LMA - Laboratoire de Mécanique et d’Acoustique

S. Forest - Strain vs. stress gradient continua and their relations to general micromorphic media

Amphithéâtre François Canac, LMA

Le 22 mai 2018 de 11h00 à 12h00

Samuel Forest
Centre des Materiaux Mines ParisTech

Generalized continua are increasingly used to account for size effects in the mechanics of materials and structures. They include higher grade media characterized by the presence of zeroth, first and second (or more) gradient of the strain tensor in the constitutive setting [1], and higher order continua that incorporate additional degrees of freedom like Cosserat rotations or micromorphic deformations [2]. In the recent years, stress gradient theories have emerged sometimes presented as special cases of strain gradient theories, or as fundamentally distinct theories [3,4]. This point of view will be developed in the presentation showing that strain and stress gradient models are NO dual theories but essentially distinct approaches to material behaviour. A similar situation is encountered in the gradient of entropy and gradient of temperature theories of rigid conductors [5,6]. Stress gradient continua include a third order tensor of additional degrees of freedom, called microdisplacement, independently of the usual displacement field. This makes them akin to general micromorphic media introduced by Eringen and Germain [7]. The relations between such gradient and micromorphic continua will be depicted based on suited internal constraints [8]. Applications will be presented dealing with the regularization of stress singularities or concentration in structures and with the dispersion of elastic waves.

  • [1] Mindlin, R.D., Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438, 1965
  • [2] Eringen, A, and Suhubi, E. Nonlinear theory of simple microelastic solids. Int. J. Eng. Sci., vol. 2, pp. 189–203, 389–404, 1964.
  • [3] S. Forest and K. Sab, Continuum stress gradient theory, Mechanics Research Communications, vol. 40, pp. 16-25, 2012. doi:10.1016/j.mechrescom.2011.12.002
  • [4] K. Sab, F. Legoll and S. Forest, Stress Gradient Elasticity Theory : Existence and Uniqueness of Solution, Journal of Elasticity, vol. 123, pp. 179—201, 2016. doi:10.1007/s10659-015-9554-1
  • [5] H. Gouin, T. Ruggeri, Mixture of fluids involving entropy gradients and acceleration waves in interfacial layers, Eur. J. Mech. B Fluids, vol. 24, pp. 596–613, 2005.
  • [6] S. Forest, M. Amestoy, Hypertemperature in thermoelastic solids , Comptes Rendus Mécanique, vol. 336, pp. 347-353, 2008. doi:10.1016/j.crme.2008.01.007
  • [7] P. Germain. The method of virtual power in continuum mechanics. Part 2 : Microstructure.
  • SIAM J Appl Math, vol. 25, pp. 556–575, 1973.
  • [8] S. Forest and K. Sab, Finite deformation second order micromorphic theory and its relations to strain and stress gradient models, Mathematics and Mechanics of Solids, in press, 2017. doi:10.1177/1081286517720844

Voir en ligne : la page personnelle de l’orateur

Articles similaires