Least Action Principle for Second Gradient Continua and Capillary Fluids: A Lagrangian Approach Following Piola’s Point of View

  • Francesco dell’Isola
  • Nicolas Auffray
  • Victor A. Eremeyev
  • Angela Madeo
  • Luca Placidi
  • Giuseppe Rosi
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 38)


As Piola would have surely conjectured, the stationary action principle holds also for capillary fluids, i.e. those fluids for which the deformation energy depends on spatial derivative of mass density (a modelling necessity which has been already remarked by Cahn and Hilliard [15, 16]). For capillary fluids it is indeed possible to define a Lagrangian density function whose corresponding Euler-Lagrange stationarity conditions once transported on the actual configuration, via a Piola’s transformation, are exactly those obtained, with different methods, in the literature. We recall that some particulat classes of second gradient fluids are sometimes also called Korteweg-de Vries or Cahn-Allen fluids. More generally those continua (which may be solid or fluid) whose deformation energy depends on the second gradient of placement are called second gradient (or Piola-Toupin or Mindlin or Green-Rivlin or Germain or second grade) continua. In the present work, following closely the procedure first conceived by Piola and carefully presented in his works translated in the present volume, a material (Lagragian) description for second gradient continua is formulated. Subsequently a Lagrangian action is introduced and by means of Piola’s transformations this action is calculated in both the material and spatial descriptions. Then the corresponding Euler-Lagrange equations and boundary conditions are calculated by using some kinematical relationships suitably established. Once an objective deformation energy volume density is assumed to depend on either C and \( {{\nabla }}C \) or on C −1 and (where C is the Cauchy-Green deformation tensor) the particular form of aforementioned Euler-Lagrange conditions and boundary conditions are established. When further particularizing the treatment to those energies which characterize fluid materials, the capillary fluid evolution conditions (see e.g. Casal [25] or Seppecher [142, 145] for an alternative deduction based on thermodynamic arguments) are recovered. Also a version of Bernoulli’s law which is valid for capillary fluids is found and, in Appendix B, all the kinematic formulas which we have found useful for the present variational formulation are gathered. Many historical comments about Gabrio Piola’s contribution to analytical continuum mechanics are also presented when it has been considered useful. In this context the reader is also referred to Capecchi and Ruta [17].


Variational Principle Virtual Work Deformation Energy Gradient Continuum Virtual Power 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alibert, J.J., Seppecher, P. and dell’Isola, F., Truss modular beams with deformation energy depending on higher displacement gradients. Mathematics and Mechanics of Solids, 8, 51-73 (2003).Google Scholar
  2. 2.
    Atai, A.A. and Steigmann, D.J., On the nonlinear mechanics of discrete networks. Archive of Applied mechanics, 67, 303-319 (1997)Google Scholar
  3. 3.
    Auriault, J.-L., Geindreau, C. and Boutin, C., Filtration law in porous media with poor separation of scales. Transport in Porous Media, 60, 89-108 (2005) .Google Scholar
  4. 4.
    Baake, E. and Georgii, H.-O., Mutation, selection, and ancestry in branching models: a variational approach. Journal of Mathematical Biology, 54, 257-303 (2007).Google Scholar
  5. 5.
    Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 63 (4), 337–403 (1976).Google Scholar
  6. 6.
    Barham, M., Steigmann, D.J., McElfresh, M. and Rudd, R.E. Limit-point instability of a magnetoelastic membrane in a stationary magnetic field. Smart Materials and Structures,17 (2008).Google Scholar
  7. 7.
    Bassanini P., Casciola C.M., Lancia M.R., Piva R., On the trailing edge singularity and Kutta condition for 3D airfoils – European journal of mechanics. B, Fluids, 15, 6, pp. 809-830 (1996)Google Scholar
  8. 8.
    Bedford, A., Hamilton’s principle in continuum mechanics. Volume 139 di Research notes in mathematics Pitman Advanced Publishing Program, 1985.Google Scholar
  9. 9.
    Berdichevsky, V., Variational principles of continuum mechanics. Voll.I,II, Springer, 2009.Google Scholar
  10. 10.
    Bleustein, J.L., A note on the boundary conditions of Toupin’s strain-gradient theory. International Journal of Solids and Structures, 3, 1053-1057 (1967).Google Scholar
  11. 11.
    Bourdin, B., Francfort, G.A. and Marigo, J.-J., The variational approach to fracture. Journal of Elasticity, 91, 1-148 (2008). (The paper also appeared as a Springer book: ISBN: 978-1-4020-6394-7).Google Scholar
  12. 12.
