On the theory of elastic shells made from a material with voids

Abstract

In this paper we present a theory for porous elastic shells using the model of Cosserat surfaces. We employ the Nunziato-Cowin theory of elastic materials with voids and introduce two scalar fields to describe the porosity of the shell: one field characterizes the volume fraction variations along the middle surface, while the other accounts for the changes in volume fraction along the shell thickness. Starting from the basic principles, we first deduce the equations of the nonlinear theory of Cosserat shells with voids. Then, in the context of the linear theory, we prove the uniqueness of solution for the boundary initial value problem. In the case of an isotropic and homogeneous material, we determine the constitutive coefficients for Cosserat shells, by comparison with the results derived from the three-dimensional theory of elastic media with voids. To this aim, we solve two elastostatic problems concerning rectangular plates with voids: the pure bending problem and the extensional deformation under hydrostatic pressure.

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

“…Some relevant studies pertaining to the application of micro-dilatation theory to the wave propagation and the numerical implementations can be also addressed in Nasedkin (2009, 2010a,b). The other relevant works in conjunction with the micro-dilatation are also available in Birsan (2003), Birsan (2006) Chirita et al (March 2006), Chirita and Ghiba (2010a,b), Singh (2011) and lately in Ramézani et al (2012b); Thurieau et al (2013), Jeong et al (2013b), Thurieau et al (2014). Fifth-rank tensor quantities D 4 u ¼ u i;jklmêi ê j ê k ê l ê m fifth-rank tensor defined as E Cauchy-Green strain tensor in [-] r u tensorial gradient of displacement vector [-] ru :¼ ðr uÞ T displacement gradient tensor m m  à or [-] r X u Lagrangian displacement gradient [-] h i…”

“…Micro-stretch Eringen (1990, 2001) 1971Micro-strain Forest and Sievert (2006) 2006Micro-morphic Eringen (2001 1965 where, Du :¼ u i;jê ê j 2 R 3 Â R 3 , D 2 u :¼ u i;jkêi ê j ê k 2 R 27 , D 3 u :¼ u i;jklêi ê j ê k ê l 2 R 81 and D 4 u :¼ u i;jklmêi ê j ê k ê l ê m 2 R 243 are second-rank tensor well known as r X u ¼ ru :¼ ðr uÞ T , third-rank tensor defined as second derivation of displacement vector, fourth-rank tensor defined as third derivation of displacement vector and fifth-rank tensor defined as fourth derivation of the displacement vector, respectively. Using a linear approximation of the above-mentioned Taylor series expansion, an approximation can be extracted as below: …”

On the theory of porous elastic rods

“…We see that the porosity field u appears only in the extension-torsion problem (62), (63). If we want to obtain a model in which the porosity is present also in the bending-shear problem (64) and (65), then we have to introduce from the beginning several porosity fields, in a similar way as in the theory of shells (Bîrsan, 2006).…”

A Nonlinear Mathematical Model of Thermoelastic Thin Plates with Voids and Its Applications

“…Scarpetta [2002] presented the minimum principle for the bending problem of elastic plates with voids. Bîrsan [2003] gave a bending theory of porous thermoelastic plates, and then, Bîrsan [2006aBîrsan [ , 2006b presented a nonlinear theory for porous elastic and thermoelastic shells. Sharma et al [2008] analyzed the 3D vibration of a thermoelastic cylindrical panel with voids.…”