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

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

5. CA-Weak Discontinuity in ΩKinematical compatibility in ΩConstitutive compatibility in ΩInternal equilibrium on  σ External equilibrium on Ω h Outer traction continuity on Ω h Inner traction continuity

6. CA-Strong Discontinuity in ΩKinematical compatibility in ΩConstitutive compatibility in Ω\SInternal equilibrium on  σ External equilibrium on S Outer traction continuity on SInner traction continuity

7. Discrete Approach in Ω\S Kinematical compatibility in Ω\SConstitutive compatibility in Ω\SInternal equilibrium on  σ External equilibrium on S Outer traction continuity on SInner traction continuity

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

9. FRAEIJS DE VEUBEKE (1951) Through That is Find the fields in Ω Kinematical compatibility in Ω Constitutive compatibility in Ω Internal equilibrium on   on  u on  u Essential BC Satisfying External equilibrium

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

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

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

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

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

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

16. FORMULATION Discrete Approach Potential Energy Functional where

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

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

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

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

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

22. MIXED MATRICES Discrete Approach Continuum Approach on S

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 Stiffness Flexibility Mixed

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.