0
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

Abstract

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.

<>

Figures

Figure 1

Generic one-dimensional periodic laminate under investigation

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

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

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

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

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.

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections