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Research Papers

A Dispersive Homogenization Model Based on Lattice Approximation for the Prediction of Wave Motion in Laminates

[+] Author and Article Information
G. Carta

Department of Structural, Infrastructural and Geomatic Engineering,  University of Cagliari, 09123 Cagliari, Italygiorgio_carta@unica.it

M. Brun1

Department of Structural, Infrastructural and Geomatic Engineering, University of Cagliari, 09123 Cagliari, Italy; Istituto Officina dei Materiali del CNR (CNR-IOM),  Unità SLACS, Cittadella Universitaria, 09042 Monserrato (CA), Italymbrun@unica.it

For a more extensive overview of both discrete and continuum approaches, refer to Maugin [10] and Engelbrecht et al.  [11].

Note that this result applies for a one-dimensional structure, while for the general case of a three-dimensional laminate E=[f(1-2ν1)(1+ν1)E1(1-ν1)+(1-f)(1-2ν2)(1+ν2)E2(1-ν2)+2·(fν11-ν1+(1-f)ν21-ν2)2/(fE11-ν1+(1-f)E21-ν2)]-1, that reduces to Eq. 2 for Poisson’s ratios ν1  = ν2  = 0 (see, for example Torquato [15]).

Note that in Eq. 24 no body forces f are assumed. For f ≠ 0, the additional term f has to be added on the right hand side of the equation. An applied acceleration a(x,t) can be expressed by the force f(x,t)=ρ a(x,t).

1

Corresponding author.

J. Appl. Mech 79(2), 021019 (Feb 24, 2012) (8 pages) doi:10.1115/1.4005579 History: Received February 10, 2011; Revised October 08, 2011; Posted February 01, 2012; Published February 13, 2012; Online February 24, 2012

The propagation of elastic waves in a periodic laminate is considered. The stratified medium is modeled as a homogenized material where the stress depends on the strain and additional higher order strain gradient terms. The homogenization scheme is based on a lattice model approximation tuned on the dispersive properties of the real laminate. The long-wave asymptotic approximation of the model shows that, despite the simplicity of the parameters identification, the proposed approach agrees well with the exact solution in a wide range of elastic impedance contrasts, also in comparison with different approximations. The effect of increasing order of approximation is also investigated. A final example of a finite structure under an impact excitation proves that the model behaves well when applied in the transient regime and that it can be considered a simple but consistent approach to build efficient algorithms for the numerical analysis of elastodynamics problems.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Generic one-dimensional periodic laminate under investigation

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Figure 2

Dispersion curves provided by the exact formulation and different approximations: (a) impedance contrast z1 /z2  = 2.5; (b) impedance contrast z1 /z2  = 10.0

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Figure 3

Comparison of the dispersion curve produced by the approximation of the lattice model with those derived from other methods: (a) impedance contrast z1 /z2  = 2.5; (b) impedance contrast z1 /z2  = 10.0

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Figure 4

Dispersion curves predicted by the approaches considered in Fig. 3 Impedance contrast z1 /z2  = 7.0; volume fraction: (a) f = 0.2; (b) f = 0.8

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Figure 5

Dependence on expansion order n of: (a) absolute value of the relative error in ω at k D = π; (b) normalized group velocity at k D = π

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Figure 6

Elastic bar subjected to an impact load. The displacement profiles are evaluated at t = 250 D/c0 and are calculated for two different values of the ratio z1 /z2 and for volume fractions f = 0.5 and f = 0.2, respectively. The black curve refers to the proposed continuum formulation, the gray curve to an explicit FEM simulation and the dashed curve to a homogeneous nondispersive wavefront.

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