FEA for Nonlinear Elastic Problems

8/17/2019 FEA for Nonlinear Elastic Problems

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3

Stress and Strain Measures

Section 3.2



4

Goals – Stress & Strain Measures

• Definition of a nonlinear elastic "ro/lem

•)nderstand t0e deformation gradient1

• 0at are Lagrangian and Eulerian strains1

• 0at is "olar decom"osition and 0o to do it1

• Ho to e"ress t0e deformation of an area and olume

• 0at are *iola-Kirc00off and auc0y stresses1



5

Mild vs !ou"# Nonlinearit$

• Mild Nonlinear *ro/lems 50a" 36

–

ontinuous7 #istor$%indeendent nonlinear relations /eteenstress and strain

– Nonlinear elasticity7 8eometric nonlinearity7 and deformation-

de"endent loads

• !ou"# Nonlinear *ro/lems 50a" 4 9 !6

– E:uality and;or ine:uality constraints in constitutie relations

– Histor$%deendent nonlinear relations /eteen stress and strain

– Elasto"lasticity and contact "ro/lems



6

'#at (s a Nonlinear Elastic Problem)

• Elastic 5same for linear and nonlinear "ro/lems6

–

Stress-strain relation is elastic – Deformation disa""ears 0en t0e a""lied load is remoed

– Deformation is 0istory-inde"endent

– *otential energy eists 5function of deformation6

• Nonlinear

– Stress-strain relation is nonlinear5* is not constant or do not eist6

– Deformation is large

• Eam"les

– <u//er material

– 'ending of a long slender mem/er

5small strain7 large dis"lacement6



!eference Frame of Stress and Strain

• $orce and dis"lacement 5ector6 are inde"endent of t0econfiguration frame in 0ic0 t0ey are defined 5!eferenceFrame (ndifference6

• Stress and strain 5tensor6 de"end on t0e configuration

• +a"ran"ian or Material Stress,Strain= 0en t0e

reference frame is undeformed configuration

• Eulerian or Satial Stress,Strain= 0en t0e referenceframe is deformed configuration

• >uestion= 0at is t0e reference frame in linear"ro/lems1



*eformation and Main"

• ?nitial domain Ω is deformed to Ω

– e can t0in@ of t0is as a main" from Ω to Ω

• -= material "oint in Ω .= material "oint in Ω

• Material "oint * in Ω is deformed to > in Ω

dis"lacement

Ω0

Ω x

X x

u

P

Q

Φ

ne-to-one ma""ingontinuously differentia/le

= +. - u = Φ = +5 7t6 5 7t6. - - u -

1, :−Φ Φ



*eformation Gradient

• ?nfinitesimal lengt0 d- in Ω deforms to d. in Ω

•

<emem/er t0at t0e ma""ing is continuousl$ differentiable

• *eformation "radient=

– gradient of ma""ing Φ

– Second%order tensor7 *eend on bot# Ω/ and Ω.

– Due to one-to-one ma""ing=

–

F includes bot# deformation and ri"id%bod$ rotation

Ω0

Ωx

u dx

dXP

Q

P'Q'

d d d d∂

= ⇒ =∂

.. - . F -

-

iiB

B

$

C

∂=

∂ -∂

= + = + ∇∂

uF 0 0 u

-

det -.≡ >F

iB

F7

7

= δ

∂ ∂

∇ = ∇ =∂ ∂

0

- .

,d d−=- F .



E.amle – 1niform E.tension

• )niform etension of a cu/e in all t0ree directions

• Continuit$ re2uirement= 0y1

• Deformation gradient=

• = uniform e"ansion 5dilatation6 or contraction

• Golume c0ange

– ?nitial olume=

– Deformed olume=

, , , 2 2 2 3 3 3 C 7 C 7 C= λ = λ = λ

,

23

- -

- -

- -

λ = λ λ

F

i λ >

, 2 3λ = λ = λ

, 2 3dG dC dC dC=

, 2 3 , 2 3 , 2 3 , 2 3 dG d d d dC dC dC dG= = λ λ λ = λ λ λ



Green%+a"ran"e Strain

• 0y different strains1

•

Lengt0 c0ange=

• !i"#t Cauc#$%Green *eformation Tensor

• Green%+a"ran"e Strain Tensor

<atio of lengt0 c0ange

dX

dx

&0e effect of rotation is eliminated

&o matc0 it0 infinitesimal strain

2 2 & &

& & &

& &

d d d d d d

d d d d

d 5 6d

− = −

= −

= −

. - . . - -

- F F - - -

- F F 0 -

& =C F F

,5 6

2= −E C 0



Green%+a"ran"e Strain cont

• *ro"erties=

–

E is s$mmetric= E&

E – No deformation= F 07 E /

– 0en 7

– E / for a ri"id%bod$ motion7 /ut

Dis"lacement gradient

Hig0er-order term

( )

& &

& & , 2

,

2

∂ ∂ ∂ ∂= + + ÷∂ ∂ ∂ ∂ = ∇ + ∇ + ∇ ∇

u u u uE

- - - -

u u u u

BiiB

B i

uu,2 C C

∂ ∂ε = + ÷ ÷∂ ∂

,∇ <<u

( )

&

,I

2≈ ∇ + ∇ =E u u

≠I



E.amle – !i"id%4od$ !otation

• <igid-/ody rotation

•

A""roac0 ,= using deformation gradient

Green%+a"ran"e strain removes ri"id%bod$ rotation from deformation

α= α − α= α + α=

, , 2

2 , 2

3 3

C cos C sin C sin C cos

C

α − α = α α

cos sin

sin cos

,

F

=

&

,

,

,

F F

= − =& ,25 6E F F 0 /



E.amle – !i"id%4od$ !otation cont

• A""roac0 2= using dis"lacement gradient

= − = α − − α= − = α + α −= − =

, , , , 2

2 2 2 , 2

3 3 3

u C C 5cos ,6 C sinu C C sin C 5cos ,6

u C

α − − α ∇ = α α −

cos , sin

sin cos ,

u

− α

∇ ∇ = − α

&

25, cos 6

25, cos 6

u u

= ∇ + ∇ + ∇ ∇ =& & ,

2

5 6E u u u u /



15

E.amle – !i"id%4od$ !otation cont

• 0at 0a""ens to engineering strain1

En"ineerin" strain is unable to ta5e care of ri"id%bod$ rotation

= − = α − − α= − = α + α −= − =

, , , , 2

2 2 2 , 2

3 3 3

u C C 5cos ,6 C sinu C C sin C 5cos ,6

u C

α − = α −

ε

cos , cos ,



16

Eulerian 6Almansi7 Strain Tensor

• Lengt0 c0ange=

• +eft Cauc#$%Green *eformation Tensor

• Eulerian 6Almansi7 Strain Tensor

!eference is deformed 6current7 confi"uration

bJ,= $inger tensor

− −

− −

−

− = −

= −= −

= −

2 2 & &

& & & ,

& & ,

& ,

d d d d d d

d d d dd 5 6d

d 5 6d

. - . . - -

. . . F F .

. 0 F F .

. 0 b .

= & b F F

−= − ,,5 6

2e 0 b



17

Eulerian Strain Tensor cont

• *ro"erties

–

Symmetric – A""roac0 engineering strain 0en

– ?n terms of dis"lacement gradient

• <elation /eteen E and e

S"atial gradient( )

∂ ∂ ∂ ∂= + − ÷∂ ∂ ∂ ∂

= ∇ + ∇ − ∇ ∇

& &

& &

,2

,

2

u u u ue. . . .

u u u u

∂∇ =∂ .

= & E F eF

∂<<

∂,

u

.



18

E.amle – +a"ran"ian Strain

• alculate F and E for deformation in t0e figure

• Ma""ing relation in Ω

• Ma""ing relation in Ω,.!

,.

C

K

)ndeformedelement

Deformed element2.

.%

=

=

= = +

= = +

∑

∑

4

? ?? ,4

? ?? ,

3C N 5s7 t6C 5s ,6

4

,

K N 5s7 t6K 5t ,62

=

=

= = − = = +

∑

∑

4

? ?? ,

4

? ?? ,

5s7 t6 N 5s7 t6 -.3!5, t6

y5s7t6 N 5s7t6y s ,



19

E.amle – +a"ran"ian Strain cont

• Deformation gradient

• 8reen-Lagrange Strain

Ω0

Ωx

u dx

dXP

QP'

Q'

Referencedomain (s, t)

5 7 6s t

-

5 7 6s t .

