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**On the plasticity of single crystals: Free energy, microforces, plastic-strain gradients.**
*(English)*
Zbl 0988.74021

Summary: This study develops a general theory of crystalline plasticity based on: classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on plastic strain-gradients. The microforce balances are shown to be equivalent to yield conditions for the individual slip systems, conditions that account for variations in free energy due to slip. When this energy is the sum of an elastic strain energy and a defect energy quadratic in the plastic-strain gradients, the resulting theory has a form identical to classical crystalline plasticity except that the yield conditions contain an additional term involving the Laplacian of the plastic strain. The field equations consist of a system of PDEs that represent the nonlocal yield conditions coupled to the classical PDE that represents the standard force balance. These are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip.

We develop a viscoplastic regularization of basic equations that obviates the need to determine the active slip systems. As a second aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. As an application of the theory, we discuss the special case of single slip. Specific solutions are presented: for a single shear band connecting constant slip-states, and for a periodic array of shear bands.

We develop a viscoplastic regularization of basic equations that obviates the need to determine the active slip systems. As a second aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. As an application of the theory, we discuss the special case of single slip. Specific solutions are presented: for a single shear band connecting constant slip-states, and for a periodic array of shear bands.

### MSC:

74E15 | Crystalline structure |

74C15 | Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) |

82D25 | Statistical mechanics of crystals |

### Keywords:

ductility; crystalline plasticity; macroforces; microforces; rate-independent constitutive theory; plastic strain-gradients; free energy; nonlocal yield conditions; force balance; viscoplastic regularization; single slip; single shear band; array of shear bands
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\textit{M. E. Gurtin}, J. Mech. Phys. Solids 48, No. 5, 989--1036 (2000; Zbl 0988.74021)

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### References:

[1] | Aifantis, E.C., The physics of plastic deformation, International journal of plasticity, 3, 211-247, (1987) · Zbl 0616.73106 |

[2] | Anand, L.; Kothari, M., A computational procedure for rate-dependent crystal plasticity, Journal of the mechanics and physics of solids, 44, 525-558, (1996) · Zbl 1054.74549 |

[3] | Asaro, R.J., Micromechanics of crystals and polycrystals, Advances in applied mechanics, 23, 1-115, (1983) |

[4] | Asaro, R.J., Crystal plasticity, Journal of applied mechanics, 50, 921-934, (1983) · Zbl 0557.73033 |

[5] | Asaro, R.J.; Needleman, A., Texture development and strain hardening in rate dependent polycrystals, Acta metallurgica, 33, 923-953, (1985) |

[6] | Asaro, R.J.; Rice, J.R., Strain localization in ductile single crystals, Journal of mechanics and physics of solids, 25, 309-338, (1977) · Zbl 0375.73097 |

[7] | Ashby, M.F., The deformation of plastically non-homogeneous alloys, Philosophical magazine, 21, 399-424, (1970) |

[8] | Ashby, M.F., The deformation of plastically non-homogeneous alloys, () |

[9] | Bassani, J.L., Plastic flow of crystals, Advances in applied mechanics, 30, 191-258, (1993) · Zbl 0803.73009 |

[10] | Cermelli, P., Gurtin, M.E., 2000. On the relation of Fe, {\bfF}p, ∇{\bfF}e and ∇{\bfF}p to geometrically necessary dislocations. (Forthcoming) |

[11] | Cermelli, P., Sellars, S., Fried, E., 2000. Finite crystal plasticity based on a microstructural model for defects. (Forthcoming) |

[12] | Coleman, B.D.; Gurtin, M.E., Thermodynamics with internal state variables, Journal of chemical physics, 47, 672-675, (1967) |

[13] | Coleman, B.D.; Noll, W., The thermodynamics of elastic materials with heat conduction and viscosity, Archive for rational mechanics and analysis, 13, 167-178, (1963) · Zbl 0113.17802 |

[14] | Cottrell, A.H., The mechanical properties of matter, (1964), Wiley New York |

[15] | Drucker, D.C., Some implications of work hardening and ideal plasticity, Quarterly of applied mathematics, 7, 411-419, (1950) · Zbl 0035.41203 |

[16] | Fleck, L.A.; Hutchinson, J.W., A phenomenological theory for strain gradient effects in plasticity, Journal of the mechanics and physics of solids, 41, 1825-1857, (1993) · Zbl 0791.73029 |

[17] | Fleck, N.A.; Muller, G.M.; Ashby, M.F.; Hutchinson, J.W., Strain gradient plasticity: theory and experiment, Acta metallurgica, 42, 475-487, (1994) |

[18] | Fried, E.; Gurtin, M.E., Continuum theory of thermally induced phase transitions based on an order parameter, Physica D, 68, 326-343, (1993) · Zbl 0793.35049 |

[19] | Fried, E.; Gurtin, M.E., Dynamic solid – solid phase transitions with phase characterized by an order parameter, Physica D, 72, 287-308, (1994) · Zbl 0812.35164 |

[20] | Gurtin, M.E., An introduction to continuum mechanics, (1981), Academic Press New York · Zbl 0559.73001 |

