VARIATIONAL FORMULATION OF THE STRAIN LOCALIZATION

PHENOMENON

GUSTAVO AYALA

OBJECTIVE

To develop a variational formulation of the strain

localization phenomenon, its implementation in a

FE code, and its application to real problems.

MATERIAL FAILURE THEORIES

2. Discrete Approach

1. Continuum Approach

Fracture process zone is concentrated along a crack

Based on a traction-displacement relationship

Inelastic deformations are concentrated over

narrow bands

Based on a stress-strain relationshipStrong Discontinuity

Approach

Variation of displacement, strain and stress fields

2. DISCRETE APPROACH (DA)

1. CONTINUUM APPROACH (CA)

x

x

x

x

x

x

[ u ]

u

k=0

k=0

k

[ u ]

[ u ]

= [ u ]1k

S

= [ u ]s

x

x

x

S S

Weak discontinuity

Strong discontinuity

a) b)

- n

h

n

k=0

+-

h

k=0

+

SS

-

+i-

i+

-

+

+-

k=0

n -+

k=0

S

SS

i-

i+

-+

a) b)

+- Sn

+

S- S+

--+

CA-Weak Discontinuity

in Ω Kinematical compatibility

in Ω Constitutive compatibility

in Ω Internal equilibrium

on σ External equilibrium

on Ωh Outer traction continuity

on Ωh Inner traction continuity

\

0

( ) 0

0

0

0S

S

S

u

n

n

b

t

n n =

n n

a) b)

- n

h

n

k=0

+-

h

k=0

+

SS

-

+i-

i+

-

+

CA-Strong Discontinuity

in Ω Kinematical compatibility

in Ω Constitutive compatibility

in Ω\S Internal equilibrium

on σ External equilibrium

on S Outer traction continuity

on S Inner traction continuity

\

0

( ) 0

0

0

0S

S

S

u

n

n

b

t

n n =

n n

+-

k=0

n -+

k=0

S

SS

i-

i+

Discrete Approach

in Ω\S Kinematical compatibility

in Ω\S Constitutive compatibility

in Ω\S Internal equilibrium

on σ External equilibrium

on S Outer traction continuity

on S Inner traction continuity

\

0

( ) 0

0

0

0

u

n

n

b

t

n n =

nS

S

ST

-+

a) b)

+- Sn

+

S- S+

--+

ENERGY FUNCTIONAL BY FRAEIJS DE VEUBEKE (1951)

ENERGY FUNCTIONAL OF THE LINEAR ELASTIC PROBLEM

V u, , , t : b u t u t u uu

u W d d d

V u, , , t U P D

1: :

2W C

u

U W d

b u t uP d d

t u u :u

uD d d

t

u

b

FRAEIJS DE VEUBEKE (1951)

Through

That is

, , , 0 V u t

: : :

0u u

u d d d

v d v d d

V b u

t u t u u u t

Find the fields

, , , , , ,t t t and tu x x σ x t x

0

( ) 0

0

u

u u

b

t

t

in Ω Kinematical compatibility

in Ω Constitutive compatibility

in Ω Internal equilibrium

on

on u

on u Essential BC

Satisfying

External equilibrium

FORMULATION WITH EMBEDDED DISCONTINUITIES

V

V

V

: b u t u t u u

: b u t u t u u

: b u t u t u u

u

h

h h h

u

u

u

u

u

W d d d

W d d d

W d d d

ENERGY FUNCTIONAL

V V V Vu, , , th

where

n

i-

-

i

(2)(1)

h

n

+

+

tt

t

b b b

VARIATION

0

( ) 0

0

u

u u

b

t

t

0

0

hi

hi

1 1 1

2 2 2

n n n =

n n n

,

,

,

h

h

u u u

h

u u u

y

y

y

First variation

, , , 0 V u tSatisfying

in Ω-, Ωh y Ω+

in Ω-, Ωh y Ω+

in Ω-, Ωh y Ω+

on

on

on

and

i

i

on

on

APPROXIMATION BY EMBEDDED DISCONTINUITIES

Functional energy of the continuum

where

Continuum Approach a) Weak discontinuity

\V \ V V V V V, , , , ,

S SSS S

u u

\V

\

: b u t u u u tu

S u

S

W d d d

V : b uuh

S

k W d

a) b)

- n

h

n

k=0

+-

h

k=0

+

SS

-

+i-

i+

-

+

WEAK DISCONTINUITIES

\ \ \

: : :

: : :

0

V

u

b u t u

b u t u

u u t n u n u

u

u

u

S S S

S S S

S S

S S

d d d v d

k d k d k d v d

d d d

First variation

Satifying

\

0

0

S S

S S

S

n n n =

n n n

n n = n = 0

.

