Plasticity, Localization, and Friction in Porous Materials

RICE UNIVERSITY

by Daniel Christopher Badders

A T h e s is S u b m i t t e d in P a r t ia l F u l f il l m e n t o f t h e

R equirements f o r t h e D e g r e e

Doctor of Philosophy

A p p r o v e d , T h e s is C o m m i t t e e :

Michael M. Carroll, Chairman Dean of the George R. Brown School of Engineering and Burton J. and Ann M. McMurtry Professor of Engineering

Mary F. Wbjeeler

and Applied Mathematics

Yves C. Angel Associate Professor of Mechanical Engineering and Materials Science

Houston, Texas May, 1995

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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Plasticity, Localization, and Friction in Porous Materials by Daniel Christopher Badders

Original variations of two-invariant rate-independent plasticity models capture

many features of the homogeneous behavior of porous materials. All of the variations

proposed are based on a simple 7 parameter critical state plasticity model proposed

by Carroll (1991). The models 'Utilize an associative flow rule along with a yield

surface dependent on Terzaghi effective pressure and shear stress with plastic volume

strain as the only hardening variable.

In Carroll’s original model, a parabolic yield surface accommodates hardening by

translating along the pressure axis while the top of the yield surface moves along a

critical state line. The base of the parabolic yield surface is of constant width, and

hardening is linear in plastic volume strain. Even in this simple form with only 5

plastic constants the model can predict dilation, shear enhanced compaction, critical

state behavior, hardening, softening, and yielding during unloading.

To test Carroll’s simplifying assumptions and to extend the constitutive formu

lation for more complex behavior, variations of the model are compared to test

data published by Terra Tek. The large groups of test data for sandstone and di-

atomite include varied stress paths for axisymmetric loading, unloading and reload

ing. Modifications proposed to Carroll’s model address nonlinear hardening, initial

elastic transverse isotropy, variation of the shape of the yield surface, variation of

elastic moduli due to pore closure, anisotropic hardening, and pore fluid pressure

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. interaction with matrix compressibility. Evaluation of these modifications indicates

they are secondary effects and points to the strength of Carroll’s original model, while

providing refinements for treating these aspects of poro-behavior.

Plastic softening creates potential for localization and and subsequent frictional

behavior. While this behavior is not represented in the constitutive model or test

data, a framework is outlined for treating formation and growth of localization bands

based on a bifurcation analysis of the plasticity model. The most practical application

of this would be using finite elements modified to include an embedded localization.

Extreme localization leading to fracture can switch the dominant form of inelastic

deformation from plasticity to frictional slip. It should be possible to accommodate

this behavior within an embedded localization finite element.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments

I thank members of my thesis committee, Dr. Carroll, Dr. Angel, and Dr. Wheeler

for being sources of inspiration and knowledge.

I thank Kirk Hansen, John Schatz, and Don Schroeder for their special assistance.

Finally, I thank my wife, Anne-Lancaster Emery Badders, for her steadfast sup

port along with my family and friends.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents

Abstract ii

Acknowledgments iv

List of Figures viii

List of Tables x

Glossary of Notation xi

1 Porous Behavior, Applications, and Background 1

1 . 1 Behavior of Rocks and Other PorousMaterials ...... 1

1.2 The Motivation for Better Porous Material Models ...... 3

1.2.1 Subsidence at Japan’s New Kansai A irport ...... 3

1.2.2 Compaction and Subsidence at Ekofisk Oil Field ...... 4

1.2.3 Testing Core Samples from Oil Reservoirs ...... 6

1.2.4 Cup-and-Cone Failure of Tensile Specimens ...... 7

1.3 Continuum M echanics...... 8

1.3.1 Kinematic and Constitutive Variables ...... 8

1.3.2 Invariance Under Rigid Body M otion ...... 11

1.4 General Plasticity Issu e s ...... 15

1.4.1 Stress and Strain Space Yield Criteria ...... 15

1.4.2 Defining Finite Plastic Deformation ...... 17

1.4.3 Associated flo w ...... 19

1.4.4 Basis of Smooth Yield Behavior ...... 21

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.4.5 Generalization to Multisurface P lasticity ...... 21

1.5 Effective S tre s s ...... 23

2 Experimental Studies of Porous Materials 26

2.1 Boise Sandstone ...... 26

2.2 Monterey D iato m ite ...... 26

3 Plasticity of Porous Media 37

3.1 Micromodel Based Volumetric P la stic ity ...... 37

3.1.1 G re e n ...... 37

3.1.2 Gurson-Tvergaard ...... 38

3.2 Cap M odels ...... 40

3.3 Cam-clay Critical State M odels ...... 43

3.4 Carroll’s Critical State Plasticity Model ...... 46

3.4.1 Extracting the Strain-Stress Relation ...... 48

3.4.2 Extracting the Stress-Strain Relation ...... 50

4 Localization and Friction 53

4.1 Localization in One Dimension ...... 53

4.2 Extending Localization Ideas to Three Dimensions ...... 54

4.3 Localization Band Analysis ...... 55

4.4 Localization Leading to Frictional Sl i p ...... 58

4.4.1 Simple Frictional L aw s ...... 59

4.4.2 Mohr Envelopes ...... 59

4.4.3 Frictional Slip Accompanied by D ila tio n ...... 60

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Combined Plasticity, Localization, and Friction 63

5.1 Towards a Finite Element Model ...... 63

5.1.1 Localization T reatm ents ...... 64

6 Model Optimization and Comparison to Experiments 67

6.1 Parameter Determination as an Optimization Problem ...... 67

6.2 Evaluating Model Variations ...... 6 8

6.2.1 Nonlinear H ardening ...... 6 8

6.2.2 Transverse Isotropy ...... 71

6.2.3 Modified Effective Stress R elations ...... 73

6.3 Summary and Conclusions ...... 73

A Visualizing Relations Between Elastic Constants 77

Bibliography 82

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figures

1.1 Features of porous material deformation ...... 2

1 . 2 Multiple length scales in porous rocks ...... 2 1.3 Four types of well failure due to compaction of weak reservoirs 5

1.4 Ductile tensile specimens fail with cup-and-cone fracture ...... 8 1.5 Depiction of a general continuum m otion ...... 9 1.6 Haigh-Westergaard stress invariants in principal stress s p a c e 12 1.7 Predicted variation of elastic moduli with porosity ...... 14 1.8 Yield surfaces depicted in strain space and stress space ...... 16 1.9 Yielding can occur during unloading ...... 18

1 .10 Maximal unloading in stress space and strain space ...... 18 1.11 Plastic flow at yield surface vertices ...... 22 1.12 Effective frictional contact stresses ...... 25

2.1 High pressure and uniaxial strain sandstone t e s t s ...... 27 2.2 Overview of drained triaxial sandstone test stresses ...... 27 2.3 Overview of drained triaxial sandstone test strains ...... 28 2.4 Overview of stress trajectories for saturated sandstone tests ...... 29 2.5 Overview of strain trajectories for saturated sandstone te s t s 29 2.6 Comparison of hydrostatic response of sandstone sam ples ...... 30 2.7 Hydrostatic loading of diatomite ...... 31 2.8 Stress trajectory for diatomite samples D7 and D 2 2 ...... 32 2.9 Overview of stress trajectories for diatomite t e s t s ...... 32 2.10 Overview of strain trajectories for diatomite t e s t s ...... 33 2.11 Stress trajectory for Diatomite ...... 33 2.12 Stress trajectory for Diatomite ...... 34 2.13 Stress trajectory for Diatomite ...... 35 2.14 Comparison of hydrostatic response of diatomite sam ples ...... 36

3.1 Cap model yield su rface ...... 41 3.2 Typical and idealized soil consolidation ...... 43 3.3 Cam-clay yield surfaces...... 45 3.4 Carroll’s critical state yield surfaces...... 47

4.1 Cap model yield su rface ...... 54 4.2 A localization band in an otherwise homogeneous infinite region. . . . 57 4.3 Hyperbolic friction envelopes with varying pc0, No, and S o...... 61 4.4 Dilation under frictional response ...... 62

6.1 Graphic interface developed in MATLAB ...... 69 6.2 Improved fit of sandstone tests with nonlinear hardening ...... 70

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3 Elastic anisotropy in sandstone ...... 75 6.4 Elastic anisotropy in diatomite ...... 76

A.l Relations between elastic moduli ...... 79 A.‘2 Constant values of Poisson’s ratio, v, in relation to E, G, and K. . . 80 A.3 Constant values of A and \ jv in relation to E,G. and K ...... 81

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tables

2.1 Sumary of Monterey diatomite test sequences ...... 30

A.l Relations between elastic constants ...... 78

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Glossary of Notation

E Young’s Modulus of Elasticity e Eulerian strain

e .4 Orthonormal basis vectors (A =1, 2, 3) of 3ft3 E, E Lagrangian strain and its rate E v, i f Plastic strain and its rate / Yield function in stress space F Deformation Gradient g Yield function in strain space K Bulk modulus P Average stress or pressure on a solid p or pt Fluid pressure for saturated porous media P The first (nonsymmetric) Piola-Kirchoff stress tensor go, g Mass densities in the reference and current configurations 5 The second (symmetric) Piola-Kirchoff stress tensor t Time T Cauchy stress tensor u Displacement vector V Specific volume (V = 1/(1 — ©)) v Particle velocity X Lagrangian position x Eulerian position e Infinitesimal strain p, Friction coefficient

Tensors and vectors are given in bold. Components of tensors and vectors are

given by indicial notation. Repeated indices in a term are summed unless otherwise

indicated. For example, u = u ii where e,- are the orthonormal basis vectors of 3ft3.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The tensor cross product, a®b, where a and b £ 3J3 is defined by (a®b)v = (b-v)a

for all v. The cross product e,- ® e;- forms the basis of second order tensors so that T = Tijki ® ej.

The gradient with respect to the Lagrangian coordinates and the Eulerian coor

dinates axe given respectively by is indicated by

d(f) Grad<£ = -r——ei = and graded = — e{ = . (0.1) uX a

The divergence with respect to the Lagrangian coordinates and the Eulerian co

ordinates are given respectively by

du-i Su2 , du3 dux du2 du3 . Div<^> = -r— + — - + -^rr = uA, A and di v(f> = — + — + — = . 0.2 0X1 a X2 aX3 ox i ox2 ox 3

The material time derivative is indicated by a superposed dot.

* = ~fo\X=con,tant • (0’3)

The components of the Kronecker delta are

1 , * = j Sij = (0.4) 0 , otherwise

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Chapter 1

Porous Behavior, Applications, and Background

1.1 Behavior of Rocks and Other Porous Materials

Pores and cracks lead to distinctive mechanical behavior in rocks and other porous

materials. Features include nonlinear elasticity, pressure sensitive plasticity, shear en

hanced compaction, induced anisotropy, softening, localization, friction, length scale

dependence, inelastic unloading, and effects due to fluid within pores. The current

study examines variations of a simplified plasticity model proposed by Carroll [Car91]

and its relevance to features of porous behavior mentioned above.

Experimental data is reviewed in chapter 2, the model is developed and contex-

tualized in chapters 3 through 4, and a parameter optimization program aids in a

comparing the model to experimental data in chapter 6 . Insight provided through

this comparison justifies simplifying assumptions proposed by Carroll and leads sig

nificant improvements of the model.

While emphasis is placed on refining constitutive formulations for homogeneous

poro-plasticity, a framework is discussed which ties plasticity in with localization,

friction, and length scale dependence. It is hoped this discussion will contribute to

future improvements in finite element methods, while the central results for modeling

homogeneous plasticity are immediately applicable.