    Boutin, C. and Auriault, J.-L., Acoustics of a bubbly fluid at large bubble concentration. European Journal of mechanics B/fluids, 12, 367-399 (1993).Google Scholar
  13. 13.
    Boutin, C., Hans, S. and Chesnais, C., Generalized beams and continua. Dynamics of reticulated structures. In Mechanics of Generalized Continua (131-141). Springer New York (2011).Google Scholar
  14. 14.
    Boutin, C. and Hans, S., Homogenisation of periodic discrete medium: Application to dynamics of framed structures. Computers and Geotechnics, 30, 303-320 (2003).Google Scholar
  15. 15.
    Cahn J.W., and Hilliard, J.E., Free Energy of a Nonuniform System. I. Interfacial Free Energy. The Journal of Chemical Physics, 28, 258-267 (1958).Google Scholar
  16. 16.
    Cahn, J.W. and Hilliard, J.E., Free energy of a non uniform system III. The Journal of Chemical Physics, 31, 688-699 (1959).Google Scholar
  17. 17.
    Capecchi, D. and Ruta, G.C., Piola’s contribution to continuum mechanics, Archive for History of Exact Sciences, 61, 303-342 (2007).Google Scholar
  18. 18.
    Carcaterra, A. and Sestieri A., Energy Density Equations and Power Flow in Structures. Journal of Sound and Vibration, 188, 269-282 (1995).Google Scholar
  19. 19.
    Carcaterra, A., E. Ciappi, A. and Iafrati, E.F., Campana, Shock spectral analysis of elastic systems impacting on the water surface. Journal of Sound and Vibration, 229, 579-605(2000).Google Scholar
  20. 20.
    Carcaterra, A., Ensemble energy average and energy flow relationships for nonstationary vibrating systems. Journal of Sound and Vibration, 288, 751-790(2005).Google Scholar
  21. 21.
    Carcaterra, A., Akay A. and Ko, I.M., Near-irreversibility in a conservative linear structure with singularity points in its modal density. Journal of the Acoustical Society of America, 119, 2141-2149 (2006) .Google Scholar
  22. 22.
    Carcaterra, A. and Akay, A., Theoretical foundations of apparent-damping phenomena and nearly irreversible energy exchange in linear conservative systems. Journal of the Acoustical Society of America, 12 1971-1982 (2007).Google Scholar
  23. 23.
    Carcaterra, A. and Akay, A., Dissipation in a finite-size bath. Physical Review E, 84, 011121 (2011).Google Scholar
  24. 24.
    Casal, P., La capillarité interne. Cahier du groupe Francais de rhéologie, 3, 31-37 (1961).Google Scholar
  25. 25.
    Casal, P., La théorie du second gradient et la capillarité. Comptes rendus de l’Académie des Sciences Série A, 274, 1571-1574 (1972).Google Scholar
  26. 26.
    Casal, P. and Gouin H., Connection between the energy equation and the motion equation in Korteweg’s theory of capillarity. Comptes rendus de l’Académie des Sciences Série II, 300, 231-234 (1985).Google Scholar
  27. 27.
    Casal, P. and Gouin H., Equations of motion of thermocapillary fluids, Comptes rendus de l’Académie des Sciences Série II, 306, 99-104 (1988).Google Scholar
  28. 28.
    Casciola C.M., Gualtieri P., Jacob B., Piva R. Scaling properties in the production range of shear dominated flows Physical review letters 95, 024503 (2005)Google Scholar
  29. 29.
    Chesnais, C., Boutin, C and Hans, S., Wave propagation and non-local effects in periodic frame materials: Generalized continuum mechanics (In preparation).Google Scholar
  30. 30.
    Chesnais, C., Boutin, C., Hans, S., Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics. Journal of the Acoustical Society of America, 132, 2873-2886 (2012).Google Scholar
  31. 31.
    Contrafatto, L. and Cuomo, M., A new thermodynamically consistent continuum model for hardening plasticity coupled with damage. International Journal of Solids and Structures, 39, 6241-6271 (2002).Google Scholar
  32. 32.
    Contrafatto, L. and Cuomo, M., A framework of elastic–plastic damaging model for concrete under multiaxial stress states, International Journal of Plasticity, 22, 2272-2300 (2006).Google Scholar
  33. 33.
    Contrafatto, L. and Cuomo, M., A globally convergent numerical algorithm for damaging elasto-plasticity based on the Multiplier method. International Journal for Numerical Methods in Engineering, 63,1089-1125 (2005).Google Scholar
  34. 34.
    Culla, A., Sestieri, A. and Carcaterra, A., Energy flow uncertainties in vibrating systems: Definition of a statistical confidence factor. Mechanical Systems and Signal Processing, 17, 635-663(2003).Google Scholar
  35. 35.
