- Text
- Strain,
- Plasticity,
- Gradient,
- Deformation,
- Finite,
- Hwang,
- Classical,
- Cylinder,
- Reference,
- Mesoscale,
- Analysis

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

240 K.C. Hwang et al. / International Journal **of** Plasticity 19 (2003) 235–251**gradient** plasticity are introduced on the mesoscale, such as the **strain** **gradient** tensor and higher-order stress , where is defined in terms **of** the mesoscale Green**strain** 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 **analysis**The 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 **analysis**The 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

- Page 1 and 2: International Journal of Plasticity
- Page 3 and 4: K.C. Hwang et al. / International J
- Page 5: K.C. Hwang et al. / International J
- Page 9 and 10: K.C. Hwang et al. / International J
- Page 11 and 12: K.C. Hwang et al. / International J
- Page 13 and 14: K.C. Hwang et al. / International J
- Page 15: K.C. Hwang et al. / International J