CH.8. PLASTICITYContinuum Mechanics Course (MMC) - ETSECCPB - UPC

Overview

Introduction Previous Notions

Principal Stress Space Normal and Shear Octahedral Stresses

Stress Invariants

Effective Stress

Principal Stress Space Normal and Shear Octahedral Stress

Stress Invariants

Projection on the Octahedral Plane

Rheological Friction Models Elastic Element

Frictional Element

Elastic-Frictional Model

2

Overview (cont’d)

Rheological Friction Models (cont’d)

Frictional Model with Hardening

Elastic-Frictional Model with Hardening

Phenomenological Behaviour Notion of Plastic Strain

Notion of Hardening

Bauschinger Effect

Elastoplastic Behaviour

1D Incremental Theory of Plasticity Additive Decomposition of Strain

Hardening Variable

Yield Stress, Yield Function and Space of Admissible Stresses

Constitutive Equation

Elastoplastic Tangent Modulus

Uniaxial Stress-Strain Curve

3

Overview (cont’d)

3D Incremental Theory of Plasticity Additive Decomposition of Strain

Hardening Variable

Yield Function

Loading - Unloading Conditions and Consistency Conditions

Constitutive Equation

Elastoplastic Constitutive Tensor

Yield Surfaces Von Mises Criterion

Tresca Criterion

Mohr-Coulomb Criterion

Drucker-Prager Criterion

4

5

Ch.8. Plasticity

8.1 Introduction

A material with plastic behavior is characterized by: A nonlinear stress-strain relationship. The existence of permanent (or plastic) strain during a

loading/unloading cycle. Lack of unicity in the stress-strain relationship.

Plasticity is seen in most materials, after an initial elastic state.

Introduction

6

12

3

1x2x

3x

1x2x

3x

1x

2x

3x

1113

1222

2321

333231

PRINCIPAL STRESSES Regardless of the state of stress, it is always possible to choose a special

set of axes (principal axes of stress or principal stress directions) so that the shear stress components vanish when the stress components are referred to this system.

The three planes perpendicular to the principle axes are the principal planes.

The normal stress components in the principal planes are the principal stresses.

Previous Notions

1

2

3

0 00 00 0

1 2 3

7

12

3

1x2x

3x

1x2x

3x

1x

2x

3x

1113

1222

2321

333231

PRINCIPAL STRESSES The Cauchy stress tensor is a symmetric 2nd order tensor so it will diagonalize

in an orthonormal basis and its eigenvalues are real numbers. Computing the eigenvalues and the corresponding eigenvectors :

Previous Notions

1 2 3

v

v v v 0 1

11 12 13

12 22 23

13 23 33

det 0not=

1 1

1 1

2 2

3 3

3 21 2 3 0I I I characteristic

equation

INVARIANTS

8

STRESS INVARIANTS Principal stresses are invariants of the stress state.

They are invariant w.r.t. rotation of the coordinate axes to which the stresses are referred.

The principal stresses are combined to form the stress invariants I :

These invariants are combined, in turn, to obtain the invariants J:

Previous Notions

1 1 2 3iiI Tr

22 1 1 2 1 3 2 3

1 :2

I I

3 detI

1 1 iiJ I

22 1 2

1 1 12 :2 2 2ij jiJ I I

33 1 1 2 3

1 1 13 33 3 3 ij jk kiJ I I I I Tr

REMARK The J invariants can be expressed the unified form:

1 1,2,3iiJ Tr i

i

REMARK The I invariants are obtained from the characteristic equation of the eigenvalue problem.

9

SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSORGiven the Cauchy stress tensor and its principal stresses, the following is defined: Mean stress

Mean pressure

A spherical or hydrostatic state of stress:

Previous Notions

1 2 31 1 13 3 3m iiTr

1 2 313mp

1 2 3 0 0

0 00 0

1

REMARK In a hydrostatic state of stress, the stress tensor is isotropic and, thus, its components are the same in any Cartesian coordinate system.As a consequence, any direction is a principal direction and the stress state (traction vector) is the same in any plane.

