1. 1/38 The virtual fields method for characterizing nonlinear behavior Dr. Stéphane AVRIL

2. 2/38 Outline General principle Damage of composites Elasto-visco-plasticity of metals

3. 3/38 Outline General principle Damage of composites Elasto-visco-plasticity of metals

4. 4/38 Measurement of displacement fields

5. 5/38 Displacement fields available all along the test

6. 6/38 Reconstruction of strain fields all along the test

7. 7/38 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)

8. 8/38 Reconstruction of stresses fields all along the test

9. 9/38 Are the stresses at equilibrium? At each measurement step, the following equation should be satisfied:

10. 10/38 Principle of the identification Iterative approch for reconstructing the stress fields until cost function J is minimized:

11. 11/38 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

12. 12/38 Graphical display: example in plasticity

13. 13/38 Outline General principle Damage of composites Elasto-visco-plasticity of metals

14. 14/38 2D Sic/Sic composite (Camus, IJSS, 2000) In-plane shear non linearity

15. 15/38 Glass epoxy laminated plate

16. 16/38 Tensile tests Shear tests non linear response Theshold Linear part Results of standard tests

17. 17/38 Shear response: Results of standard tests

18. 18/38 3 different tests Large scattering Shortcomings of standard tests

19. 19/38 Inverse problem 1 test Virtual fields method Heterogeneous stress field Displacement field measured by full- field optical technique New strategy

20. 20/38 Principle of the Virtual Fields Method Equilibrium equation Principle of virtual work :

21. 21/38 Use of four independent virtual fields During the linear response

22. 22/38 across S2 Use of only one virtual field: uniform shear. y Beyond the onset of nonlinearity

23. 23/38 Damage induces material heterogeneities. PTV : Identification of parameters driving the nonlinear behaviour

24. 24/38 K Threshold identified for the best alignment of data points Graphical interpretation

25. 25/38 G xy « dissipated » virtual work Using real measurements

26. 26/38 Reference 87,7 GPa0,006 Coeff. var (%) 12,833 Identified 83,6 GPa 8,4 0,0041 14,2 Coeff. 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.

27. 27/38 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

28. 28/38 Outline General principle Damage of composites Elasto-visco-plasticity of metals

29. 29/38 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

30. 30/38 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)

31. 31/38 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.

32. 32/38 Application onto experimental data Standard with strain gageStatically undetermined

33. 33/38 Comparison and validation

34. 34/38 Complexity of loading path has been handled

35. 35/38 Application on a heterogeneous specimen (with Prof. M. Sutton, USC) Zone of FSW

36. 36/38 Measured strain fields subimages = different time steps

37. 37/38 Experimental results on a homogeneous specimen

38. 38/38 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