Homogenized Gradient Elasticity Model for Plane Wave Propagation in Bilaminate Composites
Publication: Journal of Engineering Mechanics
Volume 144, Issue 9
Abstract
Dispersion occurs when a wave propagates through a heterogeneous medium. Such a phenomenon becomes more pronounced when the smallest wavelength of the incoming pulse approaches the size of a unit cell, as well as when the contrast in mechanical impedance of the constituent materials increases. In this contribution focusing on periodic bilaminate composites, the authors seek an accurate description of the wave propagation behavior without the explicit representation of the underlying constituent materials. To this end, a gradient elasticity model based on a novel homogenization strategy is proposed. The intrinsic parameters characterizing the microinertia effect and nonlocal interactions are fully quantified in terms of the constituent materials’ properties and volume fractions. The framework starts with suitable kinematic decompositions within a unit cell. The Hill-Mandel condition is next applied to translate the energy statements from micro to macro. The governing equation of motion and traction definitions are next extracted naturally at the macrolevel via Hamilton’s principle. The ensuing fourth-order governing equation of motion has the same form as a reference gradient model in the literature, which was derived through a fundamentally different homogenization scheme. The predictive capability of the proposed model is demonstrated through four examples, with bilaminate composites encompassing a comprehensive range of material properties and volume fractions. It is furthermore shown that the proposed model performs better than the reference model for bilaminate composites with low to moderate contrast in mechanical impedances.
Get full access to this article
View all available purchase options and get full access to this article.
References
Andrianov, I. V., V. I. Bolshakov, V. V. Danishevs’kyy, and D. Weichert. 2008. “Higher order asymptotic homogenization and wave propagation in periodic composite materials.” Proc. R. Soc. A. 464 (2093): 1181–1201. https://doi.org/10.1098/rspa.2007.0267.
Askes, H., and E. C. Aifantis. 2011. “Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results.” Int. J. Solids Struct. 48 (13): 1962–1990. https://doi.org/10.1016/j.ijsolstr.2011.03.006.
Askes, H., T. Bennett, and E. C. Aifantis. 2007. “A new formulation and -implementation of dynamically consistent gradient elasticity.” Int. J. Numer. Meth. Eng. 72 (1): 111–126. https://doi.org/10.1002/nme.2017.
Askes, H., and A. V. Metrikine. 2002. “One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 2: Static and dynamic response.” Eur. J. Mech. A. Solids. 21 (4): 573–588. https://doi.org/10.1016/S0997-7538(02)01217-2.
Bedford, A., and D. S. Drumheller. 1994. Introduction to elastic wave propagation. New York: Wiley.
Bennett, T., and H. Askes. 2009. “Finite element modelling of wave dispersion with dynamically consistent gradient elasticity.” Comput. Mech. 43 (6): 815–825. https://doi.org/10.1007/s00466-008-0347-2.
Bennett, T., I. M. Gitman, and H. Askes. 2007. “Elasticity theories with higher-order gradients of inertia and stiffness for the modelling of wave dispersion in laminates.” Int. J. Fract. 148 (2): 185–193. https://doi.org/10.1007/s10704-008-9192-8.
Berezovski, A., J. Engelbrecht, A. Salupere, K. Tamm, T. Peets, and M. Berezovski. 2013. “Dispersive waves in microstructured solids.” Int. J. Solids Struct. 50 (11–12): 1981–1990. https://doi.org/10.1016/j.ijsolstr.2013.02.018.
Biswas, R., and L. H. Poh. 2017. “A micromorphic computational homogenization framework for heterogeneous materials.” J. Mech. Phys. Solids 102: 187–208. https://doi.org/10.1016/j.jmps.2017.02.012.
Brito-Santana, H., Y. S. Wang, R. Rodríguez-Ramos, J. Bravo-Castillero, R. Guinovart-Díaz, and V. Tita. 2015. “A dispersive nonlocal model for shear wave propagation in laminated composites with periodic structures.” Eur. J. Mech. A. Solids 49: 35–48. https://doi.org/10.1016/j.euromechsol.2014.05.009.
Campbell, F. C. 2010. Structural composite materials. Materials Park, OH: ASM International.
Carta, G., T. Bennett, and H. Askes. 2012. “Determination of dynamic gradient elasticity length scales.” Proc. Inst. Civ. Eng. Eng. Comput. Mech. 165 (1): 41–47. https://doi.org/10.1680/eacm.2012.165.1.41.
Chen, W., and J. Fish. 2001. “A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales.” J. Appl. Mech. 68 (2): 153–161. https://doi.org/10.1115/1.1357165.
De Domenico, D., and H. Askes. 2016. “A new multi-scale dispersive gradient elasticity model with micro-inertia: Formulation and -finite element implementation.” Int. J. Numer. Meth. Eng. 108 (5): 485–512. https://doi.org/10.1002/nme.5222.