.

in \S and on S

on S Inner traction continuity

on S Outer traction continuity

Compatibility

Equilibrium

AC

V : uS

S

W d

Energy functional of the continuum

where

b) Strong discontinuity

\V

\

: b u t u u u tu

S u

S

W d d d

\V \ V V, , , , ,

SSS S

u u

+-

k=0

n -+

k=0

S

SS

i-

i+

STRONG DISCONTINUITY

\ \ \

: : :

: :

0

u

u

u

S S S

u

S S

S S

S S

d d d v d

d d v d

d d d

V b u t u

t u

u u t n u n u

First Variation

SatisfyingCompatibility

Equilibrium

\

0S

S

n

n = 0

in \S and on S

on S Inner traction continuity

on S Outer traction continuity

.

.

FORMULATION

Discrete Approach

Potential Energy Functional

where

\, , , Sd V V Tu t, u

d

S

d T T u

\V

\

: b u t u t u uu

S u

S

W d d d

-+

a) b)

+- Sn

+

S- S+

--+

AD

\ \ \

-

: : :

0

u u

u

S S S

S S S S

S S

d d d

v d v d d

T d T d

V b u

t u t u u u t

n u n u

First variation

Satisfying

\

0

0

S S

S S

S

T

T

n n =

n n

n n = n = 0

.

.

in \S y on S

on S Inner traction continuity

on S Outer traction continuity

Compatibility

Equilibrium

SUMMARY OF MIXED ENERGY FUNCTIONALS

Continuum Approach

VM \ \

\

, , , , , : ( )

: ( )

uS S S

S

S

S

W d d

W dS

u

u u b u t u

VM \ \

\

ˆ, , , : ( )uS S

S

S

W d d

T dS

T u u b u t u

u

Discrete Approach

VM \ \

\

, , , , , : ( )

: ( )

uS S S

S

S

S

W d d

k W dS

u

u u b u t u

b u

b) Strong discontinuity

a) Weak discontinuity

TOTAL POTENTIAL ENERGY FUNCTIONALS

VM

\

, ( ) ( )S S

W d d k W dS

uuu u b u t u

ˆ

u

u

Continuous Approach

VM

\

, ( ) ( )S S

W d d W dS

uuu u b u t u

VM

\

, ( )S S

W d d T dS

uT u u b u t u u

Discrete approach

b) Strong discontinuity

a) Weak discontinuity

Conditions satisfied a priori

u

u

u u

in \S

in \S

in u

on S

on S

TOTAL COMPLEMENTARY ENERGY FUNCTIONALS

\ \

\

, ( ) ( )u

S S S S

S S

W d d k W dS

VM n u

Continuous Approach

VM \ \

\

, ( ) ( )u

S S S S

S S

W d d W dS

n u

VM \ \

\

, ( )u

S S S S

S S

W d d dS

T n u n u

Discrete approach

b) Strong discontinuity

a) Weak discontinuity

Where\ \ \ \

1( ) : :

2S S S SW D

Conditions satisfied a priori

σ + b = 0

σ t

in \S

on σ

1. MIXED FEM

VM VM VM

1 2

0; 0; ... 0nu u u

VM VM 0 For to be stationary

Interpolation of fields

\

ˆ

· ·

\

( , ) ( ) ( )

ˆ( , ) ( ) ( )

( , ) ( ) ( )S S

c

S S

t

t

t

ee

u x N x d N x u

x N x e N x e

x N x N x

\

ˆ

·

\

( , ) ( ) ( )

ˆ( , ) ( )

( , ) ( )S

c

S

t

t

t

e

u x N x d N x u

x N x e

x N x

( )

