Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids

By Nicolas Auffray, Francesco dell'Isola, Victor A. Eremeyev, A. Madeo, Giuseppe Rosi. 2013

292 citations recorded by the source corpus.

In this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg–de Vries or Cahn–Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola–Toupin, Mindlin, Green–Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler–Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C−1 and ∇C−1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions are recovered. A version of Bernoulli’s law valid for capillary fluids is found and useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola’s contribution to analytical continuum mechanics are also presented.

Subfields: Biomedical Engineering, Materials Chemistry, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/d056d804fc7864427066dbbd9ebfe4107b8791cf

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Citations in the corpus

Second strain gradient elasticity of nano-objects (2016, methodology): “The application of symmetry conditions to the constitutive elasticity tensors in gradient continua leads to the definition of proper symmetry classes, as recently done by Olive and Auffray (2013, 2014) and Auffray et al. (2013b).”
Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions (2017, methodology): “The procedure which is used in the aforementioned papers can be summarized as follows (see [5, 19]): i) one postulates the principle of virtual work, i.”
Dynamic problems for metamaterials: Review of existing models and ideas for further research (2014, methodology): “These can be formulated through a postulation process based on the principle of least action (Auffray et al., 2013) or on the principle of virtual works (Maugin, Metrikine, 2010; Maugin, 2013).”
At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola (2013, background): “This point is also discussed in [7] as remarked already in [52].”
Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives (2017, background): “Moreover, he argues that if one wants to deal with continua more general than fluids (for a discussion of this point one can have a look at the recent paper [2]) then it may depend (in a symmetric way) also on the Lagrangian coordinates of both X̄ and X : therefore he assumes that”
Synthesis of Fibrous Complex Structures: Designing Microstructure to Deliver Targeted Macroscale Response (2015, background): “1), the set of possible external contact actions is richer and may include line forces, concentrated forces, double forces, and objects of higher order [141].”
Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof (2015, background): “, [6–14]; for a general theoretical framework in which is discussed the link between n-th gradient models and the variational formulation, see [15]).”