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Saturated porous continua in the frame of hybrid description

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Abstract

A method for modeling fluid–solid interactions in saturated porous media is proposed. The main challenge is the combination of the material and spatial descriptions. The deformation of the solid, which serves as a “container” to the fluid, is studied by following the motion of its material particles, i.e., in Lagrangian description. On the other hand, the motion of the fluid is described in spatial form, i.e., by using a Eulerian approach. However, the solid deforms and this implies a certain difference regarding the standard formulation used in spatial description of fluid mechanics where a fixed grid dissects space into elementary volumes. Here the grid is no longer fixed, and the elementary volumes will follow the deformation of the solid. Moreover, for the solid as well as for the fluid the balance equations are formulated in the current configuration, where interaction forces and couples are taken into account. By using Zhilin’s approach, entropy and temperature are incorporated in the system of equations. Constitutive equations are constructed for both elastic and inelastic components of force and couple stress tensors and interaction force and couple. The constitutive equations for elastic components are found on the basis of the energy balance equation; the constitutive equations for the inelastic components are proposed in accordance with the second law of thermodynamics. Particular emphasis is placed on the constitutive equations of the interaction force and couple, which result in a symmetric form only because of the “hybrid” approach combining the Lagrangian with the Eulerian description. Three possible examples of application of the theory have been presented. For each example, all required assumptions were first stated and discussed and then the complete set of the corresponding equations was presented.

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Authors and Affiliations

  1. Departament of Theoretical Mechanics, Peter the Great St.Petersburg Polytechnic University, Polytechnicheskaya 29, St.-Petersburg, Russia

    Olga V. Brazgina, Elena A. Ivanova & Elena N. Vilchevskaya

  2. Laboratory of Mechatronics, Institute for Problems in Mechanical Engineering of Russian Academy of Sciences (IPME RUS), Bolshoy pr. V.O. 61, St.-Petersburg, Russia

    Elena A. Ivanova

  3. Laboratory of Mathematical Methods in Mechanics of Materials, Institute for Problems in Mechanical Engineering of Russian Academy of Sciences (IPME RUS), Bolshoy pr. V.O. 61, St.-Petersburg, Russia

    Elena N. Vilchevskaya

Authors
  1. Olga V. Brazgina
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  2. Elena A. Ivanova
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  3. Elena N. Vilchevskaya
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Correspondence to Olga V. Brazgina.

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Communicated by Andreas Öchsner.

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Brazgina, O.V., Ivanova, E.A. & Vilchevskaya, E.N. Saturated porous continua in the frame of hybrid description. Continuum Mech. Thermodyn. 28, 1553–1581 (2016). https://doi.org/10.1007/s00161-016-0495-0

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