ARMA/NARMS 04-546

Unified Compaction/Dilation, Strain-Rate Sensitive, Cfor Rock Mechanics Structural Analysis Applications

A.F. Fossum and R.M. Brannon Geomechanics Department,Sandia National Laboratories1, Albuquerque, New Mexico, U

Copyright 2004, ARMA, American Rock Mechanics Association This paper was prepared for presentation at Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS): Rock MechaHouston, Texas, June 5 – 9, 2004. This paper was selected for presentation by a NARMS Program Committee following review of information contained in an abstract submitted eas presented, have not been reviewed by ARMA/NARMS and are subject to correction by the author(s). The material, as presented, does noARMA, CARMA, SMMR, their officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial puprohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstracof where and by whom the paper was presented.

ABSTRACT: Sandia’s GeoModel is a generalized plasticity model that was developed primarialso applicable to a much broader class of materials such as concretes, ceramics, and even sombeen incorporated through empirically fitted functions found to be well suited for a wide variety been generalized to include any form of inelastic material response including pore collapse aanisotropy is supported in a limited sense through kinematic hardening. Applications involvinthrough an overstress model. Inelastic deformation can be associated or non-associated, and thematerial input parameters in the rare case when all features are needed, but simpler idealizationsMises yield, or Mohr-Coulomb failure) can be replicated by simply using fewer parameters.

For computational trrelatively straightforwausing standard laborGeoModel strikes a balamicro-mechanics and genized, and semi-empThe over-arching goalgeneral-purpose constitufor any geological or predictive over a wide rrates. Being a unified simultaneously model mor (by using only a smparameters) it can duplimodels such as classicMohr-Coulomb failure.can require as many as complicated materials parameters for idealized

1. INTRODUCTION

Simulating deformation and failure of natural geological materials (such as limestone, granite, and frozen soil) as well as rock-like engineered materials (such as concrete [1] and ceramics [2]) is at the core of a broad range of applications, including exploration and production activities for the petroleum industry, structural integrity assessment for civil engineering problems, and penetration resistance and debris field predictions for the defense community. For these materials, the common feature is the presence of microscale flaws such as porosity (which permits inelasticity even in purely hydrostatic loading) and networks of microcracks (leading to low strength in the absence of confining pressure and to noticeable nonlinear elasticity, rate-sensitivity, and differences in material behavior under triaxial extension compared with triaxial compression). 2. GEOMODEL OVER1 Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE AC04-94AL85000.

The GeoModel shares work by Schwer and M

onstitutive Model

.S.A.

nics Across Borders and Disciplines, held in

arlier by the author(s). Contents of the paper, t necessarily reflect any position of NARMS, rposes without the written consent of ARMA is t must contain conspicuous acknowledgement

ly for geological materials, but is e metals. Nonlinear elasticity has of materials. The yield surface has nd growth. Deformation-induced g high strain rates are supported GeoModel can employ up to 40 (such as linear elasticity, or Von

actability and to allow rd model parameterization atory tests, the Sandia nce between first-principals phenomenological, homo-irical modeling strategies. is to provide a unified tive model that can be used rock-like material that is

ange of porosities and strain theory, the GeoModel can ultiple failure mechanisms, all subset of the available cate simpler idealized yield Von Mises plasticity and Thus, running this model 40 parameters for extremely

to only two or three simplistic materials.

VIEW

some features with earlier urry [3] in that a Pelessone

function [4] permits dilatation and compaction strains to occur simultaneously. For stress paths that result in brittle deformation, failure is associated ultimately with the attainment of a peak stress and subsequently work-softening deformation. Tensile or extensile microcrack growth dominates the micromechanical processes that result in macroscopically dilatant (volume increasing) strains even when all principal stresses are compressive. At higher pressures, these processes can undergo strain-hardening deformation associated with macroscopically compactive volumetric strain (i.e., pore collapse).

The GeoModel predicts observed material response without explicitly addressing how the material behaves as it does, and thus it reflects subscale inelastic phenomena en ensemble by phenom-enologically matching observed data to inter-polation functions. Considerations guiding the structure of the GeoModel’s material response functions are (1) consistency with microscale theory, (2) computational tractability, (3) suitability to capture trends in characterization data, and (4) physics-based judgments about how a material should behave in application domains where con-trolled experimental data cannot be obtained.

The source of inelastic deformation in geological materials (or in rock-like materials such as concrete and ceramics) is primarily growth and coalescence of microcracks and pores. Under massive confining pressures, inelasticity could include plasticity in its traditional dislocation sense or, more generally, might result from other microphysical mechanics, (internal locking, phases transformation, twinning, etc.).

