Keywords: finite element method, porous material, elasticity, voids, anisotropy, dynamic analysis.

The work presented in this paper is performed in the context of the linear theory of elastic anisotropic porous materials in accordance with the theory developed by Nunziato and Cowin [1]. This theory, which is a special branch of the general theory for micromorphic and microstretch materials, is useful for studying the manufactured materials and such materials as rock, ceramics and soils. Recently, a finite element method (FEM) has been proposed for finding numerical solutions of static [2] and dynamic [3] problems for isotropic elastic media with voids. At present we use the FEM for analyzing different problems for anisotropic elastic media with voids and suggest new damping models.

We construct new finite element schemes for investigating static, modal, harmonic and transient problems for anisotropic elastic solids with voids. In order to take into account the structural damping in elastic media, we use a new attenuation model which extends the Rayleigh models. We pose classical and generalized or weak formulations for dynamic problems with a new damping model. By applying semi-discrete FEM approximations of the solution to the governing equations in weak form we derive FEM equations with symmetrical matrices. We separately consider the cases of static, modal, harmonic and transient problems. We establish the properties of eigenvalues, resonance frequencies and eigenvectors or modes on FEM modal analysis. We formulate that the resonance frequencies for elastic body with voids are smaller than the resonance frequencies for pure elastic body. We note that for the finite element eigenvalue problems considered with partial coupling, the effective block algorithms can be applied.

We also show that the proposed new damping model with the modes superposition method enables to split the system of finite element equations into separate scalar equations for harmonic and transient problems. These approaches for dynamic problems are similar to those suggested for the theory of piezoelectricity [4]. We discuss the advantages and disadvantages of the mode superposition methods.

For direct time integration of the finite element equations of transient problems we employ the Newmark scheme in a convenient formulation, which does not explicitly use the velocities and accelerations of the nodal degrees of freedom. For each a time layers, we obtain the system of resolving equations in the symmetric form.

We consider three-dimensional, two-dimensional (plane stress and plane strain) and axisymmetric problems for anisotropic media with voids. For finding numerical results in two-dimensional problems we use unstructured meshes of quadratic triangular elements with six nodes. These elements with corresponding modifications are used for plane stress, plane strain and axially symmetric problems. The parameters of the finite element fragmentation are defined in an adaptive way by the specified relative error. For modelling three-dimensional generalized cylindrical shape structures we apply the extrusion technique of two-dimensional geometries and finite element meshes.

For the first numerical example we consider the static problem for uniaxial tension of the rectangle with a line crack under the plane strain condition. We compare the solutions for anisotropic materials with different orientations of crystallographic axes and with several values of the coupling number that characterizes porosity. As in the case of the isotropic material, the results of the calculations showed a growth in the compliance of the crack when the porosity increases. For the next examples, we investigate the axisymmetric eigenvalue problems for an anisotropic elastic porous glass with fixed foundation. We find and analyze the first resonance frequencies versus the coupling number. We also solve the harmonic problems for anisotropic porous glass by full and mode superposition methods taking into account the quality factors of material. For the last example we consider the transient problem for the anisotropic porous glass. We note that the voids can exert strong influence on the characteristics of transient process particularly for anisotropic materials.

In conclusion we formulate, that the finite element method is an effective numerical method for solving the static, modal, harmonic and transient problems for anisotropic elastic media with voids and change in porosity.

1

S.C. Cowin, J.W. Nunziato, "Linear elastic materials with voids", J. Elasticity, 13, 125-147, 1983. doi:10.1007/BF00041230

2

G. Iovane, A.V. Nasedkin, "Finite element analysis of static problems for elastic media with voids", Computers and Structures, 84, 1-2, 19-24, 2005. doi:10.1016/j.compstruc.2005.09.002

3

M. Ciarletta, G. Iovane, A.V. Nasedkin, "Finite element solutions of some problems for elastic solids with voids", Proc. XXXII Summer School-Conf. "Advanced Problems in Mechanics". St.Petersburg (Repino), June 24 - July 1, 2004. Ed. D.A. Indeitsev. St.Petersburg: Inst. Probl. Mech. Ing., 106-113, 2004.

4

A.V. Belokon, A.V. Nasedkin, A.N. Soloviev, "New schemes for the finite-element dynamic analysis of piezoelectric devices", J. Appl. Math. and Mech. (PMM), 66, 3, 481-490, 2002. doi:10.1016/S0021-8928(02)00058-8

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