Notes of Dr. Desai

7/29/2019 Notes of Dr. Desai

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NOTES FOR SHORT COURSE

Application of Finite Element and Constitutive Models

SOLID, STRUCTURE AND

SOIL-STRUCTURE INTERACTION:

STATI C, DYNAM IC, CREEP

THERMAL ANALYSES

By

Chandrakant S. Desai

2012

Tucson, AZ, USA


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PREFACE

These notes present descriptions of static and dynamic finite element method, nonlinear

techniques used, various constitutive models (elastic, plastic, creep, thermal, and disturbance-

softening , procedures for determination of parameters for the constitutive models, parametersfor typical materials and interfaces, and program features for the DSC-SST2D code.

The DSC-SST2D based on the finite element method with the DSC model is considered

to be a general purpose finite element code for analysis of a wide range of problems involvingsolids and interfaces or joints, subjected to thermomechanical static, cyclic (repetitive) and

dynamic loadings. The code permits a range of constitutive models for elastic, plastic, and creep

responses, microcracking leading to fracture, and fatigue and softening. As a result, the code canbe used for solutions in civil and geotechnical, mechanical and aerospace engineering,

engineering mechanics, and electronic packaging systems.

Although these notes mainly cover static problems, other codes are available for dynamictwo-dimensional analysis (DSC-DYN2D) and for dynamic three-dimensional analysis (DSC-

SST3D). Their brief descriptions are given below:

I. DSC-SST2D: Two-dimensional Computer code for Static, Dynamic, Creep and Thermalanalysis-Solid, Structures, and Soil-Structure Problems

1. Part I: Manual for Technical Background. The Notes for the Short Course herein havebeen adopted from this manual.

2. Part II: Users Guide3. Part III: Examples Problems-Verifications and Applications

II. DSC-DYN2D: Two-Dimensional code for Dynamic and Static Analysis-Dry and

Saturated (Porous) Materials including Liquefaction

1. Part I: Manual for Technical Background


III. DSC-SST3D: Three-Dimensional Computer code for Static and Coupled Consolidation

and Dynamic Analysis-Solid (Porous), Structures and Soil-Structure Problems:

1. Part I: Manual for Technical Background


This manual (Part I) presents the descriptions of the DSC-SST2D code. The other two are

available in separate reports.


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TABLE OF CONTENTS

TOPIC Page

Preface.......................................................................................................................................................... 2

Table of Contents..,, 3

Introduction................................................................................................................................................. 6

Finite Element Method .............................................................................................................................. 7Computational Algorithm ............................................................................................................... 8Element Library ............................................................................................................................ 10

Constitutive Models ................................................................................................................................. 14Nonlinear Analysis ........................................................................................................................ 16

Drift Correction ......................................................................................................................... 17Continuous Hardening and HISS Models ................................................................................. 17

Program Features .................................................................................................................................... 19Applied Forces ......................................................................................................................... 19

Initial orin situ Stresses ........................................................................................................... 20

Simulation of Sequences .............................................................................................................. 21Addition of Material, or Placement or Embankment ................................................................ 21Removal of Material or Excavation .......................................................................................... 24Removal of Liquid (Water) or Dewatering ............................................................................... 24

Support Systems........................................................................................................................ 26Mesh Change Option ................................................................................................................ 28Boundary Conditions ................................................................................................................ 28

Dynamic Analysis ..................................................................................................................................... 28

Newmark Method ..................................................................................................................... 30

Wilson -Method ...................................................................................................................... 30Mass Matrix ................................................................................................................................... 31

Absorbing Boundaries ................................................................................................................... 31Cyclic or Repetitive Loading ......................................................................................................... 31Creep Behavior .............................................................................................................................. 32

Material Parameters ................................................................................................................................ 32

Organization of Computer Program ...................................................................................................... 32


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Appendix I: Constitutive Models .......................................................................................................... 33Linear and Nonlinear Elastic Models ............................................................................................ 33

Linear Elastic Model.33Nonlinear Elastic Models ........................................................................................................ 33Plasticity Models ..................................................................................................................... 34

Von Mises .................................................................................................................. 35 Mohr-Coulomb ........................................................................................................... 35 Drucker Prager ........................................................................................................... 35 Modified Cam-Clay ................................................................................................... 35 Cap Model .................................................................................................................. 37 Hoek-Brown Model ................................................................................................... 39 Hierarchical Single Surface (HISS) Models .............................................................. 39

Initial Values of and ...................................................................................................... 41

Interface/Joints Element.43Cohesive and Tensile Strengths ........................................................................................... 44Creep Models44Viscoelasticplatic (vep) or Perzyna Model ............................................................................. 46

Multicomponent DSC or Overlay Models .............................................................................. 46Specializations of Overlay Model ........................................................................................... 50

Number of Overlays and Thicknesses ............................................................................... 51Layered Systems with Different Material Properties .............................................................. 51Disturbance (Disturbed State ConceptDSC) Model: Microcracking,Degradation and Softening .................................................................................................... 53

Speciaqlizations55 Thermal or Initial Strains ........................................................................................................................ 55

Elastic Behavior ...................................................................................................................... 55Plane Stress ....................................................................................................................... 56

Plain Strain ........................................................................................................................ 56Axisymmetric .................................................................................................................... 56

Thermoplastic Behavior .......................................................................................................... 57Thermoviscoplastic Behavior .................................................................................................. 58DSC Model .............................................................................................................................. 61

Cyclic or Repetitive Loading .................................................................................................................... 61

Unloading ............................................................................................................................... 63Reloading ................................................................................................................................ 66Cyclic Hardening ..................................................................................................................... 69

Appendix II: Elasto-plastic Equations .................................................................................................. 72

Appendix III: Drift Correction and DSC Computer Algorithm ........................................................ 74DSC Computer Algorithm ....................................................................................................... 75


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Appendix IV: Determination of Constants for Various Models ......................................................... 77

Elastic Constants ......................................................................................................................................... 77Plasticity Constants ..................................................................................................................................... 79

Ultimate: , ........................................................................................................................... 79Phase Change ........................................................................................................................... 81

Hardening ................................................................................................................................ 84Nonassociative......................................................................................................................... 84

Cohesive and Tensile Strengths ........................................................................................ 86

Computer Code to Find Constants for 0- and 1-Models..................................................................... 87Viscoplastic and Creep Models, 0 + vp .................................................................................... 88Mechanics of Viscoplastic Solution ........................................................................................ 88Elastoviscoplastic: Overlay Models ........................................................................................ 92Disturbance Model .................................................................................................................. 93

Cyclic Loading and Liquefaction............................................................................................................. 96Cyclic or Repetitive Loadings, Unloading and Reloading ...................................................... 96

Initial Conditions ..................................................................................................................................... 98

Environmental Effects .............................................................................................................................. 98

Interface/Joint Behavior ........................................................................................................................... 98

Material Constants.................................................................................................................................... 99

Implementation and Applications ........................................................................................................... 99

Material Constants for Typical Materials: Soils, Rock, Concrete, Solders ................................ 101-107

References ........................................................................................................................................ 108-116

PART II: USER'S GUIDE ..........................................................................................................................

PART III: EXAMPLE PROBLEMS: VERIFICATIONS AND APPLICATIONS .............................


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INTRODUCTION, FINITE ELEMENT METHOD,

CONSTITUTIVE MODELS, CONSTRUCTION SEQUENCES

INTRODUCTION

Nonlinear behavior of materials involving solids and interfaces can arise due to material

or geometric nonlinearity, or both. Material nonlinearity under mechanical, thermal and other

environmental loadings, can be due to several factors such as initial state of stress, stress path

dependent response, elastic, plastic and creep strains, change in the physical state defined by

change in the density, void ratio or water content, plastic yielding or hardening, microcracking

and damage leading to softening behavior.

Problems in solid and geomechanics can involve both types of nonlinearities. However,

in the current computer procedures, only material nonlinearity is considered with two-

dimensional (2-D) (plane stress, plane strain and axisymmetric ) and three-dimensional (3-D)

idealizations. The procedures and codes can be used for stress-deformation analysis of a wide

range of problems in solid, structural, geotechnical, and mechanical engineering and electronic

packaging involving solid materials, interfaces and joints. The loading can be static, cyclic and

repetitive and dynamic, and the material response can include elastic, plastic and creep

deformations, microcracking and damage leading to softening or degradation, fatigue failure, and

in microstructural instabilities like liquefaction. Typical examples are also presented. Part III of

the manual covers range of applications.