The GeoModel makes no explicit reference to microscale properties such as porosity, grain size, or crack density. Instead, the overall combined effects of the microstructure are modeled by casting the macroscale theory in terms of macroscale variables that are realistic to measure in the laboratory. The GeoModel presumes that there exists a convex contiguous elastic domain of stress states that are sufficiently small that the material response can be considered elastic. The boundary of the elastic domain is called the yield surface. Aside from supporting kinematic hardening, the GeoModel is isotropic, which means that the criterion for the onset of plasticity depends only on the three principal values of the stress tensor, ( )1 2 3, ,σ σ σ ,

but not on the principal directions. Consequently, as illustrated in Fig. 1, the yield surface may be visualized as a 2D surface embedded in a 3D space where the axes are the principal stresses. The elastic domain is the interior of this surface.

While the yield surface is the boundary of elastically obtainable stress states, a limit surface is

Figure 1. GeoModel continuous yield surface - (a) three-dimensional view in principal stress space with the highpressure “cap” shown as a wire frame, (b) the meridional“side” view with the cap shown on the more compressive right-hand side of the plot, and (c) the octahedral view, whichcorresponds to looking down the hydrostat showing triaxialextension (TXE) and triaxial compression (TXC) stress states.

the boundary of stresses that are quasi-statically obtainable by any means, elastic or plastic. Points outside a yield surface might be attainable through a hardening process, but points outside the limit surface are not attainable by any quasi-static process. Points on the limit surface mark the onset of material softening. Consequently, a state on the limit surface is attainable at least once, but might not be attainable thereafter. The GeoModel simulates material response only up to the limit state. It does not simulate subsequent softening, if any, because softening usually induces a change in type of the partial differential equations for momentum balance, which therefore requires a response from the host code to alter its solution algorithm (perhaps by inserting void or by activating special elements that accommodate

displacement discontinuities). Since the GeoModel does not directly model material softening, the limit surface may be regarded as fixed. Because the limit surface contains all attainable stress states, it follows that the set of all possible yield surfaces is contained within the limit surface as shown in Fig. 2. Even though developed primarily for geological applications, the GeoModel is truly a unification of

Strictly speaking, the GeoModel permits the host code to employ any definition of the strain so long as it is conjugate to the stress σ

%% in the sense that

the work rate per unit volume is given by

(3.2) :σ ε&% %% %

To satisfy the principle of material frame indifference, the host code must cast the stresses and strains in an unrotated frame. To date, all implementations of the GeoModel have approximated the strain rate by the unrotated symmetric part of the velocity gradient:

12

m nij mi nj

n m

v v R Rx x

ε ∂ ∂

≈ + ∂ ∂ & (3.3)

where %

is the velocity vector; is the current spatial position vector, and tensor

v x%R%%

is the rotation from the polar decomposition of the deformation gradient. The conjugate stress is the unrotated Cauchy stress. Henceforth, all references to the stress ijσ and the strain rate ijε& must be understood to be the unrotated stress and strain rate.

Figure 2 Distinction between a yield surface and a limit surface (This sketch shows meridional profiles of an initial yield surface that might evolve from the initial surface. All achievable stress states, and thus all possible yield surfaces, are contained within the limit surface.)

All GeoModel material parameterizations have been based on the above approximation for the strain rate. The strain rate is an approximation because, for general deformations, it is not precisely equal to the rate of any proper function of the deformation. The approximate strain rate in Eq. (3.3) exactly equals the unrotated logarithmic (Hencky) strain rate for any deformation having stationary principal stretch directions. It is an excellent approximation to the Hencky strain rate even when principal stretch directions change orientation as long as the shear strains remain small (volumetric strains may be arbitrarily large). For geological applications, material rupture generally occurs well before shear strains become large, so Eq. (3.3) is a prudent choice for the strain rate measure.

many classical plasticity models. For example, by using only a small subset of available parameters, the GeoModel can be instructed to behave like many classical models some of which are shown in Fig. 3.

Figure 3. Some other yield surface shapes supported by the GeoModel

3.1. Elasticity The GeoModel supports linear or nonlinear hypoelasticity. It presumes the material is elastically isotropic and that the elastic stiffness tensor is itself isotropic. Consequently, a rate form of Hooke’s law governs the stress:

ijklC

3. GEOMODEL THEORY

The GeoModel is founded upon an additive decomposition of the strain rate ε&

%% into separate

contributors: eε&%%

from elastic straining and pε&%%

from inelastic straining: e

ij ijkl klCσ ε= && (3.4)

(3.1) p= +eε ε ε& & &% % %% % %

Since the elastic tangent stiffness tensor, C is presumed to be isotropic, Eq. (3.4) may be written as two separate and much simpler equations, one for

ijkl

the volumetric response and the other for the deviatoric response:

evp Kε= && (3.5)

and

2 eij ijS Gγ=& & (3.6)

We have used the overbar to denote the negative of a variable in our equation for the pressure-volume response because the mean stress is typically compressive (negative) in most applications of the GeoModel and therefore p and e

vε are typically positive. G and K are the tangent shear and bulk elastic moduli; p is the pressure (negative of the mean stress); vε& is the volumetric elastic strain rate computed by the trace operation,

;ev kkε ε=& & (3.7)

ijS is the stress deviator; and eijγ& is the deviatoric

part of the elastic strain rate, defined by

1 .3

e e eij ij v ijγ ε ε δ= −& && (3.8)

