Abstract
Second gradient theories are nowadays used in many studies in order to describe in detail some transition layers which may occur in micro-structured materials and in which physical properties are sharply varying. Sometimes higher order theories are also evoked. Up to now these models have not been based on a construction of stresses similar to the one due to Cauchy, which has been applied only for simple materials. It has been widely recognized that the fundamental assumption by Cauchy that the traction depends only on the normal of the dividing surface cannot be maintained for higher gradient theories. However, this observation did not urge any author, to our knowledge, to revisit the Cauchy construction in order to adapt it to a more general conceptual framework. This is what we do in this paper for a continuum of grade N (also called N-th gradient continuum). Our construction is very similar to the one due to Cauchy; based on the tetrahedron argument, it does not introduce any argument of a different nature. In particular, we avoid invoking the principle of virtual work. As one should expect, the balance assumption and the regularity hypotheses have to be adapted to the new framework and tensorial computations become more complex.
Similar content being viewed by others
References
Aifantis E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30(10), 1279–1299 (1992)
Aifantis E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95(1–4), 299–314 (1999)
Alibert J.J., Seppecher P., dell’Isola F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8, 51–73 (2003)
Altenbach, H., Eremeyev, V.A., Lebedev, L.P.: On the existence of solution in the linear elasticity with surface stresses. ZAMM J. Appl. Math. Mechan./Zeitschrift fr Angewandte Mathematik und Mechanik., 90(3), 231–240 (2010)
Auffray, N., dell’Isola, F., Eremeyev, V. A., Madeo, A., Rosi, G. : Analytical continuum mechanics la HamiltonPiola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids. 20(4), 375–417 (2015)
Casal, P.: La theorie du second gradient et la capillarite. C. R. Acad. Sci. Paris. 274, 1571–1574 (1972)
Cauchy, A.L.: De la pression ou tension dans un corps solide. Ex. de Math. 2, 4256 (available in Gallica.bnf.fr) (1827)
Céline, C., Boutin, C., Hans, C.: Wave propagation and non-local effects in periodic frame materials: generalized continuum mechanics. Math. Mech. Solids (2013)
Chesnais, C., Boutin, C., Hans, S.: Effects of the local resonance on the wave propagation in periodic frame structures: generalized newtonian mechanics. J. Acoust. Soc. Am. 132(4), Pt. 2 (2012)
Coleman B.D., Noll W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mechan. Anal. 13(1), 167–178 (1963)
dell’Isola, F., Seppecher, P.: The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. Comptes Rendus de l’Academie de Sciences - Serie IIb: Mécanique, Physique, Chimie, Astronomie, vol. 321, 303–308 (1995)
dell’isola F., Seppecher P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)
dell’Isola, F., Seppecher, P., Madeo, A.: How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach à la D’Alembert. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 63(6), 1119–1141 (2012)
dell’Isola, F., Seppecher, P.: Hypertractions and hyperstresses convey the same mechanical information Continuum Mech. Thermodyn.2010 22: 163176 by Prof. Podio Guidugli and Prof. Vianello and some related papers on higher gradient theories. Contin. Mech. Thermodyn. 23(5), 473–478 (2011)
dell’Isola, F., Andreaus, U., Placidi, L.: A still topical contribution of Gabrio Piola to Continuum Mechanics: the creation of peri-dynamics, non-local and higher gradient continuum mechanics. In: The complete works of Gabrio Piola: Volume I, Advanced Structured Materials, vol. 38. Springer, Switzerland (2014)
dell’Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S.: The complete works of Gabrio Piola: Volume I. Advan. Struct. Mater. 38 (2014)
Del Piero G.: Non-classical continua, pseudobalance, and the law of action and reaction. Math. Mech. Complex. Syst. 2(1), 7110 (2014)
Dunn, J.E., Serrin, J.: On the thermomechanics of interstitial working. Arch. Rational Mech. Anal. 88(2), 95–133 (1985)
Eringen A.C.: Microcontinuum Field Theories. Springer-Verlag, New York (2001)
Forest, S., Cordero, N.M., Busso, E.P.: First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Comput. Mater. Sci. 50(4), 1299–1304 (2011)
Fosdick R.: Observations concerning virtual power. Math. Mech. Solids, 16, 573–585 (2011)
