Non-local elasto-viscoplastic models with dislocations and non-Schmid effect

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December 2013, 6(6): 1621-1639. doi: 10.3934/dcdss.2013.6.1621

Non-local elasto-viscoplastic models with dislocations and non-Schmid effect

Sanda Cleja-Ţigoiu ^1, and Raisa Paşcan ^1,

1.	Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei nr. 14, 010014 Bucharest, Romania, Romania

Received June 2012 Revised September 2012 Published April 2013

Abstract
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We propose a non-local model with dislocation densities and non-Schmid effect in the finite elasto-plasticity framework, which accounts for the dissipation postulate formulated through a principle of the free energy imbalance. Our goal is to characterize the scalar plastic velocities and activation condition in order to be compatible with the principle of the imbalanced free energy. The activation condition is expressed in terms of the generalized resolved stress, which is dependent not only on the Mandel stress measure, but also on the gradient of the scalar dislocation density. We analyze numerically how the model behaves for a simple shear problem into a layer when only one slip system is activated.

Keywords: non-Schmid effect., Finite elasto-viscoplastic models, dislocations, non-local behaviour.

Mathematics Subject Classification: Primary: 74C20; Secondary: 74D9.

Citation: Sanda Cleja-Ţigoiu, Raisa Paşcan. Non-local elasto-viscoplastic models with dislocations and non-Schmid effect. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1621-1639. doi: 10.3934/dcdss.2013.6.1621

References:

[1]	R. J. Asaro and J. R, Rice, Strain localization in ductile single crystals, J. Mech. Phys. Solids, 25 (1977), 309-338. doi: 10.1016/0022-5096(77)90001-1.
[2]	L. Bortoloni and P. Cermelli, Dislocation patterns and work-hardening in crystalline plasticity, Journal of Elasticity, 76 (2004), 113-138. doi: 10.1007/s10659-005-0670-1.
[3]	P. Cermelli and M. E. Gurtin, Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations, International Journal of Solids and Structures, 39 (2002), 6281-6309. doi: 10.1016/S0020-7683(02)00491-2.
[4]	S. Cleja-Ţigoiu, Bifurcations of homogeneous deformations of the bar in finite elasto-plasticity, European Journal of Mechanics A Solids, 15 (1996), 761-786.
[5]	S. Cleja-Ţigoiu, Non-local elasto-viscoplastic models with dislocations in finite elasto-plasticity. Part I: Constitutive framework, Mathematics and Mechanics of Solids, (2012), accepted. doi: 10.1177/1081286512439059.
[6]	S. Cleja-Ţigoiu, Material forces in finite elastoplasticity with continuously distributed dislocations, Int. J. Fracture, 147 (2007), 67-81.
[7]	S. Cleja-Ţigoiu, Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature, in "Recent Progress in the Mathematics of Defects," Springer, Dordrecht, (2011), 61-75.
[8]	S. Cleja-Ţigoiu and R. Paşcan, Non-local elasto-viscoplastic models with dislocations in finite elasto-plasticity. Part II: Dislocation density, International Journal of the Physics and Mechanics of Solids, (2011), submitted.
[9]	S. Cleja-Ţigoiu and E. Soós, Elastoplastic models with relaxed configurations and internal state variables, Applied Mechanics Reviews, 43 (1990), 131-151. doi: 10.1115/1.3119166.
[10]	M. Dao and R. J. Asaro, Localized deformation modes and non-Schmid effects in crystalline solids. Part I. Critical conditions of localization, Mechanics of Materials, 23 (1996), 71-102. doi: 10.1016/0167-6636(96)00012-9.
[11]	P. Grudmundson, A unified treatment of strain gradient plasticity, J. Mech. Phys. Solids, 52 (2004), 1379-1406. doi: 10.1016/j.jmps.2003.11.002.
[12]	M. E. Gurtin and L. Anand, The decomposition F=F$^e$F$^p$, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous, Int. J. Plast., 21 (2005), 1686-1719.
[13]	P. Howard, "Partial Equations in Matlab 7.0," 2005. Available from: http://www.math.tamu.edu/~phoward/m308/matbasics.pdf.
[14]	M. Kuroda, Crystal plasticity model accounting for pressure dependence of yielding and plastic expansion, Scripta Materialia, 48 (2003), 605-610. doi: 10.1016/S1359-6462(02)00465-7.
[15]	J. Mandel, "Plasticité Classique et Viscoplasticité," Course held at the Department of Mechanics of Solids, September-October 1971, International Centre for Mechanical Sciences, Udine, Courses and Lectures, No. 97, Springer-Verlag, Vienna-New York, 1972.
[16]	G. W. Recktenwald, "Numerical Methods with Matlab," Prentice Hall, New Jersey, 2000.
[17]	C. Teodosiu, A dynamic theory of dislocations and its applications to the theory of the elastic- plastic, in "Fundamental Aspects of Dislocation Theory," Vol. II (eds. J. A. Simmons, R. de Witt and R. Bullough), Conference Proceedings, National Bureau of Standards, (1970), 837-876.
[18]	C. Teodosiu and F. Sidoroff, A physical theory of finite elasto-viscoplastic behaviour of single crystal, International Journal of Engineering Science, 14 (1976), 165-176.

