A comparative discrete-dislocation/nonlocal crystal-plasticity

More documents

Recommendations

Info

typeset2:/sco3/jobs1/ELSEVIER/msa/week.17/Pmsa15088y.001 Wed May 16 07:53:37 2001 Page Wed 6 D. Columbus, M. Grujicic / Materials Science and Engineering A000 (2001) 000–000 K I I r u 1= 2 cos1−2+sin2 2 2 K I I r u 2= 2 sin2−2−cos2 (6) 2 2 where 2 2 r=x 1+x2 =tan−1x2 (7) x1 and the shear modulus is defined as =E/2(1+) and KI is the mode I stress intensity factor. The crack tip is taken to be initially at the origin (x1=x 2=0) and hence, zero-traction boundary conditions are prescribed along the crack face (the first segment of the x2=0 boundary) as: I I T 1(x1, x2)=T 2(x1, x2)=0 for x10; x2=0 (8) Along the middle segment of the x2=0 boundary, which is modeled as a cohesive-zone, and along the rightmost segment of the x2=0 boundary, the symmetry conditions are imposed as: I u 2(x1, x2)=0, I T 1(x1,x2)=0 for x10; x2=0 (9) When dislocations are present in the process window, they make contributions to the displacements and the tractions along the outer-boundary and the x2=0 crack surface, hence, the (dislocation-free) boundary conditions given above need to be corrected to account for the effect of dislocations such that the total stress and displacement fields agree with the boundary value problem shown in Fig. 1a. In the present work, the infinitemedium displacement and stress fields used are as follows: (l) (l) (l) ũ 1 (x1, x2))=cos()u 1 −sin()u 2 (l) (l) (l) ũ 2 (x1, x2)=sin()u 1 +cos()u 2 (10) (l) (l) (l) ˜ 11(x1,x2)=(cos() 11−sin() 12) cos() (l) (l) −(cos() 12−sin() 22) sin() (l) (l) (l) ˜ 22(x1, x2)=(sin() 11+cos() 12) sin() (l) (l) +(sin() 12+cos() 22)cos() (l) (l) (l) ˜ 12(x1, x2)=(cos() 11−sin() 12) sin() (l) (l) +(cos() 12−sin() 22)cos() (11) where (l) (l) b (l) (l) (l) x1 x2 u 1 (x1 ,x2 )= 2(1−) 2(r (l) ) 2+(1−) tan−1x(l) 2 x(l)n 1 (l) (l) (l) ũ2 (x1 , x2 ) = b (l) x2 4(1−) r (l)2 − 1 2 ln(r(l) ) 2 b 2 n (1−2) (12) (l) (l) 2 (l) 2 b (l) (l) (l) x2 (3(x1 ) +(x2 ) ) 11(x1 , x2 )=− 2(1−) (r (l) ) 4 (l) (l) 2 (l) 2 b (l) (l) (l) x2 ((x1 ) −(x2 ) ) 22(x1 , x2 )= 2(1−) (r n (l) ) 4 n (l) (l) 2 (l) 2 b (l) (l) (l) x1 ((x1 ) −(x2 ) ) 12(x1 , x2 )= 2(1−) (r (l) ) 4 n (13) and (l) (l) (l) x1 (x1, x2)=cos()(x1−x 1 )+sin()(x2−x 2 ) (l) (l) (l) x2 (x1, x2)=−sin()(x1−x 1 )+cos()(x2−x 2 ) (14) and r (l) (l) 2 (l) 2 (l) =(x1 ) +(x2 ) , where x m (m=1, 2) denotes the location of dislocation l. As mentioned earlier, represents the angle that the two slip systems make with the crack plane (x2=0) and is set to 60°. The updated dislocation structure determined in Section 2.1.1 can thus be used to compute the contributions of dislocations to the displacement field as: (l) ũi(t)= ũ i (t) l (i=1, 2) and to the traction fields as: 2 (l) T i(t)= ˜ ij (t)hj l j=1 (i=1,2); hj( j=1, 2); are the components of the outward surface normal) along the outer boundary of the computational domain and along the crack surface and the x2=0 symmetry boundary. The dislocation-free elastic mode I boundary conditions discussed above are then corrected for the presence of dislocations as follows: 1. −ũi (i=1, 2) displacements are added along the outer boundary; 2. −T i(i=1, 2) tractions are added along the ‘traction-free’ crack surface; 3. −ũ2 and −T 1 are prescribed along the x2=0 symmetry plane 4. zero-displacement conditions are prescribed along the bottom of the cohesive zone. In addition, the effective stiffness and the residual forces of the interfacial elements are modified to account for the contribution of dislocations to the displacements and tractions along the top portion of the cohesive zone. A detailed account of this modification is given in our recent paper [16]. 2.1.3. Solution of the boundary alue problem Once the ‘corrected’ boundary conditions are determined at the end of the time step, Abaqus/Standard is used to solve the boundary value problem. The computed stresses ˆ ij(t) are then past back to Section 2.1.1 where they are combined with the dislocation-based (k) stresses k ˜ ij (t) in accordance with Eq. (3). The effective shear stress acting on each dislocation is next computed and the procedure continued as discussed in Section 2.1.1. UNCORRECTED PROOF