Fried, E., Gurtin, M.E. : Tractions balanced and boundary conditions for non simple materials with application to liquid flow at small length scale, Arch. Rat. Mech. Anal., 182(3), 513–554 (2006)
Garajeu M., Gouin H., Saccomandi G.: Scaling Navier-Stokes equation in nanotubes. Phys. Fluids 25(8), 082003 (2013)
Germain, P.: La mthode des puissances virtuelles en mcanique des milieux continus. J. Mcanique, 12, 236–274 (1973)
Germain, P.: The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J. Appl. Math. 25(3), 556–575 (1973)
Giusteri G.G.: The multiple nature of concentrated interactions in second-gradient dissipative liquids. Zeitschrift fr angewandte Mathematik und Physik, 64(2), 371–380 (2013)
Gurtin, M.E.: Thermodynamics and the possibility of spatial interaction in elastic materials. Arch. Ration. Mechan. Anal. 195, 339–352 (1965)
Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 36, 756757 (1962)
Mindlin R.D., Tiersten H.F.: Effects of couple-stresses in linear elasticity. Arch. Rational Mech. Anal. 11, 415–448 (1962)
Mindlin, R.D.: Complex representation of displacements and stresses in plane strain with couple- stresses. 1965 Appl. Theory of Functions in Continuum Mechanics (Proc. Internat. Sympos., Tbil- isi), Mechanics of Solids (Russian), vol. I, pp. 256–259 Izdat. Nauka, Moscow (1963)
Mindlin, R.D.: Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)
Müller I.: Thermodynamics of mixtures and phase field theory. Int. J. Solids Struct. 38(6), 1105–1113 (2001)
Neff, P., Chemiki, K., Alber, H.D.: Notes on strain gradient plasticity: finite strain covariant modelling and global existence in the infinitesimal rate-independent case. Math. Models Methods Appl. Sci. 19(02), 307–346 (2009)
Noll, W.: The foundations of classical mechanics in the light of recent advances in continuum mechanics, In: The Axiomatic Method, with Special Reference to Geometry and Physics, pp 266–281. North-Holland, Amsterdam (1959)
Noll W., Virga E.: On edge interactions and surface tension. Arch. Ration. Mech. Anal. 111, 1–31 (1990)
Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9(5), 241–257 (1997)
Piola, G.: Sull’applicazione de’ principj della meccanica analitica del Lagrange ai principali problemi. Memoria di Gabrio Piola presentata al concorso del premio e coronata dall’I.R. Istituto di Scienze, ecc. nella solennita del giorno 4 ottobre 1824, Milano, Imp. Regia stamperia (1825)
Piola, G.: Memoria intorno alle equazioni fondamentali del movimento di corpi qualsivogliono considerati secondo la naturale loro forma e costituzione, Modena, Tipi del R.D. Camera (1846)
Polizzotto, C., Borino, G.: A thermodynamics-based formulation of gradient-dependent plasticity. Eur. J. Mech. A/Solids 17(5) 741–761 (1998)
Schwartz, L.: Theorie des Distributions, Hermann, Paris 1973
Sciarra, G., dell’Isola, F., Coussy, O.: Second gradient poromechanics. Int. J. Solids Struct. 44(20), 6607–6629 (2007)
Segev, R., De Botton, G.: On the consistency conditions for force systems. International Journal of Non-Linear Mechanics, Elsevier, 26(1), 47–59 (1991)
Seppecher, P.: Moving contact lines in the Cahn-Hilliard theory. Int. J. Eng. Sci. 34(9), 977–992 (1996)
Seppecher, P., Alibert, J.J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical. J. Phys. Conf. Ser. 319(1), 012018 (2011)
Thomas E.G.F.: A polarization identity for multilinear maps. Indag. Math. 25(3), 468–474 (2014)
Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
Toupin R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal., 17, 85–112 (1964)
Triantafyllidis, N., Aifantis, E.C.: A gradient approach to localization of deformation. I. Hyperelastic materials. J Elast. 16(3), 225–237 (1986)
Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, in Fliigge’s Encyclopedia of Physics, vol. III/3. Springer-Verlag, Berlin, pp. 1–662 (1965)
Truesdell, C.: Cauchy and the modern mechanics of continua. Revue d’histoire des sciences, t 45(1), 5–24 (1992)
Yang Y., Misra A.: Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int. J. Solids Struct. 49(18), 2500–2514 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by I. Fonseca
Rights and permissions
About this article
Cite this article
dell’Isola, F., Madeo, A. & Seppecher, P. Cauchy Tetrahedron Argument Applied to Higher Contact Interactions. Arch Rational Mech Anal 219, 1305–1341 (2016). https://doi.org/10.1007/s00205-015-0922-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-015-0922-6