10

SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSORThe Cauchy stress tensor can be split into:

The spherical stress tensor: Also named mean hydrostatic stress tensor or volumetric stress tensor or

mean normal stress tensor. Is an isotropic tensor and defines a hydrostatic state of stress. Tends to change the volume of the stressed body

The stress deviatoric tensor: Is an indicator of how far from a hydrostatic state of stress the state is. Tends to distort the volume of the stressed body

Previous Notions

sph

1 1:3 3sph m iiTr 1 1 1

dev m 1

11

STRESS INVARIANTS OF THE STRESS DEVIATORIC TENSOR The stress invariants of the stress deviatoric tensor:

These correspond exactly with the invariants J of the same stress deviator tensor:

Previous Notions

1 1 0J I

22 1

12

J I 2 212 :2

I I

33 1

13

J I 1 23I I 3 31 133 3 ij jk kiI I Tr

1 0I Tr 2

2 11 :2

I I 2 2 2

3 11 22 33 12 23 13 12 33 23 11 13 221det 23 ij jk kiI

12

EFFECTIVE STRESS The effective stress or equivalent uniaxial stress is the scalar:

It is an invariant value which measures the “intensity” of a 3D stress state in a terms of an (equivalent) 1D tensile stress state.

It should be “consistent”: when applied to a real 1D tensile stress, should return the intensity of this stress.

Previous Notions

'2

3 332 2ij ijJ ´: ´

13

Example

Calculate the value of the equivalent uniaxial stress for an uniaxial state of stress defined by:

x x

x

y

z

uu

E, G

0 00 0 00 0 0

u

14

Example - Solution

Mean stress:

Spherical and deviatoric parts

of the stress tensor:

1 (3 3

um Tr 0 0

30 00 0 0 0

30 0

0 03

u

mu

sph m

mu

2 0 030 0

10 0 0 03

0 0 10 03

u

u m

sph m u

m

u

23 3 4 1 1 3 2( )2 2 9 9 9 2 3ij ij u u u

0 00 0 00 0 0

u

15

16

Ch.8. Plasticity

8.2 Principal Stress Space

The principal stress space or Haigh–Westergaard stress space is the space defined by a system of Cartesian axes where the three spatial axes represent the three principal stresses for a body subject to stress:

Principal Stress Space

1 2 3

17

Any of the planes perpendicular to the hydrostatic stress axis is a octahedral plane. Its unit normal is .

Octahedral plane

1 2 3

11 13

1

n

18

Consider the principal stress space: The normal octahedral stress is defined as:

Normal and Shear Octahedral Stresses

1 2 3

1 2 3

1/ 3

3 , , 1/ 3

1/ 33 3

3

oct

m

OA OP

n

1

3oct mI

19

Consider the principal stress space: The shear or tangential octahedral

stress is defined as:

Where the is calculated from:

Normal and Shear Octahedral Stresses

3 oct AP

2 2 22 2 2 21 2 3

2 '1 2 3 2

31 23

oct AP OP OA

J

AP

1 22

23oct J

1/222 2 2

1 2 3 1 2 3

1/22 2 2

1 2 2 3 1 3

1 133

13 3

oct

oct

Alternative forms of :oct

20

In a pure spherical stress state:

In a pure deviator stress state:

Normal and Shear Octahedral Stresses

sph m 1 esf 0 2 0J

0oct

0oct ( ) ( ) 0m Tr Tr

A pure spherical stress state is located on the hydrostatic stress axis.

A pure deviator stress state is located on the octahedral plane containing the origin of the principal stress space

21

Any point in space is unambiguously defined by the three invariants: The first stress invariant characterizes the distance from the origin to

the octahedral plane containing the point.

The second deviator stress invariant characterizes the radius of the cylinder containing the point and with the hydrostatic stress axis as axis.

