Dislocation density tensor Samuel Forest Mines ParisTech / CNRS Centre des Mat´ eriaux/UMR 7633 BP 87, 91003 Evry, France Samuel.Forest@mines-paristech.fr
Plan Statistical theory of dislocations 1 The dislocation density tensor Scalar dislocation densities Continuum crystal plasticity approach 2 Incompatibility and dislocation density tensor Lattice curvature tensor Need for generalized continuum crystal plasticity 3
Plan Statistical theory of dislocations 1 The dislocation density tensor Scalar dislocation densities Continuum crystal plasticity approach 2 Incompatibility and dislocation density tensor Lattice curvature tensor Need for generalized continuum crystal plasticity 3
Plan Statistical theory of dislocations 1 The dislocation density tensor Scalar dislocation densities Continuum crystal plasticity approach 2 Incompatibility and dislocation density tensor Lattice curvature tensor Need for generalized continuum crystal plasticity 3
Resulting Burgers vector [Kr¨ oner, 1969] Resulting Burgers B s for slip system s for a closed circuit limiting the surface S „Z « B s b s = ξ ( x ) . n dS S Z = α ∼ . n dS S where ∼ ( x ) = b s ⊗ ξ ( x ) α Consider contributions of all systems and ensemble average it Burgers vector Z B = α ∼ . n dS b s ( x ) S < b s ⊗ ξ > X ∼ = α Dislocation line vector s Ergodic hypothesis: compute the dislocation density ξ ( x ) tensor by means of a volume average in DDD simula- tions Statistical theory of dislocations 5/33
Dislocation density tensor for edge dislocations Resulting Burgers vector B = nb e 1 = ∼ . e 3 S α n = S b ⊗ ξ α ∼ = − ρ G b e 1 ⊗ e 3 ρ G = n / S is the density of geometri- cally necessary dislocations according to (Ashby, 1970). 2 3 0 0 α 13 0 0 0 4 5 n edge dislocations piercing the 0 0 0 surface S out of diagonal component of α convention: ∼ z = b b × ξ diagonal component α 33 for screw disloca- tions with b = b e 3 Statistical theory of dislocations 6/33
Plan Statistical theory of dislocations 1 The dislocation density tensor Scalar dislocation densities Continuum crystal plasticity approach 2 Incompatibility and dislocation density tensor Lattice curvature tensor Need for generalized continuum crystal plasticity 3
Fourth order and scalar dislocation densities [Kr¨ oner, 1969] Two–point correlation tensor ≈ ( x , x ′ ) = < b ( x ) ⊗ ξ ( x ) ⊗ b ( x ′ ) ⊗ ξ ( x ′ ) > α The invariant quantity α ijij ( x , x ) dV = b 2 Z Z 1 χ ( x ) dV = b 2 L V = b 2 ρ V V V V where L is the total length of dislocation lines inside V and χ ( x ) equals 1 when there is a dislocation at x , 0 otherwise. ρ is the scalar dislocation density traditionally used in physical metallurgy Statistical theory of dislocations 8/33
Plan Statistical theory of dislocations 1 The dislocation density tensor Scalar dislocation densities Continuum crystal plasticity approach 2 Incompatibility and dislocation density tensor Lattice curvature tensor Need for generalized continuum crystal plasticity 3
Reminder on tensor analysis (1) The Euclidean space is endowed with an arbitrary coordinate system characterizing the points M ( q i ). The basis vectors are defined as e i = ∂ M ∂ q i The reciprocal basis ( e i ) i =1 , 3 of ( e i ) i =1 , 3 is the unique triad of vectors such that e i · e j = δ i j If a Cartesian orthonormal coordinate system is chosen, then both bases coincide. Continuum crystal plasticity approach 10/33
Reminder on tensor analysis (2) The gradient operator for a tensor field T ( X ) of arbitrary rank is then defined as grad T = T ⊗ ∇ := ∂ T ∂ q i ⊗ e i The gradient operation therefore increases the tensor rank by one. Continuum crystal plasticity approach 11/33
Reminder on tensor analysis (2) The gradient operator for a tensor field T ( X ) of arbitrary rank is then defined as grad T = T ⊗ ∇ := ∂ T ∂ q i ⊗ e i The gradient operation therefore increases the tensor rank by one. The divergence operator for a tensor field T ( X ) of arbitrary rank is then defined as div T = T · ∇ := ∂ T ∂ q i · e i The divergence operation therefore decreases the tensor rank by one. Continuum crystal plasticity approach 12/33