∂ ∂ ∂= =∂ ∂ ∂−

=

− =

- .3! 4 ; 3 -

, - - 2

- -.%4 ; 3 -

. . sF - s -

= − = −

& -.3(+ -,5 6

2 - -.2!!E F F 0



20

E.amle – +a"ran"ian Strain cont

• Almansi Strain

• Engineering Strain

'#ic# strain is consistent 8it# actual deformation)

= × = & .4+ ,.%(b F F

( )− − = − =

,,2

.!2

.22e 0 b

− − ∇ = − = −

, .%

,.33 ,u F 0

( ) −

= ∇ + ∇ = − ε

& , 2

, .32

.32 ,u u



21

E.amle – 1nia.ial Tension

• )niaial tension of incom"ressi/le material 5λ, λ > 16

•

$rom incom"ressi/ility

• Deformation gradient and deformation tensor

•

8-L Strain

−λ λ λ = ⇒ λ = λ = λ ,;2, 2 3 2 3,

−

−

λ = λ λ

,;2

,;2

F −

−

λ

= λ λ

2

,

,

C

−

−

λ −

= λ − λ −

2

,

,

, ,

, 2

,

E

= λ= λ= λ

, , ,2 2 2

3 3 3

C

C

C



22


• Almansi Strain 5b C6

• Engineering Strain

• Difference

− − λ

= − λ − λ

2, - -,

- , -2

- - ,

e

−

− λ

= λ λ

2

,

b

−

−

λ −

= λ − λ −

ε

,;2

,;2

,

,

,

−

= λ − = − λ ε = λ −2 2

,, ,, ,,

, ,

E 5 ,6 e 5, 6 ,2 2



23

Polar *ecomosition

• ant to se"arate deformation from rigid-/ody rotation

•

Similar to "rinci"al directions of strain• )ni:ue decom"osition of deformation gradient

– 9= ort#o"onal tensor 5rigid-/ody rotation6

– 1: ;= ri"#t% and left%stretc# tensor 5symmetric6

• 1 and ; 0ae t0e same eigenalues 5rincial stretc#es67

/ut different eigenectors

= =F 91 ;9



24

Polar *ecomosition cont

• Eigenectors of 1= E,7 E27 E3

• Eigenectors of ;= e,7 e27 e3

• Eigenalues of 1 and ;=λ,7 λ27 λ3

Q

Q

V

U

E1 E2

E3

λ1E1

λ2E2

λ3E3

e1

e2

e3

λ1e1

λ2e2

λ3e3

=F 91

=F ;9= × ×

= × ×

d d

d

. 9 1 -

; 9 -



25


• <elation /eteen 1 and C

– 1 and C 0ae t0e same eigenectors.

– Eigenalue of 1 is t0e s:uare root of t0at of C

• Ho to calculate 1 from C1

• Let eigenectors of C /e

• &0en7 0ere

*eformation tensor inrincial directions

= =2

1 C 1 C

, 2 3 F= E E E& = CΦ Φ

λ

Λ = λ λ

2,

22

23



26


• And

• 8eneral Deformation

,. Stretc0 in "rinci"al directions

2. <igid-/ody rotation

3. <igid-/ody translation

)seful formulas

= + = +d d d. F - b 91 - b

= Φ Λ Φ

& 1

λ Λ = λ λ

,

2

3

=

= λ ⊗∑3

i i ii ,

1 E E

== λ ⊗∑

3

i i ii ,

; e e

== ⊗∑

3

i ii ,

9 e E

=

= λ ⊗

∑

3

i i ii ,

F e E

== λ ⊗∑

32

i i ii ,C E E

== λ ⊗∑

32i i i

i ,

b e e



27

Generali<ed +a"ran"ian Strain

• 8-L strain is a s"ecial case of general form of Lagrangianstrain tensors 5Hill7 ,+#(6

( )= −2mm

,

2mE 1 0

E l P l



28

E.amle – Polar *ecomosition

• Sim"le s0ear "ro/lem


• Deformation tensor

• $ind eigenalues and eigenectors of C

C,7 ,

C27 2

C,

C2

E2

E,

60o

= + = =

, , 22 2

3 3

C @C

C

C

=2

@3

=

, @

,F

= = =

+

23&

2 %233

,, @

@ @ ,C F F

( ) ( )

λ = λ =

= = −, 2

3 3, ,

, 22 2 2 2

37 , 3

7E E

E l P l *



29

E.amle – Polar *ecomosition cont

• ?n E, J E2 coordinates

• *rinci"al Direction Matri

• Deformation tensor in "rinci"al directions

• Stretc0 tensor

′ = =

Λ

3

, 3C

, 2

, 2 3 2 F

3 2 , 2

−Φ = =

E E

= × ×Φ Φ

& C

3 , 3

=

Λ

= × × =

Φ Λ Φ

& 3 2 , 2

, 2 ! 2 3

1

E l P l * i i



30


• Ho 1 deforms a s:uare1

• <otational &ensor

– 3o cloc@ise rotation

C,7 ,

C27 2

30o

C,7 ,

C27 2

30o

, 3 2 , 2, 2 3 2

− = × = −

9 F 1

, 2, 3 2 7 ,, 2 ! 2 3 × = × =

1 1

, 2, ,.,!3 27

,, 2 ! 2 3

× = × =

9 9

! 3 # , 2

, 2 3 2

&

= × =

; F 9

E l P l * i i



31


• A straig0t line ill deform to

• onsider a diagonal line= θ 4!o

• onsider a circle

E:uation of elli"se

C,7 ,

C27 2

25o

C,7 ,

C27 2

= θ2 ,C C tan

( )θ

= − =⇒ = − θ

⇒ = +

, , 2 2 22 , 2

,, 2tan

C @ 7 C

5 @ 6tan

@

α = = α = °+

2

,

,tan 24.+

, @

+ =

− + =

− + + =

2 2 2, 2

2 2 2, 2 2

2 2 2 2, , 2 2

C C r

5 @ 6 r

2@ 5, @ 6 r

* f ti f ; l



32

*eformation of a ;olume

• ?nfinitesimal olume /y t0ree ectors

– )ndeformed=

– Deformed=

$rom ontinuum Mec0anics

d-,

d-3

d-2

d.,d.3

d.2

= × × =, 2 3 , 2 3

rst r s t

dG d 5d d 6 e dC dC dC- - -

= × × =, 2 3 , 2 3 iB@ i B @dG d 5d d 6 e d d d. . .

=

∂ ∂∂ = ÷ ÷ ÷ ÷∂ ∂ ∂ ∂ ∂∂

=∂ ∂ ∂

==

, 2 3 iB@ i B @

B, 2 3@iiB@ r s tr s t

B , 2 3@iiB@ r s t

r s t, 2 3

rst r s t

dG e d d d

e dC dC dCC C C

e dC dC dC

C C C

e dC dC dCdG

=iB@ ir Bs @t rst

e a a a e deta= = λ λ λ, 2 3 detF

* f ti f ; l t



33

*eformation of a ;olume cont

• Golume c0ange

• Golumetric Strain

• ?ncom"ressi/le condition= ,

• &ransformation of integral domain

−= −

dG dG ,dG

=

dG dG

Ω ΩΩ = Ω∫∫∫ ∫∫∫

-

d d.

f fD

E l 1 i i l * f ti f 4



34

E.amle % 1nia.ial *eformation of a 4eam

• ?nitial dimension of L00 deforms to L00

•

Deformation gradient

• onstant olume

L

0

0

L

0

0

= λ λ == λ λ == λ λ =

, , , ,

2 2 2 2

3 3 3 3

C L ; L C 0 ; 0

C 0 ; 0

λ = λ λ

,

2

3

F

= = λ λ λ

= = ÷

, 2 32

det

L 0 LA

L 0 L A

F

= ⇒ = =

L L , 0 0 A A

L L

* f ti f A



35

*eformation of an Area

• <elations0i" /eteen dS and dS

dS

x d x1

n

S x

xdS

0d X1

N

S0

X

F(X)

d X2

d x2

)ndeformed Deformed

= × == × =

, 2 , 2

- i - iB@ B @, 2 , 2

r rst s t

dS d d NdS e dC dCdS d d n dS e d d

N - -

n . .

∂ ∂=

∂ ∂ B , 2@

i - iB@ s ts t

C CNdS e d d

∂ ∂∂ ∂=

∂ ∂ ∂ ∂ B , 2@i i

i - iB@ s tr r s t

C CC CNdS e d d

∂×

∂i

r

C

* f ti f A t



36

*eformation of an Area cont

• <esults from ontinuum Mec0anics

• )se t0e second relation=

−∂ ∂∂ ∂= =

∂ ∂ ∂ ∂, B , 2 , 2@i i

i - iB@ s t rst s tr r s t

C CC CNdS e d d e d d

F

−

∂ ∂ ∂= ∂ ∂ ∂

∂ ∂∂=

∂ ∂ ∂

r s tiB@ rsti B @

B, @irst iB@

r s t

e e C C C

C CCe e .

F

F

r n dS

−= ×& dS dSn F N

−−

−

×× ⇒ =

×

& &

& ; ;

F Nn F N n

F N

−= & -dS 5 6 5 6 dSF . N -

St M



37

Stress Measures

• Stress and strain 5tensor6 de"end on t0e configuration

•

auc0y 5&rue6 Stress= $orce acts on t0e deformed config. – Stress ector at Ω=

– auc0y stress refers to t0e current deformed configuration as areference for /ot0 area and force 5true stress6

P

N

∆S0

P n ∆S

x

∆f

Undeformed congurationDeformed conguration

Cauc#$ Stress: s$m

∆ →

∆= =

∆S

limS

ft n

St ss M su s c nt



38

Stress Measures cont

• &0e same force7 /ut different area 5undeformed area6

– P refers to t0e force in t0e deformed configuration and t0e area

in t0e undeformed configuration• Ma@e /ot0 force and area to refer to undeformed config.