Stress Invariants

1I

The third deviator stress invariant characterizes the position of the point on the circle obtained from the intersection of the octahedral plane and the cylinder. It defines an angle .

2J

3J

3J

22

The projection of the principal stress space on the octahedral plane results in the division of the plane into six “sectors”: These are characterized by the different principal stress orders.

Projection on the Octahedral Plane

FEASIBLE WORK SPACE

Election of a criterion, e.g.: 1 2 3

23

Example

Determine the shape of the surface in the principal stress space corresponding to a function defined as,

21 2 , , 0aI bJ c a b c

24

Example - Solution

The relationship between the and invariants, and the and stresses is given by,

Substituting these expressions into the yield function reads,

1I 2J octoct

1

222

3 3 3 3

3 1 32 2

m oct oct

oct oct

I Tr

J

2 23 3 3

2oct octba c

25

Example - Solution

And dividing by c the yield function takes the form,

which corresponds to an axisymmetric ellipsoid with axis the hydrostatic stress axis and semi-axis:

2 23 3

123

oct oct

c ca b

3 , 2c a c b

26

38

Ch.8. Plasticity

8.4 Phenomenological Behaviour

Notion of Plastic Strain

e p PLASTIC STRAIN

eelastic limit:

LINEAR ELASTIC BEHAVIOUR

eE

39

Also known as kinematic hardening.

Bauschinger Effect

f

e

e

41

Considering the phenomenological behaviour observed, elastoplastic materials are characterized by:

Lack of unicity in the stress-strain relationship. The stress value depends on the actual strain and the previous loading

history.

A nonlinear stress-strain relationship. There may be certain phases in the deformation process with

incremental linearity.

The existence of permanent (or plastic) strain during a loading / unloading cycle.

Elastoplastic Behaviour

42

43

Ch.8. Plasticity

8.5 1D Incremental Plasticity Theory

The incremental plasticity theory is a mathematical model used to represent the evolution of the stress-strain curve in an elastoplastic material. Developed for 1D but it can be generalized for 3D problems.

Introduction

REMARK This theory is developed under the hypothesis of infinitesimal strains.

44

Total strain can be split into an elastic (recoverable) part, , and an inelastic (unrecoverable) one, :

Also,

Additive Decompositionof Strain

e

E e p

e ddE e pd d d

where

where

elastic modulus or Young modulus

ep

45

The hardening variable, , is defined as:

Such that and .

Note that is always positive and:

Then, for a monotonously increasing plastic strain process, both variables coincide:

Hardening Variable

pd sign d

0d 0

0p

REMARK The function is: sign

1

pd d sign d

pd d

0pd 0 0

p pp ppd d

46

Stress value, , threshold for the material exhibiting plastic behaviour after elastic unloading + elastic loading It is considered a material property. For

is the hardening modulus

Yield Stress and Hardening Law

f

0pf e

fd H d ( )f

H

HARDENING LAW

47

f e H ( )f

The yield function, , characterizes the state of the material:

Yield Function

, fF

,F

, 0F

ELASTO-PLASTIC STATEELASTIC STATE

: , 0F E R

0 : , 0 0eF E R

: , 0F E R

, 0F

ELASTIC DOMAIN YIELD SURFACE INITIAL ELASTIC

DOMAIN:

Space of admissible

stresses

48

Any admissible stress state must belong to the space of admissible stresses, (postulate):

Space of Admissible Stresses

Space of admissible

stresses

E

, 0F

E E E

R

, fF

( ), ( )f f E

REMARK

49

The following situations are defined: ELASTIC REGIME

ELASTOPLASTIC REGIME – UNLOADING

ELASTOPLASTIC REGIME – PLASTIC LOADING

Constitutive Equation

E

, 0dF

E

, 0dF

E

d E d

d E d

epd E d

Elastoplastictangent modulus

REMARK The situation

is not possible because, by definition, on the yield surface .

