Abstract
Topology optimization of structures and mechanisms with microstructural length-scale effect is investigated based on gradient elasticity theory. To meet the higher-order continuity requirement in gradient elasticity theory, Hermite finite elements are used in the finite element implementation. As an alternative to the gradient elasticity, the staggered gradient elasticity that requires C 0-continuity, is also presented. The solid isotropic material with penalization (SIMP) like material interpolation schemes are adopted to connect the element density with the constitutive parameters of the gradient elastic solid. The effectiveness of the proposed formulations is demonstrated via numerical examples, where remarkable length-scale effects can be found in the optimized topologies of gradient elastic solids as compared with linear elastic solids.
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References
Aifantis EC (1992) On the role of gradients in the localization of deformation and fracture. Int J Eng Sci 30(10):1279–1299
Altan B, Aifantis EC (1992) On the structure of the mode III crack-tip in gradient elasticity. Scr Metall Mater 26(2):319–324
Askes H, Aifantis EC (2011) Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 48(13):1962–1990
Askes H, Morata I, Aifantis EC (2008) Finite element analysis with staggered gradient elasticity. Comput Struct 86(11):1266–1279
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications, 2nd edn. Springer, Berlin
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158
Bruggi M, Taliercio A (2012) Maximization of the fundamental eigenfrequency of micropolar solids through topology optimization. Struct Multidiscip Optim 46(4):549–560
Bruns TE, Tortorelli DA (2001a) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459. doi:10.1016/s0045-7825(00)00278-4
Bruns TE, Tortorelli DA (2001b) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26):3443–3459
Christensen PW, Klarbring A (2008) An introduction to structural optimization, vol 153. Springer Science & Business Media
Cosserat E, Cosserat F (1909) Théorie des corps déformables. Paris
Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38
Eringen AC (1965) Linear theory of micropolar elasticity. DTIC Document
Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review*. Appl Mech Rev 54(4):331–390
Fleck N, Hutchinson J (1997) Strain gradient plasticity. Adv Appl Mech 33:296–361
Fried E, Gurtin ME (2006) Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales. Arch Ration Mech Anal 182(3):513–554. doi:10.1007/s00205-006-0015-7
Green AE, Rivlin RS (1964) Multipolar continuum mechanics. Arch Ration Mech Anal 17(2):113–147
Javili A, dell’Isola F, Steinmann P (2013) Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J Mech Phys Solids 61(12):2381–2401. doi:10.1016/j.jmps.2013.06.005
Kröner E (1963) On the physical reality of torque stresses in continuum mechanics. Int J Eng Sci 1(2):261–278
Lakes R (1995) Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. Continuum models for materials with microstructure 1–25
Li L, Khandelwal K (2014) Two-point gradient-based MMA (TGMMA) algorithm for topology optimization. Comput Struct 131:34–45. doi:10.1016/j.compstruc.2013.10.010
Li L, Khandelwal K (2015a) An adaptive quadratic approximation for structural and topology optimization. Comput Struct 151:130–147. doi:10.1016/j.compstruc.2015.01.013
Li L, Khandelwal K (2015b) Topology optimization of structures with length-scale effects using elasticity with microstructure theory. Comput Struct 157:165–177. doi:10.1016/j.compstruc.2015.05.026
Li L, Khandelwal K (2015c) Volume preserving projection filters and continuation methods in topology optimization. Eng Struct 85:144–161. doi:10.1016/j.engstruct.2014.10.052
Liu S, Su W (2010) Topology optimization of couple-stress material structures. Struct Multidiscip Optim 40(1–6):319–327
Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16(1):51–78
Mindlin R, Eshel N (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4(1):109–124
Mindlin R, Tiersten H (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11(1):415–448
Papanicolopulos SA, Zervos A, Vardoulakis I (2009) A three-dimensional C1 finite element for gradient elasticity. Int J Numer Methods Eng 77(10):1396–1415. doi:10.1002/nme.2449
Petera J, Pittman JFT (1994) Isoparametric Hermite elements. Int J Numer Methods Eng 37(20):3489–3519. doi:10.1002/nme.1620372006
Polizzotto C (2003) Gradient elasticity and nonstandard boundary conditions. Int J Solids Struct 40(26):7399–7423. doi:10.1016/j.ijsolstr.2003.06.001
Polizzotto C (2013) A second strain gradient elasticity theory with second velocity gradient inertia – part I: constitutive equations and quasi-static behavior. Int J Solids Struct 50(24):3749–3765. doi:10.1016/j.ijsolstr.2013.06.024
Polizzotto C (2016) A note on the higher order strain and stress tensors within deformation gradient elasticity theories: physical interpretations and comparisons. Int J Solids Struct 90:116–121. doi:10.1016/j.ijsolstr.2016.04.001
Rovati M, Veber D (2007) Optimal topologies for micropolar solids. Struct Multidiscip Optim 33(1):47–59
Ru C, Aifantis EC (1993) A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech 101(1–4):59–68
Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493–524
Sigmund O, Maute K (2012) Sensitivity filtering from a continuum mechanics perspective. Struct Multidiscip Optim 46(4):471–475
Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75
Stölken J, Evans A (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46(14):5109–5115
Su W, Liu S (2015) Topology design for maximization of fundamental frequency of couple-stress continuum. Struct Multidiscip Optim 1–14
Svanberg K (1987) The method of moving asymptotes- a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373
Tenek LT, Aifantis E (2002) A two-dimensional finite element implementation of a special form of gradient elasticity. Comput Model Eng Sci 3(6):731–742
Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11(1):385–414
Veber D, Taliercio A (2012) Topology optimization of three-dimensional non-centrosymmetric micropolar bodies. Struct Multidiscip Optim 45(4):575–587
Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784
Zervos A (2008) Finite elements for elasticity with microstructure and gradient elasticity. Int J Numer Methods Eng 73(4):564–595
Acknowledgements
The presented work is supported in part by the US National Science Foundation through Grant CMS-1055314. Any opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.
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Li, L., Zhang, G. & Khandelwal, K. Topology optimization of structures with gradient elastic material. Struct Multidisc Optim 56, 371–390 (2017). https://doi.org/10.1007/s00158-017-1670-z
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DOI: https://doi.org/10.1007/s00158-017-1670-z