Realistic solution procedures for engineering problems require appropriate provision for

initial conditions, non-homogeneities and interaction effects. Conventional methods based on

classical theories of elasticity and plasticity may not be capable to handle the above factors.


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Hence, the approach should be to adopt improved but simplified models that are capable to allow

for factors important for a given application. Very often it becomes necessary to resort to

numerical techniques so as to allow for these factors; the finite element method (FEM) is one of

the most powerful methods to solve engineering problems, and is used herein. The FEM code

involves the unified and general approach called the disturbed state concept (DSC), which allows

for hierarchical adoption of a wide range of constitutive models: elastic, elasto-plastic,

continuous yielding, elastoviscoplastic, and disturbance (damage), depending upon the need of

the user for specific application.

FINITE ELEMENT METHOD

In this part of the report, two-dimensional static idealization is considered. Two- and

three-dimensional static and dynamic analyses are covered in other manuals.

The finite element method has been discussed in detail in books such as Desai and Abel

(1972) and Desai (1979). The method presented here is based on the displacement approach for

2-D problems, which has been adopted in the computer code. For two-dimensional typical

element (Fig. 1), the displacement components at any point are written as

qN=u (1)

where {u}T

= [u v] is the vector of displacement components u and v at a point in the x- and y-

directions, respectively, [N} is the matrix of interpolation functions, {q}T

= [u1 v1 u2 v2 un vn]

is the nodal displacement vector , and n denotes the number of nodes.

The strain-displacement and stress-strain relations are given respectively by

qB= (2)

and


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C= (3)

where {} and {} are strain and stress vectors, respectively, [B] is the strain-displacement

transformation matrix, and [C] is the constitutive matrix.

By using the principle of minimum potential energy, the element equilibrium equations

are derived and then expressed in the incremental form as

Q=qkt (4)

where [k1] is the tangent element stiffness matrix, {Q} is the element nodal load vector, {Qr} is

the vector of unbalanced or correction loads, and denotes increment. The terms in Eq. (4) can

be expressed as

VdBCB=k tT

V

t (5)

and

SdTN+VdXN=Q T

S

T

V 1

(6)

and

dVBQ rT

r (7)

in which X is the body force vector, T is the surface traction vector, r is the unbalanced

or correction stress vector, V is the volume of the element, and S1 is the portion of surface on

which surface loads are prescribed. Equations (5) and (6) are usually integrated numerically by

using Gauss quadrature methods.

Computational Algorithm

A nonlinear problem is analyzed as a series of piecewise problems by using

incremental techniques in which the tangent constitutive matrix {C1] is updated at each load


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(-1,-1)

(-1,-1) (1,1)

(1,-1)

t

s

Local Coordinates

4

3

21

t

s

Y

X

Global Coordinates

(b)4-Node Isoparametric Element

(-1,-1)

(-1,-1) (1,1)

(1,-1)

t

s

Local Coordinates

8

7

6 5

4

31

t

s

Y

X

Global Coordinates

(a)8-Node Isoparametric Element

Figure 1. Two-dimensional Isoparametric Solid Elements


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increment, Fig. 2. A mixed procedure (Figure 2) which combines both incremental and iterative

techniques has been adopted together with improved drift correction procedure(s). In this

procedure, after applying each load increment, iterations are performed until convergence is

reached. The convergence criterion employed is based on the ratio of the norm of unbalanced

load and sum of the norm of total load and norm of equilibrating load; details are given

elsewhere (Desai, et al., 1991).

Element Library

The computer program has the provision for the following types of elements:

(i) Solid elements

(ii) Interface/joint, and

(iii) Bar elements.

(i) Solid Elements

Either 4-noded or 8-noded isoparametric finite elements as shown in Fig. 1, or infinite

elements (not operational at this time) (Damajanic and Owen, 1984) as shown in Fig. 3, can be

used. Equations (5) to (7) are used to compute element stiffness matrix and nodal load vector,

respectively. The Gauss quadrature process allows 2 or 3 point integration rules, i.e., total 4 or 9

integration points.

(ii) Joint/Interface Elements

These elements are represented by a thin layersolid element (Desai, et al., 1984; Sharma

and Desai, 1992), or zero thickness Goodman element (Goodman, et al., 1968). They can be

either 4-noded or 6-noded elements (Fig. 4) corresponding to 4-noded or 8-noded solid elements.

The shear and normal responses found from special laboratory tests are used to define the

element stiffness matrix. The constitutive laws, discussed later, are written in terms of shear


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Figure 2. Schematic of Incremental and Iterative Technique

Load

Q1

Q2

Q3

Displacement


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0

Y

X

6

5

4

3

2

1

Global coordinate

s

t

Local coordinate

(a) Biquadratic singly infinite element

0

Y

X

3

2

1

Global coordinate

s

t

Local coordinate

(b) Biquadratic doubly infinite element

Figure 3. Two-Dimensional Infinite Elements


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Figure 4. Joint/Interface Elements

y

Two-Dimensional

x

t

Body 2

Body 1

(8-noded)

Thin-Layer

Element

(4- or 6-

noded)

Body 1

Body 2


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stress, , and normal stress, n. For the thin-layer solid element, the parametric study shows that

the ratio of thickness of interface element to its width of the order of about 0.01 yield satisfactory

simulation of the interface response simulated by using the thin-layer element with finite

thickness.

(iii) Bar Elements

Two types of bar elements, 2-noded linear, and 3-noded quadrilateral elements (Fig. 5),

have been used and provide compatibility with solid and joint elements. The element stiffness

matrix and computation of axial stress are given by Desai (1979) and Lightner and Desai (1979).

CONSTITUTIVE MODELS

A number of material models have been implemented in this program. They are:

(i) Linear elastic,

(ii) Nonlinear elastic (variable moduli or hyperbolic simulation),

(iii) Elasto-plastic conventional (von Mises, Drucker-Prager, Mohr-Coulomb, and Hoek-

Brown),

(iv) Elasto-plastic continuous yielding or hardening (critical state, cap),

(v) Hierarchical Single Surface (HISS) continuous yielding (0 and 1)

(vi) Viscoelastic plastic, and

(vii) Disturbed State Concept (DSC) models; details of this general and unified approach,

from which almost all of the above models can be derived as special cases, are given

later.


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Figure 5. Bar Elements

2

1

l

X

2-node bar element

2

3

1

l

X

3-node bar element


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Each of these categories may be used for solid, structural and geologic materials and

interfaces/joints, depending upon the material behavior and users judgment. However, the most

realistic models are considered to be those based on plasticity or viscoplasticity, in particular the

HISS models, as they include other plasticity models as special cases, and provide a number of

advantages and simplifications (Desai, et al., 1986 and Desai, 2001). The disturbed state concept

(DSC) allows for the above models as special cases, and includes microcracking, damage and

degradation or softening and stiffening or healing (Desai, 1994, 1995, 2001; Desai and Toth,

1996); stiffening is not included in this code.

Descriptions of the above models are given in Appendices I and IV.

Nonlinear Analysis

A nonlinear problem is solved by using incremental-iterative procedures with required

iterative (drift) correction and convergence schemes. The basic incremental stress-strain

equations are given by

dC=d t (8)

where {d and {d} = incremental stress and strain vectors, respectively, and [C1] is the tangent

constitutive matrix. In the case of piecewise linear approximation to nonlinear elastic behavior,

[C1] =e

C1 will be composed of Et and t for solids, or knt and kst for interfaces and joints. For

elasto-plastic behavior

CC=C ptett (9)

where pC1 = tangent plasticity matrix (Appendix II).

The elastoplastic response forms a part of the creep or elastoviscoplastic and disturbance

(microcracking and softening) models in the DSC. Details of the models, elastoplastic, creep and

disturbance, and associated equations are given in Appendix I, together with the incorporation of


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thermal and cyclic hardening effects. In all cases, a drift correction procedure is used with

respect to the drift of the yield surface during incremental loading. A brief description of the drift

correction procedure is given below.