Of course, Eq. (3.5) remains valid for volumetric expansion ( 0e

vε < ) and tensile mean stresses ( 0p < ) as well. 3.2. Nonlinear Elasticity The GeoModel includes nonlinear elasticity by permitting the elastic tangent moduli to vary with the stress according to

20 1

1

exp bK b bI

= + −

(3.9)

( )1/ 2

1 2 20

1

1 exp1

g g JG g

g

− −=

−

(3.10)

where 1I is the first invariant of the Cauchy stress; is the second invariant of the deviatoric stress

tensor; and, b are material parameters. 2J

,k kg

3.3. Yield Surface In this paper, the term “plasticity” is broadened to include not only the usual flow of material by dislocations, but also any other mechanisms that lead to a marked departure from elasticity. Examples include crack growth, pore collapse, or perhaps even phase transition. Rather than explicitly tracking each of these microscale failure

mechanisms explicitly, the “yield” surface itself characterizes them all in an ensemble phenomenological manner. The GeoModel yield criterion and yield function are given by the following:

GeoModel Yield Criterion:

( ) ( )

( )1

2

,f cF I N F IJ ξ 1 κ

θ

− =Γ

(3.11)

GeoModel Yield Function:

( ) ( )

( ) ( )

22

2

1 1

, ,

,f c

f J

F I N F I

ξσ α κ θ

κ

= Γ

− −

% %% % (3.12)

The yield criterion corresponds to 0f = . Elastic states correspond to . In these equations, the material parameter N characterizes the maximum allowed translation of the yield surface when kinematic hardening is enabled, in which case

0f <

2J ξ is the second invariant of the shifted deviatoric stress tensor =ξ S - α

% %% % %

2J

%, where α is the backstress. When

kinematic hardening is disabled (i.e., when N is specified to be zero), the backstress is zero and therefore

%%

ξ is the second invariant of the deviatoric stress tensor. The function fF represents the ultimate limit on the amount of shear the material can support in the absence of pores (i.e.,

fF represents the softening transition point resulting exclusively from microcracks). By appearing as a multiple of fF , the function accommodates material weakening caused by porosity. The function

cF

( )θΓ , where θ is the Lode angle of the shifted stress, is used to account for differences in material strength in triaxial extension and triaxial compression. The yield surface evolves in time through kinematic hardening (by allowing the backstress internal state variable α

% to change)

and/or through isotropic hardening by allowing a scalar internal state variable , to change.

%

κ

For rocks and rock-like materials, the yield surface will have a shape similar to the one illustrated in Fig. 1. Figure 1 (b) shows a “side” meridional profile of the yield surface, and a family of other profiles from which the yield surface might have evolved over time by continuously varying values

of the κ internal state variable. Note from Fig. 1 (b) that very little of the yield surface exists in the tensile domain, implying that materials of this type are very weak in tension. Figure 1 (c) shows the yield surface profile from a perspective looking down onto a plane—called an octahedral plane—that is perpendicular to the hydrostatic axis and therefore represents a cross-section of the yield surface at a given pressure. Since the onset of yield must not depend on the ordering of the principal stresses, the yield surface for any isotropic yield model possesses 120 rotational symmetry about the hydrostat as well as reflective symmetry about any of the triaxial compression or triaxial extension axes labeled TXC and TXE in Fig. 1(c). Note from Fig. 1(c) that the octahedral profile is somewhat triangular in shape. This periodic asymmetry corresponds to differences in the failure limit under triaxial compression and triaxial extension. Because the yield surface is farther from the origin on a TXC axis than on a TXE axis, this materia higher strength in TXC than in TXE. The

o

l has( )θΓ function

characterizes the shape of the octahedral profile. The size of the octahedral profile at various pressures is governed by the functions fF and . cF

When the internal state variable changes, the yield surface expands or contracts isotropically on octahedral planes. The amount of isotropic expansion or contraction varies with pressure in such a manner that the family of yield surfaces corresponding to various values of is bounded by the shear limit surface,

κ

κ0.fF =

Though not used in many applications, the GeoModel supports kinematic hardening for which the symmetry axis of the yield surface is permitted to shift in stress space so that the invariants in the yield function are based on the shifted stress tensor, defined by

.ij ij ijSξ α= − (3.13)

Here, the backstress ijα is a deviatoric tensor-valued internal state that defines the origin in the octahedral plane about which the yield surface is centered. When the backstress tensor changes, the yield surface translates in stress space. The backstress is initially zero, but then evolves according to an evolution equation. The backstress allows the yield surface to translate in deviatoric stress space, thereby supporting deformation-induced anisotropy (Bauschinger effect) in a limited

capacity. The GeoModel is otherwise fully isotropic, both elastically and plastically. Consequently, the yield function is isotropic with respect to the shifted stress deviator, implying that it depends only on the invariants of the shifted stress deviator, as well as an internal state variable κ that characterizes isotropic hardening caused by pore collapse or softening caused by porosity increases:

