Non-linear elastic micro-dilatation theory: Matrix exponential function paradigm

Abstract

a b s t r a c tIn the present paper, the micro-dilatation theory or void elasticity is extended to both large displacement and large dilatation. Firstly, the deformation gradient tensor has been freshly defined by means of the matrix exponential function. The newly defined relation for the deformation gradient has painstakingly investigated for the uniqueness, decomposition issues as well as objectivity and isotropy considerations. The relation of the displacement gradient and deformation gradient tensor is brought via the matrix logarithm function. The micro-dilatation theory constitutive laws are derived using the thermodynamic principles under the zero-centrosymmetric, weakly-centrosymmetric and fully-centrosymmetric cases. These cases have been derived and scrutinized by the numerical experiments. To achieve this assignment, the basic loadings are taken into account, e.g. the hydrostatic loading, simple traction and shear. Some conclusions and outlook pertaining to the above-mentioned cases and variable bulk density have thereafter discussed.

Bending problems in the theory of elastic materials with voids and surface effects

“…Here, E PB is an effective flexural modulus of the beam with micro-dilatations in the pure bending test. The surface effects could be taken into account in (31) if we use (25), (26) instead of (22), (24) to define J. Relation (31) also can be found, assuming the equality of rotations u 1, 3 = u class 1, 3 of the beams ends or the equality of axial strains on the upper or lower beams surfaces: e 33 = e class 33 .…”

“…If there is no coupling in the media (N=0), the effective flexural modulus is equal to the material Young's modulus. In equation 31, there is an explicit relation between thickness and scale parameter h=' (see (22)- (26)). Thus, an analytical solution predicts the scale effects.…”

“…The authors provided the finite-element (FE) simulations using a solver with direct estimation algorithms. In recent papers [23][24][25][26], analyses of the different FE solutions were presented for homogenous nonlinear deformations [26], deformations of plate with a hole [23], deformation of the hollow thick cylinder under pressure [23], and for shrinkage of the rectangular porous beams [24]. An evaluation of the non-classical material constants based on the numerical analysis was then performed.…”

Static and dynamic behavior of porous elastic materials based on micro-dilatation theory: A numerical study using the MLPG method

“…The theory is founded on the balance of energy, where the presence of pores involves additional degree of freedom. Cowin-Nunziato theory (also known as micro-dilatation theory) has been successfully applied to many problems with special geometries under particular boundary conditions (see, for example, Adkin et al, 1977;Chandrasekharaiah, 1987;Scalia and Sumbatyan, 20 0 0;Scalia, 20 02 ;and recently Ramézani et al, 2012;Ramézani and Jeong, 2015 ) and showed good agreement with both experimental data and physical conclusions. The theory can also be utilized for modeling cracks in porous materials ( Ciarletta et al, 2003 ).…”

On the filtrate drilling fluid formation and near well-bore damage along the petroleum well

“…The micro-dilatation theory belongs to the generalized continuum mechanics family. The generalized continuum mechanics contains the micro-dilatation theory (Nunziato and Cowin, 1979;Cowin and Nunziato, 1983;Markov, 1995;Ciarletta et al, 2003;Iovane and Sumbatyan, 2005;Ramézani et al, 2012a;Jeong et al, 2013;Ramézani and Jeong, 2015), micro-polar theory (Cosserat and Cosserat, 1909;Eringen and Kafadar, 1976;Eringen and Suhubi, 1964;Gauthier and Jahsman, 1975;Lakes, 1986;Eringen, 2002;Sansour and Skatulla, 2008;Jeong et al, 2009;Jeong and Ramézani, 2010;Ramézani et al, 2011;Eremeyev et al, 2012;Ramézani et al, 2012b;Skatulla and Sansour, 2013), micro-stretch theory (Eringen, 1990(Eringen, , 2002Cicco, 2003;Lazar and Anastassiadis, 2006;Kirchner and Steinmann, 2007;Neff et al, 2009) and micro-morphic theory (Eringen, 2002;Neff and Forest, 2007;Forest, 2009).…”