Abstract
In this chapter the predictive theory of the motion of a solid with large deformation is described. The novel and main point of the theory is the choice of the quantities which describe the shape changes. Observation shows that the stretch matrix and the rotation matrix of the classical polar decomposition are adapted to account for part of the shape change. Moreover, an external action can be applied on the surface of the solid by a curvilinear beam. This observation and mathematics show that a third order gradient theory is mandatory (a deformation involving third order space derivatives is needed). The velocities of deformation involve the angular velocity and its gradient, the gradient of the velocity and the third gradient velocity. The free energy depends on the stretch matrix and it may be a convex function of this matrix, giving properties we are accustomed to in the small deformation theory. The volume impenetrability condition is that the three eigenvalues of the stretch matrix are larger than \(\alpha > 0, \alpha \) quantifies the resistance to crushing of the material. The theory is completely developed for a viscoelastic solid including the evolution of the temperature. The theory coherent in terms of mechanics is also coherent in terms of mathematics. Incompressibility and plasticity are taken into account within this framework. This long chapter has a classical structure: description of the shape change and of the shape change velocities, equations of motion, choice of the state quantities, choice of the quantities which describe the evolution, the laws of thermodynamics, the free energy and the pseudo-potential of dissipation, solution of the equations predicting the evolution.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.W. Ogden, Non-linear Elastic Deformations (Courier Dover Publications, 1997)
S.S. Antman, Existence of solutions of the equilibrium equations for non linearly elastic rings and arches. Indiana Univ. Math. J. 20, 281–302 (1970)
P.G. Ciarlet, Mathematical Elasticity Volume I: Three-Dimensional Elasticity (North-Holland, Amsterdam, 1988)
J. Salençon, Mécanique des milieux continus. I. (Éditions de l’École Polytechnique, Palaiseau, 2005)
M. Frémond, Grandes déformations et comportements extrêmes. C. R. Acad. Sci., Paris, Mécanique, 337(1), 24–29 (2009), published on line, http://dx.doi.org/10.1016/j.crme.2009.01.003
J.L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1 (Dunod, Paris, 1968)
H. Brezis, Analyse fonctionnelle, théorie et applications (Masson, Paris, 1983)
P. Casal, La capillarité interne. Cahier du groupe Français de rhéologie, CNRS VI 3, 31–37 (1961)
P. Casal, La théorie du second gradient et la capillarité. C. R. Acad. Sci. Paris, série A, 274, II, 1571–1574 (1972)
H. Gouin, Utilization of the second gradient theory in continuum mechanics to study motions and thermodynamics of liquid-vapor interfaces. Manuel G. Velarde. Plenum Publishing Corporation, pp. 16, 1987, Physicochemical Hydrodynamics, Series B—Physics, vol. 174, Interfacial Phenomena. \(<{\rm hal}-00614568>\) (1987)
F. Sidoroff, Sur l’équation tensorielle AX+XA=H, C. R. Acad. Sci. Paris, A 286, 71–73 (1978)
S.S. Antman, Nonlinear problems of elasticity, Second edition. Applied Mathematical Sciences, vol 107 (Springer, New York, 2005)
M. Grediac, F. Hild, Full-field measurements and identification in solid mechanics (2012), ISBN 978-1-118-57847-6 Wiley-ISTE
C. Vallée, Compatibility equations for large deformations. Int. J. Eng. Sci. 30(12), 1753–1757 (1992)
D. Fortuné, C. Vallée, Bianchi identities in the case of large deformations. Int. J. Eng. Sci. 39, 113–123 (2001)
P.G. Ciarlet, L. Gratie, O. Iosifescu, C. Mardare, C. Vallée, Another approach to the fundamental theorem of Riemannian geometry in \( \mathbb{R} ^{3}\), by way of rotation fields. J. Math. Pures Appl. 87, 237–252 (2007)
