A comparative discrete-dislocation/nonlocal crystal-plasticity

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typeset2:/sco3/jobs1/ELSEVIER/msa/week.17/Pmsa15088y.001 Wed May 16 07:53:37 2001 Page Wed 2 D. Columbus, M. Grujicic / Materials Science and Engineering A000 (2001) 000–000 crete dislocations and material separation processes was modeled by Cleveringa et al. [12,13]. Cleveringa et al. [12,13] carried out a numerical analysis of the smallscale yielding for a plane-strain mode I crack under relatively high loading rates. In their analysis, discrete dislocations, represented as line discontinuities located within a linearly elastic, isotropic medium, are allowed to interact with each other through their long-range stress fields. In addition, constitutive relations are introduced to describe several short-range events such as dislocation nucleation, their annihilation and the pinning/unpinning of the dislocations at obstacles. The velocity of dislocations is assumed to be phonon-drag controlled and thus proportional to the Peach–Koehler force. The fracture behavior of the material is described using a cohesive-zone nonlinear constitutive model [6]. Due to the nonlinear character of the cohesive-zone constitutive relation and the evolution of dislocation structure with loading, the analysis is carried out in an incremental manner. At each time step, the total stress and strain fields are computed by superposing the ones associated with the discrete dislocations and the complementary fields which enforce the imposed boundary conditions and the continuity conditions across the internal (symmetry) boundaries. The results of Cleveringa et al. [12,13] clearly demonstrated that the presence of discrete dislocations in a region surrounding the crack tip is a prerequisite for attaining sufficiently high stress levels at the crack tip to cause atomic-scale material separation. In addition, Cleveringa et al. [12,13] showed that the fracture resistance of a material is controlled by the relative magnitudes of the loading rate, rate of dislocation nucleation, and dislocation mobility. Consequently, dislocations can either increase the fracture resistance through plastic dissipation or, conversely, promote fracture at a lower level of the remotely applied stress by giving rise to high local stress concentrations in the region ahead of the crack. In the present work, plane-strain mode I fracture is analyzed using both the discrete-dislocation approach of Cleveringa et al. [12,13], and a nonlocal crystal-plasticity formulation. While the nonlocal crystal-plasticity model can not account for the discrete nature of dislocations and for various dislocation-based short-range events (e.g. dislocation nucleation, annihilation, etc.), it allows incorporation of the additional strain hardening effect that arises from the presence of geometrically necessary dislocations in the regions associated with high plastic-strain gradients. The organization of the paper is as follows: In Section 2.1, a detailed account is given of the boundary value problem and the computational procedure used UNCORRECTED PROOF in the discrete-dislocation analysis of plane-strain mode I fracture. A brief description of the nonlocal crystalplasticity model and its incorporation into the commercial finite element package Abaqus/Standard [14] is given in Section 2.2. The results of the two computational approaches are presented and discussed in Section 3. The main conclusions resulting from the present study are given in Section 4. 2. Analysis and computational procedure 2.1. Discrete-dislocation formulation The discrete-dislocation analysis of mode I fracture, presented in the present paper, closely follows the analysis of Cleveringa et al. [12,13]. A schematic of the boundary-value problem at hand is shown in Fig. 1a. The following assumptions and simplifications are made: 1. symmetry exists about the horizontal crack plane (located at x2=0) and hence only the upper or the lower half of the region around the crack tip needs to be analyzed, Fig. 1a; 2. only small-scale yielding takes place and thus loading can be applied by prescribing the elastic mode I fields on the boundary of the computational domain; 3. rotation effects can be neglected and hence a small strain formulation can be used; 4. inertial effects can be neglected and hence a quasistatic analysis can be carried out; 5. the material containing dislocations is linearly elastic and isotropic and hence its constitutive behavior is completely defined by two elastic constants: the Young’s modulus, E, and the Poisson’s ratio, . The values of E and , as well as the values of other material parameters used in the present work, are given in Table 1; and 6. the crack growth behavior can be represented using a cohesive zone crack-face traction versus crack-face separation constitutive relation. The analysis is carried out using a 1000×500 m computational domain while plastic deformation resulting from the motion of discrete dislocations is primarily confinedtoa10×8.6 m process window (Fig. 1a). The Cartesian coordinate system is also denoted in Fig. 1(a). The portion of the computational domain outside the process window is discretized using a finite element mesh consisting of 850 plane strain quadrilateral isoparametric elements (CPE4 in the Abaqus notation) (Fig. 1b). To reduce the number of degrees of freedom in the boundary value problem at hand, the size of the elements is progressively increased with their distance from the process window. The process window is discretized using a refined mesh consisting of 48×50 CPE4 elements, all of the same size (Fig. 1c). In accordance with the small-scale yielding assumption, the loading is applied by imposing displacements consistent with the elastic mode I singular field on the outside
typeset2:/sco3/jobs1/ELSEVIER/msa/week.17/Pmsa15088y.001 Wed May 16 07:53:37 2001 Page Wed D. Columbus, M. Grujicic / Materials Science and Engineering A000 (2001) 000–000 3 Fig. 1. (a) The boundary value problem of a semi-infinite region subject to elastic mode I loading containing a cohesive zone at x 2=0 to enable crack formation and growth and a process window containing discrete dislocations; (b) and (c) the finite element meshes for the outer region and the ‘process window’, respectively. Table 1 Material, cohesive-zone, dislocation and loading parameters used in the discrete-dislocation analysis Parameter Symbol Units Magnitude Equation Where Used Young’s modulus E Gpa 70 Eqs. (5), (18) and (30) Poisson’s ratio N/A 0.33 Eqs. (5), (6), (12), (13), (18) and (30) Cohesive-zone strength max Gpa 0.6 Eqs. (1) and (30) Cohesive-zone separation n nm 1.0 Eq. (1) Slip resistance s MPa 3 Eq. (2) Burger’s vector magnitude b nm 0.25 Eqs. (2), (4), (5), (12), (13), (27) and (29) nucl m 24 N/A −2 Source density 10E−4 Drag coefficient B Pa s Eq. (2) Mean nucleation shear stress ¯ nucl MPa 25 N/A S.D. of ¯ nucl 0.2¯ nucl MPa 5 N/A Nucleation time Loading rate tnucl K I s Gpa (m) 0.01 N/A Eq. (6) 1/2 s−1 50 UNCORRECTED PROOF
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A comparative discrete-dislocation/nonlocal crystal-plasticity

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