    Cuomo, M. and Ventura, G., Complementary Energy Approach to Contact Problems Based on Consistent Augmented Lagrangian regularization. Mathematical and Computer Modelling, 28, 185-204 (1998)Google Scholar
  36. 36.
    Cuomo, M. and Contrafatto, L., Stress rate formulation for elastoplastic models with internal variables based on augmented Lagrangian regularisation. International Journal of Solids and Structures, 37 3935-3964 (2000).Google Scholar
  37. 37.
    Daher, N. and Maugin, G.A., Virtual power and thermodynamics for electromagnetic continua with interfaces. Journal of Mathematical Physics, 27, 3022-3035 (1986).Google Scholar
  38. 38.
    Daher, N., Maugin, G.A., The method of virtual power in continuum mechanics. Application to media presenting singular surfaces and interfaces. Acta Mechanica, 60, 217-240 (1986) .Google Scholar
  39. 39.
    de Gennes, P.G., Some effects of long range forces on interfacial phenomena. Journal de Physique Lettres, 42, L-377, L-379 (1981).Google Scholar
  40. 40.
    dell’Isola, F. and Romano, A., On a general balance law for continua with an interface. Ricerche di Matematica, 35, 325-337 (1986).Google Scholar
  41. 41.
    dell’Isola, F. and Romano, A., On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface. International Journal of Engineering Science, 25, 1459-1468 (1987).Google Scholar
  42. 42.
    dell’Isola, F. and Romano, A., A phenomenological approach to phase transition in classical field theory. International Journal of Engineering Science, 25, 1469-1475 (1987).Google Scholar
  43. 43.
    dell’Isola, F. and Kosinski, W., Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers. Archives of Mechanics, 45, 333-359 (1993).Google Scholar
  44. 44.’Isola, Gouin, H. and Seppecher, P., Radius and surface tension of microscopic bubbles by second gradient theory, Comptes rendus de l’Académie des Sciences Série IIb, 320, 211-216, (1995).Google Scholar
  45. 45.
    dell’Isola, F. and Seppecher, P., The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power. Comptes rendus de l’Académie des Sciences Serie IIb, 321, 303-308 (1995).Google Scholar
  46. 46.
    dell’Isola, F. and Seppecher, P., Edge Contact Forces and Quasi-Balanced Power. Meccanica, 32, 33-52 (1997).Google Scholar
  47. 47.
    dell’Isola, F. and Hutter, K., What are the dominant thermomechanical processes in the basal sediment layer of large ice sheets? Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454, 1169-1195 (1972).Google Scholar
  48. 48.
    dell’Isola, F. and Vidoli, S. Damping of bending waves in truss beams by electrical transmission lines with PZT actuators. Archive of Applied Mechanics, 68, 626-636 (1998).Google Scholar
  49. 49.
    dell’Isola, F. and Vidoli, S. Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping. Archive of Applied Mechanics, 68, 1-19 (1998).Google Scholar
  50. 50.
    dell’Isola, F., Guarascio, M. and Hutter, K.A., Variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. Archive of Applied Mechanics, 70, 323-337 (2000).Google Scholar
  51. 51.
    dell’Isola, F., Madeo, A. and Seppecher, P., Boundary Conditions in Porous Media: A Variational Approach. International Journal of Solids and Structures, 46, 3150-3164 (2009).Google Scholar
  52. 52.
    dell’Isola, F., Sciarra, G. and Vidoli, S., Generalized Hooke’s law for isotropic second gradient materials. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 465, 2177-2196 (2009).Google Scholar
  53. 53.
    dell’Isola, F. and Placidi, L., Variational principles are a powerful tool also for formulating field theories. Variational Models and Methods in Solid and Fluid mechanics CISM Courses and Lectures, 535, 1-15 (2011).Google Scholar
  54. 54.
    dell’Isola, F., Seppecher, P. and Madeo, A., How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach á la D’Alembert. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 63, 1119-1141 (2012).Google Scholar
  55. 55.
    Del Piero, G., A Variational Approach to Fracture and Other Inelastic Phenomena, Journal of Elasticity, 112(1), 3–77, (2013).Google Scholar
  56. 56.
    Edwards, A.W.F., Maximisation principles in Evolutionary Biology. Philosophy of Biology, Mohan Matthen and Christopher Stephens Editors Elsevier 335-349 (2007).Google Scholar
  57. 57.
    Evans R., The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform, classical fluids. Advances in Physics, 28, 143-200 (1979).Google Scholar
  58. 58.