Drift Correction: During each increment of loading, the stress must lie on or within the yield

surface (assuming unloading is elastic). If the increments are not very small, the stress state at the

end of an increment may not lie on the relevant yield surface leading to the problem of the drift

of the currently computed stress as shown in Figure 6. The initial stress state {A} at point A lies

on the previous yield surface, F ({A}, A) = 0, where is the hardening parameter (Appendix

I). During the next increment, yielding occurs and the state of stress moves to point B. The new

yield surface is given by F ({B}, B) = 0. Owing to the tendency to drift, the stress state

represented by point B does not necessarily lie on this new yield surface, Figure 6. This

discrepancy can be cumulative and, therefore, it is important to ensure that the stresses and the

hardening parameter, , are modified so as to lie on the yield surface.

Potts and Gens (1985) examined five different methods for drift correction. They

considered subincrements of strains for each increment, and concluded that the method which

considered hardening during drift correction gave improved results. This scheme is modified and

is described in Appendix III; it is incorporated in the program. Also incorporated is a modified

version of the scheme proposed by Ortiz and Simo (1986). Details of the modified schemes are

given by Desai and Wathugala (1987), Wathugala and Desai (1993).

Continuous Hardening and HISS Models

The classical plasticity models such as von Mises, Mohr-Coulomb and Drucker-Prager do

not allow adequately for the volumetric response, and for the existence of yielding before the


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Figure 6. Schematic Showing Yield Surface Drift

J2D

J1

F({B},B)=0F({A},A)=0

Drift

B

A


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ultimate (failure) surface is reached. Hence, their use is often limited for evaluation of failure or

ultimate loads.

In the critical state and cap models, the continuous hardening or yielding parameter is

dependent only on the volumetric plastic strain, pv . However, in the hierarchical single surface

(HISS) models, hardening is dependent on both volumetric and deviatoric plastic strain

trajectories, v and D, respectively. These models, including the viscoplastic and general

Disturbed State Concept (DSC), are described in Appendix I.

The critical state and cap models allow for yielding before failure, but do not allow for

(a) hardening due to plastic shear strains,

(b) possibility of dilation before peak stress,

(c) different strengths under different stress paths (e.g., compression and extension),

(d) nonassociative behavior for frictional materials, and

(e) involve multiple (two) yield surfaces, which can cause computational difficulties.

The HISS models that involve single continuous yield surface, removes the above

limitations, are considered to be general and more powerful. A perspective and comparison of

the HISS model with such other models as critical state, cap and Lade are given by Desai, et al.,

(1986), Desai and Hashmi (1989), Desai (1992), Desai (1994), Desai (2001).

PROGRAM (DSC-SST2D) FEATURES

The computer program has the following capabilities:

(i) Applied Forces

The program allows for three types of loads, as static, repetitive and dynamic:

a) Extenal loadspoint loads and surface loads,

b) Prescribed displacements, and


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c) Prescribed temperature.

External Loads: Point loads, constant or time dependent, are prescribed at nodes,

whereas the surface loads (constant or time dependent) in the form of distributed traction or

pressure acting on the element sides, are converted to the equivalent nodal loads in the program.

Thermal Loads: Temperature increments or time-dependent temperature is applied at

nodes.

For a linear elastic analysis, total load or temperature may be applied in a single

increment, but in the case of nonlinear analysis, the total load or temperature is applied in several

increments.

Displacements: The program has an option of prescribed displacements, at nodes.

Total displacements at the nodes may be applied in a single increment for linear elastic

analysis, whereas in the case of nonlinear analysis, they are applied in several increments.

(ii) Initial orin situ Stresses

A number of options are available for computing the in situstresses (see Part II: Users

Guide). For example,

a) Prescribed in situ stress: The in situ stress is calculated using the expressions

(Chowdhury, 1978)

socniy sK=

K=

nisK+1y=

oyx

yox

2oy

(10)


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where x, y, and xy are in situ horizontal, vertical, and shear stresses, respectively, is the unit

weight of soil, Ko is the in situ ratio (x/y), y is the depth to the point of stress, and is the

slope of the side of the structure or ground surface (Figure 7).

b) Computed in situ Stresses: A finite element analysis of a soil mass is carried out for

body forces only, assuming linear elastic behavior. The computed vertical stress y is kept the

same, and the horizontal stress x and shear stress xy are computed as

nisK+1

socnis=

K=

2o

xyx

yox

(11)

For horizontal surface, xy = 0.

Simulation of Sequences

(iii) Addition of Material, or Placement Embankment

Simulation of addition of materials, which is called embankment, or placement in the

sequential construction procedure is shown in Figure 8. For each layer (lift) of embankment

placed, the equivalent nodal forces due to gravity are computed. The Youngs modulus, E, of the

material in the added lift is set to a very small value (about one percent of initial E), which

simulates a very weak material. The incremental displacements and stresses are computed

during each lift cycle and are added to those from the previous cycle; iterations are performed (if

necessary) to obtain the equilibrium for each lift. The displacements of the new surface of the

embankment are set to zero. The horizontal stress in the newly placed lift is calculated as the

vertical stress times the in situ stress ratio, Ko.

Note that in the program, the sign of the element material numbers in a newly placed lift

are set to negative, which assigns small value of Youngs modulus to those elements. At the end


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xy

xyx

yh

yv

(a) (b)

Figure 7. Initial Stresses for Inclined Surface


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Figure 8. Addition of Materials or Sequential Construction-Embankment

{o}Initial Stresses

{i}={o}+{i}Final Lift

{1}First Lift

Stress Free Surface


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of computations for the lift when equilibrium is reached, the sign of the element material

numbers is changed back to positive.

(iv) Removal of Material or Excavation

Figure 9 shows schematic of the simulation of excavation process, which is similar to

cut-outs in plates, and involves removal of material(s). The elements to be excavated (removed)

for each lift are deleted from the system and iterations are performed (if necessary) until

equilibrium is obtained. This will result in a stress free excavated surface.

The two key features of the program are:

a) Excavated elements are deleted from the initial and changing mesh.

b) Stress-free surface is established by applying equal and opposite forces on the

excavated surface and by satisfying the equilibrium equation, Eq. (4).

The above process was proposed by Goodman and Brown (1963) and Brown and King (1966).

(v) Removal of Liquid (Water) or Dewatering, Fig. 10

Dewatering causes compression or consolidation and can be modeled by using the

coupled-consolidation theory. However, in order to provide a simpler and economical

formulation, dewatering is approximated in the program by assuming uncoupled and

instantaneous response. The main effect accounted for is the increase in effective stress due to

change in the unit weight of the soil in the dewatered elements. This increase is equal to the body

force due to the weight of water within each of the elements which is dewatered. The equivalent

nodal forces are given by:

VdN=F TW

V

(12)

where {F} is the element nodal force vector and w is the unit weight of water.


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Figure 9. Removal of Materials or SequentialConstruction- Excavation

{o} Initial Stresses

{i}={o}+{i} Final Lift

Stress Free Surface

Nodal Point

Forces

{1} First Lift


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Figure 10. Dewatering

Initial Water Level

Final Water Level

1 2 3

4 5 6

97 8


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followed in the numerical procedure, bar elements will resist the tensioning forces, which is not

correct. The wrong and correct sequences are illustrated in Fig. 11.

(vii) Mesh Change Option

During any increment of the loading, the mesh can be changed, i.e., some elements can

be added or deleted, or some nodes added or deleted and/or material number of elements is

changed. This option is used to simulate embankment construction and excavation. The material

number may be changed in the case of dewatering.

(viii) Boundary Conditions

The prescribed boundary conditions (e.g., fixity) are imposed in such a manner as to

minimize the number of equations to be solved. This is achieved by not formulating equations

corresponding to degrees-of-freedom at nodal points where displacements are zero, because of

the boundary conditions.

DYNAMIC ANALYSIS

The finite element equations for dynamic analysis are given by

tQ=qK+qC+qM (13)

Where [M], [C] and [K] are the mass, damping and stiffnesses matrices, respectively, {q} is the

vector of nodal displacements, {Q(t)} is the vector of time dependent nodal forces and the

overdot denotes time derivative.

The mass matrix can be consistentwhen it is evaluated from the expression resulting

from energy considerations, while it is evaluated as lumpedwhen the mass is lumped at nodes

and appears only on the diagonals of the matrix (Desai and Abel, 1972).

