average stress (MPa)

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Dr. Stéphane AVRIL

LMPF Research Group, ENSAM Châlons en Champagne, France

THE VIRTUAL FIELDS METHOD

PRINCIPLE, RECENT ADVANCES AND APPLICATIONS

Paris

Châlons en Champagne 12th April 2007, Université de Technologie de Compiègne

OUTLINE

• Introduction and basic concepts;

• Application 1: anisotropic elasticity in composites;

• Application 2: visco-elasticity in PMMA;

• Application 3: elasto-visco-plasticity in metals;

• Application 4: heterogeneous modulus distribution;

• Conclusion and main prospects.

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Introduction and

basic concepts

Widespread use of full-field optical techniques in

experimental mechanics

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Largely used for qualitative

analyses

Largely used for qualitative

analyses

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Largely used for qualitative analyses

– Detection of shear bands or cracks ; – Model validation ;

– Verification of boundary conditions.

Interest in quantitative use for

model identification

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Interest in quantitative use for

model identification

– Identification of sophisticated models in one single test ;

– Specimen geometry is not a constraint ; – Localization effects can be handled ;

– Identification of graded properties in the same specimen.

Interest in quantitative use for

model identification

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- No standard characterization of uncertainty in full- field optical measurement techniques.

- Large amount of data to process require specific inverse approach.

Reason for poor use

for model identification

Resolution of an inverse problem Basic approach: updating

Parameters A_i

Model: f

Response M_j=f(A_i)

Measurements m_j

Matching optimization

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Particular case of full-field

measurements: alternative approach

Measurements m(x,y,z)

Direct

computation F(A_i,m)=0

A_i

Model f is derived from:

- the constitutive equations, - the conservation equations.

F(A_i,M)=0 Æ M=f(A_i)

THE VIRTUAL FIELDS METHOD

Application 1: anisotropic

elasticity in composites

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* 0

* + =

−

∫ ∫

∂V

i i V

ij

ij ε dV T u dS

σ or

I Equilibrium equations

, _j = 0

σ

III Kinematical equations (small strains/displacements)

) u u

2 ( 1

i, j j

, i

ij

= +

ε

The virtual fields method Example in linear elasticity

kl ijkl

ij

C ε

σ =

0 dS

u T dV

V

* i i V

* ij

ij

ε + =

σ

− ∫ ∫

∂

Eq. I (weak form, static)

Substitute stress from Eq. II

*

0

*

+ =

− ∫ ∫

∂V

i i ij

kl

ijkl

dV T u dS

C ε ε

kl ijkl

ij

C ε

σ =

The virtual fields method

Example in linear elasticity

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Equation valid for any KA virtual fields.

For each choice of virtual field: 1 equation.

Choice of as many VF as unknown parameters.

In elasticity:

Æ Linear system of equations to be inverted.

Æ Optimal choice of virtual fields for minimizing result uncertainty (Avril et al., Comp. Mech., 2004)

Æ More robust than updating approach (Avril et al., Int. J.

Sol. Struct., 2007)

*

0

*

+ =

− ∫ ∫

∂V

i i V

ij kl

ijkl

dV T u dS

C ε ε

The virtual fields method

Example in linear elasticity

Problem presentation

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Glass epoxy ring cut from a tube:

Example

θ

r

x y

F

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

= ⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

s r

ss r

r rr

s r

Q Q

Q

Q Q

ε ε ε σ σ σ

θ θθ

θ

θ

θ

0 0 0 0

Æ 4 parameters to identify Æ No standard test available

Need of two cameras

“back to back”

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Deformation maps

using two cameras

Polynomial fit, degree 3, transform to cylindrical and analytical differentiation:

Strain maps

ε_r ε_θ ε_s

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( )

dS M

u M T

dS e

Q

dS e

Q dS

e Q

dS e

Q

f

S s s ss

S r r

r r S

S r rr

) (

* )

( .

. .

.

