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The virtual fields method for characterizing nonlinear behavior

Dr. Stéphane AVRIL

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Outline

General principle

Damage of composites

Elasto-visco-plasticity of metals

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Outline

General principle

Damage of composites

Elasto-visco-plasticity of metals

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Measurement of displacement fields

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Displacement fields available all along the test

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Reconstruction of strain fields all along the test

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Assume constitutive equations, you can get the

stresses everywhere

σ

εε1 ε2 ε3 ε4 ε5

σ5σ4σ3σ2

σ1 σ=g(ε,X)

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Reconstruction of stresses fields all along the test

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Are the stresses at equilibrium?

0: ** V

ii

V

ijij dSuTdV

At each measurement step, the following equation should be satisfied:

0:),( ** V

ii

V

ij dSuTdVXg

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Principle of the identification

2

**:),()(

stepsall V

ii

V

ij dSuTdVXgXJ

Iterative approch for reconstructing the stress fields until cost function J is minimized:

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Choice of virtual fields in practice

Tension:

εxx* = 0

εyy* = 1

εxy* = 0

Shear:

εxx* = 0

εyy* = 0

εxy* = 1

2

1)(

stepsall S

yy eS

FLdV

SXJ

2

1)(

stepsall S

xy eS

FLdV

SXJ

L

L

2)()( stepsall

loadyy

kinyy XXJ 2)()(

stepsall

loadxy

kinxy XXJ

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Graphical display: example in plasticity

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Outline

General principle

Damage of composites

Elasto-visco-plasticity of metals

14/382D Sic/Sic composite (Camus, IJSS, 2000)

In-plane shear non linearity

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Glass epoxy laminated plate

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Tensile tests

Shear tests

non linear response

ThesholdLinear part

Results of standard tests

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Shear response:

Results of standard tests

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3 different tests

K

G

E

E

s

xy

xy

yy

xx

0

Large scattering

Shortcomings of standard tests

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Inverse problem

1 test

K

G

E

E

s

xy

xy

yy

xx

0

Virtual fields method

Heterogeneous stress fieldDisplacement field measured by full-field optical technique

New strategy

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Principle of the Virtual Fields Method

Equilibrium equation

Principle of virtual work : 0** VV

ijij dSTudV 0)(*

2

* LPudVe y

S

ijij

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BAQ

Use of four independent virtual fields

e

LPudxdyQ

dxdyQdxdyQdxdyQ

y

S

ssss

S

xyyxxy

S

yyyy

S

xxxx

)(

)(

**0

****

2

222

BAQ 1

0,,, xyxyyyxx GEE

During the linear response

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xu ; 0u *y

*x

across S2

1 ; 0 ; 0 *s

*y

*x

Use of only one virtual field: uniform shear.

y

Beyond the onset of nonlinearity

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0dSTudVV

*

V

*ijij

S

sss dSQe

e

PLdxdyKdxdyQ

S

sss

S

sss

22

00

Damage induces material heterogeneities.

PTV :

S

sss

S

sss dSKdSQe 00

Identification of parameters driving the nonlinear behaviour

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2

0

S

sss dxdyQe

PL

2

0

S

sss dxdy

K

Threshold identified for the best alignment of data points

Graphical interpretation

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Gxy

« dissipated » virtual work

Using real measurements

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K 0ss

Reference 87,7 GPa 0,006

Coeff. var (%) 12,8 33

Identified 83,6 GPa

8,4

0,0041

14,2Coeff. var (%)

Results

Chalal H., Avril S., Pierron F. and Meraghni F., Experimental identification of a damage model for composites using the grid technique coupled to the virtual fields method, Composites Part A: Applied Science and Manufacturing, vol. 37, n° 2, pp. 315-325, 2006.

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Identification of a model with six parameters in one single test

Decrease of result scattering thanks to the full-field measurements

Prospects: handling more complex models taking into account coupling effects and plasticity.

Summary

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Outline

General principle

Damage of composites

Elasto-visco-plasticity of metals

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elastxy

elastyy

elastxx

xy

yy

xxE

100

01

01

1 2

plastxy

measuredxy

plastyy

measuredyy

plastxx

measuredxx

xy

yy

xxE

100

01

01

1 2

Recursive algorithm:

At the beginning, σ=0Then, from one step to another:

Elastic properties identified during elastic regime with the virtual fields method

Irreversible strains: need a model

Implicit definition ofconstitutive equations

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xy

yy

xx

plastxy

plastyy

plastxx

N

N

N

p

Associated plasticityVon Mises criterion:

Prandtl-Reuss rulewith isotropic hardening:

Implicit definition ofconstitutive equations

)('::

::

0

pYNCN

CNp

p

eq

SN

2

3

N is the tensor of yield flow directions:

if σeq < Y(p)

if σeq = Y(p)

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Introduction of constitutive parameters in the hardening law

Bilinear model: Y(p) = Y0 + H p

Power model (JohnsonCook): Y(p) = Y0 + α pn

If the parameters are chosen, it is possible reconstruct the stress fields and to test their validity.

Iterative up to the minimization of the cost function.

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Application onto experimental data

Standard with strain gage Statically undetermined

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Comparison and validation

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Complexity of loading path has been handled

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Application on a heterogeneous specimen (with Prof. M. Sutton, USC)

Zone of FSW

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Measured strain fields

subimages = different time steps

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Experimental results on a homogeneous specimen

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Identification of model with up to 21 parameters in one single test.

Possibility of handling complex specimen geometry and strain localization.

Prospects:

1. Viscoplasticity with high-speed cameras (S53, p156)

2. Kinematic hardening, more complex loading paths

3. Software implementation with Camfit

Summary