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AN ANALYTIC SOLUTION FOR ONE-DIMENSIONAL DISSIPATIONAL STRAIN-GRADIENT PLASTICITY

Part of: Plastic materials, materials of stress-rate and internal-variable type Special kinds of problems

Published online by Cambridge University Press:  03 November 2009

ROGER YOUNG
ROGER YOUNG*
Affiliation:
Industrial Research Ltd, Box 31-310, Lower Hutt, New Zealand (email: [email protected])
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Abstract

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An analytic solution is developed for the one-dimensional dissipational slip gradient equation first described by Gurtin [“On the plasticity of single crystals: free energy, microforces, plastic strain-gradients”, J. Mech. Phys. Solids48 (2000) 989–1036] and then investigated numerically by Anand et al. [“A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results”, J. Mech. Phys. Solids53 (2005) 1798–1826]. However we find that the analytic solution is incompatible with the zero-sliprate boundary condition (“clamped boundary condition”) postulated by these authors, and is in fact excluded by the theory. As a consequence the analytic solution agrees with the numerical results except near the boundary. The equation also admits a series of higher mode solutions where the numerical result corresponds to (a particular case of) the fundamental mode. Anand et al. also established that the one-dimensional dissipational gradients strengthen the material, but this proposition only holds if zero-sliprate boundary conditions can be imposed, which we have shown cannot be done. Hence the possibility remains open that dissipational gradient weakening may also occur.

Keywords

crystal plasticitydissipational gradientsstrengthening and mechanisms

MSC classification

Secondary: 74C05: Small-strain, rate-independent theories (including rigid-plastic and elasto-plastic materials) 74M25: Micromechanics
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 3: This Special Issue is dedicated to Dr Stephen White , January 2009 , pp. 407 - 420
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Anand, L., Gurtin, M. E., Lele, S. P. and Gething, C., “A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results”, J. Mech. Phys. Solids 53 (2005) 1798–1826.CrossRefGoogle Scholar
[2]Gradshteyn, I. S. and Ryzhik, I. M, Table of integrals, series and products (Academic Press, New York, 2000).Google Scholar
[3]Gurtin, M. E., “On the plasticity of single crystals: free energy, microforces, plastic strain-gradients”, J. Mech. Phys. Solids 48 (2000) 989–1036.CrossRefGoogle Scholar
[4]Gurtin, M. E., Anand, L. and Lele, S. P., “Gradient single-crystal plasticity with free energy dependent on dislocation densities”, J. Mech. Phys. Solids 55 (2007) 1853–1878.CrossRefGoogle Scholar
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