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# Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

Computational Mechanics (Impact Factor: 2.43). 04/2012; 41(6):797-816. DOI:10.1007/s00466-007-0167-9

ABSTRACT This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing
an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use
of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary
integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior
boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method
is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on
the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated
complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the
unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is
formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic
analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time
domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous
material are then found directly from the corresponding constitutive equations for the average field values.

Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared
with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties
are compared with those obtained with the self-consistent and Mori–Tanaka schemes.

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