Lattice models in micromechanics

Applied Mechanics Reviews - Tập 55 Số 1 - Trang 35-60 - 2002
Martin Ostoja–Starzewski1
1Department of Mechanical Engineering, McGill University, 817 Sherbrooke St West, Montre´al, Que´bec, Canada H3A 2K6; [email protected]

Tóm tắt

This review presents the potential that lattice (or spring network) models hold for micromechanics applications. The models have their origin in the atomistic representations of matter on one hand, and in the truss-type systems in engineering on the other. The paper evolves by first giving a rather detailed presentation of one-dimensional and planar lattice models for classical continua. This is followed by a section on applications in mechanics of composites and key computational aspects. We then return to planar lattice models made of beams, which are a discrete counterpart of non-classical continua. The final two sections of the paper are devoted to issues of connectivity and rigidity of networks, and lattices of disordered (rather than periodic) topology. Spring network models offer an attractive alternative to finite element analyses of planar systems ranging from metals, composites, ceramics and polymers to functionally graded and granular materials, whereby a fiber network model of paper is treated in considerable detail. This review article contains 81 references.

Từ khóa


Tài liệu tham khảo

Hrennikoff A (1941), Solution of problems of elasticity by the framework method, ASME J. Appl. Mech. 8, A619–A715A619–A715.

Maxwell JC (1869), Scientific Papers II.

Askar A (1985), Lattice Dynamical Foundations of Continuum Theories, World Scientific, Singapore.

Noor AK and Nemeth MP (1980), Micropolar beam models for lattice grids with rigid joints, Comput. Methods Appl. Mech. Eng. 21, 249–263.

Triantafyllidis N and Bardenhagen S (1993), On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models, J. Elast. 33, 259–293.

Stephen NG and Wang PJ (1996), On Saint-Venant’s principle in pinjointed frameworks, Int. J. Solids Struct. 33(1) 79–97.

Noor AK (1988), Continuum modeling for repetitive lattice structures, Appl. Mech. Rev. 41(7) 285–296.

Costello GA (1997), Theory of Wire Rope, Springer-Verlag.

Lakes RS and Benedict R (1982), Noncentrosymmetry in micropolar elasticity, Int. J. Eng. Sci. 29(10) 1161–1167.

Nowacki W (1986), Theory of Asymmetric Elasticity, Oxford: Pergamon Press/Warsaw: PWN-Polish Scientific Publ.

Blouin F and Cardou A (1974), A study of helically reinforced cylinders under axially symmetric loads and application to strand mathematical modeling, Int. J. Solids Struct. 25(2) 189–200.

Samras RK , Skop RA, and Milburn DA (1974), An analysis of coupled extensional-torsional oscillations in wire rope, ASME J. Eng. Ind. 96, 1130–1135.

Love AEH (1934), The Mathematical Theory of Elasticity, Cambridge Univ Press.

Grah M , Alzebdeh K, Sheng PY, Vaudin MD, Bowman KJ, and Ostoja-Starzewski M (1996), Brittle intergranular failure in 2D microstructures: experiments and computer simulations, Acta Mater. 44(10) 4003–4018.

Kirkwood JG (1939), The skeletal modes of vibration of long chain molecules, J. Chem. Phys. 7, 506–509.

Thorpe MF and Jasiuk I (1992), New results in the theory of elasticity for two-dimensional composites Proc. R. Soc. London, Ser. A A438, 531–544.

Day AR , Snyder KA, Garboczi EJ, and Thorpe MF (1992), The elastic moduli of a sheet containing circular holes, J. Mech. Phys. Solids 40, 1031–1051.

Keating PN (1966), Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure, Phys. Rev. 145, 637–645.

Synder KA , Garboczi EJ, and Day AR (1992), The elastic moduli of simple two-dimensional composites: computer simulation and effective medium theory, J. Mech. Phys. Solids 72, 5948–5955.

Press WH, Teukolsky SA, Vetterling WT, and Flannery BP (1992), Numerical Recipes, Cambridge Univ Press.

Chung JW , Roos A, De Hosson JTh M, and van der Giessen E (1996), Fracture of disordered three-dimensional spring networks: A computer simulation methodology, Phys. Rev. B 54, 15094–15100.

Ostoja-Starzewski M and Schulte J (1996), Bounding of effective thermal conductivities of multiscale materials by essential and natural boundary conditions, Phys. Rev. B 54, 278–285.

Ostoja-Starzewski M (1998), Random field models of heterogeneous materials, Int. J. Solids Struct. 35(19) 2429–2455.

