Abstract
The most general and elegant axiomatic framework on which continuum mechanics can be based starts from the Principle of Virtual Works (or Virtual Power). This Principle, which was most likely used already at the very beginning of the development of mechanics (see e.g. Benvenuto (1981), Vailati (1897), Colonnetti (1953), Russo (2003)), became after D’Alembert the main tool for an efficient formulation of physical theories. Also in continuum mechanics it has been adopted soon (see e.g. Benvenuto (1981), Salençon (1988), Germain (1973), Berdichevsky (2009), Maugin (1980), Forest (2006)). Indeed the Principle of Virtual Works becomes applicable in continuum mechanics once one recognizes that to estimate the work expended on regular virtual displacement fields of a continuous body one needs a distribution (in the sense of Schwartz). Indeed in the present paper we prove, also by using concepts from differential geometry of embedded Riemanniam manifolds, that the Representation Theorem for Distributions allows for an effective characterization of the contact actions which may arise in N-th order strain-gradient multipolar continua (as defined by Green and Rivlin (1964)), by univocally distinguishing them in actions (forces and n-th order forces) concentrated on contact surfaces, lines (edges) and points (wedges). The used approach reconsiders the results found in the pioneering papers by Green and Rivlin (1964)–(1965), Toupin (1962), Mindlin (1964)–(1965) and Casal (1961) as systematized, for second gradient models, by Paul Germain (1973). Finally, by recalling the results found in dell’Isola and Seppecher (1995)–(1997), we indicate how Euler-Cauchy approach to contact actions and the celebrated tetrahedron argument may be adapted to N-th order strain-gradient multipolar continua.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
R. K. Abu Al-Rub Modeling the interfacial effect on the yield strength and flow stress of thin metal films on substrates Mechanics Research Communications 35 65–72 (2008)
R. Abeyaratne, N. Triantafyllidis, An investigation of localization in a porous elastic material using homogenization theory. Trans. ASME J. Appl. Mech. 51, no. 3, 481–486 (1984)
R. Abraham, J.E. Marsden, and T. Ratiu, ‘Manifolds,Tensor Analysis, and Applications’, Applied Mathematical Sciences, 75, Springer Verlag, (1988).
Alibert, J.-J., Seppecher, P., dell’Isola, F. Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8, no. 1, 51–73 (2003).
V.I. Arnold, Mathematical Methods of Classical Mechanics Springer Verlag (1979)–(1989)
C. Banfi, A. Marzocchi and A. Musesti On the principle of virtual powers in continuum mechanics Ricerche di Matematica 55: 299–310 (2006)
S. Bardenhagen, N. Triantafyllidis Derivation of higher order gradient continuum theories in 2,3-D nonlinear elasticity from periodic lattice models. J. Mech. Phys. Solids 42, no. 1, 111–139 (1994)
E. Benvenuto La scienza delle costruzioni e il suo sviluppo storico Sansoni, Firenze, (1981)
V. Berdichevsky, Variational Principles of Continuum Mechanics, Springer, (2009).
J.L. Bleustein, A note on the boundary conditions of Toupin’s strain-gradient theory International Journal of Solids and Structures, 3(6), pp. 1053–1057. (1967)
B. Bourdin, G.A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity, 91, 1–3, 2008, 1–148 (also appeared as a Springer book: ISBN: 978-1-4020-6394-7).
J.L. Borges, Pierre Menard Author of the Quixote (translation by James E. Irby) http://www.coldbacon.com/writing/borgesquixote.html
P. Casal, et H. Gouin, ‘Relation entre l’équation de l’énergie et l’équation du mouvement en théorie de Korteweg de la capillaritè’, C. R. Acad. Sci. Paris, t. 300, Série II, N. 7 231–233 (1985).
P. Casal, ‘La théorie du second gradient et la capillarité’, C. R. Acad. Sci. Paris, t. 274, Série A 1571–1574 (1972).
P. Casal, La capillarité interne, Cahier du groupe Français de rhéologie, CNRS VI, 3, pp. 31–37 (1961).
P. Casal, et H. Gouin, ‘Relation entre l’équation de l’énergie et l’équation du mouvement en théorie de Korteweg de la capillarité’, C. R. Acad. Sci. Paris, t. 300, Série II, N. 7 231–233 (1985).
