduli,

respectively. The damage-effective free-energy per unit volume is

then expressed by

ρ ˜ =

K 0

2

e e 2

v + μ0 ε

e : ε

e . (24)

The damage functions could in general be non-linear, but are taken

for the sake of simplicity and clarity here as linear,

�c := (1 − d

c ) ; �s := (1 − d

s ) , (25)

for 0 ≤ d

c < 1 and 0 ≤ d

s < 1. It is convenient to also define (consis-

tent with standard notation)

ω

c := �c −1 =

1

(1 − d

c ) ; ω

s := �s −1 =

1

(1 − d

s ) . (26)

he comprehensive damage-effective free-energy is defined as

˜ c := ω

c . (27)

he stress is then found from the hyperelastic constitutive relation

f Eq. (11) to be given by

= (1 − d

c ) ·(K 0 e

e v 1 + (1 − d

s )2 μ0 ε

e ), (28)

ith the damage-effective stress thus given by

˜ = ω

c ω

s dev [ ζ] + ω

c p ζ 1 = K 0 e e v 1 + 2 μ0 ε

e , (29)

ote too that the expression for the damage-effective Eshelby-zeta

tress is

˜ = ω

c ω

s dev [ξ]

+ ω

c p ξ 1

= dev [

˜ ζ]

+ ( p ζ − ρ ˜ c ) 1 . (30)

he expressions for the energy release rates are found according to

heir definitions within Eq. (15) to be

c =

∂( ρ)

∂d

c = −ρ ˜ c ; y s =

∂( ρ)

∂d

s = −(1 − d

c ) ρ ˜ s , (31)

hich completes the definition of all damage state variables and

otentials. A list of the damage state variables and their defini-

ions is provided in Table 1 . Having defined the damage state vari-

bles, the expressions for the stress and damage-effective stress

an be identified. Making use of the definition in Eq. (13) with

qs. (27) and (31) , the Eshelby-zeta stress is expressed as

= ζ + (1 − d

c ) y c 1 , (32)

uch that

˜ =

˜ ζ + y c 1 , (33)

hich implies

˜ =

˜ ξ − y c 1 . (34)

ithin the described framework, we note the following relations:

˜ p ζ =

˜ p ξ − y c ; ˜ q ζ ≡ ˜ q ξ . (35)

.2. Yield function and plastic potential

The Drucker-Prager/cap (DP-C) type yield function is taken as a

unction of the hyperelastic damage-effective stress ˜ ζ = J e ˜ σ, i.e.,

:=

ˆ F ( ζ) . (36)

his form is convenient because it both provides a bound for the

lastic domain E , i.e., E := { ζ ∈ S | F( ζ) ≤ 0 } , and is tenable in the

ense that its parameters can be associated with material proper-

ies obtained from measurements (e.g., friction angle, preconsoli-

ation stress, etc.). However, the yield function is required to be of

he functional form described by Eq. (16) in order to find the nec-

ssary derivatives described within Section 3 . Fortuitously, we can


Fig. 2. DP-D yield surface in ˜ q vs. −3 p damage-effective stress invariant space,

showing intersection with first stress invariant axis at X φ .

s

i

o

o

(

w

A

F

〈

X

(

h

n

t

t

o

t

s

t

e

C

n

b

f

t

l

E

G

Table 2

List of DP-D model parameters.

Type Parameter Definition

Elastic K 0 damage-effective bulk modulus

ν0 damage-effective Poisson’s ratio

Yield surface ˜ β1 damage-effective cohesion stress parameter ˜ β2 damage-effective preconsolidation stress parameter ˜ φ damage-effective friction angle ˜ ψ damage-effective dilation angle

R cap ellipticity parameter

Damage c t dilative damage parameter

c c compressive damage parameter

c s shear damage parameter

Table 3

Model parameters calibrated to measurements of Vajdova et al. (2004) .