ξ

,

( )1 2 3, , ;f f I J Jξ ξ κ= (3.14)

where

22 3

1 and2 3

J tr J trξ ξ= ξ% %% %

(3.15) 31=

The yield function in Eq. (3.14) can be cast in terms of the cylindrical Lode coordinates as

( , , ; )f f r zθ κ= (3.16)

where these Lode coordinates are defined with respect to the kinematically shifted origin in stress space. For any given values of θ , ,andz κ there must exist only one radius r that is a solution to

( , , ; ) 0f r zθ κ = . Thus, without loss in generality, elastic stress states for any isotropic yield function always can be characterized in the general functional form

( , ;r g z )θ κ< (3.17)

where ( ), ,;g zθ κ is regarded as a material function determined from experimental data. The yield function corresponding to Eq. (3.17) is written as

( 22 , ;f r g z )θ κ= − (3.18)

The GeoModel’s yield function is structured such that ( ), ;g zθ κ is separable into the product of two distinct functions, one depending only on θ and the other depending only on , and ,z κ permitting Eq. (3.17) to be written in the general form:

( ) (1 2 ;r h h z )θ κ< (3.19)

The function is scaled such that it describes the shape of the octahedral profile. The function defines the meridional profile of the yield function, and therefore this function also defines the size of the octahedral profile.

1h2h

The GeoModel has been designed to model rocks and rock-like materials. The mechanical behavior of such materials is typically driven by two underlying

mechanisms: porosity and microcracks. Figure 4(a) shows the qualitative shapes of the meridional profiles corresponding to these two mechanisms. For porosity, the meridional profile is a “cap” function that is essentially flat like a Von Mises profile for a high range of pressures and then the profile drops to zero when pressure becomes large enough to collapse pores. The meridional profile for microcracks reflects those theories that consider only the presence of microcracks without considering porosity. Microcracks lead to low strength in tension, but strength increases as pressure generates additional friction at crack faces, thereby reducing the shear load suffered by the matrix material. The intersection of these two profiles, shown shaded in Fig. 4(a), depicts the elastic region of familiar two-surface cap models. The GeoModel unifies these separate microscale theories to obtain a combined porosity and microcrack model as sketched qualitatively in Fig. 4(b). The GeoModel obtains the combined meridional yield function by multiplication of the individual compaction and microcrack profiles (and scaling appropriately to match data).

The GeoModel [Fig. 4(b)] phenomenologically permits cracks and pores to interact in a way that results in a continuously differentiable meridional profile, making the GeoModel better suited for reproducing observed data.

Figure 4. Distinction between two-surface models and the GeoModel

The GeoModel achieves a combined porous and cracked yield surface by multiplying the fracture function ( )fr z by a compaction function ( )cr z in Fig. 4(b) so that

( )r z ~ ( ) ( )f cr z r z (3.20)

The proportionality factor depends on the Lode angle θ so that the equivalent shear stress at yield, which is simply a constant multiple of r, can be

made lower in TXE than in TXC. Compaction functions depend on the porosity level (which controls where the cap curve intersects the z -axis). The curvature of a cap function controls the degree to which porosity affects the shear response. Rather than explicitly tracking porosity, the GeoModel includes an internal state variable κ and one additional material constant, R, to determine both the cap curvature and the location where the cap intersects the hydrostat (the z axis). Thus, the cap function ( )cr z implicitly depends on κ and R.

2J

1

c F

Γ

The Lode cylindrical radius r equals 2ξ , and the

Lode axial coordinate z is proportional to 1I , so that the GeoModel actually implements the idea of multiplying fracture and compaction functions by expressing Eq. (3.20) in terms of stress invariants instead of Lode coordinates so that the GeoModel yield function is of the form

( ) ( )

( )1

2f cf I f I

J ξ

θ=

Γ (3.21)

where

andf ff F N f= − = (3.22) c

The fracture function ff characterizes the cracking-related portion of the meridional yield profile. The cap function cf is normalized to have a peak value

of one. The function ( )θΓ characterizes the Lode angle dependence of the meridional profile and is normalized to equal one in triaxial compression

30θ = o . At different Lode angles, Γ usually has a value greater than one, which (because it is a divisor in Eq. (3.21)) reduces equivalent shear strength. Rather than regarding as a strength reducer, it can be alternatively interpreted as a stress intensifier.

Γ

The precise expression for the function is determined by user-specification of a parameter: the triaxial extension (TXE) to triaxial compression (TXC) strength ratio and an algebraic form, which controls the manner in which the octahedral profile radius is to vary from the value 1/

Ψ

Ψ at TXE to 1 at TXC.