A. Carcaterra, A. Akay, I.M. Koc Near-irreversibility in a conservative linear structure with singularity points in its modal density Journal of the Acoustical Society of America 1194 2141–2149 (2006)
A. Carcaterra Ensemble energy average and energy flow relationships for nonstationary vibrating systems Journal of Sound and Vibration 288,3, 751–790 (2005)
F. Collin, R. Chambon and R. Charlier A finite element method for poro mechanical modelling of geotechnical problems using local second gradient models. Int. J. Num. Meth. Engng. 65, 1749–1772 (2006).
G. Colonnetti, Scienza delle costruzioni, Torino, Edizioni scientifiche Einaudi, 3o ed., (1953–57).
E. Cosserat and F. Cosserat Note sur la théorie de l’action euclidienne. Paris, Gauthier-Villars, (1908).
E. Cosserat, and F. Cosserat Sur la Théorie des Corps Déformables, Herman, Paris, (1909).
N. Daher, G. A. Maugin Virtual power and thermodynamics for electromagnetic continua with interfaces. J. Math. Phys. 27, no. 12, 3022–3035 (1986).
N. Daher, G. A. Maugin The method of virtual power in continuum mechanics. Application to media presenting singular surfaces and interfaces. Acta Mech. 60, no. 3–4, 217–240, (1986)
F. dell’Isola and W. Kosinski “Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers,” Archives of Mechanics, vol. 45, pp. 333–359 (1993).
F. dell’Isola and P. Seppecher, “The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power”, Comptes Rendus de l’Academie de Sciences-Serie IIb: Mecanique, Physique, Chimie, Astronomie, vol. 321,, pp. 303–308 (1995).
F. dell’Isola and P. Seppecher, “Edge Contact Forces and Quasi-Balanced Power”, Meccanica, vol. 32, pp. 33–52 (1997)
F. dell’Isola, G. Sciarra, and S. Vidoli, Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465, no. 2107, 2177–2196 (2009)
F. dell’Isola, G. Sciarra, R. C.Batra, Static deformations of a linear elastic porous body filled with an inviscid fluid. Essays and papers dedicated to the memory of Clifford Ambrose Truesdell III. Vol. III. J. Elasticity 72, no. 1—3, 99–120 (2003)
F. dell’Isola, M. Guarascio, K. Hutter A variational approach for the deformation of a saturated porous solid. A secondgradient theory extending Terzaghi’s effective stress principle. Arch. Appl. Mech. 70, 323–337 (2000)
A. Di Carlo e A. Tatone (Iper-)Tensioni & Equi-Potenza AIMETA’01 XV Congresso AIMETA di Meccanica Teorica e Applicata 15th AIMETA Congress of Theoretical and Applied Mechanics 2001
M. Degiovanni, A. Marzocchi and A. Musesti Edge-force densities and second-order powers Annali di Matematica 185, 81–103 (2006)
M. Degiovanni, A. Marzocchi, and A. Musesti, A., ‘Cauchy fluxes associated with tensor fields having divergence measure’, Arch. Ration. Mech. Anal. 147 197–223 (1999).
J.E. Dunn, and J. Serrin, ‘On the thermomechanics of interstitial working’, Arch. Rational Mech. Anal., 88(2) 95–133 (1985).
J.E. Dunn ‘Interstitial working and a non classical continuum thermodynamics’, In: J. Serrin (Ed), New Perspectives in Thermodynamics, Springer Verlag, Berlin, pp. 187–222 (1986).
G. E. Exadaktylos, I. Vardoulakis, Microstructure in linear elasticity and scale effects: a reconsideration of basic rock mechanics and rock fracture mechanics, Tectonophysics, Volume 335, Issues 1–2, 25 June 2001, Pages 81–109
A.C. Fannjiang, Y.S. Chan, and G.H. Paulino Strain gradient elasticity for antiplane shear cracks: a hypersingular integrodifferential equation approach. SIAM J. Appl. Math. 62, no. 3, 1066–1091 (electronic) (2001/02).