Details of the frequency and time domain solutions for the dynamic equations are given

in Desai and Abel (1972) or in other texts on the finite element method. For the time domain


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Figure 11. Schematic of Supports or Tie Backs

2P

Wrong Sequence Correct Sequence

2P

Step 1

Step 2

2P

Physical Problem

P

P


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analysis, Equations 13 are integrated in the time domain, particularly for nonlinear analysis, by

using various time integration schemes such as Euler, Newmark Method, and Wilsons -

Method. In the present code, Newmark and Wilsons -methods are used. At time tn+1 = tn + t,

where t is the time step and tn is the previous time level at which quantities are known, Eq. (13),

are derived as

Q=qK *1+n* (14)

where (i) forNewmark Method

K+Ct

+Mt

1=K 2

*

(15a)

qt1-2

+q1-+qt

C+

q1-2

1+

t

q+

t

qM+Q=Q

nnn

n

n

2

n

1+n

*

(15b)

in which , are integration parameters in the Newmarks scheme. For conditional stability: 2

0.5.

(ii) forWilson -Method

KC

tM

tK

3

6*

2

(16a)

q2

t+q2+q

t

3C+

q2+qt

6+qt

6M+

Q-Q+Q=Q

nnn

nnn2

n1+nn

*

(16b)

in which is a parameter, usually taken as 1.4.


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It is often difficult to define the damping matrix [C]. Hence, approximate procedures are

sometimes employed; in one such method, the damping matrix is expressed as (Clough and

Penzien, 1993):

M+K=C Mk (17)

where kand M are constants adopted by the user.

In the case of cyclic material behavior, the hysteretic damping is included through the

tangent stiffness matrix, [K*], and it may not be necessary to include the damping in the

analysis.

Mass Matrix

The code allows for two options: consistent mass and lumped mass. The consistent mass

matrix is fully populated and is derived from the energy formulation. In the case of lumped mass,

the matrix is diagonal and the tributary masses are lumped at the element nodes.

Absorbing Boundaries

In dynamic analysis, the waves radiating from a structure are reflected back in the mesh

(body) from the artificial or discretized end boundaries. This can cause spurious errors in the

computed response. One way to reduce this effect is to select the end boundaries far enough such

that the waves are absorbed by internal damping of the material. However, if the end boundaries

are close to the structure, it is desirable to provide for the absorption of the waves at the end

boundaries. In this code, the viscous damping model proposed by Lysmer and Kuhlemeyer

(1969) is implemented. Since this model is not very efficient in absorbing surface waves, it is

advisable to extend the (lateral) end boundaries as far as possible away from the structure.

Cyclic or Repetitive Loading


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Details of cyclic or repetitive loading involving loading, unloading and reloading and

cyclic hardening are given in Appendix I.

Creep Behavior

The code includes the general DSC model which allows for microstructural changes

leading to fracture, failure or liquefaction and available continuum models such as elastic, plastic

and creep. For the latter, viscoelastic (ve), elasticviscoplastic (evp), and viscoelasticviscoplastic

(vevp) models can be used (Desai, 2001).

MATERIAL PARAMETERS

Appendix IV gives details for the determination of material constants for the above

models, based on appropriate laboratory tests for solids and interfaces/joints. It also gives details

of the determination of initial hardening and yield surface based on in situ stresses. Further

details for the HISS and DSC are also discussed in various references. Desai, et al. (1986), Desai

and Zhang (1987), Desai (1994, 1995, 2001), Desai, et al. (1995), Katti and Desai (1995), Desai

and Toth (1996), Desai, et al. (1997).

ORGANIZATION OF COMPUTER PROGRAM

The computer program consists of a main program and about 65 subroutines. The

program is coded in FORTRAN 90. All storage is allocated at the time of execution, and if

desired, the storage can be readily adjusted to the minimum required for the problem to be

analyzed.


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APPENDIX I

CONSTITUTIVE MODELS

This Appendix describes various constitutive models including the unified Disturbed State

Concept (DSC).

Linear and Nonlinear Elastic Models

Linear Elastic Model

It is simplest, but probably the least applicable model for the realistic simulation of

nonlinear behavior. Its main use can be for preliminary studies, and for limited situations

involving mainly the linear behavior.

The constitutive relation for the linear elastic case is given by

C= e (I.1)

where [Cc] is the elastic constitutive matrix, which, for linear elastic and isotropic material, is a

function of two elastic constants, Youngs modulus, E, and Poissons ratio, [Desai and

Siriwardane (1984); Desai (2001)].

Nonlinear Elastic Models

In the computer program, hyperbolic model proposed by Kondner (1963) and formalized

by Kulhawy, et al. (1969) and Duncan and Chang (1970) is included to represent the nonlinear

elastic behavior of solid or soil materials. The tangent modulus, E tand tangent Poissons ratio,

t, are given by (Desai and Abel, 1972)

nis2+socc2

-nis-1R-1

ppK=E

3

31f

2

a

3

n

at

(I.2)

and

A-1

p/golF-G=

2

a3

t


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where

nis2+socc2

-nis-1R-1

ppK

-d=A

3

31f

a

3

n

a

31

(I.3)

1 and 3 are major and minor principal stresses, respectively, c is cohesion, is the angle of

internal friction, pa is atmospheric pressure, Rris failure ratio, n is modulus exponent, R is

modulus number, and G, F and d are Poissons ratio parameters.

A total of eight parameters, K , n , Rf, c, , G, F and d are required to compute Et and

t. If the Poissons ratio is assumed constant, five parameters, K , n , Rf, c, and are required.

For the joint/interface elements, the normal stiffness, kn, is often assumed constant (with

a high value) for compressive normal stress and the shear stiffness, ks, is represented by the

hyperbolid model; it is expressed as (Kulhawy, et al., 1969; Desai, 1974).

nat+c

R-1

pK=k

ana

*f

2

a

n

n

w

*ts

*

(I.4)

where and n are shear and normal stresses, respectively, ca is adhesion, a is angle of interface

friction, w is unit weight of water and K*, n

*and

*

fR are constants. Thus, for the interface, six

constants, kn, K*, n

*ca and , are required.

Plasticity Models

Various plasticity models with relevant yield criteria swhave been incorporated in the

program. The details of these criteria can be found in Desai and Siriwardane (1984), Desai

(1994), Desai, et al. (1986), Desai (1995, 2001). Here, the expressions for the yield criteria are

presented with description of parameters. Compressive stresses are assumed positive.


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1. von Mises yield criterion

0=-J=F yD2 (I.5)

where J2D is the second invariant of deviatoric stress tensor, Sij, and y is the yield stress in

simple tension or compression.

2. Mohr-Coulomb yield criterion

0=socc-3

nisnis-socJ+nis

3

J-=F D2

1

(I.6)

where J1 is the first invariant of the stress tensor, ij, is the angle of internal friction, c is

cohesion, and is Lode angle given by

6

6-

J

J

2

33nis

3

1=

5.1D2

D31-

(I.7)

in which J3D is the third invariant of deviatoric stress tensor, Sij.

3. Drucker-Prager yield criterion

0=k-J-J=F 1*

D2 (I.8)

where * and k are material constants, e.g., for plane strain conditions:

nat12+9

c3=k,

nat12+9

nat=

22

* (I.9)

4. Modified Cam-clay model (Schofield and Wroth, 1968)

0=1-p

p+

ppM

q=F

oo

2

2

(I.10)

where po is the semi-major size of the ellipse, Fig. I.1, M is the slope of critical state (CS) line,

and p = (1 + 23)/3 and 2D31 J3 q . If the critical state line is considered similar to

the Mohr-Coulomb failure envelope (Eq. I.6), then


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Figure I.1 Yield Locus for Critical State Model

dpp

Critical State Line

Mcs

A

M

vp

J1/3

q=3J2D

2po


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nisnis-soc3nis3

=M

(I.11)

The size of ellipse, po, is an exponential function of the hardening parameterv = plastic

volumetric strain pv :

pxep=p voco (I.12)

where pco = initial value of po,

= hardening constant =

oe1 ,

eo = initial void ratio,

= compression index,

= swelling index, andv = trajectory or volumetric plastic strain.

5. Cap Model

The Cap model proposed by DiMaggio and Sandler (1971) has been adopted here. It

consists of a failure envelope (Ff) and a Cap surface (Fc), Figure I.2, the expressions for which

are

0=J-pxe--J=F 1///D2f (I.13)

and

0=L-J+L-X-JR=F 122

D22

c (I.14)

where /, / and / are material parameters, and R, X and L refer to the geometry of the cap

(Figure I.2) which are related as

L-pxe-R+L=X /// (I.15)

The yielding (hardening) defined by the cap is function of the plastic volumetric strain,

p

v , which is denoted by the hardening parameter =p

v . The hardening rule is expressed as


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Figure I.2 Failure and Hardening Surfaces in Cap Model

Drucker-Prager Surface

Ff

Fc

Rb

J2D

J1LZ X

von Mises Surface


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Z+W

-1nD

1-=X

(I.16)

where D and W are material parameters, and Z is related to initial cap.

