Course in Nonlinear Finite Element Analysis -...

Plasticity IComputational Mechanics, AAU, EsbjergNonlinear Finite Element Analysis

Course in Nonlinear Finite Element Analysis

Plasticity I


Types of structural nonlinearity classifications used in engineering problems

• Geometric nonlinearity• Material nonlinearity:

– time-independent behaviour such as plasticity– time-dependent behaviour such as creep– viscoelastic/viscoplastic behaviour where both

plasticity and creep effects occur simultaneously

• Contact or boundary nonlinearity


Strain RateStrain Rate effects:

If a specimen is held at constant strain, the stress will relax slowly. If the straining is resumed, the specimen will behave as though the solid were unloaded elastically. If the specimen is held at constant stress, the specimen will undergo slow, irreversible deformation, i.e. creep.


Nonlinear Material Response

1. Nonlinear Elastic2. Plastic3. Viscoelastic4. Viscoplastic


σ

εO

Loading andunloading

Nonlinear Elastic


σ

εO

Loading

Plastic A

B

Unloading

PermanentStrain


σ

εO

Loading

Viscoelastic

A

B


σ

εO

Loading

Viscoplastic

A

B

Unloading

PermanentStrain


Idealized Behavior

1. Elastic-perfectly plastic response2. Elastic-strain hardening response3. Rigid elastic -perfectly plastic response4. Rigid elastic -strain hardening response


σ

εA A ′

B C F

D

C′F′H

YElasticElastic--Perfectly Perfectly Plastic BehaviorPlastic Behavior

Loading

Loading

Unloading(elastic)

Unloading(elastic)


σ

εA A′

BC

F

D

C′F′H

YElastic Elastic

StrainStrain--HardeningHardeningBehaviorBehavior


σ

εA A′

B C F

D

C′F′H

YRigid ElasticRigid Elastic--

Perfectly Plastic Perfectly Plastic BehaviorBehavior

Continued Loading(Plastic)

Unloading(Elastic)


σ

εA A′

BC F

D

C′F′ H

YRigid ElasticRigid Elastic--

StrainStrain--Hardening Hardening Plastic BehaviorPlastic Behavior

Continued Loading(Plastic)

Unloading(Elastic)


Plasticity Theory

• Yield criterion or yield function, i.e. defines the state of stress at which material response changes from elastic to plastic.

• Flow rule, i.e. relates plastic strain increments to stress increments after the onset of initial yielding.

• Hardening rule, i.e. predicts the change in the yield surface due to plastic strains.


Yield criteria• Maximum principal stress criterion or Rankine’s criterion, i.e.

Yielding begins at a point in a member when the maximum principal stress reaches a value equal to the tensile (or compressive) yield stress in uniaxial tension (or compression)

• Maximum principal strain criterion or St. Venant’s criterion, i.e. yielding begins at a point in a member when the maximum principal strain reaches a value equal to the yield strain in uniaxial tension

• Strain energy density criterion, i.e. yielding occurs when the strain energy density is equal to strain energy density at yield for the uniaxial case

• Maximum shear-stress criterion or Tresca’s criterion, i.e. yielding occurs when the maximum shear stress reaches the value of the maximum shear stress at yield in uniaxial tension


Yield criteria• Distortional energy density or von Mises criterion

(Huber, Maxwell, Hencky), i.e. yielding occurs when the distortional energy density reaches a value equal to the distortional energy density at yield in a uniaxial case.

• Mohr-Coulomb criterion, i.e. generalized form of the Tresca criterion where the limiting shear stress is not constant, but depends on the normal stress

• Drucker-Prager yield criterion, i.e. generalization of von Mises criterion

• Hill’s criterion for orthotropic materials


Yield Criterion

Define a yield function F, which is a function of stresses {σ} and parameters {α} and Wp associated with the hardening rule.