Reminder on tensor analysis (2) The gradient operator for a tensor field T ( X ) of arbitrary rank is then defined as grad T = T ⊗ ∇ := ∂ T ∂ q i ⊗ e i The gradient operation therefore increases the tensor rank by one. The divergence operator for a tensor field T ( X ) of arbitrary rank is then defined as div T = T · ∇ := ∂ T ∂ q i · e i The divergence operation therefore decreases the tensor rank by one. The curl operator (or rotational operator) for a tensor field T ( X ) of arbitrary rank is then defined as curl T = T ∧ ∇ := ∂ T ∂ q i ∧ e i where the vector product is ∧ a ∧ b = ǫ ijk a j b k e i = ǫ ∼ : ( a ⊗ b ) The component ǫ ijk of the third rank permutation tensor is the signature of the permutation of (1 , 2 , 3). The curl operation therefore leaves the tensor rank unchanged. Continuum crystal plasticity approach 13/33
Reminder on tensor analysis (3) With respect to a Cartesian orthonormal basis, the previous formula simplify. We give the expressions for a second rank tensor T ∼ grad T = T ij , k e i ⊗ e j ⊗ e k ∼ div T = T ij , j e i ∼ We consider then successively the curl of a vector field and of a second rank vector field, in a Cartesian orthonormal coordinate frame curl u = ∂ u ∧ e j = u i , j e i ∧ e j = ǫ kij u i , j e k ∂ X j ∼ = ∂ A curl A ∧ e k = A ij , k e i ⊗ e j ∧ e k = ǫ mjk A ij , k e i ⊗ e m ∼ ∂ x k Continuum crystal plasticity approach 14/33
Reminder on tensor analysis (4) We also recall the Stokes formula for a vector field for a surface S with unit normal vector n and oriented closed border line L : � � � � u · dl = − ( curl u ) · n ds , u i dl i = − ǫ kij u i , j n k ds L S L S Applying the previous formula to u j = A ij at fixed i leads to the Stokes formula for a tensor field of rank 2: � � � � A ∼ · dl = − ( curl A ∼ ) · n ds , A ij dl j = − ǫ mjk A ij , k n m ds L S L S Continuum crystal plasticity approach 15/33
Plan Statistical theory of dislocations 1 The dislocation density tensor Scalar dislocation densities Continuum crystal plasticity approach 2 Incompatibility and dislocation density tensor Lattice curvature tensor Need for generalized continuum crystal plasticity 3
Incompatibility of elastic and plastic deformations ∼ = E F ∼ . P ∼ In continuum mechanics, the previous differential operators F are used with respect to the initial coordinates X or with respect to the current coordi- E P nates x of the material points. In the latter case, the notation ∇ , grad , div and curl are used but in the former case we adopt ∇ X , Grad , Div and Curl . F ∼ = 1 ∼ + Grad u = ⇒ Curl F ∼ = 0 The deformation gradient is a compatible field which derives from the displacement vector field. This is generally not the case for elastic and plastic deformation: Curl E ∼ � = 0 , Curl P ∼ � = 0 It may happen incidentally that elastic deformation be compatible for instance when plastic or elastic deformation is homogeneous. Continuum crystal plasticity approach 17/33
Incompatibility of elastic and plastic deformations F ∼ = E ∼ . P ∼ E ∼ relates the infinitesimal vec- F tors d ζ and dx , where d ζ re- sults from the cutting and re- leasing operations from the in- E P finitesimal current lattice vec- tor dx − 1 · dx d ζ = E ∼ If S is a smooth surface containing x in the current configuration and bounded by the closed line c , the true Burgers vector is defined as I − 1 . dx B = E ∼ c Continuum crystal plasticity approach 18/33
Dislocation density tensor in continuum crystal plasticity F ∼ = E ∼ . P ∼ E ∼ relates the infinitesimal vec- F tors d ζ and dx , where d ζ re- sults from the cutting and re- leasing operations from the in- E P finitesimal current lattice vec- tor dx − 1 · dx d ζ = E ∼ If S is a smooth surface containing x in the current configuration and bounded by the closed line c , the true Burgers vector is defined as I Z Z − 1 . dx = − − 1 ) . n ds = B = E ( curl E α ∼ · n ds ∼ ∼ c S S according to Stokes formula which gives the definition of the true dislocation density tensor − 1 = − ǫ jkl E − 1 α ∼ = − curl E ik , l e i ⊗ e j ∼ Continuum crystal plasticity approach 19/33
Dislocation density tensor in continuum crystal plasticity The Burgers vector can also be computed by means of a closed circuit c 0 ⊂ Ω 0 convected from c ⊂ Ω: � � � − 1 · dx = − 1 · F B = E E ∼ · dX = P ∼ · dX ∼ ∼ c c 0 c 0 � � T · ds = ( Curl P ∼ ) · dS = − ( Curl P ∼ ) · F − J ∼ S 0 S − T · dS has been used. We obtain the Nanson’s formula ds = J F ∼ alternative definition of the dislocation density tensor − 1 = − 1 T ∼ = − curl E J ( Curl P ∼ ) · F α ∼ ∼ Continuum crystal plasticity approach 20/33
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