6. Hoek-Brown Model Yield Criterion (Fig. I.3)

Hoek and Brown (1980) proposed a yield (failure) criterion for rock masses as

2c3c31 s+m--=F (I.17)

where 1 and 3 are major and minor principal stresses, respectively, c is uniaxial compressive

strength of intact rock material, and m and s are constants which depend upon the properties of

rock and upon the extent to which it has been broken before being subjected to stresses 1 and

3. The constant m has a finite positive value which ranges from about 0.001 for highly

disturbed rock masses to about 25 for hard intact rock. The maximum value of s is unity for

intact rock, and the minimum value is zero for heavily jointed or broken rock in which tensile

strength is reduced to zero.

In terms of stress invariants, Eq. (I.17) can be written as

0=s-3

Jm-J

3

nis+socm+

socJ4=F c

1D2

c

2D2

(I.18)

where is the Lode angle (Eq. 1.7).

7. Hierarchical Single Surface (HISS) Models (Desai, et al., 1986; Desai, 1995, 2001)

Advantages of the HISS model with respect to the foregoing models are listed in Chapter

1.

The two hierarchical models, isotropic hardening with associative behavior (0 model)

and isotropic hardening with nonassociative behavior (1 model), have been incorporated in the

program.


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Figure I.3 Hoek-Brown Model

3

1

c

Uniaxial

compresssion

Tension t

RELATIONSHIP BETWEEN

PRINCIPAL STRESS AT FAILURE

Minor principal stress or confirming pressure 3

Maorrincialstress1atfailure

CompressionUniaxial tension

Triaxial

compresion


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The continuous yield function (Fig. I.4) in the HISS plasticity Model:

0=S-1p

J+

p

J--

p

J=F r

m

a

1

2

a

1

n

2

a

D2

(I.19)

where , , m and n are material parameters, pa is atmospheric pressure, Sris the stress ratio

5.1

232

27DDJJ , and is a yield or hardening function defined as (Desai, et al. 1986; Desai and

Hashmi, 1989):

1/a= 1 (I.20a)

or

b+b-1b-pxeb= D43

D21

(I.20b)

in which a1, 1, and b1 to b4 are material constants, 2/1p

ij

p

ij dd is the trajectory of or

accumulated plastic strains, including the volumetric plastic strain (v) and deviatoric plastic

strain (D) trajectories: 2/1

D ;3/p

ij

p

ij

p

vv EdEd ; wherep

ijE = tensor of deviatoric

plastic strains.

The plastic potential function Q is expressed as

S-1p

J+

p

J--

p

J=Q r

m

a

1

2

a

1

n

Q2

a

D2

(I.21)

where

r-1-+= voQ (I.22)in which /vvr , o is value of at the beginning of shear loading, and is a nonassociative

parameter. Equations I.19 and I.21 are used for the nonassociative (1) model.

Initial Values of and Solution for in Eq. (I.19) leads to (Desai, et al., 1991; Desai, 1995, 2001)


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Figure I.4 Basic, 0, and Nonassociative, 1, Models

F/90 FQ

CaCa

J1

J2D

(a) 0 model

F/ Q/

CaCa

J1

J2D

(b) 1 model

F

Q


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where and n are shear and normal stresses, respectively, n and are related to phase change

and ultimate envelope, and and Q are hardening parameters for0 and 1, respectively. A

simple form of hardening function is given by

in which prdu andp

rdv are the incremental plastic shear and normal relative displacements,

respectively, a and b are hardening parameters, and Q is similar to that in Eq. (I.22).

Cohesive and Tensile Strengths

The yield function in the HISS model is extended to include cohesive or tensile strengths

by transforming the stress tensor as (Fig. I.4)

jiji*

ji R+= (I.28a)

where R is related to cohesive or tensile strength. Details are given in Appendix IV.

Here, R can be found from empirical relations (see Appendix IV). It can also be found as

/c=R a (I.28b)

where ac is the intercept along J2D-axis (intersection ofJ2D-axis and ultimate yield surface)

and is related to the cohesive strength, and is related to the slope of the ultimate yield envelope,

Fig. I.4.

Creep Models

Various models including elastoviscoplastic (evp) by Perzyna (1966) have been used to

characterize the creep behavior, Fig. I.5 (Cormeau, 1976; Owen and Hinton, 1980; Desai and

Zhang, 1987; Desai, et al., 1995; Samtani, et al., 1995). Overlay model for creep has been

proposed in (Zienkiewicz, et al., 1972; Pande, et al., 1977; Owen and Hinton, 1980). A general

vd=

vd+ud=

/a=

prv

pr

2pr

2 2/1

b


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Figure I.5 Schematic of Strain-Time Response Under Constant Stress

t2

h

Primary creep

a

b

c

d

e

f

g

i

t1

Secondary creep

Tertiary creep

Failure

Permanent set

Time

Strain

0


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approach called Multicomponent DSC (MDSC) has been proposed by Desai (2001). If the strains

in the component overlays, Fig. I.6, is assumed to be the same, the MDSC model specializes to

the overlay model.

Viscoelasticplatic (vep) or Perzyna Model

MDSC model contains various versions, such as elastic (e), viscoelastic (ve),

elastoviscoplastic (evp), and viscoelasticviscoplastic (vevp). Figure I.7(b) shows the general

rheological representation of MDSC model, from which various versions can be extracted

(Desai, 2001). For instance, the evp, Perzyna type model is shown in Fig. I.7(a), which is based

on the following expression for viscoplastic strain rate vector, vp

:

}{

Q}{ pv

(I.29)

N

oF

F

(I.30)

where is the fluidity parameter, is the flow function, N is the power law parameter, and Fo is

the reference value (e.g., yield stress, atmospheric constant, etc.). For associative plasticity, F

Q.

Multicomponent (MDSC) or Overlay Models

In the overlay model (Fig. I.6), the behavior of a material is assumed to be composed of

those of several overlays, each of which undergoes the same deformation (strain) and provides a

specific material characterization. The total stress field is obtained as the sum of different

contributions from each overlay. By introducing a suitable number of overlays and assigning

different material properties (parameters) to each, a variety of special models can be reproduced,

as shown below.


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occurs, an instantaneous elastic recovery, b-c, is followed by delayed elastic recovery, c-d. If the

load is continued beyond the primary creep range, secondary creep (b-e) begins which is

accompanied by irreversible deformations. Unloading at any time during b-e leaves a permanent

deformation or set (strain). On continued loading, tertiary creep begins leading to failure.

The overlay model for the two-dimensional problem is illustrated in Fig. I.6. Each

overlay can have different thicknesses and material properties. The overlays do not experience

relative motion, or they are glued together. Therefore, the overlay models exhibit the same

deformation under given loading.

In the MDSC (overlay) model developed here, a number of units are arranged in parallel,

Fig. I.7. This results in different stress fields, {j}, in each overlay (j) which contributes to the

total stress field {} according to the overlay thickness, tj; hence,

t

= jj

k

1=j

(I.31)

in which k is the total number of overlays in the model, and

1=tj

k

1=j

(I.32)

The equilibrium equations for a (finite) element become:

Q=VdtB jjk

1=j

T

V (I.33)

in which {Q} is the load vector.

From Eq. (I.33), the element stiffness matrix is obtained as

VdBtCB=k jjk

1=j

T

V

(I.34)


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where [Cj] is the constitutive matrix. This matrix will be different for each overlay, according to

the material properties.

Figure I.7 The Overlay Model in Two-Dimensional Situation (Pande, et al.,1997)

ti 1


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The solution procedure (see later) is then identical to that of standard viscoplasticity (Perzyna

type) involving time integration, with stress being calculated for each overlay (Owen and Hinton,

1980). It should be noted that the viscoplastic strain in an overlay will be different due to

differences in threshold yield values and flow rates, but the total strains in all overlays are the

same.

Specializations of MDSC (Overlay) Model

The material parameters for elastic, viscous and yield characterizations are shown in Fig.

