Micromechanics Modeling of 3D Network Smart Orthotropic Structures

Two micromechanical models for 3D smart composite
with embedded periodic or nearly periodic network of generally
orthotropic reinforcements and actuators are developed and applied to
cubic structures with unidirectional orientation of constituents.
Analytical formulas for the effective piezothermoelastic coefficients
are derived using the Asymptotic Homogenization Method (AHM).
Finite Element Analysis (FEA) is subsequently developed and used
to examine the aforementioned periodic 3D network reinforced smart
structures. The deformation responses from the FE simulations are
used to extract effective coefficients. The results from both
techniques are compared. This work considers piezoelectric materials
that respond linearly to changes in electric field, electric
displacement, mechanical stress and strain and thermal effects. This
combination of electric fields and thermo-mechanical response in
smart composite structures is characterized by piezoelectric and
thermal expansion coefficients. The problem is represented by unitcell
and the models are developed using the AHM and the FEA to
determine the effective piezoelectric and thermal expansion
coefficients. Each unit cell contains a number of orthotropic
inclusions in the form of structural reinforcements and actuators.
Using matrix representation of the coupled response of the unit cell,
the effective piezoelectric and thermal expansion coefficients are
calculated and compared with results of the asymptotic
homogenization method. A very good agreement is shown between
these two approaches.





References:
[1] A. Bensoussan, L. Lions, and G. Papanicolaou., “Asymptotic Analysis
for Periodic Structures”. 2nd ed., 2011, AMS Chelsea Publishing.
[2] E. Sanchez-Palencia, “Non-Homogeneous media and vibration theory”
Lectures Notes in Physics, 1980, 127, Verlag: Springer.
[3] N. Bakhvalov and G. Panasenko, “Homogenization: Averaging
Processes in Periodic Media: Mathematical Problems in the Mechanics
of Composite Materials” 1st ed., 1984, Moscow: Nauka.
[4] D. Cioranescu and P. Donato, “An Introduction to Homogenization” 1st
ed., 1999, Oxford University Press.
[5] A. Kalamkarov and A Kolpakov, “A new asymptotic model for a
composite piezoelastic plate” Int. J. Solids Struct., 2001, 38(34-35), pp.
6027-6044.
[6] K. Challagulla, A. Georgiades and A. Kalamkarov, “Asymptotic
homogenization modeling of smart composite generally orthotropic gridreinforced
shells: Part I-Theory, Euro. J. of Mech. A-Solids, 2010, 29(4),
pp. 530-540.
[7] A. Georgiades., K. Challagulla and A. Kalamkarov, “Asymptotic
homogenization modeling of smart composite generally orthotropic gridreinforced
shells: Part II-Applications” Euro. J. of Mech. A-Solids,
2010, 29(4), pp. 541-556.
[8] A. Kalamkarov, I. Andrianov, and V. Danishevs’kyy, “Asymptotic
homogenization of composite materials and structures” Trans. ASME,
Appl. Mech. Rev., 2009, 62(3), 030802-1 – 030802-20.
[9] P. Suquet, “Elements of homogenization theory for inelastic solid
mechanics” In: E. Sanchez-Palencia and A. Zaoui, Editors,
Homogenization Techniques for Composite Media, Lect. Notes Phys.,
272, 1987, pp. 193-278.
[10] D. Adams and D. Crane, “Finite element micromechanical analysis of a
unidirectional composite including longitudinal shear loading” Compos.
Struct., 18, 1984, pp. 1153-1165.
[11] J. Aboudi, “Micromechanical analysis of composites by the method of
cells” Appl. Mech. Rev., 42(7), 1989, pp. 193-221.
[12] M. Paley and J Aboudi, “Micromechanical analysis of composites by
the generalized method of cells” Mech. Mater., 14, 1992, pp. 127-139.
[13] J. Aboudi, “Micromechanical analysis of composites by the method of
cells – update” Appl. Mech. Rev., 49, 1996, pp. 127-139.
[14] J. Bennett and K. Haberman, “An alternate unified approach to the
micromechanical analysis of composite materials” J. Compos. Mater.,
30(16), 1996, pp. 1732-1747.
