Second order homogenization of quasi-periodic structures

Duc Trung Le, Jean-Jacques Marigo

Abstract


The paper develops a general framework to derive the effective properties of quasi-periodic elastic medium. By using the asymptotic expansion method, the solution is expanded to the second order by solving a sequence of minimization problems. The effective stiffness tensors fields entering in the expression of the macroscopic energy are obtained by solving several families of microscopic problems posed on the unit cell and which bring into play only the microstructure. As an illustrative example, we consider an anti-plane elastic case of a heterogeneous cylinder made of a bi-layer laminate and submitted to the gravity. The unit cell being one-dimensional, all the associated elementary problems can be solved in a closed form and one shows that the effective energy of the medium expanded up to the second order depends not only on the strain gradient, but also on the gradient of the volume fraction \(\theta\) characterizing the repartition of the two materials in the laminate.


Keywords


homogenization; quasi periodic; strain gradient theories; asymptotic expansions

Full Text:

PDF

References


A. Bensoussan, J. L. Lions, and G. Papanicolaou. Asymptotic analysis of periodic structures, Vol. 374. Elsevier Science, (1978).

F. Murat and L. Tartar. Calcul des variations et homogénéisation. Les méthodes del’homogénéisation: théorie et applications en physique, 57, (1985), pp. 319–369.

G. Allaire. Homogenization and two-scale convergence. SIAM Journal on Mathematical Analysis, 23, (6), (1992), pp. 1482–1518. https://doi.org/10.1137/0523084.

H. Dumontet. Study of a boundary layer problem in elastic composite materials. ESAIM: Mathematical Modelling and Numerical Analysis, 20, (2), (1986), pp. 265–286. https://doi.org/10.1051/m2an/1986200202651.

G. A. Francfort and F. Murat. Homogenization and optimal bounds in linear elasticity. Archive for Rational Mechanics and Analysis, 94, (4), (1986), pp. 307–334. https://doi.org/10.1007/bf00280908.

R. Abdelmoula and J. J. Marigo. The effective behavior of a fiber bridged crack. Journal of the Mechanics and Physics of Solids, 48, (11), (2000), pp. 2419–2444. https://doi.org/10.1016/s0022-5096(00)00003-x.

V. A. Marchenko and E. Y. Khruslov. Homogenization of partial differential equations, Vol. 46. Springer Science & Business Media, (2008).

U. Hornung. Homogenization and porous media, Vol. 6. Interdisciplinary Applied Mathematics, (1997).

B. Gambin and E. Kr¨oner. Higher-order terms in the homogenized stress-strain relation of periodic elastic media. Physica Status Solidi (B), 151, (2), (1989), pp. 513–519. https://doi.org/10.1002/pssb.2221510211.

C. Boutin. Microstructural effects in elastic composites. International Journal of Solids and Structures, 33, (7), (1996), pp. 1023–105. https://doi.org/10.1016/0020-7683(95)00089-5.

I. V. Andrianov, J. Awrejcewicz, and A. A. Diskovsky. Homogenization of quasiperiodic structures. Journal of Vibration and Acoustics, 128, (4), (2006), pp. 532–534. https://doi.org/10.1115/1.2202158.

F. Su, Z. Xu, J. Z. Cui, and Q. L. Dong. Multi-scale method for the quasi-periodic structures of composite materials. Applied Mathematics and Computation, 217, (12), (2011), pp. 5847–5852. https://doi.org/10.1016/j.amc.2010.12.068.

L. Guillot, Y. Capdeville, and J. J. Marigo. 2-D non-periodic homogenization of the elastic wave equation: SH case. Geophysical Journal International, 182, (3), (2010), pp. 1438–1454. https://doi.org/10.1111/j.1365-246x.2010.04688.x.

Y. Capdeville, L. Guillot, and J. J. Marigo. 2-D non-periodic homogenization to upscale elastic media for P-SV waves. Geophysical Journal International, 182, (2), (2010), pp. 903–922. https://doi.org/10.1111/j.1365-246x.2010.04636.x.

E. S. Palencia. Non-homogeneous media and vibration theory, Vol. 127. Springer-Verlag Berlin, (1980).

F. Devries, H. Dumontet, G. Duvaut, and F. Léné. Homogenization and damage for composite structures. International Journal for Numerical Methods in Engineering, 27, (2), (1989), pp. 285–298. https://doi.org/10.1002/nme.1620270206.

G. A. Francfort and J. J. Marigo. Stable damage evolution in a brittle continuous medium. European Journal of Mechanics Series a Solids, 12, (1993), pp. 149–149.

K. Pham and J. J. Marigo. Approche variationnelle de l’endommagement: I. Les concepts fondamentaux. Comptes Rendus Mécanique, 338, (4), (2010), pp. 191–198. https://doi.org/10.1016/j.crme.2010.03.009.

K. Pham and J. J. Marigo. Approche variationnelle de l’endommagement: II. Les modèles à gradient. Comptes Rendus Mécanique, 338, (4), (2010), pp. 199–206.

C. Dascalu, G. Bilbie, and E. Agiasofitou. Damage and size effects in elastic solids: a homogenization approach. International Journal of Solids and Structures, 45, (2), (2008), pp. 409–430. https://doi.org/10.1016/j.ijsolstr.2007.08.025.




DOI: https://doi.org/10.15625/0866-7136/13498

Refbacks

  • There are currently no refbacks.


Copyright (c) 2018 Vietnam Academy of Science and Technology

 

                      

Editorial Office of Vietnam Journal of Mechanics

3rd Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam

Tel: (+84) 24 3791 7103

Email: vjmech@vjs.ac.vn