Geo Mechanics Module Users Guide

VERSION 4.2

User s Guide

Geomechanics Module

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Geomechanics Module User’s Guide

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Version: May 2011 COMSOL 4.2

Part No. CM021801

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C o n t e n t s

C h a p t e r 1 : I n t r o d u c t i o n

Geomechanics Module Overview 6

Where Do I Access the Documentation and Model Library? . . . . . . 6

Typographical Conventions . . . . . . . . . . . . . . . . . . . 8

C h a p t e r 2 : G e o m e c h a n i c s T h e o r y

General Geomechanics Theory 12

Sign Conventions for Stress and Strain Analysis . . . . . . . . . . . 12

Defining the Stress Invariants . . . . . . . . . . . . . . . . . . 13

Defining the Yield Surface . . . . . . . . . . . . . . . . . . . 16

Defining Perfectly Plastic Material Models 17

Von Mises Criterion. . . . . . . . . . . . . . . . . . . . . . 17

Tresca Criterion . . . . . . . . . . . . . . . . . . . . . . . 18

Plasticity Models for Soils 20

Mohr-Coulomb Criterion . . . . . . . . . . . . . . . . . . . 20

Drucker-Prager Criterion . . . . . . . . . . . . . . . . . . . 21

Elliptic Cap . . . . . . . . . . . . . . . . . . . . . . . . . 23

Matsuoka-Nakai Criterion . . . . . . . . . . . . . . . . . . . 24

Lade-Duncan Criterion . . . . . . . . . . . . . . . . . . . . 25

Theory for the Cam-Clay Material Model 27

About the Cam-Clay Material Model . . . . . . . . . . . . . . . 27

Volumetric Elastic Behavior . . . . . . . . . . . . . . . . . . . 29

Hardening and Softening . . . . . . . . . . . . . . . . . . . . 30

Including Pore Fluid Pressure . . . . . . . . . . . . . . . . . . 31

Failure Criteria for Concrete, Rocks, and Other Brittle Materials 32

Bresler-Pister Criterion . . . . . . . . . . . . . . . . . . . . 32

C O N T E N T S | 3

4 | C O N T E N T S

Willam-Warnke Criterion . . . . . . . . . . . . . . . . . . . 33

Ottosen Criterion . . . . . . . . . . . . . . . . . . . . . . 34

Hoek-Brown Criterion . . . . . . . . . . . . . . . . . . . . 35

Generalized Hoek-Brown Criterion. . . . . . . . . . . . . . . . 37

Stress-Strain Theory 39

Small Strain Assumption . . . . . . . . . . . . . . . . . . . . 39

Plastic Flow . . . . . . . . . . . . . . . . . . . . . . . . . 39

Effective and Volumetric Plastic Strains. . . . . . . . . . . . . . . 41

References for the Geomechanics Module 43

C h a p t e r 3 : T h e G e o m e c h a n i c s M a t e r i a l M o d e l s

Working with the Geomechanics Material Models 46

Adding a Material Model to a Solid Mechanics Interface . . . . . . . . 46

Soil Plasticity . . . . . . . . . . . . . . . . . . . . . . . . 47

Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Cam-Clay Material Model . . . . . . . . . . . . . . . . . . . 52

Geomechanics Materials 54

The Output Materials Properties . . . . . . . . . . . . . . . . . 54

1

I n t r o d u c t i o n

Welcome to the Geomechanics Module User’s Guide. This chapter provides a Geomechanics Module Overview and the type of modeling you can achieve with its extensive set of material models for geomechanics and soil mechanics. In addition, it includes some general information about documentation and models.

The Geomechanics Module extends the capabilities of the Structural Mechanics Module. For general information about structural analysis, see the Structural Mechanics Module User’s Guide.

5

6 | C H A P T E R

Geome chan i c s Modu l e Ov e r v i ew

Note: The Geomechanics Module requires a Structural Mechanics Module license.

The Geomechanics Module is an optional package that extends the Structural Mechanics Module to the quantitative investigation of geotechnical processes. It is designed for researchers, engineers, developers, teachers, and students, and suits both single-physics and interdisciplinary studies.

The Geomechanics Module includes an extensive set of fundamental material models, such as the Drucker-Prager and Mohr-Coulomb criteria and the Cam-Clay model in soil mechanics. These material models can also couple to any new equations created, and to physics interfaces (for example, heat transfer, fluid flow, and solute transport in porous media) already built into COMSOL Multiphysics and its other specialized modules.

In this section:

• Where Do I Access the Documentation and Model Library?

• Typographical Conventions

Where Do I Access the Documentation and Model Library?

A number of Internet resources provide more information about COMSOL Multiphysics, including licensing and technical information. The electronic documentation, Dynamic Help, and the Model Library are all accessed through the COMSOL Desktop.

Note: If you are working directly from a PDF on your computer, the blue links do not work to open a model or documentation referenced in a different user guide. However, if you are using the online help desk in COMSOL Multiphysics, these links work to other modules, model examples, and documentation sets.

1 : I N T R O D U C T I O N

T H E D O C U M E N T A T I O N

The COMSOL Multiphysics User’s Guide and COMSOL Multiphysics Reference Guide describe all the interfaces included with the basic COMSOL license. These guides also have instructions about how to use COMSOL Multiphysics, and how to access the documentation electronically through the COMSOL Multiphysics help desk.

To locate and search all the documentation, in COMSOL Multiphysics:

• Click the buttons on the toolbar or

• Select Help>Documentation ( ) or Help>Dynamic Help ( ) from the main menu

and then either enter a search term or look under a specific module in the documentation tree.

T H E M O D E L L I B R A R Y

Each model comes with a theoretical background and step-by-step instructions to create the model. The models are available in COMSOL as MPH-files that you can open for further investigation. Use both the step-by-step instructions and the actual models as a template for your own modeling and applications. SI units are used to describe the relevant properties, parameters, and dimensions in most examples, but other unit systems are available.

To open the Model Library, select View>Model Library ( ) from the main menu, and then search by model name or browse under a Module folder name. If you also want to review the documentation explaining how to build a model, select the model and click Model PDF or the Dynamic Help button ( ) to reach the PDF or HTML version, respectively. Alternatively, select Help>Documentation in COMSOL and search by name or browse by Module.

If you have feedback or suggestions for additional models for the library (including those developed by you), feel free to contact us at [email protected].

C O M S O L WE B S I T E S

Main corporate web site: http://www.comsol.com/

Worldwide contact information: http://www.comsol.com/contact/

Online technical support main page: http://www.comsol.com/support/

COMSOL Support Knowledge Base, your first stop for troubleshooting assistance, where you can search for answers to any COMSOL questions:http://www.comsol.com/support/knowledgebase/

G E O M E C H A N I C S M O D U L E O V E R V I E W | 7

http://www.comsol.com/

http://www.comsol.com/contact/

http://www.comsol.com/support/

http://www.comsol.com/support/knowledgebase/

http://www.comsol.com/support/knowledgebase/

8 | C H A P T E R

Product updates: http://www.comsol.com/support/updates/

C O N T A C T I N G C O M S O L B Y E M A I L

For general product information, contact COMSOL at [email protected].

To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to [email protected]. An automatic notification and case number is sent to you by email.

C O M S O L C O M M U N I T Y

On the COMSOL web site, you find a user community at http://www.comsol.com/community/. The user community includes a discussion forum, a model exchange, news postings, and a searchable database of papers and presentations.

Typographical Conventions

All COMSOL user guides use a set of consistent typographical conventions that should make it easy for you to follow the discussion, realize what you can expect to see on the screen, and know which data you must enter into various data-entry fields.

In particular, you should be aware of these conventions:

• Click text highlighted in blue to go to other information in the PDF. When you are using the online help desk in COMSOL Multiphysics, these links also work to other Modules, model examples, and documentation sets.

• A boldface font of the shown size and style indicates that the given word(s) appear exactly that way on the COMSOL Desktop (or, for toolbar buttons, in the corresponding tooltip). For example, the Model Builder window ( ) is often referred to and this is the window that contains the model tree. As another example, the instructions might say to click the Zoom Extents button ( ), and this boldface font means that you can expect to see a button with that label (when you hover over the button with your mouse) on the COMSOL Desktop.

