Comparison of methods for evaluation of overall response of composites

1
Comparison of methods for evaluation of overall
response of composites

• Martin Wierer
• Faculty of Civil Engineering
• CTU, Prague, Czech republic

2
OUTLINE

• Introduction
• Theoretical background
• homogenization
• static condensation
• Results
• Conclusion

3
Introduction
macro
meso
micro
4
Homogenization

• homogenization based on Hashin-Shrickman
  principle
• ?(x)L0?(x)?(x)

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Homogenization

• based on periodicity of structure
• variational formulation to find u
• ??(u)L?(v)? - ?EL?(v)? ?v periodic
• ?(u) E ?(u)
• after some modifications
• vTKu vTf ?v periodic
• from Hills lemma ??(u)? ??(?)? ??(u)?(?)?
• Ku K1E 0
  
  K1u K2E ?

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Static condensation

• condensing out all inner nodes

• introduction of constraints on boundary
  nodes

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RESULTS

• meshing of PUC (RVE)

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RESULTS
n.dof the number of degrees of freedom
n.b.n. the number of boundary nodes
const.lin. linear constraints
const.par. parabolic constraints
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RESULTS

• comparison of displacements

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CONCLUSION

• two possibilities how to obtain overall response
  from mechanical loading
• static condesation should be carried out only for
  inner nodes, in the opposite case, the error is
  significant
• using homogenized properties is the least
  time-consuming method with sufficient accuracy