    Eremeev V.A., Freidin A.B. and Sharipova L.L., Nonuniqueness and stability in problems of equilibrium of elastic two-phase bodies. Doklady Physics, 48, 359-363 (2003).Google Scholar
  59. 59.
    Eremeyev V.A. and Pietraszkiewicz W., The nonlinear theory of elastic shells with phase transitions. Journal of Elasticity, 74, 67-86 (2004).Google Scholar
  60. 60.
    Eremeyev, V. A. and Pietraszkiewicz, W., Thermomechanics of shells undergoing phase transition. Journal of the Mechanics and Physics of Solids, 59, 1395-1412 (2011).Google Scholar
  61. 61.
    Eremeyev V.A. and Lebedev L.P., Existence of weak solutions in elasticity. Mathematics and Mechanics of Solids, 18, 204-217 (2013).Google Scholar
  62. 62.
    Esposito, R. and Pulvirenti, M., From particles to fluids. Handbook of mathematical fluid dynamics. Vol. III, 1–82, North-Holland, Amsterdam, 2004.Google Scholar
  63. 63.
    Fermi, E., Pasta, J. and Ulam, S., Studies of Nonlinear Problems. Document LA-1940, 1955.Google Scholar
  64. 64.
    Forest, S., Cordero, N.M. and Busso, E.P., First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Computational Materials Science, 50, 1299-1304 (2011).Google Scholar
  65. 65.
    Forest, S., Micromorphic approach for gradient elasticity, viscoplasticity, and damage. Journal of Engineering Mechanics, 135, 117-131 (2009).Google Scholar
  66. 66.
    Francfort, G.A. and Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46, 1319-1342 (1998).Google Scholar
  67. 67.
    Gatignol, R. and Seppecher, P., Modelisation of fluid-fluids interfaces with material properties. Journal de Mécanique Théorique et Appliquée, 225-247 (1986).Google Scholar
  68. 68.
    Gavrilyuk, S. and Gouin, H., A new form of governing equations of fluids arising from Hamilton’s principle. International Journal of Engineering Science, 37, 1495-1520 (1999).Google Scholar
  69. 69.
    Germain, P., La méthode des puissances virtuelles en mécanique des milieux continus. Premiére partie. Théorie du second gradient, Journal de Mécanique, 12, 235-274 (1973).Google Scholar
  70. 70.
    Germain, P., The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM, Journal of Applied Mathematics 25, 556-575 (1973).Google Scholar
  71. 71.
    Germain, P., Toward an analytical mechanics of materials, in:Nonlinear thermodynamical processes in continua (Eds.W.Muschik and G.A.Maugin), TUB-Dokumentation und Tagungen, Heft 61, Berlin, 198-212 (1992).Google Scholar
  72. 72.
    Green, A.E. and Rivlin, R.S., Multipolar continuum mechanics, Archive for Rational Mechanics and Analysis, 17, 113-147 (1964).Google Scholar
  73. 73.
    Green, A.E. aand Rivlin, R.S., Simple force and stress multipoles, Archive for Rational Mechanics and Analysis,16, 325-353 (1964).Google Scholar
  74. 74.
    Green, A.E. and Rivlin, R.S., On Cauchy’s equations of motion, Zeitschrift für Angewandte Mathematik und Physik (ZAMP),15, 290-292, (1964).Google Scholar
  75. 75.
    Green, A.E.and Rivlin, R.S., Multipolar continuum mechanics: functional theory. I, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 284, 303-324 (1965).Google Scholar
  76. 76.
    Haseganu, E.M. and Steigmann, D.J., Equilibrium analysis of finitely deformed elastic networks. Computational mechanics, 17, 359-373 (1996) .Google Scholar
  77. 77.
    Hellinger, E., Die allgemeinen Ansitze der Mechanik der Kontinua. Enz. math. Wiss. 4, 602-694 (1972).Google Scholar
  78. 78.
    Jacob B., Casciola C.M., Talamelli A., Alfredsson P.H., Scaling of mixed structure functions in turbulent boundary layers Physics of fluids 20 (4), 045101-045101-7 (2008)Google Scholar
  79. 79.
    Klimek, P., Thurner, S. and Hanel, R., Evolutionary dynamics from a variational principle, Physical Review E, 82, 011901 (2010).Google Scholar
  80. 80.
    Korteweg, D. J. and de Vries, G., On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves. Philosophical Magazine, 39, 422-443 (1895).Google Scholar
  81. 81.
    Kravchuk, A. and Neittaanmaki, P., Variational and quasi-variational Inequalities in mechanics. Springer (2007).Google Scholar
  82. 82.
