# American Institute of Mathematical Sciences

October  2020, 25(10): 3769-3805. doi: 10.3934/dcdsb.2020240

## Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

 1 Ecole polytechnique, CMAP, Route de Saclay, 91128 Palaiseau, France 2 Universidade de Lisboa, Faculdade de Ciências, Departamento de Matemática, CMAFcIO, Portugal

Received  August 2019 Revised  June 2020 Published  October 2020 Early access  July 2020

A general model of incompatible small-strain elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of the Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the limit case when the incompatibility modulus goes to infinity. Several examples are provided to illustrate this property and the physical meaning of the incompatibility modulus in connection with the dissipative nature of the processes under consideration.

Citation: Samuel Amstutz, Nicolas Van Goethem. Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3769-3805. doi: 10.3934/dcdsb.2020240
##### References:
 [1] G. Allaire, F. Jouve and N. Van Goethem, Damage and fracture evolution in brittle materials by shape optimization methods, J. Comput. Phys., 230 (2011), 5010-5044.  doi: 10.1016/j.jcp.2011.03.024. [2] C. Amrouche, Ph. G. Ciarlet, L. Gratie and S. Kesavan, On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., 86 (2006), 116-132.  doi: 10.1016/j.matpur.2006.04.004. [3] C. Amrouche, Ph. G. Ciarlet, L. Gratie and S. Kesavan, On Saint Venant's compatibility conditions and Poincaré's lemma, C. R. Math. Acad. Sci. Paris, 342 (2006), 887-891.  doi: 10.1016/j.crma.2006.03.026. [4] S. Amstutz and N. Van Goethem, The incompatibility operator: From Riemann's intrinsic view of geometry to a new model of elasto-plasticity, In J. F. Rodrigues and M. Hintermüller, editors, CIM Series in Mathematical Science (2019). Springer, (Hal report 01789190), pp 33–70. doi: 10.1007/978-3-030-33116-0_2. [5] S. Amstutz and N. Van Goethem, Analysis of the incompatibility operator and application in intrinsic elasticity with dislocations, SIAM J. Math. Anal., 48 (2016), 320-348.  doi: 10.1137/15M1020113. [6] S. Amstutz and N. Van Goethem, Incompatibility-governed elasto-plasticity for continua with dislocations, Proc. A., 473 (2017), 20160734, 21 pp. doi: 10.1098/rspa.2016.0734. [7] A. J. C. Barré de Saint-Venant and M. Navier, Première section : De la résistance des solides par Navier. - 3e éd. avec des notes et des appendices par M. Barré de Saint-Venant. tome 1, In Résumé des leçons données l'Ecole des Ponts et Chaussées sur l'application de la mécanique à l'établissement des constructions et des machines. Dunod, Paris, 1864. [8] E. Beltrami, Sull'interpretazione meccanica delle formule di Maxwell, Mem. dell'Accad. di Bologna, 7 (1886), 1-38. [9] H. Brézis, Functional Analysis. Theory and Applications. (Analyse Fonctionnelle. Théorie et Applications.), Collection Mathématiques Appliquées pour la Maȋtrise. Paris: Masson., 1983. [10] P. G. Ciarlet, Three-Dimensional Elasticity, Vol.1, Masson, Paris, 1986. [11] P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005. [12] Ph. G. Ciarlet, An introduction to differential geometry with applications to elasticity, J. Elasticity, 78-79 (2005), 1-215.  doi: 10.1007/s10659-005-4738-8. [13] Ph. G. Ciarlet and P. Ciarlet Jr., Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Models Methods Appl. Sci., 15 (2005), 259-271.  doi: 10.1142/S0218202505000352. [14] Ph. G. Ciarlet, L. Gratie and C. Mardare, Intrinsic methods in elasticity: A mathematical survey, Discrete Contin. Dyn. Syst., 23 (2009), 133-164.  