ABMs for political scenarios: the Framing theory

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• The characteristics of the Extended Drucker-Prager models:

• These models are intended for monotonic loading.

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

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

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

• The material in these models is initially isotropic.

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

• The material becomes stronger as the confining pressure increases.

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

L3.50

Extended Drucker-Prager Models

• The characteristics of the Extended Drucker-Prager models (contd.):

• These models differ in the manner in which the hydrostatic pressure dependence is introduced.

• These models include isotropic hardening or softening.

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

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

• These models can incorporate strain-rate dependent material properties.

• The material properties in these models can be made temperature dependent.

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

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

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

• Linear

• Hyperbolic

• Exponent

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

• the kind of material being modeled

• the available experimental data for calibration of the model parameters

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

L3.52

Extended Drucker-Prager Models

• The Linear Drucker-Prager model

• The yield surface of the linear model is written as

• p

is the equivalent pressure stress

• t

is the deviatoric stress measure

•

β

is the friction angle

• d

is cohesion and is related to the hardening input data

• The model allows for separate dilatation and friction angles

tan 0

F= −t p β− =d .

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• The model provides a non-circular yield surface in the deviatoric plane:

• It allows for the matching of different yield values in tension and compression.

• It provides flexibility in fitting experimental results.

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

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

L3.54

Extended Drucker-Prager Models

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

where ,

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

in uniaxial tension, and in pure shear.

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

precludes dilation angles ,

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

,

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• The flow is associated in the deviatoric plane but nonassociated in the p–tplane if

ψ ≠ β

.

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

where d^'is the hardening parameter that is related to the hardening input data as

if hardening is defined by uniaxial compression, σ_c;

if hardening is defined by uniaxial tension, σ_t;

if hardening is defined by shear (cohesion), d.

2 2

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• determines how quickly the hyperbola approaches its asymptote (see sketch).

• is the initial hydrostatic tension strength of the material, is the initial value of d^', and βis the friction angle measured at high confining pressure.

• The model treats βand l₀as constants during hardening.

0 0 _t0tan

l =d′ −p β

t0

p d ′₀

L3.58

Extended Drucker-Prager Models

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

(The Kparameter is not available for this model.)

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• Exponent model

• The general exponent form provides the most general yield criterion available in this class of models. The yield function is written as

where aand bare material parameters independent of plastic deformation and p_tis the hardening parameter that represents the hydrostatic tension strength of the material and is related to the input data as

if hardening is defined by uniaxial compression, σ_c; if hardening is defined by uniaxial tension, σ_t; and if hardening is defined by shear (cohesion), d.

b 0

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

(The parameter K is not available for this model.)

• The material parameters a, b, andp_t:

• Can be specified directly, or

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

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• Flow in the hyperbolic and exponent models

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

where,

ψ

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

ε

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

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

(

0tan

)

^{2 2} tan

G= ε σ ψ +q −p ψ,

σ 0

L3.62

Extended Drucker-Prager Models

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

• The potential is continuous and smooth.

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

• The potential is the von Mises circle in the deviatoric stress plane.

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• Associated flow is obtained in the hyperbolic model if β = ψ and

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

• The default flow potential eccentricity is ε = 0.1.

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

• Increasing the value of εprovides more roundedness to the flow potential.

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

ε < 0.1may lead to convergence problems if the material is subjected to low confining pressures because of the very sharp curvature of the flow potential near its intersection with the p-axis.

0

• These models are invoked by the

∗DRUCKER PRAGER material option.

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

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

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

compression

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

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• The yield function can be made rate dependent by using the

∗RATE DEPENDENT option or by specifying the yield stress as a function of the plastic strain rate.

• Rate dependency is rarely used for geotechnical materials.

• It may however be important for polymers.

L3.66

Extended Drucker-Prager Models

• The elasticity is defined by:

• linear elasticity, or

• porous elasticity.

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

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

–∗INITIAL CONDITIONS,

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

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

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

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

L3.68

Extended Drucker-Prager Models

• Matching experimental data

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

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

• Most common experiments performed:

• uniaxial compression (for cohesive materials)

• triaxial compression or tension tests

• shear tests for cohesive materials

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

• Record the load and displacement in the direction of loading

• Record the lateral displacements for measuring volume changes

• Triaxial test data are required for a more accurate calibration.

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

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

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

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

L3.70

Extended Drucker-Prager Models

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

• Thus, the principal stresses are all negative, with

0 ≥ σ

₁

= σ

₂

≥ σ

₃.

• The stress invariant values in triaxial compression are

• The triaxial results can, thus, be plotted in the q–pplane.

(

1 3

)

1 3

1 2

p= −3 σ σ+ , q=σ σ− , r= −q, t=q. Triaxial compression and tension tests

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

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

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

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

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

L3.72

Extended Drucker-Prager Models

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

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

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

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

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• Linear Drucker-Prager model

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

L3.74

Extended Drucker-Prager Models

• Triaxial tension test data are also needed to define K:

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

• In this case the principal stresses are 0 ≥σ₁ ≥σ₂ =σ₃.

• The stress invariants are now

K can be found by plotting qversus p.