    Korteweg, D. J., Sur la forme que prennent les équations des mouvements des fluides si l’on tient compte des forces capillaires par des variations de densité. Arch. Néer. Sci. Exactes Sér. II, 6, 1-24 (1901).Google Scholar
  83. 83.
    Kroner, E., Mechanics of Generalized Continua, Springer (1968).Google Scholar
  84. 84.
    Kupershmidt B., The variational principles of Dynamics, World Scientific (1992).Google Scholar
  85. 85.
    Landau, L.D. and Lifshitz, E.M., Quantum mechanics: Non-Relativistic Theory. Vol. 3 (3rd ed.), Pergamon Press (1977).Google Scholar
  86. 86.
    Lanczos, C., The Variational principles of mechanics. Toronto: University of Toronto (1970).Google Scholar
  87. 87.
    Lagrange, J.L., Mécanique Analytique, Editions Jaques Gabay, Sceaux (1788).Google Scholar
  88. 88.
    Lebedev, L.P., Cloud, M.J., and Eremeyev, V. A., Tensor Analysis with Applications in Mechanics. New Jersey: World Scientific (2010).Google Scholar
  89. 89.
    Leipholz, H.H.E., Six Lectures on Variational Principkes in Structural Engineering, University of Waterloo, Canada (1983).Google Scholar
  90. 90.
    Lemons, D.S., Perfect Form: Variational principles, Methods and Applications in Elementary Physics. Princeton University Press (1997).Google Scholar
  91. 91.
    Lippmann, H., Extremum and Variational principles in mechanics. CISM Springer Verlag (1972).Google Scholar
  92. 92.
    Luongo, A. and Di Egidio, A., Bifurcation equations through multiple-scales analysis for a continuous model of a planar beam. Nonlinear Dynamics, 41, 171-190 (2005).Google Scholar
  93. 93.
    Luongo, A. and Romeo, F., A Transfer-matrix-perturbation approach to the dynamics of chains of nonlinear sliding beams. Journal of Vibration and Acoustics, 128, 190-196 (2006).Google Scholar
  94. 94.
    Luongo, A., Zulli, D. and Piccardo, G., On the effect of twist angle on nonlinear galloping of suspended cables. Computers & Structures, 87, 1003-1014 (2009).Google Scholar
  95. 95.
    Madeo, A., Lekszycki, T. and dell’Isola, F., A continuum model for the bio-mechanical interactions between living tissue and bioresorbable graft after bone reconstructive surgery. Comptes rendus Mecanique, 339, 625-682 (2011).Google Scholar
  96. 96.
    Marsden, J. E., & Hughes, T. J. (1983). Mathematical foundations of elasticity. Dover Publications.Google Scholar
  97. 97.
    Maugin, G.A. and Attou, D., An asymptotic theory of thin piezoelectric plates. The Quarterly Journal of Mechanics and Applied Mathematics, 43, 347-362 (1989).Google Scholar
  98. 98.
    Maugin, G.A. and Trimarco, C., Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture. Acta Mechanica 94, 1-28 (1992).Google Scholar
  99. 99.
    Maugin, G.A., Towards an analytical mechanics of dissipative materials. Rend. Sem. Mat. Univ. Pol. Torino Etude des conditions aux limites en théorie du second gradiVol. 58, 2 (2000).Google Scholar
  100. 100.
    Maugin, G.A., The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. Continuum Mechanics and Thermodynamics, 25, 127-146 (2011).Google Scholar
  101. 101.
    Maurini, C., dell’Isola, F and del Vescovo, D., Comparison of piezoelectronic networks acting as distributed vibration absorbers. Mechanical Systems and Signal Processing, 18, 1243-1271 (2004).Google Scholar
  102. 102.
    Maurini, C., and Pouget, J. and dell’Isola, F., On a model of layered piezoelectric beams including transverse stress effect. International Journal of Solids and Structures, 4, 4473-4502 (2004).Google Scholar
  103. 103.
    McBride, A.T., Javili, A., Steinmann, P. and Bargmann, S., Geometrically nonlinear continuum thermomechanics with surface energies coupled to diffusion, Journal of the Mechanics and Physics of Solids, 59, 2116-2133 (2011).Google Scholar
  104. 104.
    McBride, A.T., Mergheim, J., Javili, A., Steinmann, P. and Bargmann, S., Micro-to-macro transitions for heterogeneous material layers accounting for in-plane stretch, Journal of the Mechanics and Physics of Solids, 60, 1221-1239 (2012).Google Scholar
  105. 105.
    Maxwell, J.C., A treatise on electricity and magnetism Voll.I,II Oxford at the Clarendon Press (1873).Google Scholar
  106. 106.
