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 8 D. Columbus, M. Grujicic / Materials Science and Engineering A000 (2001) 000–000 necessary dislocations, s G. The following function for s is chosen: s 1 k k =[(s S) +(s G) ] k (22) because it ensures that s s S whenever s S sG and that s s G whenever s S sG . Two values of k are generally investigated in the literature: (a) k=1 which corresponds to a superposition of the contributions of statistically stored and geometrically necessary dislocations to the slip resistance; and (b) k=2 which, since slip resistance scales with a square root of dislocation density, corresponds to a superposition of the densities of the two types of (statistically stored and geometrically necessary) dislocations. In the present work however, only case (a), k=1, was considered. The evolution of the component of the slip system resistance associated with statistically stored dislocations is taken to evolve as: s S= h (23) where h is an element of the matrix which describes the rate of strain hardening on the slip system due to shearing on the coplanar (self-hardening) and noncoplanar (latent-hardening) slip systems . Eq. (23) is often used within the framework of local crystal plasticity, e.g. Refs. [19,20]. Following Anand and coworkers [19,20], the following simple form for the h element of the slip system hardening matrix is adopted: h =q h (no summation on ) Here, h denotes the single-slip hardening rate while q is a matrix describing the latent hardening behavior. Following Anand and coworkers [19,20], the matrix q and the single-slip hardening rate h are given as: q 1 = q1=1.4 if and are coplanar slip systems otherwise (25) and h =h s S 0 1− (26) s 1 r where, h 0, the initial hardening rate, s 1, the saturation slip resistance, and exponent r are respectively set equal for both slip systems (h 1 0=h 2 0, s 1 1=s 2 1). The component of the slip resistance arising from the presence of geometrically necessary dislocations, s G,is assumed to be given by the following phenomenological equation [10]: s G=cbn G (27) where is the shear modulus, b the magnitude of the Burger’s vector, c a constant (set to 0.3 in accordance with Ashby [21]), and n G is the density of point obstacles (forest dislocations) associated with geometrically necessary dislocations. The short-range slip resistance arising from these point obstacles is a strong function of the type of dislocation junctions formed between mobile dislocations on a given slip system and the forest dislocations. Following Franciosi and co-workers [22,23], the dependence of the obstacle strength on the type of dislocation interaction is introduced through a set of interaction coefficients a . Consequently, the density of point obstacles, n G , can be expressed as: n G= a G (28) It is well established, e.g. Ref. [10], that the density of geometrically-necessary dislocations scales with the Nye’s dislocation tensor, the tensor which is a measure of the tensorial curl of FP . Following the procedure proposed by Dao and Parks [10], which is briefly overviewed in the Appendix, an evolution equation for the density of geometrically necessary dislocations associated with each slip system, under the plane-strain condition, is derived as: 1 d G= +F21 )] [ bp 0(3) dx2 P (F 11n 0(1) − d [ dx1 P P (F 12n 0(1) +F22n 0(2) P n 0(2) )] (29) where n 0(1) and n 0(2) are components of the slip plane normal n 0, and p 0(3) is a component of p 0, a vector defined as p 0=m 0×n 0. The geometrically necessary dislocations are of the edge character, with their line directions parallel to the x3-axis and their Burgers vectors collinear with the corresponding slip direction m 0. The stress strain relationship, Eq. (16), the flow rule, Eq. (20), and the evolution equations for the slip resistance due to statistically stored dislocations, Eq. (23), and for the geometrically-necessary dislocations, Eq. (27), constitute the non-local crystal-plasticity material constitutive model. The integration of the model along the loading path is carried out within the User Material Subroutine (UMAT) of Abaqus/Standard. With the exception of the density of geometrically-necessary dislocations, all other material state variables are integrated using a forward Euler approach. Since the integration of the density of geometrically necessary dislocations requires the computation of the gradient of state variables, which in turn depends on the material state at surrounding integration points, a backward Euler integration scheme was used to update G. Since the same integration scheme has been applied in a number of our recent studies, e.g. Ref. [24] it will not be discussed here. UNCORRECTED PROOF
