Geotech Book (2)

(1)

2

Overview

Day 1

• Lecture 1 Introduction • Lecture 2 Physical Testing

• Lecture 3 Constitutive Models: Part 1 • Lecture 4 Constitutive Models: Part 2

(2)

Analysis of Geotechnical Problems with Abaqus © Dassault Systèmes, 2008

Day 2

• Lecture 5 Analysis of Porous Media

• Workshop 2 Pore Fluid Flow Analysis: Consolidation

• Lecture 6 Modeling Aspects

• Workshop 3 Pore Fluid Flow Analysis: Wicking

4

Overview

Additional Material

• Appendix 1 Stress Equilibrium and Fluid Continuity Equations • Appendix 2 Bibliography of Geotechnical Example Problems

(3)

The Abaqus Software described in this documentation is available only under license from Dassault Systèmes and its subsidiary and may be used or reproduced only in accordance with the terms of such license.

This documentation and the software described in this documentation are subject to change without prior notice.

Dassault Systèmes and its subsidiaries shall not be responsible for the consequences of any errors or omissions that may appear in this documentation.

No part of this documentation may be reproduced or distributed in any form without prior written permission of Dassault Systèmes or its subsidiary.

Abaqus, the 3DS logo, SIMULIA and CATIA are trademarks or registered trademarks of Dassault Systèmes or its subsidiaries in the US and/or other countries.

Other company, product, and service names may be trademarks or service marks of their respective owners. For additional information concerning trademarks, copyrights, and licenses, see the Legal Notices in the Abaqus Version 6.8 Release Notes and the notices at:

http://www.simulia.com/products/products_legal.html. 6

Revision Status

Updated for 6.8 8/08 Workshop 2 Updated for 6.8 8/08 Workshop 3 Updated for 6.8 8/08 Workshop 1 Updated for 6.8 8/08 Appendix 2 Updated for 6.8 8/08 Appendix 1 Updated for 6.8 8/08 Lecture 6 Updated for 6.8 8/08 Lecture 5 Updated for 6.8 8/08 Lecture 4 Updated for 6.8 8/08 Lecture 3 Updated for 6.8 8/08 Lecture 2 Updated for 6.8 8/08 Lecture 1

(4)

© Dassault Systèmes, 2008

Lecture 1

L1.2

Overview

• Introduction

• Overview of Geotechnical Applications • Classical and Modern Design Approaches • Some Cases for Numerical (FE) Analysis • Experimental Testing and Numerical Analysis • Requirements for Realistic Constitutive Theories

(5)

L1.4

Introduction

• In this lecture we present an overview of geotechnical applications and discuss the philosophy on which the usage of numerical (finite element) analysis for geotechnical problems is based.

• Lecture 2 deals with experimental testing and how it relates to the calibration of constitutive models for geotechnical materials. • The different Abaqus constitutive models applicable to geotechnical

materials are presented in Lectures 3 and 4.

• Their usage, calibration, implementation, and limitations are discussed.

• In Lecture 5 we outline the treatment of porous media in Abaqus and discuss the coupling between fluid flow and stress/deformation. • Several modeling issues relating to geotechnical situations are

(6)

L1.6

Overview of Geotechnical Applications

(7)

• Main characteristics of soil materials • Soil is composed of solid grains and

voids; the voids may be fully or partially filled with pore fluid.

• When the soil is loaded, the pressure in the pore fluid gets modified. The pore fluid then flows and the soil deforms with time.

• The deformations and pore fluid flow will be governed by the mechanical and permeability properties of the soil, the mechanical and pore pressure boundary conditions, contact

conditions, presence of reinforcements, etc.

L1.8

Overview of Geotechnical Applications

• The following need to be considered before modeling geotechnical problems:

Select appropriate material model according to the physical material behavior.

Which material model will be appropriate for representing the physical material behavior?

In some cases detailed modeling of contact conditions is not necessary.

Is contact modeling important for the particular application?

In the coupled setting the pore fluid flow affects the deformations and the deformations affect the pore fluid flow

In an uncoupled setting the pore fluid flow does not affect the deformations, and the problem can then be modeled as a deformation problem or just a pore fluid flow problem.

Is coupled pore fluid diffusion – stress analysis necessary?

(8)

• The following need to be considered before modeling geotechnical problems:

For ascertaining equilibrium at time=0, one needs to begin with a Geostatic step, wherein known initial conditions, boundary conditions, and external loads are applied.

How detailed is the information one has about the initial conditions in the model?

Use model change, if necessary, in setting up the analysis steps. Using model change one can activate or deactivate elements and contact pairs and can model construction and excavation.

Is modeling of excavation and construction important?

Use embedded elements, if necessary, to model reinforcements. For example, for modeling of ties, soil nails, etc. Does the application contain

reinforcement?

L1.10

Overview of Geotechnical Applications

• Dams

• For the modeling of dams one needs to take into account: • Fully and partially saturated flow

• Appropriate material models for the dam material • Mechanical behavior

• Pore fluid flow behavior

(9)

• The following features also need to be taken into account for the modeling of dams:

• Sequential construction—model change • Seepage analysis—transient consolidation • Earthquake loading—dynamics

L1.12

Overview of Geotechnical Applications

• Modeling of jointed rocks • Response depends on joint

orientations and asperities of adjacent surfaces

• Joints can open and close, which can produce induce anisotropy at macro scale

• Need to take into account these features in a continuum setting

http://www.outreach.canterbury.ac.nz/resources /geology/glossary/joints.jpg

inclined joint set

(10)