doi: 10.3934/dcds.2009.23.133. [15] Ph. G. Ciarlet and C. Mardare, Intrinsic formulation of the displacement-traction problem in linearized elasticity, Math. Models Methods Appl. Sci., 24 (2014), 1197–1216. doi: 10.1142/S0218202513500814. [16] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Springer-Verlag, Berlin, 1988. Functional and variational methods, With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily, Translated from the French by Ian N. Sneddon. doi: 10.1007/978-3-642-61566-5. [17] L. Donati, Illustrazione al teorema del Menabrea, Memorie della Accademia delle Scienze dell'Istituto di Bologna, 10 (1860), 267-274. [18] G. Duvaut, Mécanique des Milieux Continus, Collection Mathématiques appliquées pour la maȋtrise. Masson, 1990. [19] Gilles A. Francfort, Alessandro Giacomini and Jean-Jacques Marigo, The elasto-plastic exquisite corpse: A Suquet legacy, J. Mech. Phys. Solids, 97 (2016), 125-139.  doi: 10.1016/j.jmps.2016.02.002. [20] G. Geymonat and F. Krasucki, Some remarks on the compatibility conditions in elasticity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 175-181. [21] G. Geymonat and F. Krasucki, Beltrami's solutions of general equilibrium equations in continuum mechanics, C. R. Math. Acad. Sci. Paris, 342 (2006), 359-363.  doi: 10.1016/j.crma.2005.12.031. [22] G. Geymonat and F. Krasucki, Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains, Commun. Pure Appl. Anal., 8 (2009), 295–309. doi: 10.3934/cpaa.2009.8.295. [23] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. (Extended version of the 1979 publ.), Springer Series in Computational Mathematics, 5. Berlin etc.: Springer-Verlag. 1986. doi: 10.1007/978-3-642-61623-5. [24] P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. [25] M. E. Gurtin, The linear theory of elasticity, In C. Truesdell, editor, Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells, pages 1–295. Springer Berlin Heidelberg, Berlin, Heidelberg, 1973. [26] H. Kozono and T. Yanagisawa, $L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J., 58 (2009), 1853-1920.  doi: 10.1512/iumj.2009.58.3605. [27] E. Kröner, Continuum theory of defects, In R. Balian, editor, Physiques des défauts, Les Houches Session XXXV (Course 3). North-Holland, Amsterdam, 1980. [28] G. Maggiani, R. Scala and N. Van Goethem, A compatible-incompatible decomposition of symmetric tensors in ${L}^p$ with application to elasticity, Math. Meth. Appl. Sci, 38 (2015), 5217-5230.  doi: 10.1002/mma.3450. [29] J. Mawhin, Les Modèles Mathématiques Sont-ils des Modèles à suivre?, Académie Royale de Belgique, 2017. [30] D. L. McDowell, Multiscale cristalline plasticity for material design, Computational Materials System Design, D. Shin and J. Saal Eds., 2018. [31] R. D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16 (1964), 51-78.  doi: 10.1007/BF00248490. [32] J. J. Moreau, Duality characterization of strain tensor distributions in an arbitrary open set, J. Math. Anal. Appl., 72 (1979), 760-770.  doi: 10.1016/0022-247X(79)90263-4. [33] P. Podio-Guidugli, The compatibility constraint in linear elasticity, In Donald E. Carlson and Yi-Chao Chen, editors, Advances in Continuum Mechanics and Thermodynamics of Material Behavior: In Recognition of the 60th Birthday of Roger L. Fosdick, pages 393–398. Springer Netherlands, Dordrecht, 2000. [34] W. Prager and P. G. Hodge Jr., Theory of Perfectly Plastic Solids, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951. [35] Ben Schweizer, On friedrichs inequality, helmholtz decomposition, vector potentials, and the div-curl lemma, In Elisabetta Rocca, Ulisse Stefanelli, Lev Truskinovsky, and Augusto Visintin, editors, Trends in Applications of Mathematics to Mechanics, pages 65–79. Springer International Publishing, 2018. doi: 10.1007/978-3-319-75940-1_4. [36] B. Sun, Incompatible deformation field and Riemann curvature tensor, Appl. Math. Mech., 38 (2017), 311-332.  