Static Deformations of a Linear Elastic Porous Body Filled with an Inviscid Fluid

Abstract

Abstract. We study infinitesimal deformations of a porous linear elastic body saturated with an inviscid fluid and subjected to conservative surface tractions. The gradient of the mass density of the solid phase is also taken as an independent kinematic variable and the corresponding higher-order stresses are considered. Balance laws and constitutive relations for finite deformations are reduced to those for infinitesimal deformations, and expressions for partial surface tractions acting on the solid and the fluid phases are derived. A boundary-value problem for a long hollow porous solid cylinder filled with an ideal fluid is solved, and the stability of the stressed reference configuration with respect to variations in the values of the coefficient coupling deformations of the two phases is investigated. An example of the problem studied is a cylindrical cavity leached out in salt formations for storing hydrocarbons.Key words: solid-fluid mixture, conservative tractions, principle of virtual power, partial tractions, fluid-filled cylindrical cavity, stability analysis. R.C. Batra dedicates this work with deep respect and admiration to

“Hypertractions and hyperstresses convey the same mechanical information Continuum Mech. Thermodyn. (2010) 22:163–176” by Prof. Podio Guidugli and Prof. Vianello and some related papers on higher gradient theories

“…-The "full set of representation formulae not only, as is relatively easy, for tractions and hypertractions in terms of stresses and hyperstresses, but also, conversely, for stresses and hyperstresses in terms of di¤ used and concentrated tractions and hypertractions ... generalizing the corresponding formulae for simple materials" invoked in [43] has been established by Paul Germain ([26], [28]) in the seventies (extending the results of Casal [9], [10]) and has su¢ ciently and with more or less succes been rephrased or exploited (see e.g. [31], [50], [8], [51], [17], [18], [25]) and even recalled in the textbook [24]. Applications of second gradient theories to the mechanics of porous media is proposed for instance (but many other references could be given) in [11], [48] [19] where many results listed as novel in [43] are exploited.…”

“…Remark that Equation (35) on pag. 173 in [43] for instance is exactly equal to Equation (18) pag. 6612 in [47] or to Equation (13) pag.107 in [49].…”

The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results

“…The aforementioned condition of scale separation is not by itself a sufficient criterion for ensuring that Cauchy theory supplies a suitable model: the best-known example is the case of deformable porous media for which both stress tensor for matrix and pressure for fluid are needed to describe its mechanical state [2][3][4][5][6][7][8]. Another example is given by the case of a periodic fibre-reinforced elastic medium with high contrast of mechanical properties.…”

Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients

“…Another possibility is to consider higher-order gradient theories, in which the deformation energy depends on second and/or higher gradients of the displacement [17,33,40]. This is done in the literature for both monophasic systems (see [14,15,19,22,24,25,44,57], in which continuous systems are investigated, and [1,26,56,64] for cases of lattice/woven structures) and for biphasic (see, e.g., [16,18,20,21,41,45,60,61]) or granular materials [72]. Unlike classical Cauchy continua [4,62,63], second-and higher-order continua can respond to concentrated forces and other generalized contact actions (highly localized stress/strain concentration effects are studied, e.g., in [10]).…”