An integrated approach for nonlinear model of composite rod (asymptotic homogenization vs. variational principles)
Instituto de Cibernética, Matemática y Física, ICIMAF
15 # 551 e/ C y D, Vedado, Habana 4, CP 10400, Ciudad de la Habana, Cuba
J.L.Gómez-Muñoz
Instituto Tecnológico de Estudios Superiores de Monterrey, Campus
Estado de México. Carretera Lago de Guadalupe Km. 3.5, Atizapán
de
Zaragoza, Estado de México, CP 52926, México
Universidad de La Habana
San Lázaro y L, Vedado, Habana 4, CP 10400, Ciudad de la Habana,
Cuba
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México. Apartado Postal 20-726, Delegación de Álvaro Obregón, 01000 México D. F., México
J.C.Sabina de Lis
Universidad
de La Laguna 38271 La Laguna, Tenerife, España
Recently, the Asymptotic
Homogenization Method (AHM) and the nonlinear Hashin-Shtrikman variational
principles have been combined in order to improve bounds on the effective
behavior of 2D and 3D nonlinear composites. The purpose of the present work is
not only to combine the AHM and variational approaches but also to compare
them. The one-dimensionality considered here provides a suitable environment to
accomplish this goal. Some applications and numerical experiments are
presented.
The exact effective response of
nonlinear materials is usually, although not always, impossible to find, due to
the dependence upon the details of the microstructure which is often known
inaccurately. It is more likely that the available information would be limited
to concentrations of the phases, mean shapes of the inclusions and some spatial
correlations.
Starting on the information at hand,
one of the ways to obtain an approximation of the effective behavior of a
nonlinear composite is to limit, over all the microgeometries consistent with
such information, the range of the possible responses. From a mathematical
point of view, such range takes the form of bounds, which include the given
information and, therefore, will not apply to all classes of material behavior,
that is, all types of composites. Several attempts to improve such bounds have
been made by using different approaches. One of them, the generalization to
nonlinear problems of the variational principles of Hashin & Shtrikman, was
originated in the work of Willis. The nonlinear Hashin-Shtrikman principles
rely on the introduction of a comparison material, which has been taken to be
linear in most studies, although it has been chosen to be nonlinear in a number
of works, where nonlinear homogeneous comparison materials are used. An
alternative, introduced by Ponte-Castañeda, consists in taking a linear
comparison composite having the same microstructure as the nonlinear composite.
This approach was incorporated to the Hashin-Shtrikman methodology by
Talbot&Willis. The advantage of using a linear comparison composite is that
any bound now includes information about the effective properties of the comparison
composite. Talbot used the linear bounds by Bruno for the linear comparison
composite employed in the nonlinear bounds for a 3D matrix-inclusion composite
dielectric, while Talbot applied the linear bounds by Bruno & Chaubell for
the linear comparison composite used in the nonlinear bounds for 2D and 3D
incompressible elastic matrix-inclusion composites. In both cases, the
nonlinear composite consists in one linear phase and one nonlinear phase and,
by alternating the roles of such constitutive behaviors for the matrix and the
inclusion improved bounds are provided: an upper bound for a linear matrix
containing nonlinear inclusions, and a lower bound for a nonlinear matrix
containing linear inclusions.
A different way to approximate the
effective behavior of a nonlinear composite is to obtain an estimate in a
direct approach such as the Asymptotic Homogenization Method (AHM) which has
been successfully applied to problems of linear elasticity and
piezoelectricity, for instance, closed-form expressions for the effective
coefficients of fiber-reinforced composites with transversely isotropic
constituents in the context of linear elasticity and piezoelectricity for
square and hexagonal symmetry were obtained, and exact formulae for the
effective stiffness of fiber-reinforced composites with isotropic components
including the limit cases in which fibers are empty or infinitely rigid were
obtained which have proven to lie between available linear bounds. References
to text books and many other linear applications of AHM can be found in our
early works.
Recently, we have used AHM estimates
instead of known bounds O.P.Bruno, for the effective behavior of the linear
comparison material, in order to improve the nonlinear bounds applied to 2D
composites. Moreover, the AHM
approach and the Finite Elements Method have been combined
(A.M.León-Mecías, J.Bravo-Castillero, A.Mesejo-Chiong,
L.D.Pérez-Fernández) to obtain an improved estimate for the
effective behavior of a linear comparison composite with the same microgeometry
as the 3D composite and used it to improve those results.
Other interesting approaches in predicting and
bounding the effective properties of linear and nonlinear materials are
presented in another works. Also here it is possible to find a wide variety of
results including generalizations of the Hashin-Shtrikman variational
principles and applications to piezocomposites, as well as those concerning
classical theories.
The purpose of this work
is not only to combine the AHM and variational approaches, but also to compare
them in order to validate the effectiveness of the direct application of the
AHM to nonlinear composites. Although the one-dimensionality considered here
might seem to be lacking of realism and generality, it represents a comfortable
environment to accomplish this first attempt, even when the constitutive
potentials employed here generalize those in D.R.S.Talbot papers. This work is
structured as follows: section 2 is devoted to comment the theoretical
foundations of the AHM and the variational principles used here; in section 3,
some applications and examples, including numerical calculations as well as
experimental data, are presented and discussed; and section 4 is dedicated to
some concluding remarks.
The one-dimensional case has proved to be a suitable environment to address the study of the effective behavior of a nonlinear composite. Two different approaches to accomplish this goal were presented and described: the Asymptotic Homogenization Method and some variational bounding procedures. Application to power-law composites yields as a result that the AHM estimate coincides with all the variational bounds except the elementary upper bound. In this case, the importance of an adequate election of the comparison material is noticed. In the second and third applications, for particular two-phase composites, it is obtained that the AHM estimate and the lower bounds coincide for every choice of the parameters taken as independent variables, even when, in the third example, two different (linear and power-law) comparison composites were used.
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