Type Parameter Value Units Maximum variation

Elastic K 0 29.6 GPa ± 7.1%

ν0 0.12 - ± 3.0%

Yield surface ˜ β1 140.0 MPa ± 14.2% ˜ β2 -5.5 MPa ± 7.1% ˜ φ 0.6 - ± 7.1% ˜ ψ 0.16 - ± 7.1%

R 5.7 - ± 0%

Damage c t 10 0 0 - ± 15%

c c 55 - ± 20%

c s 55 - ± 36%

w

E

e

G

T

m

m

m

e

a

w

f

p

t

e

I

t

b

S

S

w

s

a

c

w

d

e

b

fi

m

ee by examining Eqs. (34) and (35) that this expression is read-

ly available, since F( ζ) ≡ F( ξ − y c 1 ) . Performing the substitution

f p ξ − y c for p ζ provides the alternative but equivalent expression

f the DP-C yield criteria, which we call a Drucker-Prager/Damage

DP-D) yield criteria:

F := ‖ dev [

˜ ζ]‖

2 − F φ

cap

(A

φ ˜ β1 − B

φ ˜ p ζ)2

= ‖ dev

[ ˜ ξ] ‖

2 − F φ

cap

(A

φ ˜ β1 − B

φ( p ξ − y c ) )2 ≤ 0 ,

(37)

here,

φ :=

2

√

6 cos ˜ φ

3 + r sin

˜ φ, B

φ :=

2

√

6 sin

˜ φ

3 + r sin

˜ φ, −1 ≤ r ≤ 1 , (38a)

φcap := 1 − 〈 β2 − 3( p ξ − y c ) 〉 ˜ β2 − 3( p ξ − y c )

(X

φ − ˜ β2 ) 2 , (38b)

β2 − 3( p ξ − y c ) 〉 :=

1

2

[| β2 − 3( p ξ − y c ) | + ( β2 − 3( p ξ − y c )) ],

(38c)

φ :=

˜ β2 − R

(A

φ ˜ β1 − B

φ ˜ β2

). (38d)

The yield function is defined in damage-effective stress space

see Fig. 2 ). The material parameters of the yield function are

ence damage-effective parameters, and are so denoted with tilde

otation. The parameters ˜ φ and

˜ β1 are the (damage-effective) fric-

ion angle and the cohesion parameter, respectively. The parame-

er ˜ β2 is associated with the preconsolidation stress, the position

f the cap along the ˜ I ζ1

= 3 p ζ axis being given by X

φ . The parame-

er R controls the ellipticity of the cap, and the shape of the yield

urface on the octahedral plane is controlled by −1 ≤ r ≤ 1 , such

hat r = 1 and r = −1 coincide with the intersection of the triaxial

xtension (TE) and triaxial compression (TC) corners of the Mohr–

oulomb yield surface, respectively. The Frobenius norm is de-

oted by ‖ ( •) ‖ , |( · )| is the absolute value, and 〈 · 〉 is the Macaulay

racket. For non-associative plasticity, we have similar functional

orm for the plastic potential function G p as for the yield func-

ion F , the only difference being that the damage-effective di-

atation angle ˜ ψ replaces the damage-effective friction angle ˜ φ in

q. (37) such that

p := ‖ dev

[ ˜ ξ] ‖

2 − F ψ

cap

(A

ψ ˜ β1 − B

ψ ( p ξ − y c ) )2

, (39)

here the superscripts denote ˜ ψ is used in place of ˜ φ in

qs. (38a) –(38d) .

The shear damage potential is chosen to be similar to the gen-

ral form proposed by Lemaitre and Chaboche (1994) ,

s :=

Sω

s

c α(s + 1)

(−c αω

c y s

S

)s +1

. (40)

he positive material parameters S and s can generally be deter-

ined from calibration with measurements for various types of

aterials (e.g., as described in Lemaitre and Chaboche, 1994 ). The

aterial parameter c α is added here to the expression in consid-

ration of keeping proportionality with the comprehensive dam-

ge rate ( Eq. (19 )). The form of Eq. (40) is convenient because it is

idely used and has been shown to be applicable to many dif-

erent types of materials under varying loading conditions with

roper choice of the material parameters. In adapting this equa-

ion for the modeling of damage in porous rock, it is necessary to

xamine the choice of material parameters carefully here as well.

n particular, two distinctive damage material behaviors of geoma-

erials are considered: (i) pressure dependence, and (ii) differences

etween dilative and compressive regimes. In order to address (i),

is taken to be pressure dependent by

:= c −p/p c s , (41)

here c s is a positive material constant, p is the mean Cauchy

tress, and p c is the (Cauchy) preconsolidation stress. In order to

ddress (ii), c α is defined as

α :=

(1 − 〈 sgn [ e p v ] 〉

)c c + 〈 sgn [ e p v ] 〉 c t , (42)

here c c and c t are positive material constants for compaction and

ilation, respectively. Table 2 lists all model parameters.