I-6. By adopting different values of the parameters, the overlay model can specialize to various

versions. For instance, consider the overlay model with two viscoplastic units; such a two-

overlay model is commonly adopted; Table 1 gives examples of specializations.

Table I.1: Specializations of MDSC (Overlay) Models

Specialization

Plasticity

Model

No. of

Overlays Thickness Parameters

Elastic (e)1

von Mises 1 1.0E, , , N and veryhigh y

Viscoelastic (ve)2

von Mises 2 0.5, 0.5E1, 1, 1, N1, y1 = 0;E2, 2, 2, N2, y2 =very high

Elastoviscoplastic

(evp)3

(Perzyna type)

Any 1 1.0 E, , , N, y or F

Viscoelasticviscoplastic

(ve vp)4

von Mises

Any

1

= 21

0.5

0.5

E1, 1, 1, N1, y1 = 0

E2, 2, 2, N2, y2 or F

1-4The following notes show resultant models with the specific choice of parameters.


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Notes:1Here, as y is high, only the elastic spring will be operational because the dashpot slider

unit will be essentially not operational.

2Here, for overlay 1 as yl = 0, only the spring and dashpot will operate, as y2 > > , only the

spring will operate in overlay 2.

3Here, with one overlay, all units are operational.

4Here, the first overlay (with y1 = 0), leads to the spring and dashpot, and, in the second

overlay, all units are operational.

Number of Overlays and Thicknesses

Usually, two overlays are sufficient and the thickness of each overlay is prescribed as 0.5.

Layered Systems with Different Material Properties

When a problem with layered material (e.g., pavement) is to be analyzed, some materials

may behave as viscoelasticviscoplastic (vevp), and others are elastic or elasto-plastic, the

following procedure can be used:

(i) For the material with vevp response, two overlays (Table I.1) can be used.


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(ii) For the elastic response, the material is considered with one overlay and infinitely large

yield strength (Table I.1).

(iii) For the elasto-plastic response of the material, one overlay is used and the fluidity

parameter, , is taken to be very small, approximately 1/600 of fluidity parameter prescribed for

the vevp material, and N = 1.

DISTURBED STATE CONCEPT (DSC)

The DSC is considered as the culmination of various models developed previously. It is

general and unified from which most of the other models can be obtained as special cases. Its

hierarchical nature allows formulation of general constitutive matrix in computer (finite element)

procedures; hence, a chosen model can be achieved by inserting material parameters for that

model, say, elastic or continuous yield plasticity.

The DSC has been covered in a number of publications (Desai and Ma, 1992; Desai,

1995, 2001; Desai and Toth, 1996; Katti and Desai, 1995; Desai, et al., 1998a,b). Hence, brief

description is given below.

In the DSC, a deforming material element is assumed to consist of various components.

For instance, for a dry material, it is assumed to contain two components: continuum orrelative

intact(RI) and discontinuum orfully adjusted(FA) phases. These components interact and

merge into each other, transforming the initial RI phase to the ultimate FA phase. The

transformation occurs due to continuous modifications in the microstructure of the material. The

disturbance or microstructural changes act as a coupling mechanism between the RI and FA

phases.

The incremental constitutive equations for the DSC can be expressed as follows:


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iccc

iia

dD

dCD

dCDd

1

(I.35a)

where a,i, and c denote observed, RI and FA states, respectively, {} and {} are the stress and

strain vectors, and dD the increment (or rate) of disturbance, D.

Degradation and Softening

The disturbance can be defined on the basis observed (laboratory and/or field) behavior

in terms of stress-strain, volumetric strain, pore water pressure, ultrasonic properties as P- and S-

waves, e.g., shear wave velocity (Desai, 2001). For instance, D can be expressed (Fig. I.8) as

ci

ai

D

(I-36a)

Disturbance can be expressed in terms of an internal variable such as accumulated deviatoric

plastic strain (D) or worki:

zDA

u eDD

1 (I-36b)

where Du, A, and Z are parameters determined by using Eq. (I-35).

The continuum or RI phase can be characterized by using models based on continuum

elasticity, plasticity or viscoplasticity. For instance, the constitutive matrix [Ci] can be defined by

the HISS plasticity or conventional plasticity model. The FA part can be modeled in various

ways by assuming that FA part (i) has no strength like conventional damage model by Kachanov

(1986), (ii) has hydrostatic strength like in classical plasticity, and (iii) has strength

corresponding to the critical state (Schofield and Wroth, 1968), at which the material deforms

without change in volume or density. For instance, if we assume that the FA part has only

hydrostatic strength, defined by bulk modulus, K, Eq. (I-35a) reduces to:


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Figure I.8 Schematic of Elastoplastic and softening (DSC) Responses

Elastoplastic(virgin)

(a) Elastoplastic Response with Unloading and Reloading

Elastoplastic(i)

(b) DSC Softening with Unloading and Reloading

Softening: Observed(a)

Fully Adjusted(c)

D


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iii

iia

SdD

ID

dCDd

3

1(I-35b)

where {I} is the unit vector and {S} is the vector of shear stress components. Here, it is assumed

that the mean pressure p (= J i/3 = ii/3) and the strains are the same in the RI and FA parts. In

that case, eq. (I-35a) can be written as

dCd DSCa (I-35c)

where [CDSC

] is the general constitutive matrix and dD = {R}T

{di}, R is derived on the basis of

the adopted yield function (Desai, 2001). The constitutive matrix is given by

icTciDSC

R

CDCDC

1(I-35d)

Specializations

If D = 0, that is, the material is considered as a continuum, Eq. (I-35a) reduces to

ii dCd (I-35e)

where [Ct

] can be elastic, elastoplastic, or elastoviscoplastic model.

THERMAL OR INITIAL STRAINS

Thermal and mechanical (loading) cycles are available in the finite element code. The

implementation aspects for various characterizations and cyclic (loading-unloading-reloading)

are described below.

Elastic Behavior

In the case of elastic behavior, the effect of known temperature change causing initial

strains, are given below for various two-dimensional idealizations:


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Plane Stress

0.0

T

dTT

dTT

xy

Ty

Tx

(I.37)

where is the coefficient of thermal expansion and dT is the temperature change = T To, To is

initial (previous) temperature and T is the current temperature.

Plane Strain

dTET

T

dTT

dTT

Tz

xy

Ty

Tx

0.0

1

1

(I.38)

where E and are the elastic parameters.

Axisymmetric

0.0

T

dTT

dTT

dTT

rz

T

Tz

Tr

(I.39)

Then the incremental elastic constitutive relation is given by

TddC

dCd

e

ee

(I.40)

where [Ce] is the elastic (tangent) constitutive matrix, and {d }, [d e} and {d (T)} are the

vectors of total, elastic and thermal strains, respectively.


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If the parameters E and vary with temperature, they can be expressed in terms of

temperature as (Desai, et al., 1997; Desai, 2001):

TC

t

rT

TEE

(I.41a)

C

r

rT

T

(I.41b)

where Erand rare values at reference temperature, Tr(e.g., room temperature = 300 K), and cT

and c are parameters found from laboratory tests.

Thermoplastic Behavior

The normality rule gives the increment of plastic strain vector {dp(T)} as

TQTd p

,,(I.42)

where Q is the plastic potential function; for associative rule, Q F, where F is the yield

function. Now, the total incremental strain vector {dt} is given by

TdTdTdTd pet (I.43)

where {d(T)} is the strain vector due to temperature change. Hence,

TdQdTd e

(I.44a)

and

dTIQ

TC

TCd

T

e

e

0

1

e

e

d

d

(I.44b)

where 0]11[01 I for two-dimensional case and [1 1 1 0 0 0] for three-dimensional case.Now, the consistency condition gives

0T,, dF (I.45)


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Therefore,

dTT

Fd

Fd

FdF

T

(I.46)

Then, use of Eqs. (I.44) and I.46) gives

2/1

o

Id

QQFQTC

F

dTTCF

dTT

FTC

T

F

T

e

T

e

T

T

e

T

(I.47a)

Therefore,

dT

QQFQTC

F

Q

T

FITC

FQ

ITC

dQQFQTC

F

TCFQ

TCd

T

e

T

oe

T

T

oe

T

e

T

e

T

e

2/1T

2/1

I

(I.47b)

The parameters in the elastoplastic model, e.g., HISS-0. can be expressed as function of

temperature as

c

r

rT

TPTP

(I.48)

where P is any parameter such as E, , Eq. (I.40); , , R, n, Eq. (I.19); a1, 1, Eq. (I.20); Pris its

value at reference temperature Tr, and c is parameter found from laboratory tests.