The compaction function, cf . Under compression, the pores in a material can irreversibly collapse, thereby resulting in permanent (plastic) volume

changes when the load is removed. Plastic volume changes can occur for porous media even if the matrix material is plastically incompressible. Permanent volume changes can also occur if a material undergoes an irreversible phase transformation. The compaction function cf accounts for the presence of pores in a material by controlling where the yield function will intersect the 1I axis in compression. Porosity also degrades material shear strength, because, from Eq. (3.21), the compaction function effectively reduces the nonporous yield strength, defined previously by the fracture function ff . The function cf is defined as

1 o wise

=

X

1

4bExp

g E

where pequivγ is the equivalent plastic shear strain

(which, for proportional loading, is conjugate to the equivalent shear stress, 2J ), and p

vε is the plastic compaction volume strain. Mathematically,

2pequiv dtγ = ∫ γ&

%%

p (3.27)

The plastic compaction volume strain, pvε , is

computed indirectly through a “crush curve” to be discussed later in the paper.

-600

-500

-400

-300

-200

-100

0-0.1-0.09-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.010

Volume Strain

Hyd

rost

atic

Str

ess

(Mpa

)Elastic-Plastic Coupling

( )1

2 21

1 if;

therc

If I I

X

κκ κ

κ

<

− − −

(3.23)

where κ and are state variables determined from evolution equations designed to enforce consistency with user-supplied parameters obtained from hydrostatic compression testing. Note that the equation of the elliptical portion of the compaction curve (see Fig. 4(a)) is

Figure 5. Elastic-plastic coupling: deformation-induced changes in elastic moduli

2

2 1c

IfX

κκ

− + − (3.24)

Meridional shear limiter function, fF . Numerous microphysical analyses (as well as a preponderance of data) suggest that, for microcracked media, the onset of yield depends on all three stress invariants, which implies that the “yield” function for microcracked media must depend on all three cylindrical Lode coordinates. Given the wide variety of microscale predictions for the meridional profile shape, a four-parameter exponential spline is used that is capable of replicating many microphysical idealized theories, as well as actual observed material yield and rupture response at low and moderate pressures (i.e., at pressures well below the cap elastic limit so that observed data primarily reflect microcrack damage, rather than combined cracking with pore collapse). The shear limit function has the form,

=

Elastic-plastic coupling. A cap model is used when the material being studied contains enough porosity (or highly compliant second phase inclusions) so that inelastic volume reduction becomes possible through irreversible reduction of pore space. Intuitively, one might expect the elastic moduli to stiffen as pores collapse, but the material might actually become more elastically compliant as shown in Fig. 5 (a phenomenon that might be explained, for example, by rubblization of a ligament network). Regardless of its microphysical origins, the elastic moduli of a porous material are permitted to vary with plastic strain by generalizing the nonlinear elastic moduli expressions in Eqs. (3.9) and (3.10) to ( ) ( )1 1 3 2 1fF I a a Exp a I a I = − − + 4 1 (3.28)

where the are user-specified material parameters determined from experimental data.

ka 2

0 1 31

exp pv

bK b b bI ε

= + − − −

(3.25)

( )1/ 21 2 2 4

0 31

1 exp1 p

equiv

g g J gG g xpg γ

− − = − −

(3.26) −

The limit surface in Fig. 2 is the boundary of all stress states that the material is capable of supporting. Mathematically the limit surface is defined by ( ) 0F =σ

%%, where

( ) ( ) ( )( )

21

2 2

fF IF J

θ

= − Γ

σσ σ

σ%%

% %% %%%

(3.29)

The limit function %

depends only on σ%

, not

on any internal state variables. A yield function ( )F σ% %

( ), , ,f κσ α% %

κ

% % on the other hand, depends on the

backstress tensor %

and on the scalar internal state variable . The yield function is presumed to share some qualitative features with the shear limit surface, but depends on internal variables, viz:

α%

( ) ( ) ( ){ } ( )( )

2

1

2

2

,, ,

where and

f c

c c

F I N Ff J

F f

κκ

θ

− = − Γ

≡ − =

ξ ξσ α ξ

ξ

ξ S α

% %% %% % %% % %

%%

% %% % %%

(3.30)

In using the GeoModel it is assumed that the user will first determine fF

−

cF

; then, the initial yield surface is simply , reduced perhaps by a compaction function if the material initially contains pores. The size of the yield octahedral profile is smaller than the limit surface profile because of the multiplier compaction function . The yield surface origin is also offset from the limit surface origin by an amount governed by the kinematic hardening backstress tensor α .

fF N

cF

%%When the yield surface reaches the limit surface and when the stress lies on the limit surface, the material will begin to soften. A mesh independent, constitutive-level description of material response no longer remains possible; the host code must intervene by inserting void or by invoking special elements capable of supporting displacement discontinuities.

The Lode-angle function. The Lode-angle function, ( )θΓ , controls the shape of the octahedral yield profile. We normalize this function so that its value is unity in TXC ( 30θ = o ). To ensure convexity of the octahedral yield profile, the Lode angle function must satisfy

( ) ( )

( ) ( ) ( ) ( )''

' 0cos sin

θ θ

θ θ θ θ

Γ + Γ≥

Γ − Γ (3.31)

Three Lode-angle functions have been formulated: Gudehus, Willam-Warnke, and Mohr-Coulomb.