Having now explicit expressions for the potential functions, the

xpressions for the evolution equations provided in Section 3 can

e found. For expressing the flow rule, it is convenient to first de-

ne

ˆ :=

∂G p ∂ ξ

=

∂G p

∂ dev [ξ] +

1

3

∂G p ∂ p ξ

1 = 2 ω

c ω

s dev

[ ˜ ξ] + ω

c C ψ 1 , (43)


h

m

T

d

w

q

A

w

5

o

c

l

b

d

V

D

a

i

e

s

p

a

T

s

p

p

a

s

g

A

n

i

i

F

a

5

t

T

m

c

t

d

h

w

where

C ψ := 2(A

ψ ˜ β1 − B

ψ ˜ p ζ )

[

F ψ

cap B

ψ

3

− (A

ψ ˜ β1 − B

ψ ˜ p ζ ) 〈 β2 − 3

p ζ 〉 (X

ψ − ˜ β2 ) 2

]

.

(44)

The flow rule of Eq. (21) is then expressed as

d

p =

˙ λ ˆ m . (45)

The deviatoric and volumetric parts of the plastic deformation rate

can be distinguished as

dev [d

p ]

= 2

λω

c ω

s dev [

˜ ζ], vol

[d

p ]

= 3

λω

c C ψ . (46)

Similarly, the damage rates of Eqs. (19) and (20) are found to be

expressed by

˙ d

c = 3 c α ˙ λC ψ , ˙ d

s =

˙ λω

s ω

c

(−c αω

c y s

S

)s

. (47)

The intermediate equations necessary for finding these solutions

are provided in Appendix A .

4.3. Implicit integration

Backward Euler implicit integration making use of the exponen-

tial map is utilized in order to obtain the integrated flow rule in

terms of the Hencky strain (cf. Simo, 1992; 1998 , among others).

This allows the classical predictor/corrector return mapping algo-

rithm of the classical small strain theory to be preserved (cf. Borja

and Tamagnini, 1998; de Souza Neto et al., 2011 , among others).

The detailed procedure involving the exponential map is not re-

peated here, as it has been well documented in the literature, and

we refer also to the details provided in Bennett et al. (2016) for a

description of the kinematics relevant to the current work.

The integrated flow rule is hence given by

ε

p n +1

= ε

p n + �λ ˆ m n +1 , (48)

which provides the elastic strain at a given increment of total

strain �ε (i.e., the predictor/corrector equation),

ε

e n +1 = ε

e,tr n +1

− �λ ˆ m n +1 , (49)

where the superscript ( •) tr denotes a trial value, and is for the elas-

tic strain taken to be at each time step as ε e,tr n +1

= ε e n + �ε . The ex-

pressions for the volumetric and deviatoric elastic strains can also

be respectively expressed by

e e v ,n +1 = e e,tr v ,n +1

− 3�λω

c n +1 C

ψ

n +1 , (50)

and

ε

e n +1 = ε

e,tr n +1 − 2�λω

c n +1 ω

s n +1 dev

[˜ ζ]. (51)

It is useful also to define

ε e s := ‖ ε

e ‖ , (52)

such that

ε e s,n +1 = ε e,tr s,n +1

− 2�λω

c n +1 ω

s n +1 ‖ dev

[˜ ζ]‖ . (53)

The update equations for the damage variables are found from

Eq. (47) , making use of Eqs. (50) and (53) to find

d

c n +1 = c α

d

c n + �e p v

1 + �e p v ; d

s n +1 = d

s n +

3

2

√

6

�ε p s

˜ q ζn +1

(−c αω

c n +1 y

s n +1

S

)s

.

(54)

Having the updated elastic strains allows the updated stress to

be calculated according to the constitutive equation (28) . We note

ere also the update equations for the mean and deviatoric stress

easures. The mean stress update is given by

p ζn +1

= (1 − d

c n +1 ) ·

(K 0 e

e v ,n +1

)= (1 − d

c n +1 ) · K 0

(tr [ε

e,tr n +1

]− 3�λω

c n +1 C

ψ

n +1

)=

˜ p ζ ,tr n +1

/ω

c n +1 − 3 K 0 �λC ψ

n +1 . (55)

he deviatoric stress update is similarly found as

ev [ξn +1

]≡ dev

[ζn +1

]= (1 − d

c n +1 ) · (1 − d

s )2 μ0 ε

e n +1

= (1 − d

c n +1 ) · (1 − d

s )2 μ0

(´ε

e,tr n +1 − 2�λω

c n +1 ω

s n +1 dev

[˜ ζn +1

]).