• Modeling of pile foundations • Contact modeling is important • Modeling of friction between the

pile and the soil

• Cohesive elements with damage characteristics

• Soil plasticity

• Coupled consolidation analysis for modeling short and long term behavior

Distribution of pore pressure immediately after a pile is

driven

L1.14

Stability analysis using Abaqus/Explicit

Overview of Geotechnical Applications

• Slope failure analysis

• Unreinforced and reinforced slopes • Soil plasticity

• Material failure • Embedded elements

Stability analysis of unreinforced and reinforced slope at working load levels using

(11)

• Retaining walls • Contact

• Soil plasticity models • Coupled consolidation • Embedded elements

• Landfills

• Quasi-static analysis • Creep modeling

• Geomembranes and geo-synthetics • Embedded elements

• Contact

http://www.versa-loc.biz/versatech/images/wallcomp_soils.jpg

http://www.keepingavclean.com/images/landfill105.jpg

L1.16

Overview of Geotechnical Applications

• Mining and tunneling

• Excavations and construction • Liners and ties

• Material needs to be added or removed

• Contact conditions need to be taken into account

• Transient material degradation needs to be represented in the model

Bench Top heading Invert and Concrete floor Shotcrete liners layer 1 layer 2 layer 3 layer 4 Gr avity Gr avity

(12)

• Nuclear waste disposal

• Sequentially coupled thermo-hydromechanical (THM) analysis • Ground water seepage modeling

• Cross fluid flow, heat flow, and mechanical contact interactions between soil and non-soil regions

• Examine worst-case scenarios, provide data on environmental impact

Heater (diameter 0.9) Bentonite blocks Steel liner Granite Heaters Bentonite barrier Concrete plug

Service zone, control and data acquisition system

4.54 1.004.54 4.34 17.4 2.7 70.4 (Dimensions in meters) 2. 2 8

Principal access tunnel to KWO Granite

Courtesy: SIMULIA/Scandinavia

Independent thermal expansion properties for:

• Soil matrix • Pore fluid Void ratio-dependent permeability

L1.18

Overview of Geotechnical Applications

• Abaqus provides the following features for geotechnical applications: • Full coupling between stress and pore fluid diffusion

• Anisotropic and void ratio-dependent permeability, velocity-dependent permeability

• Wide range of soil plasticity models including dilation and compaction available in Abaqus/Standard and Abaqus/Explicit

• Jointed material model for modeling jointed rocks

• Saturated and unsaturated media and automatic computation of phreatic surface and capillary zone

(13)

• Abaqus features for geotechnical applications (contd.):

• Contact between permeable/permeable and permeable/impermeable regions

• Stress equilibrium as well as fluid continuity maintained • Embedded elements for modeling reinforcement

• Elements can be added or deleted for modeling construction and excavation

• Thermal expansion for pore fluid and solid matrix

• Independent bulk moduli for soil grains, pore fluid, and elastic properties for soil matrix

• Moisture and gel swelling

Classical and Modern

Design Approaches

(14)

• In the classical approach two basic types of calculations are done: failure estimates and deformation estimates.

• Failure estimates are based on rigid perfectly plastic stress-strain assumptions:

• Examples:

L1.22

Classical and Modern Design Approaches

• Examples (cont.):

• The result is a factor of safety, which is evaluated based on experience (design code).

Bearing capacity of a foundation

(15)

• Deformation estimates assume linear elastic behavior with average elastic properties:

• Foundation settlement example: settlement

p

is bearing pressure

b

is width of foundation

E

,

v

are average elastic properties

f

is shape factor (based on small scale tests) 2

1 v

w

pb

f

E

⎛

−

⎞

=

_⎜

_⎟

⎝

⎠

, L1.24

Classical and Modern Design Approaches

• In the modern approach failure and deformation characteristics are obtained from the same analysis. The analysis requires:

• a complete constitutive model, which can include the loading and unloading behavior, and

(16)

• Numerical (finite element) analysis can handle arbitrary geometries. • When finite elements are used, we have the option of allowing different

shear strains along the failure surface, and thus the stress state along the failure surface can change from point to point.

Some Cases for Numerical

(FE) Analysis

(17)

• Slope stability and deformation:

• Classical limit failure calculations can predict ultimate stability of a slope.

• Numerical finite element analysis is necessary for the calculation of deformations and detailed soil behavior.

L1.28

Some Cases for Numerical (FE) Analysis

• Construction of an earth dam and subsequent filling of reservoir: • Detailed soil behavior plays an important role.

• Local soil failure may trigger overall collapse or hamper functionality.

• The event sequence - construction, filling of reservoir, and long-term consolidation - must be considered using numerical modeling.

(18)

• Tunneling:

• The initial state of stress of the soil or rock mass is important.

• Stability depends on: • Initial state of stress • Sequence of excavation

• Stabilizing aids such as liners and rock bolts

Experimental Testing and

Numerical Analysis

(19)

Laboratory testing

Constitutive model

Calibration

Finite element model

Small or large scale testing

Comparison of model predictions with test results

Design prototype

L1.32

Experimental Testing and Numerical Analysis

• Experimental testing and constitutive model calibration:

• Laboratory tests should represent the load or deformation conditions of the physical problem.

• Choose a constitutive model that represents the major characteristics; minor features may be ignored.

• Obtain test measurements suitable for the specific constitutive model. • Calibrate the model parameters using measured values of repeatable

(20)

• Finite Element Analysis:

• Numerical model should capture the important features of the physical problem without irrelevant detail.

• Use of an appropriate constitutive model is critical, although simplifications are often justifiable.

• Provides a tool to aid design, supplementing engineering judgment and experience.