First Piola%=irc##off StressNot s$mmetric

= <elation /eteen σ and P

∆ →∆= =

∆

&

S

limSfT P N

= = & d dS dSf n P N −= ×&

dS dSn F N

−= ,P F σ

−= =& & d 5 dS 6 dSf F N P N




39


• )nsymmetric "ro"erty of P ma@es it difficult to use

– <emem/er e used t0e symmetric "ro"erty of stress 9 strainseeral times in linear "ro/lems

• Ma@e P symmetric /y multi"lying it0 F-&

– ust conenient mat0ematical :uantities

• $urt0er sim"lification is "ossi/le /y 0andling differently

Second Piola%=irc##off Stress: s$mmetric

=irc##off Stress: s$mmetric

− − −= × = × ×& , & S P F F F = × × & ,

F S F

= = × × & F S Fσ




40


• Eam"le

• /seration

– $or linear "ro/lems 5small deformation6=

– $or linear "ro/lems 5small deformation6=

– S and E are conBugate in energy

– S and E are inariant in rigid-/ody motion

(nte"ration can be done inΩ/

Ω Ω Ω

Ω = Ω = Ω

∫∫∫ ∫∫∫ ∫∫∫

= d = d = dε σ ε τ ε

≈ ≈ ≈P Sτ

≈ ≈E e

E.amle 1nia.ial Tension



41


• auc0y 5true6 stress= 7 σ22 σ33 σ,2 σ23 σ,3

•

Deformation gradient=

• $irst *-K stress

• Second *-K stress

L

0

0

L

0

0$

σ =,,$

A

−

− −

−

λ

= λ = λ

,,

, ,2

,3

7 ,

F

− −= × × = = = =λλ

2, &

,, ,, 2 2 2 ,,

$ , $ A $A $S 5 6

A A AA AF F

−= = =λ

,,, ,,

, - -

$ , $ A $* 5

A A A AF σ

Summar$



42

Summar$

• Nonlinear elastic "ro/lems use different measures ofstress and strain due to c0anges in t0e reference frame

• Lagrangian strain is inde"endent of rigid-/ody rotation7/ut engineering strain is not

• Any deformation can /e uni:uely decom"osed into rigid-

/ody rotation and stretc0• &0e determinant of deformation gradient is related to t0e

olume c0ange7 0ile t0e deformation gradient andsurface normal are related to t0e area c0ange

•$our different stress measures are defined /ased on t0ereference frame.

• All stress and strain measures are identical 0en t0edeformation is infinitesimal



43

Nonlinear Elastic Anal$sis

Section 3.3

Goals



44

Goals

• )nderstanding t0e "rinci"le of minimum "otential energy

– )nderstand t0e conce"t of ariation

• )nderstanding St. Genant-Kirc00off material

• Ho to o/tain t0e goerning e:uation for nonlinear elastic"ro/lem

• 0at is t0e total Lagrangian formulation1

• 0at is t0e u"dated Lagrangian formulation1

• )nderstanding t0e lineariation "rocess

Numerical Met#ods for Nonlinear Elastic Problem



45

Numerical Met#ods for Nonlinear Elastic Problem

• e ill o/tain t0e ariational e:uation using t0e rincileof minimum otential ener"$

– nly "ossi/le for elastic materials 5"otential eists6

• &0e N-< met0od ill /e used 5need aco/ian matri6

• Total +a"ran"ian 6material7 formulation uses t0e

undeformed configuration as a reference7 0ile t0eudated +a"ran"ian 6satial7 uses t0e currentconfiguration as a reference

• &0e total and u"dated Lagrangian formulations are

mat#ematicall$ e2uivalent /ut 0ae different as"ects incom"utation

Total +a"ran"ian Formulation



46

Total +a"ran"ian Formulation

• )sing incremental force met#od and N%! met#od

– &otal No. of load ste"s 5N67 current load ste" 5n6

• Assume t0at t0e solution 0as conerged u" to tn

• ant to find t0e e:uili/rium state at tn,

0Ω

nΩ

X x

nu

∆

u

Undeformed conguration(known)

Last conerged conguration(known)

!urrent conguration(unknown)

0P

nP

+ = + ∆n , n nf f f

Total +a"ran"ian Formulation cont



47

Total +a"ran"ian Formulation cont

• ?n &L7 t0e undeformed confi"uration is t0e reference

•

2

nd

*-K stress 5S6 and 8-L strain 5E6 are t0e natural c0oice• ?n elastic material7 strain ener"$ densit$ ' eists7 suc0

t0at

• e need to e"ress in terms of E

∂=

∂

stressstrain



St ;enant%=irc##off Material



49

St ;enant%=irc##off Material

• Strain energy density for St. Genant-Kirc00off material

• $ourt0-order constitutie tensor 5isotro"ic material6

– LameOs constants=

– ?dentity tensor 52nd order6=

– ?dentity tensor 54t0 order6=

– &ensor "roduct=

ontraction o"erator== ,

2

5 6 = =E E * E =iB iB

= a /a b

= λ ⊗ + µ2* 0 0 ( ν

λ = µ =

+ ν − ν + ν

E E

5, 65, 2 6 25, 6

= δiB F0

= δ δ + δ δ,iB@l i@ Bl il B@2

? 5 6

= ∀= = = + +ii ,, 22 33

= 7 2nd-order sym.

= tr5 6 a a a a

( a a a

0 a a

⊗ = iB @la a 54t0-order6a a

St ;enant%=irc##off Material cont



50

St ;enant%=irc##off Material cont

• Stress calculation

– differentiate strain energy density

– Limited to small strain /ut large rotation

– <igid-/ody rotation is remoed and only t0e stretc0 tensorcontri/utes to t0e strain

–

an s0o

Deformation tensor

∂= = = λ + µ

∂5 6

= tr5 6 2E

S * E E 0 EE

= − = − = −& & & 2, , ,2 2 25 6 5 6 5 6E F F 0 1 9 9u 0 1 0

∂ ∂= =∂ ∂ 2SE C

E.amle



51

E.amle

• E 37 and ν .3

• 8-L strain=

• LameOs constants=

• 2nd *-K Stress=

,.!

,.

C

K

)ndeformedelement

Deformed element2.

.%

= − -.3(+ -- -.2!!E

νλ = = µ = =+ ν − ν + ν

E E,%73( ,,7!3(5, 65, 2 6 25, 6

= λ + µ = λ − + µ −

= −

, .3(+

tr5 6 2 5.3(+ .2!!6 2 , .2!!,,72+#

37!#!

S E 0 E

− = =

& ,7(%2 -,

- 2,7!,#FSF

E.amle – Simle S#ear Problem



52

E.amle Simle S#ear Problem

• Deformation ma"

•

Material "ro"erties

• 2nd *-K stress

C,7 ,

C27 2

-0.4 -0.2 0.0 0.2 0.4

20

10

0

-10

-20

Cauchy

2nd P-K

Shear parameter k

S h e a r s t r e

s s

= + = =, , 2 2 2 3 3 C @C 7 C 7 C

= − =

&

2

@, ,5 6

2 2 @ @E F F 0

=

, @

,F

νλ = = µ = =+ ν − ν + ν

E E4M*a 4M*a5, 65, 2 6 25, 6

= λ + µ =

2

2@ 2@tr5 6 2 2- M*a2@ 3@

S E 0 E

+ += =

+

2 4 3&

3 2

!@ 3@ 2@ 3@,2- M*a

2@ 3@ 3@FSF



;ariational Formulation



54

;ariational Formulation

• e ant to minimie t0e "otential energy 5e:uili/rium6

Πint= stored internal energy

Πet= "otential energy of a""lied loads

• ant to find u t0at minimies t0e "otential energy

– *ertur/ u in t0e direction of > "ro"ortional to τ

– ?f u minimies t0e "otential7 Π5u6 must /e smaller t0an Π5uτ6 for all"ossi/le >

Ω Ω Γ

Π = Π + Π

= Ω − Ω − Γ ∫∫ ∫∫ ∫ s o

int et

& / &

5 6 5 6 5 6

5 6d d d

u u u

E u f u t

τ = + τu u u

;ariational Formulation cont



55


• ;ariation of Potential Ener"$ 5Directional Deriatie6

– Π de"ends on u only7 /ut Π de"ends on /ot0 u and >

– Minimum otential ener"$ #aens 8#en its variation becomes<ero for ever$ ossible >

– ne-dimensional eam"le

e ill use Poer-/arQ for ariation

Π(u)

RuR

At minimum: all directional

derivatives are <ero

τ=Π = Π + ττ

d5 7 6 5 6du u u u

E.amle – +inear Srin"



56

E.amle +inear Srin"

• *otential energy=

• *ertur/ation=

• Differentiation=

•

Ealuate at original state=

@

f

u

;ariation is similar to differentiation ???

