VARIATIONAL FORMULATION OF THE STRAIN LOCALIZATION PHENOMENON GUSTAVO AYALA

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

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

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

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

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

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

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

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

FORMULATION WITH EMBEDDED DISCONTINUITIES ENERGY FUNCTIONAL where b b b

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

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

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

AC Energy functional of the continuum where b) Strong discontinuity

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

FORMULATION Discrete Approach Potential Energy Functional where

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

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

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

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

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

MIXED MATRICES Discrete Approach Continuum Approach on S

DISPLACEMENT FEM Interpolation of fields Stiffness matrix Continuum Approach Discrete Approach

FORCE FEM Interpolation of fields Flexibility matrix Continuum Approach Discrete Approach

TENSION BAR PROBLEM Properties Geometry

MATRICES FOR THE LINEAR ELEMENT Stiffness Flexibility Mixed

RESULTS Load-displacement diagram Stress-jump diagram

2D IDEALIZATION

RESULTS Load – displacement diagram Stress – Jump diagram

EVOLUTION TO FAILURE

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.

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