show all references

References:

[1]	R. J. Asaro and J. R, Rice, Strain localization in ductile single crystals, J. Mech. Phys. Solids, 25 (1977), 309-338. doi: 10.1016/0022-5096(77)90001-1.
[2]	L. Bortoloni and P. Cermelli, Dislocation patterns and work-hardening in crystalline plasticity, Journal of Elasticity, 76 (2004), 113-138. doi: 10.1007/s10659-005-0670-1.
[3]	P. Cermelli and M. E. Gurtin, Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations, International Journal of Solids and Structures, 39 (2002), 6281-6309. doi: 10.1016/S0020-7683(02)00491-2.
[4]	S. Cleja-Ţigoiu, Bifurcations of homogeneous deformations of the bar in finite elasto-plasticity, European Journal of Mechanics A Solids, 15 (1996), 761-786.
[5]	S. Cleja-Ţigoiu, Non-local elasto-viscoplastic models with dislocations in finite elasto-plasticity. Part I: Constitutive framework, Mathematics and Mechanics of Solids, (2012), accepted. doi: 10.1177/1081286512439059.
[6]	S. Cleja-Ţigoiu, Material forces in finite elastoplasticity with continuously distributed dislocations, Int. J. Fracture, 147 (2007), 67-81.
[7]	S. Cleja-Ţigoiu, Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature, in "Recent Progress in the Mathematics of Defects," Springer, Dordrecht, (2011), 61-75.
[8]	S. Cleja-Ţigoiu and R. Paşcan, Non-local elasto-viscoplastic models with dislocations in finite elasto-plasticity. Part II: Dislocation density, International Journal of the Physics and Mechanics of Solids, (2011), submitted.
[9]	S. Cleja-Ţigoiu and E. Soós, Elastoplastic models with relaxed configurations and internal state variables, Applied Mechanics Reviews, 43 (1990), 131-151. doi: 10.1115/1.3119166.
[10]	M. Dao and R. J. Asaro, Localized deformation modes and non-Schmid effects in crystalline solids. Part I. Critical conditions of localization, Mechanics of Materials, 23 (1996), 71-102. doi: 10.1016/0167-6636(96)00012-9.
[11]	P. Grudmundson, A unified treatment of strain gradient plasticity, J. Mech. Phys. Solids, 52 (2004), 1379-1406. doi: 10.1016/j.jmps.2003.11.002.
[12]	M. E. Gurtin and L. Anand, The decomposition F=F$^e$F$^p$, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous, Int. J. Plast., 21 (2005), 1686-1719.
[13]	P. Howard, "Partial Equations in Matlab 7.0," 2005. Available from: http://www.math.tamu.edu/~phoward/m308/matbasics.pdf.
[14]	M. Kuroda, Crystal plasticity model accounting for pressure dependence of yielding and plastic expansion, Scripta Materialia, 48 (2003), 605-610. doi: 10.1016/S1359-6462(02)00465-7.
[15]	J. Mandel, "Plasticité Classique et Viscoplasticité," Course held at the Department of Mechanics of Solids, September-October 1971, International Centre for Mechanical Sciences, Udine, Courses and Lectures, No. 97, Springer-Verlag, Vienna-New York, 1972.
[16]	G. W. Recktenwald, "Numerical Methods with Matlab," Prentice Hall, New Jersey, 2000.
[17]	C. Teodosiu, A dynamic theory of dislocations and its applications to the theory of the elastic- plastic, in "Fundamental Aspects of Dislocation Theory," Vol. II (eds. J. A. Simmons, R. de Witt and R. Bullough), Conference Proceedings, National Bureau of Standards, (1970), 837-876.
[18]	C. Teodosiu and F. Sidoroff, A physical theory of finite elasto-viscoplastic behaviour of single crystal, International Journal of Engineering Science, 14 (1976), 165-176.

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