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Completing Biot theory
Published in J.-L. Auriault, C. Geindreau, P. Royer, J.-F. Bloch, C. Boutin, J. Lewandowska, Poromechanics II, 2020
T.J.T. Spanos, N. Udey, M.B. Dusseault
The motion of an individual particle in a mechanical system is described by its position, velocity and acceleration. The principle of least action, properly applied in conjunction with the basic conservation principles (conservation of mass, energy, momentum, and the entropy production inequality), gives the most general formulation of the law governing the motion of mechanical systems. For large-scale (megascopic) description of a porous medium comprising a porous elastic matrix filled with a viscous fluid, one observes two degrees of freedom for a material point in the context of a specific physical process (e.g. the propagation of a sound wave). Hence, a separate Euler-Lagrange equation and an energy momentum tensor are obtained for the fluid and solid. Furthermore the interaction between fluid and solid motions plays a role in the megascopic thermodynamic equations, and porosity must be treated at this scale as a dynamic quantity, although the relationship between porosity and the average strains of the phases can only be specified in the context of a specific physical process. This dynamic role for porosity is independent of temperature, so theories of poromechanics and porodynamics are developed which are analogous to thermomechanics and thermodynamics (Spanos, 2001).
An Introductory Review of Classical Mechanics
Published in Ramaswamy Jagannathan, Sameen Ahmed Khan, Quantum Mechanics of Charged Particle Beam Optics, 2019
Ramaswamy Jagannathan, Sameen Ahmed Khan
to first order in δx¯(t), any arbitrary small deviation in the path between the fixed initial and final points, x¯(ti) and χ x¯(tf), respectively. Usually, along the actual path, the action takes the least value, and hence Hamilton’s principle is often called the principle of least action. Nature seems to have chosen such variational principles in formulating its basic laws.
Wave optics
Published in Timothy R. Groves, Charged Particle Optics Theory, 2017
The path taken by a ray of light can be found from Fermat’s principle, which states that the physical path represents the shortest possible transit time through a medium. The path taken by a particle can be found from the principle of least action, which states that the physical path represents the minimum of the action integral. These two principles are strikingly similar. They arose from a classical description, but as we shall see in the following, each has a wave-optical analog as well. No classical analog exists for the quantum mechanical description. However, quantum mechanical motion of particles and photons approaches classical behavior in the high-energy limit. The analogy between particle optics and light optics is deep and pervasive.
Bending of a Thin Rectangular Isotropic Micropolar Plate
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2019
In the year 1986, the Cosserat brothers, Eugen and Francois, published a monograph [14], where they presented a new variant of continuum mechanics as well as the mechanics of rods and shells. They have attempted to unify the field theories of mechanics, optics, and electrodynamics through a common principle of least action. They postulated that the invariance of energy under Euclidean transformation was able to derive the balance of force and balance of momentum. However, they never wrote down the constitutive relations. Later on, this continuum theory of elasticity is named as Cosserat or micropolar continuum solid. The Cosserat theory of elasticity is one of the most prominent extended continuum models that feature three additional degrees of freedom which are related to the rotation of particles and they need not coincide with the macroscopic rotation of continuum.