( , ) 0dF

E

, 0F 50

Consider the elastoplastic regime in plastic loading,

Since the hardening variable is defined as:

Elastoplastic Tangent Modulus

pd sign d

HARDENING LAW

fd H d 1pd dH

Efor

E

, 0 , 0fF dF

( )

, 0f

sign H

dF d d

1d sign dH

51

Elastoplastic Tangent Modulus

1

1

1 1( )

11 1

e

p

e p

epE

d dE

d dH

d d d dE H

EHd d dE H

E H

strain

strain

Additive strain decomposition :

Elastic

Plastic

ep EHEE H

ELASTOPLASTIC

TANGENT MODULUSepd E d

52

Following the constitutive equation defined

Uniaxial Stress-Strain Curve

ELASTIC REGIME

ELASTOPLASTIC REGIME

d E d

epd E d

REMARK Plastic strain is generated only during the plastic loading process.

53

The value of the hardening modulus, , determines the following situations:

Role of the Hardening Modulus

H

ep EHEE H

Linear elasticity

Perfect plasticity

Plasticity with strain hardening

0H

Plasticity with strain softening

0H

55

In real materials, the stress-strain curve shows a combination of the three types of hardening modulus.

Plasticity in Real Materials

0H 0H

0H

56

57

Ch.8. Plasticity

8.6 3D Incremental Theory

The 1D incremental plasticity theory can be generalized to a multiaxial stress state in 3D.

The same concepts are used: Additive decomposition of strain Hardening variable Yield function

Plus, additional ones are added: Loading - unloading conditions Consistency conditions

Introduction

58

Total strain can be split into an elastic (recoverable) part, , and an inelastic (unrecoverable) one, :

Also,

Additive Decompositionof Strain

1 :e Ce p

1 :ed d Ce pd d d

where

where

ep

constitutive elastic (constant) tensor

59

1

e p

eD

E

The hardening variable, , is a scalar:

Where is known as the plastic multiplier.

The flow rule is defined as:

Where is the plastic potential function

Hardening Variable

d

, pf

,p Gd

0,

,G

with

60

1 p

dD d sign

The yield function, , is a scalar defined as:

Yield Function

, fF

,F

, 0F

ELASTOPLASTIC STATEELASTIC STATE

: , 0F E

0 : , 0 0F E

: , 0F E

, 0F

ELASTIC DOMAIN YIELD SURFACE

INITIAL ELASTIC DOMAIN: E E E

Space of admissible

stresses

Equivalent uniaxial stress

Yield stress

61

1 , fD F

Loading/unloading conditions (also known as Karush-Kuhn-Tucker conditions):

Consistency conditions:

Loading-Unloading Conditions and Consistency Condition

0 ; , 0 ; , 0F F

For , 0 , 0F dF

,0; 0 0;

,0;

0; 0,

0;

0; 0

p

p

p

GF dF d

Gd

F dFG

d

F dF

0

0

0

ELASTIC LOADING/UNLOADING

ELASTOPLASTIC NEUTRAL LOADING

IMPOSSIBLE62

ELASTOPLASTICLOADING

The following situations are defined: ELASTIC REGIME

ELASTOPLASTIC REGIME – ELASTIC UNLOADING

ELASTOPLASTIC REGIME – PLASTIC LOADING

Constitutive Equation

E

, 0dF

E

, 0dF

E

:d d C

:d d C

:epd d CELASTOPLASTIC

CONSTITUTIVE TENSOR

( 0 and , 0)F dF

( 0)F

( 0 and , 0)F dF

63 04/12/2015MMC - ETSECCPB - UPC

The elastoplastic constitutive tensor is written as:

Elastoplastic Constitutive Tensor

: :

: :

ep

G F

F GH

C CC C

C

, , , , , , , 1, 2,3ijpq rskl

pq rsepijkl ijkl

pqrspq rs

G F

i j k l p q r sF GH

C CC C

C

REMARK When the plastic potential function and the yield function coincide, it is said that there is associated flow:

, ,G F

64

65

Ch.8. Plasticity

8.7 Failure Criteria: Yield Surfaces

The initial yield surface, , is the external boundary of the initial elastic domain for the virgin material The state of stress inside the yield surface is elastic for the virgin material. When in a deformation process, the stress state reaches the yield surface, the

virgin material looses elasticity for the first time: this is considered as a failure criterion for design. Subsequent stages in the deformation process are not considered.

