EVALUACI

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 O Kinematical compatibility in O Constitutive
compatibility in O Internal equilibrium on
Gs External equilibrium on Oh Outer traction
continuity on Oh Inner traction continuity
6
CA-Strong Discontinuity
in O Kinematical compatibility in O Constitutive
compatibility in O\S Internal equilibrium on
Gs External equilibrium on S Outer traction
continuity on S Inner traction continuity
7
Discrete Approach
in O\S Kinematical compatibility in
O\S Constitutive compatibility in O\S Internal
equilibrium on Gs External equilibrium on S
Outer traction continuity on S Inner traction
continuity
8
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 O Kinematical compatibility in O Constitutive
compatibility in O Internal equilibrium on Gs on
Gu on Gu Essential BC
External equilibrium
10
FORMULATION WITH EMBEDDED DISCONTINUITIES
ENERGY FUNCTIONAL
where
11
VARIATION

• First variation

• Satisfying

in O-, Oh y O in O-, Oh y O in O-, Oh y
O 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 Gs

• 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
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