S. Forest, M. Amestoy, S. Cantournet, G. Damamme, S. Kruch Mécanique des Milieux Continus ECOLE DES MINES DE PARIS Année 2005–2006
S. Forest Mechanics of generalized continua: construction by homogenization J.Phys. IV France 8 1998
S. Forest Homogenization methods and the mechanics of generalized continua-part 2 Theoretical and applied Mechanics vol. 28–29, pp. 113–143 (2002)
S. Forest. Milieux continus généralisés et matériaux hétérogènes. Les Presses de l’Ecole des Mines de Paris, ISBN: 2-911762-67-3, 200 pages, (2006).
S. Forest. Generalized continua. In K.H.J. Buschow, R.W. Cahn, M.C. Flemings, B. Ilschner, E.J. Kramer, and S. Mahajan, editors, Encyclopedia of Materials: Science and Technology updates, pages 1–7. Elsevier, Oxford, (2005).
S. Forest and M. Amestoy. Mécanique des milieux continus. Cours de l’Ecole des Mines de Paris n25e6 3121, 264 pages, (2004, 2005, 2006).
S. Forest. Milieux continus généralisés et matériaux hétérog`enes. Mémoire d’habilitation à diriger des recherches, (2004).
E. Fried and M.E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small length scales. Archive for Rational Mechanics and Analysis 182, 513–554 (2006).
E. Fried and M.E. Gurtin, A continuum mechanical theory for turbulence: a generalized Navier-Stokes-equation with boundary conditions. Theoretical and Computational Fluid Dynamics 182, 513–554 (2008)
P. Germain Cours de M`ecanique des Milieux Continus, tome I, Masson, Paris, (1973).
P. Germain La méthode des puissances virtuelles en mécanique des milieux continus. Première partie. Théorie du second gradient. J. Mécanique 12, 235–274 (1973)
P. Germain The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)
P. Germain ‘Sur l’application de la méthode des puissances virtuelles en mécanique des milieux continus’, C. R. Acad. Sci. Paris Série A-B 274 A1051-A1055.(1972)
A. E. Green, R. S. Rivlin, Multipolar continuum mechanics, Arch. Rational Mech. Anal., 17, 113–147 (1964)
A. E. Green, R. S. Rivlin, Simple force and stress multipoles, Arch. Rational Mech. Anal.,16, 325–353 (1964)
A. E. Green, R. S. Rivlin,. On Cauchy’s equations of motion, Z. Angew. Math. Phys.,ZAMP. 15,, 290–292, (1964)
A. E. Green, R. S. Rivlin Multipolar continuum mechanics: Functional theory. I, Proc. Roy. Soc. Ser. A, 284, 303–324 (1965)
M.E. Gurtin, A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations, International Journal of Plasticity 50, 809–819 (2002).
N. Kirchner, P. Steinmann, On the material setting of gradient hyperelasticity. (English summary) Math. Mech. Solids 12 (2007)
O.D. Kellogg, Foundations of Potential Theory Springer, Berlin, (1929)
W. Kosinski, Field Singularities and Wave Analysis in Continuum Mechanics. Ellis Horwood Series: Mathematics and Its Applications, Wiley & Sons, PWN — Polish Scientific Publishers, Warsaw 1986
R. Larsson, S. A Diebels, Second-order homogenization procedure for multi-scale analysis based on micropolar kinematics. Internat. J. Numer. Methods Engrg. 69, no. 12, 2485–2512 (2007).
M. Lazar, G. A. Maugin A note on line forces in gradient elasticity Mechanics Research Communications 33 674–680 (2006)
M. Lucchesi ·M. Silhavý ·N. Zani On the Balance Equation for Stresses Concentrated on Curves J Elasticity 90:209–223 (2008).
A. Madeo, F. dell’Isola, N. Ianiro, G. Sciarra, “A second gradient poroelastic model of consolidation”, SIMAI 2008, Rome 15–19 September (2008).
A. Madeo, F. dell’Isola, N. Ianiro, and G. Sciarra, “A variational deduction of second gradient poroelasticity II: An application to the consolidation problem,” Journal of Mechanics of Materials and Structures, vol. 3,, pp. 607–625 (2008)
A. Marzocchi,, A. Musesti, Balanced virtual powers in Continuum Mechanics. Meccanica 38, 369–389 (2003)
A. Marzocchi, A. Musesti,, ‘Decomposition and integral representation of Cauchy interactions associated with measures’, Cont. Mech. Thermodyn. 13 149–169 (2001).