• The names of other items on the COMSOL Desktop that do not have direct labels contain a leading uppercase letter. For instance, we often refer to the Main toolbar— the horizontal bar containing several icons that are displayed on top of the user interface. However, nowhere on the COMSOL Desktop, nor the toolbar itself, includes the word “Main”.


http://www.comsol.com/support/updates/

http://www.comsol.com/community/

• The forward arrow symbol > indicates selecting a series of menu items in a specific order. For example, Options>Results is equivalent to: From the Options menu, select Results.

• A Code (monospace) font indicates you are to make a keyboard entry in the user interface. You might see an instruction such as “Enter (or type) 1.25 in the Current

density edit field.” The monospace font also is an indication of programming code. or a variable name. An italic Code (monospace) font indicates user inputs and parts of names that can vary or be defined by the user.

• An italic font indicates the introduction of important terminology. Expect to find an explanation in the same paragraph or in the Glossary. The names of other user guides in the COMSOL documentation set also have an italic font.

The Difference Between Nodes, Buttons, and Icons• Node: A node is located in the Model Builder and has an icon image to the left of it.

Right-click a node to open a Context Menu and to perform actions.

• Button: Click a button to perform an action. Usually located on a toolbar (the Main toolbar or the Graphics window toolbar, for example), or in the upper right corner of a Settings window.

• Icon: An icon is an image that displays on a window (for example, the Model Wizard or Model Library) or displays in a Context Menu when a node is right-clicked. Sometimes selecting an icon from a node’s Context Menu adds a node with the same image and name, sometimes it simply performs the action indicated (for example, Delete, Enable, or Disable).

G E O M E C H A N I C S M O D U L E O V E R V I E W | 9

10 | C H A P T E R

2

G e o m e c h a n i c s T h e o r y

The Geomechanics Module contains new materials which are used in combination with the Structural Mechanics Module to account for the plasticity of soils and failure criteria in rocks, concrete and other brittle materials.

In this chapter:

• General Geomechanics Theory

• Defining Perfectly Plastic Material Models

• Plasticity Models for Soils

• Theory for the Cam-Clay Material Model

• Failure Criteria for Concrete, Rocks, and Other Brittle Materials

• Stress-Strain Theory

• References for the Geomechanics Module

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12 | C H A P T E R

Gene r a l G eome chan i c s T h e o r y

Note: See also the Theory for the Solid Mechanics Interface in the Structural Mechanics User’s Guide for more information.

In this section:

• Sign Conventions for Stress and Strain Analysis

• Defining the Stress Invariants

• Defining the Yield Surface

Sign Conventions for Stress and Strain Analysis

The theory of plasticity was initially developed in continuum mechanics where tensile stresses are usually considered to be positive quantities and compressive stresses are negative quantities. Under this convention, force and displacement components are considered positive if directed in the positive directions of the coordinate axes. Tensile normal strains and tensile normal stresses are treated as positive (Ref. 11).

Engineering analysis and design for soil and rock structures, however, are in most cases concerned with compressive stresses (Ref. 1). Therefore, in geotechnical applications the opposite sign convention is usually adopted because compressive normal stresses are more common than tensile ones (Ref. 5).

The Geomechanics Module follows the sign convention of continuum mechanics as stated in Ref. 1 (positive tensile stresses). This is consistent with the notation used in the Structural Mechanics Module. This means the principal stresses often have a negative sign, and are sorted as 123.

The convention used in Ref. 1 refers to the hydrostatic pressure (trace of the stress Cauchy tensor) with a positive sign. The use of the first invariant of Cauchy stress tensor I1() is preferred through this document, in order to avoid misunderstandings with the convention in the Structural Mechanics Module (where pressure is positive under compression, or equivalently, it has the opposite sign of the Cauchy stress tensor’s trace).

2 : G E O M E C H A N I C S T H E O R Y

Defining the Stress Invariants

The starting point for defining the theory of plastic deformation of soils is the definition of the first, second, and third invariants for the Cauchy stress tensor . These invariants are defined as follows:

(2-1)

The first invariant I1 is the trace of the tensor, the second invariant I2 is the sum of the principal two-rowed minors of the determinant of , and the third invariant I3 is the determinant of . (Ref. 1).

When 1, 2, and 3, represent the principal components of the stress tensor, these invariants can be written as

The principal components of the stress tensor are the roots of the characteristic equation (Cayley–Hamilton theorem)

which comes from calculating the eigenvalues of the tensor .

Note: The invariants I1, I2, and I3 can be called in user-defined yield criteria by referencing the variables solid.I1s, solid.I2s and solid.I3s.

Defining the deviatoric stress as the traceless tensor

introduces us to the first, second, and third deviatoric stress invariants

I1 ii trace ==

I2 12--- iijj ijji– 1

2--- I1

2 :– ==

I3 det 13--- : I1

3– I1I2+==

I1 1 2 3+ +=

I2 12--- I1

2 12 2

2 32

+ + – =

I3 1 2 3 =

12 23 13+ +=

3 I12

– I2 I3–+ 0=

det ij ij– 0=

ijD ij

13---I1ij–=

G E N E R A L G E O M E C H A N I C S T H E O R Y | 13

14 | C H A P T E R

(2-2)

As defined above J20. In soil plasticity, the most relevant invariants are I1, J2, and J3. I1 represents the effect of mean stress, J2 represents the magnitude of shear stress, and J3 is the direction of the shear stress.

Note: The invariants J2 and J3 can be called in user-defined yield criteria by referencing the variables solid.II2s and solid.II3s.

O T H E R I N V A R I A N T S

It is possible to define other invariants in terms of the primary invariants. One common auxiliary invariant is the Lode angle

(2-3)

The Lode angle is bounded to 03 when the principal stresses are sorted as 123 (Ref. 1).

Following this convention, =corresponds to the tensile meridian, and =3corresponds to the compressive meridian. The Lode angle is part of a cylindrical coordinate system (the Haigh–Westergaard coordinates) with height (hydrostatic axis)

and radius .

Note: The Lode angle is undefined at the hydrostatic axis, where all three principal stresses are equal (1=2=3=I1/3) and J2=0. To avoid division by zero, the Lode angle is computed from the inverse tangent function atan2, instead of the inverse cosine, as stated in Equation 2-3.

J1 iiD = 0=

J2 12---ij

DjiD =

13---I1

2 I2–=

J3 det D 13--- D D :D 2

27------I1

3 13---I1I2– I3+===

3cos 3 32

-----------J3

J23 2

------------=

I1/ 3= r 2J2=


Note: The Lode angle and the effective (von Mises) stress can be called in user-defined yield criteria by referencing the variables solid.thetaL and solid.mises.

The octahedral plane (also called -plane) is defined perpendicular to the hydrostatic axis in the Haigh–Westergaard coordinate system. The stress normal to this plane is oct=I1/3, and the shear stress on that plane is defined by .

The functionals described in Equation 2-1 and Equation 2-2 enter into expressions that define various kind of yield and failure surfaces. A yield surface is a surface in the 3D space of principal stresses which circumscribe an elastic state of stress.

P R I N C I P A L S T R E S S E S

The principal stresses 1, 2, and 3) are the eigenvalues of the stress tensor, and when sorted as they can be written by using the invariants I1 and J2 and the Lode angle 03 (Ref. 1):

The principal stresses can be obtained by using the expressions in Table 2-1.

TABLE 2-1: PRINCIPAL STRESSES SORTED AS IN THE STRUCTURAL MECHANICS MODULE

PRINCIPAL STRESS

EXPRESSION

1 solid.I1s/3+sqrt(4/3*solid.II2s)*cos(solid.thetaL)

2 solid.I1s/3+sqrt(4/3*solid.II2s)*cos(solid.thetaL-2*pi/3)

3 solid.I1s/3+sqrt(4/3*solid.II2s)*cos(solid.thetaL+2*pi/3)

oct 2/3J2=

1 2 3

1 =13---I1

4J23

---------- cos+

2 =13---I1

4J23

---------- 23

------– cos+

3 =13---I1

4J23

---------- 23

------+ cos+

G E N E R A L G E O M E C H A N I C S T H E O R Y | 15

16 | C H A P T E R

Defining the Yield Surface

A yield criterion serves to define the stress condition under which plastic deformation occurs. Stress paths within the yield surface result in purely recoverable deformations (elastic behavior), while paths intersecting the yield surface produces both recoverable and permanent deformations (plastic strains).