Garboczi E (1998), Finite Element Programs and Finite Difference Programs for Computing the Linear Electric and Elastic Properties of Digital Images of Random Materials, NISTIR 6269, NIST, Gaithersburg MD.

Torquato S (1997), Effective stiffness tensor of composite media, J. Mech. Phys. Solids 45, 1421–1448.

Keller JB (1964), A theorem on the conductivity of a composite medium, J. Math. Phys. 5, 548–549.

Mendelson KS (1975), Effective conductivity of two-phase material with cylindrical phase boundaries, J. Appl. Phys. 46, 917–918.

Cherkaev AV , Lurie KA, and Milton GW (1992), Invariant properties of the stress in plane elasticity and equivalence classes of composites, Proc. R. Soc. London, Ser. A A438, 519–529.

Ostoja-Starzewski M and Jasiuk I (1995), Stress invariance in planar Cosserat elasticity, Proc. R. Soc. London, Ser. A 451, 453–470.

errata: 452, 15031503 (1995).

Wozniak C (1970), Surface Lattice Structures (in Polish), Polish Sci Publ, Warsaw.

Gulati in 32.

Gibson LJ and Ashby MF (1988), Cellular Solids, Pergamon Press.

Ostoja-Starzewski M , Sheng PY, and Alzebdeh K (1996), Spring network models in elasticity and fracture of composites and polycrystals, Comput. Mater. Sci. 7(1,2) 82–93.

Jasiuk I , Chen J, and Thorpe MF (1994), Elastic moduli of two dimensional materials with polygonal holes, Appl. Mech. Rev. 47 (1, Pt 2) S18–S28S18–S28.

Jasiuk I (1995), Polygonal cavities vis-a`-vis rigid inclusions: Effective elastic moduli of materials with polygonal inclusions, Int. J. Solids Struct. 32, 407–422.

Stalne K and Gustafson PJ (2001), A three dimensional finite element fibre model for composite material stiffness and hygroexpansion analysis, Proc. 2nd Eur. Conf. Comp. Mech. ECCM-2001, Cracow, Poland.

Wozniak C (1997), Internal variables in dynamics of composite solids with periodic microstructure, Arch. Mech. 49(2) 421–441.

Wozniak C (1966), Load carrying structures of dense lattice type, Arch. Mech. Stos. 18(5) 581–597.

Cielecka I , Wozniak C, and Wozniak M (1998), Internal variables in macrodynamics of two-dimensional periodic cellular media, Arch. Mech. 50(1) 3–19.

Pshenichnov GI (1993), A Theory of Latticed Plates and Shells, World Scientific, Singapore.

Cioranescu D and Saint Jean Paulin J (1999), Homogenization of Reticulated Structures, Springer Verlag, New York.

Holnicki-Szulc J and Rogula D (1979a), Non-local, continuum models of large engineering structures, Arch. Mech. 31(6) 793–802.

Holnicki-Szulc J and Rogula D (1979b), Boundary problems in non-local, continuum models of large engineering structures, Arch. Mech. 31(6) 803–811.

Bardenhagen S , and Triantafyllidis N (1994), Derivation of higher order gradient continuum theories in 2,3-D non-linear elasticity from periodic lattice models, J. Mech. Phys. Solids 42, 111–139.

Wan XL and Stronge WJ (1999), Micropolar theory for two-dimensional stresses in elastic honeycomb, Proc. R. Soc. London, Ser. A 455, 2091–2116.

Chen JY , Huang Y, and Ortiz M (1998), Fracture analysis of cellular materials: A strain gradient model, J. Mech. Phys. Solids 46, 789–828.

Crapo H and Whiteley W (1989), The geometry of rigid structures, in Encyclopedia of Mathematics and its Applications, Cambridge Univ Press.

Laman (1970), On graphs and rigidity of plane skeletal structures, Eng. Math. 4, 331–340.

Asimov L and Roth B (1978), The rigidity of graphs, Trans. Am. Math. Soc. 245, 279–289.

Feng S , Thorpe MF, and Garboczi E (1985), Effective-medium theory of percolation on central-force elastic networks, Phys. Rev. B 31, 276–280.

Boal DH (1993), Rigidity and connectivity percolation in heterogeneous polymer-fluid networks, Phys. Rev. E 47, 4604–4606.

Hansen JC , Chien S, Skalak R, and Hoger A (1996), An elastic network model based on the structure of the red blood cell membrane skeleton, Biophys. J. 70, 146–166.

