Analytical and numerical studies on Penalized Micro-Dilatation (PMD) theory: Macro-micro link concept

In the present study, the micro-dilatation theory has been investigated. The analytical investigations on the penalized cases using a novel total energy density concept have been achieved via quadratic, affine and linear vector-to-vector and vector-to-scalar mappings. The total energy density has divided into four counterparts which are explicitly linked into the constitutive parameters. This novel energy density concept leads to four explicit constitutive laws. By taking advantage of the strongly ellipticity for the total strain energy density, a freshly defined coupling number was introduced. This issue yields two interesting outcomes, the first one is that the micro-dilatation coupling modulus β is bounded β2 < Kξ and negative (β < 0) and the second one is that the pore pressure must be less than a specific threshold (P2 < φ2Kξ). As a matter of fact, the micro-dilatation theory holds a rigorous restriction on the pore-fluid pressure comparing to Biotʹs theory in which there is no restriction. We have analytically scrutinized the case limits and particularly bulk (case1) and conformal case (case2) in which the heterogeneous deformations occur. The numerical simulations of the micro-dilatation theory have been brought afterwards using Galerkin FEM under an Augmented Lagrangian–Eulerian (ALE) setting. The aforementioned numerical simulations substantiate that one can obtain the physically acceptable results for moderate and low coupling numbers. Based upon these results about the micro-dilatation theory some new routes in determining the micro-dilatation theory constants have been triggered.