In general, the yield surface can be described as

where fc can be a constant value (for perfectly plastic materials), or a variable for strain-hardening materials. The yield surface F is a surface in the space of principal stresses, for which the elastic regime (F<0) is enclosed.

For brittle materials, the yield surface represents a failure surface, which is a stress level at which the material collapses instead of deforms plastically.

Note: Some authors define the yield criterion as f ()= fc, while the yield surface is a isosurface in the space of principal stresses F=0, which can be chosen for numerical purposes as .

F f fc– 0= =

F f 2= fc2

– 0=


De f i n i n g P e r f e c t l y P l a s t i c Ma t e r i a l Mode l s

For perfect elastoplastic materials, the yield surface is fixed in the 3D space of principal stress, and therefore, plastic deformations occur only when the stress path moves on the yield surface (the regime inside the yield surface is elastic, and stress paths beyond the yield surface are not allowed).

In general, the yield criterion depends on various parameters. Most of the plasticity models are based on isotropic assumptions, which require the yield function to be independent of the chosen coordinate system. This introduces the concept of using the stress invariants previously defined in Equation 2-1, Equation 2-2, and Equation 2-3.

In this section:

• Von Mises Criterion

• Tresca Criterion

Von Mises Criterion

The von Mises criterion suggests that the yielding of the material begins when the second deviatoric stress invariant J2 reaches a critical value. This criterion can be written in terms of the elements of Cauchy’s stress tensor (Ref. 1)

or equivalently .

The von Mises criterion is implemented in the Structural Mechanics Module as , where ys is a user-defined yield stress level (yield stress in uniaxial

tension).

Note: The effective or von Mises stress ( ) is available in the variable solid.mises.

J216--- 11 22– 2 22 33– 2 33 11– 2+ + 12

2 232 13

2+ + + k2= =

J2 k=

3J2 ys=

mises 3J2=

D E F I N I N G P E R F E C T L Y P L A S T I C M A T E R I A L M O D E L S | 17

18 | C H A P T E R

Tresca Criterion

Tresca yield surface is normally expressed in terms of the principal stress components

Tresca criterion is a hexagonal prism with axis equally inclined to the three principal stress axes. When the principal stresses fulfill 123, this criterion can be written as

By using the representation of principal stresses in term of the invariants J2 and the Lode angle 03, this criterion can alternatively be written as

or equivalently .

The maximum shear stress is reached at the meridians (=0 or =). Tresca criterion can be circumscribed by setting the Lode angle =0, or equivalently, by a von Mises criterion .

The minimum shear is reached at=, so the Tresca criterion can be inscribed by setting a von Mises criterion . When dealing with soils, the parameter k is also called undrained shear strength.

12---max 1 2– 1 3– 2 3– k=

12--- 1 3– k=

12---

4J23

---------- 23

------+ cos–cos

J2 3---+

sin k= =

J2 6---–

cos k=

3J2 2k=

J2 k=


Figure 2-1: The upper and lower limits of Tresca criterion.

Note: Tresca effective stress is implemented in the Structural Mechanics Module with the variable solid.tresca.

Figure 2-2: Classical yield criteria for metals. Left: Tresca criterion. Right: von Mises criterion.

The von Mises and Tresca criteria are independent of the first stress invariant I1 and are mainly used for the analysis of plastic deformation in metals and ductile materials, though some researchers use these criteria for describing fully saturated cohesive soils (that is, clays) under undrained conditions.

Upper limit

Lower limit

D E F I N I N G P E R F E C T L Y P L A S T I C M A T E R I A L M O D E L S | 19

20 | C H A P T E R

P l a s t i c i t y Mode l s f o r S o i l s

In this section:

• Mohr-Coulomb Criterion

• Drucker-Prager Criterion

• Elliptic Cap

• Matsuoka-Nakai Criterion

• Lade-Duncan Criterion

Mohr-Coulomb Criterion

Mohr-Coulomb criterion is the most popular criterion in soil mechanics. It was developed by Coulomb before Tresca and von Mises criteria for metals, and it was the first criterion to account for the hydrostatic pressure. The criterion states that failure occurs when the shear stress and the normal stress acting on any element in the material satisfy the equation

here, is the shear stress, c the cohesion, and denotes the angle of internal friction.

With the help of Mohr’s circle, this criterion can be written as

Mohr-Coulomb criterion defines an irregular hexagon pyramid in the space of principal stresses, which generates singularities in the derivatives of the yield function.

tan c–+ 0=

12--- 1 3– 1

2--- 1 3+ c– cossin+ 0=


Figure 2-3: The Mohr-Coulomb criterion.

The Mohr-Coulomb criterion can be written in terms of the invariants I1 and J2 and the Lode angle 03 (Ref. 1, Ref. 9) when the principal stresses are sorted as 123,

The tensile meridian is defined when and the compressive meridian when =.

Rearranging terms, Mohr-Coulomb criterion reads

where , , and .

In the special case of frictionless material, (=0, =0, k=c), the Mohr-Coulomb criterion reduces to Tresca’s maximum shear stress criterion, 13=2k or equivalently .

Drucker-Prager Criterion

Mohr-Coulomb criterion causes numerical difficulties when treating the plastic flow at the corners of the yield surface. The Drucker-Prager model neglects the influence of the invariant J3 (introduced by the Lode angle) on the cross-sectional shape of the yield surface. It can be considered as the first attempt to approximate the Mohr-Coulomb criterion by a smooth function based on the invariants I1 and J2

Tens

ile m

erid

ian

Compressiv

e meri

dian

F 13---I1

J23------ 1 sin+ 1 sin– 2

3------+

cos–cos c cos–+sin 0= =

F J2m I1 k–+ 0= =

m 6– cos 1 3 sin 6– sin–= 3sin=

k c cos=

F J2 6---–

cos k– 0= =

P L A S T I C I T Y M O D E L S F O R S O I L S | 21

22 | C H A P T E R

together with two material constants (which can be related to Mohr-Coulomb’s coefficients)

This is sometimes also called extended von Mises criterion, since it is equivalent to von-Mises criterion for metals when setting =

Figure 2-4: The Drucker-Prager criterion.

The coefficients in the Drucker-Prager model can be matched to the coefficients in the Mohr-Coulomb criterion by

and

The symbol ± is related to either matching the tensile meridian (positive sign) or the compressive meridian (negative sign) of Mohr-Coulomb’s pyramid.

The matching at the tensile meridian (=) comes from setting in Mohr-Coulomb criterion, and the matching at the

compressive meridian (=) from setting .

F J2 I1 k–+ 0= =

23

------- sin3 sin

--------------------------= k 2 3c cos3 sin

---------------------------=

m 0 3 sin+ 2 3 =

m 3 3 sin– 2 3 =


Figure 2-5: The Drucker-Prager criterion showing the tensile and compressive meridians (inner and outer circles), and the Lode angle compared to the cross section of Mohr-Coulomb criterion in the -plane.

In the special case of frictionless material, (=0, =0, ), the Drucker -Prager criterion reduces to von Mises criterion .

When matching Drucker-Prager criterion to Mohr-Coulomb criterion in 2D plane-strain applications, the parameters are

and

and when matching both criteria in 2D plane-stress applications, the matching parameters are:

and

Elliptic Cap

Both Mohr-Coulomb and Drucker-Prager criteria portrait a conic yield surface which opens in the hydrostatic axis direction. Normally, these soil models are not accurate above a given limit pressure, since real life materials can not bear infinite loads and still

Tensile meridianCompressive meridian

k 2c 3=

J2 2c 3=

= tan

9 12 2tan+------------------------------------ k = 3c

9 12 2tan+------------------------------------

= 13

------- sin k = 23

------- c cos


24 | C H A P T E R

behave elastically. A simpler way to overcome this problem is to add an elliptical end-cap to these soil models.

The elliptic cap is an elliptic yield surface of semi axes as shown in Figure 2-6. The initial pressure pa (SI units: Pa) denotes the pressure at which the elastic range circumscribed by either Mohr-Coulomb pyramid or Drucker-Prager cone is not valid any longer, so a cap surface is added. The limit pressure pb gives the curvature of the ellipse, and denotes the maximum admissible hydrostatic pressure for which the material start deforming plastically. Pressures higher than pb are not allowed

Figure 2-6: Elliptic cap model in Haigh–Westergaard coordinate system.