Introduction

0E

0E

66 04/12/2015MMC - ETSECCPB - UPC

The yield surface is usually expressed in terms of the following invariants to make it independent of the reference system (in the principal stress space):

Where:

The elastoplastic behavior will be isotropic.

Yield (Failure) Criteria

1 2 3, , 0eF I J J

F

1 1 2 3iiI Tr

22 1

12

J I 2 212 :2

I I

33 1

13

J I 1 23I I 3 31 133 3 ij jk kiI I Tr

with 1 2 3

REMARK Due to the adopted principal stress criteria, the definition of yield surface only affects the first sector of the principal stress space.

67

The yield surface is defined as:

Where is the effective stress.

(often termed the Von-Mises stress)

The shear octahedral stress is, by definition, . Thus, the effective stress is rewritten:

And the yield surface is given by:

Von Mises Criterion

( ) 0eF

23J

1 22

23oct J

1 22

32 octJ 3 33

2 2oct oct

3( ) 02 oct eF

REMARK The Von Mises criterion depends solely on the second deviator stress invariant.

68

0eF

The octahedral stresses characterizes the radius of the cylinder containing the point and with the hydrostatic stress axis as axis.

Von Mises Criterion

2 233 3oct e oct e

REMARK The Von Mises Criterion is adequate for metals, where hydrostatic stress states have an elastic behavior and failure is typically due to deviatoric stress components.

2 eF J F

69

3( ) 02 oct eF

Example

Consider a beam under a composed flexure state such that for a beam section the stress state takes the form,

Obtain the expression for Von Mises criterion.

x x 0

0 00 0 0

x xy

xy

70

Example - Solution

The mean stress is:

The deviator part of the stress tensor is:

The second deviator stress invariant is given by,

13 3

xm Tr

23

1esf 3

13

0 00 0

0 0 0 0

x m xy x xy

xy m xy x

m x

2 2 2 2 2 2 22

1 1 4 1 1 1:2 2 9 9 9 3x x x xy xy x xyJ

71

Example - Solution

The uniaxial effective stress is:

Finally, the Von Mises yield surface is given by the expression:

(Criterion in design codes for metal beams)

2 223 3x xyJ

2( ) 3 0eF J 2 23x xy e

co

(comparison stress)

72

Also known as the maximum shear stress criterion, it establishes that the elastic domain ends when:

It can be written univocally in terms of invariants and :

Tresca Criterion

1 3max 2 2

e 1 3 0eF

1 3 2 3( ) ,e eF J J F2J 3J

Plane parallel to axis 2

73

0eF

Tresca Criterion

REMARK The Tresca yield surface is appropriate for metals, which have an elastic behavior under hydrostatic stress states and basically have the same traction/compression behavior.

2 3, 0eF J J F

74

1 3 0eF

Von Mises and Tresca Criteria

75

Example

Obtain the expression of the Tresca criterion for an uniaxial state of stress defined by:

x x

x

y

z

uu

E, G0 0

0 0 00 0 0

u

76

Example - Solution

Consider:

The Tresca criterion is expressed as:

0u 1 3( ) e u e u e

u

F

1

3 0u

0u 1 3( ) e u e u e

u

F

1

3

0

u

( ) 0eF u e Note that it coincides with the Von Mises criterion for an uniaxial state of stress.

