1. 1/20 The virtual fields method for elasto- plastic behaviour Dr. Stéphane AVRIL

2. 2/20 Measurement of displacement fields

3. 3/20 Displacement fields available all along the test

4. 4/20 Reconstruction of strain fields all along the test

5. 5/20 Assume constitutive equations, you can get the stresses everywhere σ ε ε1ε1 ε2ε2 ε3ε3 ε4ε4 ε5ε5 σ5σ5 σ4σ4 σ3σ3 σ2σ2 σ1σ1 σ=g(ε,X)

6. 6/20 Reconstruction of stresses fields all along the test

7. 7/20 Are the stresses at equilibrium? At each measurement step, the following equation should be satisfied:

8. 8/20 Principle of the identification Iterative approch for reconstructing the stress fields until cost function J is minimized:

9. 9/20 Choice of virtual fields in practice Tension: ε xx * = 0 ε yy * = 1 ε xy * = 0 Shear: ε xx * = 0 ε yy * = 0 ε xy * = 1 L L

10. 10/20 Graphical display: example in plasticity

11. 11/20 Recursive algorithm: At the beginning, σ=0 Then, from one step to another: Elastic properties identified during elastic regime with the virtual fields method Irreversible strains: need a model Implicit definition of constitutive equations

12. 12/20 Associated plasticity Von Mises criterion: Prandtl-Reuss rule with isotropic hardening: Implicit definition of constitutive equations N is the tensor of yield flow directions: if σ eq < Y(p) if σ eq = Y(p)

13. 13/20 Introduction of constitutive parameters in the hardening law Bilinear model: Y(p) = Y 0 + H p Power model (JohnsonCook): Y(p) = Y 0 + α p n 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.

14. 14/20 Application onto experimental data Standard with strain gageStatically undetermined

15. 15/20 Comparison and validation

16. 16/20 Complexity of loading path has been handled

17. 17/20 Application on a heterogeneous specimen (with Prof. M. Sutton, USC) Zone of FSW

18. 18/20 Measured strain fields subimages = different time steps

19. 19/20 Experimental results on a homogeneous specimen

20. 20/20 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