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This note addresses finite plasticity under the constraint that plastic deformations are compatible. In this case, the total elastoplastic deformation of the medium is decomposed as y = ye ○ yp, where the plastic deformation yp is defined on the fixed reference configuration and the elastic deformation ye is a mapping from the varying intermediate configuration yp(Ω). Correspondingly, the energy of the medium features both Lagrangian (plastic, loads) and not Lagrangian contributions (elastic).
We present a variational formulation of the static elastoplastic problem in this setting and show that a solution is attained in a suitable class of admissible deformations. Possible extensions of the result, especially in the direction of quasistatic evolutions, are also discussed.
Stefanelli, Ulisse 1
@article{COCV_2019__25__A21_0,
author = {Stefanelli, Ulisse},
title = {Existence for dislocation-free finite plasticity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018014},
zbl = {1442.35449},
mrnumber = {3982967},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2018014/}
}
TY - JOUR AU - Stefanelli, Ulisse TI - Existence for dislocation-free finite plasticity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2018014/ DO - 10.1051/cocv/2018014 LA - en ID - COCV_2019__25__A21_0 ER -
Stefanelli, Ulisse. Existence for dislocation-free finite plasticity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 21. doi : 10.1051/cocv/2018014. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2018014/
[1] and , An incremental energy minimization state update algorithm for 3D phenomenological internal-variable SMA constitutive models based on isotropic flow potentials. Int. J. Numer. Methods Eng. 105 (2016) 197–220. | Zbl | MR | DOI
[2] , Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1976) 337–403. | Zbl | MR | DOI
[3] , Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinburgh Sect. A 88 (1988) 315–328. | Zbl | MR | DOI
[4] and , Frank energy for nematic elastomers: a nonlinear model. ESAIM: COCV 21 (2015) 372–377. | Zbl | MR | mathdoc-id
[5] , and , Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity. Arch. Ration. Mech. Anal. 224 (2017) 743–816. | Zbl | MR | DOI
[6] , and , A Note on Locking Materials and Gradient Polyconvexity. Preprint (2017). | arXiv | Zbl | MR
[7] , Mechanics of Incremental Deformations: Theory of Elasticity and Viscoelasticity of Initially Stressed Solids and Fluids, Including Thermodynamic Foundations and Applications to Finite Strain. John Wiley & Sons, Inc., New York, London, Sydney (1965). | MR
[8] , and , Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. Edinburgh Sect. A 458 (2002) 299–317. | Zbl | MR | DOI
[9] and , On the characterization of geometrically necessary dislocations in finite plasticity. J. Mech. Phys. Solids 49 (2001) 1539–1568. | Zbl | DOI
[10] , Mathematical Elasticity. Vol. 1 of Three Dimensional Elasticity. Elsevier (1988). | Zbl
[11] and , Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97 (1987) 171–188. | Zbl | MR | DOI
[12] , On the equivalence of FeFp and FpFe. J. Appl. Mech. 39 (1972) 287–230. | DOI
[13] , and , Relaxation of a model in finite plasticity with two slip systems. Math. Models Methods Appl. Sci. 23 (2013) 2111–2128. | Zbl | MR | DOI
[14] , Direct Methods in the Calculus of Variations, 2nd edn. Vol. 78 of Applied Mathematical Sciences. Springer, New York (2008). | Zbl | MR
[15] and , A minimization problem involving variation of the domain. Commun. Pure Appl. Math. 45 (1992) 871–897. | Zbl | MR | DOI
