The virtual fields method for elasto-plastic behaviour

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The virtual fields method for elasto-plastic
behaviour

• Dr. Stéphane AVRIL

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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
s
s5
s4
s3
s2
sg(e,X)
s1
e
e1
e2
e3
e4
e5
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Reconstruction of stresses fields all along the
test
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Are the stresses at equilibrium?
At each measurement step, the following equation
should be satisfied
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Principle of the identification
Iterative approch for reconstructing the stress
fields until cost function J is minimized
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Choice of virtual fields in practice
Shear
Tension
L
exx 0 eyy 0 exy 1
exx 0 eyy 1 exy 0
L
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Graphical display example in plasticity
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Implicit definition of constitutive equations
Recursive algorithm At the beginning, s0 Then,
from one step to another
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Implicit definition of constitutive equations
Associated plasticity Von Mises criterion
N is the tensor of yield flow directions
if seq lt Y(p)
Prandtl-Reuss rule with isotropic hardening
if seq 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 a 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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Summary

• Identification of model with up to 21 parameters
  in one single test.
• Possibility of handling complex specimen geometry
  and strain localization.
• Prospects
• Viscoplasticity with high-speed cameras (S53,
  p156)
• Kinematic hardening, more complex loading paths
• Software implementation with Camfit