*

*

*

*

*

∫

∫ ∫ ∫

∫ + =

+ +

+

S

r r

ε ε

ε ε ε

ε ε

ε ε

ε

[ ] ^{A × ( ) ( ) Q = B}

4 virtual fields

dS M

u M T

dS

f

) (

* ) (

* ∫

∫

: ⁼

e r r

ε σ

PVW Constitutive equations

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

= ⎡

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

s r

ss r

r rr

s r

Q Q

Q

Q Q

ε ε ε σ σ σ

θ θθ

θ

θ

θ

0 0 0 0

&

Identification

Qrr Q_θθ Q_rθ Q_ss

Reference* (GPa) 10 40 3 4

Identified (GPa) 11.4 45.4 2.62 6.78

Coeff. var (%) – 9 tests 29 10 29 4

Moulart R., Avril S., Pierron F., Identification of the through-thickness rigidities of a thick laminated composite tube, Composites Part A: Applied Science and Manufacturing, vol. 37, n° 2, pp. 326-336, 2006.

Results

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Application 2: visco-elasticity in

polycarbonate

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

− +

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

−

=

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

xy yy xx

i xy i

xx i

xx i

xy

i xy i

xx

xy yy xx

r xy r

xx r

xx r

xy

r xy r

xx

xy yy xx

Q Q

Q Q

Q Q

i Q

Q Q

Q

Q Q

ε ε ε ω

ε ε ε σ

σ σ

0 0

0 0 0

0

0 0

)]

, ( cos[

) , ( )

, (

) cos(

) , ( )

, (

y x t

y x y

x

t y

x y

x

ϕ ω

ε ε

ω σ

σ

+

=

=

)]

, (

) [

, ( )

, (

) , ( )

, (

y x t

i t i

e y x y

x

e y x y

x

ϕ ω

ω

ε ε

σ σ

= +

=

Constitutive equations and assumptions

Inertial excitation with steady state and linear response.

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Experimental arrangements

Polycarbonate panel 200mm x 150mm h = 3mm

CCD

O

M P

δ Q

λ

dα

δ = 2λ . dα

The deflectometry technique

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Out of resonance 80 Hz Near resonance 100 Hz

Slope measurements

Out of resonance 80 Hz Near resonance 100 Hz

Deduced curvatures

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⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎜⎜

⎝

⎛

⎥⎥

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢⎢

⎢

⎣

⎡

− +

⎥⎥

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢⎢

⎢

⎣

⎡

−

⎪ =

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

s y x

xy xx

yy xy

xy xx

xy xx

yy xy

xy xx

s y x

k k k B

B B

B

B B

i D

D D

D

D D

M M M

0 2 0

0 0

0 2 0

0 0

ω

xy xy xy

xx xx xx

xx xy 2

xx

D ; B

D B

D ; D

) 1

( E D

= β

= β

= ν ν

= −

Four parameters to identify

Stiffness Damping

( ^j ) ^{VF y} ( ^j )

x

VF

=

1 + ,

=

1 +

Identification

E (GPa)

3 3,2 3,4 3,6 3,8 4 4,2 4,4 4,6 4,8 5

80 Hz 90 Hz 100 Hz Reference

Results

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ν

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

80 Hz 90 Hz 100 Hz

Results

β_xx (10^-4 s)

0 0,2 0,4 0,6 0,8 1 1,2

80 Hz 90 Hz 100 Hz Reference

Results

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β_xy (10^-4 s)

0 0,2 0,4 0,6 0,8 1 1,2 1,4

80 Hz 90 Hz 100 Hz

Results

Giraudeau A., Guo B., Pierron F., Stiffness and Damping Identification from Full Field Measurements on Vibrating Plates, Experimental Mechanics, Vol. 46, N°6, pp. 777-787, 2006.

- Improvement of the measurement temporal accuracy.