Π = × − ×2,2

5u6 @ u f u

Π + τ = × + τ − × + τ2,25u u6 @ 5u u6 f 5u u6

Π + τ = × + τ × − × τd

5u u6 @ 5u u6 u f ud

τ=

Π + τ = × × − × = τ

d5u u6 @ u u f u

d




57


• Gariational E:uation

– $rom t0e definition of stress

– Note= load term is similar to linear "ro/lems

– Nonlinearit$ in t#e strain ener"$ term

• Need to rite LHS in terms of u and >

for all >

;ariational e2uation in T+ formulation

Ω Ω Γ

∂

Π = Ω − Ω − Γ =∂∫∫ ∫∫ ∫ s o

& / & 5 6

5 7 6 = d d d

E

u u E u f u tE

Ω Ω Γ Ω = Ω + Γ ∫∫ ∫∫ ∫ s

o

& / & = d d dS E u f u t






59



• Linear in terms of strain if St. Genant-Kirc00off material

is used• Also linear in terms of >

• Nonlinear in terms of u /ecause dis"lacement-strainrelation is nonlinear

for all >

Energy form Load form

Ω Ω Γ Ω = Ω + Γ ∫∫ ∫∫ ∫ s

o

& / & = d d dS E u f u t

a5 7 6u u $ 5 6u

= ∀ ∈$a5 7 6 5 67u u u u #

+ineari<ation



60

+ near <at on

• e are still in continuum domain 5not discretied yet6

• <esidual

• e ant to linearie <5u6 in t0e direction of ∆u

– $irst7 assume t0at u is "ertur/ed in t0e direction of ∆u using aaria/le τ. &0en lineariation /ecomes

– <5u6 is nonlinear .r.t. u7 /ut L<5u6F is linear .r.t. ∆u

– ?teration @ did not conerged7 and e ant to ma@e t0e residual atiteration @, ero

= − $<5 6 a5 7 6 5 6u u u u

τ=

∂ + τ∆ ∂ = = ∆ ∂τ ∂

&

<5 6 <L<5 6F

u uu u

u

+ ∂≈ ∆ + = ∂

& @@ , @ @<5 6

<5 6 <5 6 u

u u uu

+ineari<ation cont



61

+

• &0is is N-< met0od 5see 0a"ter 26

• )"date state

• e @no 0o to calculate <5u@67 /ut 0o a/out 1

– nly lineariation of energy form ill /e re:uired

– e ill address dis"lacement-de"endent load later

∂ ∆ = − ∂

& @

@ @<5 6 <5 6u u uu

+

+ +

= + ∆

= +

@ , @ @

@ , @ ,

u u u

. - u ∂ ∂

@<5 6u

u

∂ ∂

= −∂ ∂ $<5 6F a5 7 6 5 6Fu u u uu u

+ineari<ation cont



62

• Lineariation of energy form

– Note t0at t0e domain is fied 5undeformed reference6

– Need to e"ress in terms of dis"lacement increment ∆u

•

Stress increment 5St. Genant-Kirc00off material6

• Strain increment 58reen-Lagrange strain6

Ω Ω = Ω = ∆ + ∆ Ω ∫∫ ∫∫ La5 7 6F L = d = = Fdu u S E S E S E

∂∆ = ∆ = ∆

∂= =

SS E * E

E

∆ = ∆ + ∆& & ,25 6E F F F F

∂ ∂ + ∂∆ ∆ = ∆ = ∆ = = ∇ ∆ ÷ ÷∂ ∂ ∂

5 6. - u uF u

- - -

+ineari<ation cont



63

• Strain increment

• ?nc. strain ariation

• +ineari<ed ener"$ form

– (mlicitl$ deends on u: but bilinear 8rt u and >

– First term@ tan"ent stiffness

– Second term@ initial stiffness

??? +inear 8rt u

??? +inear 8rt u

∆ = ∆ + ∆

= ∇ ∆ + ∇ ∆

= ∇ ∆

& & ,2

& & ,

2 &

5 6

5 6

sym5 6

E F F F F

u F F u

u F

∆ = ∆ ∇

= ∇ ∆

= ∇ ∇ ∆

& &

&

sym5 6F

sym5 6

sym5 6

E u F

u F

u u

Ω= ∆ + ∆ Ω ≡ ∆

∫∫

La5 7 6F = = = Fd a 5 T 7 6u u E * E S E u u u

+ineari<ation cont



64

• N-< ?teration it0 ?ncremental $orce

– Let tn /e t0e current load ste" and 5@,6 /e t0e current iteration

– &0en7 t0e N-< iteration can /e done /y

–

)"date t0e total dis"lacement

• ?n discrete form

• 0at are and 1

∆ = − ∀ ∈$ n @ @ n @a 5 T 7 6 5 6 a5 7 67u u u u u u u #

+ = + ∆n @ , n @ @u u u

∆ =& n @ @ & n @

& U V FU V U V U Vd = d d !n @

& F= n @U V!

E.amle – 1nia.ial 4ar



65

• Kinematics

• Strain ariation

• Strain energy density and stress

• Energy and load forms

•

L0%1

m

1 2F %1&&

x

= =2 2du du

u 7 udC dC

= + = + ÷

2

2,, 2 2du , du ,E u 5u 6dC 2 dC 2

= × 2,,, ,,2

5E 6 E 5E 6 ∂ = = × = + ÷∂

2,, ,, 2 2

,,

,S E E E u 5u 6

E 2

= + = +,, 2 2

du du duE u 5, u 6

dC dC dC

= = +∫ L

,, ,, ,, 2 2a5u7u6 S E AdC S AL 5, u 6u =$ 25u6 u $

( )= + − = ∀2 ,, - 2 2< u S AL 5, u 6 $ -7 u




66

• Lineariation

• N-< iteration

∆ = ∆ = + ∆,, ,, 2 2

S E E E5, u 6 u ∆ = ∆,, 2 2

E u u

( )∆ = × × ∆ + × ∆

= + ∆ + ∆

∫ -L,, ,, ,, ,,-

2- 2 2 2 ,, - 2 2

a 5uT u7u6 E E E S E AdC

EAL 5, u 6 u u S AL u u

+ + ∆ = − +@ 2 @ @ @ @2 ,, - 2 ,, 2 -E5, u 6 S FAL u $ S 5, u 6AL

+ = + ∆@ , @ @2 s 2u u u




67

(a) with initia stiffness!teration u "train "tress con#

0 0$0000 0$0000 0$0000 9$999%&011 0$5000 0$6250 125$00 7$655%&01

2 0$3478 0$4083 81$664 1$014%&02

3 0$3252 0$3781 75$616 4$236%&06

(') withot initia stiffness!teration u "train "tress con#

0 0$0000 0$0000 0$0000 9$999%&01

1 0$5000 0$6250 125$00 7$655%&01

2 0$3056 0$3252 70$448 6$442%&03

3 0$3291 0$3833 76$651 3$524%&04

4 0$3238 0$3762 75$242 1$568%&05

5 0$3250 0$3770 75$541 7$314%&07

1dated +a"ran"ian Formulation



68

" "

• &0e current configuration is t0e reference frame

– <emem/er it is un@non until e sole t0e "ro/lem

– Ho are e going to integrate if e donOt @no integral domain1

• '#at stress and strain s#ould be used)

– $or stress7 e can use auc0y stress 5σ6

– $or strain7 engineering strain is a "air of auc0y stress – 'ut7 it must /e defined in t0e current configuration

∂ ∂= + = ∇ ÷

÷∂ ∂

&

,

sym5 62

u uu

. .

;ariational E2uation in 1+



69

2

• ?nstead of deriing a ne ariational e:uation7 e illconert from &L e:uation

Similarly

− −

= × ×

⇒ = × ×

&

, &

,

F S F

S F F

σ

σ

− −

∂ ∂= + ÷

÷∂ ∂

∂ ∂= + ÷ ÷∂ ∂ ∂ ∂ ∂ ∂

= + ÷ ÷∂ ∂ ∂ ∂

∂ ∂= + ÷ ÷∂ ∂

= × ×

& &

&

& & ,

& & &

& &

&

,

2

,2

,

2

,2

u uE F F

- -

u uF F F F- -

- u u -F F

. - - .

u uF F. .

F F

∆ = × ∆ ×

∂∆ ∂∆∆ = + ÷

÷∂ ∂

&

& ,

2

E F F

u u

. .