=

dev

[ ˜ ζ

tr

n +1

] ω

c n +1

ω

s n +1

(1 + 4 μ0 �λω

c n +1

ω

s n +1

) , (56)

hich provides (see also further details provided in Appendix A ),

ζn +1

= (1 − d

c n +1 ) · ˜ q

ζ ,tr n +1

/(1 + 4�λω

c n +1 ω

s n +1 ) . (57)

n iterative solution of the integrated equations is required. In this

ork we employ a Newton–Raphson iteration.

. Comparison of model predictions with measurements

The model is evaluated by calibration and comparison to lab-

ratory and in-situ measurements. Two different materials and

orresponding sets of measurements are considered: (1) Tavel

imestone after Vajdova et al. (2004) , and (2) a horizontal well-

ore through 30% porosity limestone off the coast of Brazil as

escribed by Coelho et al. (2005) . The experimental data of

ajdova et al. (2004) were chosen as a first example for the DP-

model calibration for a number of reasons: it consists of tri-

xial compression (TC) at a wide range of confining pressures,

t includes isotropic compression data, and it is widely refer-

nced as representative for describing the pressure dependent

o called “brittle–ductile transition” and “shear enhanced com-

action/dilation” behavior of rocks ( Rudnicki and Rice, 1975; Borja

nd Regueiro, 2001; Wong and Baud, 2012; Tjioe and Borja, 2015 ).

he wellbore study was chosen because it is accompanied by sub-

tantial site-characterization measurements and is through high

orosity limestone. The model parameters were calibrated inde-

endently to each of the materials and corresponding data sets,

nd the calibration methods along with calibrated values are pre-

ented independently in each of the following sub-sections.

It is noted that the shear stress intercept in q vs. p space is

iven by q ′ = 2 c ′ , where c ′ is the Mohr–Coulomb (MC) intercept.

lthough the MC friction angle is a parameter of the model, it

eeds to be evaluated in damage-effective space to do so. Examin-

ng Eq. (29) , it can be seen that the damage-effective friction angle

s related to the (measured) friction angle by ˜ φ = tan

−1 (ω

s tan φ) .

or each calibration procedure, the initial yield surface is provided

long with the corresponding damage-effective yield surface.

.1. Tavel limestone triaxial compression simulations

The DP-D model was calibrated to the triaxial compression (TC)

est data on Tavel limestone presented by Vajdova et al. (2004) .

he isotropic compression data allowed the damage-effective bulk

odulus K 0 , the preconsolidation stress parameter ˜ β2 and the

ompressive hardening parameter c c to be determined from the

est data (see Fig. 3 ). The TC measurements also allowed for the

etermination of the Mohr-Coulomb (MC) friction angle φ and co-

esion intercept c ′ (corresponding to the β1 parameter), which

ere directly reported by the authors as c ′ = 60 MPa and φ = 27 ◦,


Fig. 3. Calibration (solid line) of bulk modulus K , preconsolidation pressure p c , and

compression damage parameter c c to isotropic compression test data (symbols) of

Vajdova et al. (2004) .

Fig. 4. Drucker-Prager/Damage initial yield surface calibrated to measurements

showing load path for 100 MPa confining pressure (solid line) and damage-effective

stress load path (dashed line), as well as damage-effective yield surface (dotted).

r

t

i

p

s

i

k

t

c

a

e

S

m

i

i

d

s

h

t

c

i

l

s

s

Table 4

Variation of parameters from those reported in Table 3 required to best fit measure-

ments of Vajdova et al. (2004) .

Confining pressure Parameter(s) Variation

10 MPa K 0 , β2 + 1.4%

ν0 , φ, ˜ ψ + 1.4%

20 MPa ν0 , φ -3.0%

30 MPa K 0 , β2 -1.4%

ν0 , φ, ˜ ψ + 3.0%

50 MPa K 0 , β2 -4.5%

ν0 , φ, ˜ ψ + 2.9%

c t -15%

c s -9.1%

100 MPa K 0 , β2 -7.1%

ν0 , φ, ˜ ψ -7.1%

150 MPa ν0 , φ, ˜ ψ -2.9%

c c + 20%

c s + 36%

200 MPa ν0 , φ, ˜ ψ + 2.9%

240 MPa ν0 , φ, ˜ ψ + 2.9%

Table 5

Least squares linear regressions of curves in Fig. 7 for back calculating α.