77

0 00 0 00 0 0

u

Example

Consider a beam under a composed flexure state such that for a beam section the stress state takes the form,

Obtain the expression for Tresca yield surface.

x x 0

0 00 0 0

x xy

xy

78

Example - Solution

The principal stresses are:

Taking the definition of the Tresca yield surface,

2 2 2 21 3

1 1 1 1,2 4 2 4x x xy x x xy

1 3 0eF

2 2 2 21 3

1 1 1 12 4 2 4e x x xy x x xy

2 24x xy e

co

s(comparison stress)

79

It is a generalization of the Tresca criterion, by including the influence of the first stress invariant.

In the Mohr circle’s plane, the Mohr-Coulomb yield function takes the form,

Mohr-Coulomb Criterion

tanc internal friction anglecohesion

REMARK The yield line cuts the normal stress axis at a positive value, limiting the materials tensile strength.

80

Consider the stress state for which the yield point is reached:

Mohr-Coulomb Criterion

1 3

cos

sin2

A

A

R

R

1 3 1 3 1 3

1 3 1 3

tg 0 cos sin tg 02 2 2

sin 2 cos 0

A A c c

c

1 3 1 3 sin 2 cos 0F c

REMARK For and , the Tresca criterion is recovered.

0 / 2ec

81

Mohr-Coulomb Criterion

REMARK The Mohr-Coulomb yield surface is appropriate for frictional cohesive materials, such as concrete, soils or rocks which have considerably different tensile and compressive values for the uniaxial elastic limit.

1 2 3, , 0F I J J F

82

It is a generalization of the Von Mises criterion, by including the influence of the first stress invariant.

The yield surface is given by the expression:

Where:

It can be rewritten as:

Drucker-Prager Criterion

1/223 0mF J

1 2 3 12sin 6 cos; ;

3 33 3 sin 3 3 sin mIc

1/21 2 1 2

33 ,2oct octF I J I J

REMARK For and , the Von Mises criterion is recovered.

0 / 2ec

83

Drucker-Prager Criterion

REMARK The Drucker-Prager yield surface, like the Mohr-Coulomb one, is appropriate for frictional cohesive materials, such as concrete, soils or rocks which have considerably different tensile and compressive values for the uniaxial elastic limit.

1 2,F I J F

84

Mohr-Coulomb and Drucker-Prager Criteria

85

86

Ch.8. Plasticity

Summary

12

3

1x2x

3x

1x2x

3x

1x

2x

3x

1113

1222

2321

333231

The principal stress directions correspond to the set of axes that make the shear stress components vanish when the stress components are referred to this system. The normal stress components in the principal planes are the principal stresses.

Mean stress:

Mean pressure:

A spherical or hydrostatic state of stress:

Summary

1

2

3

0 00 00 0

1 2 3

1 2 31 1 13 3 3m iiTr

mp

1 2 3 1

87

The Cauchy stress tensor can be split into:

Stress invariants:

Summary (cont’d)

sph :sph m 1dev m 1

spherical stress tensor:stress deviator tensor:

1 1 2 3iiI Tr

22 1 1 2 1 3 2 3

1 :2

I I

3 detI

1 1 iiJ I

22 1 2

1 1 12 :2 2 2ij jiJ I I

33 1 1 2 3

1 1 13 33 3 3 ij jk kiJ I I I I Tr

1 0I Tr 2

2 11 :2

I I 3

1det3 ij jk kiI

1 1 0J I

22 1

12

J I 2 212 :2

I I

33 1

13

J I 1 23I I 3 31 133 3 ij jk kiI I Tr

88

Effective stress:

Principal stress space orHaigh–Westergaard stress space:

Normal octahedral stress:

Shear octahedral stress:

Summary (cont’d)

'2

3 332 2ij ijJ ´: ´

1 2 3

1 1, 1, ,13

Tn

1

3oct mI

1 22

23oct J

89

Any point in space is unambiguously defined by the three invariants:

Summary (cont’d)

The projection of the principal stress space on the octahedral plane results in the division of the plane into six spaces:

FEASIBLE WORK SPACE

90

Phenomenological behaviour: Plastic strain Hardening Bauschinger Effect

Elastoplastic materials are characterized by: Lack of unicity in the stress-strain relationship. A nonlinear stress-strain relationship. The existence of permanent (or plastic) strain during a loading / unloading

cycle.