Note that the sign convention for the pressure is taken from the Structural Mechanics Module: positive sign under compression, so pa and pb are positive parameters. Figure 2-6 shows the cap in terms of the variables and .

Matsuoka-Nakai Criterion

Matsuoka and Nakai (Ref. 3) discovered that the sliding of soil particles occurs in the plane in which the ratio of shear stress to normal stress has its maximum value, which they called the mobilized plane. They defined the yield surface as

where the parameter =/nSTP equals the maximum ratio between shear stress and normal stress in the spatially mobilized plane (STP-plane), and the invariants are applied over the effective stress tensor (this is the Cauchy stress tensor minus the fluid pore pressure).

Matsuoka-Nakai criterion circumscribes Mohr-Coulomb criterion in dry soils, when

ppbpa

q

q 3J2 = p I– 1 3=

F 9 92+ I3 I1I2– 0= =


and denotes the angle of internal friction in Mohr-Coulomb criterion.

Figure 2-7: The Matsuoka-Nakai criterion and Mohr-Coulomb criterion in the principal stress space.

Lade-Duncan Criterion

The Lade-Duncan criterion was originally developed to model a large volume of laboratory sample test data of cohesionless soils. This criterion is defined as

where I1 and I3 are the first and third stress invariants respectively, and k is a parameter related to the direction of the plastic strain increment in the triaxial plane. The parameter k can vary from 27 for hydrostatic stress conditions (1=2=3), up to a critical value kc at failure. In terms of the invariant I1, J2 and J3, this criterion can be written as

Lade-Duncan criterion can be fitted to the compressive meridian of Mohr-Coulomb surface, by choosing

2 23

----------- tan=

F kI3 I13

– 0= =

F J313---I1J2–

127------ 1

k---–

I13

+ 0= =

k 3 sin– 3/ 2 1 sin– cos =


26 | C H A P T E R

with as the angle of internal friction in Mohr-Coulomb criterion.

Figure 2-8: Comparing the Mohr-Coulomb, Matsuoka-Nakai, and Lade-Duncan criteria when matching the tensile meridian.

Note: The Lade-Duncan criterion does not match Mohr-Coulomb criterion (nor Matsuoka-Nakai criterion) at the tensile meridian.

Mohr-Coulomb

Matsuoka-NakaiLade-Duncan


Th eo r y f o r t h e C am-C l a y Ma t e r i a l Mod e l

In this section:

• About the Cam-Clay Material Model

• Volumetric Elastic Behavior

• Hardening and Softening

• Including Pore Fluid Pressure

About the Cam-Clay Material Model

The Cam-Clay material model was developed at the University of Cambridge in the 1970s, and since then it has experienced different modifications. The modified Cam-Clay model is the most commonly used due to the smooth yield surface, and it is the one implemented in the Geomechanics Module.

The Cam-Clay model is a so-called critical state model, where the loading and unloading of the material follows different trajectories in stress space. The model also features hardening and softening of clays. Different formulations can be found in textbooks about these models (see Ref. 13, 14, and 15).

The yield function is written in terms of the variables and , following the Structural Mechanics Module sign convention:

(2-4)

This is an ellipse in p-q plane, with a cross section independent of Lode angle and smooth for differentiation. Note that p, q and pc are always positive variables.

The material parameter M0 defines the slope of a line in the p-q space called critical state line, and it can be related to the angle of internal friction in Mohr-Coulomb criterion

q 3J2 =

p I– 1 3=

F q2 M2 p pc– p+ 0= =

M 6 sin3 sin–--------------------------=

T H E O R Y F O R T H E C A M - C L A Y M A T E R I A L M O D E L | 27

28 | C H A P T E R

Figure 2-9: Modified Cam-Clay surface in the p-q plane. The ellipse circumscribes a non-linear elastic region.

In the Cam-Clay model, hardening is controlled by the consolidation pressure pc, which depends exponentially on the volumetric plastic strain pvol (see Effective and Volumetric Plastic Strains).

(2-5)

Note: The volumetric plastic strain is available in the variable solid.epvol and the consolidation pressure in the variable solid.Pc.

Here, the parameter pc0 is the initial consolidation pressure, and the exponent Bp is a parameter which depends on the initial void ratio e0, the swelling index and the compression index

The initial void ratio, the compression index and the swelling index are all positive parameters and should fulfill , so .

q

ppCpC/2

softeninghardening

Critical state line

non-linear elastic region

q = Mp

pc pc0eBppvol–

=

Bp1 e0+

–---------------=

0 Bp 0


Note: The void ratio e is the ratio between pore volume and solid volume. It can be written in terms of the porosity as e=/(1).

In this formulation, the compression index is the slope of the virgin isotropic consolidation line, and is the slope of the rebound -reloading line (also called loading-reloading line) in the e vs ln(p) plane.

Figure 2-10: Slopes of the virgin isotropic consolidation line, and rebound-reloading line in the e vs ln(p) plane.

Note: If you add an Initial Stress and Strain sub-feature to your Cam-Clay material, you have to secure that the initial consolidation pressure pc0 is equal or bigger than one third of minus the trace of the initial stress tensor, otherwise your initial stress state will be outside the Cam-Clay ellipse.

Volumetric Elastic Behavior

The stress-strain relation beyond the elastic range is of great importance in soil mechanics. For additive decomposition of strains, Cauchy’s stress tensor is written as

here, is the total strain tensor, 0 and 0 are the initial stress and strain tensors, th is the thermal strain, p is the plastic strain, and C is the fourth-order elasticity tensor.

ln(p)

N

ln(prefN) ln(pC0)

e

ij ij0 C+ ijkl kl kl

0– kl

th– kl

p– =


30 | C H A P T E R

For a linear elastic material, the trace of the Cauchy’s stress tensor is linearly related to the volumetric elastic strain (the trace of the elastic strain tensor) by the elastic bulk modulus

here p0=trace0/3 is the trace of the initial stress tensor 0, and K is the bulk modulus, a constant parameter independent of the stress or strain.

Note: The elastic volumetric strain is available in the variable solid.eelvol.

The modified Cam-Clay model introduces a non-linear relation between stress and volumetric elastic strain

with

and K0 a reference bulk modulus. This formulation gives a tangent bulk modulus KTBep. The reference bulk modulus is calculated from the initial consolidation pressure pc0, and the void ratio at reference pressure N.

Hardening and Softening

The yield surface for the modified Cam-Clay model reads

The associated flow rule (Q=F) and the yield surface written in terms of these two invariants, F(I1, J2), give a rate equation for the plastic strain tensor calculated from the derivatives of F with respect to the stress tensor

Note: Here, p means the plastic multiplier, see Plastic Flow.

The plastic strain rate tensor includes both deviatoric and isotropic parts. Note that

p I– 1 3 p0 Kelvol–= =

p p0– K– 0eBeelvol–

= Be1 e0+

---------------=

F q2 M2 p pc– p+ 0= =

· ijp

pQij---------- p

FI1--------

I1ij---------- F

J2---------

J2ij----------+

= =

· ijp


and . We can use these relations for writing the plastic flow as

The trace of the plastic strain rate tensor (the volumetric plastic strain rate ) then reads

This relation explains the reason why there is isotropic hardening for p>pc/2 and isotropic softening for p<pc/2.

In the Cam-Clay model, the hardening is controlled by the consolidation pressure variable as a function of volumetric plastic strain, as written in Equation 2-5.

Hardening will introduce changes in the shape of the Cam-Clay ellipse, since its mayor semi axis depends on the value of pc.

Including Pore Fluid Pressure

When you add a pore fluid pressure pf to the Cam-Clay material, the yield surface is shifted on the p axis

The quantity ppf is normally regarded as the effective pressure, or effective stress.

I1 ij ij= J2 ij ijD

=

· ijp

pFp-------–

ij

3------ F

q------- 3

2q-------ij

D+

p M2 2p pc– –ij

3------ 3ij

D+

= =

·volp

·volp

trace · ijp

p M2 2p pc– – = =

pc volp

F q2 M2 p pf– pc– p pf– + 0= =


32 | C H A P T E R

F a i l u r e C r i t e r i a f o r C on c r e t e , Ro c k s , a nd O t h e r B r i t t l e Ma t e r i a l s

In this section:

• Bresler-Pister Criterion

• Willam-Warnke Criterion

• Ottosen Criterion

• Hoek-Brown Criterion

• Generalized Hoek-Brown Criterion

Bresler-Pister Criterion

The Bresler-Pister criterion (Ref. 2) was originally devised to predict the strength of concrete under multiaxial stresses. This failure criterion is an extension of the Drucker-Prager criterion to brittle materials and can be expressed in terms of the stress invariants as

(2-6)

here, k1, k2, and k3 are material parameters.