Confining pressure Slope Coefficient of determination α

10 MPa 635.649 0.998975 709.430

20 MPa 602.050 0.997152 671.931

30 MPa 647.793 0.997677 722.983

50 MPa 604.175 0.994839 674.302

100 MPa 46.045 0.999199 51.390

150 MPa 60.165 0.994861 67.148

200 MPa 50.458 0.995565 56.315

240 MPa 52.437 0.994642 58.523

Table 6

DP-D model parameters established from mea-

surements of Coelho et al. (2005) .

Type Parameter Value Units

Elastic K 0 571.0 MPa

ν0 0.15 -

Yield Surface ˜ β1 25.0 MPa ˜ β2 -30.0 MPa ˜ φ 0.42 - ˜ ψ 0.1 -

r 0 -

R 2 -

Damage c t 10 -

c c 10 -

c s 40 -

1

t

a

a

c

f

q

u

u

d

d

i

(

V

o

u

d

s

espectively. The damage parameter c t was calibrated to best fit

he TC measurements that were on the dilative side (at low confin-

ng pressure), and the c s parameter calibrated to the high confining

ressure curves. Fig. 4 shows the initial DP-D yield surface along-

ide the damage-effective yield surface. A critical aspect of calibrat-

ng the model was to adjust the ellipticity parameter R in order to

eep the brittle–ductile transition point (peak of the plastic po-

ential surface in p –q space) near the intersection of the 100 MPa

onfining pressure stress paths, consistent with the measurements

nd description provided by the authors (op. cit.). A low damage-

ffective dilation angle was chosen as typical of rock ˜ ψ = 0 . 16 .

ome minor adjustments of parameters were needed to best fit the

easurements, while attempting to keep as many fixed as possible.

A comparison of the model simulations with the measurements

s provided in Fig. 5 . All calibrated model parameters are reported

n Tables 3 , and 4 reports variations of model parameters for in-

ividual tests necessary to best fit the measurements. The results

how that the model is able to simulate well both the distinctive

ardening and softening evident in the measurements, as well as

he pressure dependent transition between these regimes, the so-

alled brittle-ductile transition. We emphasize that this behavior

s modeled as changes in porosity and microfracture through evo-

ution of the damage state variables in a thermodynamically con-

istent way made possible by recognizing the role of the Eshelby

tress tensor as described in Sections 2 and 3 .

Evolution of damage parameters is provided for the 50 MPa and

00 MPa confining pressure simulations in Fig. 6 . These tests mark

he transition from brittle to ductile behavior as described by the

uthors ( op. cit. ) and apparent in Fig. 5 . The comprehensive dam-

ge parameter, d

c , in the 50 MPa simulation transitions from de-

reasing to increasing, which roughly corresponds to the transition

rom hardening to softening apparent in the 50 MPa stress ratio

/ p curve on the right side ( Fig. 6 b); whereas, the 100 MPa sim-

lation predicts only increasing d

c . Note that negative total vol-

me strain is plotted, showing that the 50 MPa specimen is pre-

icted to dilate during shear and the 100 MPa specimen is pre-

icted to contract. Recalling that d

c is proportional to the poros-

ty through the plastic part of volume strain only (see Eqs. (6) and

54) ), the 50 MPa d

c curve exhibits the phenomenon described by

ajdova et al. (2004) (see also Lubarda et al., 1996 , among others)

f an initial decrease in porosity followed by a subsequent increase

nder dilation. The figure shows that the shear damage parameter,

s , is predicted to monotonically increase in both simulations, but

ignificantly more in the 50 MPa simulation, where softening is ev-


Fig. 5. Comparison of model simulations (solid lines) with measurements (symbols) of Tavel limestone in triaxial compression ( Vajdova et al., 2004 ) with confining pressures

indicated on the figures.

Fig. 6. Model predictions of change in damage during simulated TC at confining pressures of 50 MPa and 100 MPa: (a) change in comprehensive damage, �d c , alongside

(negative) volume strain, e v , and (b) change in shear damage, d s , alongside (negative) stress ratio, q / p .

Fig. 7. Model predictions with parameters calibrated to measurements of

Vajdova et al. (2004) for relation between changes in d c and plastic volumetric de-

formation as described in Eq. (8) . The slope of each line corresponds to α(1 − n 0 ) =

0 . 896 α in Eq. (8) , and linear regression values are reported in Table 5 .

Fig. 8. Plane strain finite element model mesh and boundary conditions for bore-

hole simulation.

ident. We emphasize that both damage parameters contribute to

the overall amount of damage as apparent for example in Eq. (28) .