Requirements for Realistic

Constitutive Theories

(21)

• Realistic constitutive models should be:

• able to represent the macro behavior as dictated by micromechanics

• suitable for numerical implementation

• able to represent material behavior in any relevant spatial situation: • one-dimensional

• plane strain • axisymmetric • three-dimensional

• able to reasonably extrapolate to conditions that cannot be reproduced with laboratory testing equipment

(22)

Lecture 2

L2.2

Overview

• Physical Testing

• Basic Experimental Observations

(23)

L2.4

Physical Testing

• Geotechnical materials:

• are generally voided and sensitive to volume changes.

• Volume changes are closely tied to the magnitude of the hydrostatic pressure stress:

• Hence, it is important to test the materials over the range of hydrostatic pressure of interest.

• Standard tests:

• Hydrostatic (or isotropic) compression tests • Oedometer (or uniaxial strain) tests • Triaxial compression and extension tests

• Uniaxial compression tests (a special case of triaxial compression) • Shear tests

(24)

• A practical constitutive model should require for calibration only the information generated by these standard tests.

• Assumptions need to be made regarding tensile and true triaxial behavior as direct tensile tests and true triaxial tests are difficult to perform.

• The diversity of geotechnical materials means that a wide range of behaviors is possible.

• What follows are some very general observations for frictional materials.

(25)

• Hydrostatic or isotropic compression test:

L2.8

Basic Experimental Observations

(26)

• Triaxial compression tests:

L2.10

• The “critical state” concept:

• Casagrande defined “critical state” as the state (for monotonic loading) at which continued shear deformation can occur without further change in effective stress and volume (void ratio) of the material.

Basic Experimental Observations

(27)

• Cyclic tests:

• Isotropic compression or uniaxial strain (oedometer) tests

• Triaxial or uniaxial compression tests provide the deviatoric behavior

L2.12

Basic Experimental Observations

(28)

• Essential aspects of behavior of voided frictional materials:

+ Nonlinear stress-strain behavior + Irreversible deformations

+ Influence of hydrostatic pressure stress on “strength” + Influence of hydrostatic pressure on stress-strain behavior + Influence of intermediate principal stress on “strength” + Shear stressing-dilatancy coupling

L2.14

Basic Experimental Observations

• Essential aspects of behavior of voided frictional materials (contd.):

+ Influence of hydrostatic pressure stress on volume changes + Hardening/softening related to volume changes

+ Stress path dependency + Effects of small stress reversals

– Effects of large stress reversals (hysteresis)

– Degradation of elastic stiffness after large stress reversals + Included in Abaqus

(29)

L2.16

• Basic requirements for laboratory testing of geotechnical materials: • The specimens must be tested under the assumption that they

represent an average material behavior:

• In the ground there will be some variation within the same soil or rock mass, and the spatial scale of such variations may be large compared to laboratory test specimens.

• The tests must simulate the in situ conditions as closely as possible: • drainage conditions

• density of the material • range of stresses

• For deep mining cases these may be difficult to obtain.

• All stresses and strains must be measured throughout the stress-strain response:

• to allow complete characterization of the constitutive behavior.

Testing Requirements and Calibration of

Constitutive Models

(30)

• Laboratory testing should be guided by a previously proposed constitutive model:

• The understanding of this model is necessary for the correct interpretation of laboratory tests.

• The model parameters should be physical and measurable in practicable experiments.

L2.18

• Laboratory tests for calibrating relevant components of constitutive models:

• Isotropic compression test and uniaxial strain (oedometer) test:

• One test is required to calibrate hydrostatic behavior.

• One unloading is necessary to calibrate the elastic part of this behavior. • Triaxial compression tests:

• Two (preferably more) tests are required to calibrate the shear behavior and its hydrostatic pressure dependence. • One unloading in each test is necessary

to calibrate the elastic part of this behavior.

Isotropic compression

Triaxial compression

Testing Requirements and Calibration of

(31)

• Triaxial extension tests:

• The triaxial extension test is performed in a triaxial machine.

• In the presence of confining lateral pressures the vertical stress is gradually decreased till failure.

• Two (preferably more) such tests are required to calibrate the intermediate principal stress dependence of the shear behavior.

• Direct tension test:

• The direct tension test or the uniaxial tension test is performed by applying uniaxial strain along a particular direction.

• One test is required to calibrate the tensile behavior of cohesive materials like rocks or soils with cohesion.

L2.20

• Truly triaxial or cubical tests: • Many tests are required to calibrate the behavior of the material when subjected to different stresses in all directions.

Testing Requirements and Calibration of

Constitutive Models

(32)

• Shear box tests and indirect tension (Brazilian) test: • These are useful in calibrating the cohesive properties of

materials.

Shear box test Indirect tension test

• Multiple unloading-reloading cycles in any of above tests: • These are useful for calibrating the effects of large stress

(33)

Lecture 3

L3.2

Overview

• Stress Invariants and Spaces • Overview of Constitutive Models • Elasticity

• Plastic Behavior of Soils • Mohr-Coulomb Model

(34)

L3.4

Stress Invariants and Spaces

• Stress in three dimensions: three direct and three shear components.

• Symmetry of stress tensor:

• Abaqus convention: tensile stress is positive.

12 21 13 31 23 32

(35)

• Principal stresses: stresses normal to planes in which shear stresses are zero.

L3.6

Stress Invariants and Spaces

• In two dimensions (Mohr’s circle):

2 2 11 22 11 22 1, 2 12 12 11 22 2 2 2 tan 2

σ

τ

θ

σ

+ ⎛ − ⎞ = ± _⎜ _⎟ + ⎝ ⎠ = −

(36)

• Stress decomposition: deviatoric plus hydrostatic:

• Abaqus invariants:

p

= −

S

I

.

σ

(

)

1 3 1 trace( ) 3 3 : 2 9 : 2 p q r = − = ⎛ ⎞ =_⎜ ⋅ _⎟ ⎝ ⎠ S S S S S pressure stress, ,

Mises equivalent stress, ,

third invariant, .