[21] | Gurtin, M.E., Generalized ginzburg – landau and cahn – hilliard equations based on a microforce balance, Physica D, 92, 178-192, (1996) · Zbl 0885.35121 |

[22] | Gurtin, M.E., 1998. On a gradient theory of crystalline plasticity. Res. Rept. 98-CNA-004, Math. Dept., Carnegie-Mellon U., Pittsburgh |

[23] | Gurtin, M.E., 1999. On the plasticity of single crystals: free energy, microforces, plastic strain gradients. Res. Rept. 99-CNA-009, Math. Dept., Carnegie-Mellon U., Pittsburgh |

[24] | Havner, K.S., Finite deformation of crystalline solids, (1992), Cambridge University Press London · Zbl 0774.73001 |

[25] | Hill, R.; Rice, J.R., Constitutive analysis of elastic-plastic crystals at arbitrary strain, Journal of the mechanics and physics of solids, 20, 401-413, (1972) · Zbl 0254.73031 |

[26] | Lee, E.H., Elastic-plastic deformation at finite strains, Journal of applied mechanics, 36, 1-6, (1969) · Zbl 0179.55603 |

[27] | Lucchesi, M.; Owen, D.R.; Podio-Guidugli, P., Archive for rational mechanics and analysis, 117, 53-96, (1992) |

[28] | Lucchesi, M.; Podio-Guidugli, P., International journal of plasticity, 11, 1-14, (1995) · Zbl 0819.73017 |

[29] | Mandel, J., Generalisation de la theorie de la plasticite de W T Koiter, International journal of solids and structures, 1, 273-295, (1965) |

[30] | Muhlhaus, H.B.; Aifantis, E.C., A variational principle for gradient plasticity, International journal of solids and structures, 28, 845-857, (1991) · Zbl 0749.73029 |

[31] | Muhlhaus, H.B.; Aifantis, E.C., The influence of microstructure-induced gradients on the localization in viscoplastic materials, Acta mechanica, 89, 217-231, (1991) · Zbl 0749.73033 |

[32] | Naghdi, P.M.; Srinivasa, A.R., A dynamical theory of structured solids. I. basic developments, Philosophical transactions of the royal society of London, 345A, 425-458, (1993) · Zbl 0807.73001 |

[33] | Naghdi, P.M.; Srinivasa, A.R., Characterisation of dislocations and their influence on plastic deformation in single crystals, International journal of solids and structures, 7, 1157-1182, (1994) · Zbl 0899.73456 |

[34] | Nye, J.F., Some geometrical relations in dislocated crystals, Acta metallurgica, 1, 153-162, (1953) |

[35] | Peirce, D.; Asaro, R.J.; Needleman, A.I., An analysis of nonuniform and localized deformation in ductile single crystals, Acta metallurgica, 30, 1087-1119, (1982) |

[36] | Rice, J.R., Inelastic constitutive relations for solids: an internal-variable theory and its applications to metal plasticity, Journal of the mechanics and physics of solids, 19, 443-455, (1971) · Zbl 0235.73002 |

[37] | Schwartz, A.J., Stölken, J.S., King, W.E., Campbell, G.H., Lassila, D.H., Sun, S., Adams, B.L., 1999. Analysis of compression behavior of a [011] Ta single crystal with orientation imaging microscopy and crystal plasticity. Rept. UCRL-JC-133402, Lawrence Livermore National Laboratory, Livermore |

[38] | Shu, J.Y.; Fleck, N.A., Strain gradient crystal plasticity: size-dependent deformation of bicrystals, Journal of the mechanics and physics of solids, 47, 297-324, (1999) · Zbl 0956.74006 |

[39] | Taylor, G.I.; Elam, C.F., The distortion of an aluminum crystal during a tensile test, Proceedings of the royal society of London, 102A, 643-667, (1923) |

[40] | Taylor, G.I.; Elam, C.F., The plastic extension and fracture of aluminum crystals, Proceedings of the royal society of London, 108A, 28-51, (1925) |

[41] | Taylor, G.I., Plastic strain in metals, Journal of the institute of metals, 62, 307-325, (1938) |

[42] | Taylor, G.I., Analysis of plastic strain in a cubic crystal, () · Zbl 0022.08604 |

[43] | Teodosiu, C., 1970, A dynamical theory of dislocations and its application to the theory of the elasto-plastic continuum, Simmons, J.A., de Wit, R., Bollough, R., Proceedings of the Conference on Fundamental Aspects of Dislocation Theory, 1969. Natl. Bur. Stand. Spec. Publ., 317(2), 837-876 |

[44] | Teodosiu, C.; Sidoroff, F., A physical theory of the finite elasto-viscoplastic behaviour of single crystals, International journal of engineering science, 14, 165-176, (1976) · Zbl 0329.73037 |

[45] | Truesdell, C.A.; Noll, W., The non-linear field theories of mechanics, () · Zbl 0779.73004 |

[46] | Zbib, H.M.; Aifantis, E.C., On the gradient-dependent theory of plasticity and shear banding, Acta mechanica, 92, 209-225, (1992) · Zbl 0751.73022 |

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