Summary (cont’d)

Perfectly Plastic Material

94

1D Incremental Theory: Additive decomposition of strain:

Hardening variable:

Yield stress, :

Yield function:

Summary (cont’d)

e p e

E where

pd sign d such that and0d 0

0p

ffd H d f e H HARDENING LAW

hardening parameter

, fF , 0F

ELASTOPLASTIC STATE

ELASTIC STATE

, 0F

95

1D Incremental Theory (cont’d): Space of admissible stresses

Constitutive Equation: ELASTIC REGIME:

ELASTOPLASTIC REGIME: – UNLOADING

ELASTOPLASTIC REGIME: – PLASTIC LOADING

Summary (cont’d)

Space of admissible

stresses

, 0F

E E E

R

E d E d

; , 0dF E d E d

, 0F

, 0F

, 0F

; , 0dF E epd E d

Elastoplastictangent modulus

ep EHEE H

96

1D Incremental Theory (cont’d): Uniaxial Stress-Strain Curve

Summary (cont’d)

ELASTIC REGIME

ELASTOPLASTIC REGIME

d E d

epd E d

97

1D Incremental Theory (cont’d): Role of the hardening parameter

Summary (cont’d)

Plasticity with strain hardening

Lineal elasticity

Perfect plasticity

Perfect plasticity

Plasticity with strain softening

98

3D Incremental Theory: generalization of the 1D incremental theory.

The same concepts are used: Additive decomposition of strain:

Hardening variable:

Yield function:

Summary (cont’d)

1 :e Ce p where

d 0, with

,p Gd

FLOW RULE: plastic potential function

plastic multiplier

, fF

, 0F

ELASTOPLASTIC STATE

ELASTIC STATE

, 0F

0

equivalent uniaxial stress

99

3D Incremental Theory (cont’d):

Plus, additional ones are added: Loading - unloading conditions (Kuhn-Tucker conditions):

Consistency conditions: If , then

Summary (cont’d)

0, 0, 0

FF

0 00 0

FF

, 0F , 0F

0; 0 00; 0 0; 00; 0

p

F dFF dFF dF

LOADING

PLASTIC LOADING

impossible

100

3D Incremental Theory (cont’d): Constitutive Equation:

ELASTIC REGIME:

ELASTOPLASTIC REGIME: – UNLOADING

ELASTOPLASTIC REGIME: – PLASTIC LOADING

Summary (cont’d)

E

; , 0dF E

, 0F

; , 0dF E Elastoplastic constitutive tangent tensor

, 0F

, 0F

:d d C

:d d C

:epd d C

: :

: :

ep

G F

F GH

C CC C

C

101

The yield surface is the external boundary of the elastic domain.

Summary (cont’d)

1 2 3, ,F I J J F

1 1 2 3iiI Tr

22 1

12

J I 2 212 :2

I I

33 1

13

J I 1 23I I 3 31 133 3 ij jk kiI I Tr

with 1 2 3

102

Yield criteria: Von Mises Criterion

Tresca Criterion

Summary (cont’d)

23 0eF J 32e oct

1 3 0eF 2 3( ) ,F J J F

2F J F

103

Yield criteria: Mohr-Coulomb Criterion

Drucker-Prager Criterion

Summary (cont’d)

1 3 1 3 sin 2 cos 0F c

internal friction angle

cohesion 1 2 3, ,F I J J F

1/223 0mF J

1 2 3 12sin 6 cos; ;

3 33 3 sin 3 3 sin mIc

1 2,F I J F

3 cotanc 104