σ

L3.8

Stress Invariants and Spaces

• For all models except the Mohr-Coulomb, also define the deviatoric stress measure:

so that

t = q/K

in triaxial tension

(r = q)

and

t = q

in triaxial compression

(r = − q)

. If

K = 1

,

t = q

.

K

is typically between

0.8

and

1.0

.

• t

= constant is a “rounded” surface in the deviatoric plane. 3 1 1 1 1 2 q r t K K q ⎡ _⎛ _⎞_{⎛ ⎞} ⎤ ⎢ ⎥ = + − −_⎜ _{⎟⎜ ⎟} ⎝ ⎠ ⎢ ⎝ ⎠ ⎥ ⎣ ⎦ ,

(37)

• Useful planes:

L3.10

Stress Invariants and Spaces

(38)

• Deviatoric (or

Π

) plane:

(39)

• Elasticity models: • Linear, isotropic

• Porous, isotropic (nonlinear)

• Damaged, orthotropic (nonlinear; used in concrete, jointed material) • Plasticity models:

• Open surface, pressure independent (Mises)

• Open surface, pressure dependent (Drucker-Prager, Mohr-Coulomb) • Closed surface (Cam-clay, Drucker-Prager with Cap)

• Multisurface (jointed material) • Nested surfaces (bounding surface*)

L3.14

Overview of Constitutive Models

• Other inelastic models:

• Continuum damage theories* • Endochronic theories*

None of the available models (with the possible exception of the jointed material model) is capable of accurately handling large stress reversals such as those occurring during cyclic loading or severe dynamic events.

(40)

L3.16

Elasticity

• Linear isotropic elasticity: Young’s modulus and Poisson’s ratio

Input file usage:

*ELASTIC E,ν

(41)

• Porous elasticity: • Nonlinear, isotropic

• Pressure stress varies as an exponential function of volumetric strain:

or

where

J

el

− 1

_{is the nominal volumetric elastic strain.}

J = dV/dV

° is the ratio of current volume to reference volume. = ln(Jel₎ _and_Jel_{= exp (} _).

is the logarithmic measure of volumetric elastic strain. is the elastic tensile stress limit, can be zero or non zero.

(

)

0

(

( )

)

0 1 exp 1 exp el el el t t vol e p p p p

ε

(deviatoric stiffness dependent on the pressure stress)

where the instantaneous shear modulus, , is

are material parameters,

p

0is the

initial value of hydrostatic pressure stress, and

e

0

ε

−

+

=

+

. ˆ G el t p G v

κ

, , ,

(42)

•

ε

volel has an arbitrary origin, defined so that 0 0 el vol p

=

p at

ε

=

. Porous elasticity L3.20

Elasticity

• Usage *POROUS ELASTIC, SHEAR=POISSON κ, ν, ptel

*POROUS ELASTIC, SHEAR=G

(43)

• ∗INITIAL CONDITIONS, TYPE=RATIO is required to define the initial void ratio (porosity) of the material when porous elasticity is used.

• User subroutine VOIDRIcan be used to specify complex initial void ratio distributions.

(44)

• Plasticity models are characterized by the following: • Yield surface

• It is a surface defining the yield criterion for the plasticity model. • Flow rule

• The condition determining the plastic strain after yielding.

• Plastic flow can be associated or nonassociated.

• Associated flow means that the plastic strain direction is along the direction of yielding, or normal to the yield surface.

• Nonassociated flow means that the plastic strain direction is not along the direction of yielding.

L3.24

Plastic Behavior of Soils

• Dilation

• Dilation during plastic flow means that the material changes its volume while undergoing yielding.

Yielding without dilation

(ductile metals) Yielding with dilation (soils)

Volume increases

(45)

• Yielding in soils is associated with internal friction between soil grains. • Internal friction between soil grains results from interference at the grain

level.

• Dilation angle

ψ =

arctan(

v

/

u

).

• The plastic flow will be associative when

ψ = φ

.

Dilation angleψ Friction angle φ Volume increases v u Direction of yielding L3.26

• Yielding in ductile metals is generally not accompanied by volume change.

• The yielding is due to dislocation.

• The flow is associative as there is no internal friction and the material does not dilate during yielding.

• Both

φ

and

ψ

are zero.

• Yield stress does not depend on pressure.

Pressure Shear

stress

Yield surface

Plastic Behavior of Soils

(46)

• Yielding in soils is generally accompanied by volume change. • Existence of internal friction (nonzero

φ

)

• Existence of cohesion

• similar to resistance against dislocation in ductile metals • The material may dilate during yielding in shear.

• The flow can be associative (

ψ = φ

), or nonassociative. • Yield stress depends on pressure.

Pressure Shear

stress

Yield surface

(47)

• The characteristics of the Mohr-Coulomb plasticity model in Abaqus: • The model is intended for granular materials like soils under monotonic

loading.

• It does not consider rate dependence.

• The linear isotropic elastic response is followed by non-recoverable response idealized as being plastic.

• The yield behavior depends on the hydrostatic pressure:

• The material becomes stronger as the confining pressure increases. • The yield behavior may be influenced by the magnitude of the

intermediate principal stress.

L3.30

Mohr-Coulomb Model

• The model includes isotropic hardening or softening.

• The inelastic behavior is generally accompanied by volume change. • The flow rule may include:

• inelastic dilation as well as • inelastic shearing

• The plastic flow potential is smooth and nonassociated. • Material properties can be temperature dependent.

