iMechanica - periodic materials
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enIdentification of higher-order continua equivalent to a Cauchy elastic composite
http://imechanica.org/node/21468
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/609">homogenization</a></div><div class="field-item odd"><a href="/taxonomy/term/11736">Higher-order continuum</a></div><div class="field-item even"><a href="/taxonomy/term/1782">size-effect</a></div><div class="field-item odd"><a href="/taxonomy/term/11737">Non-local elasticity</a></div><div class="field-item even"><a href="/taxonomy/term/7324">periodic materials</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>Dear Mechanician,</span></p>
<p><span>A novel method for the identification of higher-order continua equivalent to a Cauchy composite has been published, as a result of the collaboration between the following two ERC projects:</span></p>
<p><span><a href="http://erc-instabilities.unitn.it">http://erc-instabilities.unitn.it</a></span></p>
<p><span><a href="http://musam.imtlucca.it/CA2PVM.html">http://musam.imtlucca.it/CA2PVM.html</a></span></p>
<p><strong><span>Full paper:</span></strong></p>
<p><span><a href="https://doi.org/10.1016/j.mechrescom.2017.07.002">https://doi.org/10.1016/j.mechrescom.2017.07.002</a></span></p>
<p><strong><span>Abstract:</span></strong></p>
<p><span>A heterogeneous Cauchy elastic material may display micromechanical effects that can be modeled in a homogeneous equivalent material through the introduction of higher-order elastic continua. Asymptotic homogenization techniques provide an elegant and rigorous route to the evaluation of equivalent higher-order materials, but are often of difficult and awkward practical implementation. On the other hand, identification techniques, though relying on simplifying assumptions, are of straightforward use. A novel strategy for the identification of equivalent second-gradient Mindlin solids is proposed in an attempt to combine the accuracy of asymptotic techniques with the simplicity of identification approaches. Following the asymptotic homogenization scheme, the overall behaviour is defined via perturbation functions, which (differently from the asymptotic scheme) are evaluated on a finite domain obtained as the periodic repetition of cells and subject to quadratic displacement boundary conditions. As a consequence, the periodicity of the perturbation function is satisfied only in an approximate sense, nevertheless results from the proposed identification algorithm are shown to be reasonably accurate.</span></p>
</div></div></div>Wed, 02 Aug 2017 18:29:00 +0000marco.paggi21468 at http://imechanica.orghttp://imechanica.org/node/21468#commentshttp://imechanica.org/crss/node/21468Journal Club Theme of April 2012: Phononics: Structural Dynamics of Materials
http://imechanica.org/node/12210
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/437">video</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/6923">phononic crystals</a></div><div class="field-item odd"><a href="/taxonomy/term/6924">acoustic metamaterials</a></div><div class="field-item even"><a href="/taxonomy/term/7322">Phononics</a></div><div class="field-item odd"><a href="/taxonomy/term/7323">phononic materials</a></div><div class="field-item even"><a href="/taxonomy/term/7324">periodic materials</a></div><div class="field-item odd"><a href="/taxonomy/term/7325">band gaps</a></div><div class="field-item even"><a href="/taxonomy/term/7326">Bragg scattering</a></div><div class="field-item odd"><a href="/taxonomy/term/7327">Local resonance</a></div><div class="field-item even"><a href="/taxonomy/term/7328">Multiscale dispersive design</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Engineering structures are made out of materials and as such there is a natural hierarchy in which a material’s intrinsic properties contribute to shaping up the structure’s response. It is possible however to reverse this hierarchy and engineer materials that are made out of structures. In this case, the intrinsic properties of a material are shaped by the structural response. Such a configuration can only be realized with a repeated structure, forming an array of identical unit cells. Of course the structural components in the unit cell will themselves be formed out of a constituent material (or more), but this is at another level and the properties of the overall material will be different from those of this constituent material. The emerging field of phononics is mostly concerned with this type of reverse construction, whereby one can conduct structural dynamics analysis and design towards the design of materials with desired properties in contrast to the conventional activity of direct design of structures with a desired response. There is various terminology in the literature for the description of this type of material, the most common are phononic materials, lattice materials and periodic materials. Figure 1 illustrates the concept of a “structure made out of a material” versus a “material made out of a structure”. </p>
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<br /><img src="http://www.spacedavis.com/Downloads/Fig_01_IMechanica.png" alt=" Demonstration of material-structure reversal in the formation of a phononic material unit cell. " title=" Demonstration of material-structure reversal in the formation of a phononic material unit cell. " width="624" height="554" /><br />
Figure 1: Demonstration of material-structure reversal in the formation of a phononic material unit cell.