This criterion can also be written (Ref. 17) in term of the uniaxial compressive strength fc and the octahedral normal and shear stresses

and

so Equation 2-6 simplifies to

here, the parameters a, b, and c are obtained from the uniaxial compression, uniaxial tension and biaxial compression tests. The octahedral normal stress octis considered positive when tensile, and fc is taken positive.

F J2 k1I12 k2I1 k3+ + + 0= =

oct = I1/3 oct 2J2/3=

Foctfc

--------- a– boctfc

---------- coct

2

fc2

----------–+ 0= =


Willam-Warnke Criterion

The Willam-Warnke criterion (Ref. 10) is used to predict failure in concrete and other cohesive-frictional materials such as rock, soil, and ceramics. The same as for the Bresler-Pister criterion, it depends only on three parameters. It was developed to describe initial concrete failure under triaxial conditions. The failure surface is convex, continuously differentiable and it is fitted to test data in the low compression range. The material is considered perfect elastoplastic (no hardening), augmented by a brittle failure condition (tension cut-off) in the tensile regime.

The original “three-parameter” Willam-Warnke failure criterion was defined as

(2-7)

here, fc is the uniaxial compressive strength, ft is uniaxial tensile strength, and fb is obtained from the biaxial compression test. All parameters are positive. The octahedral normal and shear stresses are defined as usual , and .

This can be written as

with .

The function r( describes the segment of an ellipse on the octahedral plane when . By using the Lode angle , the dimensionless function r( is defined as

Here, the tensile and compressive meridian rt and rc are defined in terms of the positive parameters fc, fb, and ft:

F 35---octfc

--------- r 1ft--- 1

fb----–

oct 1– 0=+=

oct = I1/3 oct 2J2/3=

F J252---r 1

3------I1 fc– 0=+=

fbft/ fbfc ftfc– =

0 3

r 2rc rc

2 rt2– rc 2rt rc– 4 rc

2 rt2– 5rt

2 4– rcrt+2

cos+cos

4 rc2 rt

2– rc 2rt– 2+2

cos---------------------------------------------------------------------------------------------------------------------------------------------------------------=

rt =65---

fbft2fbfc ftfc+----------------------------

rc =65---

fbft3fbft f+ bfc ftfc–-------------------------------------------

F A I L U R E C R I T E R I A F O R C O N C R E T E , R O C K S , A N D O T H E R B R I T T L E M A T E R I A L S | 33

34 | C H A P T E R

The function r( can be interpreted as the friction angle which depends on the Lode angle (Ref. 10).

Figure 2-11: The deviatoric section of Willam-Warnke failure criterion.

Ottosen Criterion

The Ottosen criterion is a four-parameter failure criterion proposed for short-time loading of concrete. It corresponds to a smooth convex failure surface with curved meridians, which open in the negative direction of the hydrostatic axis. The trace in the deviatoric plane changes from almost triangular to a more circular shape with increasing hydrostatic pressure. The criterion is in agreement with experimental results over a wide range of stress states, including both triaxial tests along the tensile and the compressive meridian and biaxial tests (Ref. 18).

Ottosen criterion is commonly written as (Ref. 17, Ref. 18):

In this formulation, the parameters a>0, b>0, k1>0 and k2>0 are dimensionless, fc>0 is the uniaxial compressive strength of concrete (with a positive sign), and the function

(dimensionless) is defined as

rc

rtr

F afc----J2 J2 bI1 fc–+ + 0= =

0


The parameter k1 is called the size factor. The parameter k2 (also called shape factor) is positive and bounded to 0k21(Ref. 17, Ref. 18).

Typical values for these parameters are obtained by curve-fitting the uniaxial compressive strength fc, uniaxial tensile strength ft, and from the biaxial and triaxial data (for instance, typical biaxial compressive strength of concrete is 16% higher than the uniaxial compressive strength). The parameters fc, fb, and ft are positive.

The compressive and tensile meridians (as defined in Willam-Warnke criterion) are

For concrete, the ratio lies between 0.54~0.58.

Note: Ottosen criterion is equivalent to Drucker-Prager when a=0 and =constant.

Hoek-Brown Criterion

Hoek-Brown criterion is an empirical type of model which is commonly used when dealing with rock masses of variable quality. The Hoek-Brown criterion is widely used

TABLE 2-2: TYPICAL PARAMETER VALUES FOR OTTOSEN FAILURE CRITERION (Ref. 18).

ft/ fc a b k1 k2 t c

0.08 1.8076 4.0962 14.4863 0.9914 14.4725 7.7834

0.10 1.2759 3.1962 11.7365 0.9801 11.7109 6.5315

0.12 0.9218 2.5969 9.9110 0.9647 9.8720 5.6979

k113--- k2 3 cos acos J3 0cos

k13--- 1

3---– k– 2 3 cos acos

J3 0,cos

=

rc = 1c----- = 1

/3 -------------------

rt = 1t----- = 1

0 -----------

c/t rt/rc=


36 | C H A P T E R

within civil engineering and is popular due to the fact that the material parameters can be estimated based on simple field observations together with knowledge of the uniaxial compressive strength of the intact rock material. The Hoek–Brown criterion is one of the few non-linear criteria widely accepted and used by engineers to estimate the yield and failure of rock masses. The original Hoek-Brown failure criterion states (Ref. 5)

where 1230 are the principal stresses at failure (as defined in geotechnical engineering, this is, absolute value), c is the uniaxial compressive strength of the intact rock (positive parameter), and m and s are positive material parameters.

If the expression is converted into to the sign convention for principal stresses in the Structural Mechanics Module, it becomes

with c, m, and s positive material parameters. (In this case, note that 1 sc/m).

As developed originally, there is no relation between the parameters m and s and the physical characteristics of a rock mass measured in laboratory tests. However, for intact rock, s1 and mmi, which is measured in a triaxial test.

For jointed rock masses, 0s1 and mmi. The parameter m usually lies in the range 5m30 (Ref. 7).

Hoek-Brown criterion can be written in terms of the invariants I1 and J2 and the Lode angle 03, so

TABLE 2-3: CHARACTERISTIC VALUES FOR DIFFERENT ROCK TYPES

m ROCK TYPE

5 Carbonate rocks, dolomite, limestone

10 Consolidated rocks, mudstone, shale

15 Sandstone

20 Fine grained rocks

25 Coarse grained rocks

1 3 mc3 sc2++=

1 3 mc1– sc2++=

F 2 J2 3---+

cm1c

------------– s+ 0=–sin=


Generalized Hoek-Brown Criterion

The generalized Hoek-Brown criterion was developed in order to fit the Geological Strength Index (GSI) classification of isotropic rock masses (Ref. 6). A new relationship between GSI, m, s and the newly introduced parameter a was developed, to give a smoother transition between very poor quality rock masses (GSI<25) and stronger rocks

In term of the invariants J2 and the Lode angle this equals

where 123 are the principal stresses (using the Structural Mechanics Module conventions) of the effective stress tensor (this is, the Cauchy tensor minus the fluid pore pressure).

The positive parameter mb is a reduced value of the material constant mi:

s and a are positive parameters for the rock mass given by the following relationships:

The disturbance factor D was introduced to account for the effects of stress relaxation and blast damage, and it varies from 0 for undisturbed in-situ rock masses to 1 for very damaged rock masses.