(48)

• Description of the Mohr-Coulomb Model: • The model requires linear isotropic elasticity. • The Mohr-Coulomb yield function is given by:

where

R

_mc

(Θ,φ)

is a measure of the shape of the yield surface in the deviatoric plane,

φ

is the slope of the Mohr-Coulomb yield surface in the

R

mc

q

− p

stress plane, which is commonly referred to as the friction angle of the material,

tan

0

mc

F

=

R q

−

p

φ

− =

c

,

1

1 sin

cos

tan

3

3 3 cos

mc

R

π

φ

⎛

⎞

⎛

⎞

=

_⎜

Θ +

_⎟

+

_⎜

Θ +

_⎟

⎝

⎠

⎝

⎠

,

0 ≤ <

φ

90

; L3.32

Mohr-Coulomb Model

c

is the cohesion of the material; and

Θ

is the deviatoric polar angle defined as

• The Mohr-Coulomb model assumes that the hardening is defined in terms of the material’s cohesion,

c

.

• The cohesion can be defined as a function of plastic strain, temperature, or field variables.

• The hardening is isotropic.

3

cos(3 )

r

q

(49)

(a) (b)

Yield surface in the meridionial plane (a) and the deviatoric plane (b)

pl

0 c

=

c

ε

=

(50)

ε

defines the rate at which

G

approaches the asymptote.

• The flow potential tends to a straight line in the meridional stress plane as the meridional eccentricity tends to zero.

Mohr-Coulomb flow potential in the meridional plane

L3.36

Mohr-Coulomb Model

• R

_mw

(Θ, e,φ)

controls the shape of

G

in the deviatoric plane:

• The deviatoric eccentricity,

e

, describes the “out-of-roundedness” of the deviatoric section in terms of the ratio between the shear stress along the extension meridian

(Θ = 0)

and the shear stress along the compression meridian

(Θ =

π / 3)

.

• The default value of the deviatoric eccentricity is calculated as and allows the Abaqus Mohr-Coulomb model to match the

behavior of the classical Mohr-Coulomb model in triaxial compression and tension.

(

)

(

) (

)

(

)

(

)

(

)

(

)

2 2 2 2 2 2 2

4 1

cos

2

1 ,

3 2 1

cos

2

1 4 1

cos

5

4

mw mc

e

R

e

π φ

−

Θ +

−

_⎛

_⎞

=

_⎜

_⎟

⎝

⎠

−

Θ +

−

Θ +

−

3 sin 3 sin e

φ

− = +

(51)

• The deviatoric eccentricity may have the following range:

½

≤ e ≤ 1.0

. • The plastic flow in the deviatoric plane is always nonassociated.

L3.38

Mohr-Coulomb Model

• Usage and calibration

• Linear, isotropic elasticity must be used. • The ∗MOHR COULOMB option is used

to define

ψ

,

φ

,

e

, and

ε

.

• The ECCENTRICITY parameter is used to define

ε

. The default value is 0.1.

• The DEVIATORIC ECCENTRICITY parameter

can be used to define

e

. The deviatoric eccentricity may have the following range:

• If

e

is defined directly, Abaqus matches the classical Mohr-Coulomb model only in triaxial compression.

• The friction angle,

φ

, and the dilation angle,

ψ

, can be functions of temperature and field variables.

1 1.0 2< ≤e .

φ

ψ

e

ε

(52)

• The *MOHR-COULOMB option must always be accompanied by the *MOHR-COULOMB HARDENING material option:

• It defines the evolution of the cohesion,

c

: • It can be made temperature and/or

field dependent.

• *EXPANSION can be used to introduce thermal volume change effects.

L3.40

Mohr-Coulomb Model

• The plastic flow in the deviatoric plane is always nonassociated;

• Needs the unsymmetric solver (∗STEP, UNSYMM=YES).

(53)

• Calibration of the Mohr-Coulomb model:

• Use the critical stress states from several different triaxial tests. • The critical stress states are plotted in the meridional plane to

provide an estimate of: • friction angle,

φ

, and • cohesion,

c

.

• The dilation angle,

ψ

, is chosen so that the volume change during the plastic deformation matches that seen experimentally.

• Hardening data is obtained from one of the triaxial tests.

φ

c

Pressure mc R q p L3.42

Mohr-Coulomb Model

• The Abaqus Mohr-Coulomb model uses a smooth plastic flow potential: • It does not always provide the same plastic behavior as the

classical (associated) Mohr-Coulomb model, which has a faceted flow potential.

• With the default value of deviatoric eccentricity,

e

, Abaqus does match classical Mohr-Coulomb behavior under triaxial extension or compression.

• Benchmark Problem 1.14.5, Finite deformation of an elastic-plastic

granular material, shows how to match the Abaqus Mohr-Coulomb model to the classical Mohr-Coulomb model for plane strain deformation.

(54)

• Example: Limit Load Analysis of a Strip Foundation

• This example involves the limit load analysis of a strip foundation (Benchmark Problem 1.14.4).

• A rigid footing 10 feet wide on a granular soil material of 12 feet depth is loaded and the average pressure vs. displacement of the footing is computed.

• The model is assumed to be in a plane strain condition.

• CPE8R finite elements and CINPE5R infinite elements are used. • The footing is driven by prescribed displacement.

Symmetry plane

Right half of footing

Soil

L3.44

Mohr-Coulomb Model

• Boundary conditions:

• Symmetry boundary conditions along the symmetry plane • Fixed at the bottom

• Rough friction assumed between the footing and the soil

• Soil nodes lying on the footing are constrained to move vertically in unison and with zero horizontal displacements using linear constraint equations.

• Mohr-Coulomb model:

• Elastic modulus 30,000 psi and Poisson’s ratio 0.3 • Friction angle 20o

• Cohesion 10 psi • Dilation angles used:

• 0o _{(nondilatant flow)} • 20o _{(dilatant flow)}

(55)

• The Mohr-Coulomb material data is specified as follows: *MATERIAL,NAME=A1 *ELASTIC 30000.,0.3 *MOHR COULOMB 20.,0.

