Finite deformation analysis of mechanism-based strain gradient ...

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240 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251gradient plasticity are introduced on the mesoscale, such as the strain gradient tensor and higher-order stress , where is defined in terms of the mesoscale Greenstrain E by IJK ¼ E IK;J þ E JK;I E IJ;K ; ð17Þand is the work conjugate of . The microscale strain within the mesoscale cell(Fig. 1) is related to the mesoscale strain measures by the Taylor expansion,E~ IJ ¼ E IJ þ E IJ;K X~ K þ 0 X~ 2¼ E IJ þ 1 ð2 KIJ þ KJI ÞX~ K þ 0 X~ 2; ð18Þwhere X~ K is the local coordinate origined at the center of the mesoscale cell.4.1. Microscale analysisThe stress T~ and strain E~ on the microscale satisfy the constitutive relation (10)except the stress ~, which is governed by the Taylor dislocation model in (13),pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~ ¼ ref "~ 2N þ l;ð19Þwhere the microscale effective strain "~ is related to the microscale Green strain E~ inthe same way as in (8), and is the mesoscale effective strain gradient given in (15).The microscale constitutive relation now becomes 2 0T~ ¼ J~ dUV J~CdJ~~1 þ 2~ J~ 2 ~3 E "~ 2 þ E ~ 1 34 @KKAC 3"~3~1 5 ð20Þwhere the microscale variables can be expressed in terms of the mesoscale ones viaTaylor expansion up to the first order, such asJ~ ¼ J þ dJ1X~ K ¼ J þ JC MNKMNX~ K ; ð21ÞdX KC~ 1 IJ ¼ C 1 IJ C 1 IM C 1 JNð KMN þ KNM ÞX~ K ; ð22ÞE ~ IJ þ E IJ þ 1 ð2 KIJ þ KJI ÞX~ K : ð23Þ4.2. Mesoscale analysisThe mesoscale constitutive relations are derived from the work equality betweenthe micro- and mesoscales,
K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 241ðT~ IJ E~ IJ dV ¼ ðT IJ E IJ þ IJK IJK ÞV cell ; ð24ÞVcellwhere the integration is over the mesoscale cell V cell in the reference configuration(Fig. 1), and stands for the virtual variation. Using the kinematics relation (18)between the strain measures on theses two scales, we obtain the mesoscale stress T IJand higher-order stress IJK in terms of microscale stress T~ IJ ,T IJ ¼ 1 T~ IJ dV; ð25ÞV cellðVcell IJK ¼ 1 T~ KI X~ J þ T~ KJ X~ I dV2V cellðVcellSubstituting the microscale constitutive relation (20) into (25) and (26), we obtainthe following finite deformation mesoscale constitutive relations for MSG plasticity," ! #T IJ ¼ KJðJ 1ÞC 1 IJ þ 23" J 2 3 E IJ " 2 þ E KKC 1 IJ ; ð27Þ3ð26Þ IJK þ l 2 "K6 V IJK þ " IJK þ 2 ref f ðÞf " 0 ðÞ "" IJK; ð28Þwhere is given in (13), l " =10 Yb and is less than 100 nm for typical metallicmaterials (Gao et al., 1999; Huang et al., 2000a,b), Y is the initial yield stress,V IJK ¼ 1 4 J ð2J 1ÞC 1 MN C 1 JK IMN þ C 1 IK JMNðJ 1Þ C 1 JM C 1 KNð IMN þ INM Þþ C 1 IM C 1 KNð JMN þ JNM Þ ; IJK ¼ 1 7243 J 4 13 C MNð IMN C JK þ IMN C IK ÞþJ 4 3ð2IJK þ IKJ þ JKI Þð Þ J 8 3 E MN IMN C 1 JK þ JMN C 1 IKþ 2 3 J 2 3 C1MN IMN JK þ IMN IK!23 IPP C 1 2JK3 JPP C 1 IK þ 2 " 2 þ E PP3!þ 2 " 2 þ E PPC 1 IM C 1 KNð JMN þ JNM Þ ;3C 1 JM C 1 KNð IMN þ INM Þð29Þð30Þ
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