    Mindlin, R.D., Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 51-78 (1964).Google Scholar
  107. 107.
    Mindlin, R.D., Second gradient of strain and surface tension in linear elasticity. International Journal of Solids and Structures, 1, 417-438 (1965).Google Scholar
  108. 108.
    Mindlin, R.D. and Eshel, N.N. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures, 4, 109-124 (1968).Google Scholar
  109. 109.
    Misra, A. and Chang, C.S., Effective Elastic Moduli of Heterogeneous Granular Solids. International Journal of Solids and Structures, 30, 2547-2566 (1993).Google Scholar
  110. 110.
    Misra, A. and Yang, Y.,. Micromechanical model for cohesive materials based upon pseudo-granular structure. International Journal of Solids and Structures, 47, 2970-2981 (2010) .Google Scholar
  111. 111.
    Misra, A. and Singh, V., Micromechanical model for viscoelastic-materials undergoing damage. Continuum Mechanics and Thermodynamics, 25, 1-16 (2013).Google Scholar
  112. 112.
    Misra, A. and Ching, W.Y., Theoretical nonlinear response of complex single crystal under multi-axial tensile loading, Scientific Reports, 3 (2013).Google Scholar
  113. 113.
    Moiseiwitsch, B.L., Variational principles. Dover (2004).Google Scholar
  114. 114.
    Nadler, B. and Steigmann, D.J., A model for frictional slip in woven fabrics. Comptes Rendus Mecanique, 331, 797-804 (2003).Google Scholar
  115. 115.
    Nadler, B., Papadopoulos, P. and Steigmann, D.J., Multiscale constitutive modeling and numerical simulation of fabric material, International Journal of Solids and Structures, 43, 206-221 (2006).Google Scholar
  116. 116.
    Noll, W. Foundations of mechanics and Thermodynamics, Selected Papers. Springer-Verlag, New York (1974).Google Scholar
  117. 117.
    Noll, W. and Truesdell, C. The Non-Linear Field Theories of mechanics, Encyclopie of Phisics, vol. III/3, Springer-Verlag, New York (1965).Google Scholar
  118. 118.
    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, 1825Google Scholar
  119. 119.
    Piola, G., La meccanica de’ corpi naturalmente estesi: trattata col calcolo delle variazioni, Milano, Giusti, (1833).Google Scholar
  120. 120.
    Piola, G., Nuova analisi per tutte le questioni della meccanica molecolare - del Signor Dottore Don Gabrio Piola - Ricevuta adí 21 Marzo 1835, Memorie di Matematica e di Fisica della Società Italiana delle Scienze residente in Modena, 21, pp. 155-321, (1836).Google Scholar
  121. 121.
    Piola, G., Intorno alle equazioni fondamentali del movimento di corpi qualsivogliono, considerati secondo la naturale loro forma e costituzione - Memoria del Signor Dottor Gabrio Piola - Ricevuta adí 6 Ottobre 1845, Memorie di Matematica e di Fisica della Società Italiana delle Scienze residente in Modena, 24, pp. 1-186, (1848). Translated in this volume.Google Scholar
  122. 122.
    Piola, G., Di un principio controverso della Meccanica analitica di Lagrange e delle molteplici sue applicazioni - Memoria postuma di Gabrio Piola - (pubblicata per cura del prof. Francesco Brioschi), Memorie dell’I.R. Istituto Lombardo di Scienze, Lettere ed Arti, 6, pp. 389-496, (1856). Translated in this volume.Google Scholar
  123. 123.
    Poisson, S.-D., Mémoire sur l’équilibre et le mouvement des Corps solides élastiques. Mémoires de l’Institut de France T. VIII. pag. 326, 400;Google Scholar
  124. 124.
    Poisson, S.-D., Mémoire sur les Equations générales de l’équilibre et du mouvement des Corps solides, élastiques et fluides. Journal de l’Ecole Polytechnique, 13, 1-174 (1829).Google Scholar
  125. 125.
    Poisson, S.-D., Nouvelle Théorie de l’Action Capillaire. Bachelier, Paris (1831)Google Scholar
  126. 126.
    Pietraszkiewicz, W., Eremeyev, V.A. and Konopinska, V., Extended non-linear relations of elastic shells undergoing phase transitions. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 87, 150-159 (2007).Google Scholar
  127. 127.
    Quiligotti, S., Maugin, G.A. and dell’Isola, F., An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures, Acta Mechanica, 160, 45-60 (2003).Google Scholar
  128. 128.