*MOHR COULOMB HARDENING 10.,

• A prescribed displacement of 5 inches is applied and the reaction is computed.

Elastic modulus and Poisson’s ratio

Friction angle and dilation angle

Cohesion L3.46

Mohr-Coulomb Model

• Results Terzaghi (175 psi) Prandtl (143 psi) Mohr Coulomb with no dilation Mohr Coulomb with dilation

(56)

• Results

• The Abaqus results lie close to the Terzaghi solution for strip foundations on granular soils.

• The solutions by the Prandtl theory and the Terzaghi theory are from: • Chen, W. F., Limit Analysis and Soil Plasticity, Elsevier,

Amsterdam, 1975.

• The Prandtl solution considers the stresses due to the weight of the soil as small compared to the soil strength.

• This theory was often used for predicting the behavior of metals indented by rigid materials.

• The Terzaghi solution takes into account the soil and overburden weight and also the material cohesion.

(57)

• The characteristics of the Extended Drucker-Prager models: • These models are intended for monotonic loading.

• For example, the limit load analysis of a soil foundation.

• These models are the simplest available models for simulating frictional materials.

• The elastic response in these models is followed by a non-recoverable response, which is idealized as being plastic.

• The material in these models is initially isotropic.

• The yield behavior of these models depends on the hydrostatic pressure.

• The material becomes stronger as the confining pressure increases. • The yield behavior may be influenced by the magnitude of the

intermediate principal stress.

L3.50

Extended Drucker-Prager Models

• The characteristics of the Extended Drucker-Prager models (contd.): • These models differ in the manner in which the hydrostatic pressure

dependence is introduced.

• These models include isotropic hardening or softening.

• The inelastic behavior in these models is generally accompanied by volume change:

• The flow rule may include inelastic dilation as well as inelastic shearing.

• These models can incorporate strain-rate dependent material properties. • The material properties in these models can be made temperature

dependent.

• Either linear elasticity or nonlinear porous elasticity can be used with these models.

(58)

• Three Extended Drucker-Prager models based on the yield surface shape in the meridional plane are available in Abaqus:

• Linear • Hyperbolic • Exponent

• The particular form of the material model can be chosen based on: • the kind of material being modeled

• the available experimental data for calibration of the model parameters

• the range of pressure stress values that the material is likely to see

L3.52

Extended Drucker-Prager Models

• The Linear Drucker-Prager model

• The yield surface of the linear model is written as

• p

is the equivalent pressure stress

• t

is the deviatoric stress measure •

β

is the friction angle

• d

is cohesion and is related to the hardening input data

• The model allows for separate dilatation and friction angles

tan 0

(59)

• The model provides a non-circular yield surface in the deviatoric plane: • It allows for the matching of different yield values in tension and

compression.

• It provides flexibility in fitting experimental results.

• However, the surface is too smooth to be a close approximation to the Mohr-Coulomb surface.

Kis the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression.

L3.54

Extended Drucker-Prager Models

• We assume a (possibly) nonassociated flow rule, where the direction of the inelastic deformation vector is normal to a linear plastic potential,

G

:

where ,

c

is a constant that depends on the type of hardening data, in uniaxial compression,

in uniaxial tension, and in pure shear.

ψ is the dilation angle in the p–tplane. This flow rule definition

precludes dilation angles ,

which is not likely to be a limitation for real materials.

,

pl pl

d

G

d

c

ε

σ

∂

=

∂

tan

G

= −

t

p

ψ

11 pl pl

d

ε

=

d

ε

11 pl pl

d

ε

=

d

ε

3

pl pl

d

ε =

γ

(

)

71.5 tan 3 ψ > ° ψ >

(60)

• The flow is associated in the deviatoric plane but nonassociated in the

p–t

plane if

ψ ≠ β

.

• For

ψ = 0

, the material is nondilatational. • If

ψ = β

, the model is fully associated.

L3.56

Extended Drucker-Prager Models

• Hyperbolic model

• The hyperbolic yield criterion is a continuous combination of the maximum tensile stress condition of Rankine (tensile cut-off) and the linear Drucker-Prager condition at high confining stress. It is written as

where

d

'_{is the hardening parameter that is related to the hardening input} data as

if hardening is defined by uniaxial compression,

σ

_c; if hardening is defined by uniaxial tension,

and

l

₀as constants during hardening.

0 ₀ t₀tan l =d′ −p

β

0 t p d ′₀ L3.58

Extended Drucker-Prager Models

• The yield surface is a von Mises circle in the deviatoric stress plane. (The

K

parameter is not available for this model.)

is not available for this model.)

• The material parameters

a

,

b

, and

p

t: • Can be specified directly, or

• Abaqus will determine them from specified triaxial test data using a least squares fit.

(63)

• Flow in the hyperbolic and exponent models

• Flow potential governing the plastic flow in the hyperbolic and general exponent models:

where,

ψ

is the dilation angle in the meridional plane at high confining pressure, is the initial yield stress,

ε

is the eccentricity defining the rate at which the function approaches its asymptote

• The flow potential tends to a straight line as the eccentricity tends to zero.

(

)

2 ₂ 0tan tan G=

ε σ

ψ

+q −p

ψ

, 0

σ

L3.62

Extended Drucker-Prager Models

• The function approaches the linear Drucker-Prager flow potential asymptotically at high confining pressure and intersects the hydrostatic pressure axis at 90°.

• The potential is continuous and smooth.

• It ensures that the flow direction is always defined uniquely.

(64)

• Associated flow is obtained in the hyperbolic model if

β = ψ

and

• In the general exponent model the flow is always nonassociated in the meridional plane.

