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Finite Deformation based Analysis of Strain Gradient Plasticity

Finite Deformation based Analysis of Strain Gradient Plasticity. Background: Size Effects in Plasticity. Flow Stress ( MPa ). Beam thickness (micron). Results of the bending of micro sized single crystal copper beams. 1. Geometrically Necessary Dislocations.

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Finite Deformation based Analysis of Strain Gradient Plasticity

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  1. Finite Deformation based Analysis of Strain Gradient Plasticity Strain Gradient Plasticity

  2. Background: Size Effects in Plasticity Flow Stress (MPa) Beam thickness (micron) Results of the bending of micro sized single crystal copper beams.1 1. C. Motz et al. / Acta Materialia 53 (2005) 4269–4279

  3. Geometrically Necessary Dislocations • Accumulation of dislocations during plastic flow is responsible plastic hardening. • Size effect is attributed to the accumulation of Geometrically necessary dislocations(GNDs) . Strain Gradient Plasticity

  4. Density of GNDs • In case of an uniform strain distribution no GNDs are necessary. • For a uniform shear the displacement field is given by – u1 = kx2 , u2 = u3 = 0 • The number of dislocation slipping in each block is same. Strain Gradient Plasticity

  5. Density of GNDs • Slip distance in the first cell : • No of GNDs at the boundary : n1b n1 n2 δx2 δx1 Strain Gradient Plasticity

  6. Incremental virtual work • Conventional theory • Gradient theory Where - work conjugate to

  7. Principle of virtual work • Conventional theory • Gradient Plasticity theory Micro-stress or Generalized effective stress Q = Effective stress (σe) in case of conventional theory Strain Gradient Plasticity

  8. Equilibrium Equations and BCs • Conventional Eqm. equation and BCs: • Additional consistency equation and BCs: Strain Gradient Plasticity

  9. Finite Deformation Formulation • s,ρ: First P-K stress. • Virtual work equation in the reference configuration: Tio Fij ni Ni to Vo V Ti t Vo V Strain Gradient Plasticity

  10. Incremental Principle of Virtual Work • Rate of first P-K Stress in terms of Jaumann rate (Assuming F=1, J=1 in case of updated Lagrangian framework): • Rate of higher order first P-K Stress in terms of convected rate: Strain Gradient Plasticity

  11. Updated Lagrangian Framework • The principle of VW equation becomes: • M has all the symmetry. Strain Gradient Plasticity

  12. Constitutive Equations Strain Gradient Plasticity

  13. Finite Element Formulation Linear interpolation Element Strain Gradient Plasticity

  14. Element Stiffness Matrix 1 2 3 4 5 Strain Gradient Plasticity

  15. Element Stiffness Matrix • The second term in the virtual work equation • Finally, Strain Gradient Plasticity

  16. THANK YOU! Strain Gradient Plasticity

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