Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids
TLDR
In this article, a Lagrangian action is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments.Abstract:
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 materia...read more
Citations
More filters
Journal ArticleDOI
General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach
TL;DR: In this paper, a variational macroscopic theory of porous media (VMTPM) is proposed to derive a general biphasic formulation at finite deformations, which proceeds from the inclusion of the intrinsic volumetric strain among the kinematic descriptors, aside to macro-scopic displacements, and uses the Hamilton least action principle as the unique primitive concept of mechanics invoked to derive momentum balance equations.
Book ChapterDOI
Circuit Analogies in the Search for New Metamaterials: Phenomenology of a Mechanical Diode
TL;DR: The Pantographic metamaterial as mentioned in this paper is a particular metammaterial, composed of two orthogonal families of fibers, with remarkable deformation properties, which recall by analogy the law characteristic of diodes, in the theory of electrical circuits.
Journal ArticleDOI
Verification of strain gradient elasticity computation by analytical solutions
Journal ArticleDOI
Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials
TL;DR: In this article, the authors consider the time evolution of a one dimensional gradient continuum and construct and analyze discrete approximations in terms of physically realizable mechanical systems, called microscopic because they are living on a smaller space scale.
Book ChapterDOI
Linear Dynamics of 2D Pantographic Metamaterials: Numerical and Experimental Study
Marco Laudato,Luca Manzari +1 more
TL;DR: In this article, the results of numerical and experimental studies on the linear time-invariant dynamics of a 2D pantographic material are presented, and the outcomes of a linear second gradient model enforcing the same symmetry of the microstructure is compared to experimental observations obtained via Digital Image Correlation (DIC).
References
More filters
Book
A Treatise on Electricity and Magnetism
TL;DR: The most influential nineteenth-century scientist for twentieth-century physics, James Clerk Maxwell (1831-1879) demonstrated that electricity, magnetism and light are all manifestations of the same phenomenon: the electromagnetic field as discussed by the authors.
Journal ArticleDOI
Free Energy of a Nonuniform System. I. Interfacial Free Energy
John W. Cahn,John E. Hilliard +1 more
TL;DR: In this article, it was shown that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2.
Book
Théorie des distributions
TL;DR: The merite as discussed by the authors is a date marque une date dans le progres des mathematiques and de la physique en levant l'ambiguite que constituait le succes des methodes de calcul symbolique aupres des physiciens and l'inacceptabilite de leurs formules au regard de la rigueur mathematiques.
Book
A comprehensive introduction to differential geometry
TL;DR: Spivak's comprehensive introduction to differential geometry as discussed by the authors takes as its theme the classical roots of contemporary differential geometry, and explains why it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely to rigorize the concepts of classical differential geometry.
Journal ArticleDOI
XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves
D. J. de Korteweg,G. de Vries +1 more
TL;DR: In this article, the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves were discussed, and a new model of long wave propagation was proposed.