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 9 Fig. 2. The variation of the normalized mode I stress intensity factor with crack extension for the discrete dislocation analysis (Discrete Dislocation) and two crystal plasticity analyses (Model-I and Model- II). 3. Results and discussion 3.1. Discrete-dislocation analysis In this section, selected results are shown for the discrete-dislocation analysis of plane-strain mode I crack growth. The material, cohesive-zone, dislocation, and loading parameters used in this analysis are summarized in Table 1. The variation of crack extension with the applied stress intensity factor (the curve denoted ‘Discrete Dislocation’) is shown in Fig. 2. The stress intensity factor is normalized with respect to its critical value corresponding to unstable crack growth without any dislocation activity, K I0, defined as [25]: KI0= En 1− 2 (30) where n=e max n is the work of separation. The results shown in Fig. 2 indicate that the crack begins to extend at the normalized stress intensity factor of K I/K I0=1.10. The initial crack extension (up to ca. 0.3 m) is characterized with a very slow rise in the normalized stress intensity factor. This regime of crack extension is followed by one in which the normalized stress intensity factor rises very steeply. When K I/K I0 reaches a value of approximately 2.0, the crack extension regime characterized by a slow increase in K I/K I0 The dislocation structure, a contour plot of the total normal stress, 22, and the distorted finite element mesh in the process window at the onset of crack extension are shown in Fig. 3a–c, respectively. To distinguish between positive and negative dislocations residing on the two slip systems shown in Fig. 3a, solid and open symbols are used to represent dislocations on the =60° and =120° slip systems, respectively; while, delta and gradient symbols denote, respectively, positive and negative dislocations on both slip systems. The results shown in Fig. 3a can be summarized as follows: 1. the dislocation activity is localized on slip systems originating from the region near the crack tip; 2. a substantially larger number of dislocations are associated with the =60° slip system than with the =120° slip system; 3. there is some evidence of dislocation polarization. For example, mostly negative dislocations are found near the crack tip on both slip systems. Conversely, near the outer-boundary of the process window the dislocations are mostly of the positive sign. In the total 22 stress field contour plot shown in Fig. 3b, the darkest shade of gray is used to denote the highest positive stress values. It should be noted that despite the fact that the dislocation-based 22 stress is singular, no stress singularities are present in Fig. 3b since the stresses had to be extrapolated to the nodal points for plotting purposes. The results shown in Fig. 3b can be summarized as follows: 1. the stress field is roughly partitioned into three regions, with little stress variation within each region. The region ahead of the crack tip is characterized by relatively large tensile 22 stress, while the region in the crack wake contains small (mostly negative) 22 stress values; 2. at many places 22 is dominated by the dislocationbased ˜ 22 stress which gives rise to small (diamondshaped) regions of significantly different value of 22 than that of its surroundings. The distorted finite element mesh in the process window of the computational domain at the onset of crack extension is shown in Fig. 3c. To enhance clarity, a magnification factor of 30 is applied to the nodal displacements. The location of the crack tip is denoted by an arrow. A careful examination of Fig. 3c reveals the presence of two not fully developed deformation bands emanating from the crack tip and aligned with the respective slip systems. These bands are clearly the result of the localized dislocation structure observed in Fig. 3a. The dislocation structure, the contour plot of the total normal stress, 22, and the distorted finite element mesh in the process window at the normalized stress intensity factor KI/KI0=1.48 are shown respectively in Fig. 4a–c. Based on the results shown in Fig. 2, at UNCORRECTED PROOF resumes. This period of crack growth is again pro- ceeded by the regime within which the crack-extension resistance (change of K I/K I0 per change in crack extension) is relatively high.
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A comparative discrete-dislocation/nonlocal crystal-plasticity

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