Optimal design of manufacturable three-dimensional composites with multifunctional characteristics

We present an optimization method to design three-dimensional composite microstructures with multifunctional characteristics. To illustrate the fascinating types of microstructures that can arise in multifunctional optimization, we apply our methodology to the study the simultaneous transport of heat and electricity in three-dimensional, two-phase composites. We assume that phase 1 has a high thermal conductivity but low electrical conductivity and phase 2 has a low thermal conductivity but high electrical conductivity. The objective functions consist of different combinations of the dimensionless effective thermal and electrical conductivities. When the sum of the effective thermal and electrical conductivities is maximized, we find that the optimal three-dimensional microstructures are triply periodic bicontinuous composites with interfaces that are the Schwartz primitive (P) and diamond (D) minimal surfaces. Maximizing the effective thermal conductivity and minimizing the effective electrical conductivity results in a special dispersion of inclusions in a connected matrix. The effective properties of both the bicontinuous and singly connected microstructures lie on known optimal cross-property bounds. When the sum of the effective thermal and electrical conductivities is minimized, the result is the three-dimensional checkerboard, which is the optimal single-scale microstructure. It is important to note that current fabrication techniques enable one to manufacture all of the aforementioned optimal single-scale composites.

[1]  S. Shtrikman,et al.  A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials , 1962 .

[2]  M. Beran.  Use of the vibrational approach to determine bounds for the effective permittivity in random media , 1965 .

[3]  On a phase interchange relationship for composite materials , 1976 .

[4]  David J. Bergman,et al.  The dielectric constant of a composite material—A problem in classical physics , 1978 .

[5]  Graeme W. Milton,et al.  Bounds on the transport and optical properties of a two‐component composite material , 1981 .

[6]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[7]  D. Kinderlehrer,et al.  Homogenization and effective moduli of materials and media , 1986 .

[8]  Gilles A. Francfort,et al.  Homogenization and optimal bounds in linear elasticity , 1986 .

[9]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[10]  Andrej Cherkaev,et al.  On the effective conductivity of polycrystals and a three‐dimensional phase‐interchange inequality , 1988 .

[11]  James G. Berryman,et al.  Microgeometry of random composites and porous media , 1988 .

[12]  Salvatore Torquato,et al.  Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media , 1991 .

[13]  V. Nesi.  Multiphase interchange inequalities , 1991 .

[14]  S. Torquato,et al.  Connection between the conductivity and bulk modulus of isotropic composite materials , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  Holyst,et al.  Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  S. Torquato,et al.  Design of materials with extreme thermal expansion using a three-phase topology optimization method , 1997 .

[17]  R. Lipton,et al.  Optimal material layout for 3D elastic structures , 1997 .

[18]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[19]  C. Brinker,et al.  Self-assembly of mesoscopically ordered chromatic polydiacetylene/silica nanocomposites , 2001, Nature.

[20]  Salvatore Torquato,et al.  Effective-medium approximation for composite media: Realizable single-scale dispersions , 2001 .

[21]  Shmuel Vigdergauz,et al.  The effective properties of a perforated elastic plate Numerical optimization by genetic algorithm , 2001 .

[22]  Anthony G. Evans,et al.  Lightweight Materials and Structures , 2001 .

[23]  Salvatore Torquato,et al.  Designing composite microstructures with targeted properties , 2001 .

[24]  G. Milton.  The Theory of Composites , 2002 .

[25]  S. Nemat-Nasser.  Micromechanics of actuation of ionic polymer-metal composites , 2002 .

[26]  S. Torquato,et al.  Multifunctional composites: optimizing microstructures for simultaneous transport of heat and electricity. , 2002, Physical review letters.

[27]  J. Cesarano,et al.  Directed colloidal assembly of 3D periodic structures , 2002 .

[28]  C. Julien,et al.  New trends in intercalation compounds for energy storage , 2002 .

[29]  B. Smarsly,et al.  Self‐Assembly and Characterization of Mesostructured Silica Films with a 3D Arrangement of Isolated Spherical Mesopores , 2003 .

[30]  T I Zohdi,et al.  Genetic design of solids possessing a random–particulate microstructure , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[31]  S. Torquato,et al.  Random Heterogeneous Materials: Microstructure and Macroscopic Properties , 2005 .