Unlike solid materials, pressure, shear, or combined loads can cause large per

manent volume changes in porous materials. Changes in loading can cause voids

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a (compression) . softening nonlinear elasticity \ tensile nonlinear failure (■— unloading or reverse yielding (tension)

Figure 1.1 Deformation of porous materials often introduces nonlinear elastic response, yielding, localization and softening, and frictional behavior.

and cracks to grow or coalesce which can result in significant softening. Even under

uniform conditions, softening can lead to highly non-uniform Tesponse such as the

formation of shear bands. In extreme cases, fractures form and subsequent response

is dominated by frictional slip along these fractures. Chapters 3, and 4 address these

issues of volumetric plasticity, localization, and friction.

microcracks pores and relatively joints and faults, joints, and pores pore clusters homogeneous bedding and bedding

sub-pore scale pore scale core scale wellbore scale formation scale

Figure 1.2 Multiple length scales in porous rocks effect structural behavior (adapted from [Sch93]).

Localization and friction behavior depend on length scales arising from the mate

rial or loading. As shown in figure 1.2, cracks and pores may occur at many distinct

scales in rocks. The behavior at one scale results from the behavior at progressively

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smaller scales. Practical limits prohibit resolving all scales within one analysis, creat

ing a need for strategies to deal with the scale cascade. Chapter 5 addresses combined

modelling of length scales, plasticity, localization, and friction.

1.2 The Motivation for Better Porous Material Models

Before discussing the framework for modelling porous materials, it is helpful to review

the need for these models. While models for inelastic behavior of porous materials

have improved dramatically over the last few decades, significant room for improve

ment remains. A wide range of applications depend on aspects of poro-mechanical

behavior. Examples include

• water wells, oil production, and ground water clean up,

• energy absorbing packaging, crash barriers, and car bumpers,

• ductile fracture of metals,

• material processing including ceramics and powder metallurgy,

• foundations, excavations, mines, and tunnels, and

• ground subsidence, seismology, and faulting.

A few specific examples below demonstrate the significance of these applications.

1.2.1 Subsidence at Japan’s New Kansai Airport

Neufville [Neu91] describes subsidence problems at Japan’s new Kansai airport. It is

being constructed on a 1,300 acre man made island sitting 3 miles off Japan’s coast.

The project was initially expected to cost twelve billion dollars, most of which went

into dumping 250 million cubic yards of fill on the ocean floor to create the island.

Based on models of the fill and underlying clay, the island was expected to grad

ually settle 23 feet during its first 50 years. However, the island has already settled

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more than 23 feet, requiring an additional 10 feet of fill to be added over the entire

1300 acre island at a cost of over four billion dollars. The weight of this extra fill may

contribute to further subsidence and additional measures may be required to keep

the island above high tides. If more accurate material models had been available to

predict compaction of the fill and underlying clay, it would have significantly reduced

construction costs.

1.2.2 Compaction and Subsidence at Ekofisk Oil Field

Porous rock compaction profoundly effects production at oil fields such as Ekofisk in

the North Sea. Sulak |S.ul91,] describes the history of Ekofisk between 1969 and 1989.

During this time, production totalled 900 million barrels of oil and 2 trillion cubic

feet of gas. While reservoir compaction contributed to production rates, it also led

to subsidence and enormous associated costs.

The Ekofisk reservoir consists of a 1000 foot thick layer of porous chalk covering

12,000 acres at a depth of about 10,000 feet. The weight of overlaying rocks com

presses the 30% to 50% porous rock with a vertical stress of 9,000 psi. Fluid pressure

within the reservoir supports part of this overburden, but oil and gas extraction have

dropped the fluid pressure from 7,000 to 4,000 psi. With the lower fluid pressures,

the effective vertical stress on the chalk has increased from 2,000 to 5,000 psi causing

significant compaction of the reservoir and subsidence above the reservoir.

In weak reservoirs rocks, compaction usually closes pores and faults and thereby

reduces permeability and oil production rates. Oddly, Ekofisk production rates have

continued to climb. Rhett and Teufel [RT92] suggested one possible explanation.

Their tests of Ekofisk chalk samples showed permeability decreases for hydrostatic

compaction but increases under strongly deviatoric loads. It is also known that per

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meability at Ekofisk is due primarily to a network of open fractures and perhaps a

better understanding of the fracturing process will reveal how the permeability has

remained high.

While compaction apparently hasn’t reduced permeability at Ekofisk, declining

fluid pressures and resulting compaction have caused problems. Subsidence necessi

tated roughly one billion dollars in repairs and protective measures for the platforms

and tanks between 1987 and 1989. In addition from 1986 to 1991 nearly two billion

dollars were spent injecting water into the reservoir to boost fluid pressure, reduce

compaction, and increase production rates. Further financial losses were caused by

compaction induced well failures such as those studied by Hamilton, Mailer, and

Prins [HMP92] and diagrammed in figure 1.3.

1) bedding slip.., • 3) tensile failure 2) fault slip...

Weak porous reservoir rock / 4) buckling compacts because declining../ fluid pressure supports less \ of the overburden.

Figure 1.3 Four types of well failure due to compaction of weak reservoirs (adapted from [HMP92]).

Problems encountered at Ekofisk and other oil fields such as the Lost Hills field

in California (see Bruno [Bru92]) have prompted extensive studies of porous rock

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behavior. Such studies provided much of the experimental data discussed in chapter 2

and motivated features of the material models in chapter 3.

1.2.3 Testing Core Samples from Oil Reservoirs

The production potential of a given reservoir depends largely on the permeability,

porosity, and compressibility of the reservoir rock. Measurements of these properties

are often made by testing core samples recovered during drilling.

The rock and fluid within reservoirs are generally under extreme pressure due

to the weight of the overburden. Extracting a core sample from a well releases the

in-situ loads. The sample may be reloaded to measure in situ properties, but the

effects of the unloading and reloading are generally unknown. Errors in estimates

of the permeability, stiffness, and strength of reservoir rock can significantly affect

estimates of production potential and lead to financial disasters.

Holt and Kenter [HK92] simulated core damage on synthetic sandstones. They

prepared samples by consolidating a mixture of sand and sodium silicate and then

injecting a hardening agent. By adjusting consolidation pressures and the proportions

of the mixture, they obtained 30% porous synthetic rocks resembling oil reservoir

sandstones. The consolidation stresses could be maintained following cementation

to model undisturbed reservoir rock. Some samples were tested in this undisturbed

state while others were first unloaded and reloaded.

To simulate coring, samples were unloaded axially and then radially. During this

unloading, the samples expanded about about 1.4% and during subsequent reloading

compacted about 1.9%. Once reloaded, the stiffness of these samples in uniaxial com

paction was found to be roughly half that of undisturbed samples. The net effect of

coring was a loss of stiffness, a 0.5% volume reduction, and a 2% drop in permeability.

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Interestingly, under further triaxial compaction, there were no discernible differences

between the undisturbed samples and those that had been unloaded and reloaded.

To simulate a more gentle coring process, some samples were unloaded hydro-

statically. Again, unloading and reloading resulted in reduced stiffness, volume, and

permeability, but these changes were smaller than samples first unloaded axially.

Popular material models such as Mohr-Coulomb, Gurson-Tvergaard, critical-state,

and cap models do not allow for damage during hydrostatic unloading. The model

emphasized in the current work does allow for damage during pressure unloading.

1.2.4 Cup-and-Cone Failure of Tensile Specimens

The formation and rupture of pores governs ductile fracture behavior. The most

familiar example of ductile fracture is the cup-and-cone fracture of tensile specimens

as shown in figure 1.4. In a typical round tensile specimen, necking creates maximum

hydrostatic tension at the center of the specimen causing voids to form. The voids

coalesce into a crack which propagates in a direction perpendicular to the tensile load.

Near the surface the crack turns and propagates as a shear fracture.

Tvergaard and Needleman [TN84] simulated the process of cup-and-cone fracture

using finite elements with a porous material model. While their method qualitatively

demonstrated cup-and-cone fracturing, further developments are needed to improve

predictive capabilities. Improved models of porous material behavior should enable

much more accurate modeling of ductile fracture under triaxial and inhomogeneous

loading conditions. Such a fundamental engineering capability would have far reach

ing significance.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1.4 A typical round tensile specimen undergoes necking, formation of central voids and cracks, and finally shear failure on a lip near the surface.

(This figure adapted from [Smi 8 6 ]).

1.3 Continuum Mechanics

At the microscale, porous materials are clearly discontinuous. None-the-less, con

tinuum models apply when considering the average response over a sufficiently large

region. Aspects of a suitable continuum formulation are reviewed below and are

further developed in chapters 3, 5, and 6 .

1.3.1 Kinematic and Constitutive Variables

Using the notation of Naghdi [Nag90], consider the motion of a body B from a refer

ence configuration kq to a configuration k at time t as shown in figure 1.5. A particle

in the reference configuration is identified by its position, X, in a three-dimensional

Euclidean space. A motion of a body can be given by a mapping % °f the position of

particles in configuration kq to their position in configuration k at time t.

Xi = Xi(X,t). ( 1-1)

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deformed configuration, k at time, t

reference configuration, kq

Figure 1.5 Depiction of a general continuum motion

The displacement of a particle is given by

ui(X,t) = Xi(X ,t)-X A6iA. (1.2)

The deformation gradient for the motion x is

F iA = ■?§-, det F > 0 . (1.3) OX. A

F may be factored

F = RU = V R . (1.4)

where R, U, and V are the rotation, right stretch, and left stretch tensors respec

tively. R is an orthogonal rotation so that R R T = R TR = 1. Both U and V are

symmetric positive-definite. These factorizations are unique according to the polar

decomposition theorem.

The Lagrangian and Eulerian strains, E and e, have components given by

E a b = \ { F iAF lB - M and e{] = - F ^ F ^ ) . (1 .5 )

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The Cauchy stress tensor or true stress tensor, T = Tije,- ® ej, represents force

per unit area in the current configuration acting in the direction of et- on a surface

normal to ej. For a surface perpendicular to the unit normal n the components of

the traction are given in terms of the Cauchy stress by f,- = Tijfij.

While the true stress refers to area in the current configuration, engineering stress

refers to area in the reference configuration. A component of the engineering stress

or first Piola-Kirchoff stress, Pi a , represents a force in the direction of e,- per unit

undeformed area on a surface normal to e. 4 . The second Piola-Kirchoff stress, S can

also be introduced and the three stress tensors related.

(det F ) T = F S Ft = P Ft . (1.6)

Conservation of angular momentum leads to the result that T, S, and PFT are

symmetric.

Conservation of linear momentum leads to the equations of motion 1 in either the

Lagrangian or Eulerian description.

PiA,A + Qok = QoVi or Tijj + gb{ = gvi , (1.7)

where g0 and g are the mass densities in the reference and current configurations,

respectively, b is a body force per unit mass and, v = x is the particle velocity field.

The distinctions between E and e disappear for infinitesimal motions as do the

distinctions between T, P, and S.

e _ e = e and

1A sign convention for stress is implicit in these equations of motion. Here the equations correspond to the general engineering convention where stresses and strains are positive in tension. In rock mechanics, stresses and strains are usually taken to be positive in compression. This later convention is used where appropriate in later chapters.

