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

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

248 K.C. Hwang et al. /

248 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251HRR field (Hutchinson, 1968; Rice and Rosengren, 1968). The numerical resultsalso showed that the stress around the crack tip in MSG plasticity is not only moresingular than the HRR field, but also more singular than the classical elastic K field(square-root singularity). This provides a means to explain the cleavage fractureobserved in ductile materials (e.g. Elssner et al., 1994).We use the same loading, material properties, and finite element mesh as Jiang etal. (2001) but the finite deformation effect is accounted for. We take a circular domainof radius 10 3 l centered at the crack tip in our plane-strain finite element analysis,where l is the intrinsic material length in (14). The origin of the coordinate systemcoincides with the crack tip, while the traction-free crack faces coincide with thenegative x 1 axis. The classical mode-I elastic K field is imposed on the outer boundaryof the domain (10 3 l) with the elastic stress intensity K I increasing monotonically.The stress intensity factor K I is normalized by Y l 1/2 in the numerical analysis,where Y =0.2% E is the initial yield stress, E is the Young’s modulus, and l is theintrinsic material length in (14). The stress is normalized by Y , while the position(coordinate) is normalized by l. Other material properties include the plastic workhardening exponent N=0.2, Poisson’s ratio =0.3, and the reference stress in (13).The empirical material constant in the Taylor dislocation model and the Burgersvector b only appear through the intrinsic material length l in (14), and it is thereforenot necessary to specify the values of and b for the above normalizations.Fig 5 shows the distribution of the normalized stress ahead of the mode-I cracktip, i.e. / Y , versus the normalized distance r/l to the crack tip, where is theCauchy (true) stress along =1.014 ( =180 being the crack faces), and r is thedistance to the crack tip in the reference (undeformed) configuration. The remoteapplied stress intensity factor is K I =10 Y l 1/2 . The distribution of for infinitesimaldeformation MSG plasticity theory (Jiang et al., 2001) and the correspondingcurves for classical plasticity (i.e. without the strain gradient effect) are also shownfor comparison. For r>10l, all curves become the same straight line with the slopeof 1/2, corresponding to the elastic K I field (since the slope represents the order ofsingularity in the log–log plot). For r between 0.3l and 10l, all curves still coincidebut they deviate significantly from the straight line (of the slope 1/2). This indicatesthat the material undergoes significant plastic deformation, though neither thestrain gradient effect nor the finite deformation effect is important in this range.Once the distance r to the crack tip is less than 0.3l, the normal stresses predictedby MSG plasticity (for both finite and infinitesimal deformation) increase muchquicker than their counterparts in classical plasticity. A representative value of theintrinsic material length is l=5 mm [Fleck et al., 1994; Stolken and Evans, 1998; Eq.(14) with =0.5 and b=0.3nm]. At a distance r=0.02l=0.1 mm to the crack tip,which is still within the intended range of applications for MSG plasticity (Gao etal., 1999; Huang et al., 2000b), the normal stress predicted by MSG plasticity isapproximately twice of that for classical plasticity. For infinitesimal deformationtheories, the curve for classical plasticity has the slope of N/(N+1) and correspondsto the HRR field (Hutchinson, 1968; Rice and Rosengren, 1968), while thecurve for MSG plasticity has the slope higher than 1/2 (in absolute value), indicatingthe crack tip field in MSG plasticity is more singular than not only the HRR field,

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 249Fig. 5. The Cauchy (true) stress normalized by the initial yield stress Y versus the normalized distanceto the crack tip, r/l, ahead of the crack tip (polar angle =1.014 ), where l is the intrinsic materiallength for MSG plasticity, the plastic work hardening exponent N=0.2, Poisson’s ratio =0.3, the ratioof yield stress to elastic modulus Y /E=0.2%, and the remotely applied elastic stress intensity factor K I / Y l 1/2 =10. The results are presented for both finite and infinitesimal deformation of MSG plasticity andclassical plasticity.but also the classical elastic K field. The effect of finite deformation does not seem tobe significant in Fig. 5, as seen from the comparison between curves for infinitesimaland finite deformation. This is because we have taken a relatively small appliedstress intensity factor (K I =10 Y l 1/2 ). In fact, the strain is only a few percent (3%)at a distance of 0.1 mm (r=0.02l) to the crack tip. It can be established from Fig. 4that the strain level needs to be significantly higher (>10%) in order to display thedifference between the finite and infinitesimal deformation. We have indeed imposeda much larger applied stress intensity factor, but the finite deformation analysis hassome numerical difficulties in convergence.6. SummaryWe have developed a finite deformation theory of mechanism-based strain gradient(MSG) plasticity based on the Taylor dislocation model. The theory ensuresthe proper decomposition of deformation into volumetric and deviatoric parts suchthat the volumetric deformation does not contribute to the deviatoric strain gradient,which represents the density of geometrically necessary dislocations. We haveconducted the analytic study for a thin cylinder under large torsion, and establishedthat the effect of finite deformation is significant when R 0 >0.1, where is the twist

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