ε

ε

;ariational E2uation in 1+ cont



70

2

• Energy $orm

– e Bust s0oed t0at material and s"atial forms aremat0ematically e:uialent

• Alt0oug0 t0ey are e:uialent7 e use different notation=


0at 0a""ens to load form1

(s t#is linear or nonlinear)

− −

Ω Ω

= Ω = Ω

∫∫ ∫∫ σ ε

, & & a5 7 6 = d 5 6 = 5 6du u S E F F F F

− −σ ε = δ δ σ ε = σ ε, ,i@ @l Bl mi mn nB m@ nl @l mn mn mn$ $ $ $

Ω Ω ΩΩ = Ω = Ω∫∫ ∫∫ ∫∫

= d = d = dS E σ ε σ ε

Ω= Ω∫∫ σ ε

a5 7 6 = du u

= ∀ ∈$a5 7 6 5 67u u u u #

+ineari<ation of 1+



71

• Lineariation of ill /e c0allenging /ecause e donOt@no t0e current configuration 5it is function of u6

• Similar to t0e energy form7 e can conert t0e lineariedenergy form of &L

• <emem/er

•

?nitial stiffness term

η ∆@l5 7 6u u

a 5 7 6u u

Ω∆ = ∆ + ∆ Ω∫∫ a 5 T 7 6 = = = Fdu u u E * E S E

− −

− −

∂ ∂∆ ∂∆ ∂∆ = + ÷

÷∂ ∂ ∂ ∂

∂ ∂∆ ∂∆ ∂= σ + ÷ ÷∂ ∂ ∂ ∂ ∂ ∂∆ ∂∆ ∂

≡ σ + ÷∂ ∂ ∂ ∂

& & , &

m m m m, ,i@ @l Bli B i B

m m m m@l

@ l @ l

,= 5 6 =

2

u u u u,$ $2 C C C C

u u u u,

2

u u u uS E F F

- - - -

+ineari<ation of 1+ cont



72

4t0-order s"atialconstitutie tensor

• ?nitial stiffness term

• &angent stiffness term

0ere

∆ = ∆= = 5 7 6S E u uη ∆ = ∇ ∇ ∆& 5 7 6 sym5 6u u u u

∆ = × × × ∆ ×

= ε ∆ε = ε ∆ε

ε ε

& &

@i @l lB iBmn "m ": :n

@l @i lB iBmn "m :n ":

5 = = 6 5 6 = = 5 6

$ $ D $ $,

$ $ D $ $

E * E F F * F F

∆ = ∆ε= = = =E * E c

=iB@l ir Bs @m ln rsmn,

c $ $ $ $ D

Satial Constitutive Tensor



73

• $or St. Genant-Kirc00off material

• ?t is "ossi/le to s0o

• /seration – * 5material6 is constant7 /ut c 5s"atial6 is not

–

= λ ⊗ + µ = λδ δ + µ δ δ + δ δrsmn rs mn rm sn rn sm

5 6 2 5 6* 0 0 ( *

= λ + µ + iB@l iB @l i@ Bl il B@,

c 8 8 58 8 8 8 6 .

= ≠ ε= 7 =S * E c

+ineari<ation of 1+ cont



74

• $rom e:uialence7 t0e energy form is linearied in &L andconerted to )L

• N-< ?teration

• /serations

–

&o formulations are t0eoretically identical it0 differente"ression

– Numerical im"lementation ill /e different

– Different constitutie relation

Ω= ∆ + Ω∫∫ ε ε σ η

La5 7 6F = = = Fdu u c

Ω∆ = ∆ + Ω∫∫ ε ε σ η

a 5 T 7 6 = = = Fdu u u c

∆ = − ∀ ∈$ n @ @ n @a 5 T 7 6 5 6 a5 7 67u u u u u u u #




75

• Kinematics

• Deformation gradient=

• auc0y stress=

• Strain ariation=

•

Energy 9 load forms=

• <esidual=

L0%1

m

1 2F %1&&

x = =+ +

2 2

2 2

u udu du7

d , u d , u

= = + = +,, 2 2d

$ , u 7 , udC

σ = = + +2

,, ,, ,, ,, 2 2 2

, ,$ S $ E5u u 65, u 6

2

− −ε = =+

& , 2,, ,, ,, ,,

2

u5u6 $ E $

, u

= σ ε = σ∫ L

,, ,, ,, 2a5u7u6 5u6Ad Au =$ 25u6 u $

( )= σ − = ∀2 ,, 2< u A $ -7 u




76

• S"atial constitutie relation=

• Lineariation=

!teration u "train "tress con#

0 0$0000 0$0000 0$000 9$999%&01

1 0$5000 0$3333 187$500 7$655%&01

2 0$3478 0$2581 110$068 1$014%&02

3 0$3252 0$2454 100$206 4$236%&06

= = + 3,,,, ,, ,, ,, ,, 2

,c $ $ $ $ E 5, u 6 E

ε ε ∆ = + ∆∫ L 2,, ,,,, ,, 2 2 2-

5u6c 5 u6Ad EA5, u 6 u u

σσ η ∆ = ∆

+∫ L ,,

,, ,, 2 2-2

A5 u7u6Ad u u

, u

( )∆ = ε ε ∆ + σ η ∆σ

= + ∆ + ∆+

∫ L

,, ,,,, ,, ,,

2 ,,2 2 2 2 2

2

a 5uT u7u6 5u6c 5 u6 5 u7u6 Ad

EA5, u 6 u u Au u, u



77

H$erelastic Material Model

Section 3.!

Goals



78

• )nderstand t0e definition of 0y"erelastic material

• )nderstand strain energy density function and 0o to useit to o/tain stress

• )nderstand t0e role of inariants in 0y"erelasticity

• )nderstand 0o to im"ose incom"ressi/ility

• )nderstand mied formulation and "ertur/ed Lagrangianformulation

• )nderstand lineariation "rocess 0en strain energydensity is ritten in terms of inariants

'#at (s H$erelasticit$)



79

• Hy"erelastic material - stress-strain relations0i" deriesfrom a strain energy density function

– Stress is a function of total strain 5inde"endent of 0istory6

– De"ending on strain energy density7 different names are used7suc0 as Mooney-<ilin7 gden7 eo07 or "olynomial model

• 8enerally comes it0 incom"ressi/ility 5 ,6

– &0e olume "reseres during large deformation

– Mied formulation J com"letely incom"ressi/le 0y"erelasticity

– *enalty formulation - nearly incom"ressi/le 0y"erelasticity

• Eam"le= ru//er7 /iological tissues – nonlinear elastic7 isotro"ic7 incom"ressi/le and generally

inde"endent of strain rate

• Hy"oelastic material= relation is gien in terms of stress

and strain rates

Strain Ener"$ *ensit$



80

• e are interested in isotro"ic materials

– Material frame indifference= no matter 0at coordinate system is

c0osen7 t0e res"onse of t0e material is identical – &0e com"onents of a deformation tensor de"ends on coord. system

– &0ree inariants of C is inde"endent of coord. system

•

?nariants of C

– ?n order to /e material frame indifferent7 material "ro"ertiesmust /e e"ressed using inariants

– $or incom"ressi/ility7 ?3 ,

No deformation?, 3

?2 3

?3 ,

= = + + = λ + λ + λ2 2 2, ,, 22 33 , 2 3? tr5 6 C

= − = λ λ + λ λ + λ λ 2 2 2 2 2 2 2 2,

2 , 2 2 3 3 ,2? 5tr 6 tr5 6C C

= = λ λ λ2 2 23 , 2 3? detC

Strain Ener"$ *ensit$ cont



81

• Strain Energy Density $unction

– Must /e ero 0en C 07 i.e.7 λ, λ2 λ3 ,

– $or incom"ressi/le material

– E= Neo-Hoo@ean model

–

Mooney-<ilin model

∞

+ + == − − −∑ m n @

, 2 3 mn@ , 2 3m n @ ,

5? 7? 7? 6 A 5? 36 5? 36 5? ,6

∞

+ == − −∑ m n

, 2 mn , 2m n ,

5? 7? 6 A 5? 36 5? 36

= −, , ,5? 6 A 5? 36

= − + −, 2 , , , 25? 7? 6 A 5? 36 A 5? 36

µ=,A 2

Strain Ener"$ *ensit$ cont



82

• Strain Energy Density $unction

– eo0 model

– gden model

– 0en N , and a , ,7 Neo-Hoo@ean material

– 0en N 27 α, 27 and α2 & 27 Mooney-<ilin material

= − + − + −2 3, , ,- , 2- , 3- , 5? 6 A 5? 36 A 5? 36 A 5? 36

( )α α α

=

µλ λ λ = λ + λ + λ −

α

∑ i i i

Ni

, , 2 3 , 2 3

i , i

5 7 7 6 3

=

µ = α µ∑N

i i

i ,

,

2

E.amle – Neo%Hoo5ean Model



83

• )niaial tension it0 incom"ressi/ility

• Energy density

• Nominal stress

-0.8 -0.4 0 0.4 0.8-250

-200

-150

-100

-50

0

50

omina strain

) o m i n a s

t r e s s

eo*+ooean

-inear eastic

λ = λ λ = λ = λ, 2 3 , ;

= − = λ + λ + λ − = λ + −λ

2 2 2 2,- , ,- , 2 3 ,-

2 A 5? 36 A 5 36 A 5 36

∂ = = λ − = µ + ε − ÷ ÷∂λ λ + ε ,- 2 2

, ,* 2A , 5, 6

E.amle – St ;enant =irc##off Material



84

• S0o t0at St. Genant-Kirc00off material 0as t0e folloingstrain energy density

• $irst term

• Second term

λ= + µ 2 25 6 tr5 6 tr5 6

2E E E

∂= =

∂tr5 6

tr5 6 =E

E 0 E 0E

∂ ∂ ∂= = λ + µ

∂ ∂ ∂

25 6 tr5 6 tr5 6tr5 6

E E ES E

E E E

∂λ = λ = λ ⊗

∂

tr5 6tr5 6 5 = 6 5 6 =

EE 0 0 E 0 0 E

E

∂= δ δ + δ δ = + =

∂iB Bi

i@ Bl Bi iB B@ il l@ l@ l@

@l

E EE E E E 2E

E

E.amle – St ;enant =irc##off Material cont



85

• &0erefore

*

∂ ∂= λ + µ∂ ∂= λ ⊗ + µ= λ ⊗ + µ

2tr5 6 tr5 6tr5 65 6 = 2

5 6 2 =

E E

S E E E0 0 E E

0 0 ( E

Nearl$ (ncomressible H$erelasticit$



86

• ?ncom"ressi/le material

– annot calculate stress from strain. 0y1

• Nearly incom"ressi/le material

– Many material s0o nearly incom"ressi/le /e0aior

– e can use t0e /ul@ modulus to model it

• )sing ?, and ?2 enoug0 for incom"ressi/ility1 – No7 ?, and ?2 actually ary under 0ydrostatic deformation

– e ill use reduced inariants= ,7 27 and 3

• ill , and 2 /e constant under dilatation1

− −= = = =,;3 2;3 ,;2, , 3 2 2 3 3 3 ? ? ? ? ?