Thermoviscoplastic Behavior

The total temperature dependent strain rate vector, , is assumed to be the sum of

thermoelastic strain rate, )(Te , thermoviscoplastic strain rate, )(Tvp , and the thermal strain

rate due to temperature change dT, )(T , as

TTT vpe (I.49)

Here, the thermoviscoplastic strain is contributed by rheologic or creep and temperature effects.


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With Perzynas (1966) viscoplastic theory, Eq. (I.29), Eq. (I.49) can be written as

TFF

TFTT

e

o

e

(I.50)

where and are temperature dependent fluidity parameter and flow function, respectively.

Then the constitutive equations are given by

T

F

F

TFTTC

o

e

(I.51)

Viscous or creep behavior requires integration in time. The thermoviscoplastic strain rate

is evaluated from Eq. (I.29) at time step n, Fig. I.9. Then the strain rate at step (n + 1) can be

expressed by using Taylor series expansion as (Desai, et al., 1995); Owen and Hinton, 1980)

IdTGdGT

IdTT

T

dT

TT

nnnnnvp

n

nvp

n

nvpnvpnvp

21

~

~

~

1

(I.52a)

where

n

d ~ is the stress increment, dT

n

is the temperature increment, and [G1]

n

, [G2]

n

denote

gradient matrices at time step, n.

The increment of viscoplastic strain, nvp Td )( , can be found during the time interval

tn = tn+1- tn, Fig. I.9, as

11 nvpnvpnvp TTtTd (I.53)

where 0- 1. For = 0, Eq. (I.53) gives the Euler scheme, for = 0.5 the Crank-Nicolson

scheme and so on. The present code allows for = 0 and 0.5.

Now, Eq. (I.51) can be written in the incremental form as


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Figure I.9 Time Integration for Viscoplastic Strains

nvp

tn tn+1 t

tn

nvp

vpn+1vp


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TdTd

dTCd

nvp

nen

~ (I.54)

Use of Eqs. (I.52) and (I.54), leads to

TdTtG

tTdTCd

n

n

n

2

n

nvp

n~

evpn

~

(I.55)

where

11

nneeevp tGTCITCTC

DSC Model

In the case of the DSC model, Eq. (I.35), the RI response can be simulated as elastic, Eq.

(I.40), elastoplastic, Eq. (I.47b), or elastoviscoplastic, Eq. (I.55), which include the temperature

dependence.

With the general DSC model, Eq. (I.35), the disturbance parameters, Du, A and Z, Eq.

(I.36b) can be expressed as functions of temperature, by using Eq. (I.48). Their values

determined from tests at different temperatures, which are used to define the function in Eq.

(I.48).

CYCLIC AND REPETITIVE LOADING

Cyclic and repetitive loading, involving loading, unloading and reloading, occur in many

problems such as dynamics and earthquakes, thermomechnical response such as in electronic

packaging and semiconductor systems, and pavements. If the simulated behavior involves

continuing increase in stress along the same loading path, without unloading and reloading, Fig.

I.10, it is often referred to as monotonic orvirgin loading. The unloading and reloading are often

referred to as nonvirgin loading. Loading in the opposite side, i.e., negative side of the (stress)

response, is sometimes referred to as reverse (reloading) loading. Cyclic loading without stress


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62

Figure I.10 Schematic of Loading, Unloading, and Reloading

Reloading(Reverse)

Unloading

Reloading

Loading

Virgin

A

Unloading


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63

reversal is often referred to one-way, while with stress reversals, it is referred to as two-way. In

the case of degradation or softening, decrease in stress beyond the peak occurs, but it is

considered different from unloading.

For the virgin loading, the constitutive equations, Eq. (I.35), apply. For nonvirgin

loading, it is required to consider additional and separate, often approximate, simulations.

In the case of elastoplastic model (e.g., HISS-0), the simulated virgin response allows

for the effect of plastic strains and plastic hardening or yielding, Fig. I.11(a). In the case of the

softening behavior, the plasticity model can simulate the RI behavior, and the use of DSC allows

for the degradation, Fig. I.11(b).

Plastic deformations can occur during unloading and reloading, and can influence the

overall response, Fig. I.11. Although models to allow for such behavior have been proposed in

the context of kinematic hardening plasticity (Mroz, et al., 1978); Somasundaram and Desai,

1988), they are often relatively complex and may involve computational difficulties. Hence,

approximate schemes that are simple but can provide satisfactory simulation have often been

used; one such method implemented in the present code, is described below.

Unloading

As indicated in Fig. I.10, the unloading response is usually nonlinear. However, as a

simplification, it is often treated as linear. Here, both linear and nonlinear elastic simulations are

included. For the nonlinear case, of which the linear simulation is a special case, the procedure

proposed by Shao and Desai (1998a,b) is used. During unloading, the following incremental

stress-strain equation is used:

dCd UL (I.56)


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Figure I.11 Unloading Interpolation Function for CTC and SS Tests

Ge

Gb

p e (b) Simple Shear Test

Ee

Eu

Eb

Current

1p 1e 1

A

(a) CTC Test

1-3


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65

where [CUL

] is the elastic constitutive matrix with variable elastic unloading modulus, Eu, Fig.

I.11, and the Poissons ratio, , is assumed to be constant. The modulus Eu is given by

pbu EEE

111 (I.57)

where Eb

is the slope of the unloading curve (response) at the point (A) of unloading, Fig.

I.11(a), and Ep

is the plastic modulus, which is evaluated by using the following equation:

2

22

1

K

D

b

D

aa

p

JJ

pKpE

(I.58)

where K1 and K2 are constants, pa is the atmospheric pressure (used for nondimensionalization,

and bDJ2 and J2D and the second invariants of the deviatoric stress tensor, Sij, at the start of

unloading (point A), and at the current state during unloading, respectively.

The values of K1 and K2 are found from laboratory tests. For triaxial compression CTC:

1 > 2 = 3) and simple shear (SS) tests, their values are derived as follows:

Triaxial Compression (CTC) Test

0.1

113

1

22

2

bep

D

b

D

EE

JJK

(I.59a)

1

22

12

1

2

1

3

K

a

D

b

D

p p

JJ

KK

(I.59b)

where Ee

is the elastic modulus (slope) at the end of unloading and p1 is the plastic strain, Fig.

I.11(a).

Simple Shear (SS) Test, Fig. I.11(b)

The relation between the elastic (Youngs) and shear moduli (G) are given by

bb GE 12 (I.60a)


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66

ee GE 12 (I.60b)

Substitution of Eq. (I.60) into Eq. (I.59) and replacing DJ2 by (shear stress) andp

1

by 12/3 p , where pis the plastic shear strain, Fig. I.11(b), leads to

0.111

2

GG

Kep

b

(I.61a)

1

2

1

2

1

1

K

a

b

p pKK

(I.61b)

where b and are the shear stresses at the point of unloading, and during unloading,

respectively.

The values of p1 and p are evaluated by using the following equations:

be

b

p

EE

JD 11

2

32

1 (I.62a)

and

be

bp

GG

11

2

(I.62b)

Reloading

Figure I.12 shows two cases of reloading, for the one-way and two-way. In both cases,

the following constitutive equation is used:

dCRdCRd eDSCa 1 (I.63)where R is the interpolation parameter such that 0 R 1; R = 0 for the beginning of

reloading and R = 1 at the end of reloading. Thus, at the beginning of reloading, the behavior is

elastic, given by

dCd ea (I.64a)

at the end of reloading, virgin response resumes:


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67

Figure I.12 Two Reloading Cases

Eb

Ebr=Eb

E

Unloading

Reloading

A

B

(a)Reloading case 1: A B (one-way)

Unloading

Ebr=Ee

Reloading(Reverse)Unloading

A

B

(b)Reloading case 2: A B (two-way)


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68

dCd DSCa (I.64b)

The elastic modulus, ER, for the two cases, Fig. I.12, is different. For case 1, the elastic

modulus at the start of reloading, Ebr

, is given by

bbr EE (I.65a)

where Eb

is the unloading slope at the beginning of unloading, Fig. I.13(a). For case 2,

ebr EE (I.65b)

where Ec

is the slope at the end of unloading, Fig. I.13(b).

