Theoretical and computational comparison of models for dislocation dissociation and stacking fault/core formation in fcc crystals

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• 作者：J.R. Mianroodi^a ; ^{jaber.rezaeimianroodi@rwth-aachen.de" class="auth_mail" title="E-mail the corresponding author} ; A. Hunter^b ; I.J. Beyerlein^b ; B. Svendsen^a ; ^c
• 关键词：Phase field ; Dislocation ; Dissociation ; Core ; Stacking fault ; Molecular statics
• 刊名：Journal of the Mechanics and Physics of Solids
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：95
• 期：Complete
• 页码：719-741
• 全文大小：10786 K

文摘

The purpose of the current work is the theoretical and computational comparison of selected models for the energetics of dislocation dissociation resulting in stacking fault and partial dislocation (core) formation in fcc crystals as based on the (generalized) Peierls–Nabarro (GPN: e.g., 51 and 38), and phase-field (PF: e.g., 40, 17, 18 and 23), methodologies (e.g., Wang and Li, 2010). More specifically, in the current work, the GPN-based model of Xiang et al. (2008) is compared theoretically with the PF-based models of Shen and Wang (2004), 17 and 18, and Mianroodi and Svendsen (2015). This is carried out here with the help of a unified formulation for these models via a generalization of the approach of Cahn and Hilliard (1958) to mechanics. Differences among these include the model forms for the free energy density mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0055.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=d19be057be565b39c3f648a16d8a9e11" title="Click to view the MathML source">ψ_elamathContainer hidden">mathCode"><math altimg="si0055.gif" overflow="scroll">ψelamath> of the lattice and the free energy density mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0056.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=0c5e4666b3731feedf20878f846c91e2" title="Click to view the MathML source">ψ_slimathContainer hidden">mathCode"><math altimg="si0056.gif" overflow="scroll">ψslimath> associated with dislocation slip. In the PF-based models, for example, mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0057.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=03b79b33e9b2d710e981071e368bdb7b" title="Click to view the MathML source">ψ_elamathContainer hidden">mathCode"><math altimg="si0057.gif" overflow="scroll">ψelamath> is formulated with respect to the residual distortion mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0058.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=248f63820cf7b0256d237fcd51fc9d37" title="Click to view the MathML source">H_RmathContainer hidden">mathCode"><math altimg="si0058.gif" overflow="scroll">mathvariant="bold-italic">Hmathvariant="normal">Rmath> due to dislocation slip (e.g., 20 and 28), and with respect to the dislocation tensor mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0059.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=0bfe94ccce209b4f688f2df409672b09">mathContainer hidden">mathCode"><math altimg="si0059.gif" overflow="scroll">curlmathvariant="bold-italic">Hmathvariant="normal">Rmath> in the GPN model (e.g., Xiang et al., 2008). As shown here, both model forms for mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0060.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=f16236f216fdcb1d8732375971565e8a" title="Click to view the MathML source">ψ_elamathContainer hidden">mathCode"><math altimg="si0060.gif" overflow="scroll">ψelamath> are in fact mathematically equal and so physically equivalent. On the other hand, model forms for mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0061.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=5d57b41686416cd6d2907dc3e150d900" title="Click to view the MathML source">ψ_slimathContainer hidden">mathCode"><math altimg="si0061.gif" overflow="scroll">ψslimath> differ in the assumed dependence on the phase or disregistry fields mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0062.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=713927fed361d5d8e805ff85f69fdd75" title="Click to view the MathML source">ϕmathContainer hidden">mathCode"><math altimg="si0062.gif" overflow="scroll">mathvariant="bold-italic">ϕmath>, whose spatial variation represents the transition from unslipped to slipped regions in the crystal. In particular, Xiang et al. (2008) and 17 and 18 work with mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0063.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=309a9cd419f2be71f285f04f8bab166c" title="Click to view the MathML source">ψ_sli(ϕ)mathContainer hidden">mathCode"><math altimg="si0063.gif" overflow="scroll">ψsli(mathvariant="bold-italic">ϕ)math>. On the other hand, Shen and Wang (2004) and Mianroodi and Svendsen (2015) employ mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0064.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=ca4aad13542223c7d07eafb8ea632dd5" title="Click to view the MathML source">ψ_sli(ϕ,∇ϕ)mathContainer hidden">mathCode"><math altimg="si0064.gif" overflow="scroll">ψsli(mathvariant="bold-italic">ϕ,∇mathvariant="bold-italic">ϕ)math>. To investigate the consequences of these differences for the modeling of the dislocation core, dissociation, and stacking fault formation, predictions from the models of 17 and 18 and Mianroodi and Svendsen (2015) are compared with results from molecular statics (MS) for the deformation field of dissociated edge and screw dipoles in Al and Au. Particularly notable is the agreement of the MS and PF strain field results for the case of perfect screw dissociation which, in contrast to the edge case, are characterized by asymmetric displacement and strain fields. The degree of this asymmetry is apparently related to the corresponding anisotropy ratio. As well, comparison of MS and PF disregistry fields implies that the gradient dependence of mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022509616300369&_mathId=si0065.gif&_user=111111111&_pii=S0022509616300369&_rdoc=1&_issn=00225096&md5=ad361cf272bc36688d2f7080930d868f" title="Click to view the MathML source">ψ_slimathContainer hidden">mathCode"><math altimg="si0065.gif" overflow="scroll">ψslimath> results in a broadening of the (otherwise too narrow) disregistry profile to the form predicted by MS.