+oc5in"



87

• 0at is loc@ing – Elements do not ant to deform een if forces are a""lied –

Loc@ing is one of t0e most common modes of failure in NL analysis – ?t is ery difficult to find and solutions s0o strange /e0aiors

• &y"es of loc@ing – S0ear loc@ing= s0ell or /eam elements under transerse loading –

Golumetric loc@ing= large elastic and "lastic deformation• 0y does loc@ing occur1

– ?ncom"ressi/le s"0ere under 0ydrostatic "ressure

s.here"

Golumetric strain

* r e s s u r e No uni:ue "ressurefor gien dis"l.

Ho8 to solve loc5in" roblems)



88

• Mied formulation 5incom"ressi/ility6

– anOt inter"olate "ressure from dis"lacements

– *ressure s0ould /e considered as an inde"endent aria/le

– 'ecomes t0e Lagrange multi"lier met0od

– &0e stiffness matri /ecomes "ositie semi-definite

4, formulation

Dis"lacement

*ressure

Penalt$ Met#od



89

• ?nstead of incom"ressi/ility7 t0e material is assumed to /e nearlyincom"ressi/le

•

&0is is closer to actual o/seration• )se a large /ul@ modulus 5"enalty "arameter6 so t0at a small olume

c0ange causes a large "ressure c0ange

• Large "enalty term ma@es t0e stiffness matri ill-conditioned

• ?ll-conditioned matri often yields ecessie deformation• &em"orarily reduce t0e "enalty term in t0e stiffness calculation

• Stress calculation use t0e "enalty term as it is

Golumetric strain

* r e s s u r e

)ni:ue "ressurefor gien dis"l.%

,

,KF

,

,

=

E.amle – H$drostatic Tension



90

• ?nariants

• <educed inariants

?, and ?2 are not constant

, and 2 are constant

= α

= α = α

, ,

2 2

3 3

C

C C

α

= α α

F

α

= α α

2

22

C

= α = α = α2 4 #, 2 3? 3 ? 3 ?

−

−

= =

= == = α

,;3, , 3

2;32 2 3,;2 3

3 3

? ? 3

? ? 3 ?

Strain Ener"$ *ensit$



91

• )sing reduced inariants

– D5,7 26= Distortional strain energy density

– H536= Dilatational strain energy density

•

&0e second terms is related to nearly incom"ressi/le/e0aior

–K= /ul@ modulus for linear elastic material

/'as:

= +, 2 3 D , 2 H 3

5 7 7 6 5 7 6 5 6

= − 2H 3 3

K 5 6 5 ,6

2

= λ + µ23

2H 3 3

, 5 6 5 ,6

2D= −

Moone$%!ivlin Material



92

• Most "o"ular model

– ?nitial s0ear modulus W 25A, A,6

–

?nitial oungOs modulus W #5A, A,6 53D6 or (5A, A,6 52D6 – 'ul@ modulus K

• Hydrostatic "ressure

– Numerical insta/ility for large K 5olumetric loc@ing6

– *enalty met0od it0 K as a "enalty "arameter

= += − + − + −

, 2 3 D , 2 H 3

2, , , 2 3

5 7 7 6 5 7 6 5 6K

A 5 36 A 5 36 5 ,62

∂∂= = = −∂ ∂H 3

3 3" K5 ,6

Moone$%!ivlin Material cont



93

• Second *-K stress

– )se c0ain rule of differentiation

∂∂ ∂∂ ∂ ∂ ∂= = + +∂ ∂ ∂ ∂ ∂ ∂ ∂= + + −

3, 2

, 2 3

, ,7 , 27 3 37

A A K5 ,6E E E

S E E E E ∂=∂7aaE E

− −

− −

−

= −

= −

=

,;3 4;3,,7 3 ,7 , 3 373

2;3 !;3227 3 27 2 3 373

,;2,37 3 372

5? 6? ? 5? 6?

5? 6? ? 5? 6?

5? 6?

E E E

E E E

E E

=

= + −

= + − +

,7

27

+37 imn Brs mr ns4

? 2

? 45, tr 6 4

? 52 4tr 6 4 e e E E F

E

E

E

0

E 0 E

E 0 E

−

−

=

=

=

,;3, , 3

2;32 2 3

,;23 3

? ?

? ?

?

−

=

= −

=

,7

27 ,

,37 3

? 2

? 25? 6

? 2?

E

E

E

0

0 C

C

E.amle



94

• S0o

• Let

• &0en

• Deriaties

and

−= = − = ,,7 27 , 37 3? 2 7 ? 25? 67 ? 2?E E E0 0 C C

= = =, ,, 2 32 3

? tr5 67 ? tr5 67 ? tr5 6C CC CCC

= = − = + −2 3, ,, , 2 , 2 3 3 , , 22 #

? ? 7 ? ? ? 7 ? ? ? ? ?

∂∂ ∂

= δ = =∂ ∂ ∂

3, 2

iB Bi B@ @iiB iB iB

?? ?

7 7

−∂∂ ∂= δ = δ − =

∂ ∂ ∂,3, 2

iB , iB Bi 3 BiiB iB iB

?? ?7 ? 7 ?

∂ ∂=

∂ ∂2

C E

Mi.ed Formulation



95

• )sing /ul@ modulus often causes insta/ility

– Selectiely reduced integration 5$ull integration for deiatoric

"art7 reduced integration for dilatation "art6

• Mied formulation= ?nde"endent treatment of "ressure

–

*ressure " is additional un@non 5"ure incom"ressi/le material6 – Adantage= No numerical insta/ility

– Disadantage= system matri is not "ositie definite

• *ertur/ed Lagrangian formulation

– Second term ma@e t0e material nearly incom"ressi/le and t0esystem matri "ositie definite

= −H 3 3 5 7"6 "5 ,6

= − − 2H 3 3

, 5 7"6 "5 ,6 "

2K

;ariational E2uation 6Perturbed +a"ran"ian7



96

• Stress calculation

• Gariation of strain energy density

• ?ntroduce a ector of un@nons=

Golumetric strain

= + +, ,7 , 27 37A A "E E ES

= +

= + − −

7 7"

3

"

"= 5 , 6"

K

EE

S E

= 5 7"6r u

Ω = + Ω ∫∫ a 5 7 6 = "H dr r S E

= − −3"

H ,K

= − + − + − + 2, 2 3 , , , 2 3

,5 7 7 6 A 5 36 A 5 36 "5 ,6 "

2K

E.amle – Simle S#ear



97

• alculate 2nd *-K stress for t0e sim"le s0ear deformation – material "ro"erties 5A,7 A,7 K6

C,7 ,

C27 2

45o

= = =

& , , , , , , 2

, ,

F C F F

−

=

− = − = −

−

= = −

,7

27 ,

,37 3

? 2

# 2

? 25? 6 2 4

#4 2

? 2? 2 2

2

E

E

E

0

0 C

C

= = =, 2 3? 47 ? 47 ? ,

E.amle – Simle S#ear cont



98

Note= S,,7 S22 and S33 are not ero

−

= − = − − −

= − = −

,7 ,7 37

(27 27 373

! 4 4 2

? ? 4 , 3 3 ,

% ! 2

? ? ! 2 3

,

E E E

E E E

= + + −

− − +

= + − − − +

, ,7 , 27 3 37

, , , ,

, , , ,

, ,

A A K5 ,6

!A %A 4A !A

2 4A !A A 2A 3

A A

E E ES

−

−

−

= =

= =

= =

,;3, , 3

2;32 2 3

,;23 3

? ? 4

? ? 4

? ,

Stress Calculation Al"orit#m



99

• 8ien= UEV UE,,7 E227 E337 E,27 E237 E,3V& 7 U"V7 5A,7 A,6

$or "enalty met0od7 useK53 J ,6 instead of "

= = +& U V U, , , - - -V U V 2U V U V0 C E 0

= + += + + − − −= − + − + −

, , 2 3

2 , 2 , 3 2 3 4 4 ! ! # #

3 , 2 4 4 3 4 # , ! ! 4 ! 2 # #

?

?