M. Frémond, Non-smooth Thermomechanics (Springer, Heidelberg, 2002)
M. Frémond, Phase change in mechanics. UMI-Springer Lecture Notes Series no 13 ( 2012). ISBN 978-3-642-24608-1, doi:10.1007/978-3-642-24609-8, http://www.springer.com/mathematics/book/978-3-642-24608-1
M. Frémond, Collisions, Edizioni del Dipartimento di Ingegneria Civile (Università di Roma “Tor Vergata”, 2007), ISBN 978-88-6296-000-7
P. Germain, Mécanique des milieux continus (Masson, Paris, 1973)
E. Bonetti, P. Colli, M. Frémond, A phase field model with thermal memory governed by the entropy balance. M3AS Math. Models Appl. Sci. 13, 231–256 (2003)
E. Bonetti, M. Frémond, A phase transition model with the entropy balance. Math. Meth. Appl. Sci. 26, 539–556 (2003)
M. Frémond, Positions d’équilibre de solides en grandes déformations. C. R. Acad. Sci., Paris, Ser. I 347, 457–462 (2009), published on line, http://dx.doi.org/10.1016/j.crma.2009.02.001
G. Amendola, M. Fabrizio, J.M. Golden, Thermodynamics of Materials with Memory (Springer, Theory and Applications, 2012)
J. Ball, Convexity conditions and existence theorems in non linear elasticity. Arch. Rational Mech. Anal. 63, 337–403 (1977)
M.D.P. Monteiro Marques, J.J. Moreau, Isotropie et convexité dans l’espace des tenseurs symétriques, Séminaire d’Analyse Convexe, Éxposé n\( {{}^\circ } 6\) (Université de Montpellier II, 1982)
E. Bonetti, P. Colli, M. Frémond, The motion of a solid with large deformations. C. R. Acad. Sci., Paris, Ser. I, 351, 579–583 (2013), published on line 20/08/2013, http://dx.doi.org/10.1016/j.crma.2013.06.012
E. Bonetti, P. Colli, M. Frémond, The 3D Motion of a Solid with Large Deformations, C. R. Acad. Sci., Paris, published on line February 3, 2014, doi:10.1016/j.crma.2014.01.007, C. R. Acad. Sci., Paris, 352(3), 183–187 (2014)
E. Bonetti, P. Colli, M. Frémond, 2D Motion with Large Deformations, Bollettino dell’Unione Matematica Italiana, 7, 19–44 ( 2014), published on line February 25, 2014, doi:10.1007/s40574-014-0002-0
J.J. Moreau, Sur la naissance de la cavitation dans une conduite. C. R. Acad. Sci Paris 259, 3948–3950 (1965)
J.J. Moreau, Principes extrémaux pour le problème de la naissance de la cavitation. J. de Mécanique 5, 439–470 (1966)
J.J. Moreau, Fonctionnelles convexes, Edizioni del Dipartimento di Engegneria Civile, Università di Roma “Tor Vergata”, Roma (2003) and Séminaire sur les équations aux dérivées partielles (Collège de France, Paris, 1966), ISBN 978-88-6296-001-4
I. Ekeland, R. Temam, Convex Analysis and Variational Problems (North Holland, Amsterdam, 1976)
P.M. Suquet, Existence et régularité des solutions des équations de la plasticité parfaite. C. R. Acad. Sci Paris, A 286, 1201–1204 (1978)
B. Halphen, Salençon, Élastoplasticité (Presses de l’École nationale des Ponts et Chaussées, Paris, 1987)
Nguyen Quoc Son, Problèmes de plasticité et de rupture, cours d’option D.E.A. d’Analyse Numérique et Applications, 1980–81 (Université de Paris-Sud, Dép. de mathématique, 1982), p. 154
H. Maitournan, Mécanique des structures anélastiques (École Polytechnique, 2013)
B. Halphen, Nguyen Quoc Son, Sur les matériaux standards généralisés. J. Mécanique 14(1), 39–63 (1975)
A. Friaa, La loi de Norton-Hoff généralisée en plasticité et en viscoplasticité (thèse de l’Université Pierre et Marie Curie, Paris, 1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Frémond, M. (2017). There Is Neither Flattening nor Self-contact or Contact with an Obstacle. Smooth Evolution. In: Virtual Work and Shape Change in Solid Mechanics. Springer Series in Solid and Structural Mechanics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-40682-4_30
Download citation
DOI: https://doi.org/10.1007/978-3-319-40682-4_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40681-7
Online ISBN: 978-3-319-40682-4
eBook Packages: EngineeringEngineering (R0)