Abstract
The objective of the present article is to assess the well-posedness of the stress gradient linear elastic problems recently introduced by Forest and Sab (Mech. Res. Commun. 40:16–25, 2012) and to formulate the corresponding existence and uniqueness theorems. In particular, we show that such theorems can be established in the case of the boundary value problems formulated in (Forest and Sab in Mech. Res. Commun. 40:16–25, 2012) with the corresponding boundary conditions.
References
Adams, R.A.: Sobolev Spaces. Academic Press, San Diego, CA (1975)
Amrouche, C., Ciarlet, P.G., Gratie, L., Kesavan, S.: On the characterizations of matrix fields as linearized strain tensor fields. J. Math. Pures Appl. 86, 116–132 (2006)
Auffray, N.: Analytical expressions for odd-order anisotropic tensor dimension. C. R., Méc. 342, 284–291 (2014)
Auffray, N., Le Quang, H., He, Q.C.: Matrix representations for 3D strain-gradient elasticity. J. Mech. Phys. Solids 61, 1202–1223 (2013)
Bleustein, J.L.: A note on the boundary conditions of Toupin’s strain-gradient theory. Int. J. Solids Struct. 3, 1053–1057 (1967)
Chakravarthy, S.S., Curtin, W.A.: Stress-gradient plasticity. Proc. Natl. Acad. Sci. USA 108, 15716–15720 (2011)
Dell’Isola, F., Seppecher, P.: The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. C. R. Acad. Sci. Paris IIb 321, 303–308 (1995)
Dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32, 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”. Z. Angew. Math. Phys. 63, 1119–1141 (2012)
Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
Forest, S., Aifantis, E.C.: Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. Int. J. Solids Struct. 47, 3367–3376 (2010)
Forest, S., Sab, K.: Stress gradient continuum theory. Mech. Res. Commun. 40, 16–25 (2012)
Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus, première partie: théorie du second gradient. J. Méc. 12, 235–274 (1973)
Iesan, D.: Thermoelasticity of bodies with microstructure and microtemperatures. Int. J. Solids Struct. 44, 8648–8662 (2007)
Iesan, D., Quintanilla, R.: Existence and continuous dependence results in the theory of microstretch elastic bodies. Int. J. Eng. Sci. 32, 991–1001 (1994)
Iesan, D., Quintanilla, R.: On a strain gradient theory of thermoviscoelasticity. Mech. Res. Commun. 48, 52–58 (2013)
Jeong, J., Neff, P.: Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids 15, 78–95 (2010)
Lebée, A., Sab, K.: A bending-gradient model for thick plates, Part I: Theory. Int. J. Solids Struct. 48, 2878–2888 (2011)
Lebée, A., Sab, K.: A bending-gradient model for thick plates, Part II: Closed-form solutions for cylindrical bending of laminates. Int. J. Solids Struct. 48, 2889–2901 (2011)
Mindlin, R.D.: Second gradient of strain and surface–tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)
Mindlin, R.D., Eshel, N.N.: On first strain gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)
Moreau, J.J.: Duality characterization of strain tensor distributions in an arbitrary open set. J. Math. Anal. Appl. 72, 760–770 (1979)
Necas, J.: Les Méthodes Directes en Théorie des Equations Elliptiques. Masson, Paris (1967)
Neff, P., Forest, S.: A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elast. 87, 239–276 (2007)
Olive, M., Auffray, N.: Symmetry classes for even-order tensors. Math. Mech. Complex Syst. 1, 177–210 (2013)
Olive, M., Auffray, N.: Symmetry classes for odd-order tensors. Z. Angew. Math. Mech. 94, 421–447 (2014)
Polizzotto, C.: Stress gradient versus strain gradient constitutive models within elasticity. Int. J. Solids Struct. 51, 1809–1818 (2014)
Polizzotto, C.: A unifying variational framework for stress gradient and strain gradient elasticity theories. Eur. J. Mech. A, Solids 49, 430–440 (2015)
Ru, C.Q., Aifantis, E.C.: A simple approach to solve boundary value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993)
Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sab, K., Legoll, F. & Forest, S. Stress Gradient Elasticity Theory: Existence and Uniqueness of Solution. J Elast 123, 179–201 (2016). https://doi.org/10.1007/s10659-015-9554-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-015-9554-1
Keywords
- Stress gradient elasticity
- Higher order elasticity
- Boundary conditions
- Existence and uniqueness of solution