    Rinaldi, A. and Lai, Y.-C., Statistical damage theory of 2D lattices: Energetics and physical foundations of damage parameter. International Journal of Plasticity, 23, 1796-1825(2007).Google Scholar
  129. 129.
    Rinaldi, A., Krajcinovic, D., Peralta, P. and Lai, Y.-C., Lattice models of polycrystalline microstructures: A quantitative approach. Mechanics of Materials, 40, 17-36 (2008).Google Scholar
  130. 130.
    Rivlin, R.S. Forty years of nonlinear continuum mechanics Proc.IX Intl. Congress on Rheology Mexico (1984) reprinted In Barenblatt G.I. and Joseph D.D. Eds. Collected Papers of R.S. Rivlin Volume II Springer (1996)Google Scholar
  131. 131.
    Rivlin, R.S. Red herrings and sundry unidentified fish in nonlinear continuum mechanics In Barenblatt G.I. and Joseph D.D. Eds. Collected Papers of R.S. Rivlin Volume II Springer (1996)Google Scholar
  132. 132.
    Rorres, C., Completing Book II of Archimedes’s On Floating Bodies.The mathematical intelligencer, 26, 32-42 (2004).Google Scholar
  133. 133.
    Russo, L., The Forgotten Revolution. Springer Verlag (2003).Google Scholar
  134. 134.
    Santilli, R., Foundations of theoretical mechanics II. Birkhoffian generalization of Hamiltonian mechanics. Springer (1982).Google Scholar
  135. 135.
    Schwartz, L., Théorie des Distributions, Hermann Paris, (1973).Google Scholar
  136. 136.
    Sciarra G., dell’Isola, F. and Hutter, K., A solid-fluid mixture model allowing for solid dilatation under external pressure. Continuum Mechanics and Thermodynamics, 13, 287-306 (2001).Google Scholar
  137. 137.
    Sciarra, G., dell’Isola, F. and Coussy, O., Second gradient poromechanics. International Journal of Solids and Structures, 44,6607-6629 (2007).Google Scholar
  138. 138.
    Sciarra, G., dell’Isola, F., Ianiro, N. and Madeo A., A variational deduction of second gradient poroelasticity part I: General theory. Journal of Mechanics of Materials and Structures, 3, 507-526 (2008).Google Scholar
  139. 139.
    Sedov, L.I., Models of continuous media with internal degrees of freedom, Journal of Applied Mathematics and Mechanics, 32, 803-819 (1972)Google Scholar
  140. 140.
    Sedov, L.I., Variational Methods of constructing Models of Continuous Media. Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids. Springer Vienna, 346-358 (1968).Google Scholar
  141. 141.
    Seppecher, P., Etude d’une Modelisation des Zones Capillaires Fluides: Interfaces et Lignes de Contact, Thése de l’Universitá Paris VI, Avril (1987).Google Scholar
  142. 142.
    Seppecher, P., Thermodynamique des zones capillaires, Annales de Physique, 13, 13-22 (1988).Google Scholar
  143. 143.
    Seppecher, P., Etude des conditions aux limites en théorie du second gradient : cas de la capillarité, Comptes rendus de l’Académie des Sciences, 309, 497-502 (1989).Google Scholar
  144. 144.
    Seppecher, P., Equilibrium of a Cahn and Hilliard fluid on a wall: Influence of the wetting properties of the fluid upon the stability of a thin liquid film, European Journal of mechanics B/fluids, 12, 69-84 (1993).Google Scholar
  145. 145.
    Seppecher, P., A numerical study of a moving contact line in Cahn-Hilliard theory, International Journal of Engineering Science, 34, 977-992 (1996).Google Scholar
  146. 146.
    Seppecher, P., Les Fluides de Cahn-Hilliard. Mémoire d’Habilitation á Diriger des Recherches, Universitá du Sud Toulon Var (1996).Google Scholar
  147. 147.
    Seppecher, P., Second-gradient theory : application to Cahn-Hilliard fluids, in Continuum Thermomechanics, Springer Netherlands, 379-388 (2002).Google Scholar
  148. 148.
    Seppecher, P., Line Tension Effect upon Static Wetting, Oil and Gas Science and Technology- Rev. IFP, vol 56, 77-81 (2001).Google Scholar
  149. 149.
    Davison, E., Soper Classical Field Theory. Dover Publications (2008).Google Scholar
  150. 150.
    Soubestre, J. and Boutin, C., Non-local dynamic behavior of linear fiber reinforced materials, Mechanics of Materials, 55, 16-32 (2012).Google Scholar
  151. 151.
    Sunyk, R. and Steinmann, P., On Higher Gradients in continuum-Atomistic Modelling. International Journal of Solids and Structures, 40, 6877-6896 (2003).Google Scholar
  152. 152.
