Static Plane-Symmetric Nonlinear Spinor and Scalar Fields in GR
The virtual fields method for characterizing nonlinear behavior
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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
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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