doi: 10.1007/s10483-017-2176-8. [37] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1979. Studies in Mathematics and its Applications, Vol. 2. [38] T. W. Ting, St Venant's compatibility conditions and basic problems in elasticity, Rocky Mountain J. Math., 7 (1977), 47-52.  doi: 10.1216/RMJ-1977-7-1-47. [39] N. Van Goethem, Strain incompatibility in single crystals: Kröner's formula revisited, J. Elast., 103 (2011), 95-111.  doi: 10.1007/s10659-010-9275-4. [40] N. Van Goethem, Direct expression of incompatibility in curvilinear systems, ANZIAM J., 58 (2016), 33-50.  doi: 10.1017/S1446181116000158. [41] N. Van Goethem, Incompatibility-governed singularities in linear elasticity with dislocations, Math. Mech. Solids, 22 (2017), 1688-1695.  doi: 10.1177/1081286516642817. [42] N. Van Goethem and F. Dupret, A distributional approach to $2{D}$ Volterra dislocations at the continuum scale, European J. Appl. Math., 23 (2012), 417-439.  doi: 10.1017/S0956792512000010. [43] N. Van Goethem and F. Dupret, A distributional approach to the geometry of $2{D}$ dislocations at the continuum scale, Ann. Univ. Ferrara Sez. VII Sci. Mat., 58 (2012), 407-434.  doi: 10.1007/s11565-012-0149-5. [44] V. Volterra, Sur l'équilibre des corps élastiques multiplement connexes, Ann. Sci. École Norm. Sup., 24 (1907), 401–517. doi: 10.24033/asens.583. [45] W. von Wahl, Estimating $\nabla u$ by div $u$ and curl $u$, Math. Methods Appl. Sci., 15 (1992), 123-143.  doi: 10.1002/mma.1670150206. [46] M. Xavier, E. A. Fancello, J. M. C. Farias, N. Van Goethem and A. A. Novotny, Topological derivative-based fracture modelling in brittle materials: A phenomenological approach, Engineering Fracture Mechanics, 179 (2017), 13-27.  doi: 10.1016/j.engfracmech.2017.04.005. [47] A. Yavari and A. Goriely, Riemann–Cartan geometry of nonlinear dislocation mechanics, Arch. Ration. Mech. Anal., 205 (2012), 59-118.  doi: 10.1007/s00205-012-0500-0.

show all references

##### References:
 [1] G. Allaire, F. Jouve and N. Van Goethem, Damage and fracture evolution in brittle materials by shape optimization methods, J. Comput. Phys., 230 (2011), 5010-5044.  doi: 10.1016/j.jcp.2011.03.024. [2] C. Amrouche, Ph. G. Ciarlet, L. Gratie and S. Kesavan, On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., 86 (2006), 116-132.  doi: 10.1016/j.matpur.2006.04.004. [3] C. Amrouche, Ph. G. Ciarlet, L. Gratie and S. Kesavan, On Saint Venant's compatibility conditions and Poincaré's lemma, C. R. Math. Acad. Sci. Paris, 342 (2006), 887-891.  doi: 10.1016/j.crma.2006.03.026. [4] S. Amstutz and N. Van Goethem, The incompatibility operator: From Riemann's intrinsic view of geometry to a new model of elasto-plasticity, In J. F. Rodrigues and M. Hintermüller, editors, CIM Series in Mathematical Science (2019). Springer, (Hal report 01789190), pp 33–70. doi: 10.1007/978-3-030-33116-0_2. [5] S. Amstutz and N. Van Goethem, Analysis of the incompatibility operator and application in intrinsic elasticity with dislocations, SIAM J. Math. Anal., 48 (2016), 320-348.  doi: 10.1137/15M1020113. [6] S. Amstutz and N. Van Goethem, Incompatibility-governed elasto-plasticity for continua with dislocations, Proc. A., 473 (2017), 20160734, 21 pp. doi: 10.1098/rspa.2016.0734. [7] A. J. C. Barré de Saint-Venant and M. Navier, Première section : De la résistance des solides par Navier. - 3e éd. avec des notes et des appendices par M. Barré de Saint-Venant. tome 1, In Résumé des leçons données l'Ecole des Ponts et Chaussées sur l'application de la mécanique à l'établissement des constructions et des machines. Dunod, Paris, 1864. [8] E. Beltrami, Sull'interpretazione meccanica delle formule di Maxwell, Mem. dell'Accad. di Bologna, 7 (1886), 1-38. [9] H. Brézis, Functional Analysis. Theory and Applications. (Analyse Fonctionnelle. Théorie et Applications.), Collection Mathématiques Appliquées pour la Maȋtrise. Paris: Masson., 1983. [10] P. G. Ciarlet, Three-Dimensional Elasticity, Vol.1, Masson, Paris, 1986. [11] P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005. [12] Ph. G. Ciarlet, An introduction to differential geometry with applications to elasticity, J. Elasticity, 78-79 (2005), 1-215.  