    Steigmann, D.J., Equilibrium of prestressed networks, IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 48, 195-215 (1992).Google Scholar
  153. 153.
    Steigmann, D.J. and Ogden, R.W., Elastic surface-substrate interactions (1999). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,455, 437-474 (1982).Google Scholar
  154. 154.
    Steigmann, D.J. The variational structure of a nonlinear theory for spatial lattices, Meccanica, 31, 441-455(1996).Google Scholar
  155. 155.
    Steigmann, D.J. and Faulkner, M.G. Variational theory for spatial rods. Journal of Elasticity, 33, 1-26(1993).Google Scholar
  156. 156.
    Steeb H. and Diebels S., Modeling thin films applying an extended continuum theory based on a scalar-valued order parameter – Part I: Isothermal case. International Journal of Solids and Structures, 41 5071-5085(2004).Google Scholar
  157. 157.
    Steinmann, P., Elizondo, A. and Sunyk, R., Studies of validity of the Cauchy-Born rule by direct comparison of continuum and atomistic modelling. Modelling and Simulation in Materials Science and Engineering, 15 (2007).Google Scholar
  158. 158.
    Steinmann, P., McBride, A.T., Bargmann, S. and Javili, A., A deformational and configurational framework for geometrically nonlinear continuum thermomechanics coupled to diffusion. International Journal of Non-Linear mechanics, 47, 215-227 (2012) .Google Scholar
  159. 159.
    Spivak, M., A comprehensive introduction to differential geometry, Vol. I and II. Second edition. Publish or Perish, Inc., Wilmington, Del. (1979).Google Scholar
  160. 160.
    Toupin R.A., Elastic Materials with couple-stresses. Archive for Rational Mechanics and Analysis, 11, 385-414 (1962)Google Scholar
  161. 161.
    Toupin R.A., Theories of elasticity with couple-stress. Archive for Rational Mechanics and Analysis, 17 85-112 (1964).Google Scholar
  162. 162.
    Truesdell, C., Essays in the Hystory of mechanics Springer Verlag (1968).Google Scholar
  163. 163.
    Truesdell, C.and Toupin R.A., The Classical field Theories Handbuch der Physic Band III/1 Springer (1960).Google Scholar
  164. 164.
    Van Kampen, N.G., Condensation of a classical gas with long range attraction, Physical Review, 135, A362-A369 (1964)Google Scholar
  165. 165.
    Vailati, G., Il principio dei lavori virtuali da Aristotele a Erone d’Alessandria, Scritti (Bologna, Forni, 1897), 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. 1897Google Scholar
  166. 166.
    Vujanovic, B.D. and Jones S.E., Variational Methods in Nonconservative Phenomena. Academic Press (1989).Google Scholar
  167. 167.
    Yang, Y., and Misra, A., Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation. Computer Modeling in Engineering and Sciences, 64, 1-36 (2010).Google Scholar
  168. 168.
    Yang, Y., and Misra, A., Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. International Journal of Solids and Structures, 49, 2500-2514 (2012) .Google Scholar
  169. 169.
    Yang, Y., Ching, W.Y. and Misra A., Higher-order continuum theory applied to fracture simulation of nano-scale intergranular glassy film. Journal of Nanomechanics and Micromechanics, 1, 60-71 (2011) .Google Scholar
  170. 170.
    Yeremeyev, V.A., Freidin, A.B. and Sharipova, L.L. The stability of the equilibrium of two-phase elastic solids. Journal of Applied Mathematics and mechanics (PMM), 71, 61-84 (2007).Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Francesco dell’Isola
    • 2
  • Nicolas Auffray
    • 1
  • Victor A. Eremeyev
    • 3
    • 4
  • Angela Madeo
    • 5
  • Luca Placidi
    • 6
  • Giuseppe Rosi
    • 7
  1. 1.Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRSUniversité Paris-EstMarne-la-ValléeFrance
  2. 2.Dipartimento di Ingegneria Strutturale e GeotecnicaUniversità di Roma La SapienzaRomaItaly
  3. 3.Institut für MechanikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
  4. 4.South Scientific Center of RASci and South Federal UniversityRostov on DonRussia
  5. 5.Laboratoire de Génie Civil et Ingénierie EnvironnementaleUniversité de Lyon–INSAVilleurbanne CedexFrance
  6. 6.International Telematic University UninettunoRomeItaly
  7. 7.International Center MeMOCS “Mathematics and Mechanics of Complex System”Università degli studi dell’AquilaCisterna di LatinaItaly

Personalised recommendations