G. Maugin MathReview MR1437786 (98d:73003) 73A05 (73B18 73S10) on the paper F.dell’Isola P. Seppecher (1997).
G. Maugin MathReview MR1600928 (99e:73005) on the paper G. Capriz and G. Mazzini Invariance and balance in continuum mechanics. Nonlinear analysis and continuum mechanics (Ferrara,1992), 27–35, Springer, New York, 1998.
G. A. Maugin A.V. Metrikine Editors Mechanics of Generalized Continua, One Hundred Years After the Cosserats, Springer (2010)
R.D. Mindlin, Second gradient of strain and surface tension in linear elasticity Int. J. Solids and Struct. 1,4,417–438 (1965)
R.D. Mindlin, Micro-structure in linear elasticity, Arch. Rat. Mech. Analysis, vol. 16, pp. 51–78 (1964)
R. D. Mindlin, Complex representation of displacements and stresses in plane strain with couple-stresses. 1965 Appl. Theory of Functions in Continuum Mechanics (Proc. Internat. Sympos., Tbilisi), Vol. I, Mechanics of Solids (Russian) pp. 256–259 Izdat. ”Nauka”, Moscow (1963)
R. D. Mindlin, H. F. Tiersten, Effects of couple-stresses in linear elasticity. Arch. Rational Mech. Anal. 11 415–448 (1962)
R. D. Mindlin Influence of couple-stresses on stress concentrations Main features of cosserat theory are reviewed by lecturer and some recent solutions of the equations, for cases of stress concentration around small holes in elastic solids, are described EXPERIMENTAL MECHANICS 3, 1, 1–7, THE WILLIAM M. MURRAY LECTURE, (1962)
R. D. Mindlin N. N. Eshel On first strain-gradient theories in linear elasticity International Journal of Solids and Structures 4,1 1968, 109–124
R.D. Mindlin,. Stress functions for a Cosserat continuum International Journal of Solids and Structures, 1(3), pp. 265–271 (1965)
R.D. Mindlin On the equations of elastic materials with microstructure International Journal of Solids and Structures, 1(1), pp. 73–78 (1965).
M. Muntersbjom, Francis Bacon’s Philosophy of Science: Machina intellectus and Forma indita Philosophy of Science, 70 (December 2003) pp. 1137–1148.
W. Noll, E.G. Virga, ‘On edge interactions and surface tension’, Arch. Rational Mech. Anal., 111(1) 1–31 (1990).
W. Noll ‘The foundations of classical mechanics in the light of recent advances in continuum mechanics’ Proceeding of the Berkeley Symposium on the Axiomatic Method, Amsterdam,, pp. 226–281 (1959).
W. Noll ‘Lectures on the foundations of continuum mechanics and thermodynamics’, Arch. Rational Mech. Anal. 52 62–92 (1973).
W. Noll ‘The geometry of contact separation and reformation of continuous bodies’, Arch. Rational Mech. Anal., 122(3) 197–212 (1993).
C. Pideri and P. Seppecher A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9, no. 5, 241–257 (1997).
P. Podio-Guidugli A virtual power format for thermomechanics Continuum Mech. Thermodyn. 20: 479–487 (2009)
P. Podio-Guidugli, Contact interactions, stress, and material symmetry, for nonsimple elastic materials. (English, Serbo-Croatian summary) Issue dedicated to the memory of Professor Rastko Stojanovic (Belgrade, 2002). Theoret. Appl. Mech. 28/29 (2002), 261–276.
P. Podio-Guidugli and M. Vianello Hypertractions and hyperstresses convey the same mechanical information Continuum Mech. Thermodyn. 22:163–176 (2010)
C. Polizzotto, Strain-gradient elastic-plastic material models and assessment of the higher order boundary conditions. Eur. J. Mech. A Solids 26, no. 2, 189–211 (2007).
Rorres C. Completing Book II of Archimedes’s On Floating Bodies THE MATHEMATICAL INTELLIGENCER 26,3, Pages 32–42 (2004)
L. Russo The Forgotten Revolution Springer Verlag (2003)
J. Salençon Mécanique des milieux continus Ed. Ellipses (1988)–(1995) Handbook of Continuum Mechanics Ed. Springer (Berlin, 2001) Mécanique des milieux continus. Tome I. Éd. École polytechnique, Palaiseau; Ellipses, Paris, (2002)–(2005)
L. Schwartz, Théorie des Distributions, Hermann Paris, (1973).
G. Sciarra, F. dell’Isola and O. Coussy Second gradient poromechanics. Internat. J. Solids Structures 44, no. 20, 6607–6629 (2007).
