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

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

246 K.C. Hwang et al. /

246 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251(47) asT ¼ 2r 2 0 hT Z;ð48Þwhere h is the initial thickness of the cylinder and r 0 is the mean cylinder radius inthe current configuration.Fig 3 shows the normalized torque, T/(2R 2 0 h Y), versus the normalized twist,R 0 , for l/R 0 =1, 0.5, 0.1 and 0, where R 0 and h are the initial mean cylinder radiusand thickness in the reference configuration, l is the intrinsic material length in (14),the initial yield stress Y is 0.2% times the Young’s modulus E, andl/R 0 =0 correspondsto classical plasticity theory (without strain gradient effect). Other materialproperties include the Possion’s ratio =0.3,plastic work hardening exponentN.EN=0.2, and the reference stress ref = Y Y The empirical material constant inthe Taylor dislocation model and the Burgers vector b only appear through theintrinsic material length l in (14), and it is therefore not necessary to specify thevalues of and b for a given ratio l/R 0 . For a small twist, R 0

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 247effect becomes significant. For example, the classical plasticity theory (l/R 0 =0) predictsa maximum torque that occurs approximately at R 0 =0.5, but there is nomaximum torque for MSG plasticity (l/R 0 50). The curves for cylinder radius beingone or two times the intrinsic material length l (i.e. l/R 0 =1,0.5) are much higherthan that predicted by classical plasticity, which is clearly due to the strain gradientleffect. Even the curve for cylinder radius being ten times l (i.e.R 0=0.1) shows significantsize effect.Fig 4 shows the torque-twist relation for both infinitesimal and finite deformationtheories of classical plasticity (l/R 0 =0) and MSG plasticity (l/R 0 =1). The materialproperties and the normalizations are identical to those in Fig. 3. The curvesaccounting for finite deformation are significantly lower than those for infinitesimaldeformation, indicating the finite deformation effect is significant for R 0 >0.1.5.2. Mode-I fracture analysisJiang et al. (2001) used the infinitesimal deformation MSG plasticity theory (Gaoet al., 1999; Huang et al., 2000a,b) to investigate fracture around a stationary mode-I crack tip field. Due to the strain gradient effect, stress level around the crack tip inMSG plasticity is significantly higher than that in the calssical plasticity, i.e. theFig. 4. The normalized torque, T/(2R 0 2 h Y ), versus the normalized twist, R 0 , for both finite and infinitesimaldeformation theories of MSG plasticity (l/R 0 =1) and classical plasticity (l/R 0 =0), where h and R 0are the thickness and mean radius of the cylinder in the reference configuration, respectively; Y is theinitial yield stress, l is the intrinsic material length for MSG plasticity. Plasticity work hardening exponentN=0.2, Young’s modulus E=500 Y , and Poisson’s ratio =0.3.

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