[16] , A proposal for a continuum theory of defective crystals. Arch. Ration. Mech. Anal. 96 (1986) 295–317. | Zbl | MR | DOI
[17] and , On the defect-preserving deformations in crystals. Int. J. Plast. 5 (1989) 337–369. | Zbl | DOI
[18] and , A critical revisiting of finite elasto-plasticity. SIAM J. Math. Anal. 47 (2015) 526–565. | Zbl | MR | DOI
[19] and , Inequalities in Mechanics and Physics. Springer, Berlin (1976). | Zbl | MR | DOI
[20] and , Local invertibility of Sobolev functions. SIAM J. Math. Anal. 26 (1995) 280–304. | Zbl | MR | DOI
[21] and , Equilibrium configurations of defective crystals. Arch. Ration. Mech. Anal. 120 (1992) 245–283. | Zbl | MR | DOI
[22] , and , Weak continuity and lower semicontinuity results for determinants. Arch. Ration. Mech. Anal. 178 (2005) 411–448. | Zbl | MR | DOI
[23] and , Existence results for a class of rate- independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55–91. | Zbl | MR
[24] , and , Gradient theory for plasticity via homogenization of discrete dislocations. J. Eur. Math. Soc. (JEMS) 12 (2010) 1231–1266. | Zbl | MR | DOI
[25] and , Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials. Proc. R. Soc. Edinburgh Sect. A 138 (2008) 1019–1041. | Zbl | MR | DOI
[26] and , Quasiconformal Mappings and Sobolev Spaces, Vol. 54. Kluwer Academic Publishers, Dordrecht, Germany (1990). | Zbl | MR | DOI
[27] and Finite plasticity in P⊤ P. Part I: constitutive model. Contin. Mech. Thermodyn. 29 (2017) 97–116. | Zbl | MR | DOI
[28] and , Finite plasticity in P⊤ P. Part II: quasistatic evolution and linearization. SIAM J. Math. Anal. 49 (2017) 1356–1384. | Zbl | MR | DOI
[29] , and , Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta. Mater. 53 (2005) 1821–1830. | DOI
[30] , Variational principles in the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 13 (1963) 179–191. | Zbl | MR | DOI
[31] , Variational principles for linear elastodynamics. Arch. Ration. Mech. Anal. 16 (1964) 34–50. | Zbl | MR | DOI
[32] , and , The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010). | MR | DOI
[33] and , Lectures on Mappings of Finite Distortion. Vol. 2096 of Lecture Notes in Mathematics. Springer, Cham (2014). | Zbl | MR | DOI
[34] , Dislocation-free plastic deformation under high stress. Mater. Sci. Eng. A 350 (2003) 1–7. | DOI
[35] and , Microstructural pattern formation in finite-deformation single-slip crystal plasticity under cyclic loading: relaxation vs. gradient plasticity. Comput. Methods Appl. Mech. Eng. 278 (2014) 765–793. | Zbl | MR | DOI
[36] , Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Ration. Mech. Anal. 4 (1959) 273–334. | Zbl | MR | DOI
[37] and , Mathematical Methods in Continuum Mechanics of Solids. Interaction of Mechanics and Mathematics. Springer, Cham/Heidelberg (2018). | Zbl | MR
[38] and , Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discret. Contin. Dyn. Syst. Ser. S 5 (2012) 591–604. | Zbl | MR
[39] , and , Existence results for incompressible magnetoelasticity. Discret. Contin. Dyn. Syst. 35 (2015) 2615–2623. | Zbl | MR | DOI
[40] , Elastic-plastic deformation at finite strains. J. Appl. Mech. 36 (1969) 1–6. | Zbl | DOI
[41] , A First Course in Sobolev Spaces. Vol 105 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, USA (2009). | Zbl | MR
[42] , Duality in constitutive formulation of finite-strain elastoplasticity based on F = Fe Fp and F = FpFe decompositions. Int. J. Plast. 15 (1999) 1277–1290. | Zbl | DOI
[43] , Plasticity Theory. Macmillan Publishing Company, New York (1990). | Zbl
[44] , Plasticité Classique et Viscoplasticité. Vol. 97 of CISM Courses and Lectures. Springer-Verlag, Berlin (1972). | Zbl | MR