Miles RE (1964), Random polygons determined by random lines in a plane, Proc. Natl. Acad. Sci. U.S.A. 52, 901–907.

Cox HL (1952), The elasticity and strength of paper and other fibrous materials, Br. J. Appl. Phys. 3, 72–79.

Page DH, Tydeman PA, and Hunt M (1961), A study of fibre-to-fibre bonding by direct observation, The Formation and Structure of Paper—Trans. Oxford Symp1, 171–193.

Ostoja-Starzewski M , Quadrelli MB, and Stahl DC (1999), Kinematics and stress transfer in quasi-planar random fiber networks, C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron. 327, 1223–1229.

Stahl DC and Cramer SM (1988), A three-dimensional network model for a low density fibrous composite, ASME J. Eng. Mater. Technol. 120(2) 126–130.

Chung JW , Roos A, De Hosson JThM, and van der Geissen E (1996), Fracture of disordered three-dimensional spring networks: A computer simulation methodology, Phys. Rev. B 54(21) 15094–15100.

Sastry AM , Cheng X, and Wang CW (1998), Mechanics of stochastic fibrous networks, J. Thermoplast. Compos. Mater. 11, 211–296.

Cheng X , and Sastry AM (1999), On transport in stochastic, heterogeneous fibrous domains, Mech. Mater. 31, 765–786.

Cook RD, Malkus ME and Plesha ME (1989), Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York.

Stoyan D, Kendall WS, and Mecke J (1987), Stochastic Geometry and its Applications, John Wiley & Sons, New York.

Kuznetsov EN (1991), Underconstrained Structural Systems, Springer-Verlag, New York.

Kellomaki M , A¨stro¨m J, and Timonen J (1996), Rigidity and dynamics of random spring networks, Phys. Rev. Lett. 77(13) 2730–2733.

Raisanen VI , Alava MJ, and Nieminen RM (1997), Failure of planar fiber networks, J. Appl. Phys. 82(8) 3747–3753.

Ostoja-Starzewski M , and Stahl DC (2000), Random fiber networks and special elastic orthotropy of paper, J. Elast. 60(2), 131–149.

Cundall PA , and Strack ODL (1979), A discrete numerical model for granular assemblies, Geotechnique 29(1) 47–65.

Bathurst RJ , and Rothenburg L (1988), Micromechanical aspects of isotropic granular assemblies with linear contact interactions, ASME J. Appl. Mech. 55, 17–23.

Bathurst RJ , and Rothenburg L (1989), Note on a random isotropic granular material with negative Poisson’s ratio, Int. J. Eng. Sci. 26, 373–383.

Jagota A and Benison SJ (1994), Spring-network and finite-element models for elasticity and fracture, in Non-linearity and Breakdown in Soft Condensed Matter, KK Bardhan, BK Chakrabarti, and A Hansen (eds), Lecture Notes in Physics437, Springer-Verlag, NY, 186–201.

Satake M (1976), Constitution of mechanics of granular materials through graph representation, Proc. 26th Japan Natl. Congr. Theor. Appl. Mech., 257–266.

Satake M (1978), Constitution of mechanics of granular materials through the graph theory, Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, SC Cowin and M Satake (eds), 47–62.

Ziman JM (1979), Models of Disorder, Cambridge University Press.

Torquato S (1991), Random heterogeneous media: microstructure and improved bounds on effective properties, Appl. Mech. Rev. 44(2) 37–76.

Ostoja-Starzewski M , Alzebdeh K, and Jasiuk I (1995), Linear elasticity of planar Delaunay networks-III: Self-consistent approximations, Acta Mech. 110, 57–72.

Alzebdeh K and Ostoja-Starzewski M (1999), On a spring network model and effective elastic moduli of granular materials, ASME J. Appl. Mech. 66, 172–180.

Kruyt NP and Rothenburg L (1996), Micromechanical definition of the strain tensor for granular materials, ASME J. Appl. Mech. 63, 706–711.

Kruyt NP and Bathurst RJ (1998), Statistical theories for the elastic moduli of two-dimensional assemblies of granular materials, Int. J. Eng. Sci. 36, 1127–1142.

Kruyt NP and Rothenburg L (2001), Statistics of the elastic behaviour of granular materials, Int. J. Solids Struct. 38, 4879–4899.

Rothenburg L and Bathurst RJ (1996), Micromechanical features of granular assemblies with planar particles, Geotechnique 42, 79–95.

Cherkaev A (2000), Variational Methods for Structural Optimization, Springer-Verlag.