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1.3.2 Invariance Under Rigid Body Motion

Consider two distinct motions of a body, x and ^ +, which differ only by a rigid

rotation and rigid displacement. This can be expressed as

X+(X,t) = QX(X,t) + q , (1.9)

where the tensor Q is an orthogonal rotation and q is a displacement vector. The

intuitive necessity that these superposed rigid body motions should cause changes in

material response leads to invariance conditions on the kinematic and constitutive

variables. Thus,

F + = QF , R+ = Q R , U+ = U, V+ = QVQT ,

E+ = E, E+= E , S + = S , S+ = S, T+ = QTQt . (1.10)

Models for continuum materials must also satisfy invariance under superposed rigid

body motions. One way to ensure proper behavior is to formulate models in terms of

invariant quantities2. Common stress invariants are

(first invariant of stress) J\ =

(second invariant of the deviatoric stress) , ( d l )

(third invariant of deviatoric stress) J 3 = | Sipspmsmi ,

where the components of deviatoric stress, sare given as

(deviatoric stress) s,j = — - bij(?kk • ( 1 -1 2 ) O

While these invariants have been given here in terms of

in terms of S , T , or P and the distinction should be made where relevant. Other

2Where models are formulated in terms of stress and stress rates as in most plasticity models, the stress rate must also be invariant. Many publications describe invariant or objective stress rates , but they are not addressed here.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. forms of invariants can also be constructed from J\, J'2, and J'z. For example, the

Haigh-Westergaard invariants described by Jain [Jai89] which can be expressed as

r _ i ( 3\/3 JI e (1.13) ' “ 3 ‘ arCC0S \ 2 { J $ 2

p J i

These invariants are convenient because of their clear meaning particularly in refer

ence to principal stress space as depicted in figure 1 .6 .

Figure 1 . 6 Haigh-Westergaard stress invariants have a clear meaning in principal stress space.

The principal stresses can be be given in terms of the Haigh-Westergaard invariants

as P cos#

. = < (1.14) 0 2 P • + COS (# - |7r)

03 P COS (6 + |7 r )

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In axisymmetric experiments where o\ > u2 = <7 3 , the invariants can be expressed

more simply as

J\ = 01 + 2(72 )

J2 = ^( 0 - i - 0 2 )2 ; (1.15)

J3 = 2j(ai ~ l72)3 >

P = + 2cr2 ) ,

r = ^ '16^ ■fl = -0,

Elastic Properties of Porous Materials

Many authors have examined elastic behavior of porous materials. A primary goal

is to describe the elastic moduli in terms of the properties of the constituents and

arrangement of the pores. One successful approach has been to analyze a micromodel

which approximates the structure.

The hollow sphere micromodel has been found particularly useful for describing

porous materials. Torre (1948) first proposed the use of a hollow sphere model and

used it to predict hydrostatic yield strength. Mackenzie [Mac50] extended the hol

low sphere model to estimate the shear and bulk moduli of porous materials at low

porosity. Christensen [Chr79] gave an analytical result for the effective bulk modulus

of a hollow sphere

+ for0<^<1’ (L17)

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where Km and Gm are the bulk and shear modulus of the matrix material. However,

Christensen [Chr79] indicated no corresponding results are available for the effective

shear modulus.

Badders [Bad91] applied finite element analysis to the hollow sphere to approxi

mate effective shear moduli for the special case of nearly incompressible matrix mate

rials. That approach has been extended for a wide range of matrix compressibilities

and the effective elastic properties are shown in figure 1.7. Notably, the effective

UJ0.5-

0.5 1 0.5 num phi

0.5 num

Figure 1.7 Elastic moduli predicted by finite element analysis of hollow sphere micromodel.

Young’s and shear moduli vary little with the Poisson’s ratio of the matrix material.

In contrast, the effective bulk modulus depends strongly on the Poisson’s ratio of the

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matrix material. Of some related interest, a graphical means for viewing relations

between elastic constants is presented in Appendix A.

Elastic properties also vary with the shape and arrangement of pores so the hol

low sphere can only be considered as one approximation of a range of possibilities.

Variations of the hollow sphere micromodel have been used to examine effects of

various shapes of pores and interactions between pores.

1.4 General Plasticity Issues

The unresolved framework for general theories of finite plasticity presents a special

difficulty in planning experiments and developing specialized models for porous ma

terials. While most of the unresolved issues are beyond the scope of the current

work, they are addressed below briefly while establishing the necessary background

for later chapters. Naghdi [Nag90] provides an excellent review of issues in general

finite plasticity.

1.4.1 Stress and Strain Space Yield Criteria

The choice of yield criteria is one differentiating factor among plasticity theories3.

Traditionally, these criteria have been formulated in terms of stress, such as the yield

stress in uniaxial tension. The common Tresca and Mises yield criteria represent

extensions of the tensile yield criteria to general stress states. These yield criteria

can be visualized as yield surfaces in principal stress space which bound the stress

states that can be reached without yielding. Given an elastic stress-strain relation,

a yield surface in terms of stress has a corresponding yield surface in strain space as

3Notably, some plasticity theories lack a yield surface. The Bodner-Partom model [BP75] is one sucessful example. However, these models primarily address rate dependence and will not be dis cussed here.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16

pictured in figure 1.8. For notational convenience, yield criteria are frequently given

using yield functions, /(

formulations respectively. By convention, the yield functions are negative inside the

yield surface, zero on the yield surface, and positive outside the yield surface.

The yield surface.can.be offset from the strain origin even for isotropic hardening. stress space strain space

Figure 1.8 Yield surfaces can be plotted in both strain space and stress space.

In most elastic-plastic formulations, the stress-space and strain-space yield crite

ria can be used interchangeably. However, important distinctions arise for perfect

plasticity and for strain softening. To clarify the distinctions, Naghdi [Nag90] defined

the quantities g and / by

(U8) For plastic states where / = g = 0, hardening, softening, and perfect plasticity can

be identified as

f/g > 0 <-*• hardening,

f/g < 0 *-+ softening, and (1-19)

f/g = 0 +-> perfect plasticity.

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For plastic states where g = 0, g can be used to distinguish loading, neutral

loading4, and unloading.

g > 0 loading,

g < 0 «-► unloading, and ( 1 .2 0 )

<) = 0 <-> neutral loading.

In contrast, for plastic states where / = 0,

f > 0 —> loading,

/ < 0 —*■ unloading or loading, and ( 1 -2 1 )

/ = 0 —> loading or neutral loading.

Because of this ambiguity that arises from the stress-space yield criteria during plastic

softening and perfect plastic yielding, Naghdi suggests the strain space criteria should

be taken as primary. The stress-space yield criteria should always follow from the

strain space criteria.

1.4.2 Defining Finite Plastic Deformation

If a point in a body can be unloaded elastically to zero stress, then plastic strain is

equal to the total strain at zero stress. If the yield surface in stress space does not

contain the origin, then yielding occurs during unloading as in figure 1.9 and identify

ing the plastic component of strain becomes more difficult. Casey and Naghdi [CN92]

proposed defining plastic strain using a maximal unloading condition as shown in fig

ure 1.10. The maximal unloading from any state of stress and strain, ( E,S ), can

be defined as the state ( E P,S P) which minimizes ||S|| in the region / < 0. At the

4Neutral loading here refers to a trajectory along the yield surface

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(L-Lo)/L,

Figure 1.9 Yielding can occur during unloading.

maximal unloading state, Casey and Naghdi define the plastic strain as being equal to

total strain. This definition has the advantage that the plastic strain is not changed

strain space stress space

Figure 1 . 1 0 Casey uses a maximal unloading (min|[5(| (within the current yield surface in stress space to define plastic strain (adapted from [CN92]).

by attempts to measure it. Additionally, when the stress space origin is within the

yield surface this definition agrees with the classical definition. However, the max

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imal unloading definition appears inconsistent with the usual idea that the elastic

component of the strain will be non-zero for non-zero stresses.

Lacking a measurable definition of plastic strain, it can simply be considered a

state variable which follow from the plasticity model. The usual assumption is that

the total strain can be expressed as a sum of elastic and plastic components and that

the elastic stress is a function of the elastic component of strain. While this additive

decomposition of strain is favored by many, it has its opponents.

Lee (1969) first proposed the use of a multiplicative decomposition of the deforma

tion gradient into elastic and plastic components. Moran, Ortiz, and Shih [MOS90]

recognized opposing views, but favor Lee’s multiplicative decomposition for its close

tie to crystalline plasticity. Naghdi [Nag90] has pointed out inconsistencies in some

multiplicative formulations.

1.4.3 Associated flow

Drucker [Dru52] postulated for hardening materials that external work during a closed

stress cycle is positive if and only if the cycle is accompanied by plastic deformation.

This postulate leads to the important conclusion that yield surfaces in stress space

must be convex and that plastic flow is in a direction given perpendicular to the yield

surface. This is generally applied as a normal flow equation

. = $ (1 .2 2 )

where the consistency parameter, A, is found by the consistency condition f = f = 0

during yielding.

Il’iushin [I1’61] strengthened Druckers postulate to include softening materials by

instead considering closed strain cycles rather than stress cycles. Other researchers

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have tried to extend proofs of normality to finite plasticity. The clearest proof is pro

vided by Naghdi [Nag90] and is based on a work assumption. Specificially Naghdi’s

proof rests on the assumption that “the external work done on any body by surface

tractions and by body forces in any sufficiently smooth spatially homogeneous cy

cle during the time interval (to,tj) is nonnegative.” Requiring a “smooth spatially

homogeneous cycle” is particularly important in distinguishing plasticity from fric

tional slip on a discrete plane which can be shown not to fit normality assumptions.

While Naghdi’s proof is given in the context of an addititive decomposition of strain,

Lubliner [Lub 8 6 ] proves normality in the context of a multiplicative decomposition

of strain using convex analysis. Simo [Sim 8 8 ] also advances this formulation in a

somewhat broader context. It is not evident what the implications of these proofs are

for plasticity formulations using additive decompositions.

Verification of Normality

Experimentally verifying normality requires an accurate estimate of the normal to the

yield surface and an accurate estimate of the elastic response. Some carefully planned

experiments such as [CC81] suggest normal flow rules are valid for porous rocks

undergoing homogeneous deformation. Various claims to the contrary can probably

be attributed to inaccurate approximations of the yield surface or elastic behavior.

Another approach to justifying normal flow rules was examined by Carroll and

Carman [CC85]. Using finite element analysis of a hollow sphere under homogenous

strain boundary conditions, they found plastic flow directions roughly normal to an

approximate yield surface. This was re-examined by Badders [Bad91], and by refining

the yield surface approximation, even greater agreement was obtained.

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1.4.4 Basis of Smooth Yield Behavior

Most materials are formed of many randomly oriented grains or crystals and no single

slip system dominates plastic behavior. Overall yield criteria and plastic flow response

are smoothed by the averaging of randomly oriented grains. The reasonable expecta

tion of smooth behavior simplifies measuring or modeling the mechanical properties

of such materials.

In contrast, single crystal plasticity is governed by activation of individual slip

systems. Each slip system has its own yield criteria, and vertices or corners may form

at the intersections of these limit surfaces. Also as a result, a small change in the

applied stress could change the slip system activated and cause a large change in strain

response. Similar unsmooth response can be expected for rocks with preferentially

oriented joints. Behavior of single crystals and jointed rocks are mentioned here only

as exceptions to the type of plastic behavior to be considered.

1.4.5 Generalization to Multisurface Plasticity

Some plasticity models incorporate two or more intersecting yield surfaces. These

models provide an expedient means to fit experimental data, but often are avoided

on theoretical grounds. Where yield surfaces intersect in an unsmooth fashion, nor

mality proofs are not applicable and normal flow rules break down — there may

be a range of possible flow directions as pictured in figure 1 .1 1 (a). This represents

a potential for bifurcation of the stress-strain relation where uniform loading may

lead to non-uniform response. As has been observed before, a yield vertex may arise

when fractures or weakening occur on two crossed planes as pictured in figure 1 .1 1 (b).

Distinct yield criteria will exist for each plane. At the yield vertex, yielding may occur

on either plane or both.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1.11 At a yield vertex(a), normality proofs do not lead to a flow rule, and there may be a range of possible flow directions. A yield vertex may arise from weakness or fractures (b) on intersecting planes in a material.

Where the multisurface formulation is intended to approximate a smooth yield

surface it is desirable to extend normal flow rules to the yield vertices. Sometimes the

vertex is bisected to define the direction for the normal flow rule. Hofstetter [HST93],

Engelmann [EH91], and Simo [SKG 8 8 ], favor a solution proposed by Koiter [Koi53].