The Gudehus and Willam-Warnke options correspond to fully differentiable yield functions. The Mohr-Coulomb option is differentiable everywhere except at triaxial states. Precise functional forms of available Lode angle functions are the following:

i. Gudehus:

( ) ( ) ( )1 11 sin 3 1 sin 32

θ θ θ Γ = + + − Ψ

ii. Willam-Warnke:

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

22 2 *

2 * 2 2 * 2

*

4 1 cos 2 1

2 1 cos 2 1 4 1 cos 5 4

where6

αθ

α α

πα θ

− Ψ + Ψ −Γ =

− Ψ + Ψ − − Ψ + Ψ −

= +

Ψ

iii. Mohr-Coulomb:

( )( )

2 3 sin sincos3 sin 31

sin 31

φ θθ θφ

φ

Γ = − −

− Ψ=

+ Ψ

For the Gudehus case, the convexity requirement,

Eq. (3.31), requires 79 7

< Ψ <

Ψ

9 . The shape of the

octahedral yield profile is described by user specification of the parameter , the triaxial extension/compression (TXE/TXC) strength ratio at a given pressure. Options i.-iii. are distinguished by how the octahedral yield profile varies in stress space from TXE to pure shear to TXC at a fixed pressure. Figure 6 illustrates the shape of these profiles for various values of .

Ψ

Figure 6. Octahedral yield profiles, plotted at allowable values of the strength ratio (The comparison is made for a strength ratio of 0.8Ψ = .)

3.4. Flow Rule In the associative case the direction of the plastic strain rate is parallel to the normal to the yield surface and thus the normal to the yield surface can be obtained by the gradient of the yield function

,

pij

ij

f

κ

ε λσ

∂= ∂ α

%%

&& (3.32) Theκ evolution law. Guided by trends in observed data and by microphysical theories, the isotropic hardening modulus is constructed as

( )

'1

1'

/3 ,1 ( )

min/

1 ( )

pv

f

pv

f

dX dI RF

hI dX d

R RF

κ

φ εκ

κ εκ

∂ ∂ − =

− −

(3.37)

where λ& is a multiplier called the consistency parameter determined by requiring the stress to remain on the yield surface during inelastic loading. The subscripts on the partial derivative indicate that the internal state variables are held constant. For non-normality the user specifies a flow function

( ), ,φ κσ α% %% %

such that

,

pij

ij κ

φε λσ

∂= ∂ α

%%

&& (3.33) ε

where R is a shape parameter; X represents the intersection of the cap function cf with the 1I axis, and p

v represents pore-collapse volume strain determined empirically in which porosity in a material is plotted as a function of the applied pressure. The mathematical form of the plastic volumetric strain is given by

Plastic strain rate points normal to surfaces of constant φ . If the flow function is associative, the plastic potential function equals the yield function.

The functional form of φ is the same as that of f, but with different values for constant material parameters, i.e., , , 2

pfa 4pfa pfR , and Ψ differ

from their counterpart parameters ( )

pf

2 4, ,a a ,R Ψ used to define the yield surface.

( ){ }3 1 2

0

1 exppv p p p

X p

ε ξ ξ

ξ

= − − + = −

(3.38)

where the kp are constants. The shape parameter R relating X to κ is given by

( )

( )f

XR

Fκ

κ−

= (3.39) The consistency parameter requires the stress to be on the yield surface during plastic loading (f = 0) and must remain on the yield surface throughout a plastic loading interval. The yield function f depends on the stress, the isotropic hardening internal state variable, and the kinematic hardening state variable tensor

κijα . Thus,

The first term of Eq. (3.37) is in effect when the stress state falls on the “compaction dominated” part of the yield surface, labeled in Fig. 7, while the second term dominates in the dilatation regime. As the stress point passes through the critical-state point the hardening modulus transitions from a compaction-dominated branch to a dilatation-dominated branch.

0ij ijij ij ij

f f ff σ κ ασ κ α∂ ∂ ∂

= + +∂ ∂ ∂

& & && = (3.34)

To use Eq. (3.34) to solve for the consistency parameter λ& , evolution equations must be supplied.

3.5. Evolution Equations Explicit expressions are now presented for an

isotropic hardening modulus and a kinematic hardening tensor,

hκ

αH%%, such that the evolution of the

internal state variables may be written in the forms

Figure 7. meridiona

(3.35) hκκ = && α%

λ

λ= αα H &&%% %%

(3.36)

The%

eusing a

function, is coα

%%

Compaction and dilatation dominated regions in the l plane

volution law. Kinematic hardening entails shifted stress tensor

% in the yield

instead of the actual stress. The backstress, mputed with the use of evolution equations.