{ } { }( )( ) 0W,,F

0W,,F

p

p

=ασ

=ασ


Yield Criterion

Possible values of F:F < 0 - elastic rangeF = 0 - yieldingF > 0 - impossible

Possible values of dF:dF < 0 - unloadingdF = 0 - continued yieldingdF > 0 - impossible


Yield Criterion

1. Defines the onset of yielding2. |σ| = σy

3. σy - yield stress in uniaxial tension4. Tresca5. von Mises


Flow Rule

Define a plastic potential Q, which is a function of

stresses {σ} and parameters {α} and Wp

associated with the hardening rule. Also define

a scalar dλ that may be called a “plastic

multiplier.” Plastic strain increments are given by:

{ } { }( )

{ } { } λ⎭⎬⎫

⎩⎨⎧

σ∂∂

=ε

ασ=

dQd

W,,QQ

p

p


Flow Rule

λσ∂∂=γ

λσ∂∂=ε

λσ∂∂=ε

dQd

dQd

dQd

xz

pxz

y

py

x

px

M


Flow Rule

(soils) materials granular rule - flow tednonassociamaterialsductile rule - flow associated

rule. flow tednonassociarule. flow associated

FQFQ

≠=


Flow Rule

1. Relates stress increment {dσ} to strain increment {dε} after yielding.

2. Uniaxial case: dσ = Et dε3. Prandtl-Reuss often used.4. Associated - ductile materials.5. Non-associated - soil or granular

materials.


Hardening Rule• If an unloading is followed by a reversed loading, e.g.

tension is followed by compression, metals exhibit yielding at lower load than was the original yield limit. This is termed the Bauschinger effect.

• In a general multi-axial stress state, the hardening phenomena correspond to change in the size/shape and/or translation of the original elastic domain. This phenomenon is often simplified by assuming that the elastic domain does not change in shape, but only uniformly expands (isotropic hardening) or translates (kinematic hardening) or expands and translates (mixed hardening) in the stress space.


Hardening Rule

Basically two hardening rules exist:


Hardening Rule

The parameter {α} locates the center of the yield surface in stress space. Before any yielding occurs {α} = 0. In kinematic hardening the yield surface moves in the direction of plastic straining, so {α}≠ 0. The parameter Wp describes how the yield surface grows. For isotropic hardening, {α} = 0 throughout the analysis and Wp ≠ 0.


Hardening Rule

The parameters {α} and Wp are given by:

{ } { }{ } { }∫∫

εσ=

ε=α

pTp

p

dW

dC

Where C can be assumed to be a material constant.


Hardening Rules

1. Kinematic1. Yield surface retains size and shape and

translates in stress space.2. Isotropic

1. Yield surface retains shape but increases in size.


ε

σ

B

YσE E

Y2σ

Bσ

Bσ

tE

tE

tE

Kinematic

Isotropic


1σ

3σ

2σ

Yield Surface


1σ

3σ

2σ

Yield Surface

Isotropic Hardening


1σ

3σ

2σ

Yield Surface

Kinematic Hardening


Maximum Principal Strain Criterion

Also known as St. Venant’s criterion.

Yielding begins at a point in a member when the maximum principal strain reaches a value equal to the yield strain in uniaxial tension.

EYεY =


1σ

1σ

Assume a uniaxial case where σ1 is the only non-zero principal stress. Yielding occurs when :

1 Y 1Y YE

ε = ε = ⇒ σ =


1σ

1σ

2σ2σ

Assume a case where two principal stresses σ1 and σ2 both act. Yielding occurs when ε1=Y

21 σ≥σ

⎟⎠⎞

⎜⎝⎛ σν−⎟

⎠⎞

⎜⎝⎛ σ=ε

EE21

1


1σ

1σ

2σ2σ

σ2 >0 Yielding occurs at σ1 >Y

σ2 <0 Yielding occurs at σ1 <Y

21 σ≥σ

⎟⎠⎞

⎜⎝⎛ σν−⎟

⎠⎞

⎜⎝⎛ σ=ε

EE21

1


( )

YY

E1ε

ε

321

321

3211

1

±=νσ−νσ−σ

−νσ−νσ−σ=

νσ−νσ−σ=

1f

strain principal largestthe is Assume


( )

( )

( ) YYE1ε

YYE1ε

YYE1ε

2132132133

3123123222

3213213211

±=νσ−νσ−σ−νσ−νσ−σ=νσ−νσ−σ=



3

2

1

f

f

f

unorderedare strain principalAssume


Y

max

−σ=

νσ−νσ−σ=σ≠≠

e

kjie

f

kji


1σ

2σ

Yσ1 =

Yσ1 −=

Yσ2 =

Yσ2 −=

Yσ 12 =νσ−

Yσ 21 =νσ−

Yσ 21 −=νσ−

Yσ 12 −=νσ−A

BC

D

Maximum Principal Strain


Strain Energy Density Criterion

Proposed by Beltrami states that yielding occurs when the strain energy density is equal to strain energy density at yield for the uniaxial case.

