# American Institute of Mathematical Sciences

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

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

 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

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.
Citation: Sanda Cleja-Ţigoiu, Raisa Paşcan. Non-local elasto-viscoplastic models with dislocations and non-Schmid effect. Discrete & 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. doi: 10.1016/0022-5096(77)90001-1. Google Scholar [2] L. Bortoloni and P. Cermelli, Dislocation patterns and work-hardening in crystalline plasticity,, Journal of Elasticity, 76 (2004), 113. doi: 10.1007/s10659-005-0670-1. Google Scholar [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. doi: 10.1016/S0020-7683(02)00491-2. Google Scholar [4] S. Cleja-Ţigoiu, Bifurcations of homogeneous deformations of the bar in finite elasto-plasticity,, European Journal of Mechanics A Solids, 15 (1996), 761. Google Scholar [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). doi: 10.1177/1081286512439059. Google Scholar [6] S. Cleja-Ţigoiu, Material forces in finite elastoplasticity with continuously distributed dislocations,, Int. J. Fracture, 147 (2007), 67. Google Scholar [7] S. Cleja-Ţigoiu, Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature,, in, (2011), 61. Google Scholar [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). Google Scholar [9] S. Cleja-Ţigoiu and E. Soós, Elastoplastic models with relaxed configurations and internal state variables,, Applied Mechanics Reviews, 43 (1990), 131. doi: 10.1115/1.3119166. Google Scholar [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. doi: 10.1016/0167-6636(96)00012-9. Google Scholar [11] P. Grudmundson, A unified treatment of strain gradient plasticity,, J. Mech. Phys. Solids, 52 (2004), 1379. doi: 10.1016/j.jmps.2003.11.002. Google Scholar [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. Google Scholar [13] P. Howard, "Partial Equations in Matlab 7.0,", 2005. Available from: \url{http://www.math.tamu.edu/~phoward/m308/matbasics.pdf}., (). Google Scholar [14] M. Kuroda, Crystal plasticity model accounting for pressure dependence of yielding and plastic expansion,, Scripta Materialia, 48 (2003), 605. doi: 10.1016/S1359-6462(02)00465-7. Google Scholar [15] J. Mandel, "Plasticité Classique et Viscoplasticité,", Course held at the Department of Mechanics of Solids, (1971). Google Scholar [16] G. W. Recktenwald, "Numerical Methods with Matlab,", Prentice Hall, (2000). Google Scholar [17] C. Teodosiu, A dynamic theory of dislocations and its applications to the theory of the elastic- plastic,, in, (1970), 837. Google Scholar [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. Google Scholar

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. doi: 10.1016/0022-5096(77)90001-1. Google Scholar [2] L. Bortoloni and P. Cermelli, Dislocation patterns and work-hardening in crystalline plasticity,, Journal of Elasticity, 76 (2004), 113. doi: 10.1007/s10659-005-0670-1. Google Scholar [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. doi: 10.1016/S0020-7683(02)00491-2. Google Scholar [4] S. Cleja-Ţigoiu, Bifurcations of homogeneous deformations of the bar in finite elasto-plasticity,, European Journal of Mechanics A Solids, 15 (1996), 761. Google Scholar [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). doi: 10.1177/1081286512439059. Google Scholar [6] S. Cleja-Ţigoiu, Material forces in finite elastoplasticity with continuously distributed dislocations,, Int. J. Fracture, 147 (2007), 67. Google Scholar [7] S. Cleja-Ţigoiu, Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature,, in, (2011), 61. Google Scholar [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). Google Scholar [9] S. Cleja-Ţigoiu and E. Soós, Elastoplastic models with relaxed configurations and internal state variables,, Applied Mechanics Reviews, 43 (1990), 131. doi: 10.1115/1.3119166. Google Scholar [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. doi: 10.1016/0167-6636(96)00012-9. Google Scholar [11] P. Grudmundson, A unified treatment of strain gradient plasticity,, J. Mech. Phys. Solids, 52 (2004), 1379. doi: 10.1016/j.jmps.2003.11.002. Google Scholar [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. Google Scholar [13] P. Howard, "Partial Equations in Matlab 7.0,", 2005. Available from: \url{http://www.math.tamu.edu/~phoward/m308/matbasics.pdf}., (). Google Scholar [14] M. Kuroda, Crystal plasticity model accounting for pressure dependence of yielding and plastic expansion,, Scripta Materialia, 48 (2003), 605. doi: 10.1016/S1359-6462(02)00465-7. Google Scholar [15] J. Mandel, "Plasticité Classique et Viscoplasticité,", Course held at the Department of Mechanics of Solids, (1971). Google Scholar [16] G. W. Recktenwald, "Numerical Methods with Matlab,", Prentice Hall, (2000). Google Scholar [17] C. Teodosiu, A dynamic theory of dislocations and its applications to the theory of the elastic- plastic,, in, (1970), 837. Google Scholar [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. Google Scholar
 [1] Michael Herty, Reinhard Illner. Coupling of non-local driving behaviour with fundamental diagrams. Kinetic & Related Models, 2012, 5 (4) : 843-855. doi: 10.3934/krm.2012.5.843 [2] Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks & Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015 [3] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [4] Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks & Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107 [5] Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036 [6] Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511 [7] Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037 [8] José A. Carrillo, Raluca Eftimie, Franca Hoffmann. Non-local kinetic and macroscopic models for self-organised animal aggregations. Kinetic & Related Models, 2015, 8 (3) : 413-441. doi: 10.3934/krm.2015.8.413 [9] Jan Friedrich, Oliver Kolb, Simone Göttlich. A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks & Heterogeneous Media, 2018, 13 (4) : 531-547. doi: 10.3934/nhm.2018024 [10] Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029 [11] Chiu-Yen Kao, Yuan Lou, Wenxian Shen. Random dispersal vs. non-local dispersal. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 551-596. doi: 10.3934/dcds.2010.26.551 [12] Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319 [13] Matteo Focardi. Vector-valued obstacle problems for non-local energies. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 487-507. doi: 10.3934/dcdsb.2012.17.487 [14] Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475 [15] Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701 [16] Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012 [17] Rafael Abreu, Cristian Morales-Rodrigo, Antonio Suárez. Some eigenvalue problems with non-local boundary conditions and applications. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2465-2474. doi: 10.3934/cpaa.2014.13.2465 [18] Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71 [19] Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105 [20] Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653

2018 Impact Factor: 0.545