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Finite Deformation based Analysis of Strain Gradient Plasticity

VIEWS: 3 PAGES: 16

									   Finite Deformation based Analysis of
          Strain Gradient Plasticity




20/04/2009       Strain Gradient Plasticity   1
        Background: Size Effects in Plasticity




                                                       Flow Stress (MPa)
                                                                           Beam thickness (micron)
                                                       Results of the bending of micro sized single crystal copper
                                                       beams.1




20/04/2009       1. C. Motz et al. / Acta Materialia 53 (2005) 4269–4279                                      2
  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) .



20/04/2009          Strain Gradient Plasticity       3
                 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.
 20/04/2009              Strain Gradient Plasticity       4
                         Density of GNDs
                   n1b     n1 n2



   δx2



             δx1

• Slip distance in the first cell : n1b   1 x2
• No of GNDs at the boundary : n  n1  n2
                                        
                                            x1 x2
                    x2   x2  x1
               1 2
                    b            b           b


20/04/2009                    Strain Gradient Plasticity   5
        Incremental virtual work
• Conventional theory
                w   ij ij
• Gradient theory
           w   ij ij   ijk ijk
Where
         ijk  uk ,ij
          ijk - work conjugate to ijk
                  Principle of virtual work
• Conventional theory

         dV   T u dS
      V
             ij   ij
                        
                           S
                                i     i


• Gradient Plasticity theory

 
 V
          ij  (Q   e ) P   i ,P dV   Tiui  t P dS
      ij                         i                  
                                                                 S
                                                                        
   Micro-stress or
   Generalized effective       Q = Effective stress (σe) in case of
   stress                      conventional theory
20/04/2009                          Strain Gradient Plasticity               7
       Equilibrium Equations and BCs
• Conventional Eqm. equation and BCs:
          ij , j  0
          ij n j  Ti
• Additional consistency equation and BCs:
       Q   e   i ,i  0
        i ni  t



20/04/2009                    Strain Gradient Plasticity   8
     Finite Deformation Formulation
• s, ρ: First P-K stress.                                                 Tio
                                                                                                  ni
                                                                                Ni t
                                                                                    o
         s  JF 1    F 1
                                                                                        Fij
           JF 1   F 1
                                                                     Vo                       V
      T  N s
          o


      to  N  

• Virtual work equation in the reference
                            V                                                                 V
  configuration:                                                o




     s  F
    Vo
              ij   ij  (q   e ) P  i ,P dVo   (Ti o ui  to P )dSo
                                               i
                                                                 So

20/04/2009                              Strain Gradient Plasticity                                     9
  Incremental Principle of Virtual Work

  
  Vo
     sij Fij  (q   e ) P  i oP,i dVo          
                                                          So
                                                             Ti o ui  to P dSo


• Rate of first P-K Stress in terms of Jaumann rate (Assuming
  F=1, J=1 in case of updated Lagrangian framework):
                    ˆ
               s      D  D      LT

• Rate of higher order first P-K Stress in terms of convected
  rate:          
                   



20/04/2009                        Strain Gradient Plasticity                          10
      Updated Lagrangian Framework
 • The principle of VW equation becomes:
   ˆ                                                                               P 
                                                                                                                 
                                                                               

 ij ij  ( kjik   ik  jk )ij   ik L jkij  (q   e )   i o,i dVo  S Toiui  to dSo
                                                                            P                                      P
                                                                                                      
Vo                                                                                              o




                        ik Dkj 
                                       1
                                          ik Dkj   il Dlj   1  ik  lj   il kj Dkl
                                       2                          2

                        kj Dki 
                                       1
                                          kj il   lj ik Dkl
                                       2

                          ik   Dkj   kj Dki Dij 
                                                          1
                                                             ik lj   il kj   kj il   lj ik DklDij
                                                          2
                                                         ˆ
                                                        M ijkl DklDij

 • M has all the symmetry.
 20/04/2009                                             Strain Gradient Plasticity                                    11
                       Constitutive Equations
              ˆ  R    P m 
             ij   ijkl kl    kl


             q   e  f  h,  P ,  ,P , mij , mij ,k   mijij
                                       i
                                                                ˆ

                   
                 f '  h,  P ,  ,P , mij , mij ,k 
                                       i



               ˆ
                   W  W  
               
                   L 
20/04/2009                               Strain Gradient Plasticity   12
             Finite Element Formulation

             Element                                    Linear interpolation


                       (u1 , u2 ) ,  P
                                                             6
                                                         ui   N in D n
                                                         
                                                               n 1


                                                                 3
                                                           M n nP
                                                         P
                                                                 
                                                                n 1




20/04/2009                 Strain Gradient Plasticity                          13
                  Element Stiffness Matrix
              ˆ 
             ij ij  ( kj ik   ik  jk ) ij   ik L jk  ij
                                                               

 ˆ 
 ij ij  Rijkl ( kl   P mkl ) ij
                                                                                     1

                   B RBD 
Rijkl  kl ij  D n
       
                             T         T             n
                                                                                        2
Rijkl  P   m    D  B  R mM  
                                n T
                                          T
            kl
                 ij
                                                                   n
                                                                                        3

   ik Dkj   kj Dki  Dij  M ijkl DklDij  D
                              ˆ                   n
                                                         B M BD 
                                                                   ˆ
                                                                   T        T       n
                                                                                        4

 ik L jk  ij   D n   11P    22
                                                  P    P D 
                         T
                                1                    2         12     3
                                                                                n
                                                                                        5

                                                 Dn 
                         K                             
                                           K ep  P   F1              
                                                 n 
                                  e
                                                  
20/04/2009                                     Strain Gradient Plasticity                   14
             Element Stiffness Matrix
• The second term in the virtual work equation
                        (q   e )   i  oP,i
                                    P




                                  Dn 
               K                       
                            K ep  P   F2              
                                  n 
                    e
                                   

• Finally,                                      
                Ke           K ep   Dn   F1 
               K T                  P     
                              K p   n   F2 
                ep
20/04/2009                        Strain Gradient Plasticity   15
20/04/2009   Strain Gradient Plasticity   16

								
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