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So what are the implications and benefits of undergoing such a material-structure reversal, or having a material system for which we can analyze and control the dynamics rather than a conventional structural system? The answer is threefold, if not more:</p>
<p>-Unlike a conventional structure, which is finite in its spatial extent, a phononic material is in principle infinite and has local intrinsic properties. These properties, which are defined point-wise, include familiar static quantities, such as Young’s modulus and density, which can be obtained by a process of homogenization. This brings rise to the notions of “elasticity of a structure” and “density of a structure”. Not any less intriguing, a phononic material also exhibits frequency-dependent dynamic properties and this has recently been mathematically formulated by <a href="http://dx.doi.org/10.1016/j.mechmat.2009.01.010"><font color="#0000FF">Willis, 2009</font></a> , <a href="http://rspa.royalsocietypublishing.org/content/early/2011/01/12/rspa.2010.0389.full"><font color="#0000FF">Norris, 2011</font></a> and <a href="http://prb.aps.org/abstract/PRB/v83/i10/e104103"><font color="#0000FF">Nemat-Nasser et al., 2011</font></a>. In <font color="#000000">Nemat-Nasser et al., 2011</font>, an elegant scheme is proposed to uniquely obtain the effective dynamic density and effective dynamic modulus of a one-dimensional phononic material. The derived effective properties are shown to recover the dispersion relation, that is, the dispersion curves can be obtained using these effective properties alone.</p>
<p>-For a conventional structure, a dynamical characterization involves obtaining a single set of natural frequencies and mode shapes – which depend on many factors including the size of the structure and its boundary conditions. In a phononic material, we obtain an infinite number of sets of natural frequencies and mode shapes, one for each specific wavenumber (or wave vector) within what is known as the Brillouin zone (BZ). (The BZ describes an irreducible unit cell in the space of wavenumbers as opposed to the unit cell shown in Fig. 1 which is described in real space.) The implication of this difference is significant as it implies a much larger space of dynamic properties and hence much more opportunities for dynamical functionalities. For an engineer, this in turn provides a much richer design problem as now the dynamics is described in terms of both spatial and temporal frequencies as opposed to only temporal frequencies in a conventional structure. This space-time frequency space is what is commonly known as the frequency band structure (as shown in Fig. 1). It is possible to design the structural features of the unit cell in a manner that allows one to control the shape of this frequency band structure (and hence control the dynamics in this wider space of variables). See for example, the papers by <a href="http://rsta.royalsocietypublishing.org/content/361/1806/1001"><font color="#0000FF">Sigmund and Jensen, 2003</font></a> and <a href="http://pre.aps.org/abstract/PRE/v84/i6/e065701"><font color="#0000FF">Bilal and Hussein, 2011</font></a> which have tackled this problem through unit cell topology optimization.</p>
<p>-It is possible to link the two entities shown in Fig. 1 and form structures composed of a phononic material or more than one phononic material. In this manner a hierarchical structure-material-structure-material construction may be realized. (One more “material” is added at the end of this four-term expression because as mentioned above the structure from which a phononic material unit cell is composed is in itself made out of a material, or more). This process of designing structures using phononic materials, which is referred to as multiscale dispersive design (see <a href="http://www.sciencedirect.com/science/article/pii/S0022460X07005627"><font color="#0000FF">Hussein et al., 2007</font></a>), allows the designer to dynamically “patch” different parts of a structure to allow a remarkable level of dynamical functionality and control. For example, in <a href="http://www.sciencedirect.com/science/article/pii/S0022460X07005627"><font color="#0000FF">Hussein et al., 2007</font></a>, a rod is considered that is subjected to two excitations at different locations and different frequencies. Using this design approach, the rod may consist primarily of a regular homogeneous material but also partially from two distinct types of phononic materials (see Fig. 2). Each of these two phononic materials in turn is designed to exhibit a band gap at a specific frequency of excitation and is located as a patch at/around the location of the corresponding excitation source. (A band gap is a frequency range in which no wave propagation is permitted). The result is that the forced response of the structure as a whole is significantly less compared to a statically equivalent fully homogenous rod. One can envisage the application of this methodology to many problems in vibration control in which there are multiple excitation sources and/or a desire to spatially tailor the dynamical response of an engineering structure.