Koiter generalized the flow at a vertex by summing contributions from the intersecting

surfaces.

E ' a/i i- 1 where

n is the number of independent yield surfaces,

fi is the ith yield surface, and

Ai is the plastic consistency parameter for the ith surface.

The consistency parameters as described by Hofstetter [HST93] can be given in the

Kuhn-Tucker complimentary form.

A; > 0 , fi < 0 , and A,/; = 0 (no sum on i, i= l, 2 ,...,n). (1-24)

This scheme has been incorporated into various models as a reasonable and useful

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. way to approximate smooth yield surfaces with multiple intersecting surfaces.

1.5 Effective Stress

Terzaghi 5 suggested saturated soil is loaded by a total stress which can be separated

into two parts. He identified one part as the neutral stress (or pore fluid pressure, p*)

which acts uniformly in every direction throughout the fluid and solid. He identified

the remainder as the effective stress which causes deformation and failure of the solid.

= 4 0tal - 8ijVf . (1.25)

While many credit Terzaghi with pioneering this concept, a historical review by

de Boer and Ehlers [dBE90] emphasizes earlier work by Fillunger. Between 1915 and

1935, Fillunger performed a series of experiments and published a series of articles on

the subject. Only following a dispute with Fillunger did Terzaghi publish his concept

of effective stress in 1936.

Biot [Bio41] also deserves recognition for providing a more complete elastic treat

ment and incorporating Darcy’s law to account for viscous fluid flow through porous

media. Later, Biot [Bio55, Bio56a, Bio56b] further extended his formulation to treat

anisotropy and wave propagation in saturated media.

More recently, various authors have recast effective stress formulations in terms

of mixture theories. Katsube and Carroll [KC87a, KC87b] use this approach and

compare their resulting formulation with Biot’s formulation. Unlike Biot’s treatment,

their formulation includes terms representing viscous fluid shear forces on the solid.

Their formulation reduces to that of Biot where fluid velocity gradient terms are

ignored and can thus be seen as a useful extension or generalization of Biot’s work.

5Excerpts and translations of Terzaghi’s work [vT36] were given by de Boer and Ehlers [dBE90], but Terzaghi’s work was not reviewed directly.

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While accurate treatment of viscous and dynamic effects are important in some

applications, the current work emphasizes rate independent behavior and requires

only a minor adjustment to Terzaghi’s formula,

= 4 ?tal ~ aS':PJ’ (L26)

where the scalar a adjusts for compressibility of the solid material of the porous

matrix. A value of a = ( 1 — K jK , ) follows from the formulations of Biot [Bio41] or

Katsube and Carroll [KC87a, KC87b] where K is the effective bulk modulus of the

porous material and Ks is the bulk modulus of the solid forming the matrix of the

porous material. Notably, where the matrix material yield strength is independent

of pressure6, the effective stress of equation 1.26 also appears suitable for plastic

behavior of porous materials.

While traditional effective stress formulations can be applied to plastic behav

ior, frictional behavior may require a different approach. Many researchers apply

Terzaghi’s effective stress to frictional conditions. A simple argument suggests a dif

ferent effective stress formulation may govern frictional behavior depending on what

portion of the slip surface is wetted by pressurized fluid.

If the entire frictional surface is wetted by pressurized fluid, the effective normal

stress should equal the total normal stress minus the fluid pressure. If the contact

surface is unwetted, the effective and total normal stress should be equal. In general

where the wetted fraction of the slip plane is w, the effective normal stress, iV', can

be computed from the total normal stress, N , as N1 — N — (wpf). If the slip plane

crosses many randomly located pores it is reasonable to assume w > . If significant

sliding has occurred, it is reasonable to expect that most of the slip plane would be

6If there are isolated pores within the matrix material, then the effective stress may not be suitable for predicting plastic behavior.

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For wetted contact, fluid pressure supports part of the matrix stress.

For unwetted contact, the matrix stress is supported only by contact forces.

Figure 1.12 Effective frictional stresses depend on whether pressurized fluid wets the contacts surfaces.

wetted so that w « 1. In this case the use of Terzaghi’s effective stress will give

equivalent results.

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Chapter 2

Experimental Studies of Porous Materials

The model introduced in section 3.4 was largely motivated by series of triaxial ex

periments on diatomite and sandstone. Those experiments are reviewed below in

sections 2.1 and 2.2. Chapter 6 compares these experiments to various models.

2.1 Boise Sandstone

Ingebretsen and Schatz [IS84] performed a series of triaxial tests on saturated samples

of Boise sandstone. The sandstone was chosen as a relatively homogeneous, perme

able, and representative sample of weak reservoir rock. A thorough background char

acterization included petrographic analysis, electron microscope analysis, ultrasonic

measurements of elastic properties, and permeability measurements. Initial drained

testing included high pressure hydrostatic compression, hydrostatic creep, triaxial

compression at three distinct confining pressures, and uniaxial strain. Finally, ten

diverse test sequences were run which combined loading and unloading including

varying pore fluid pressure, confining pressure, and axial loads.

2.2 Monterey Diatomite

Moss and Torres [MT 8 8 ] present plots and tabulated data from a series of triaxial

tests on saturated Monterey Diatomite. The experimental aparatus and procedures

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Drained High Pressure Test Drained Uniaxial Strain Test 60 120

50 100

Q. 40 s? 80

j= 10 axial ----- radial 1 ----- radial 2 ......

0 1 2 3 4 5 6 0 10 20 30 40 50 60 strain (%) radial stress (psi)

Figure 2.1 Drained high pressure hydrostatic and uniaxial strain sandstone tests

20 co a § 15

g o 10 co

tos 13 x „ C3 0 S 1 2 S13 S14 5 10 15 radial stress (100$osi) ^

Figure 2.2 Overview of stress trajectories for drained triaxial sandstone tests

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0 S 1 2 S 13 S14

0 0.5 0 0.5 radial strain

Figure 2.3 Overview of strain trajectories for drained triaxial sandstone tests

were similar to that used by Ingebretsen and Schatz [IS84], but the saturating fluid

was at atomospheric pressure.

The loading and unloading test sequences are sumarized in table 2.1.

As shown in figure 2.14, good consistency was obtained between all diatomite

samples during initial hydrostatic loading. However, figure 2.14(b) does show highly

anisotropic response. Under uniform hydrostatic loading, the samples compressed

more in the axial direction then in the radial direction. Geomaterials formed by

sedimentation are often transversely isotropy.

While the models introduced in chapter 3 focus on isotropic behavior, section 6.2.2

1 Moss and Torres [MT88] give no explanation for the apparent omission of tests D1-D6 and D8- D ll from the published report. In fact, the report only provides plots and tabulated data with no accompanying background information, material characterization, explanations, observations, or sumary. In these respects, Ingebretsen and Schatz [IS84] provided a far more complete report.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29

o 0 SOI

501 J V so? \L 505 20 0 10 20 0 radial stress (1000 psi)

Figure 2.4 Overview of stress trajectories for sandstone tests.

.S 0 _SHL 502 5 0 2 5 M 501 u,CS 5 6 • x . CB 4

0 .506- _SQ2 _S02 ^ 0 9 JLHL

0 1 0 1 radial &rainl(%) 0 1 0 1

Figure 2.5 Overview of strain trajectories for sandstone tests.

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- 2 0 1 K a o o 1c 0.8 o 15

i-> 0.6

0.4

0.2

0 0.5 volume strain (%) (a) GO

Figure 2.6 The initial hydrostatic loading repsonse shows some variation between sandstone samples. In a plot of volume strain versus effective

pressure (a), sandstone samples 6 and II standout from the rest of the samples. In a plot of the difference of axial and radial strains versus effective

pressure (b), sandstone samples 4, 6 , 9, and 10 show the largest anisotropy.

Test loading/unloading ID # sequence D7 2-B-2 D12 1-5-I-E loading unloading description D13 1-5-1-E type type of loading type D14 1-5 1 A hydrostatic dz = dT D15 1-5 2 B uniaxial strain eT = 0 D16 1-3 3 C uniaxial stress dT = 0 D17 1-3-C-A 4 D biaxial stress d, = 0 D18 1-3-C-A 5 E deviatoric stress dz = —2 dr D19 1 -A D20 1-3-C-A D21 1-A D22 2-D-C

Table 2.1 Sumary of Monterey diatomite test

sequences 1 reported by Moss and Torres [MT 8 8 ].

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350 radial strain 300 axial strain

250

I 150 a 100

0 1 2 3 4 5 6 % strain

Figure 2.7 Diatomite sample D21 under hydrostatic loading shows nonlinear elastic-plastic response which is much softer in the axial direction.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. poue ih emiso o h cprgt we. ute erdcin rhbtd ihu pr ission. perm without prohibited reproduction Further owner. copyright the of ission perm with eproduced R

axial stress (1000 psi) O W ^ O \ OO o tO ON oo O to 4 ^ On oo Figure 2.8 Comparison of stress paths for uniaxial strain loading of loading strain uniaxial for paths of stress Comparison 2.8 Figure 5 1 ^ ^ « & .9. 1 9 s o u 250 “ Figure 2.9 Overview of stress trajectories for diatomite tests. diatomite for trajectories Overviewstressof 2.9 Figure 300 400 200 450 350 samples D7 and D22 shows inconsistency of the diatomite. D22showsthe of inconsistency and D7 samples 100 150

0 D20 4 2 0 4 2 0 S . 100 2 1 D ^ ^ radial stressradial (1000 psi)

6 1 0 D21 200 pressure (psi) D17 > 3 1 D ^ / 0 0 500 400 300 D22 D22 ^ 4 1 D ^ / ^ D7 D18 32 33

20 10

0 _DQ2 _ J212_ J213_ 1214

1215. J216 1212 J 2 1 &.

s m D21 J222_

0 1 rdSial strain (%) 0 1

Figure 2.10 Overview of strain trajectories for diatomite tests.

500 D12 450 D13 400

u 350 u a 2 300 £-4U ^5 250 ■ C/3 2 200 55 150 100

0 50 100 150 200 250 300 350 400 450 500 Pressure

Figure 2 . 1 1 Diatomite samples D12 and D13 underwent similar stress trajectories.

Stress difference 400 200 300 500 600 100 Figure 2.12 Diatomite samples D14 and D15 and D14samples Diatomite 2.12 Figure 0 newn iia tes trajectories. stress similar underwent 0 0 10 0 20 0 30 400 350 300 250 200 150 100 50 Pressure D15 D14 34 poue ih emiso o h cprgt we. ute erdcin rhbtdwtot emission. perm without prohibited reproduction Further owner. copyright the of ission perm with eproduced R

Stress difference 200 250 350 400 450 300 500 150 100 50 iue .3 itmt ape 1-1 and D16-D18 samples Diatomite 2.13 Figure 0 5 10 5 20 5 30 5 400 350 300 250 200 150 100 50 0 2 newn iia tes trajectories. stress similar D20underwent D20 D17 D16 D18 Pressure 35 poue ih emiso o h cprgt we. ute erdcin rhbtd ihu pr ission. perm without prohibited reproduction Further owner. copyright the of ission perm with eproduced R

effective pressure ( 1 0 0 psi) 4 0 consistency between diatomite samples. In a plot of volume strain versus strain volume of plot a In samples. diatomite between consistency effective pressure (a) consistent nonlinearity is apparent. In a plot of the the of plot a In apparent. is nonlinearity consistent (a) effectivepressure difference of axial and radial strains versus effective pressure (b), all (b), effectivepressure versus strains radial and differenceaxial of Figure 2.14 The initial hydrostatic loading repsonse showsgood repsonse loading hydrostatic initial The 2.14 Figure volume strain (%) strain volume diatomite samples show very significant anisotropy. significantshow samples very diatomite (a) 5 10 0 strain difference (%) difference strain G>) 2 4 36 37

Chapter 3

Plasticity of Porous Media

Many plasticity models motivated by experiments and theory have been proposed for

porous materials. The micromodel based plasticity models of Green, Gurson, and

Tvergaard as well as the cap and Cam-clay models are reviewed below. These models

were selected not only for their wide use, but also for their similarities to the plasticity

model discussed in section 3.4.