• The default flow potential eccentricity is

ε = 0.1

.

• It provides an almost constant dilatational angle over a wide range of confining pressure stress values.

• Increasing the value of

ε

provides more roundedness to the flow potential.

• The dilation angle increases smoothly as the confining pressure decreases.

ε < 0.1

may lead to convergence problems if the material is subjected to low confining pressures because of the very sharp curvature of the flow potential near its intersection with the

p

-axis.

0 0tan l

ε

σ

ψ

= . L3.64

Extended Drucker-Prager Models

• Usage

• These models are invoked by the ∗DRUCKER PRAGER material option.

• The SHEAR CRITERION parameter is set to LINEAR, HYPERBOLIC, or EXPONENT to define the yield surface shape.

• The ∗DRUCKER PRAGER HARDENING option should also be used.

• This option defines the evolution of the yield stress in uniaxial

compression

(TYPE=COMPRESSION), in uniaxial tension (TYPE=TENSION), or in pure shear (TYPE=SHEAR).

(65)

• The yield function can be made rate dependent by using the ∗RATE DEPENDENT option or by specifying the yield stress as a function of the plastic strain rate.

• Rate dependency is rarely used for geotechnical materials.

• It may however be important for polymers.

L3.66

Extended Drucker-Prager Models

• The elasticity is defined by: • linear elasticity, or • porous elasticity.

• All material parameters in these models can be made temperature or field dependent.

– Thermal expansion can be used to introduce thermal volume change effects.

–∗INITIAL CONDITIONS,

TYPE=RATIO is required to define the initial void ratio of the material if porous elasticity is used.

(66)

• Analyses using a nonassociated flow version of the model may require the use of the unsymmetric solver because of the resulting unsymmetric plasticity equations.

• If the default symmetric solver is used when the flow is nonassociated, Abaqus may not find a converged solution.

L3.68

Extended Drucker-Prager Models

• Matching experimental data

• The simplest modified Drucker-Prager model requires at least two experiments for calibration.

• Simplest model: linear, rate independent, temperature independent, and yielding independent of the third stress invariant.

• Most common experiments performed:

• uniaxial compression (for cohesive materials) • triaxial compression or tension tests

• shear tests for cohesive materials

• The uniaxial compression test involves compressing the sample between two rigid platens.

• Record the load and displacement in the direction of loading • Record the lateral displacements for measuring volume changes • Triaxial test data are required for a more accurate calibration.

(67)

• A triaxial machine enables application of a confining pressure and a differential stress.

• Several tests covering the range of confining pressures of interest are usually performed.

• The stress and strain in the direction of loading are recorded, together with the lateral strain, to enable calibration of volume changes.

L3.70

Extended Drucker-Prager Models

• In a triaxial compression test the specimen is confined by pressure and an additional compression stress is superposed in one direction.

• Thus, the principal stresses are all negative, with

0 ≥

σ

1

= σ

2

≥

σ

3.

• The stress invariant values in triaxial compression are

• The triaxial results can, thus, be plotted in the

q–p

plane.

(

1 3

)

1 3

1 2 3

p= −

σ σ

+ , q=

σ σ

− , r= −q, t=q.

(68)

• The stress state corresponding to some user-chosen critical level provides one data point for calibrating the yield surface material parameters.

• Choose the stress at the onset of inelastic behavior or the ultimate yield stress.

• Additional data points are obtained from triaxial tests at different levels of confinement.

• These data points define the shape and position of the yield surface in the meridional plane.

L3.72

Extended Drucker-Prager Models

• Defining the shape and position of the yield surface is adequate to define the model if it is to be used as a failure surface.

• To incorporate isotropic hardening, one of the stress-strains curves from the triaxial tests can be used to define the hardening behavior.

• The curve that represents hardening most accurately over a wide range of loading conditions should be chosen.

• Unloading measurements in these tests are useful to calibrate the elasticity, particularly in cases where the initial elastic region is not well defined.

(69)

• Linear Drucker-Prager model

• Fitting the best straight line through the results provides the friction angle

β

.

L3.74

Extended Drucker-Prager Models

• Triaxial tension test data are also needed to define

K

:

• Confine the specimen by pressure and then reduce the pressure in one direction.

• In this case the principal stresses are

0 ≥

σ

₁

≥

σ

₂

=

σ

₃. • The stress invariants are now

K

can be found by plotting

q

versus

p

.

– K

is the ratio of the values of

q

for triaxial tension and compression at the same value of

p

.

• The dilation angle

ψ

is obtained from shear tests, and it must be chosen such that a reasonable match of the volume changes during yielding is obtained. • Generally,

0 ≤

ψ ≤ β

.

(

1 3

)

1 3 1 2 3 q p q r q t K

σ

σ σ

= − + , = − , = , = .

(70)

• Hyperbolic model

• Use the triaxial compression results at high confining pressures to obtain

β

and

d

'_{for the hyperbolic model.}

• Hydrostatic tension,

p

_t, is also needed to complete the calibration.

tan d′ β − −p_t p d ′ q β

(

)

2 2 0 t 0tan tan 0 F= d′ −p β +q −p β−d′= Hyperbolic : L3.76

Extended Drucker-Prager Models

• Exponent model

• Abaqus provides a capability to determine the material parameters

a

,

b

, and

p

_trequired for the exponent model from triaxial data:

• A “best fit” of the triaxial test data at different levels of confining stress is performed.

(71)

• The data points obtained from triaxial tests are specified using the ∗TRIAXIAL TEST DATA option. • The TEST DATA parameter is

required on the ∗DRUCKER PRAGER option to use this feature. • The ∗TRIAXIAL TEST DATA option must be used with the ∗DRUCKER PRAGER option.

