material nonlinearity i - aalborg...

Computational Mechanics, AAU, EsbjergNonlinear FEM

Course inNonlinear FEMMaterial nonlinearity I

Material nonlinearity I 2Computational Mechanics, AAU, EsbjergNonlinear FEM

Outline

Lecture 1 – IntroductionLecture 2 – Geometric nonlinearityLecture 3 – Material nonlinearityLecture 4 – Material nonlinearity continuedLecture 5 – Geometric nonlinearity revisitedLecture 6 – Issues in nonlinear FEALecture 7 – Contact nonlinearityLecture 8 – Contact nonlinearity continuedLecture 9 – DynamicsLecture 10 – Dynamics continued


Nonlinear FEMLecture 1 – Introduction, Cook [17.1]:

– Types of nonlinear problems– Definitions

Lecture 2 – Geometric nonlinearity, Cook [17.10, 18.1-18.6]:– Linear buckling or eigen buckling– Prestress and stress stiffening– Nonlinear buckling and imperfections– Solution methods

Lecture 3 – Material nonlinearity, Cook [17.3, 17.4]:– Plasticity systems– Yield criteria

Lecture 4 – Material nonlinearity revisited, Cook [17.6, 17.2]:– Flow rules– Hardening rules– Tangent stiffness


Nonlinear FEMLecture 5 – Geometric nonlinearity revisited, Cook [17.9, 17.3-17.4]:

- The incremental equation of equilibrium- The nonlinear strain-displacement matrix- The tangent-stiffness matrix- Strain measures

Lecture 6 – Issues in nonlinear FEA, Cook [17.2, 17.9-17.10]:– Solution methods and strategies– Convergence and stop criteria– Postprocessing/Results– Troubleshooting


Nonlinear FEMLecture 7 – Contact nonlinearity, Cook [17.8]:

– Contact applications– Contact kinematics– Contact algorithms

Lecture 8 – Contact nonlinearity continued, Cook [17.8]:– Issues in FE contact analysis/troubleshooting

Lecture 9 – Dynamics, Cook [11.1-11.5]:– Solution methods– Implicit methods– Explicit methods

Lecture 10 – Dynamics continued, Cook [11.11-11.18]:– Dynamic problems and models– Damping– Issues in FE dynamic analysis/troubleshooting


References• [ANSYS] ANSYS 10.0 Documentation (installed

with ANSYS):– Basic Analysis Procedures– Advanced Analysis Techniques– Modeling and Meshing Guide– Structural Analysis Guide– Thermal Analysis Guide– APDL Programmer’s Guide– ANSYS Tutorials

• [Cook] Cook, R. D.; Concepts and applications of finite element analysis, John Wiley & Sons


Small-Strain Elasticity Relations

Use engineering definition of shear strain:

Do not use the tensor definition of shear strain:

y,x,xy uv +=γ

2uv y,x,

xy

+=ε


Limitations on Uniaxial Stress-Strain Data

1. Rate of Loading2. Temperature

a. Lower than Room Temperatureb. Higher than Room Temperature

3. Unloading and Load Reversal4. Multiaxial States of Stress


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

1. Yield Criterion2. Flow Rule3. Hardening Rule


Inelastic Material Behavior

Inelastic:Inelastic:

Material response that is characterized by a stress-strain diagram that is nonlinear and retains a permanent strain or returns slowly to an unstrained state on complete unloading.


Inelastic Material Behavior

Plasticity:Plasticity:

Inelastic behavior of materials that retain a permanent set on complete unloading.


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


Comparison of Yield Stress in Shear

Y3

1

Y21

Y54Y

Mises von

Tresca

Strain Principal Maximum

Stress Principal Maximum


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


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

⎟⎠⎞

⎜⎝⎛ σ

ν−⎟⎠⎞

⎜⎝⎛ σ

=εEE

211


1σ

1σ

2σ2σ

σ2 >0 Yielding occurs at σ1 >Y

σ2 <0 Yielding occurs at σ1 <Y

21 σ≥σ

⎟⎠⎞

⎜⎝⎛ σ

ν−⎟⎠⎞

⎜⎝⎛ σ

=εEE

211


( )

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

IIJJ

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 =⎣ ⎦


( )

( ) ( ) ( )