The interpolation parameter, R , for both cases is found as

b

D

D

J

JR

2

2 (I.66)

where bDJ2 and J2D are the second invariants of the shear stress tensor at the beginning of the last

unloading and current level, respectively.

In computer (finite element) analysis, the reloading stress path may be between the above

two cases. Then, a parameter, S, is defined as an indicator of the direction of reloading:

d

b

Tb dS (I.67)

where -1 S 1, {b}, {} and {d} are the stress vectors before unloading, the current stress

vector and the next stress increment respectively. S = -1 indicates case 1 reloading, while S = 1

indicates case 2 reloading. Now, E

br

is interpolated between E

b

and E as

ebbrE

S

E

S

E 2

1

2

11

(I.68a)

Then, the modulus for reloading, ER, is found as

E

R

E

R

EbrR

11

(I.68b)


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69

where E is the elastic modulus of the material, which is often found as (average) slope of the line

joining the unloading and end of unloading points or the initial slope, Fig. I.13(a). Then at the

beginning of reloading when R = 0, ER

= Ebr

, which ensures smooth transition from unloading to

reloading, Fig. I.12(b). At the end of reloading (R = 1), E-R

- = E, which ensures smooth transition

from reloading to the virgin loading.

Cyclic Hardening

In the case of elastoplastic behavior, there exists a yield surface (Fo) corresponding to the

initial or past state of stress experienced by the material before the present cyclic or repetitive

load is applied, Fig. I.13. When unloading occurs, the plastic strains can change (increase or

decrease), and hence, for the reloading after the unloading, the yield surface that defines the

elastic limit usually expands from Fo to the initial surface, Fi, corresponding to each cycle N (= 1,

2, ). As a result, the magnitudes of plastic strains decrease from one cycle to the next, which is

often referred to as cyclic hardening.

For a given load or stress (increment), the final or bounding surface, Fb, can be defined

by solving the incremental constitutive equations, (I-35). In the case of repetitive loading under

constant amplitude of load (stress), Fig. I.13(b), the maximum load (Pmax) will be the amplitude

of the load (stress). In the case of cyclic (one-way) loading, Fig. I.13(c), the bounding surface,

Fb, would change for each stress increase. Note that in the repetitive load analysis, here, the time

effects are not included.

Mroz, et al. (1978) proposed a model for cyclic hardening, which was adopted by

Bonaquist and Witczak (1997) for materials in pavement structures. The approximate (modified)

method for cyclic hardening implemented in the present code is similar, and is described below.


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70

Figure I.13 Cyclic Hardening Under Repeated Loading

O

B

A

C

B

F0F1

F2

Fb

J1

J2D

(a)Cyclic hardening

B

A C

B

O

(c)Loading-Unloading-Reloading

O B

A

Pmax

P

Time

(b)Repeated wheel load


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71

For the given load or stress increment, two bounding surfaces are defined, F o and Fb, Fig.

I.13, and the corresponding hardening functions and parameters are o and b, Eq. (I.20), and o

and b, respectively. Here, denotes the accumulated plastic strains:

2/1

pTpdd (I.69)

where {dp} is the vector of incremental plastic strains. Then the initial yield surface parameter,

i, for a given cycle, i, is expressed as

ohoi cN

b1

1 (I.70a)

where hc is the cyclic hardening parameter, determined from laboratory repetitive tests. It

controls the rate of expansion of the initial yield surface, Fi, at the end of unloading for a given

cycle, N. If hc = 0, no cyclic hardening occurs.

Bonaquist and Witczak (1997) considered repeated tests involving the same stress

(amplitude) to an initially unstrained material specimen, o = 0. Then, Eq. (I.70a) becomes

bhib cN 1 (I.70b)

orch

b N

1

where is the plastic strain trajectory up to cycle N. Plots of normalized trajectory /b vs

number of cycles are used to find hc through a least square procedure. For the granular material,

hc = 1.06 was found (Bonaquist and Witczak, 1997).

With the above formulation, the value ofi, Eq. (I.70) is used to evaluate the hardening

function, i, Eq. (I.20). It is used to define the elastoplastic constitutive matrix [Cep

] = [Ci], Eq.

(I.36e), the general DSC matrix [DDSC

], Eqs. (I-35e), and (I.63), when reloading occurs


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72

APPENDIX II

ELASTO-PLASTIC EQUATIONS

The incremental total strain vector {d} is the sum of incremental elastic, {de} and

plastic, {p} strain vectors, i.e.,

d+d=d pe (II.1)The incremental elastic strain is related to the incremental stress as

dC=d ee (II.2)where {d} is the incremental stress vector and [Ce] is the elasticity matrix.

Using the theory of plasticity, the incremental plastic strain vector is given by the flow

rule

Q=d 'p (II.3)

where / is the scalar constant of proportionality.

The consistency condition is

dF = 0 (II.4)Equations (II.1) to (II.4) are combined to obtain the incremental stress-strain relation

dC=d pe (II.5)where [C

ep] is the elasto-plastic constitutive matrix.

The expression for/ and [Cep] are derived as

H-QCF

dCF

=e

T

e

T

'

(II.6)

and


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H-QC

F

CF

Q

C

-C=Ce

T

e

T

e

epe

(II.7)

where H is the term due to hardening. For non-hardening yield function, H = 0 and for hardening

yield functions, H is defined as follows:

(i) Critical State and Cap model

F

F=H

(II.8)

where

F

F=

jiji

2/1

F

(II.9)

(ii) HISS model (non-assdociative)

DQ

D

Q

F+

F=H

(II.10)

where

Q

Q=

jiji

2/1

Q

(II.11)

and

Q

Q=

ji Dji D

2/1

DQ

(II.12)

where D denotes deviatoric part. For associative model Q F.

The elastoplastic constitutive matrix [Cep

] represents the response of the material in the

relative intact (RI) state and forms a part of the general DSC matrix, Eq. (I.35c), when

disturbance (softening or degradation) is considered.

Derivations for creep and DSC models are given by Desai (2001).


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74

APPENDIX III

DRIFT CORRECTION AND DSC COMPUTER ALGORITHM

Under a given stress increment, {d}, the stresses at point B do not lie on the yield

surface, Fig. 6 (in the main text), i.e., F ({B}, B) > 0, where is the hardening function. The

stress vector {B} and B are to be corrected so that F ({B}, B) 0. The method, designated as

correction method by Potts and Gens (1985) and modified by Desai, et al. (1991), is described

below.

The correction is carried out by an iteration procedure. At the nth iteration, the stresses

and hardening parameters are given by

QC

e/

1nn (III.1)

dnn 1 (III.2)

vvvn dn 1 (III.3)

DDDn dn 1 (III.4)

where

HC

F

e

T

n

Q

F

, 1-n1/ (III.5)

Fd / (III.6)

FVvd

/ (III.7)

FDDd / (III.8)


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75

0in which

F

F=

jiji

2/1

F

(III.9)

3/

F=

ii

VF

(III.10)

F

F=

ji Dji D

2/1

DF

(III.11)

For non-hardening, yield function, H = 0 in Eq. III.5 and for hardening yield function, H

is given by Eq. II.8 or Eq. II.10 of Appendix II.

The derivatives

QF, and

D

FF

, are evaluated at the stress point {n-1}. The

iterations are performed until the yield function is satisfied, i.e., F ({n}, n) 0 within the

tolerance of 10-6

or less. For the first iteration, {o} is taken as {B} and o as B.

DSC Computer Algorithm

According to Eq. (I.35a), the DSC incremental finite element equations are given by

~~o

~~~Q- QdQqdk

iDSC (III.12)

whereDSC

k~

is the nonsymmetrical stiffness matrix,i

qd~

is the vector of nodal increment

displacements,~

Q is the applied load vector,

~oQ is the balanced load vector. Incremental

iterative solution of Eq. (III.12) involves negative definite stiffness matrix in the softening zone

(Desai and Toth, 1996). However, a number of approximate but simplified strategies can be used

(Desai and Woo, 1993; Desai, et al., 1999; Desai, 2001). One such scheme is to first solve for the


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76

RI response by considering only the symmetric part ofDSC

k~

that defines the RI behavior. Hence,

the following RI equations are first solved:

~ 1~~

i

n

i

n

i

n Qdqdk (III.13)

wherei

Documents

Notes of Dr. Desai