TABLE 2-4: DISTURBANCE FACTOR IN ROCK MASSES

D DESCRIPTION OF ROCK MASS

0 Undisturbed rock mass

0~0.5 Poor quality rock mass

1 3– cmb1c

--------------– s+ a

=

0 3

F 2 J2 3---+

cmb1c

--------------– s+ a

–sin 0= =

mb miexp GSI 100–28 14D–-------------------------- =

s = exp GSI 100–9 3D–

--------------------------

a = 12--- 1

6--- exp GSI–

15------------- exp 20–

3--------- –

+


38 | C H A P T E R

Hoek-Brown generalized criterion can be written in terms of the invariants I1, J2 and the Lode angle 03 to

0.8 Damaged rock mass

1.0 Severely damaged rock mass

TABLE 2-4: DISTURBANCE FACTOR IN ROCK MASSES

D DESCRIPTION OF ROCK MASS

F 2 J2 3---+

sin c–mb

c--------

I13-----

4J23

---------- cos+ – s+

a

0==


S t r e s s - S t r a i n Th eo r y

In this section:

• Small Strain Assumption

• Plastic Flow

• Effective and Volumetric Plastic Strains

Small Strain Assumption

The stress-strain relation beyond the elastic range is of great importance in soil mechanics. In the case of small strains, the strain tensor can be additively decomposed into elastic and inelastic components. For this case, Cauchy’s stress tensor is written as

where is the total strain tensor, 0 and 0 are the initial stress and strain tensors, th is the thermal strain, p is the plastic strain, and C is the fourth-order elasticity tensor.

In the special case of zero initial strain and no thermal expansion, the elastic part of the strain tensor simplifies to , so the strain tensor is normally additively decomposed into elastic and plastic (inelastic) components

The total strain increment (also called total strain rate) is written as

where and represent the elastic and plastic strain rates. If elor p are large, the additive decomposition might produce incorrect results, however, the additive decomposition of strains is widely use for metal and soil plasticity.

Plastic Flow

The flow rule defines the relationship between the increment of plastic strain and the current state of stress ijfor a yielded element subject to further loading. The direction of the plastic strain increment is defined by the flow rule

ij ij0

– Cijkl kl kl0

– klth

– klp

– =

ijel ij ij

p–=

ij ijel ij

p+=

· ij · ijel

· ijp

+=

· ijel

· ijp

· ijp

S T R E S S - S T R A I N T H E O R Y | 39

40 | C H A P T E R

Here is a positive multiplier (also called the consistency parameter or plastic multiplier) dependent on the current state of stress and the load history, and Q is the plastic potential.

Is it important to notice that the “dot” means the rate at which the plastic strain tensor changes with respect to , and it does not represent a “time derivative.”

Figure 2-12: Plastic potential, plastic strain increment, and plastic multiplier.

The direction of the plastic strain increment is perpendicular to the surface (in the principal stresses space) defined by the plastic potential at the current state of stress (see Figure 2-12).

The plastic multiplier is determined by the complementarity conditions

(2-8)

where F is the user-defined yield surface, so the elastic regime is defined by F<0.

If the plastic potential and the yield surface coincide with each other (Q=F), the flow rule is called associated, and the rate in Equation 2-9 is solved together with the conditions in Equation 2-8.

(2-9)

· ijp

Qij----------=

Q ij

Q

· ijp

Qij----------=

· ijp

Q ij ijp

F

F

=

000

· ijp

Fij----------=


For non-associated flow rule, the yield function does not coincide with the plastic potential, and together the conditions in Equation 2-8 the rate in Equation 2-10 is solved for a user-defined plastic potential Q (normally, a smoothed version of F).

(2-10)

The evolution of the plastic strain tensor (either Equation 2-9 or Equation 2-10) is implemented at Gauss points in the plastic element ELPLASTIC.

Effective and Volumetric Plastic Strains

For isotropic, perfectly plastic materials, the plastic potential can be written in terms of the invariants of Cauchy’s stress tensor (Ref. 1),

so the increment of the plastic strain tensor can be decomposed into

The increment in the plastic strain tensor includes both deviatoric and volumetric parts, and it is symmetric given the following properties

(2-11)

The trace of the incremental plastic strain tensor, which is called the volumetric plastic strain rate , relies only on the dependency of the plastic potential to the first invariant I1(), since and are traceless

For metals, under von Mises yield criterion and associated flow rule, the volumetric plastic strain is always zero, since the plastic potential is independent of the invariant

· ijp

Qij----------=

· ijp

Q Q I1 J2 J3 =

· ijp

· ijp

Qij---------- Q

I1--------

I1ij---------- Q

J2---------

J2ij---------- Q

J3---------

J3ij----------+ +

= =

· ijp

I1ij---------- ij=

J2ij---------- ij

D=

J3ij---------- ik

DkjD 2

3---J2ij–=

·volp

J2 ij J3 ij

·volp

trace · ijp

trace Qij---------- 3Q

I1--------= = =

S T R E S S - S T R A I N T H E O R Y | 41

42 | C H A P T E R

I1(). This is not the case for most materials used in geotechnical applications. For instance, a nonzero volumetric plastic strain is explicitly used in the Cam-Clay material.

Another common measure of inelastic deformation is the effective plastic strain rate, which is defined as

(2-12)

In the special case when the plastic potential Q is written in terms of the invariants I1 and J2, the effective plastic strain rate reads

(2-13)

The effective plastic strain rate as defined in Equation 2-13, has both a deviatoric and a volumetric components, compared to the effective (von Mises) stress

, which is independent of the hydrostatic component of stress.

Note: The effective plastic strain and the volumetric plastic strain are available in the variables solid.epe and solid.epvol.

·ep 2

3---· ij

p· ji

p=

·ep

23--- 3 Q

I1-------- 2

2 QJ2--------- 2

J2+ =

·ep

e 3J2=


Re f e r e n c e s f o r t h e Geome chan i c s Modu l e

1. W.F. Chen and E. Mizuno, Nonlinear Analysis in Soil Mechanics: Theory and Implementation (Developments in Geotechnical Engineering), 3rd ed., Elsevier Science, 1990.

2. B. Bresler and K.S. Pister, “Strength of Concrete Under Combined Stresses,” ACI Journal, vol. 551, no. 9, pp. 321–345, 1958.

3. H. Matsuoka and T. Nakai, “Stress-deformation and Strength Characteristics of Soil Under Three Different Principal Stresses,” Proc. JSCE, vol. 232, 1974.

4. H. Matsuoka and T. Nakai, “Relationship Among Tresca, Mises, Mohr-Coulomb, and Matsuoka-Nakai Failure Criteria,” Soils and Foundations, vol. 25, no. 4, pp.123–128, 1985.

5. H.S. Yu, Plasticity and Geotechnics, Springer, 2006.

6. V. Marinos, P. Marinos, and E. Hoek, “The Geological Strength Index: Applications and Limitations,” Bull. Eng. Geol. Environ., vol. 64, pp. 55–65, 2005.

7. J. Jaeger, N. G. Cook, and R. Zimmerman, Fundamentals of Rock Mechanics, 4th ed., Wiley-Blackwell, 2007.

8. G. C. Nayak and O. C. Zienkiewicz, “Convenient Form of Stress Invariants for Plasticity,” J. Struct. Div. ASCE, vol. 98, pp. 949–954, 1972.

9. A.J. Abbo and S.W. Sloan, “A Smooth Hyperbolic Approximation to the Mohr-Coulomb Yield Criterion,” Computers and Structures, vol. 54, no. 3, pp. 427–441, 1995.

10. K. J. Willam and E. P. Warnke, “Constitutive Model for the Triaxial Behavior of Concrete,” IABSE Reports of the Working Commissions, Colloquium (Bergamo): Concrete Structures Subjected to Triaxial Stresses, vol. 19, 1974.

11. B. H. G. Brady and E. T. Brown, Rock Mechanics for Underground Mining, 3rd ed., Springer, 2004.

12. H.A. Taiebat and J.P. Carter, “Flow Rule Effects in the Tresca Model,” Computer and Geotechnics, vol. 35, pp. 500–503, 2008.

R E F E R E N C E S F O R T H E G E O M E C H A N I C S M O D U L E | 43

44 | C H A P T E R

13. A. Stankiewicz and others, “Gradient-enhanced Cam-Clay Model in Simulation of Strain Localization in Soil,” Foundations of Civil and Environmental Engineering, no.7, 2006.

14. D. M. Wood, Soil Behaviour and Critical State Soil Mechanics, Cambridge University Press, 2007.

15. D. M. Potts and L. Zadravkovic, Finite Element Analysis in Geothechnical Engineering, Thomas Telford, 1999.

16. W. Tiecheng and others, Stress-strain Relation for Concrete Under Triaxial Loading, 16th ASCE Engineering Mechanics Conference, 2003.