1 1oct 1 2 3 13 2

2 2 21oct 1 2 2 3 3 19

21 2

σ σ σ σ I

τ σ σ σ σ σ σ

2I 6I

= + + =

⎡ ⎤= − + − + −⎣ ⎦= +

Octahedral Stresses


Maximum Octahedral Shear Stress Criterion

( ) ( ) ( )

( ) ( ) ( )

2 2 2 21 11 2 2 3 3 16 3

2 2 21oct 1 2 2

oct

3 3 19

Y

τ σ σ σ σ σ

2τ3

σ

f

f

⎡ ⎤= σ − σ + σ − σ + σ − σ −⎣ ⎦⎡ ⎤= − +

=

+ −⎣

−

− ⎦


1σ

2σ

Y

D


Y

Y−

Y−

Von Mises


1σ

2σ

Yσ1 =

Yσ1 −=

Yσ2 =

Yσ2 −=

A

B

Y

D


Yσ 21 =σ−

Y

Y−

Y−

Yσ 21 −=σ−

TrescaVon Mises


1σ

2σ

3σ

TrescaVon Mises

π - Plane


OC 1.15OB

=


Mohr-Coulomb

( )1 2 3

1 3 1 3

1 3

sin 2 cos

1 sin 1 sin 12 cos 2 cos

f c

c c

σ > σ > σ

= σ − σ + σ + σ φ − φ

+ φ − φσ − σ =

φ φ

Generalized form of the Tresca criterion where the limiting shear stress is not constant, but depends on the normal stress.


Mohr-Coulomb

( )1 2 3

1 3 1 3 sin 2 cosf c

c = cohesion = internal friction angle

σ > σ > σ

= σ − σ + σ + σ φ − φ

φ

Generalized form of the Tresca criterion where the limiting shear stress is not constant, but depends on the normal stress.


Mohr-Coulomb

( )1 2 3

1 3 1 31 3

1 3

sin 2 cos2 2

1 sin 1 sin 12 cos 2 cos

c

c c

σ > σ > σσ − σ σ − σ

+ σ + σ φ = φ −

⎛ ⎞ ⎛ ⎞+ φ − φσ − σ =⎜ ⎟ ⎜ ⎟φ φ⎝ ⎠ ⎝ ⎠


Mohr-Coulomb

C T

1 3

T C

2 cos 2 cosY Y1 sin 1 sin

1Y Y

c cφ φ= =

− φ + φ

σ σ− =


Mohr-Coulomb

1 T 2 3

T T

T

Y 0Y Y sin 2 cos 0

2 cosY1 sin

Uniaxial Tension

f cc

σ = σ = σ == + φ − φ =

φ=

+ φ


Mohr-Coulomb

1 2 3 C

C C

C

0 YY Y sin 2 cos 0

2 cosY1 sin

Uniaxial Compression

f cc

σ = σ = σ = −= − φ − φ =

φ=

− φ


Mohr-Coulomb

T CT

T

1 TCC

Y Y2 cos cY 2 Y1 sin2 cos YY 2 tan1 sin 2 Y

c

c −

⎧φ ⎫ == ⎪⎪+ φ⎪ ⎪⇒⎬ ⎨φ ⎛ ⎞π⎪ ⎪= φ = − ⎜ ⎟⎪ ⎪− φ⎭ ⎝ ⎠⎩


Mohr-Coulomb

C

T

Y 1 sinmY 1 sin

+ φ= =

− φ


Mohr-Coulomb

1 3

T C

1 3 C

1Y Y

m Y

σ σ− =

σ − σ =


Drucker-Prager Yield Criterion

Generalization of von Mises Criterion

( )

( )

1 2I J K

2sin3 3 sin

6 cosK3 3 sin

f =

c

α + −

φα =

− φ

φ=

+ φ


Hill’s Criterion for Orthotropic Materials

( ) ( ) ( )( ) ( ) ( )

2 2 222 33 33 11 11 22

2 2 2 2 2 223 32 13 31 12 21

F G H

L M N 1

f = σ − σ + σ − σ + σ − σ

+ σ + σ + σ + σ + σ + σ −


Hill’s Criterion for Orthotropic Materials

2 2 2

2 2 2

2 2 2

2 2 223 13 12

1 1 12FZ Y X1 1 12G

Z X Y1 1 12H

X Y Z1 1 12L 2M 2N

S S S

= + −

= + −

= + −

= = =

material nonlinearity i - aalborg...

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