VARIATIONAL FORMULATION OF THE STRAIN LOCALIZATION PHENOMENON

1. VARIATIONAL FORMULATION OF THE STRAIN LOCALIZATION PHENOMENON GUSTAVO AYALA

2. OBJECTIVE • To develop a variational formulation of the strain localization phenomenon, its implementation in a FE code, and its application to real problems.

3. MATERIAL FAILURE THEORIES 1. Continuum Approach • Inelastic deformations are concentrated over narrow bands • Based on a stress-strain relationship Strong Discontinuity Approach 2. Discrete Approach • Fracture process zone is concentrated along a crack • Based on a traction-displacement relationship

4. Variation of displacement, strain and stress fields 1. CONTINUUM APPROACH (CA) Weak discontinuity Strong discontinuity 2. DISCRETE APPROACH (DA)

5. CA-Weak Discontinuity in Ω Kinematical compatibility in Ω Constitutive compatibility in Ω Internal equilibrium on Gσ External equilibrium on Ωh Outer traction continuity on Ωh Inner traction continuity

6. CA-Strong Discontinuity in Ω Kinematical compatibility in Ω Constitutive compatibility in Ω\S Internal equilibrium on Gσ External equilibrium on S Outer traction continuity on S Inner traction continuity

7. Discrete Approach in Ω\S Kinematical compatibility in Ω\S Constitutive compatibility in Ω\S Internal equilibrium on Gσ External equilibrium on S Outer traction continuity on S Inner traction continuity

8. b ENERGY FUNCTIONAL BY FRAEIJS DE VEUBEKE (1951) ENERGY FUNCTIONAL OF THE LINEAR ELASTIC PROBLEM

9. FRAEIJS DE VEUBEKE (1951) Find the fields • Through • That is • Satisfying in Ω Kinematical compatibility in Ω Constitutive compatibility in Ω Internal equilibrium on Gs on Gu on GuEssential BC External equilibrium

10. b b b FORMULATION WITH EMBEDDED DISCONTINUITIES ENERGY FUNCTIONAL where

11. VARIATION • First variation • Satisfying in Ω-, Ωh y Ω+ in Ω-, Ωh y Ω+ in Ω-, Ωh y Ω+ on on on and on on

12. APPROXIMATION BY EMBEDDED DISCONTINUITIES • Continuum Approach a) Weak discontinuity Functional energy of the continuum where

13. WEAK DISCONTINUITIES • First variation • Satifying in W\S and on S on S Inner traction continuity on S Outer traction continuity • Compatibility • Equilibrium • . • .

14. AC • b) Strong discontinuity Energy functional of the continuum where

15. STRONG DISCONTINUITY • First Variation • Satisfying in W\S and on S on S Inner traction continuity on S Outer traction continuity • Compatibility • Equilibrium • . • .

16. FORMULATION • Discrete Approach • Potential Energy Functional where

17. AD • First variation • Satisfying in W\S y on S on S Inner traction continuity on S Outer traction continuity • Compatibility • Equilibrium • . • .

18. SUMMARY OF MIXED ENERGY FUNCTIONALS Continuum Approach a) Weak discontinuity b) Strong discontinuity Discrete Approach

19. TOTAL POTENTIAL ENERGY FUNCTIONALS Conditions satisfied a priori in W\S in W\S in Gu on S on S Continuous Approach a) Weak discontinuity b) Strong discontinuity Discrete approach

20. TOTAL COMPLEMENTARY ENERGY FUNCTIONALS Conditions satisfied a priori in W\S on Gσ Continuous Approach a) Weak discontinuity b) Strong discontinuity Discrete approach Where

21. 1. MIXED FEM • Interpolation of fields • DA • CA • Dependent fields • For to be stationary

22. on S MIXED MATRICES • Continuum Approach • Discrete Approach

23. DISPLACEMENT FEM • Interpolation of fields • Stiffness matrix • Continuum Approach • Discrete Approach

24. FORCE FEM • Interpolation of fields • Flexibility matrix • Continuum Approach • Discrete Approach

25. TENSION BAR PROBLEM Properties • Geometry

26. MATRICES FOR THE LINEAR ELEMENT • Mixed • Flexibility • Stiffness

27. RESULTS • Load-displacement diagram • Stress-jump diagram

28. 2D IDEALIZATION

29. RESULTS • Load – displacement diagram • Stress – Jump diagram

30. EVOLUTION TO FAILURE

31. CONCLUSIONS • A general variational formulation of the strain localization phenomenon and its discrete approximation were developed. • With the energy functionals developed in this work, it is possible to formulate Displacement, Flexibility and Mixed FE matrices with embedded discontinuities. • The advantage of this formulation is that the FE matrices are symmetric, with the stability and convergence of the numerical solutions, guaranteed at a reduced computational cost. • There is a relationship between the CA and DA in the Strong Discontinuity formulation not only in the Damage models, but also in their variational formulations.

32. FUTURE RESEARCH • Implement 2 and 3D formulations in a FE with embedded discontinuities code to simulate the evolution of more complex structures to collapse.

33. Thank you