? 5 6 5 6 5 6

=

= + + + − − −

= − − −

− − −

,727 2 3 3 , , 2 4 ! #

2 2 237 2 3 ! 3 , # , 2 4

! # 3 4 # 4 , ! 4 ! 2 #

U? V 2U, , , V

U? V 2U V

U? V 2U

V

EE

E

− −

− −

−

= −= −

=

,;3 4;3,,7 3 ,7 , 3 373

2;3 !;3227 3 27 2 3 373

,;2,37 3 372

U V ? U? V ? ? U? VU V ? U? V ? ? U? V

U V ? U? V7

E E E

E E E

E E

= + +,- ,7 -, 27 37U V A U V A U V "U VE E ES

+ineari<ation 6Penalt$ Met#od7



100

• Stress increment

• Material stiffness

• Linearied energy form

Ω ∆ = ∆ + ∆ Ω ∫∫

-

a 5 T 7 6 = = = du u u E * E S E

∆ = ∆ = ∆7 7 = =E ES E * E

∂= = + + − + ⊗

∂ ,- ,7 -, 27 3 37 37 37A A K5 ,6 K EE EE EE E E

S*

E

+ineari<ation cont



101

• Second-order deriaties of reduced inariants

− − − −

− − − −

− −

= − ⊗ + ⊗ + ⊗ −

= − ⊗ + ⊗ + ⊗ −

= − ⊗ +

4 % 4,3 3 3 3

! ( !23 3 3 3

3 ,2 2

,7 ,7 ,7 37 37 ,7 , 37 37 , 373 3 3 3

27 27 27 37 37 27 2 37 37 2 373 3 3 3

37 37 37 373 3

, 4 , ? ? ? 5? ? ? ? 6 ? ? ? ? ? ? ?

3 + 32 , 2

? ? ? 5? ? ? ? 6 ? ? ? ? ? ? ?3 + 3

, , ? ? ? ? ?

4 2

EE EE E E E E E E EE

EE EE E E E E E E EE

EE E E EE

− − − −

=

= ⊗ −

= ⊗ −

,7

27

, , , ,37 3 3

?

? 4

? 4? ?

EE

EE

EE

/

0 0 (

C C C (C

MAT+A4 Function Moone$

l l d f d f d



102

• alculates S and * for a gien deformation gradient

%

% 2nd PK stress and material stiffness for Mooney-Rivlin material

%

function [Stress D] = Mooney(F !"# !#" K ltan$

% n&uts'

% F = Deformation radient [)*)]

% !"# !#" K = Material constants% ltan = # +alculate stress alone,

% " +alculate stress and material stiffness

% ut&uts'

% Stress = 2nd PK stress [S"" S22 S)) S"2 S2) S")],

% D = Material stiffness [.*.]

%

Summar$

H l l d 0



103

• Hy"erelastic material= strain energy density eists it0incom"ressi/le constraint

• ?n order to /e material frame indifferent7 material"ro"erties must /e e"ressed using inariants

• Numerical insta/ility 5olumetric loc@ing6 can occur 0enlarge /ul@ modulus is used for incom"ressi/ility

• Mied formulation is used for "urely incom"ressi/ility5additional "ressure aria/le7 non-*D tangent stiffness6

• *ertur/ed Lagrangian formulation for nearly

incom"ressi/ility 5reduced integration for "ressure term6



104

Finite Element Formulation forNonlinear Elasticit$

Section 3.#

;oi"t Notation

ill t0 ; i t t ti / t0 t



105

• e ill use t0e ;oi"t notation /ecause t0e tensornotation is not conenient for im"lementation –

2nd

-order tensor ector – 4t0-order tensor matri

• Stress and strain ectors 5Goigt notation6

– Since stress and strain are symmetric7 e donOt need 2, com"onent

= & ,, 22 ,2U V UE E 2E VE

= &

,, 22 ,2

U V US S S VS

%Node 9uadrilateral Element in T+

ill l t i 4 d d il t l l t t



106

• e ill use "lane-strain7 4-node :uadrilateral element todiscuss im"lementation of nonlinear elastic $EA

• e ill use &L formulation• )L formulation ill /e discussed in 0a"ter 4

inite $ement atundeformed

domain

*eference$ement

X 1

X 2

1 2

3+

s

t

(,1-,1)

(1-,1)

(1-1)

(,1-1)

(nterolation and (soarametric Main"

Di l t i t l ti



107

• Dis"lacement inter"olation

• (soarametric main"

– &0e same inter"olation function is used for geometry ma""ing

Nodal dis"lacement ector 5u?7 ?6

?nter"olation function

Nodal coordinate 5C?7 ?6

(nterolation 6s#ae7 function

• Same for all elements

• Ma""ing de"ends of geometry

== ∑

eN

? ?? , N 5 6

u s u

== ∑

eN

? ?? ,

N 5 6- s -

= − −= + −

= + +

= − +

,

, 4,

2 4,

3 4,

4 4

N 5, s65, t6N 5, s65, t6

N 5, s65, t6

N 5, s65, t6

*islacement and *eformation Gradients

Di l t di t



108

• Dis"lacement gradient

– Ho to calculate


– 4ot# dislacement and deformation "radients are not s$mmetric

=

∂∂

=∂ ∂∑

eN?

?? ,

N 5 6su

u- - == ∑

eN

i7B ?7B ?i? ,u N 5 6us

∇ = & - ,7, ,72 27, 272Uu u u u Vu

= = + +& &

,, ,2 2, 22 ,7, ,72 27, 272U V U$ $ $ $ V U, u u u , u VF

∂

∂

?N 5 61

s

-

Green%+a"ran"e Strain

• 8 L t i



109

• 8reen-Lagrange strain

– Due to nonlinearity7

– $or St. Genant-Kirc00off material7

+ + = = + + + + +

,,7, ,7, ,7, 27, 27,2,,,

22 272 ,72 27, 272 2722

,2 ,72 27, ,72 ,7, 27, 272

u 5u u u u 6EU V E u 5u u u u 6

2E u u u u u u

E

≠U V FU VE 4 d

=U V FU VS * E

λ + µ λ = λ λ + µ µ

2 - F 2 -

- -

*

;ariation of G%! Strain

• Alt0 0 E5 6 is li is li5 6E u u



110

• Alt0oug0 E5u6 is nonlinear7 is linear

$unction of uDifferent from linear strain-dis"lacement matri

= N

U V FU VE 4 d= ∇ &

5 7 6 sym5 6E u u u F

5 7 6E u u

= + + + + + +

L

L

L

,, ,7, 2, ,7, ,, 27, 2, 27, ,, 47, 2, 47,

N ,2 ,72 22 ,72 ,2 272 22 272 ,2 472 22 472

,, ,72 2, ,72 ,, 272 2, 272 ,, 472 2, 472

,2 ,7, 22 ,7, ,2 27, 22 27, ,2 47, 22 47,

$ N $ N $ N $ N $ N $ N

F $ N $ N $ N $ N $ N $ N

$ N $ N $ N $ N $ N $ N

$ N $ N $ N $ N $ N $ N

4

;ariational E2uation

• Ener y form



111

• Energy form

• Load form

• <esidual

Ω

Ω

= Ω

≈ Ω

≡

∫∫ ∫∫

-

-

& & N

& int

a5 7 6 = d

U V F U V d

U V U V

u u S E

d 4 S

d F

Ω Γ

Ω Γ=

= Ω + Γ

≈ Ω + Γ

≡

∫∫ ∫

∑ ∫∫ ∫

$S

- -

e

S- -

& / &

N& /? ? ?

? ,

& et

5 6 d d

N 5 6 d N 5 6 d

U V U V

u u f u t

u s f s t

d F

= ∀ ∈ #& int & et

0U V U 5 6V U V U V7 U Vd F d d F d

+ineari<ation – Tan"ent Stiffness

• ?ncremental strain ∆ ∆U V FU VE 4 d



112

• ?ncremental strain

• Lineariation

∆ = ∆NU V FU VE 4 d

Ω Ω ∆ Ω = Ω ∆ ∫∫ ∫∫

- -

& & N N= = d U V F F Fd U VE * E d 4 * 4 d

Ω Ω ∆ Ω = Ω ∆ ∫∫ ∫∫

- -

& & 8 8= d U V F F F d U VS E d 4 4 d

=

,, ,2

,2 22

,, ,2

,2 22

S S - -

S S - - F

- - S S

- - S S

Σ

=

,7, 27, 37, 47,

,72 272 372 4728

,7, 27, 37, 47,

27, 272 372 472

N - N - N - N -

N - N - N - N - F

- N - N - N - N

- N - N - N - N

4

+ineari<ation – Tan"ent Stiffness

• &angent stiffness



113

• &angent stiffness

• Discrete incremental e:uation 5N-< iteration6

– K& F c0anges according to stress and strain

– Soled iteratiely until t0e residual term anis0es

Ω

= + Ω ∫∫

-

& & -

& N N 8 8 F F F F F F F d= 4 * 4 4 4

∆ = − ∀ ∈ #& & et int

& 0U V FU V U V U V7 U Vd = d d F F d

Summar$

• $or elastic material t0e ariational e:uation can /e



114

• $or elastic material7 t0e ariational e:uation can /eo/tained from t0e "rinci"le of minimum "otential energy

• St. Genant-Kirc00off material 0as linear relations0i"/eteen 2nd *-K stress and 8-L strain

• ?n &L7 nonlinearity comes from nonlinear strain-dis"lacement relation

• ?n )L7 nonlinearity comes from constitutie relation andun@non current domain 5aco/ian of deformationgradient6

•

&L and )L are mat0ematically e:uialent7 /ut 0aedifferent reference frames

• &L and )L 0ae different inter"retation of constitutierelation.



