Engineering Transactions, 65, 1, pp. 31–38, 2017
10.24423/engtrans.759.2017

### Two-Phase Isotropic Composites of Extremal Moduli. The Inverse Homogenization Problem

Tomasz ŁUKASIAK
Warsaw University of Technology
Poland

This work deals with the inverse homogenization problem: for given two well-ordered elastic and isotropic materials characterized by the bulk and shear moduli ($\kappa_1$, $\mu_1$), ($\kappa_{12}$, $\mu_2$) and the volume fraction $\rho$ of the second material reconstructing the layout of the most second- rank orthogonal laminates within a hexagonal 2D periodicity cell $\mathit{Y}$ corresponding to the predefined values of moduli ($\kappa^*$, $\mu^*$) of the effective isotropic composite. The used algorithm follows from imposing the finite element (FE) approximation on the solution to the basic cell problems of the homogenization theory [7] along with periodicity assumptions. The material properties of each element are described by three independent parameters. Thus, the formulated inverse problem is solved numerically by the gradient method. The adopted cell structure, i.e., the hexagonal cell with the rotational symmetry of $120^\circ$ angle guarantees the isotropic effective properties of the composite, and thus the optimization problem is greatly simplified. Isotropic constraints do not appear in the formulated optimization problem.
Keywords: laminates; isotropic composites; inverse-homogenization
Full Text: PDF

#### References

Bendsøe M.P., Kikuchi N., Generating optimal topologies in optimal design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, 71(2): 197–224, 1988, doi: 10.1016/0045-7825(88)90086-2.

Bendsøe M.P., Sigmund O., Material interpolation schemes in topology optimization, Archive of Applied Mechanics, 69(9): 635–654,1999, doi: 10.1007/s004190050248.

Cherkaev A.V., Gibiansky L.V., Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite, Journal of the Mechanics and Physics of Solids, 41(5): 937–980, 1993, doi: 10.1016/0022-5096(93)90006-2.

Czarnecki S., Isotropic material design, Computational Methods in Science and Technology, 21(2): 49–64, 2015, doi: 10.12921/cmst.2015.21.02.001.

Dzierżanowski G., On the comparison of material interpolation schemes and optimal composite properties in plane shape optimization, Structural and Multidisciplinary Optimization, 46(5): 693–710, 2012, doi: 10.1007/s00158-012-0788-2.

Dierżanowski G.: Optimization of material layout in thin elastic plates [in Polish], Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa, 2010.

Hassani B., Hinton E.. Homogenization and structural topology optimization: theory, practice and software, Springer, New York, 1998.

Lewiński T., Telega J.J.: Plates, laminates, and shells: asymptotic analysis and homogenization, World Scientific, Singapore, 2000.

Łukasiak T., Two-phase isotropic composites with prescribed bulk and shear moduli, [in:] Recent Advances in Computational Mechanics, Łodygowski T., Rakowski J., Litewka P. (Eds.), pp. 213–222, CRC Press, London, 2014, doi: 10.1201/b16513-29.

Łukasiak T., HSρ – an isotropic interpolation scheme based on Hashin-Shtrikman variational bounds, [in:] Advances in Mechanics: Theoretical, Computational and Interdisciplinary Issues, Proceedings of the 3rd Polish Congress of Mechanics (PCM) and 21st International Conference on Computer Methods in Mechanics (CMM), Gdansk, Poland, 8-11 September 2015, Kleiber M., Burczyński T., Wilde K., Górski J., Winkelmann K., Smakosz Ł. (Eds.), pp. 355–360, CRC Press, London 2016, doi: 10.1201/b20057-77.

Sigmund O., A new class of extremal composites, Journal of the Mechanics and Physics of Solids, 48(2): 397–428, 2000, doi: 10.1016/S0022-5096(99)00034-4.

Stolpe M., Svanberg K., An alternative interpolation scheme for minimum compliance optimization, Structural and Multidisciplinary Optimization, 22(2): 116–124, 2001, doi: 10.1007/s001580100129.

DOI: 10.24423/engtrans.759.2017