= −ξ S α% %% % %

Initially, the backstress is zero and upon the onset of yielding it evolves in proportion to the deviatoric part of the plastic strain rate:

( ) pHGα=α α γ&% % %% % %

& (3.40)

where

p pdev dev φλ ∂= = ∂

γ εξ

&& &%% %% %%

(3.41)

Hence, comparing with Eq. (3.36), the kinematic hardening modulus tensor is given by

( )HG devαα

φ ∂= ∂

H αξ% %% %%%

(3.42)

where H is a material constant and ( )Gα α%

0→

( )1fF I

% is a

scalar-valued decay function designed to limit the kinematic hardening such that as α

%

approaches the shear limit surface, . Since the yield function itself is defined in terms of

, the maximum kinematic translation that can occur before reaching the limit surface equals the model-offset parameter N. The decay function is given by

G%

( )1fF I N−

( ) 222

11 , where2

JG J

N

αα α= − =α α% %% %

tr (3.43)

Kinematic hardening causes the octahedral profile to translate so that it no longer remains centered at the origin.

With the use of Eqs. (3.41), (3.36), (3.34), (3.33), (3.32), (3.4), and (3.1), the consistency parameter, λ& can be found as

: :

: : :

f

f f f hα κ

λ φκ

∂∂

=∂ ∂ ∂ ∂− −∂ ∂ ∂ ∂

C εσ

C Hσ σ α

&% %% %%& %%%

%%%%% % %% % %

(3.44)

4. RATE DEPENDENCE

Under high strain rates, elastic material response occurs almost instantaneously, but the physical mechanisms that give rise to observable inelasticity cannot proceed instantaneously. Materials have inherent “viscosity” that retards the rate at which damage accumulates. For example, cracks grow at a

finite speed. If a stress level induces crack growth, the quasi-static solution for material damage will not be realized unless sufficient time elapses to permit the cracks to change length. Likewise, pore collapse takes finite time. When the cracks are growing towards the quasi-static solution, the stress will also decrease toward the quasi-static solution. Until sufficient time has elapsed the stress state will lie outside the yield surface. If the applied strain is released during this damage accumulation period, the total damage will be ultimately lower than it would have been under quasi-static loading through the same strain path.

A generalized Duvaut-Lions [5] rate-sensitive formulation is used in which the user specifies a “relaxation” parameter governing the characteristic speed at which the material can respond inelastically. The Duvaut-Lions equations are the following:

( )

( )1

: :

1 :

1 1:

1 1

1 1

e vp

e e e vp

vp

vp e low

e lo

low

low

g

γ

γη

τ

τ τ

τ τ

κ κ κτ τ

−

= +

= = −

=

=

= −

+ = +

+ =

+ =

ε ε ε

σ C ε C ε ε

ε g

ε C σ σ

σ σ C ε σ

α α α

& & &% % %% % %

& &&% % % % % %% % % % % %% %% %& &% %% %

&

&% % % %% % % %%%

&&% % % % %% % % % %%%

&% % %% % %

&

w

&

(3.45)

where the term “low” denotes quasi-static solution; “vp” denotes “visco-plastic”; and η is the viscosity coefficient. It is assumed that the inviscid solutions and the total strain rates are constant over a time interval. The solutions of Eqs. (3.45) are given by

( ) ( )( )

( ) ( )( ) ( )

/ /

/

/ /

/ /

0 1

: 1

0 1

0 1

t low t

e t

t low t

t low t

e e

e

e e

e e

τ τ

τ

τ τ

τ τ

τ

κ κ κ

− −

−

− −

− −

= + −

+ −

= + −

= + −

σ σ σ

C ε

α α α

% % %% % %&

% %% %%%

% % %% % %

(3.46)

The characteristic time was determined empirically to provide flexibility in matching high strain-rate data for a wide range of rock types and is given by

( )

( ) ( )

2

1

24 5 3 1

1 0

1 0

Teqiv p

veqiv

eqiv pv

f T for

T f T T I for

ε εετ

ε ε

= > = + − <

&&

&

(3.47)

-50500

0

50

100

150

200

250

300

350

400

-3000-2500-2000-1500-100000

-I1 (MPa)

J 21/

2 (MPa

)

TXC DataTXE DataModel

Compression

Extension

where the Tk are constants. Figure 9. GeoModel prediction versus measured triaxial compression and triaxial extension limit states Figure 10 shows the GeoModel prediction of an unconfined compression test and a triaxial compression test conducted at 20 MPa confining pressure. The model accurately captures the initial compaction caused by pore collapse and then transitions to dilation-dominated deformation.

5. MODEL VERSUS DATA

We illustrate now the capability of the model by showing some comparisons of the model with real data obtained from experiments conducted on Salem Limestone. A single set of parameters was used for all of the simulations. Figure 8 shows the GeoModel prediction versus measured results for

-150

-130

-110

-90

-70

-50

-30

-10

-0.006-0.005-0.004-0.003-0.002-0.00100.0010.0020.003

Strain

DataModel

Confining Pressure =20 Mpa

Confining Pressure = 0 Mpa

Radial Strain Volumetric Strain Axial Strain

Pa)

-700

-600

-500

-400

-300

-200

-100

0-0.-0.08-0.06-0.04-0.0200.02

Volume Strain

Axi

al S

tres

s (M

Pa)

DataModel

Uniaxial Strain

Hydrostatic Compression

(M

inin

g Pr

essu

re)