The evolution of d

c for all simulations is shown in Fig. 7 plotted

against plastic volumetric deformation as 1 − 1 /J p . This plot corre-


Fig. 9. Model predictions of stress distribution around borehole.

s

α

a

i

d

T

d

t

A

n

t

c

r

a

t

d

o

t

c

e

p

f

m

v

t

c

r

d

y

r

t

c

d

i

s

a

i

w

s

p

5

ponds to the relation of Eq. (8) , where the slope of each curve is

(1 − n 0 ) . The average initial porosity was reported by the authors

s 10.4%, allowing for the model predictions of the proportional-

ty factor α at each confining pressure to be calculated. The pre-

icted linearized values for α calculated in this way are reported in

able 5 . Notably, α is predicted to increase approximately by an or-

er of magnitude on the dilative (brittle) side of the brittle-ductile

ransition (i.e, for the 10, 20, 30, and 50 MPa confining pressures).

s can be seen in Fig. 7 and also by the coefficient of determi-

ation for the least squares linear regression reported in Table 5 ,

he model predicts a near-linear relationship in Eq. (8) (i.e., near-

onstant alpha) at any specific confining pressure. Furthermore, the

esults suggest that the proportionality between volumetric dam-

ge and porosity ( α) is binary, with approximately constant dis-

inct values for dilative and compactive regimes.

It should be emphasized that the material model considers only

uctile damage and does not address the onset of brittle fracture

bserved by the authors at low confining pressures and evident in

he measurements of Fig. 5 (a) as the cataclysmic failure of the low

onfining pressure (below 100 MPa in this case) specimens. How-

ver, the dilative volumetric damage predicted at low confining

ressures may be considered to be indicative of such a cataclysmic

ailure, i.e., comparison of the model predictions to the measure-
r
ents suggests that the threshold for an increase in d

c may be

ery low for this material. As is evidenced by the comparison with

he measurements and is perhaps what we may intuitively expect,

onsistent with a description of the rock as being able to sustain

elatively little shear induced dilation in comparison to shear in-

uced compaction.

It should also be pointed out that shear band bifurcation anal-

sis was not conducted on the specimen responses. The softening

esponses predicted by the constitutive model resulted from ma-

erial softening, and not from localized deformation. Evidently, lo-

alized deformation in the form of a deformation band could have

ominated most of the softening responses exhibited by the spec-

mens. However, material and geometric imperfections on these

pecimens were so pervasive that any bifurcation analysis from

n initially homogeneous response would not be meaningful, since

nitial specimen responses were not truly homogeneous to begin

ith. Nevertheless, we have checked the constitutive tangent ten-

or for the 10 MPa simulation and observed that this tensor lost

ositive-definiteness right at the onset of plasticity.

.2. Simulated borehole through limestone

The model was implemented in an Abaqus UMAT material sub-

outine. A horizontal borehole in a porous limestone reservoir off


Fig. 10. Model predictions of damage around borehole.

c

S

a

t

c

p

a

t

fl

6

c

i

c

n

t

s

r

t

B

f

d

t

d

h

p

m

m

f

d

p

c

t

d

t

t

n

T

b

the coast of Brazil was simulated in order to examine the model

predictions of stress and damage around the borehole. The ma-

terial properties from laboratory measurements and the far-field

stress field from in situ measurements, along with a more detailed

description of the reservoir are reported by Coelho et al. (2005) .

The elastic material properties were taken from those reported

by Coelho et al. (2005) , as were the MC friction angle, cohesion

intercept, and yield point on the hydrostatic axis ( X

φ determin-

ing ˜ β1 and

˜ β2 ). The intermediate principal stress model param-

eter r was taken as the median value of zero, and the elliptic-

ity parameter R was assigned a typical value of 2. The only re-

maining model parameters that needed to be calibrated were the

damage parameters, c t , c c , and c s , which were kept equivalent and

calibrated to best mimic the Drucker-Prager/Cap model simulated

stress field results of Spiezia et al. (2016) , while reaching final

damage states within reasonable threshold values (i.e., to not pre-

dict breakout at the walls, which is consistent with the observa-

tions). The mean stress value for evaluating Eq. (41) was taken as

the average mean stress value of 18 MPa. All model parameters are

reported in Table 6 .

The borehole radius is 4.25 in. = 107.95 mm. The in-situ geo-

static stresses were reported by Coelho et al. (2005) as vertical

stress σv = 32.1 MPa and major and minor horizontal stresses, re-

spectively, σH = σh = 9.0 MPa. The finite element model was con-

structed as plane strain, making use of the two-fold symmetry (see

Fig. 8 ). The analysis simulated the drilling of the borehole by ap-

plying the far-field stresses and pressure at the borehole wall si-

multaneously.