[45] and . Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differ. Equ. 22 (2005) 73–99. | Zbl | MR | DOI
[46] , , and , Dynamic observation of dislocation-free plastic deformation in gold thin foils. Mater. Sci. Eng. A 350 (2003) 8–16. | DOI
[47] , The Thermomechanics of Plasticity and Fracture. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1992). | Zbl | MR
[48] , Finite elastoplasticity, Lie groups and geodesics on SL(d), in Geometry, Dynamics, and Mechanics, edited by , and . Springer-Verlag (2002) 61–90. | Zbl | MR
[49] , Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15 (2003) 351–382. | Zbl | MR | DOI
[50] , Existence of minimizers in incremental elasto-plasticity with finite strains. SIAM J. Math. Anal. 36 (2004) 384–404. | Zbl | MR | DOI
[51] and , Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity. Z. Angew. Math. Mech. (ZAMM) 86 (2006) 233–250. | Zbl | MR | DOI
[52] and , Rate-independent systems, in Theory and Application. Vol. 193 of Applied Mathematical Sciences. Springer, New York (2015). | Zbl | MR
[53] , Multiple Integrals in the Calculus of Variations. Springer-Verlag, Berlin (1966). | Zbl | MR | DOI
[54] , and , On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11 (1994) 217–243. | MR | Zbl | mathdoc-id | DOI
[55] , A critical review of the state of finite plasticity. J. Appl. Math. Phys. 41 (1990) 315–394. | MR | Zbl
[56] , and , Numerical approximation of incremental infinitesimal gradient plasticity. Int. J. Numer. Methods Eng. 77 (2009) 414–436. | MR | Zbl | DOI
[57] , Decomposition of strain measures and their rates in finite deformation elastoplasticity. Int. J. Solids Struct. 15 (1979) 155–166. | Zbl | DOI
[58] and , Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47 (1999) 397–462 | MR | Zbl | DOI
[59] , and , A theory of subgrain dislocation structures. J. Mech. Phys. Solids 48 (2000) 2077–2114. | MR | Zbl | DOI
[60] and , Kinematic description of crystal plasticity in the finite kinematic framework: a micromechanical understanding of F = FeFp. J. Mech. Phys. Solids 67 (2014) 40–61. | MR | Zbl | DOI
[61] , and , Derivation of F = FeFp as the continuum limit of crystalline slip. J. Mech. Phys. Solids 89 (2016) 231–254. | MR | Zbl | DOI
[62] , Weak convergence and completely additive vector functions on a set. Sibir. Math. 9 (1968) 1039–1045. | Zbl | DOI
[63] , Nonlinear Partial Differential Equations With Applications, 2nd edn. Vol. 153 of International Series of Numerical Mathematics. Birkhäuser/Springer, Basel AG, Basel (2013). | MR | Zbl | DOI
[64] and , A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis. Z. Angew. Math. Phys. 69 (2018) 55. | MR | Zbl | DOI
[65] and , Existence of energy minimizers for magnetostrictive materials. SIAM J. Math. Anal. 36 (2005) 2004–2019. | MR | Zbl | DOI
[66] , and , Plasticity of bcc micropillars controlled by competition between dislocation multiplication and depletion. Acta Mater. 61 (2013) 3233–3241. | DOI
[67] , , , , , et al., Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism. Science 300 (2003) 464–467. | DOI
[68] , Regularity properties of deformations with finite energy. Arch. Ration. Mech. Anal. 100 (1988) 105–127. | MR | Zbl | DOI
[69] , Almost-everywhere injectivity in nonlinear elasticity. Proc. R. Soc. Edinburgh Sect. A 109 (1988) 79–95. | MR | Zbl | DOI
[70] and , The Nonlinear Field Theories Handbuch der Physik, Band III/3. Springer-Verlag, Berlin (1965).
[71] , , and , Sample dimensions influence strength and crystal plasticity. Science 305 (2004) 986–989. | DOI
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