- Extension to anisotropic plates underway

Prospects

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Application 3: elasto-visco-

plasticity of metals

Constitutive equations and assumptions

) ,

,

( X

g σ ε

σ _& = _&

Constitutive parameters to identify

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Elasto-plasticity with plane stress and Von Mises criterion

⎪ ⎪

⎪

⎭

⎪ ⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪ ⎪

⎪

⎨

⎧

−

−

−

⎥ ⎥

⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎣

⎡

− −

⎪ =

⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

s xy s

s xy y

s xx x

s y x

p s p s p s

E

ε σ ε σ ε σ

ν ν ν

ν σ ν

σ σ

&

&

&

&

&

&

&

&

&

3 2 3 2 3

2 0 1

0

0 0 1

1

)

( ^p

Q ε ε

σ & = −

Perzyna’s model for viscoplasticity

⎪⎪

⎪⎪

⎭

⎪⎪

⎪⎪

⎬

⎫

⎪⎪

⎪⎪

⎩

⎪⎪

⎪⎪

⎨

⎧

− −

−

− −

−

− −

−

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

− −

⎪ =

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

s xy n s s

s xy n y s

s xx n x s

s y x

p s H

p s H

s p

H

E

σ σ

γ σ ε

σ σ

γ σ ε

σ σ

γ σ ε

ν ν ν

ν σ ν

σ σ

) 1 3 (

) 1 ( 2

3

) 1 ( 2

3

2 0 1

0

0 0 1

1

0 0 0

2

&

&

&

&

&

&

Constitutive parameters to identify: E, ν, γ, σ , n, dH/dp.

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Mechanical arrangement

L=28mm

x y

w=22mm

20 mm

Average strain rate: 1s^-1 Images of the specimen recorded with high speed cameras

Deformation deduced by digital image correlation.

Fields of strain rate

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Virtual field

y u

u ^* _x = 0 ; ^* _y = −

Across the measurement area:

0 ;

1 ;

0 ^{* *}

* = _yy = _xy =

xx ε ε

ε

Identification

( ) ∑ ∫∫

= ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

= ^N

l

l S

t

t y

t Sh

L t dtdS P

t y x E S n

E n

F

l

1

2

0

0 0

) ) (

, , , , , , 1 (

, ,

, γ σ σ γ

σ _&

Æ minimization of a cost function defined from the principle of virtual work:

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Identification

0 5 10 15 20 25

0 50 100 150 200 250 300 350 400

time (ms)

average stress (MPa)

from displacement fields from resultant load

- Numerical issues

- Larger strain rates Æ Hopkinson

- Heterogeneities: microscopic scale, welds...

Prospects

Pannier Y., Avril S., Rotinat R., Pierron F., Identification of elasto-plastic constitutive parameters from statically undetermined tests using the virtual fields method, Experimental Mechanics, Vol. 46, N°6, pp. 735-755, 2006.

Avril S., Pierron F., Yan J., Sutton M., Identification of viscoplastic parameters using DIC

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Heterogeneous modulus

distribution

*

0

*

+ =

− ∫ ∫

∂V

i i V

ij kl

ijkl

dV T u dS

C ε ε

The virtual fields method for heterogeneous solids in linear

elasticity

*

0

*

+ =

− ∑ ∫ ∫

∂V

i i V

ij kl n

n

ijkl

dV T u dS

C

n

ε

ε

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V₁ V₂ V₃

V_n

Discretization of the solid

Cube with a stiff inclusion buried in it.

Silicone gel materials mimicking human tissue containing a tumour.

Experimental arrangements

1/8

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Displacement fields measured by MRI

Scanning tomographic method:

Æ 3D bulk measurements!!

Principle of the identification

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3D Results

Avril S., Huntley J., Pierron F. and Steele D., 3D-Heterogeneous stiffness

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Conclusions

- Huge potential for some difficult and important engineering issues (heterogeneous materials, high strain rate, micro-scale…);

- Training required on optical FFMT: not a black box, primary influence on identification results

- Full-field processing: noise filtering, displacements to strains…

Prospects

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Measurements m(x,y,z)

KEY POINT F(A_i,m)=0

A_i

Key point:

field reconstruction

Model f is derived from:

- the constitutive equations, - the conservation equations.

F(A_i,M)=0

Comparison of two approaches

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Comparison of two approaches

Comparison of two approaches

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…

Thank you for attention

…