3.1 Micromodel Based Volumetric Plasticity

3.1.1 Green

Green [Gre72] proposed a plasticity theory for porous solids based on micromechanical

models. Assuming the yield surface should be smooth, convex, and symmetric in

tension and compression, Green suggested a yield function of the form

/(Jx, J', ) = 3J' + - m Y o , (3-1)

where Yo is the initial yield strength of the matrix material.

To determine a{), Green considered a hollow sphere of perfectly plastic material

subjected to uniform external pressure. In his analysis, he duplicated the earlier

work by Torre [Tor48] and found complete yielding occurs for Ji = 2T0 In . Thus,

the expression for &(),

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38

To determine /?(), Green considered a simple cubic cell containing a spherical

void loaded in pure shear where the yield condition of equation 3.1 reduces to dJ2 =

P{4>)Yq. Using approximate analytic methods, he obtained

' (3'3) Many researchers have subsequently proposed yield surfaces similar to Green’s.

Carman [Car 8 6 ] reviews nine distinct models and casts them in the form of equa

tion 3.1 with unique expressions for a(

can be attributed to variations in the porous composition and microstructure and to

variations in experimental and analytical approaches. The Gurson-Tvergaard model

appears to be one of the more flexible variations and is discussed next.

3.1.2 Gurson-Tvergaard

Gurson [Gur77] proposed a yield condition for ductile porous materials based on an

analytical bounding approach. Unlike previous researchers, Gurson did not assume

a shape for the yield surface, but rather determined the shape through his analytical

approximation. He examined both cylindrical and spherical voids under different

assumptions regarding the plastic flow. For a fully plastic hollow sphere model, he

obtained the yield function

/( Ju 4 <£) = + 2(j> cosh ( | 0 - (1 + A (3.4)

Tvergaard (1981) proposed allowing for various void geometries by adding ad

justable parameters to Gurson’s model

f( Ji,J2,) = ^ + 2 <^cosh ( ^ J - ( 1 + q22) , (3.5)

Based on finite element analysis of an array of cylindrical holes, Tvergaard proposed

qi = 1.5 and q2 = q\. For q\ — q2 = 1, Tvergaard’s model reduces to Gurson’s model.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39

Variations of Gurson-Tvergaard Models

Many other variations of the models of Gurson and Tvergaard have been proposed.

Egelmann [EH91] implemented a variation by Horn and McMeeking in the finite

element program NIKE2D. The modified model includes isotropic hardening by re

placing Yq in equation 3.5 with a matrix yield strength which is a function of a scalar

effective plastic strain.

Y(ep) = Yo + Ep£*, (3.6)

where the effective plastic strain is given by

/<§*&*&)»- m

Goya, Nagaki, and Sowerby [GNS92, NSG91] proposed modifications of Gurson’s

model to account for anisotropy due to void shape and distribution.

= ^ + ( 2 ^ cosh ( A ) _ 2qAn - 1 - ?3/ (2n)) • (3.8)

They use a finite element micromodel of ellipsoid voids in rectangular cells to evaluate

the parameters qi, q2, 9 3 .

Mear and Hutchinson [MH85] extended Gurson’s model to include mixed isotropic

and kinematic hardening.

/ ( jj, 4 «7 F, 0) = | | + 2 ^ cosh - (1 +

where

Ji and J'2 are defined in terms of flow stress,

B = T — A, is the flow stress,

A is the back stress (the center of the yield surface),

crF = (1 — b)Y0 + ba e, and

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For 6 = 1 isotropic hardening is obtained, for 6 = 0 pure kinematic hardening is

obtained, and for 0 < b < 1 mixed hardening is obtained. Various forms of evolutions

equations for the back stress, A, are possible, but Mear and Hutchinson carefully chose

one so that their model coincides with Gurson’s model for proportional loading. While

this choice served their purpose of examining the effect of yield surface curvature on

localization behavior, it may not be the best choice for modelling real materials.

With a more flexible approach, this mixed hardening variation of Gurson’s model

could capture the asymmetric yield behavior in tension and compression found in

most materials.

3.2 Cap Models

Simo, Ju, Pister, and Taylor [SJPT 8 8 ] and Hofstetter, Simo, and Taylor [HST93]

review, assess, and modify the inviscid two invariant cap model originally proposed

by DiMaggio and Sandler in 1971. The original cap model is outlined here based on

their review, while various modifications will be summarized briefly at the end of this

section.

The original cap model assumed a linear elastic stress-strain relation, additive

decomposition of strain into elastic and plastic components, and normal flow. The

model was formulated to overcome the problems of excessive dilation which follows

from applying normal flow to Mohr-Coulomb type yield surfaces. As pictured in

figure -3.1 this was done by adding a cap to model permanent crushing and by replacing

the Mohr-Coulomb surface with a pair of surfaces to fit dilative behavior.

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compressive comer region

failure mode

cap mode tensile v comer regioir tensile 11 cutoff mode,,

T 0 J

Figure 3.1 A cap model yield surface (adapted from [SJPT 8 8 , HST93]).

Thus, the cap model yield surface is comprised of three separate surfaces with

yield functions given by

= \fti~ [ a - 7 exp(-/?Ji) + 9Ji] >

/ 2 (Ji,J',K) = /j* - j -Jx (k) - L(k) ] 2 - [Jx - L(ft ) ] 2 , and (3.10)

/3(Ji) = T -Jl

where a, /?, and 0 are material constants, and k is a hardening parameter. The

function L(k) is given by the McAuley bracket defined as

K if K > 0 L{k) = (k) = (3.11)

0 if ft < 0

The intersection of /i and / 2 occurs at Ji = L{n) which leads to a relation for X( k)

X( k) — ft — R[a — 7 exp(—/?k) + 0ft] . (3.12)

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Equation 3.12 is used as an implicit equation for k while X(k) is related directly to

volumetric plastic strain, e{J, as

el(X) = W[l-exp(-DX(K))} or X(k) = ^ l n ( l - &) (3.13)

where W and D are material constants found from hydrostatic compression tests.

Modified Cap Models

Simo, Hofstetter, et al [SJPT 8 8 , HST93] proposed modifications to the original cap

model and to the original algorithmic treatments. Within time stepping algorithms,

it is necessary to evaluate the yield functions for trial states where J\ > X(k) but

in equation 3.11 this leads to square roots of negative numbers. To overcome this

dificulty, Simo Hofstetter, et al propose a modified cap surface,

MJuJZ'K) = + [ a - 7 e x p (-M ) + 9Ji] • (3.14)

This is functionally similar equation 3.11, but avoids algorithmic problems. They also

provide a rigorous algorithmic treatment based on an unconditionally stable radial

return method and consistent tangent moduli which provides quadratic convergence

with Newton type iterative schemes. Notably in their formulation, they exclude

the possibility of contraction of the cap and state that accurately incorporating cap

contraction would require including a dependence on k in the yield functions for the

tensile cutoff and failure envelope modes.

Whirley and Engelmann [EH91] implemented several extensions of the cap model

for their NIKE2D implementation. Following a proposal by Krauss, they allowed arbi

trary functions of relative volume VfVo to model variations in bulk modulus K, shear

modulus G, and cap axis ratio R. They also adopted a kinematic hardening model

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43

following Isenberg, Vaughn, and Sandler. The additional parameters and arbitrary

functions provide extreme freedom to fit broad sets of experimental data.

Zaman [ZRAA94] proposed a cap-type model with special attention to hardening

in the post pore collapse regime.

3.3 Cam-clay Critical State Models

Britto and Gunn [BG87] review the classical formulation of the Cam-clay and modified

Cam-clay critical state material models and present their implemention of the models

in a finite element program called CRISP, outline of development below - i.e. classical

formulation, strain state formulation and tangent •modulus, brief review of variations

of the models accounting for anisotropy, etc.

V V

slope= -K

In P In P (a) typical soil (b) Cam-clay idealization

Figure 3.2 Plots of logarithmic pressure versus specific volume for hydrostatic consolidation of typical soil and idealized Cam-clay material (adapted from [BG87]).

Cam-clay models idealize the hydrostatic response of typical soils as shown in

figure 3.2. The elastic volumetric response is given by V = VK — ft In P . This

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corresponds to a varying bulk modulus given by

VP I<= — - (3.15) A»

Using the relations V = 1/(1 — ) and = o — e„, the specific volume can be

expressed in terms of total volumetric strain. This implies a choice of a reference state

where = fa, V = Vo, and P = Pq > 0., and e„ = e®+e£ choose a reference configura

tion with specific volume, Vo = 1 /( 1 —o) and pressure, Pa take the plastic volumetric

strain, e£, to be zero in that configuration, between porosity and volumetric strain,,

it Porosity and specific volume are related by V = 1/(1 — ) = {(pa — e„)/(l — e„)

The program CRISP The stress-strain formulation

Britto and Gunn present the models in the context of those state variables, but

within CRISP they implement the models in a stress-strain formulation.

terms of the state variables P, t , and evu.

The Cam-clay model

/(P,r,e„) = ^ - l n P c + lnP (3.16)

where,

• M is the slope of the critical state line, r = MP,

• Pc is the current yield strength in hydrostatic compression and

The the hardening function, Pc, is given by the isotropic compression line as

V = N — A In Pc where N and A are the intercept and slope on a semilog plot.

For yielding with t/P > M the value of Pc increases while for yielding with

r/P < M the value of Pc decreases.

The material is assumed rigid-plastic in shear and nonlinear elastic-plastic in bulk

compression. Normally the bulk elastic compression is given as a line of slope — k on a

plot ov V versus In P. This can be expressed as a varying bulk modulus K = VP/re.

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o ' Iuie7 ' o state/ state. c critical a critical/ MPc £ MPc -3 subsequent. C/3 5^4 surface modified ^ C/3 initial P Cam-clay N Cam-clay yield surfaci yield surface i

Pc/2.72 Pc 0 Pc/2 Pc effective pressure effective pressure

Figure 3.3 Cam-clay and modified Cam-clay yield surfaces (adapted from [BG87]).

Other values associated with the Cam-clay model are

• V = 1/(1 — ) = (d>o — e„)/( 1 — e„) is the specific volume and

• k, A, M, and T are material constants:

• k is the slope the elastic V vs ln(P) line,

• A is the slope of the plastic V vs ln(P) line,

• T is the value of V for ln(P) = 0 on the critical state line

Britto and Gunn [BG87] give the elasto-plastic tangent stiffness matrices for the

Cam-clay and modified Cam-clay models in the form

DeaaT Dep = 1 - De (3.17) aTDea - cTHa

where De is the elastic stiffness matrix. The variables a, c, and H depend on the

model and are given by

a = df/dcr ,

c = df/dh where h is the hardening parameter, and

H = dh/deP.

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Thevanayagam and Chameau [TC92] review developments of Cam-clay models

and introduce a fabric tensor as a state variable to aid in modeling anisotropy and

effects of various stress paths.

Zienkiewicz, Humpheson, and Lewis [ZHL75] formulated a generalization of criti

cal state and Mohr-Coulomb models.

[DAS87] Desai provides a framework for constitutive models that resembles a

critical state model but admits various levels of complexity in experimental response

including effects such as:

associative isotropic hardening,

nonassociative isotropic hardening,

anisotropic hardening, and

strain softening.