Fig. 9 shows the predicted stress distribution around the bore-

hole with (excess) pressure on the borehole wall absent. Fig. 10

shows the corresponding model predictions of damage distribu-

tion. Fig. 10 (a) plots the volumetric damage parameter d

c , and

Fig. 10 (b) plots the shear damage parameter d

s . Localization of

both volumetric and shear damage occur at the borehole wall

in the orientation coinciding with the direction of least principal

stress, but with opposite trends, i.e., shear damage accompanied by

volumetric hardening (healing). In other words, the simulation pre-

dicts shear induced compaction in this region, where overall hard-

ening is accompanied by microfracture. This is indicative of the

well known tendency for “borehole breakout,” where stress con-

centrations at the borehole wall cause microfracture and eventually

collapse of sections of the wall, effectively elongating the borehole

in that direction (cf. Read and Martin, 1996; Zoback, 2010 ). This is
u
onsistent also with the Drucker-Prager/cap model predictions of

piezia et al. (2016) , who predicted similar trends of volumetric

nd shear strain localization around the borehole wall (although

heir model did not include damage). We emphasize that the in-

lusion of volumetric and shear damage in the model not only

rovides for a prediction of damage around the borehole wall, but

lso provides for a prediction of the stress field associated with

he state of damage, i.e., the stress field prediction of Fig. 9 is in-

uenced by the damage predicted in Fig. 10 .

. Conclusion

Relations between inelastic volume change (bulk plasticity),

hanges in porosity, and evolution of damage have been exam-

ned and incorporated into a unified hyper-elastoplasticity and

ontinuum damage framework. This has allowed for a thermody-

amically consistent description of the damage of porous rocks

hat separates volumetric and isochoric contributions to damage,

uch that the isochoric damage part describes distributed microc-

acks exclusive of their associated apertures. An Eshelby-like stress

ensor we call the Eshelby-zeta stress, which was identified by

ennett et al. (2016) as being energy-conjugate to the plastic de-

ormation rate, has been shown herein to play a valuable role in

istinguishing between volumetric and isochoric contributions to

he overall damaged state of the material while ensuring thermo-

ynamic consistency.

This theory has been developed and implemented in a novel

yper-elastoplastic damage constitutive model appropriate for

orous rocks, what we call a Drucker-Prager/Damage (DP-D)

odel. The model is based on a Drucker-Prager/Cap plasticity

odel previously presented by Regueiro and Ebrahimi (2010) and

urther modified by Bennett et al. (2016) , but the addition of

amage and associated damage potentials is novel. Furthermore,

reviously phenomenological hardening/softening rules are ex-

luded, such that hardening/softening behavior is modeled en-

irely through the described damage evolution making use of the

amage-effective stress concept. The DP-D model is provided in

erms of a current configuration representation of finite deforma-

ion hyper-elastoplasticity, and an implicit integration scheme for

umerical implementation has also been provided.

The DP-D model has been applied to simulation of confined

C measurements on Tavel limestone exemplifying the so called

rittle-ductile transition behavior ( Vajdova et al., 2004 ). The sim-

lations show that the model is capable of simulating the pressure


d

s

c

p

s

a

l

f

a

fi

g

e

a

w

m

a

T

b

a

s

s

s

d

f

r

t

(

t

t

f

p

d

d

p

m

o

s

c

c

c

p

A

m

G

0

w

n

G

A

a

T

s

w

s

T

G

s

m

T

f

A

R

A

A

A

A

A

ependent hardening/softening behavior exhibited in these mea-

urements entirely through damage and damage-effective stress

oncepts related to porosity and microfracture changes and made

ossible by recognizing the role of the Eshelby stress tensor in en-

uring thermodynamic consistency. In particular, marked softening

t low confining pressures is captured well by modeling the evo-

ution of the damaged state with dilative strain. The model results

urther allow for evaluation of the relationship between damage

nd porosity established on kinematic grounds, where an identi-

ed proportionality factor ( α) has been examined. The results sug-

est that for the measurements considered, such proportionality is

ssentially binary, with the factor α increasing by approximately

n order of magnitude between compressive and dilative regimes,

hile remaining approximately constant in each.

