L13. FEM for a non-linear material mechanics problem

Weak formulation - Static virtual work

Major aim: Evaluate momentum balance in weak form via FE-approximations

àVΣ

t : l âV = àVf × u âV + à

G

t × u âG " u Î U

Ý Reduce to a 2D computational domain

V ® t W, âV ® t âW, G ® t S, âG ® t âS

Ý Plane strain (for the in-plane directions 8Α, Β< = 1, 2)

Σ ® ΣΑΒ eΑ Ä eΒ + Σ33 e3 Ä e3 ; Ε ® ÎΑΒ eΑ Ä eΒ

Þ Consider 2D problem

àW

Σt : Ε âW = à

W

f × u âW + àSt × u âS " u Î U

FE discretization

� Displacement and strain fields

Consider triangulation We, e = 1, ... NEL. On each We introduce interpolation

u = âI=1

NODE

NI@Ξ1, Ξ2D uI, x = âI=1

NODE

NI@Ξ1, Ξ2D xI, u = âI=1

NODE

NI@Ξ1, Ξ2D uI Þ

l = âI=1

NODE

uI Ä gI Þ Ε =1

2âI=1

NODE IuI Ä gI + gI Ä uIM8NI@Ξ1, Ξ2D<I=1,2,NODE = element interpolation functions

8gI<I=1,2,3 = gradients of NI; gI =¶NI

¶x

Restrict to CST elements:

N1 = 1 - HΞ1 + Ξ2L, N2 = Ξ1, N3 = Ξ2 Þ

gI =¶NI

¶x= â

i=1

2 ¶NI

¶Ξi

Gi with Gi =¶Ξi

¶x

Ý Basis vector 9Gi=i=1,2

may be assessed explicitly, cf. details in lecture notes L13!

:0.0

0.5

1.0

Ξ10.0

0.5

1.0

Ξ20.0

0.5

1.0

,

0.0

0.5

1.0Ξ1

0.0

0.5

1.0

Ξ20.0

0.5

1.0

,

0.0

0.5

1.0Ξ1

0.0

0.5

1.0

Ξ20.0

0.5

1.0 >

� Gradient of basis function - Convective basis

Assess the gradients gI Þ introduce isoparametric mapping

x = âI=1

3

NI@Ξ1, Ξ2D xI

xI = are element nodal positions in W.

Ý 8gI<I=1,2,3 specified in terms of the contra-variant basis vector 9Gi=i=1,2

gI =¶NI

¶Ξi

Gi with Gi = Gij Gj and Gi =¶x

¶Ξi

Ý Gij = IGijM-1and Gij = Gi × Gj = J G1 × G1

G2 × G1

G1 × G2

G2 × G2N

Assess the basis vector Gi via isoparametric mapping of geometry:

x = âI=1

3

NI@Ξ1, Ξ2D xI Þ dx = Gi dΞi with Gi = âI=1

3 ¶NI

¶Ξi

xI

Explicit expressions

x = H1 - HΞ1 + Ξ2LL x1 + Ξ1 x2 + Ξ2 x3

Þ G1 =¶x

¶Ξ1= x2 - x1 = Dx2; G2 =

¶x

¶Ξ2= x3 - x1 = Dx3Þ

Gij = Gji =Dx2 × Dx2

Dx3 × Dx2

Dx2 × Dx3

Dx3 × Dx3; Gij = IGijM-1

=1

D

Dx3 × Dx3

-Dx3 × Dx2

-Dx2 × Dx3

Dx2 × Dx2

Þ

gI =¶NI

¶Ξ1

G1 +¶NI

¶Ξ2

G2 , G1 = G11 G1 + G12 G2 , G2 = G21 G1 + G22 G2

Ý Area of the element

2 Ae = HG1 ´ G2L × Ez = IDx2 ´ Dx3M × Ez =.. = Dx2 × Dx3 - Dx3 × Dx2

Static virtual work - Voight matrix formulation

� Voight matrix formulation

The static virtual work is rewritten as

àW

Σ : Ε âW = àW

f × u âW + àSt × u âS " u Î U

2 L13.nb

Let us immediately reformulate in matrix form - named "Voight matrix formulation"!