17. W.F. Chen, Plasticity in Reinforced Concrete, McGraw-Hill, 1982.

18. N. Ottosen, “A Failure Criterion for Concrete,” Journal of the Engineering Mechanics Division, ASCE, vol. 103, no. 4, pp. 527–535, 1977.


3

T h e G e o m e c h a n i c s M a t e r i a l M o d e l s

The Geomechanics Module has materials which are used in combination with the Solid Mechanics interface to account for the plasticity of soils and failure criteria in rocks, concrete, and other materials used in geotechnical applications.

In this chapter:

• Working with the Geomechanics Material Models

• Geomechanics Materials

45

46 | C H A P T E R

Work i n g w i t h t h e Geome chan i c s Ma t e r i a l Mode l s

Note: See Geomechanics Theory for background information about these material models.

In this section:

• Adding a Material Model to a Solid Mechanics Interface

• Soil Plasticity

• Concrete

• Rocks

• Cam-Clay Material Model

• The Output Materials Properties

Also see

Adding a Material Model to a Solid Mechanics Interface

Note: The Geomechanics Module requires a Structural Mechanics Module license.

1 Add a Solid Mechanics interface from the Structural Mechanics branch ( ) in the Model Wizard.

2 In the Model Builder, right-click the Solid Mechanics node ( ) to add an Elastoplastic

Soil Model, Brittle Material Model, or Cam-Clay Material Model.

Note: Except for the settings described in this section, see The Solid Mechanics Interface in the Structural Mechanics User’s Guide for details.

3 : T H E G E O M E C H A N I C S M A T E R I A L M O D E L S

Soil Plasticity

The Soil Plasticity node adds the equations for linear elasticity and plasticity, and the interface for defining the elastic material properties and yield surface. Right-click to add Thermal Expansion and Initial Stress and Strain features. The yield criteria are described in the theory section:

• Drucker-Prager Criterion

• Mohr-Coulomb Criterion

• Matsuoka-Nakai Criterion

• Lade-Duncan Criterion

To display additional features for the physics interface feature nodes (and the physics interfaces), click the Show button ( ) on the Model Builder and then select the applicable option.

S H O W O R H I D E O P T I O N S F O R P H Y S I C S F E A T U R E N O D E S

For most physics interface feature nodes, the Equation and Override and Contribution sections are displayed on a feature node Settings window by default. You can also click the Expand Sections button on the Model Builder to always show some sections in an expanded view, or go to these menus to hide options as required. Click the Show

button ( ) on the Model Builder and then select Equation View to display the Equation

View node under all physics interface nodes in the Model Builder.

See the description for each physics interface for more links or go to Showing and Expanding Advanced Feature Nodes and Sections for more information.

D O M A I N S E L E C T I O N

Select the domains where you want apply the feature and compute the displacements, stresses, and strains.

M O D E L I N P U T S

Define model inputs, for example, the temperature field if the material model uses a temperature-dependent material property. If no model inputs are required, this section is empty.

To use a mixed formulation by adding the negative mean pressure as an extra dependent variable to solve for, select the Nearly incompressible material check box.

W O R K I N G W I T H T H E G E O M E C H A N I C S M A T E R I A L M O D E L S | 47

48 | C H A P T E R

Specification of Elastic Properties for Isotropic MaterialsFrom the Specify list, select a pair of elastic properties for an isotropic material. Select:

• Young’s modulus and Poisson’s ratio to specify Young’s modulus (elastic modulus) E (SI unit: Pa) and Poisson’s ratio (dimensionless). For an isotropic material Young’s modulus is the spring stiffness in Hooke’s law, which in 1D form is

where is the stress and is the strain. Poisson’s ratio defines the normal strain in the perpendicular direction, generated from a normal strain in the other direction and follows the equation

• Bulk modulus and shear modulus to specify the bulk modulus K (SI unit: Pa) and the shear modulus G (SI unit: Pa). The bulk modulus is a measure of the solid’s resistance to volume changes. The shear modulus is a measure of the solid’s resistance to shear deformations.

• Lamé constants to specify the Lamé constants (SI unit: Pa) and (SI unit: Pa).

• Pressure-wave and shear-wave speeds to specify the pressure-wave speed (longitudinal wave speed) cp (SI unit: m/s) and the shear-wave speed (transverse wave speed) cs (SI unit: m/s).

For each pair of properties, select from the applicable list to use the value From material or enter a User defined value or expression.

Each of these pairs define the elastic properties and it is possible to convert from one set of properties to another.

DensityThe default Density (SI unit: kg/m3) uses values From material. If User defined is selected, enter another value or expression.

P L A S T I C I T Y M O D E L

Select a Yield criterion—Drucker-Prager, Mohr-Coulomb, Matsuoka-Nakai, or Lade-Duncan.

Drucker-PragerIf required, select the Match to Mohr-Coulomb criterion check box (see Mohr-Coulomb Criterion). If this check box is selected, the default values for Cohesion c (SI unit: Pa)

E=

l l–=


and the Angle of internal friction (SI unit: rad) are taken From material. If User defined is selected, then enter other values or expressions.

If the Match to Mohr-Coulomb criterion check box is NOT selected, then the default Drucker-Prager alpha coefficient (dimensionless) and Drucker-Prager k coefficient k (SI unit: Pa) are taken From material. Select User defined to enter other values or expressions.

If required, select the Include elliptic cap check box (see Elliptic Cap). Enter a value or expression for the semi-axes of the elliptic cap (SI unit: Pa).

Mohr-CoulombThe default Angle of internal friction (SI unit: rad) and Cohesion c (SI unit: Pa) are taken From material. Select User defined to enter other values or expressions.

Under Flow potential select either Drucker-Prager matched at compressive meridian, Drucker-Prager matched at tensile meridian or Associated.

If required, select the Include elliptic cap check box (see Elliptic Cap). Enter a value or expression for the semi-axes of the elliptic cap (SI unit: Pa).

Matsuoka-NakaiIf required, select the Match to Mohr-Coulomb criterion check box. If this check box is selected, the default Angle of internal friction (SI unit: rad) is taken From material. Select User defined to enter other values or expressions.

If the Match to Mohr-Coulomb criterion check box is NOT selected, then the default Matsuoka-Nakai mu coefficient (dimensionless) is taken From material. Select User

defined to enter other values or expressions.

Lade-DuncanIf required, select the Match to Mohr-Coulomb criterion check box. If this check box is selected, then enter a value or expression for the Angle of internal friction (SI unit: rad). Alternatively, select From material.

If the Match to Mohr-Coulomb criterion check box is NOT selected, then the default Lade-Duncan k coefficient k(dimensionless) is taken From material. Select User defined

to enter other values or expressions.

G E O M E T R I C N O N L I N E A R I T Y

To add the equations for a geometrically nonlinear elastic solid, select the Include

geometric nonlinearity check box. This is useful for modeling large deformations.


50 | C H A P T E R

Concrete

The Concrete node adds the equations for linear elasticity, and the interface for defining the elastic material properties and failure surface. Right-click to add the Thermal

Expansion and Initial Stress and Strain features. The failure criterion are described in the theory section:

• Bresler-Pister Criterion

• Willam-Warnke Criterion

• Ottosen Criterion

Note: The settings for Domain Selection, Equations, Model Inputs, Linear Elastic Model, and Geometric Nonlinearity are the same as for the Soil Plasticity.







C O N C R E T E M O D E L

Select a Concrete criterion—Bresler-Pister, Willam-Warnke, or Ottosen.

Bresler-PisterThe defaults for the Uniaxial tensile strength t, Uniaxial compressive strength c, and Biaxial compressive strength b (SI unit: Pa) are taken From material. Select User defined



Willam-WarnkeThe defaults for the Uniaxial tensile strength t, Uniaxial compressive strengthc, and Biaxial compressive strength b (SI unit: Pa) are taken From material. Select User defined


OttosenThe defaults for the Uniaxial tensile strengthc, Ottosen’s parameters a and b, Size factor k1, and Shape factor k2 are taken From material. Select User defined to enter other values or expressions.

