Approximation of displacement and mixed formulations of solids with embedded discontinuities using the Meshless Element Free Galerkin Method

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Approximation of displacement and mixed
formulations of solids with embedded
discontinuities using the Meshless Element Free
Galerkin Method

• Felix Saucedo, Gustavo Ayala and Gelacio Juárez

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Outline

• Motivation
• Objective
• Variational formulation
• Isotropic continuum damage model
• Isotropic discrete damage model
• Approximation by Element Free Galerkin Method
• Weight functions
• Numerical integration
• Example
• Conclusions

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Motivation

• Finite element strain localization in elastic
  damage
• The finite element solutions are very sensitive
  to mesh size and orientation.
• Distortion of mesh.

Chen, Wu Belytschko 2000
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Objective

• Develop a displacement and mixed variational
  formulations and approximations by a meshless
  method for a solid with embedded discontinuities.
• This is advance of my PhD work.

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Variational formulation

• The functional energy proposed by Fraeijs de
  Veubeke for a continuum solid
• for a solid with embedded discontinuities develop
  by Juarez and Ayala

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Displacement variational formulation

• Derived formulations
• the functional with two independent fields is
  given by
• and for the Discrete Discontinuity approach

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Strain-Displacement mixed formulation

• Now, in this case the displacement, traction and
  strain fields are considered independent
• for Discrete Discontinuity approach

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Field approximation

• The independent fields for the formulations are
  approximated as
• Dependent field

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Approximation of the displacement formulation

• The solution to the continuum problem are
  functions u and t which make P stationary. That
  is for a solution to the continuum problem, the
  variation is dP 0, then we have
• To approximate the stationary value of the energy
  functional of the displacement approach the
  following matrix is obtained

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Discrete discontinuity approximation

• The first variation for Discrete Discontinuity
  approach
• replacing the approximation

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Lagrange multipliers

• The variational (or weak) form of the equilibrium
  equation is posed as follows. Consider trial
  functions and Lagrange
  multipliers for all test functions
  and . Then if (
  Dolbow and Belytschko, 1996)

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Approximation of the mixed formulation

• In this case, the first variation
• To approximate it, the following matrix is
  obtained

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Discrete discontinuity approximation

• The first variation
• To approximate it, the following matrix is
  obtained

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Isotropic continuum damage model

• Free energy Helmholtz
• Damage variable
• Constitutive equation
• Law of evolution of damage
• Damage criterion
• Rule hardening
• Loading/unloading conditions
• Consistency condition

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Continuum damage model
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Isotropic discrete damage model

• Free energy Helmholtz
• Damage variable
• Constitutive equation
• Law of evolution of damage
• Damage criterion
• Rule hardening
• Loading/unloading conditions
• Consistency condition

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Discrete damage model
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Approximation by Element Free Galerkin Method

• Approximation of u(x) can be expressed as
• In order to determine a(x), a weighted discrete
  error norm can be constructed as
• Minimize quadratic form, leads to linear
  equations for a can be written in shape function
  form

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Weight function

• Cubic spline

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Numerical integration Gaussian
Background mesh
Gauss point
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How do I use EFGM?
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Example Tension bar

• Properties

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Solution 1D

• Displacement d vs. load P

• Jump u vs. stress sS

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Conclusions

• The presented variational formulations impose in
  a natural way the essential boundary conditions
  in the Meshless Method approximation.
• The use of continuum damage models in Meshless
  Methods does not need tracking to propagate
  damage.

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Thank you