G. Sciarra, F. dell’Isola and K. Hutter A solid-fluid mixture model allowing for solid dilatation under external pressure. Contin. Mech. Thermodyn. 13, no. 5, 287–306 (2001).
G. Sciarra, F. dell’Isola, N. Ianiro, and A. Madeo, “A variational deduction of second gradient poroelasticity part I: General theory”, Journal of Mechanics of Materials and Structures, vol. 3, pp. 507–526 (2008).
P. Seppecher ‘Etude des conditions aux limites en théorie du second gradient: cas de la capillarité, C. R. Acad. Sci. Paris, t. 309, Série II 497–502 (1989).
P. Seppecher Etude d’une Modélisation des Zones Capillaires Fluides: Interfaces et Lignes de Contact, Thèse de l’Université Paris VI, Avril (1987).
M. Šilhavý ‘The existence of the flux vector and the divergence theorem for general Cauchy fluxes’, Arch.Ration. Mech. Anal. 90 195–211 (1985).
M. Šilhavý ‘Cauchy’s stress theorem and tensor fields with divergences in Lp’, Arch. Ration. Mech. Anal. 116 223–255 (1991).
M. Sokolowski, Theory of couple-stresses in bodies with constrained rotations. In CISM courses and lectures, vol. 26. Berlin, Germany: Springer (1970).
M. Spivak A comprehensive introduction to differential geometry. Voll. I and II. Second edition. Publish or Perish, Inc., Wilmington, Del. (1979)
A. S. J. Suiker and C. S. Chang, Application of higher-order tensor theory for formulating enhanced continuum models. Acta Mech. 142, 223–234 (2000).
R.A. Toupin Elastic Materials with couple-stresses, Arch. Rat. Mech. Analysis, vol. 11, pp. 385–414 (1962)
R. A. Toupin, Theories of elasticity with couple-stress. Arch. Rational Mech. Anal. 17 85–112 (1964).
N. Triantafyllidis and S. Bardenhagen, On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models. J. Elasticity 33 (1993)
N. Triantafyllidis and S. Bardenhagen The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models. J. Mech. Phys. Solids 44, no. 11, 1891–1928 (1996)
C.A. Truesdell A First Course in Rational Continuum Mechanics, Vol. I General Concepts, Academic Press, New York, 1977.
N. Triantafyllidis and E.C. Aifantis A gradient approach to localization of deformation. I. Hyperelastic materials. J. Elasticity 16, no. 3, 225–237 (1986).
Y. Yang, and A. Misra “Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation,” Computer Modeling in Engineering and Sciences, Vol. 64, No. 1, 1–36 (2010)
Y. Yang, W.Y. Ching and Misra, A., “Higher-order continuum theory applied to fracture simulation of nano-scale intergranular glassy film,” Journal of Nanomechanics and Micromechanics, (in print). (2011)
G. Vailati, Il principio dei lavori virtuali da Aristotele a Erone d’Alessandria, Scritti (Bologna, Forni, 1987), vol. II, pp. 113–128, Atti della R. Accademia delle Scienze di Torino, vol. XXXII, adunanza del 13 giugno 1897, quaderno IG (091) 75 I–III (1897).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
This work is dedicated to Professor Antonio Romano in occasion of his 70-th birthday.
Rights and permissions
Copyright information
© 2011 CISM, Udine
About this chapter
Cite this chapter
dell’Isola, F., Seppecher, P., Madeo, A. (2011). Beyond Euler-Cauchy Continua: The structure of contact actions in N-th gradient generalized continua: a generalization of the Cauchy tetrahedron argument. In: dell’Isola, F., Gavrilyuk, S. (eds) Variational Models and Methods in Solid and Fluid Mechanics. CISM Courses and Lectures, vol 535. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0983-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-7091-0983-0_2
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-0982-3
Online ISBN: 978-3-7091-0983-0
eBook Packages: EngineeringEngineering (R0)