Dingreville, R., J. Robbins, and T. E. Voth. 2013. “Multiresolution modeling of the dynamic loading of metal matrix composites.” JOM 65 (2): 203–214. https://doi.org/10.1007/s11837-012-0508-9.
Dingreville, R., J. Robbins, and T. E. Voth. 2014. “Wave propagation and dispersion in elasto-plastic microstructured materials.” Int. J. Solids Struct. 51 (11–12): 2226–2237. https://doi.org/10.1016/j.ijsolstr.2014.02.030.
Dontsov, E. V., R. D. Tokmashev, and B. B. Guzina. 2013. “A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion.” Int. J. Solids Struct. 50 (22–23): 3674–3684. https://doi.org/10.1016/j.ijsolstr.2013.07.012.
Engelbrecht, J., and A. Berezovski. 2015. “Reflections on mathematical models of deformation waves in elastic microstructured solids.” Math. Mech. Complex Syst. 3 (1): 43–82. https://doi.org/10.2140/memocs.2015.3.43.
Engelbrecht, J., A. Berezovski, F. Pastrone, and M. Braun. 2005. “Waves in microstructured materials and dispersion.” Philos. Mag. 85 (33–35): 4127–4141. https://doi.org/10.1080/14786430500362769.
Fish, J., and W. Chen. 2001. “Higher-order homogenization of initial/boundary-value problem.” J. Eng. Mech. 127 (12): 1223–1230. https://doi.org/10.1061/(ASCE)0733-9399(2001)127:12(1223).
Fleck, N. A., and J. W. Hutchinson. 1997. “Strain gradient plasticity.” Adv. Appl. Mech. 33: 295–361. https://doi.org/10.1016/S0065-2156(08)70388-0.
Gonella, S., M. S. Greene, and W. K. Liu. 2011. “Characterization of heterogeneous solids via wave methods in computational microelasticity.” J. Mech. Phys. Solids 59 (5): 959–974. https://doi.org/10.1016/j.jmps.2011.03.003.
Hu, R., and C. Oskay. 2017. “Nonlocal homogenization model for wave dispersion and attenuation in elastic and viscoelastic periodic layered media.” J. Appl. Mech. 84 (3): 031003. https://doi.org/10.1115/1.4035364.
Hui, T., and C. Oskay. 2014. “A high order homogenization model for transient dynamics of heterogeneous media including micro-inertia effects.” Comput. Method. Appl. Mech. Eng. 273: 181–203. https://doi.org/10.1016/j.cma.2014.01.028.
Lim, C. W., G. Zhang, and J. N. Reddy. 2015. “A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation.” J. Mech. Phys. Solids 78: 298–313. https://doi.org/10.1016/j.jmps.2015.02.001.
Mallick, P. K. 2007. Fibre-reinforced composites—Materials, manufacturing and design. Boca Raton, FL: CRC Press.
Metrikine, A. V., and H. Askes. 2002. “One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 1: Generic formulation.” Eur. J. Mech. A. Solids 21 (4): 555–572. https://doi.org/10.1016/S0997-7538(02)01218-4.
Papargyri-Beskou, S., D. Polyzos, and D. E. Besko. 2009. “Wave dispersion in gradient elastic solids and structures: A unified treatment.” Int. J. Solids Struct. 46 (21): 3751–3759. https://doi.org/10.1016/j.ijsolstr.2009.05.002.
Pichugin, A. V., H. Askes, and A. Tyas. 2008. “Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories.” J. Sound Vib. 313 (3–5): 858–874. https://doi.org/10.1016/j.jsv.2007.12.005.
Poh, L. H. 2013. “Scale transition of a higher order plasticity model—A consistent homogenization theory from meso to macro.” J. Mech. Phys. Solids 61 (12): 2692–2710. https://doi.org/10.1016/j.jmps.2013.09.004.
Polyzos, D., and D. I. Fotiadis. 2012. “Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models.” Int. J. Solids Struct. 49 (3–4): 470–480. https://doi.org/10.1016/j.ijsolstr.2011.10.021.
Sun, G., and L. H. Poh. 2016. “Homogenization of intergranular fracture towards a transient gradient damage model.” J. Mech. Phys. Solids 95: 374–392. https://doi.org/10.1016/j.jmps.2016.05.035.
Valkó, P. P., and J. Abate. 2004. “Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion.” Comput. Math. Appl. 48 (3–4): 629–636. https://doi.org/10.1016/j.camwa.2002.10.017.
Wautier, A., and B. B. Guzina. 2015. “On the second-order homogenization of wave motion in periodic media and the sound of a chessboard.” J. Mech. Phys. Solids 78: 382–414. https://doi.org/10.1016/j.jmps.2015.03.001.
Information & Authors
Information
Published In
Copyright
©2018 American Society of Civil Engineers.
History
Received: Aug 15, 2017
Accepted: Feb 21, 2018
Published online: Jun 18, 2018
Published in print: Sep 1, 2018
Discussion open until: Nov 18, 2018
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.