( )2 2 20 1 2 3 1 2 1 3 2 3

1U 2 02E

⎡ ⎤= σ +σ +σ − ν σ σ +σ σ +σ σ >⎣ ⎦



( )[ ]

[ ]E2

YE2

1U

0Y

2E2

1U

2210

321

32312123

22

210

=σ=

=σ=σ=σ

σσ+σσ+σσν−σ+σ+σ=

:case Uniaxial



( )

( )

22 2 21 2 3 1 2 1 3 2 3

2 2 2 21 2 3 1 2 1 3 2 3

1 Y2 02 E 2 E

2 Y 0

σ + σ + σ − ν σ σ + σ σ + σ σ − =

σ + σ + σ − ν σ σ + σ σ + σ σ − =



( )

( )

2 2e

2 2 2e 1 2 3 1 2 1 3 2 3

f σ Y

σ 2

= −

= σ + σ + σ − ν σ σ + σ σ + σ σ


Biaxial Case

( ) 0Y2 221

22

21 =−σσν−σ+σ


Maximum Shear-Stress Criterionor Tresca Criterion

Yielding occurs when the maximum shear stress reaches the value of the maximum shear stress at yield in uniaxial tension.


Uniaxial Loading

2Y

20Y

0σ0σYσ

max

3

2

1

=−

=τ

===


Tresca Criterion

maxe

e

σ2Yσ

τ=

−=f


Tresca Criterion

( )321max

213

132

321

,,max2

2

2

τττ=τ

σ−σ=τ

σ−σ=τ

σ−σ=τ


Tresca Criterion

YYY

21

13

32

±=σ−σ±=σ−σ±=σ−σ


1σ

2σ

Yσ1 =

Yσ1 −=

Yσ2 =

Yσ2 −=

Y

D

Yσ 21 =σ−

Y

Y−

Y−

Yσ 21 −=σ−

Tresca

0σ3 =


Distortional Energy Density von Mises Criterion

Yielding occurs when the distortional energy density reaches a value equal to the distortional energy density at yield in a uniaxial case.


Strain Energy Density

( )2 2 20 1 2 3 1 2 1 3 2 3

0 V D

1U 22E

U U U

⎡ ⎤= σ + σ + σ − ν σ σ + σ σ + σ σ⎣ ⎦

= +



( )

( ) ( ) ( )

21 2 3

V

2 2 21 2 2 3 3 1

D

σ σ σU

18K

U12G

+ +=

σ − σ + σ − σ + σ − σ=



( )EK

3 1 2 ν

Bulk Modulus

=−



( )EG

2 1 ν

Shear Modulus

=+


Distortional StrainEnergy Density

( ) ( ) ( )G12

213

232

221 σ−σ+σ−σ+σ−σ

=DU


Uniaxial Loading

( ) ( ) ( )1 2 3

2 2 2

D

2

D

σ Y σ 0 σ 0

Y 0 0 0 0 YU

12GYU6G

= = =

− + − + −=

=


Deviatoric Stress

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

−

=

m

m

m

dT

σσσσ

σσσσ

σσσσ

zzzyzx

yzyyyx

xzxyxx


( ) ( ) ( )[ ]

( )( )( )213132321271

3127

2213

133

213

232

2216

1

213

122

1

σσσ2σσσ2σσσ2

σσσσσσ

0

−−−−−−=

++=

−+−+−=

+=

=

IIIIJ

IIJ

J

Deviatoric Stress Invariants


von Mises Criterion

( ) ( ) ( )G2

JG12

22

132

322

21 =σ−σ+σ−σ+σ−σ

=DU


von Mises Criterion

( ) ( ) ( )2 2 2 21 2 2 3 3 1 Y

12G 6G

22

1J Y3

σ − σ + σ − σ + σ − σ=

=


Von Mises Criterion

( ) ( ) ( )

( ) ( ) ( )

2 2 2 21 11 2 2 3 3 16 3

2 2e

2 2 21e 1 2 2 3 3 1 22

Y

σ Y

3J

f

f

⎡ ⎤= σ − σ + σ − σ + σ − σ −⎣ ⎦

= −

⎡ ⎤σ = σ − σ + σ − σ + σ − σ =⎣ ⎦

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