[15] D. Allen and J Boyd, “Convergence rates for computational predictions
of stiffness loss in metal matrix composites” In: Composite Materials
and Structures (ASME, New York) AMD 179/AD, 37, 1993, pp. 31-45.
[16] C. Bigelow, “Thermal residual stresses in a silicon-carbide/titanium
(0/90) laminate” J. Compos. Tech. Res., 15, 1993, pp. 304-310.
[17] J. Bystrom, N. Jekabsons and J. Varna, “An evaluation of different
models for prediction of elastic properties of woven composites” Comp.
Part B, 31(1), 2000, pp. 7-20.
[18] X. Wang, X., Wang, X., G. Zhou and X. Zhou, “Multi-scale analyses of
3D woven composite based on periodicity boundary conditions” J.
Compos. Mater., 41(14), 2007, pp. 1773-1788.
[19] P. Bossea, K. Challagulla and T. Venkatesh, “Effect of foam shape and
porosity aspect ratio on the electromechanical properties of 3-3
piezoelectric foams” Acta Materialia, 60 (19), 2012, pp. 6464-6475.
[20] S. Li, “On the unit cell for micromechanical analysis of fiber-reinforced
composites” Proc. R. Soc. London, Ser. A 455, 1999, pp. 815-838.
[21] S. Li and A. Wongsto, “Unit cells for micromechanical analyses of
particle-reinforced composites” Mech. Mater., 36(7), 2004, pp. 543-572.
[22] C. Sun and R. Vaidya, “Prediction of composite properties from a
representative volume element” Compos. Sci. Technol., 56, 1996, pp.
171-179.
[23] H. Pettermann and S. Suresh, “A comprehensive unit cell model: a study
of coupled effects in piezoelectric 1-3 composites” Int. J. Solids Struct.,
37(39), 2000, pp. 5447-5464.
[24] J. Michel, H. Moulinec and P. Suquet, “Effective properties of
composite materials with periodic microstructures: a computational
approach” Comput. Meth. Appl. Mech. Eng., 172(1-4), 1999, pp. 109-
143.
[25] C. Miehe, J. Schroder and C. Bayreuther, “On the homogenization
analysis of composite materials based on discretized fluctuation on the
micro-structure” Acta Mech., 155(1-2), 2002, pp. 1-16.
[26] Z. Xia, Y. Zhang and F Ellyin, “A unified periodical boundary
conditions for representative volume elements of composites and
applications” Int. J. Solids Struct., 40(8), 2003, pp. 1907-1921.
[27] Z. Xia, Z. Chuwei, Y. Qiaoling and W. Xinwei, “On selection of
repeated unit cell model and application of unified periodic boundary
conditions in micro-mechanical analysis of composites” Int. J. Solids
Struct., 43(2), 2006, pp. 266-278.
[28] J. Oliveira, J. Pinho-da-Cruz and F. Teixeira-Dias, “Asymptotic
homogenisation in linear elasticity. Part II: Finite element procedures
and multiscale applications” Comput. Mater. Sci., 45(4), 2009, pp. 1081-
1096.
[29] M. Würkner, H. Berger and U. Gabbertm, “On Numerical evaluation of
effective material properties for composite structure with rhombic fiber
arrangements” Int. J. of Eng. Sci., 49(4), 2011, pp. 322-332.
[30] E. Hassan, A. Kalamkarov, A. Georgiades and K. Challagulla, “An
asymptotic homogenization model for smart 3D grid-reinforced
composite structures with generally orthotropic constituents”. Smart
Mater. and Struct., 18(7), 2009, 075006 (16pp).
[31] E. Hassan, A. Georgiades, M. Savi and A. Kalamkarov, “Analytical and
numerical analysis of 3D grid-reinforced orthotropic composite
structures” Int. J. of Eng. Science, 49(7), 2011, pp. 589-605.
[32] F. Cote, P. Masson and N. Mrad, “Dynamic and static assessment of
piezoelectric embedded composites”. Proc. SPIE 4701, 2002, pp. 316-
325.
[33] P. Mallick, “Fiber-Reinforced Composites: Materials, Manufacturing
and Design” 2nd ed., 2007, Boca Raton: CRC Press.