• This capability allows for all three parameters,

a

,

b

, and

p

t, to be calibrated, or, if some of the parameters are known, to calibrate only the unknown parameters.

L3.78

Extended Drucker-Prager Models

• Example: Limit Load Analysis of a Strip Foundation

• This is a continuation of the limit load analysis of a strip foundation example previously analyzed with the Mohr-Coulomb material model. • Review of the model:

• A rigid footing 10 feet wide on a granular soil material of 12 feet depth is loaded and the average pressure vs. displacement of the footing is computed.

• The model is assumed to be in a plane strain condition, and uses CPE8R and CINPE5R elements.

• The footing is driven by prescribed displacement.

Symmetry plane

Right half of footing

(72)

• Extended Drucker-Prager model:

• Elastic modulus 30,000 psi and Poisson’s ratio 0.3

• Friction angle 30.64o _{for dilatant flow and 30.16 for nondilatant flow}

• Stress ratio 1.0

• Yield stress 20.2 psi for dilatant flow and 19.8 psi for nondilatant flow • Dilation angles used: 0o _{(nondilatant flow) and 30.16}o _{(dilatant flow)}

• These material parameters have been selected such that they match with the Mohr-Coulomb material data under plane strain conditions.

• See “Matching Mohr-Coulomb parameters to the Drucker-Prager model,” Section 18.3.1 of the Abaqus Analysis User’s Manual.

L3.80

Extended Drucker-Prager Models

• The Drucker-Prager material data is specified as follows:

*MATERIAL,NAME= A1 *ELASTIC

30000.,0.3

*DRUCKER PRAGER,SHEAR CRITERION=LINEAR 30.64,1.0,0.

*DRUCKER PRAGER HARDENING 20.2,0.

• A prescribed displacement of 5 inches is applied and the reaction is computed.

Elastic modulus and Poisson’s ratio

Friction angle, stress ratio, and dilation angle Yield stress and plastic strain

(73)

Analysis of Geotechnical Problems with Abaqus © Dassault Systèmes, 2008 • Results Terzaghi (175 psi) Prandtl (143 psi) Mohr Coulomb with no dilation Mohr Coulomb with dilation Drucker Prager with dilation Drucker Prager with no dilation L3.82

Extended Drucker-Prager Models

• Results

• It is observed that the Drucker-Prager results for the dilatant and nondilatant cases are close to the respective Mohr-Coulomb solutions. • The nondilatant Drucker-Prager and Mohr-Coulomb models lead to a

softer response and a lower limit load than the corresponding dilatant versions.

• Presence of dilation results in stiffening of a volumetrically compressed material.

• The limit load values obtained for the nondilatant Drucker-Prager and Mohr-Coulomb models lie between the Prandtl and Terzaghi solutions.

(74)

Lecture 4

L4.2

Overview

• Modified Drucker-Prager/Cap Model • Critical State (Clay) Plasticity Model • Jointed Material Model

• Soil Plasticity Models - Summary

(75)

L4.4

Modified Drucker-Prager/Cap Model

• The characteristics of the Modified Drucker-Prager/Cap model:

• This model is intended to simulate the constitutive response of cohesive geological materials.

• It adds a “cap” yield surface to the linear Drucker-Prager model • to bound the model in hydrostatic compression, and • to help control volume dilatancy when the material yields in

shear.

• The elastic response is followed by a non-recoverable response idealized as being plastic.

(76)

• The yield behavior depends on the hydrostatic pressure. • There are two distinct regions of behavior.

• On the failure surface the material is perfectly plastic. • On the cap yield surface it hardens and also stiffens.

The hardening/softening behavior is a function of the volumetric plastic strain.

• The yield behavior may be influenced by the magnitude of the intermediate principal stress.

L4.6

Modified Drucker-Prager/Cap Model

• The inelastic behavior is generally accompanied by volume changes.

• On the failure surface the material dilates. • On the cap surface it compacts.

• At the intersection of these surfaces, the material can yield indefinitely at constant shear stress without changing volume.

• Under large stress reversals the model provides reasonable material response on the cap region;

• however, on the failure surface region the model is acceptable only for essentially monotonic loading.

(77)

• Description

• The model can use linear elasticity or nonlinear porous elasticity. • The model uses two main yield surface segments.

• a linearly pressure-dependent Drucker-Prager shear failure surface • a compression cap yield surface

• The Drucker-Prager failure surface itself is perfectly plastic (no hardening), but plastic flow on this surface produces inelastic volume increase, which causes the cap to soften.

• The Drucker-Prager failure surface is

•

β

is the angle of friction;

d

is the cohesion of the material.

•

t

is the measure of the deviatoric stress, and it allows matching of different stress values in tension and compression in the deviatoric plane.

tan

0

s

F

= −

t

p

β

− =

d

.

L4.8

(78)

• The cap yield surface is

where

R

is a material parameter that controls the shape of the cap,

α

is a small number, and

is an evolution parameter that represents the volumetric plastic strain driven hardening/softening.

(

)

₍

₎

(

)

2 2

tan

0

1 cos

c a a

Rt

F

p

R d

p

β

α α

β

⎡

⎤

=

−

+

_⎢

_⎥

−

+

=

+ −

⎣

⎦

,

( )

pl a vol

p

ε

L4.10

Modified Drucker-Prager/Cap Model

• The cap yield surface

• is elliptical with constant eccentricity in the meridional (

p–t

) plane. • includes dependence on the third stress invariant in the deviatoric

plane.

• hardens or softens as a function of the volumetric plastic strain. • Volumetric plastic compaction (when yielding on the cap)

causes hardening.

• Volumetric plastic dilation (when yielding on the shear failure surface) causes softening.