1

k

u

u

d -B x u

u n

Dependent fields

CA DA

MIXED MATRICES

\

\

\

\ \\

ˆ ˆ ˆ

ˆ ·

\

·

ˆ

S

S S

S

S

S SS

SS

S

S

K

d

ext

u u

ee e

ee e

d eu

eu

d0 0 0 0 K 0F

0 0 0 0 K K u0

0 0 K 0 K 0 e 00 0 0 K 0 K 0e

K K 0 0 0 0

00 K 0 K 0 0

Discrete Approach

Continuum Approach

\

\

\

\ \\

ˆ ˆ ˆ

ˆ ·

\

ˆ

S

S

S

S SS

S

S

K

d ext

u

ee e

d eu

d0 0 0 K F0 0 0 K u

F0 0 K K

0eK K 0 0

u on S

DISPLACEMENT FEM

S

F

Interpolation of fields

Stiffness matrix

Continuum Approach

Discrete Approach

( ) ( ) Cu d +N x N x u

\ \

\ \

S S

S

S S

d d

d d

T TC

ext

T TC C C

B C B B C BFd

uB C B B C B F

S

S

S

S

d

d

n

T

1S

S

nk

u

T u

FORCE FEM

SFF

Interpolation of fields

Flexibility matrix

Continuum Approach

Discrete Approach

\\

· ·

( , ) ( ) ( )S SS St

x N x N x

\ \ \ \ \ \

\ \ \

·

\\ \

·

\ \

S S S S S S S

u

S S S S S S

SS S

S SS S

d d d

d d

T T T

T

N D N N D N n N u

N D N N D NFF

S

S

S

d

d

n u

:

:

CS S

S

DS S

S

d

d

D

n D

TENSION BAR PROBLEM

2

1000

1

0.005 /

2.0

1.0

u

E MPa

f MPa

G MN m

L m

A m

Properties

Geometryu

y

x

1

L

E, A2

2f

d2d1

MATRICES FOR THE LINEAR ELEMENT

1 1

2 2

11 1

1 1

L f dHf dEA

StiffnessFlexibility

1 1

2 2

1 1

1 1 1

d fEA

d fLH

1 1

2 2

1

2

1

2

0 0 0 0 02 2

0 0 0 0 02 2

0 0 0 0 02 2

00 0 03 6

00 0 0 06 3

00 0

2 2 2 3 6

0 02 2 2 6 3

A A

A A

d fA A d f

u EAHEA EA AL AL

eL L

eEA EA AL AL

L LA A A AL AL

A A A AL AL

Mixed

RESULTS

Load-displacement diagram Stress-jump diagram

0.002 0.004 0.006 0.008 0.010

0.2

0.4

0.6

0.8

1.0

00

[ u ] (m)

(

MP

a)

[ u ] =HEmáx

B

C

D

S

u

Gf

0.002 0.004 0.006 0.008 0.010

0.2

0.4

0.6

0.8

1.0

00

d (m)

P (

MN

)

B

A

C

D

Gf

2D IDEALIZATION

P

x

y

L

h

P

x

y

L

h

RESULTS

Load – displacement diagram Stress – Jump diagram

0.002 0.004 0.006 0.008 0.010

0.2

0.4

0.6

0.8

1.0

00

d (m)

P (

MN

)

0.002 0.004 0.006 0.008 0.010

0.2

0.4

0.6

0.8

1.0

00

[ u ] (m)

(M

Pa)

[ u ] = HEmáx

S

u

2

u

EVOLUTION TO FAILURE

CONCLUSIONS

A general variational formulation of the strain localization phenomenon and its discrete approximation were developed.

With the energy functionals developed in this work, it is possible to formulate Displacement, Flexibility and Mixed FE matrices with embedded discontinuities.

The advantage of this formulation is that the FE matrices are symmetric, with the stability and convergence of the numerical solutions, guaranteed at a reduced computational cost.

There is a relationship between the CA and DA in the Strong Discontinuity formulation not only in the Damage models, but also in their variational formulations.

FUTURE RESEARCH

Implement 2 and 3D formulations in a FE with embedded discontinuities code to simulate the evolution of more complex structures to collapse.