Keywords: porous material, buckling, axial compression, critical load.

This paper considers an isotropic porous shell loaded by axial compression shown in Figure 53.1. The effects of varying porous material and geometric parameters of the cylindrical shell, on the elastic buckling load are studied. The shell is made of an isotropic porous material which varies only across the thickness of the shell wall. Banhart [1] reviewed the possibilities of manufacturing metal foams or other porous metallic structures as well as ways for characterizing the properties of cellular metals. In the present paper the porous shell is a generalization of a sandwich construction.

A general overview of modelling and stability of sandwich structures is presented in survey paper [2]. The material model used in the current paper is as described in references [3,4]. The basic assumption of porosity varying in the transverse direction is defined as

where is the coefficient of the panel porosity; , are values of the Young's modulus for the neutral and outer surfaces (shown in Figure 53.2); is the dimensionless coordinate; is the thickness of the panel.

A non-linear hypothesis of the deformation of a plane cross section of the shell is assumed. The geometric and physical relationships (according to Hooke's law) are linear. The above stated problem gives the system of differential equations and boundary conditions written in a cylindrical coordinates system. In a second step, basic equations of elastic buckling of an isotropic porous shell are obtained. They are formulated on the grounds of the principle of stationary total potential energy of the compressed shell, , where is the energy of elastic strain and is the work of load. A simply supported cylindrical porous panel carries a uniformly distributed axial load of intensity over its span (the length of the curve). In a particular case a shell is made of an isotropic non-porous material, and the elasticity coefficient does not depend on the coordinate . In this case a Lorenz-Timoshenko-Southwell formula of the following form was obtained: . . In addition to analytical treatment, the problem was also analyzed using the FEM method by means of the ANSYS computer system. Some graphical outputfrom this system is shown in Figure 53.3. The analytical and numerical analyses were carried out for following porous shells: Young's modulus MPa, MPa, MPa and MPa, and suitably , , . The material properties varying through the thickness of the cross-section were discretized with 41 layers of constant properties.

The FEM critical parameter values for a family of shells are 3.5% smaller than values obtained by means of the analytical solution. Comparison of analytical solution with the FE results and Lorenz-Timoshenko-Southwell formula shows that the error is within the range +8.6% and -0.4%.

1

J. Banhart, "Manufacture, characterisation and application of cellular metals and metal foams", Progress in Materials Science, 46, 559-632, 2001. doi:10.1016/S0079-6425(00)00002-5

2

L. Librescu, T. Hause, "Recent developments in the modelling and behaviour of advanced sandwich constructions: a survey", Composite Structures, 48, 1-17, 2000. doi:10.1016/S0263-8223(99)00068-9

3

K. Magnucki, P. Stasiewicz, "Elastic bending of an isotropic porous beam", Int. Journal of Applied Mechanics and Engineering, 9(2), 351-360, 2004.

4

K. Magnucki, P. Stasiewicz, "Elastic buckling of a porous beam", Journal of Theoretical and Applied Mechanics, 42(4), 859-868, 2004.

5

J. Mielniczuk, "Plasticity of porous materials. Theory and the limit load capacity", Publishers of Poznan University of Technology, Poznan (in Polish), 2000.

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