115

MAT+A4 Code forH$erelastic Material Model

Section 3.%

HBPE!3*m

• 'uilding t0e tangent stiffness matri =F and t0e residual



116

• 'uilding t0e tangent stiffness matri7 =F7 and t0e residualforce ector7 U!V7 for 0y"erelastic material

• ?n"ut aria/les for H*E<3D.m

aria'e /rra sie eanin

! !nteer ateria !dentification o$ (3) (ot sed)

R (3,1) ateria .ro.erties (/10, /01, )

/;% -oica #aria'e !f tre, sa#e stress #aes

-;/ -oica #aria'e !f tre, cacate the o'a stiffness matri<

% !nteer ;ota nm'er of eements

= !nteer imension of .ro'em (3)

>?@ (3,%) Aoordinates of a nodes

-% (8,%) %ement connecti#it

function /0P1R)D(MD PRP PD!31 43!5 51 5DF 607 41$%88888888888888888888888888888888888888888888888888888888888888888888888

% M!5 PR9R!M +MP359 94:!4 S3FF51SS M!3R6 !5D R1SD!4 FR+1 FR

% /0P1R14!S3+ M!31R!4 MD14S

%88888888888888888888888888888888888888888888888888888888888888888888888



117

%

%%

lo;al DSP3D FR+1 9KF S9M!

%

% nteration &oints and <eits

69=[-#>?@@)?#2.A"BA.)D# #>?@@)?#2.A"BA.)D#],

C93=[">##############D# ">##############D#],

%

% nde* for istory varia;les (eac interation &t$

535=#,

%

%4P 1R 141M153S 3/S S M!5 4P 3 +MP31 K !5D F

for 1="'51

% 5odal coordinates and incremental dis&lacements

1460=607(41(1'$'$,

% 4ocal to lo;al ma&&in

DF=Eeros("2$,

for ="'B

=(-"$85DFG",

DF('G2$=(41(1$-"$85DFG"'(41(1$-"$85DFG),

end

DSP=DSP3D(DF$,

DSP=resa&e(DSP5DFB$, %

%4P 1R 5319R!35 P53S

for 46="'2 for 40="'2 for 47="'2

1"=69(46$, 12=69(40$, 1)=69(47$,

535 = 535 G ",

%

% Determinant and sa&e function derivatives

[H S/PD D13] = S/!P14([1" 12 1)] 1460$,

F!+=C93(46$8C93(40$8C93(47$8D13,



% % Residual forces

FR+1(DF$ = FR+1(DF$ - F!+8:MI8S3R1SS,

%

% 3anent stiffness



119

% 3anent stiffness

if 43!5

S9=[S3R1SS("$ S3R1SS($ S3R1SS(.$,

S3R1SS($ S3R1SS(2$ S3R1SS(?$,

S3R1SS(.$ S3R1SS(?$ S3R1SS()$],

S/1!D=Eeros(A$,

S/1!D("')"')$=S9,

S/1!D('.'.$=S9,

S/1!D(@'A@'A$=S9,

%

1KF = :MI8D3!58:M G :9I8S/1!D8:9,

9KF(DFDF$=9KF(DFDF$GF!+81KF,

end

end, end, end,

end

end

H$erelastic Material Anal$sis 1sin" A4A91S

• ELEMEN& &*E3D(<H ELSE&NE



120

ELEMEN&7&*E3D(<H7ELSE&NE – (-node linear /ric@7 reduced integration it0 0ourglass control7

0y/rid it0 constant "ressure

• MA&E<?AL7NAMEMNEH*E<ELAS&?7 MNE-<?GL?N(.7 2.7 – Mooney-<ilin material it0 A, ( and A, 2

• S&A&?7D?<E& – $ied time ste" 5no automatic time ste" control6

y

H$erelastic Material Anal$sis 1sin" A4A91SHEAD?N8



121

- ?ncom"ressi/le 0y"erelasticity 5Mooney-<ilin6 )niaial tensionNDE7NSE&ALL,7

27,.37,.7,.747.7,.7!7.7.7,.#7,.7.7,.%7,.7,.7,.

(7.7,.7,.NSE&7NSE&$AE,,727374NSE&7NSE&$AE3,727!7#NSE&7NSE&$AE42737#7%

NSE&7NSE&$AE#47,7(7!ELEMEN&7&*EC3*!H7ELSE&NE,7,7273747!7#7%7(SL?D SE&?N7 ELSE&NE7 MA&E<?AL MNEDMATE!(A+:NAMEMNEBDHBPE!E+AST(C: MNEB%!(;+(N/ /


• Analytical solution "rocedure



122

Analytical solution "rocedure – 8radually increase t0e "rinci"al stretc0 λ from , to # –

Deformation gradient

– alculate ,7E and 27E

– alculate 2nd *-K stress

– alculate auc0y stress

– <emoe t0e 0ydrostatic com"onent of stress

λ

= λ λ

, ;

, ;

F

= +, ,7 , 27A A E ES

= × × & ,F S F

σ = σ − σ,, ,, 22


• om"arison it0 analytical stress s numerical stress



123

om"arison it0 analytical stress s. numerical stress



Elastomer Test Procedures

• Elastomer tests



125

Elastomer tests – sim"le tension7 sim"le com"ression7 e:ui-/iaial tension7 sim"le

s0ear7 "ure s0ear7 and olumetric com"ression

0 1 2 3 4 5 6 70

10

20

30

40

50

60

70

Nominal strain

N o m i n a l s t r e s s

uni-axial

bi-axialpure shear

Elastomer Tests• Data ty"e= Nominal stress s. "rinci"al stretc0



126

FF L

Simple tension test

F

F

L

Pure shear test

L

F

Equal ia!ial test

F

L

"olumetric compression test

y" " "

*ata Prearation

• Need enoug0 num/er of inde"endent e"erimental data



127

Need enoug0 num/er of inde"endent e"erimental data – No ran@ deficiency for cure fitting algorit0m

•

All tests measure "rinci"al stress and "rinci"le stretc0

Experiment Type Stretch Stress

Uniaxial tension Stretch ratio λ = L/L0Nominal stress TE = F/A0

Equi-iaxialtension

Stretch ratioλ

= L/L0 in y-

!irection

Nominal stress TE = F/A0

in y-!irection

"ure shear test Stretch ratioλ

= L/L0Nominal stress TE = F/A0

#olumetric test $ompression ratioλ

= L/L0"ressure TE = F/A0

*ata Prearation cont

• )ni-aial test λ = λ λ = λ = λ, 2 37 , ;



128

)ni aial test

• E:ui-/iaial test

• *ure s0ear test

, 2 37 ;

−∂

= = − λ λ +∂λ3

,- -,

)

& 25, 65A A 6− − λ = = λ − λ − λ

,-& 2 3,- -,

-,

A&5A 7A 7 6 U V U V 25 6 25, 6

A. b

λ = λ = λ λ = λ2, 2 37 , ;

−∂= = λ − λ + λ

∂λ! 2

,- -,, )

& 25 65A A 62

λ = λ λ = λ = λ, 2 37 ,7 , ;−∂

= = λ − λ +∂λ

3,- -,

)& 25 65A A 6

*ata Prearation cont

• Data *re"aration



129

Data *re"arat on

• $or Mooney-<ilin material model7 nominal stress is alinear function of material "arameters 5A,7 A,6

+

+

λ λ λ λ λ λ λ. .

. .

. .

, 2 3 i i , ND&

E E E E E E E, 2 3 i i , ND&

&y"e , , , 4 4 4

& & & & & & &

Curve Fittin" for Moone$%!ivlin Material

• Need to determine A, and A, /y minimiing error



130

N m , , y m m g/eteen test data and model

• $or Mooney-<ilin7 &5A,7 A,7 l@6 is linear function – Least-s:uares can /e used

( )=

− λ∑,- -,

ND& 2E@ ,- -, @

A 7A @ ,

minimie & &5A 7A 7 6

λ

λ = = = λ

/ /

& ,,

& 2 ,

& ND& ND&

5 6&

& 5 6U V U V FU V

& 5 6

.

.T b - b

.

=

,

,

AU VA

b

=

/

E,E

E 2

END&

&

& U V

&

T

Curve Fittin" cont

• Minimie error5s:uare6



131

5 : 6

• Minimiation Linear regression e:uation

= − −

= − −

= − +

& E & E

E & E

E & E & & E & &

U V U V U V U V

U V U V

U V U V 2U V F U V U V F FU V

e e T T T T

T -b T -b

T T b - T b - - b

=& & E F FU V F U V- - b - T

Stabilit$ of Constitutive Model

• Sta/le material= t0e slo"e in t0e stress-strain cure is



"alays "ositie 5*ruc5er stabilit$6

•

Sta/ility re:uirement 5Mooney-<ilin material6

• Sta/ility c0ec@ is normally "erformed at seeral s"ecified

deformations 5"rinci"al directions6

• ?n order to /e *.D.

>εd = = d -*

σ ε + σ ε >, , 2 2d d d d -

ε

ε ε > ε

,, ,2 ,

, 2 2, 22 2

D D d

d d D D d

Documents

FEA for Nonlinear Elastic Problems