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

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** **gradient**leffect. 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 **deformation**theories **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 infinitesimal**deformation**, indicating the finite **deformation** effect is significant for R 0 >0.1.5.2. Mode-I fracture **analysis**Jiang 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 infinitesimal**deformation** 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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