– Kis the ratio of the values of q for triaxial tension and compression at the same value of p.

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

• Generally,

0 ≤ ψ ≤ β

.

(

1 3

)

1 3

1 2

3

p q r q t q

σ σ σ σ K

= − + , = − , = , = .

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• Hyperbolic model

• Use the triaxial compression results at high confining pressures to obtain βand d^'for the hyperbolic model.

• Hydrostatic tension, p_t , is also needed to complete the calibration.

tan d′ β

− −p_t p

d ′ q

β

( _{0 t 0}tan )^{2 2} tan 0 F= d′ −p β +q −p β−d′= Hyperbolic :

L3.76

Extended Drucker-Prager Models

• Exponent model

• Abaqus provides a capability to determine the material parameters a, b, and p_trequired for the exponent model from triaxial data:

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

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• The data points obtained from triaxial tests are specified using the

∗TRIAXIAL TEST DATA option.

• The TEST DATA parameter is required on the ∗DRUCKER PRAGER option to use this feature.

• The ∗TRIAXIAL TEST DATA option must be used with the ∗DRUCKER PRAGER option.

• This capability allows for all three parameters, a, b, and p_t, to be calibrated, or, if some of the parameters are known, to calibrate only the unknown parameters.

L3.78

Extended Drucker-Prager Models

• Example: Limit Load Analysis of a Strip Foundation

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

• Review of the model:

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

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

• The footing is driven by prescribed displacement.

Symmetry plane

Right half of footing

Soil

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• Extended Drucker-Prager model:

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

• Friction angle 30.64^o for dilatant flow and 30.16 for nondilatant flow

• Stress ratio 1.0

• Yield stress 20.2 psi for dilatant flow and 19.8 psi for nondilatant flow

• Dilation angles used: 0^o (nondilatant flow) and 30.16^o (dilatant flow)

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

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

L3.80

Extended Drucker-Prager Models

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

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

Elastic modulus and Poisson’s ratio

Friction angle, stress ratio, and dilation angle

Yield stress and plastic strain

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• Results

Terzaghi (175 psi)

Prandtl (143 psi)

Mohr Coulomb with no dilation

Mohr Coulomb with dilation Drucker Prager with dilation Drucker Prager with no dilation

L3.82

Extended Drucker-Prager Models

• Results

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

• The nondilatant Drucker-Prager and Mohr-Coulomb models lead to a softer response and a lower limit load than the corresponding dilatant versions.

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

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

© Dassault Systèmes, 2008

Lecture 4

L4.2

Overview

• Modified Drucker-Prager/Cap Model

• Critical State (Clay) Plasticity Model

• Jointed Material Model

• Soil Plasticity Models - Summary

• Comments on the Numerical Implementation

© Dassault Systèmes, 2008

L4.4

Modified Drucker-Prager/Cap Model

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

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

• It adds a “cap” yield surface to the linear Drucker-Prager model

• to bound the model in hydrostatic compression, and

• to help control volume dilatancy when the material yields in shear.

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

• The material is initially isotropic.

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• The yield behavior depends on the hydrostatic pressure.

• There are two distinct regions of behavior.

• On the failure surface the material is perfectly plastic.

• On the cap yield surface it hardens and also stiffens.

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

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

L4.6

Modified Drucker-Prager/Cap Model

• The inelastic behavior is generally accompanied by volume changes.

• On the failure surface the material dilates.

• On the cap surface it compacts.

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

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

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

• The material properties can be temperature dependent.

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• Description

• The model can use linear elasticity or nonlinear porous elasticity.

• The model uses two main yield surface segments.

• a linearly pressure-dependent Drucker-Prager shear failure surface

• a compression cap yield surface

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

• The Drucker-Prager failure surface is

•

β

is the angle of friction; dis the cohesion of the material.

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

tan 0

Fs = −t p β− =d .

L4.8

Modified Drucker-Prager/Cap Model

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• The cap yield surface is

where

Ris a material parameter that controls the shape of the cap, αis a small number, and

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

( ) ( ) ( )

• The cap yield surface

• is elliptical with constant eccentricity in the meridional ( p–t ^{) plane.}

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

• hardens or softens as a function of the volumetric plastic strain.

• Volumetric plastic compaction (when yielding on the cap) causes hardening.

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

Analysis of Geotechnical Problems with Abaqus

© Dassault Systèmes, 2008

• The hardening/softening law is a user-defined piecewise linear function relating the hydrostatic compression yield stress, p_b, and the

corresponding volumetric plastic strain,

(

0

)

pl pl

b b vol vol

p = p ε +ε .

L4.12

• The volumetric plastic strain axis in the hardening curve has an arbitrary origin.

• is the position on this axis corresponding to the initial state of the material when the analysis begins, thus defining the position of the cap (p_b) at the start of the analysis.

• The evolution parameter p_ais then given as

• The parameter αis a small number (typically 0.01 to 0.05) used to define a transition yield surface

so that the model provides a smooth intersection between the cap and the failure surfaces.

In document Nuevos M´ etodos basados en Soft Computing para Calibraci´ on de Modelos Basados en Agentes: (página 34-37)