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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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020240
References:
[1]

G. AllaireF. 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.  Google Scholar

[2]

C. AmrouchePh. G. CiarletL. 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.  Google Scholar

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C. AmrouchePh. G. CiarletL. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[8]

E. Beltrami, Sull'interpretazione meccanica delle formule di Maxwell, Mem. dell'Accad. di Bologna, 7 (1886), 1-38.   Google Scholar

[9]

H. Brézis, Functional Analysis. Theory and Applications. (Analyse Fonctionnelle. Théorie et Applications.), Collection Mathématiques Appliquées pour la Maȋtrise. Paris: Masson., 1983.  Google Scholar

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P. G. Ciarlet, Three-Dimensional Elasticity, Vol.1, Masson, Paris, 1986.  Google Scholar

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P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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Ph. G. CiarletL. 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.  Google Scholar

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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.  Google Scholar

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G. Duvaut, Mécanique des Milieux Continus, Collection Mathématiques appliquées pour la maȋtrise. Masson, 1990. Google Scholar

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Gilles A. FrancfortAlessandro 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[27]

E. Kröner, Continuum theory of defects, In R. Balian, editor, Physiques des défauts, Les Houches Session XXXV (Course 3). North-Holland, Amsterdam, 1980. Google Scholar

[28]

G. MaggianiR. 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.  Google Scholar

[29]

J. Mawhin, Les Modèles Mathématiques Sont-ils des Modèles à suivre?, Académie Royale de Belgique, 2017. Google Scholar

[30]

D. L. McDowell, Multiscale cristalline plasticity for material design, Computational Materials System Design, D. Shin and J. Saal Eds., 2018. Google Scholar

[31]

R. D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16 (1964), 51-78.  doi: 10.1007/BF00248490.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

B. Sun, Incompatible deformation field and Riemann curvature tensor, Appl. Math. Mech., 38 (2017), 311-332.  doi: 10.1007/s10483-017-2176-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

N. Van Goethem, Direct expression of incompatibility in curvilinear systems, ANZIAM J., 58 (2016), 33-50.  doi: 10.1017/S1446181116000158.  Google Scholar

[41]

N. Van Goethem, Incompatibility-governed singularities in linear elasticity with dislocations, Math. Mech. Solids, 22 (2017), 1688-1695.  doi: 10.1177/1081286516642817.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

M. XavierE. A. FancelloJ. M. C. FariasN. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. AllaireF. 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.  Google Scholar

[2]

C. AmrouchePh. G. CiarletL. 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.  Google Scholar

[3]

C. AmrouchePh. G. CiarletL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[8]

E. Beltrami, Sull'interpretazione meccanica delle formule di Maxwell, Mem. dell'Accad. di Bologna, 7 (1886), 1-38.   Google Scholar

[9]

H. Brézis, Functional Analysis. Theory and Applications. (Analyse Fonctionnelle. Théorie et Applications.), Collection Mathématiques Appliquées pour la Maȋtrise. Paris: Masson., 1983.  Google Scholar

[10]

P. G. Ciarlet, Three-Dimensional Elasticity, Vol.1, Masson, Paris, 1986.  Google Scholar

[11]

P. G. Ciarlet, An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Ph. G. CiarletL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. Donati, Illustrazione al teorema del Menabrea, Memorie della Accademia delle Scienze dell'Istituto di Bologna, 10 (1860), 267-274.   Google Scholar

[18]

G. Duvaut, Mécanique des Milieux Continus, Collection Mathématiques appliquées pour la maȋtrise. Masson, 1990. Google Scholar

[19]

Gilles A. FrancfortAlessandro 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[27]

E. Kröner, Continuum theory of defects, In R. Balian, editor, Physiques des défauts, Les Houches Session XXXV (Course 3). North-Holland, Amsterdam, 1980. Google Scholar

[28]

G. MaggianiR. 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.  Google Scholar

[29]

J. Mawhin, Les Modèles Mathématiques Sont-ils des Modèles à suivre?, Académie Royale de Belgique, 2017. Google Scholar

[30]

D. L. McDowell, Multiscale cristalline plasticity for material design, Computational Materials System Design, D. Shin and J. Saal Eds., 2018. Google Scholar

[31]

R. D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16 (1964), 51-78.  doi: 10.1007/BF00248490.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

B. Sun, Incompatible deformation field and Riemann curvature tensor, Appl. Math. Mech., 38 (2017), 311-332.  doi: 10.1007/s10483-017-2176-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

N. Van Goethem, Direct expression of incompatibility in curvilinear systems, ANZIAM J., 58 (2016), 33-50.  doi: 10.1017/S1446181116000158.  Google Scholar

[41]

N. Van Goethem, Incompatibility-governed singularities in linear elasticity with dislocations, Math. Mech. Solids, 22 (2017), 1688-1695.  doi: 10.1177/1081286516642817.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

M. XavierE. A. FancelloJ. M. C. FariasN. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  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)
Figure 2.  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)
Figure 3.  Functions $ \varphi $, $ \psi $ and $ u $ for $ \ell = -1000 $ (blue), $ \ell = -100 $ (red), $ \ell = -10 $ (yellow), elastic solution (dashed)
Table 1.  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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