doi: 10.1007/s10659-005-4738-8. [13] Ph. G. Ciarlet and P. Ciarlet Jr., Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Models Methods Appl. Sci., 15 (2005), 259-271.  doi: 10.1142/S0218202505000352. [14] Ph. G. Ciarlet, L. Gratie and C. Mardare, Intrinsic methods in elasticity: A mathematical survey, Discrete Contin. Dyn. Syst., 23 (2009), 133-164.  doi: 10.3934/dcds.2009.23.133. [15] Ph. G. Ciarlet and C. Mardare, Intrinsic formulation of the displacement-traction problem in linearized elasticity, Math. Models Methods Appl. Sci., 24 (2014), 1197–1216. doi: 10.1142/S0218202513500814. [16] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Springer-Verlag, Berlin, 1988. Functional and variational methods, With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily, Translated from the French by Ian N. Sneddon. doi: 10.1007/978-3-642-61566-5. [17] L. Donati, Illustrazione al teorema del Menabrea, Memorie della Accademia delle Scienze dell'Istituto di Bologna, 10 (1860), 267-274. [18] G. Duvaut, Mécanique des Milieux Continus, Collection Mathématiques appliquées pour la maȋtrise. Masson, 1990. [19] Gilles A. Francfort, Alessandro Giacomini and Jean-Jacques Marigo, The elasto-plastic exquisite corpse: A Suquet legacy, J. Mech. Phys. Solids, 97 (2016), 125-139.  doi: 10.1016/j.jmps.2016.02.002. [20] G. Geymonat and F. Krasucki, Some remarks on the compatibility conditions in elasticity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 175-181. [21] G. Geymonat and F. Krasucki, Beltrami's solutions of general equilibrium equations in continuum mechanics, C. R. Math. Acad. Sci. Paris, 342 (2006), 359-363.  doi: 10.1016/j.crma.2005.12.031. [22] G. Geymonat and F. Krasucki, Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains, Commun. Pure Appl. Anal., 8 (2009), 295–309. doi: 10.3934/cpaa.2009.8.295. [23] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. (Extended version of the 1979 publ.), Springer Series in Computational Mathematics, 5. Berlin etc.: Springer-Verlag. 1986. doi: 10.1007/978-3-642-61623-5. [24] P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. [25] M. E. Gurtin, The linear theory of elasticity, In C. Truesdell, editor, Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells, pages 1–295. Springer Berlin Heidelberg, Berlin, Heidelberg, 1973. [26] H. Kozono and T. Yanagisawa, $L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J., 58 (2009), 1853-1920.  doi: 10.1512/iumj.2009.58.3605. [27] E. Kröner, Continuum theory of defects, In R. Balian, editor, Physiques des défauts, Les Houches Session XXXV (Course 3). North-Holland, Amsterdam, 1980. [28] G. Maggiani, R. Scala and N. Van Goethem, A compatible-incompatible decomposition of symmetric tensors in ${L}^p$ with application to elasticity, Math. Meth. Appl. Sci, 38 (2015), 5217-5230.  doi: 10.1002/mma.3450. [29] J. Mawhin, Les Modèles Mathématiques Sont-ils des Modèles à suivre?, Académie Royale de Belgique, 2017. [30] D. L. McDowell, Multiscale cristalline plasticity for material design, Computational Materials System Design, D. Shin and J. Saal Eds., 2018. [31] R. D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16 (1964), 51-78.  