# ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

Copyright © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Editor: Holm Altenbach

Impact Factor: 1.008

ISI Journal Citation Reports © Ranking: 2013: 76/251 (Mathematics Applied); 79/139 (Mechanics)

Online ISSN: 1521-4001

Associated Title(s): GAMM-Mitteilungen, Mathematische Nachrichten, Mathematical Logic Quarterly, PAMM

### Recently Published Issues

Current Issue:March 2015

Volume 95, Issue 3

Volume 95, Issue 2

Volume 95, Issue 1

Volume 94, Issue 12

Volume 94, Issue 11

Special Issue: ICoVIS 2013: Vibro-Impact Systems a...

### Recently Published Articles

- The influence of hydrostatic stress on the frequency equation of flexural waves in a magnetoelastic transversly isotropic circular cylinder
Abo-el-nour N. Abd-alla, Aishah Raizah and Luca Placidi

Article first published online: 21 MAR 2015 | DOI: 10.1002/zamm.201400059

In this paper, we investigated the influence of initial stress on the frequency equation of flexural waves in a transversely isotropic circular cylinder permeated by a magnetic field. The problem is represented by the equations of elasticity taking into account the effect of the magnetic field as given by Maxwell's equations in the quasi-static approximation. The free stress conditions on the inner and outer surfaces of the hollow circular cylinder were used to form a frequency equation in terms of the wavelength, the cylinder radii, the initial stress and the material constants. The frequency equations have been derived in the form of a determinant involving Bessel functions and its roots given the values of the characteristic circular frequency parameters of the first three modes for various geometries. These roots, which correspond to various modes, have been verified numerically and represented graphically in different values for the initial stress. It is recognized that the flexural elastic waves in a solid body propagated under the influence of initial stress can be differentiated in a clear manner from those propagated in the absence of an initial stress. We also observed the initial stress has a great effect on the propagation of magnetoelastic flexural waves. Therefore this research is theoretically useful to convey information on electromagnetic properties of the material: for example through a precise measurement of the surface current induced by the presence of the magnetic field.

- The effective solution of two-dimensional integro-differential equations and their applications in the theory of viscoelasticity
Nugzar Shavlakadze

Article first published online: 21 MAR 2015 | DOI: 10.1002/zamm.201400091

The effective solutions for integro-differential equations related to problems of interaction of an elastic thin finite inclusion with a plate, when the inclusion and plate materials possess the creep property are constructed. If the geometric parameter of the inclusion is measured along its length according to the parabolic and linear law we have managed to investigate the obtained boundary value problems of the theory of analytic functions and to get exact solutions and establish behavior of unknown contact stresses at the ends of an elastic inclusion.

- The spin, the nutation, and the precession of the Earth's axis revisited from a (numerical) mechanics perspective
Wolfgang H. Müller

Article first published online: 16 MAR 2015 | DOI: 10.1002/zamm.201400252

Modeling the motion of the Earth's axis,

*i.e*., its spin, nutation and its precession, is a prime example of our ongoing effort to simulate the behavior of complex mechanical systems. In fact, models of increasing complexity of this motion have been presented for more than 400 years leading to an increasingly accurate description. The objective of this paper is twofold namely, first, to provide a review of these efforts and, second, to provide an improved analysis, if possible, based on today's numerical possibilities. Newton himself treated the problem of the precession of the Earth, a.k.a. the precession of the equinoxes, in Liber III, Propositio XXXIX of his Principia . He decomposed the duration of the full precession into a part due to the Sun and another part due to the Moon, which would lead to a total duration of 25,918 years. This agrees fairly well with the experimentally observed value. However, Newton does not really provide a concise rational derivation of his result. This task was left to Chandrasekhar in Chapter 26 of his annotations to Newton's book . He follows an approach suggested by Scarborough starting from Euler's equations for the gyroscope and by calculating the torques due to the Sun and to the Moon on a tilted spheroidal Earth. These differential equations can be solved approximately in an analytic fashion, yielding something close to Newton's more or less fortuitous result. However, the equations can also be treated more properly in a numerical fashion by using a Runge-Kutta approach allowing for a study of their general non-linear behavior. This paper will show how and discuss the outcome of the numerical solution. A comparison to actual measurements will also be attempted. When solving the Euler equations for the aforementioned case numerically it shows that besides the precessional movement of the Earth's axis there is also a nutational one present. However, as we shall show, if Scarborough's procedure is followed, the period of this nutation turns out to be roughly half a year with a very small amplitude whereas the observed (main) nutational period is much longer, namely roughly nineteen years, and much more intense amplitude-wise. The reason for this discrepancy is based on the assumption that the torques of both the Sun and the Moon are due to gravitational actions within the equinoctial plane. Whilst this is true for the Sun, the revolution of the Moon around the Earth occurs in a plane, which is inclined by roughly 5° w.r.t. the equinoctial. Moreover, this plane rotates such that the ascending and descending nodes of the moon precede with a period of roughly 18 years. If all of this is taken into account the analytically predicted nutation period will be of the order of the observed value . As in the case of the precession we will provide a more stringent analysis based on a numerical solution of the Euler equations, which leads beyond the results presented in Sect.12.10 of . - Geometrically nonlinear FEM analysis of 6-parameter resultant shell theory based on 2-D Cosserat constitutive model
S. Burzyński, J. Chróścielewski and W. Witkowski

Article first published online: 12 MAR 2015 | DOI: 10.1002/zamm.201400092

We develop the elastic constitutive law for the resultant statically and kinematically exact, nonlinear, 6-parameter shell theory. The Cosserat plane stress equations are integrated through-the- thickness under assumption of the Reissner-Mindlin kinematics. The resulting constitutive equations for stress resultant and couple resultants are expressed in terms of two micropolar constants: the micropolar modulus and the micropolar characteristic length

*l*. Based on FEM simulations we evaluate their influence on the behaviour of shell models in the geometrically nonlinear range of deformations. - On perturbation method in mechanical, thermal and thermo-mechanical loadings of plates: cylindrical bending of FG plates
F. Fallah, A. Nosier, M. Sharifi and F. Ghezelbash

Article first published online: 4 MAR 2015 | DOI: 10.1002/zamm.201400136

The performance of perturbation method in nonlinear analyses of plates subjected to mechanical, thermal, and thermo-mechanical loadings is investigated. To this end, cylindrical bending of FG plates with clamped and simply-supported edges is considered. The governing equations of Mindlin's first-order shear deformation theory with von Kármán's geometric nonlinearity are solved using one- and two-parameter perturbation methods and the results are compared with the results of an analytical solution. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. It is shown that the accuracy of any-order expansion in perturbation method depends not only on the perturbation parameter, but also on the location chosen for the perturbation parameter and, in general, the solution becomes more accurate when the perturbation parameter is specified at the location where its corresponding response quantity is a maximum. Under thermal loading the possibility of using different parameters as the perturbation parameter for various boundary conditions is investigated. It is observed that, instead of a one-parameter perturbation method, a two-parameter perturbation method must be used in the thermal analysis of FG plates. Also, buckling and post-buckling behavior of FG plates in cylindrical bending is investigated. It is shown that under thermal loading, a bifurcation-type buckling occurs in clamped FG plates. In addition, a snap-through buckling may occur in simply-supported FG plates under thermo-mechanical loading.