3.4 Carroll’s Critical State Plasticity Model

Carroll [Car91] proposed 1 a simple critical state plasticity model. The model com

bines isotropic linear elasticity and associative flow with a yield function dependent

only on effective pressure, shear stress, and plastic volume strain. In its simplest

form 2, the yield function can be written as

/(P,r,e;)=|—«n »> lP~W)Y— j +^2,-1. | M , (3.18)

Carroll first presented this model to industrial sponsors in 1986 and later at technical conferences, but it wasn’t published until 1991. 2Equation 3.18 is equivalent to the yield function given by Carroll [Car91], but has been parame terized slightly differently for convenience in generalizations to follow.

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where Ps is a material parameter and where Pc{tpv) and rs(eP) are linear hardening

functions,

Pc(epv) = Pco + Epcepv and Ts ( e pv) = t s0 + ETstp . (3.19)

shear stress

effective pressure

Figure 3.4 Initial and subsequent parabolic yield surfaces for Carroll’s critical state plasticiy model (figure adapted from [Car91]).

Carroll’s critical state model has 7 material constants including the bulk modulus

K, the shear modulus G, and the 5 plastic parameters Ps, Pco, Epc, rs0, and ETS.

Carroll points out the minimum of 5 plastic parameters are necessary to capture

the initial yield strength in tension,

the initial yield strength in compression,

the initial shear strength,

the slope of the critical state line, and

the relative rate of volume change and hardening.

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The rest of the model follows from simplifying assumptions based on observations of

experimental data, micromodels, or theory. To test simplifying assumptions leading

to equation 3.18, the prediction of Carroll’s model can be compared to experimental

data and to predictions using a more generalized yield function,

+ ( ^ ^ n ’ - 1 - (3-2o)

The additional hardening functions Ps (ep), and rs(ep) provide extra flexibility in fitting

hardening behavior. The parameter k was suggested by Carroll [Car91] to vary the

shape of the yield surface. For k = 0 the yield surface is parabolic while k = 1

gives an ellipsoidal surface. For k > 0, the surface is smooth at the midplane where

r = rc(ep) The hardening functions Ps(ep) and rc(ep) could take various forms but the

linear form of Pc{(-P) and rc(ep) are addopted here.

Ps(epv) = PaQ + Epsepv rc(epv) = To + ETCepv (3.21)

To further generalize the yield function, the shape parameter, k, and the hard

ening function Ps(ep) may be allowed to take on different values on the dilative and

compactive sides of the yield surface,

Ki if P < Pa Psi(e) if P < Pco Ps = (3.22)

k2 if P > Pa Ps2 (e) if P > Pco

For Psi(e) = Pc(e), a type of modified Cam-Clay model is recovered.

3.4.1 Extracting the Strain-Stress Relation

A convenient stress-strain relation follows from the yield surface of equation 3.20

or 3.18 when combined with linear elasticity and a normal flow rule.

P = (ev - el)K r = {7 - 7P)26 ' linear elasticity (3.23)

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/(P , t, £y) =0 yield condition (3.24)

kP-Xd_l Y = h d-l flow rule (3.25) v dP 7 2 dr The yield conditon also implies / = 0 during yielding. This consistency condition

can be expanded using the chain rule of differentiation. dj_ / = + ~ t + = 0 • (3.26) dP' dr' ' del Rearranging this gives the volumetric plastic strain rate as

,, s - aj_ (3.27) Set Substituting this into the flow rule of equation 3.25i gives A, €p ZL.p + Zlj- \ _ S _ _ _dPj_±_3j__ (3.28) ZL~ ZL ZL dP 34 dP

With this expression for A, the flow rule of equation 3 . 2 5 2 becomes

3 + 7 P = — (fp ^ f f p If (3.29) 7 2 dj_ di 34 dP In matrix form, equations 3.27 and 3.29 become

- _ ZL ZL r 2E. Sr1 ep 34„ (3.30) •<- ft.

1 1 " p T T z l z l l 34 3P 84 Note for P = Pc, that df/dP, el = 0, e* — 0, and isochoric perfect plasticity results.

This plastic compliance matrix of equation 3.30 has a determinant of zero and

the equation cannot be inverted to give stress rates in terms of plastic strain rates.

This can be overcome by adding terms representing the elastic strain rate to obtain

an equation for the total strain rate.

ZL _ ZL r 1 1 1T ZL i t €y 34 84 (3.31) 7 ■») ~dP 84 J

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3.4.2 Extracting the Stress-Strain Relation

Pande, Beer, and Williams [PBW90] demonstrate an approach for construction the

stress-strain relations for general plasticity models. The general approach is adapted

here for a formulation in terms of volume strain and a shear strain.

Expressing the elastic relation of equation 3.23 in rate form and substituting

expressions for ep and 7 P from the flow rule gives

P = I<\ev - and f = 2G fy - (3.32)

Rearranging this gives

Kev = P + \ K ^ - and 2 ( ? 7 = f + 3AG^. (3.33) OP OT

Multiplying the first equation by dffdP and adding the result to the second

equation times df/dr gives

df ■ df : — P + —T (3.34) dP dr

Substituting equation 3.25i into equation 3.26 provides an alternate expression

for the bracketed term,

df h df : — P + — T (3.35) dP dr _ del dp

Substituting this expression for the bracketed term into equation 3.34 and rearranging

gives an expression for A,

A = i ( ^ e . + '2G|(i| , (3.36)

where

(3.37) P [dPJ [dr) deldP'

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Finally, substituting the expression for X into equation 3.32 gives the desired

relation, P 2GK df df p \dPJ (3 dP dr (3.38) 3GK df df T 0 dPdr 2 0 (Iif 7 This equation for stress rates in terms of the total strain rates is convenient for

simulating strain controlled tests.

Critical State Model Limitations

The behavior represented by Carroll’s critical state plasticity model is quite broad

given its simple form. This is exemplified in test data comparisons presented in

chapter 6 . However, many types of behavior are not explicitly accounted for in

this model. Among the behaviors beyond the scope of this model are inhomogeneous

response, complex fluid interaction, and shear strain induced hardening or anisotropy.

Extending this model to treat complex fluid interaction is an active topic for many

researchers, but is not treated here. Favored numerical methods for coupling fluid

and structural response include finite element and finite difference formulations based

on time step splitting.

Large shear strains can induce hardening or softening behavior in many porous

materials. This may be particularly true in porous metals where the metal matrix is

capable of strain hardening. For rocks shear strain hardening is small compared to

hardening due to compaction in porous rocks, except under uncommonly large shear

strains.

For yielding on the dilantant side of the critical state yield surface, softening can

occur leading to localization. If not treated separately, the homogeneous plasticity

model develops an instability leading to localization in an infinitessimally thin band

with no net energy disappation contradicting behavior of real materials. This is

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considered the most important limitation for applying the model to rock behavior.

For this reason, the next three chapters outline approaches to handling inhomogeous

behavior before returning to the central issue of the homogeneous plasticity in the

final chapter.

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Chapter 4

Localization and Friction

Dilatant plasticity can lead to softening behavior which in turn leads to localization.

For continuum models without inherent length scales, such as nearly all stress-strain

type constitutive relations, localization occurs in a region of zero thickness. Thus

for continuum models without localization limiters, the volume of material being

deformed is vanishingly small and the process proceeds with no energy disapation.

While this may be a reasonable approximation of the behavior of extremely brittle

materials, most materials behave quite differently. Various approaches have been

proposed to mend or bipass tradditional continuum models.

4.1 Localization in One Dimension

Bazant [Baz89] proposed a series coupling model for localization in one dimension.

This model is significant in its elegant simplicity and as a basis for formulating a

complete three dimenstional model. In tensile extension of a bar, localization occurs

in a finite region of the bar. In post peak behavior, the localization region of the bar

elongates plastically, while the remainder of the bar unloads elastically. The series

coupling model merely recognizes that the tensile load is the same across both sections

of the bar and that total elongation of the bar is equal to the sum of the elongation

of the indivual reagions. This is pictured in figure 4.1

The length of the localization region at bifurcation represents a characteristic

length for the bar. By testing bars of different legnths, the characteristic length can

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(L-L ^/L 0

•Figure 4.1 Bazant’s series coupling model.

be computed using the series coupling model. While originally envisioned for ductile

necking, this same process can be used to backout a characteristic length for other

types of localizations.

4.2 Extending Localization Ideas to Three Dimensions

There are several difficulties in extending localization models from one to three di

mensions:

• Localization in one dimension is characterized by a single characteristic length

and a single plot of uniaxial stress versus uniaxial strain. Localization in three

dimensions involves the shape and size of the localization region, as well as a

nonlocal continuum stress strain relation.

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• There is very little data available for characterizing the problem of three dimen

sional localization. The one dimensional models are essentially empirical, but

there is insufficient data to build a fully empirical three dimensional model.

• In three dimensions, there are many more independent variables and the local

ization is much more complex. The localization is likely to break the symmetry

of the material response and make the material highly anisotropic.

• More diverse boundary conditions are possible in three dimensions.

Despite these difficulties, the problem is significantly simpler under relatively uni

form loading conditions. In this case, experimental and mathematical examination

lead to recognition of shear band formation as the dominant mode of localization.

4.3 Localization Band Analysis

Hadamard, Thomas, Hill, and Mandel were among the first to analyze bifurcation

into localization bands. Rudnicki and Rice [RR75] first extended the analysis to

specific forms of pressure sensitive plasticity. In doing so, they acknowledged that

accurate modelling of localization may depend on micromechanical response that

isn’t captured by typical continuum models. Indeed, subsequent studies[Ric80, MH85,

OLN87, BN 8 6 , Baz8 8 b, XN92] have found localization depends strongly on details

of the constitutive model such as presence of vertices in the yield surface, choice

of kinematic versus isotropic hardening, deviation from normality, rate dependence,

and introduction of length scales to limit localization. In effect, the continuum model

must accurately reflect the micromechanical response in order to accurately model

localization. In this regard, the ability to model localization can be a test of the

adequacy of a continuum model.

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Ortiz [OLN87] presented a simplified analysis of localization based on infinitesi

mal, rate independent, thermally decoupled deformation. As a first cut, this appears

to be an appropriate localization analysis to apply to the volumetric plasticity model

of section 3.4. In standard fashion, one considers a uniformly loaded infinite homoge

neous region and the admissibility of a bifurcation into regions of distinct response.

For simplicity the interface between regions is assumed to be a plane with unit normal

n. Tractions and displacements remain continuous, but bifurcation leads to a jump

in the displacement gradient. Maxwell’s compatibility conditions gives the form of

the jump in the displacement gradient, | a s

M = uh - uli = 9ini (4-1)

where ufj and u~j are values of the displacement gradient on opposite sides of the

interface. The vectors g and n are unknown at this point, but follow from the analysis.

While the magnitude of the jump is given by \g\, the nature of the bifurcation is given

by n and m = g/\g\. The corresponding jump in infinitesimal strain is

1 1 Ieu] = + gm ) = - —^m kni + m(nfc) (4.2)

Because of microstructural effects, bifurcations often arise in pairs and form a

localization band as shown in figure 4.2. For g 1 n, a shear band is formed. For

g || n, a separation band is formed. Between these extremes, are a range of mixed

mode localizations.

Up to the point of bifurcation, stresses and strains remain continuous. Thus,

(T) incremental stress-strain relations can be given in terms of the tangent stiffness, C

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Figure 4.2 A localization, band formed by jumps in the displacement gradient given by ±g on pair of planes normal to n in an otherwise homogeneous infinite region (adapted from [BZ93]).

The earliest bifurcation is determined by considering plastic loading on both sides

of the bifurcation plane, so that the relevant stiffness is the elastoplastic tangent

modulus.

Tractions must remain continuous at bifurcation, leading to a jump condition on

the stress,

Jn,-(T{j]l = n,-[

Combining equations 4.3 and 4.4 gives

n,-C£,Iefc/J = 0 (4.5)

Finally, combining this with equation 4.2, it follows that

= 0 (4.6)

For this to have a nontrivial solutions requires

det(A(n)) = 0 , where A(n) = . (4.7)

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Thus, bifurcation is possible for a plane normal to (n) when A(n) has a zero eigen

value, and the corresponding eigenvector, m gives the nature of the bifurcation.