C

onf

ress

-

Axi

al S

t

(

Figure 10. GeoModel prediction versus measured results for an unconfined compression test and a triaxial compression test conducted at 20 MPa

Figure 8. GeoModel prediction versus measured data from hydrostatic compression tests and uniaxial strain tests Figure 11 shows the GeoModel prediction versus

measured results for a triaxial compression test conducted at 400 MPa confining pressure. The load path is shown as well as a number of yield envelopes that correspond to various points of the axial stress versus volume strain curve recorded during the test. Again, the model accurately mimics the micromechanical processes that occur during the loading process, including pore collapse resulting in inelastic volumetric compaction, microcrack and microvoid growth resulting in inelastic volumetric dilation. The model allows these deformation mechanisms to occur simultaneously. When the pore space becomes depleted the dilational

hydrostatic compression experiments. This figure demonstrates the GeoModel’s ability to model nonlinear elasticity, compaction from pore collapse and elastic-plastic coupling as seen by the slope at the end of the unloading segment compared with the initial slope of the loading segment. Also shown in Figure 8 is the GeoModel prediction versus measured results for uniaxial strain tests.

Figure 9 shows the GeoModel’s prediction of experimental triaxial compression and triaxial extension limit-states. It was found that the triaxial extensile strength is approximately 70% of the triaxial compressive strength.

mechanisms begin to dominate. Figure 11 shows where these phenomena occur on the evolving yield surface.

40

60

80

100

120

140

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03Strain Rate (s-1)

Uco

nfin

ed C

ompr

essi

ve S

teng

th (M

Pa)

DataGeoModel

-140

-120

-100

-80

-60

-40

-20

0-0.010-0.008-0.006-0.004-0.0020.0000.0020.0040.006

Strain

Axi

al S

tres

s (P

a)

RadialAxialVolumetric

100/s

10/s

1/s0.1/s

0.01/s

10E-5/s

-1 2 0 0

-1 0 0 0

-8 0 0

-6 0 0

-4 0 0

-2 0 0

0-0 .1 4-0 .1 2-0 .1-0 .0 8-0 .0 6-0 .0 4-0 .0 200 .0 2

V o lu m e tr ic S tra in

Axia

l Stre

ss (M

Pa)

D a taG e o M o d e l

0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

4 0 0

4 5 0

-5 0 0 0-4 0 0 0-3 0 0 0-2 0 0 0-1 0 0 001 0 0 0

I1 (M P a )

J 2^1/

2 (M

Pa)

L a b o ra to ry L o a d P a thG e o M o d e lP e a k S h e a r-S tre s s D a ta

Figure 12. GeoModel prediction versus measured results of unconfined compressive strengths as a function of strain rate from Kolsky-bar strain-rate tests

Figure 11. GeoModel prediction versus measured results for a triaxial compression test, the load path followed, the evolving yield loci in the meridional plane. Acknowledgement. The authors gratefully acknowledge the

support from the National Nuclear Security Administration’s Advanced Simulation and Computing Program and a Laboratory Directed Research and Development Program at Sandia National Laboratories.

Figure 12 shows the GeoModel prediction versus measured results for a series of Kolsky-bar strain-rate tests conducted on unconfined compression specimens [6]. Unconfined compressive strength increases with increasing strain rate and, as seen in Fig. 12, it doubles from 1x10-5 s-1 to 100 s-1. REFERENCES

6. SUMMARY 1. Warren, T.L., A.F. Fossum, and D.J. Frew. 2004.

Penetration into low strength (23 MPa) concrete: target characterization and simulations. Int J. Impact Engr. (accepted).

A generalized plasticity model has been developed for a wide range of natural and man-made materials and has been implemented in finite element computer codes residing on massively parallel platforms at Sandia National Laboratories to solve rock mechanics structural analysis problems in energy and defense related problems. Computationally tractable and accurate predictions can be made on broad temporal and spatial scales because the GeoModel captures salient microphysical phenomena en ensemble through phenomenological governing equations that can be parameterized from standard laboratory scale experiments. Future development will focus on the inclusion of ubiquitous jointing and thermally activated mechanisms.

2. Arguello, J.G., A.F. Fossum, D.H. Zeuch and K.G. Ewsuk. 2001. Continuum-based FEM modeling of alumina powder compaction. KONA, 19: 166-177.

3. Schwer, L.E., Y.D. Murry. 1994. A three-invariant smooth cap model with mixed hardening, Int. J. for Num. And Anal. Mech. In Geomech. 18: 657-688.

4. Pelessone, D. 1989. A modified formulation of the cap model, Gulf Atomics Report GA-C19579, prepared for the Defense Nuclear Agency under Contract DNA-001086-C0277, January.

5. Duvaut, G. and J.L. Lions. 1972. Les Inequations en Mecaniquie et en Physique, Dunod, Paris.

6. Frew, D.J., M.J. Forrestal, and W. Chen. 2001. A split Hopkinson Pressure Bar Technique to determine compressive stress-strain data for rock materials, Experimental Mechanics, 41: 40-46.