Rocks

The Rocks node adds the equations for linear elasticity, and the interface for defining the elastic material properties and failure surface. Right-click to add the Thermal

Expansion and Initial Stress and Strain features. The failure criterion are described in the theory section:

• Hoek-Brown Criterion

• Generalized Hoek-Brown Criterion

Note: The settings for Domain Selection, Equations, Model Inputs, Linear Elastic Model, and Geometric Nonlinearity are the same as for the Soil Plasticity.








52 | C H A P T E R

R O C K M O D E L

Select a Rock criterion—Original Hoek-Brown or Generalized Hoek-Brown.

Original Hoek-BrownThe defaults for the Uniaxial compressive strengthc, and Hoek-Brown’s parameters m and s are taken From material. Select User defined to enter other values or expressions.

Generalized Hoek-BrownIf required, select the Use generalized Hoek-Brown check box (see Generalized Hoek-Brown Criterion).

The defaults for the Uniaxial compressive strengthc, Geological strength index GSI, Disturbance factor D, and Intact rock parameter mi are taken From material. Select User

defined to enter other values or expressions.

Cam-Clay Material Model

The Cam-Clay Material Model adds the equations and interface for defining the material properties for the modified Cam-Clay material model (see Theory for the Cam-Clay Material Model). Right-click to add the Thermal Expansion and Initial Stress and Strain features.

Note: The settings for Domain Selection, Equations, Model Inputs, and Geometric

Nonlinearity are the same as for the Soil Plasticity.








C A M - C L A Y M A T E R I A L M O D E L

To use a mixed formulation by adding the negative mean pressure as an extra dependent variable to solve for, select the Nearly incompressible material check box.

From the Specify list, define the elastic properties either in terms of Poisson’s ratio or Shear modulus.

The defaults for the Poisson’s ratio (unitless) or Shear modulus G (SI unit: Pa), Density (SI unit: kg/m3), M parameter M, Swelling index , Compression index , and Void

ratio at reference pressure N are taken From material. Select User defined to enter other values or expressions.

You can also match the slope of the virgin consolidation line (the Cam-Clay M

parameter) to the Angle of internal friction, in that case, select Match to Mohr-Coulomb

criterion, and then select the Angle of internal friction (SI unit: rad) either or From

material or User defined.

Enter a value or expression for the Reference pressure for N PrefN (SI unit: Pa), the Initial void ratio e0 and the Initial consolidation pressure Pc0 (SI unit: Pa).


54 | C H A P T E R

Geome chan i c s Ma t e r i a l s

The Output Materials Properties

The material property groups (including all associated properties) listed in Table 3-1can be added to models from the Material page. See Material Properties Reference in the COMSOL Multiphysics User’s Guide for information about other material properties, including those for Solid Mechanics.

TABLE 3-1: GEOMECHANICS MODELS MATERIALS

PROPERTY GROUP AND PROPERTY NAME/VARIABLE SI UNIT

MOHR-COULOMB

Cohesion cohesion Pa

Angle of internal friction internalphi rad

DRUCKER-PRAGER

Drucker-Prager alpha coefficient alphaDrucker 1

Druker-Prager k coefficient kDrucker Pa

MATSUOKA-NAKAI

Matsuoka-Nakai mu coefficient muMatsuoka 1

LADE-DUNCAN

Lade-Duncan k coefficient kLade 1

OTTOSEN

Ottosen a parameter aOttosen 1

Ottosen b parameter bOttosen 1

Size factor k1Ottosen 1

Shape factor k2Ottosen 1

HOEK BROWN

Hoek-Brown m parameter mHB 1

Hoek-Brown s parameter sHB 1

Geological strength index GSI 1

Disturbance factor Dfactor 1

Intact rock parameter miHB 1

YIELD STRESS PARAMETERS

Uniaxial tensile strength sigmaut Pa

Uniaxial compressive strength sigmauc Pa


Biaxial compressive strength sigmabc Pa

CAM-CLAY MATERIAL MODEL

Swelling index kappaSwelling 1

Compression index lambdaComp 1

Initial void ratio e0 1

TABLE 3-1: GEOMECHANICS MODELS MATERIALS

PROPERTY GROUP AND PROPERTY NAME/VARIABLE SI UNIT

G E O M E C H A N I C S M A T E R I A L S | 55

56 | C H A P T E R

I n d e x

A absolute values 36

additively decomposed 39

alpha coefficient 49

angle of internal friction 20, 26, 49

B biaxial compression 33

biaxial data 35

biaxial tension 50

Bresler-Pister criterion 32

bulk modulus 48

C calcite 36

Cam-Clay material model (node) 52

Cam-Clay model 27

carbonate rocks 36

Cauchy stress tensor 13

Cayley-Hamilton theorem 13

ceramics 33

circle, Mohr 20

clays 19

cohesion 20, 49

cohesionless soils 25

cohesive-frictional materials 33

complementarity conditions 40

compressive meridians 14, 21–22, 33–34

compressive stresses 12

concrete 32–34

concrete (node) 50

consistency parameters 40

contacting COMSOL 8

continuum mechanics 12

crystal cleavage 36

D deviatoric plane changes 34

deviatoric stress 13, 17

disturbance factor 37

documentation, location 7

dolomite 36

Drucker-Prager criterion 21

ductile materials 19

E effective plastic strain rate 42

effective stress tensor 37

elastic modulus 48

elastic properties, specifying 48

elastic volumetric strain variable 30

elastoplastic materials 16, 33

emailing COMSOL 8

F failure surfaces 16

flow rule 39

fluid pore pressure 24

friction, angle 20, 26, 49

frictionless materials 21, 23

G generalized Hoek-Brown criterion 37

geological strength index (GSI) 37

geometric nonlinearity 49

H Haigh–Westergaard coordinates 14

hexagonal prism 18

Hoek-Brown criterion 35

Hooke’s law 48

hydrostatic axis 14

hydrostatic pressure 12, 20

hydrostatic stress 25

I Internet resources 6

isotropic materials 17, 48

isotropic rocks 37

K k coefficient 49

knowledge base, COMSOL 7

L Lade-Duncan criterion 25

Lamé constants 48

limestone 36

Lode angle 14, 18, 21, 33, 36

I N D E X | 57

58 | I N D E X

M marble 36

Matsuoka-Nakai criterion 24

meridians, tensile and compressive 14,

21, 33–34

metal plasticity 39

metals 19, 22, 41

mixed formulation 47, 53

mobilized planes 24

Model Library, accessing in COMSOL 7

Mohr-Coulomb criterion 20

MPH-files 7

mu coefficient 49

multiaxial stress states 32

N nearly incompressible material 47, 53

O octahedral normal stress 32

octahedral plane 15

opening the Model Library 7

Ottosen criterion 34

over-consolidation pressure 28

P perfectly elastoplastic materials 17

plastic multiplier 40

plastic strain tensor variables 41

plasticity 39

plasticity, theory 12

Poisson’s ratio 48

pressure, wave speed 48

principal stresses 15

R rock mass 37

rock types 36

rock yield criteria 20

rocks 35

rocks (node) 51

S saturated cohesive soils 19

shape factors 35

shear modulus 48

shear stresses 14, 20, 32

shear-wave speed 48

sign convention 12

size factors 35

soil deformation theory 29, 39

soil plasticity 14, 39

soil plasticity (node) 47

soil yield criteria 20

spatially mobilized planes (STP) 24

stress invariants 13

T technical support, COMSOL 8

tensile meridian 14

tensile meridians 21–22, 26, 33–34

tensile normal strains and stresses 12

The Cam-Clay Material Model node adds

the equations and interface for de-

fining the material properties for the

modified Cam-Clay material model.

Right-click to add the Thermal Ex-

pansion and Initial Stress and Strain

features. 52

Tresca effective stress variable 19

Tresca yield criterion 18

triaxial conditions 33

triaxial data 35

typographical conventions 8

U undrained shear strength 18

uniaxial compression 34, 36

uniaxial compressive strength 32

uniaxial tension 17, 50

user community, COMSOL 8

V variables

elastic volumetric strain 30

Tresca effective stress 19

volumetric plastic strain 28

volumetric plastic strain rate 41

volumetric plastic strain variable 28

von Mises criterion 17–18, 22, 41

W web sites, COMSOL 7

Willam-Warnke criterion 33

Y yield functions 17, 20

yield surfaces 16, 21, 40

Young’s modulus 48

I N D E X | 59

60 | I N D E X

www.comsol.com

Documents

Geo Mechanics Module Users Guide