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<img src="http://www.spacedavis.com/Downloads/Fig_02_IMechanica2.png" alt=" Structure Composed of Phononic Materials (Top) and Phononic Materials and a Homogenous Material (Bottom) " title=" Structure Composed of Phononic Materials (Top) and Phononic Materials and a Homogenous Material (Bottom) " width="738" height="263" /></p>
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Figure 2: Demonstration of the Multiscale Dispersive Design methodology
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Another aspect of interest in phononic materials is the nature of the band gap opening mechanism, which may be based on Bragg scattering (like the cases considered above) or local resonance. For further reading, there are numerous references in the literature, e.g., <a href="http://prl.aps.org/abstract/PRL/v71/i13/p2022_1"><font color="#0000FF">Kushwaha et al., 1993</font></a>, <a href="http://www.sciencedirect.com/science/article/pii/003810989390888T"><font color="#0000FF">Sigalas and Economou, 1993</font></a> and <a href="http://www.sciencemag.org/content/289/5485/1734.abstract"><font color="#0000FF">Liu et al., 2000</font></a>. Also of interest is the development of models of phononic materials that incorporate damping (<a href="http://jap.aip.org/resource/1/japiau/v108/i9/p093506_s1?isAuthorized=no"><font color="#0000FF">Hussein and Frazier, 2010</font></a> , <a href="http://dx.doi.org/10.1115/1.4003943"><font color="#0000FF">Farzbod and Leamy, 2011</font></a>) and nonlinearity (<a href="http://pre.aps.org/abstract/PRE/v77/i1/e015601"><font color="#0000FF">Porter et al., 2008</font></a> , <a href="http://dx.doi.org/10.1115/1.4004661"><font color="#0000FF">Narisetti et al., 2011</font></a>). The December 2007 edition of the iMechanica Journal club (posted by Biswajit Bannerjee) discusses the related topic of “<font color="#000000"><a href="node/2381">Elastodynamic band gaps and metamaterials</a></font>”, and the upcoming May 2012 edition (to be posted by Alessandro Spadoni) will also discuss this area of research.
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<strong>References</strong>
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Phani, A.S., Woodhouse, J. and Fleck, N.A., “<a href="http://www.ncbi.nlm.nih.gov/pubmed/16642813">Wave propagation in two-dimensional periodic lattices</a> ,” <em>Journal of the Acoustical Society of America</em>, <strong>119</strong>, 1995-2005, 2006.
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Willis J. R., “<a href="http://dx.doi.org/10.1016/j.mechmat.2009.01.010">Exact effective relations for dynamics of a laminated body</a> ,” <em>Mechanics of Materials</em>, <strong>41</strong>, 385-393, 2009.
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Norris, A.N., “<a href="http://rspa.royalsocietypublishing.org/content/early/2011/01/12/rspa.2010.0389.full">Effective Willis constitutive equations for periodically stratified anisotropic elastic media</a> ,” <em>Proceedings of the Royal Society A-Mathematical, Physical and Engineering Sciences</em>, <strong>467</strong>, 1749-1769, 2011.
