On two elastodynamic homogenization methods for periodic composites
•A constructive link between Willis’ method and the two-scale asymptotic method.•Effective impedance tensor characterizing Willis’ effective elastodynamic law.•Recursion formula for solving the hierarchical motion equations issuing from asymptotic analysis. Two-scale asymptotic method and Willis’ me...
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| Published in | Applied mathematical modelling Vol. 113; pp. 109 - 128 |
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| Format | Journal Article |
| Language | English |
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01.01.2023
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| Abstract | •A constructive link between Willis’ method and the two-scale asymptotic method.•Effective impedance tensor characterizing Willis’ effective elastodynamic law.•Recursion formula for solving the hierarchical motion equations issuing from asymptotic analysis.
Two-scale asymptotic method and Willis’ method are jointly examined for the elastodynamic homogenization of periodic composites. The effective elastodynamic constitutive law given by Willis’ method, non-local both in time and space, is first reformulated so as to make appear a compact expression for the effective impedance tensor characterizing it. The two-scale asymptotic method is then revisited by exploiting the fact that the results obtained by it constitute an approximation to the general ones delivered by Willis’ method. The solutions for the hierarchical motion equations issuing from asymptotic analysis are shown to admit a compact recursive representation. A generic compact expression is derived for the effective impedance tensor associated with the nth-order approximation of asymptotic analysis. Remarkably, this expression turns out to be formally identical to the one of the effective impedance tensor obtained via Willis’ method. As an example of illustration, the elastodynamic homogenization of a layered composite undergoing anti-plane shear is investigated in detail. |
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| AbstractList | •A constructive link between Willis’ method and the two-scale asymptotic method.•Effective impedance tensor characterizing Willis’ effective elastodynamic law.•Recursion formula for solving the hierarchical motion equations issuing from asymptotic analysis.
Two-scale asymptotic method and Willis’ method are jointly examined for the elastodynamic homogenization of periodic composites. The effective elastodynamic constitutive law given by Willis’ method, non-local both in time and space, is first reformulated so as to make appear a compact expression for the effective impedance tensor characterizing it. The two-scale asymptotic method is then revisited by exploiting the fact that the results obtained by it constitute an approximation to the general ones delivered by Willis’ method. The solutions for the hierarchical motion equations issuing from asymptotic analysis are shown to admit a compact recursive representation. A generic compact expression is derived for the effective impedance tensor associated with the nth-order approximation of asymptotic analysis. Remarkably, this expression turns out to be formally identical to the one of the effective impedance tensor obtained via Willis’ method. As an example of illustration, the elastodynamic homogenization of a layered composite undergoing anti-plane shear is investigated in detail. |
| Author | Le Quang, Hung Luo, Wei-Zhi He, Qi-Chang |
| Author_xml | – sequence: 1 givenname: Wei-Zhi surname: Luo fullname: Luo, Wei-Zhi – sequence: 2 givenname: Qi-Chang orcidid: 0000-0003-2311-1929 surname: He fullname: He, Qi-Chang email: qi-chang.he@u-pem.fr – sequence: 3 givenname: Hung surname: Le Quang fullname: Le Quang, Hung |
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| Keywords | Homogenization Composites Asymptotic analysis Dynamics Dispersion relation |
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