doi: 10.1007/BF00248490. [32] J. J. Moreau, Duality characterization of strain tensor distributions in an arbitrary open set, J. Math. Anal. Appl., 72 (1979), 760-770.  doi: 10.1016/0022-247X(79)90263-4. [33] P. Podio-Guidugli, The compatibility constraint in linear elasticity, In Donald E. Carlson and Yi-Chao Chen, editors, Advances in Continuum Mechanics and Thermodynamics of Material Behavior: In Recognition of the 60th Birthday of Roger L. Fosdick, pages 393–398. Springer Netherlands, Dordrecht, 2000. [34] W. Prager and P. G. Hodge Jr., Theory of Perfectly Plastic Solids, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951. [35] Ben Schweizer, On friedrichs inequality, helmholtz decomposition, vector potentials, and the div-curl lemma, In Elisabetta Rocca, Ulisse Stefanelli, Lev Truskinovsky, and Augusto Visintin, editors, Trends in Applications of Mathematics to Mechanics, pages 65–79. Springer International Publishing, 2018. doi: 10.1007/978-3-319-75940-1_4. [36] B. Sun, Incompatible deformation field and Riemann curvature tensor, Appl. Math. Mech., 38 (2017), 311-332.  doi: 10.1007/s10483-017-2176-8. [37] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1979. Studies in Mathematics and its Applications, Vol. 2. [38] T. W. Ting, St Venant's compatibility conditions and basic problems in elasticity, Rocky Mountain J. Math., 7 (1977), 47-52.  doi: 10.1216/RMJ-1977-7-1-47. [39] N. Van Goethem, Strain incompatibility in single crystals: Kröner's formula revisited, J. Elast., 103 (2011), 95-111.  doi: 10.1007/s10659-010-9275-4. [40] N. Van Goethem, Direct expression of incompatibility in curvilinear systems, ANZIAM J., 58 (2016), 33-50.  doi: 10.1017/S1446181116000158. [41] N. Van Goethem, Incompatibility-governed singularities in linear elasticity with dislocations, Math. Mech. Solids, 22 (2017), 1688-1695.  doi: 10.1177/1081286516642817. [42] N. Van Goethem and F. Dupret, A distributional approach to $2{D}$ Volterra dislocations at the continuum scale, European J. Appl. Math., 23 (2012), 417-439.  doi: 10.1017/S0956792512000010. [43] N. Van Goethem and F. Dupret, A distributional approach to the geometry of $2{D}$ dislocations at the continuum scale, Ann. Univ. Ferrara Sez. VII Sci. Mat., 58 (2012), 407-434.  doi: 10.1007/s11565-012-0149-5. [44] V. Volterra, Sur l'équilibre des corps élastiques multiplement connexes, Ann. Sci. École Norm. Sup., 24 (1907), 401–517. doi: 10.24033/asens.583. [45] W. von Wahl, Estimating $\nabla u$ by div $u$ and curl $u$, Math. Methods Appl. Sci., 15 (1992), 123-143.  doi: 10.1002/mma.1670150206. [46] M. Xavier, E. A. Fancello, J. M. C. Farias, N. Van Goethem and A. A. Novotny, Topological derivative-based fracture modelling in brittle materials: A phenomenological approach, Engineering Fracture Mechanics, 179 (2017), 13-27.  doi: 10.1016/j.engfracmech.2017.04.005. [47] A. Yavari and A. Goriely, Riemann–Cartan geometry of nonlinear dislocation mechanics, Arch. Ration. Mech. Anal., 205 (2012), 59-118.  doi: 10.1007/s00205-012-0500-0.
In-plane strain ($\varphi(z)$, top left) and vertical strain ($\psi(z)$, top right) with $z$ on the horizontal axis, for $\ell = -10$ (blue), $\ell = -100$ (red), $\ell = -1000$ (yellow). Value of $u(h)$ as a function of $\ell$ (bottom right)
Strain components in cylindrical coordinates, as functions of $r$, for $\ell = -1000$ (blue), $\ell = -100$ (red), $\ell = -20$ (yellow), classical plane strain elastic solution (dashed)
Functions $\varphi$, $\psi$ and $u$ for $\ell = -1000$ (blue), $\ell = -100$ (red), $\ell = -10$ (yellow), elastic solution (dashed)
External work
 $\ell$ $-1000$ $-100$ $-20$ $W$ $0.3006$ $0.4616$ $1.0620$
 $\ell$ $-1000$ $-100$ $-20$ $W$ $0.3006$ $0.4616$ $1.0620$
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