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Nemat-Nasser, S., Willis, J.R., Srivastava, A. and Amirkhizi, A.V., “<a href="http://prb.aps.org/abstract/PRB/v83/i10/e104103">Homogenization of periodic elastic composites and locally resonant sonic materials</a> ,” <em>Physical Review B</em>, <strong>83</strong>, 104103, 2011.
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Sigmund, O. and Jensen, J.S., “<a href="http://rsta.royalsocietypublishing.org/content/361/1806/1001">Systematic design of phononic band-gap materials and structures by topology optimization</a> ,” <em>Philosophical Transactions of the Royal Society of London, series A-Mathematical, Physical and Engineering Sciences</em>, <strong>361</strong>, 1001-1019, 2003.
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Bilal, O.R. and Hussein, M.I., “<a href="http://pre.aps.org/abstract/PRE/v84/i6/e065701">Ultrawide phononic band gap for combined in-plane and out-of-plane waves</a> ,” <em>Physical Review E</em>, <strong>84</strong>, 065701(R), 2011.
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Hussein, M.I., Hulbert, G.M. and Scott, R.A., “<a href="http://www.sciencedirect.com/science/article/pii/S0022460X07005627">Dispersive elastodynamics of 1D banded materials and structures: Design</a> ,” <em>Journal of Sound and Vibration</em>, <strong>307</strong>, 865-893, 2007.
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Kushwaha, M.S., Halevi, P., Dobrzynski, L. and Djafari-Rouhani, B., “<a href="http://prl.aps.org/abstract/PRL/v71/i13/p2022_1">Acoustic band-structure of periodic elastic composites</a> ,” <em>Physical Review Letters</em>, <strong>71</strong>, 2022-2025, 1993.
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Siglas, M. and Economou, E.N., “<a href="http://www.sciencedirect.com/science/article/pii/003810989390888T">Band-structure of elastic waves in 2-dimensional systems</a> ,” <em>Solid State Communications</em>, <strong>86</strong>, 141-143, 1993.
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Liu, Z.Y., Zhang, X.X., Mao, Y.W., Zhu, Y.Y., Yang, Z.Y., Chan, C.T. and Sheng, P., “<a href="http://www.sciencemag.org/content/289/5485/1734.abstract">Locally resonant sonic materials</a> ,” <em>Science</em>, <strong>289</strong>, 1734-1736, 2000.
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Hussein, M.I. and Frazier, M.J., “<a href="http://jap.aip.org/resource/1/japiau/v108/i9/p093506_s1?isAuthorized=no">Band structure of phononic crystals with general damping</a> ,” <em>Journal of Applied Physics</em>; <strong>108</strong>, 093506, 2010.
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Farzbod, F. and Leamy, M.J., "<a href="http://dx.doi.org/10.1115/1.4003943">Analysis of Bloch's method in structures with energy dissipation</a> ," <em>Journal of Vibration and Acoustics</em>, <strong>133</strong>, 051010, 2011.
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Porter, M.A., Daraio, C., Herbold, E.B., Szelengowicz, I. and Kevrekidis, P.G., “<a href="http://pre.aps.org/abstract/PRE/v77/i1/e015601">Highly nonlinear solitary waves in periodic dimer granular chains</a> ,” <em>Physical Review E</em>, <strong>77</strong>, 015601, 2008.
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Narisetti, R.K., Ruzzene, M. and Leamy, M.J., 2011, "<a href="http://dx.doi.org/10.1115/1.4004661">A perturbation approach for analyzing dispersion and group velocities in two-dimensional nonlinear periodic lattices</a> ," <em>Journal of Vibration and Acoustics</em>, <strong>133</strong>, 061020, 2011.
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</div></div></div>Sun, 01 Apr 2012 02:43:29 +0000mihussein12210 at http://imechanica.orghttp://imechanica.org/node/12210#commentshttp://imechanica.org/crss/node/12210