Þ Consider the Voight matrix form of Σt and Ε via Σt : Ε ® Σt

Ε Þ

Σt : Ε = Σij Εij ® Ε

tΣ =

HΕ11, Ε22, Ε33 = 0, Γ12, Γ13 = 0, Γ23 = 0LΣ11

Σ22

Σ33 ¹ 0Σ12

Σ13 = 0Σ23 = 0

where the situation at plane strain was introduced. In this case we may thus reduce the involved Voight arrays as

Εt

= HΕ11, Ε22, 0, Γ12L with Γ12 = Ε12 + Ε21,

Σt = HΣ11 Σ22 Σ33 Σ12L

� The B-matrix relation

B-matrix relation Ε = Be ue in terms of components of the gradients 8gI<I=1,2,3 as

Ε11

Ε22

0

Γ12

=

gx1 0 gx

2 0 gx3 0

0 gy1 0 gy

2 0 gy3

0 0 0 0 0 0

gy1 gx

1 gy2 gx

2 gy3 gx

3

ux1

uy1

ux2

uy2

ux3

uy3

The motivation for the planar CST element considered simply comes from

Ε =1

2âI=1

3 IuI Ä gI + gI Ä uIM =1

2

Igx1 + gx

1M ux1 ux

1 gy1 + uy

1 gx1

uy1 gx

1 + ux1 gy

1 Igy1 + gy

1M uy1

+

1

2

Igx2 + gx

2M ux2 ux

2 gy2 + uy

2 gx2

uy2 gx

2 + ux2 gy

2 Igy2 + gy

2M uy2

+1

2

Igx3 + gx

3M ux3 ux

3 gy3 + uy

3 gx3

uy3 gx

3 + ux3 gy

3 Igy3 + gy

3M uy3

We may further note the properties of the CST approximation in terms of the volumetric strain written as

Εv = 1 : Ε =1

2âI=1

3 I1 : IuI Ä gIM + 1 : IgI Ä uIMM = âI=1

3

uI × gI

In particular, let us consider the situation when Εv = 0 corresponding to pure deviatoric deformation as depicted in the

Figure below.

L13.nb 3

� Identity and tangent stiffness tensor representations

Fourth order: obtained via Ε : Ε = Ε : HI : ΕL = Ε : H1 Ä� 1L : Ε = Εt I Ε Þ

I =

1 0 0 00 1 0 00 0 1 0

0 0 0 1

2

Second order: via Εv = 1 : Ε ® Εv = Εt 1 Þ 1 =

1110

Deviatoric projection: Εdev = II -1

31 1tM Ε = Idev ΕÞ

Idev =

2

3-

1

3-

1

30

-1

3

2

3-

1

30

-1

3-

1

3

2

30

0 0 0 1

2

Algorithmic tangent: Ea ® Ea = Ee - 2 G J 2 G

3 G+H

nΚ+Σy

Σetr Ν

tr HΝtrLt

+ Μ3 G

Σetr IdevN, Νtr =

3

2

Σdevtr

Σetr

where it appears that the elastic stiffness modulus tensor Ee = 2 G Idev + K 1 1t can be represented as

Ee = 2 G

2

3-

1

3-

1

30

-1

3

2

3-

1

30

-1

3-

1

3

2

30

0 0 0 1

2

+ K

1 1 11 1 11 1 10 0 0

0000

In view of the elastic stiffness properties

G =E

2 H1 + ΝL , K =E

3 H1 - 2 ΝLwe thus retrieve the well known matrix representation of the isotropic elastic behavior at plane strain

Σ11

Σ22

Σ12

=E

H1 + ΝL H1 - 2 ΝL1 - Ν Ν 0

Ν 1 - Ν 0

0 0 1

2H1 - 2 ΝL

Ε11

Ε22

Γ12

where the out-of-plane component was omitted in this represenation.

� Static virtual work - Nodal FE-forces

Internal virtual work Wi:

Wi = àW

Εt

Σ âW = utA

e=1

NEL àWe

Bet

Σ âW = ut b with b = A

e=1

NELbe and be = à

We

Bet

Σ âW

8be<e=1,NEL =internal nodal forces

External virtual work We:

4 L13.nb

We = àW

f × u âW + àSt × u âS = ut

A

e=1

NELfe = ut f with f = A

e=1

NELfe

8fe<e=1,NEL=applied element nodal forces.

Equilibrium, i.e. Wi = We Þ g = b@DuD - f = 0

Ý Assembly operator Ae defines topology between ue and u, e.g. ue = Ae u.

� Linearized Static virtual work - Stiffness relationships

Consider the linearization

dg = A

e=1

NEL àWe

Bet d Σ âW = 8d Σ = Ea d Εe = Ea Be due< = A

e=1

NELA

e=1

NELKe du = K du

K = ¶b � ¶u =tangent stiffness; Ke =element stiffness

Ke = àWe

Bet Ea Be âW

Ý Also the algorithmic stiffness tensor Ea is formulated in Voight form.

� Solution procedure - Newton-Raphson's method

Solve for the balance between internal and external finite element nodal forces:

g@uD = b@uD - f@tD = 0

Þ Newton iteration procedure: given uHiL; Compute improved solution uHi+1L:

uHi+1L = uHiL + Ξ

Ξ is the iterative improvement computed from

ut Hg + K ΞL = 0 Þ Ξ = -K-1 g

L13.nb 5