The model has been implemented within an Abaqus UMAT

aterial subroutine, and nonlinear finite element simulations of

reservoir borehole through limestone have also been provided.

hese simulations were carried out in order to demonstrate the ro-

ustness of the model and to investigate the predictions of stress

nd damage distribution around the borehole wall. They demon-

trate the usefulness of the model for the analysis of borehole

tability and the prediction of both damaged zones and resulting

tress fields around the borehole wall.

The separation of volumetric and isochoric damage variables

escribed in this work is especially significant because it allows

or the possibility of a thermodynamically consistent tensorial rep-

esentation of isochoric damage to be constructed from the con-

inuum distribution of microcracks without the need to disregard

or otherwise make any simplifying assumptions about) their aper-

ures. Incorporating a second order isochoric damage tensor into

he DP-D model presented here is currently underway as part of

uture work with the goal of providing a description of elasto-

lastic anisotropy that ascribes the relative weakness of the bed-

ing plane observed in many geomaterials to the anisotropy pro-

uced by the microcracks. We note that further applications of the

roposed model could also include extension to coupled hydro-

echanical problems where hydraulic conductivity is dependent

n the damaged state. The phenomenological distinction empha-

ized in this work between the contribution of porosity and mi-

rofracture to damaged could be advantageous in this regard, espe-

ially in distinguishing changes (possibly anisotropic) in hydraulic

onductivity attributed to microfracture from those attributed to

orosity.

cknowledgments

This material is based upon work supported by the U.S. Depart-

ent of Energy , Office of Science, Office of Basic Energy Sciences,

eosciences Research Program, under Award number DE-FG02-

3ER15454 . Support for materials and additional student hours

ere provided by the National Science Foundation under Award

umber CMMI-1462231 , and by the Stanford University James M.

ere Research Fellowship.

ppendix A

The evolution of the damage variables are found from

∂G c ∂y c

=2 c α(A

ψ ˜ β1 − B

ψ ˜ p ζ)[3 〈 β2 − 3

p ζ 〉 (X

ψ − ˜ β2 ) 2

(A

ψ ˜ β1 − B

ψ ˜ p ζ)− F

ψ

cap B

ψ

]= −3 c αC ψ ,

(58)

nd

∂G s ∂y s

=

∂G d ∂y s

= −ω

s ω

c

(−c αω

c y s

S

)s

.

(59)

he detailed solution of q n +1 is provided by first noting that

:= dev [ ζ] =

√

2

3

q n = 2 με

e , (60)

here ˆ n := ε e / ‖ ε e ‖ ≡ ε e,tr / ‖ ε e,tr ‖ =: ˆ n

tr . Eq. (60) also reveals that

/ ‖ s ‖ ≡ ˆ n by,

‖

s ‖ =

√

2

3

˜ q = 2 μ‖ ε

e ‖

‖

s ‖

ε

e

‖ ε

e ‖

=

√

2

3

˜ q ε

e

‖ ε

e ‖

= 2 με

e

‖

s ‖

n =

√

2

3

˜ q n = 2 με

e

⇒ ‖

s ‖

n =

s

⇒

ˆ n =

˜ s

‖

s ‖

=

s

‖ s ‖

.

(61)

he plastic potential function can then be written as

p :=

2

3

˜ q 2 − F ψ

cap

(A

ψ ˜ β1 − B

ψ ˜ p ζ)2 ≤ 0 , (62)

uch that

ˆ :=

∂G p ∂ ξ

=

√

3

2

∂G p ∂q

ˆ n +

1

3

∂G p ∂ p ξ

1 =

2

√

6

3

ω

c ω

s ˜ q n + ω

c C ψ 1 . (63)

he evolution and update equation for ε e s,n +1

and q n +1 are then

ound by noticing

ε

e n +1 = ε e s,n +1 n = ε e,tr

s,n +1 n − �λ

√

3

2

∂G p ∂q

∣∣∣n +1

ˆ n

⇒ ε e s,n +1 = ε e,tr s,n +1

− �λ

√

3

2

∂G p ∂q

∣∣∣n +1

ε e s,n +1 = ε e,tr s,n +1

− �λ2

√

6

3

˜ q n +1 ω

c n +1 ω

s n +1 .

(64)

nd then from

˜ q n +1 = 2

√

3

2

με e s,n +1

=

√

6 μ

(

ε e,tr s,n +1

− �λ

√

3

2

∂G p ∂q

)

=

√

6 με e,tr s,n +1

− 4�λω

c n +1 ω

s n +1 q n +1

⇒

˜ q n +1 =

˜ q tr n +1

/(1 + 4�λω

c n +1 ω

s n +1 ) .

(65)

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