Since this bifurcation criteria does not have a closed form solution, it must be

examined repeatedly at each step for a given loading path. Further discussion of this

procedure is given by Ortiz [OLN87].

Limitations of Bifurcation analysis

One limitation of shear band bifurcation analysis must be noted. The form of lo

calization given by equation 4.1 only represents one possible bifurcation mode. This

analysis does not exclude the possibility of earlier bifurcation modes. Drucker [Dru93]

suggested a rotating shear band is one possible bifurcation which can preceed the fix

band bifurcation in some circumstances. An experimental study by McClintock and

Zheng [MZ93] and a mathematical analysis by Bazant [Baz93] also suggest that lack

ing sufficient constraint from boundary conditions, a secondary bifurcation mode exist

that is asymmetric.

4.4 Localization Leading to Frictional Slip

In extreme cases, localization leads to fractures. Once fractures form, the dominant

form of inelastic deformation may switch from distributed plastic straining to localized

frictional slip on the fracture surfaces. Because the fracture is assumed to have formed

as a limitting response of localization, the potential slip plain is known. Thus, the

problem become one of merely selecting a criterion under which sliding occurs. A few

tradditioal formulations are discussed below along with relative merits.

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4.4.1 Simple Frictional Laws

For planar contact between two solids, experiments suggest a simple linear relation

between the normal contact stress, N, and shear stress, 5, necessary to initiate fric

tional sliding.

m = pW . (4.8)

This simple friction law remains the basis of most formulations of frictional deforma

tion. Coulomb proposed that frictional shear failure of rocks is described by a linear

relationship between the shear stress and normal stress

Ill'll = S0 + fiN (4.9)

where So is the cohesion and fi is the coefficient of friction.

4.4.2 Mohr Envelopes

Mohr generalized the Coulomb frictional limit by considering the failure shear stress

as an empirical function of normal stress.

||S|| =/(*). (4.10)

For f{N) linear, the Mohr envelope corresponds to Coulomb friction, which is why

people often refer to it as Mohr-Coulomb friction.

Two specific non-linear forms of f(N) given below may be usefulfor treating

sliding on rough surface. At low stresses, asperities on rough surfaceseffectively

increase the angle of friction, while at high stresses, these asperities are sheared off.

Parabolic Mohr’s Friction

A general parabolic form for f(N) may be given as:

f(N) = fiiy/N - No . (4.11)

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Where No is the hydrostatic strength (negative in tension) and /zx is a frictional

coefficient. The parameter jii can also be expressed in terms of the cohesive shear

strength So as: c Hi = —j== for N q < 0 and So > 0 (4-12) v No Parabolic friction limits should better account for limits shear strength at high normal

forces.

Hyperbolic Mohr’s Friction

A general hyperbolic form for f{N) may be given as:

f(N) . = h

Where No, and a are adjustable parameters. The parameter a can also be

expressed in terms of the cohesive shear strength So as:

a = '~T~ to \ for JVo < 0 and S0 > /*« N0 (4.14) 2 (IhIoNo)

Hyperbolicfriction limits is allows for effects of cohesion and roughness at lower

normal pressures and converges to the idealized linear response at higher normal

stress.

4.4.3 Frictional Slip Accompanied by Dilation

While frictional slip is usually thought of as purely a shear deformation, in can also

lead to dilation. Nemat-Nasser and Obata [NN088] considered microcracks as a

source of dilatancy in brittle materials. As shown in figure 4.4(a), sliding on microc

racks can introduce dilatancy. Li and Nordlund [LN93] extended this concept.

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|i= 1.5

\l=l Nn=-1 S„ = 8 |X= 0.5 jn= 0.5 S« =6

S0 = 4

Figure 4.3 Hyperbolic friction envelopes with varying N0, and So.

Sulak [Sul91] suggested dilation can also accompany slip on rough surfaces as

shown in figure 4.4(b), but the dilation may reverse as asperities on opposing faces

are soon sheared off and gaps close.

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fractured rock

fractured rock after shear displacement

Figure 4.4 Multiple mechanisms have been proposed for dilation under fricitonal response. (a)Pre-existing cracks under compression can lead to dilation via wing cracks (figure adapted from [LN93]). (b)Surface roughness can cause dilation during slip (This figure adapted from [Sul91]).

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Chapter 5

Combined Plasticity, Localization, and Friction

While volumetric plasticity, softening, localization, and frictional deformation have

been treated in the literature, it appears no treatment links all these phenomena

within one model.

5.1 Towards a Finite Element Model

The methods for solving volumetric plasticity models are fairly well established. A

particularly useful framework for these models is established in [SKG 8 8 ] by Simo,

Kennedy, and Govindjee. They address plastic and viscoplastic models with multi

ple yields surfaces which may intersect with non-smooth vertices. They present an

unconditionally stable algorithm based on a closest-point-projection return mapping.

Even softening within certain constraints on the effective moduli is permissible.

Their methods are proposed within the framework of an Eulerian formulation

where nodal displacements at step n + 1 are given implicitly in terms of a secant

stiffness matrix K and nodal forces / ,

K u n+1 = f . (5.1)

The components of K and f depend nonlinearly on u so that solutions at each

time step involve iterating until sufficient accuracy is obtained. The convergence

of this iteration is guaranteed using the radial return algorithm of Simo, Kennedy,

and Govindjee [SKG 8 8 ]. The time stepping algorithm is exact only in the limit as

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time step size go to zero, but is stable for large time steps and good convergence is

guaranteed by proper formulation of the expressions K and f . Specifically, Simo,

Kennedy, and Govindjee present a method of deriving a algorithmic secant stiffness

referred to as the consistent tangent modulus. Using this consistent moduli, quadratic

convergence is obtained as time step size is reduced.

5.1.1 Localization Treatments

Many methods have been proposed to treat localization within finite element models.

The methods can be categorized depending on whether finite element thickness is

•intended to be equal to. greater than, or less than the thickness of the localization to

be modeled.

• sub-h These approaches depend on element sizes smaller than the size scale of

the localizations to capture the physics of the localization.

• h-order These approaches depend being the approximate thickness of the local

ization

• super-h These approaches attempt to capture the effects of localization that are

much thinner than the element.

Approaches to sub-h modelling include imbricate finite elements, automatic mesh

refinement, and non-local continuum formulations. The primary difficulty with these

methods arises from the scale cascade mentioned in chapter 1. The localization often

occurs at a scale much smaller than the scale of interest in a particular analysis.

While automatic mesh refinement deals with this scale cascade, it often leads to an

unecessarilly large number of degrees of freedom. Of course where the localization is

of the same scale as the analysis, these methods are ideal.

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Methods that are h-order can use the element size to limit localization. However,

grid orientation can significantly affect the formation of localization bands. Where

band orientations are known in advance, this can be used in constructing a grid to

properly capture localization.

Methods that are super-h capture the effect of the localization without requiring

excess detail. This can significantly help in dealing with scale cascades. Examples

of super-h methods include discrete elements, fictitious crack models, and embedded

localization elements.

Embedded Localization Elements

Ortiz, Leroy, and Needleman [OLN87] seem to be the first to have implemented a

finite element with embedded localization. To embed localization within an element,

they:

• perform a bifurcation analysis to determine whether localization will occur,

• insert additional shape functions to admit a displacement discontinuity, and

• eliminate the amplitude of the localization from the global equations via static

condensation.

Belytschko, Fish, and Engelmann [BFE 8 8 ] proposed an element with internal

localization band based on a 3D variational formulation. Unlike the formulation

of Ortiz et al [OLN87] their localization could occur under constant strain and the

localization was assumed to have finite width.

One drawback common to embedded localization formulations proposed in by

Ortiz [OLN87], Belytschko[BFE 8 8 ], Dvorkin[DCG90, DA91], and Klisinksi [KRS91]

is that localization bands are discontinuous across elements. Adjacent elements are

likely to form localizations, but the locations are independently determined in each

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. element. This is physically inaccurate and might be expected to have significant

implications.

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Chapter 6

Model Optimization and Comparison to Experiments

For complex models, it can be difficult to determine values for constitutive parameters

directly from experimental measurements. By treating parameter determination as

an optimization problem, a model can be accurately and objectively matched to a

representative group of experiments. A MATLAB program for constitutive modeling

and parameter optimization is discussed below in section 6 .1 .

While parameter optimization can be viewed as merely curve fitting, it can also be

applied as a tool to examine constitutive assumptions or to compare one model with

another. This process can also help focus attention on otherwise obscure experimental

behavior. In section 6.2 below, several variations of Carroll’s critical state plastic

ity model are compared to laboratory test data on Boise sandstone and Monterey

diatomite. Section summarizes improvements to the constitutive models and conclu

sions regarding constitutive assumptions.

6.1 Parameter Determination as an Optimization Problem

The idea of treating parameter determination as an optimization problem is not new.

For example, Simo [SJPTS 8 ] used parameter optimization to assess a cap model in

relation to extensive experimental data on concrete. In particular Simo applied a

robust variation of the Marquardt-Levenberg algorithm to optimize parameters in a

strain driven cap model.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the current analysis, optimization speed was not as important as providing a

convenient tool to explore variations of constitutive models in relation to experimental

data. Both MATLAB and FORTRAN programs were developed, but the graphic

capabilities and flexibility of MATLAB were preferred over the faster results possible

in FORTRAN. While quasi-Newton methods were considered, a simplex search was

found to be more robust.

While both stress driven and strain driven constitutive implementations were de

veloped, the former was used most extensively in evaluating experimental data. The

stress driven constitutive model was implemented in incremental form following equa

tion 3.31..

The graphic interface created in MATLAB and shown in figure 6.1 provided for

flexible selection of test data, constitutive models, constitutive parameters, optimiza

tion techniques, and interactive plotting of results.

6.2 Evaluating Model Variations

Several extensions of Carroll’s 7 parameter critical state mode1, are addressed below.

In all a total of 13 additional parameters were considered as well as changes in the

significance of several of the original parameters. The motivation, effectiveness, and

further potential refinements are addressed below.

6.2.1 Nonlinear Hardening

Under extreme compaction, pore closure leads to a rapidly increasing rate of plastic

hardening. This behavior resembles compaction of a perfectly plastic hollow sphere

as studied by Torre [Tor48]. To incorporate this behavior in Carroll’s model, a re-

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Figure 6.1 A graphic interface was developed in MATLAB for parameter optimization and for comparing variations of constitutive models with experimental data.

placement for the linear hardening function Pc{ep) can be given as

Pc(epv) = Pco + Epc0ln • (6-1)

This form retains the initial yield point and hardening rate of Carroll’s model, while

incorporating the singular behavior as eg approaches o corresponding to complete

pore closure1.

1 Where eg is taken as an engineering volume strain and ignoring matrix compressibility, A — Aeg. In this context, complete pore closure occurs when eg = o where o is taken as the initial porosity. If eg is taken as a logarithmic volume strain, the form of equation torre.hardening:eq should be changed or or the parameters reinterpreted.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70

To retain the constant slope of the critical state line, the critical shear stress should

be related linearly to Pc(e%)- This can be done by expressing rs linearly in terms of

Pc, or by using the equation

o rs(e£) = t s0 + ETSo In (6 .2) ^0 — £v /

As seen in figure 6.2, the modified hardening functions of equations 6.1 and 6.2

accurately match experimental behavior for large deformations. The most significant

remaining discrepancy is the sudden onset of plasticity. This could be addressed by

incorporating Carroll and Holt’s [CH72] exact solution for compaction of an elastic

perfectly plastic .hollow sphere. Another alternative to address this problem follows

from the compaction solution for a hollow sphere of Mohr-Coulomb material provided

by Bhatt, Carroll, and Schatz.

hydrostatic compaction uniaxial strain compaction 25 C/3 Cu o 20 o o ixial 5 1 5 C£

0 0 0 10 20 -5 0 5 10 volumetric strain (%) strain (%)

Figure 6.2 High pressure hydrostatic and uniaxial strain compaction of Boise sandstone is fit very accurately by modified hardening functions.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71

6.2.2 Transverse Isotropy

Sedimentation can create layered materials that are transversely isotropic. Typical

of transversely isotropic materials, figure 2.14 shows very pronounced anisotropy be

tween axial and radial response under uniform hydrostatic loads. Transverse isotropy

can also be induce during axisymmetric deformation where there is a large differ

ence in axial and radial strain. This type of induced anisotropy can be expected in

sandstone even though figure 2 . 6 shows samples where relatively isotropic initially.