typeset2:/sco3/jobs1/ELSEVIER/msa/week.17/Pmsa15088y.001 Wed May 16 07:53:37 2001 Page Wed 2.2. Nonlocal crystal-plasticity formulation D. Columbus, M. Grujicic / Materials Science and Engineering A000 (2001) 000–000 7 In this section, the same plane strain mode I crack boundary value problem shown in Fig. 1a is analyzed. However, while the crack behavior is again represented using the cohesive zone formalism, the bulk material is modeled using a rate-dependent, isothermal, elastic-viscoplastic, finite-strain, nonlocal crystal-plasticity formulation. The continuum mechanics and materials science foundations for this model can be traced to the work of Anand and coworkers [19,20], and Dao and Parks [10]. Within the present model, the (initial) reference configuration is assumed to consist of a perfect, stressfree crystal lattice and the embedded material. The position of each material point in the reference configuration is given by its position vector X, while each material point in the current configuration is described by its position vector, x. The mapping of the reference configuration into the current configuration is described by the deformation gradient, F=x/X, a second-order tensor. Typically, in order to reach the current configuration, the reference configuration must be deformed both elastically and plastically. The total deformation gradient can be multiplicatively decomposed into its elastic, Fe , and plastic, Fp , parts as F=F e ·Fp .Inother words, the deformation of a single-crystal material point is considered to be the result of two independent atomic-scale processes: (i) an elastic distortion of the crystal lattice corresponding to the stretching of atomic bonds and; (ii) a plastic deformation associated with atomic plane slippage which leaves the crystal lattice undistorted. The present constitutive model is based on the following governing variables: (i) the Cauchy stress, T; (ii) the deformation gradient, F; (iii) crystal slip systems, labeled by integers . Each slip system is specified by a unit slip-plane normal n 0, and a unit vector m 0 aligned in the slip direction, both defined in the reference configuration; (iv) the plastic deformation gradient, Fp , with det (Fp )=1 (plastic deformation by slip does not give rise to a volume change); (v) the slip system deformation resistance s 0 which has the units of stress; and (vi) the density of geometrically-necessary dislocations associated with the slip systems , G. Based on the aforementioned multiplicative decomposition of the deformation gradient, the elastic deformation gradient Fe which describes the elastic distortions and rigid-body rotations of the crystal lattice, can be defined by: Fe =FFp−1 , det (Fe )0 (15) The plastic deformation gradient, Fp , on the other UNCORRECTED PROOF hand, accounts for the cumulative effect of shearing on all slip systems in the crystal. Since elastic stretches in metallic materials are generally small, the constitutive equation for stress under isothermal conditions can be defined by the linear relation: T*=C[Ee ] (16) where C is a fourth-order elasticity tensor, and E e and T * are, respectively, the Green elastic strain measure and the second Piola–Kirchoff stress measure relative to the intermediate, isoclinic configuration obtained after plastic shearing of the stress-free lattice as described by F p . E e and T * are respectively defined as: E e =(1/2){F eT F e −I}T*det(F e )F e−1 TF e−T (17) where I is the second order identity tensor. An isotropic-elasticity fourth-order tensor C, defined as: C= E 1+ I+ 1+ IIn (18) is used in order to ensure consistency in the elastic materials response in the two (discrete-dislocation and crystal-plasticity) formulations. I in Eq. (18) is the fourth-order identity tensor and denotes the tensorial product. The evolution equation for the plastic deformation gradient is defined by the flow rule: Lp =F pFp−1 = S0 S0m0n0 (19) where Lp is the plastic velocity gradient and S0 the Schmid tensor. To obtain consistency in the plastic responses of the material in the two formulations, the same two slip systems discussed in the previous sections are used. The plastic shearing rate on a slip system is described using the following simple power-law relation: =˜ sign( ) (20) s 1/m where ˜ is a reference plastic shearing rate, and s are the resolved shear stress and the deformation resistance on slip system , respectively, and m is the material rate-sensitivity parameter. Since elastic stretches in metallic materials are generally small, the resolved shear stress on slip system can be defined as: =T* S0 (21) where the raised dot denotes the scalar product between two second order tensors. To complete the materials constitutive model discussed above, an evolution equation for the slip resistance, s , should be defined. Toward that end, the overall slip resistance of a given slip system is to have both a component arising from statistically stored dislocations, s S , and a component due to geometrically
Page 1 and 2: typeset2:/sco3/jobs1/ELSEVIER/msa/w
Page 5: typeset2:/sco3/jobs1/ELSEVIER/msa/w

A comparative discrete-dislocation/nonlocal crystal-plasticity

Create successful ePaper yourself

Delete template?

Save as template?