Transverse isotropic theories for elastic solids are well developed, but do not pre

dict potential is best understood in the case of linear elasticity, but can also enter

into plastic behavior. Both elastic and plastic transverse isotropy are treated below

as extensions to Carroll’s critical state model and are compared to experimental data.

Linear Elastic Transverse Isotropy

To fully represent transversely isotropic linear elasticity requires 5 elastic constants

compared to 2 for the fully isotropic case. The stress-strain relations for transverse

isotropy as given 2 by Lekhnitskii [Lek63] are

For diatomite and sandstone experiments addressed here, < 7 i3 = a12 =

so that Grz does not enter the measured response. Also, for these axisymmetric

experiments

2A sign error in one coefficient given by Lekhnitskii [Lek63, equation 3.14] and repeated else where [Zie71, equation 4.22] has been corrected here.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72

terms of volumetric and shear components as

1 c 7 p K (6.4) 3 n T 7 2 7 • where the compliance coefficients can be given in terms of Ez, Er, i/r, and vz as

( l- 4 i/z) 2(1 — vr) (6.5) K Ez + Er _L_ 2(l + 2i/,) (I-ur) ,and (6 .6 ) 2 G 3£, 3Er 2(1 + !/,) 2(1 - Vr) (6.7) 3 Ez 3 Er

This elastic behavior as given in equation 6.7 was added to the MATLAB pro

gram for constitutive modeling and optimization. As a result, overall agreement

between the constitutive model and the experimental data improved as exemplified

in figures 6.3 and 6.4.

Quite notably in figure 6.3, the agreement with experimental behavior at small

strains is degraded while agreement at large strains is dramatically improved. Clearly

the transverse isotropic elastic coefficients change during plastic strain representing

induced transverse isotropy. The variation of these coefficients during plastic straining

could be derived from a hollow ellipsoid micromodel similar to the isotropic results

for a hollow sphere presented in figure 1.7. The appearance of induced anisotropy is

particularly strong in sandstone tests 5, 8 , and 10 and in diatomite tests 13 and 22.

However, the level of induced isotropy varies from test to test and is not fit acceptably

using the same parameter values for all tests of a given material.

Plastic Anisotropy by Translating the Yield Surface

In equation 3.18, the yield surface remains centered on the pressure axis in stress

space. By relaxing this constraint and allowing the yield surface to translate in

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73

shear, anisotropic hardening can be captured. This was done in equation 3.20 by

introducing a shear offset function rc(ep). The offset is assumed to be a linear function

of volumetric strain,

rcK ) = tcQ + Ercepv . (6 .8 )

For sandstone, only slightly better fits are obtained by permitting yield surface

offsets. Furthermore, the optimum fit is obtained with T& fs 0 and Erc ~ 0. For these

sandstone test, the added complexity introduced by these parameters is unjustified.

Similar results are seen with the experimental data for diatomite. Improvement in

the agreement with experimental data is insignificant and once again the optimum fit

is obtained with tc 0 rH 0 cLH(i Erc ~ 0. This strongly supports assumption in 'Carroll’s

model that the yield surface remains centered on the pressure axis.

6.2.3 Modified Effective Stress Relations

The experimental sandstone data exhibits minor inconsistencies with the model dur

ing fluid pressure cycling following large straining. This suggests the effective stress

formulation is not capturing all the relevant effects of the pore pressure. The param

eter a of equation 1.26 which normally corrects for matrix compressibility can also

be used to explore possible improvements in the effective stress treatment.

When implemented within the constitutive modelling program, the effects of vari

ations of the parameter a are hidden by other minor discrepancies of the model.

6.3 Summary and Conclusions

Discrepancies between Carroll’s model and test data suggests induced isotropy is the

most significant effect which cannot be conveniently be incorporated into the model.

This effect would require a dependence on plastic shear strain which is not in keeping

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74

with the simple form of Carroll’s model. Also, as discussed in section 3.4.2, the shear

strain hardening and complex fluid interaction are not accounted for by the model.

For yielding on the dilatant side of the yield surface, critical state plasticity mod

els develop instabilities leading to localization. Without a special treatment of this

condition, the localization will form in an infinitely thin region and failure occurs with

zero energy dissipation. To correct this non-physical result, a bifurcation analysis can

be coupled with an embedded localization finite element. This approach was briefly

outlined in chapter 5. Implementation of a generalized version of Carroll’s plasticity

model within such a framework could be an extremely powerful model.

The simple critical state plasticity model proposed by Carroll appears to be a

powerful means for capturing homogeneous behavior of porous materials. In con

trast to existing models, Carroll model is particularly effective at capturing yield

ing during unloading. Refinements to account for nonlinear hardening, variation of

the shape of the yield surface, initial elastic transverse isotropy, variation of elas

tic moduli due to pore closure, variation of Terzaghi effective stress with matrix

compressibility, and anisotropic kinematic hardening have been examined. While

these are seen as secondary effects, the refinements may offer significant advantages

over Carroll’s simple model for some applications. Adding nonlinear hardening into

Carroll’s model appears to be the most significant refinement for matching sandstone

test data. Incorporation of transverse elasticity was the most significant refinement

in recreating tests on Monterey diatomite.

shear stress (1000 psi) effective pressure (1000 psi)isotropy transverse elastic with model / model isotropic / experiment / 15 10 10 Figure 6.3 The addition of two parameters to model transverse isotropic isotropic transverse model to parameters twoof addition The 6.3 Figure 4 0 5 2 6 8 0 0 0 elasticity significantly improves agreement with experimental data as data experimental with agreement improves significantly elasticity volume strain (%) volume strain (%)strain volume (%) strain volume 2 shear strain (%) shear strain (%) strain shear (%) strain shear 2 Sandstone test #7 versus 11 and 1311 modelsand parameter versus test Sandstone #7 xmlfe eefrsnsoets $7. test sandstone for here exemplified 4 4 6 6 8 C o 8 o o o o o o . s- to 0 1 2 4 0 6 0 0 2 2 4 4 6 6 75 8 poue ih emiso o h cprgt we. ute erdcin rhbtd ihu pr ission. perm without prohibited reproduction Further owner. copyright the of ission perm with eproduced R j o A •H JG o effective pressure (100 psi) eprmn / isotropic model / experiment co r o co C/3 u S 0.5 elasticity slightly improves agreement with experimental data, particular for particular data, experimental with agreement improves slightly elasticity Figure 6.4 The addition of two parameters to model transverse isotropic isotropic transverse model to parameters oftwo addition The 6.4 Figure 2 A 0 1 initial shear strain response, as exemplified here for diatomite test #17. test diatomite for here exemplified as response, strain shear initial 0 volumestrain (%) ha tan() shear strain (%) shear strain (%) 15 5 Diatomite test #17 versus 11 and 1311Diatomite test parameter and versus #17 models 10 [ model with transversewith elasticmodel isotropy o a. 2 0 0 volumestrain (%) 5 10 15 76 77

Appendix A

Visualizing Relations Between Elastic Constants

The isothermal linear elastic relation between stress and strain can be given as

a = Ce where C is a tensor of 81 coefficients. For isotropic materials, energy and

symmetry considerations allow C to be expressed in terms of only two coefficients.

For example, Cijki = + 2G6ikSji- Equivalent expressions can be written in

terms of any suitable pair of elastic constants. Table A.l gives five common elastic

constants and relations between them.

The smooth surface given by E{G, K) = (9KG)/(-SK + G) embodies the relations

between E, G, and K. The first law of thermodynamics requires these moduli to be

positive for simple linear elastic materials, so the plot in figure A.l is limited to the

positive octant.

Figure A.2 demonstrates the relationship of Poisson’s ratio to E, G, and K.

Constant values of Poisson’s ratio correspond to straight lines from the origin. The

lower limit, v = —1, coincides with the G axis where K = E = 0 - since K , E, and

v are not independent on that line, relations in terms of ( K,E ), (K, v), and ( K,E)

break down along that line. Similarly, the upper limit, v = .5, coincides with the

I( axis where G = E = 0 and relations in terms of (E, G), (E , i/), and (G, u) break

down along that line.

1 Shear modulus, sometimes denoted by fi, together with A are referred to as the Lame constants. Here, we refer to A as Lame’s constant and denote shear modulus by G.

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Young’s shear bulk, Poisson’s Lame’s modulus, E modulus, G modulus, K ratio, v constant1, A GE G (E -2 G ) (E,G) - - E 1 3(3 G-E) 2 G ~ l ' 3G-E 3 KE 3/C - E 3/C(3/C - E) (E,K) - - 9 K-E 6 K 9fC — E E E Eu (E ,r) - - 2 ( 1 + v) 3(1 - 2v) ( 1 + j/ ) ( 1 - 2u) 9KG - - 3/C - 2G K - j G (G,K) 3K + G 2(3 K + G)

2 G( 1 + u) (G,u) 2G{l + v) - - 2 Gv 3(1-2!/) 1 - 2v G( 3A + 2 G) A (G, A) - A + §G - A+ 6 -' 2(A + G) 3/C(l - 2v) (K,u) 3/C(l - 2u) - - ZKv 2 ( 1 + u) l + v K(I< - A) (/C, A) \{K-X) - A - (3/C - A) ZK - A A(1 + i/)(l - 2v) A(1 — 2v) A(1 + v) (i/,A) - - V 2v •iu

Table A .l Relations between elastic constants (adapted from Sokolnikoff [Sok56, page 71]).

Negative values of Poisson’s ratio describe materials that expand in the transverse

direction under uniaxial tension. Engineers often consider such behavior unrealistic

and restrict Poisson’s ratio to the range 0 < v < 0.5. This range corresponds to the

shaded region in figure A.l. For this lower limit of v — 0, the expressions for K(E, v)

and G(E, v) lead to E = 2 G = 3K.

Figure A.3 partially reveals the relation of A to u, E, G, and K. Noting that

E(v, A), G(v, A), and /C(z/, A) all contain the factor A/V, lines of constant A jv are

shown in figure A.3. For constant values of v ^ 0, those curves make it clear that E,

G, and K are proportional to A which represents a generalized modulus. Curves given

by constant values of A are also shown in figure A.3 and reveal that A = 0 coincides

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0 0.5

1.5

2 0 G

Figure A .l The relationship between elastic moduli can be plotted as a surface in three dimensions.

with the line for v = 0. Positive and negative regions of A and v coincide, and so A

may be restricted to non-negative values for the same reason and v. Figure A.3 also

show that for v = .5, K = A, and a similar statement can be made for v = — 1 where

A = 2G/3,

It can be noted that the relations G(v, A), E{v, A), and K(i/, A) breakdown for

v = 0 where v is not independent of A. Similarly, A(i/, I(), G(v,E), and G(v,K)

break down at v = — 1

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0.5 fv'=0.4 ...

/v=a

2 0 G

Figure A.2 Constant values of Poisson’s ratio, v, in relation to E, G, and K.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e A .3 C< VAhies aad^ i Q * b * « O S , e > &nd /;'

fright 0 l^/7er er Kpr^ c i0r Pmhlbited, ■ d w'thout Psrmj, 'lssion. 82

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