Abstract
Topology optimization (TO) methods for fracture resistance offer new possibilities for designing stronger structures or materials with lower masses than conventional designs. This article presents an overview of TO techniques for fracture resistance, from pioneering works to the most recent developments at the time of writing. We first review stress-based methods, which were the forerunners of crack resistance methods, producing optimal designs that prevent any damage or crack initiation. Other works followed, taking into account the presence of defects or cracks in structures, but using classical approaches aimed at minimizing compliance in an elastic framework. TO methods for fatigue damage are also an important branch of these approaches and are reviewed. We then present more recent methodologies, including non-linear effects in structural design, such as plasticity and damage. Finally, we describe the latest methods of TO design for fracture resistance, including an explicit description of crack propagation during loading, from initiation to failure of structures and materials. In particular, the design of two-phase materials that are more resistant to cracking and that can be manufactured by 3D printing is discussed. The article concludes with some challenges and promising avenues for the coming years in this field.
Similar content being viewed by others
Notes
By critical, we mean the stress concentration directly contributing to crack initiation.
References
Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–390
Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49:1–38
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4(3–4):250–252
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654
Querin OM, Young V, Steven GP, Xie YM (2000) Computational efficiency and validation of bi-directional evolutionary structural optimisation. Comput Methods Appl Mech Eng 189(2):559–573
Li Y, Xie YM (2021) Evolutionary topology optimization for structures made of multiple materials with different properties in tension and compression. Compos Struct 259:113497
Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246
Allaire G, Jouve F, Toader A (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393
Takezawa Akihiro, Nishiwaki Shinji, Kitamura Mitsuru (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229(7):2697–2718
Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech 81(8):081009
Liu C, Du Z, Zhu Y, Zhang W, Zhang X, Guo X (2020) Optimal design of shell-graded-infill structures by a hybrid MMC-MMV approach. Comput Methods Appl Mech Eng 369:113187
Da D, Xia L, Li G, Huang X (2018) Evolutionary topology optimization of continuum structures with smooth boundary representation. Struct Multidiscip Optim 57:2143–2159
Rozvany GIN (2009) A critical review of established methods of structural topology optimization. Struct Multidiscip Optim 37(3):217–237
van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472
Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23(4):595–622
Wang C, Zhao Z, Zhou M, Sigmund O, Zhang XS (2021) A comprehensive review of educational articles on structural and multidisciplinary optimization. Structural and Multidisciplinary Optimization 64:2827–2880
Wang Y, Li X, Long K, Wei P (2023) Open-source codes of topology optimization: a summary for beginners to start their research. CMES Comput Model Eng Sci 137(1):1–34
Sigmund O, Maute K (2013) Topology optimization approaches: a comparative review. Struct Multidiscip Optim 48(6):1031–1055
James KA, Waisman H (2015) Topology optimization of structures under variable loading using a damage superposition approach. Int J Numer Methods Eng 101(5):375–406
Lu G, Yu TX (2003) Energy absorption of structures and materials. Elsevier, Amsterdam
Ngo TD, Kashani A, Imbalzano G, Nguyen KTQ, Ds Hui (2018) Additive manufacturing (3D printing): a review of materials, methods, applications and challenges. Composites B Eng 143:172–196
Wang X, Jiang M, Zhou Z, Gou J, Hui D (2017) 3D printing of polymer matrix composites: a review and prospective. Composites B Eng 110:442–458
Kao YT, Zhang Y, Wang J, Tai BL (2016) Loading-unloading cycles of 3D-printing built bi-material structures with ceramic and elastomer. In: International manufacturing science and engineering conference, vol 49910. American Society of Mechanical Engineers, New York, p V003T08A008
Amin AR, Kao YT, Tai BL, Wang J (2017) Dynamic response of 3D-printed bi-material structure using drop weight impact test. In: International manufacturing science and engineering conference, vol 50732. American Society of Mechanical Engineers, New York, p V002T01A021
Cai M, Kaiser PK (2004) Numerical simulation of the Brazilian test and the tensile strength of anisotropic rocks and rocks with pre-existing cracks. Int J Rock Mech Min Sci 41:478–483
Ludwig C, Rabold F, Kuna M, Schurig M, Schlums H (2020) Simulation of anisotropic crack growth behavior of nickel base alloys under thermomechanical fatigue. Eng Fract Mech 224:106800
Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–156
Daux C, Moës N, Dolbow J, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng 48:1741–1760
Gravouil A, Moës N, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets. Part II: level set update. Int J Numer Methods Eng 53(11):2569–86
Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342
Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48:797–826
Amor H, Marigo J-J, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229
Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199:2776–2778
Xia L, Da D, Yvonnet J (2018) Topology optimization for maximizing the fracture resistance of quasi-brittle composites. Comput Methods Appl Mech Eng 332:234–254
Desmorat B, Desmorat R (2008) Topology optimization in damage governed low cycle fatigue. CR Mec 336(5):448–453
Lew AJ, Buehler MJ (2021) A deep learning augmented genetic algorithm approach to polycrystalline 2D material fracture discovery and design. Appl Phys Rev 8(4):041414
Khimin D, Steinbach MC, Wick T (2022) Space–time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems. J Comput Phys 470:111554
Yang RJ, Chen CJ (1996) Stress-based topology optimization. Struct Optim 12(2–3):98–105
Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43(8):1453–1478
Holmberg E, Torstenfelt B, Klarbring A (2013) Stress constrained topology optimization. Struct Multidiscip Optim 48:33–47
Emmendoerfer H Jr, Fancello EA (2014) A level set approach for topology optimization with local stress constraints. Int J Numer Methods Eng 99(2):129–156
Jeong SH, Yoon GH, Takezawa A, Choi D-H (2014) Development of a novel phase-field method for local stress-based shape and topology optimization. Comput Struct 132:84–98
Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidiscip Optim 36:125–141
Luo Y, Kang Z (2012) Topology optimization of continuum structures with Drucker–Prager yield stress constraints. Comput Struct 90:65–75
Verbart A, Langelaar M, van Keulen F (2016) Damage approach: a new method for topology optimization with local stress constraints. Struct Multidiscip Optim 53:1081–1098
Long K, Wang X, Liu H (2019) Stress-constrained topology optimization of continuum structures subjected to harmonic force excitation using sequential quadratic programming. Struct Multidiscip Optim 59:1747–1759
Giraldo-Londono O, Paulino GH (2021) Polystress: a Matlab implementation for local stress-constrained topology optimization using the augmented Lagrangian method. Struct Multidiscip Optim 63:2065–2097
Nguyen SH, Kim HG (2020) Stress-constrained shape and topology optimization with the level set method using trimmed hexahedral meshes. Comput Methods Appl Mech Eng 366:113061
McBane S, Choi Y, Willcox K (2022) Stress-constrained topology optimization of lattice-like structures using component-wise reduced order models. Comput Methods Appl Mech Eng 400:115525
Kundu RD, Li W, Zhang XS (2022) Multimaterial stress-constrained topology optimization with multiple distinct yield criteria. Extreme Mech Lett 54:101716
Norato JA, Smith HA, Deaton JD, Kolonay RM (2022) A maximum-rectifier-function approach to stress-constrained topology optimization. Struct Multidiscip Optim 65(10):286
Allaire G, Jouve F, Maillot H (2004) Topology optimization for minimum stress design with the homogenization method. Struct Multidiscip Optim 28:87–98
Allaire G, Jouve F (2008) Minimum stress optimal design with the level set method. Eng Anal Bound Elem 32(11):909–918
Xia Q, Shi T, Liu S, Wang MY (2012) A level set solution to the stress-based structural shape and topology optimization. Comput Struct 90:55–64
Cai S, Zhang W, Zhu J, Gao T (2014) Stress constrained shape and topology optimization with fixed mesh: a b-spline finite cell method combined with level set function. Comput Methods Appl Mech Eng 278:361–387
Lian H, Christiansen AN, Tortorelli DA, Sigmund O, Aage N (2017) Combined shape and topology optimization for minimization of maximal von Mises stress. Struct Multidiscip Optim 55:1541–1557
Picelli R, Townsend S, Brampton C, Norato J, Kim HA (2018) Stress-based shape and topology optimization with the level set method. Comput Methods Appl Mech Eng 329:1–23
Xia L, Zhang L, Xia Q, Shi T (2018) Stress-based topology optimization using bi-directional evolutionary structural optimization method. Comput Methods Appl Mech Eng 333:356–370
Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41:605–620
Qian X (2017) Undercut and overhang angle control in topology optimization: a density gradient based integral approach. Int J Numer Methods Eng 111(3):247–272
Wang C, Qian X (2018) Heaviside projection-based aggregation in stress-constrained topology optimization. Int J Numer Methods Eng 115(7):849–871
Pereira JT, Fancello EA, Barcellos CS (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidiscip Optim 26(1–2):50–66
da Silva G, Aage N, Beck AT, Sigmund O (2021) Three-dimensional manufacturing tolerant topology optimization with hundreds of millions of local stress constraints. Int J Numer Methods Eng 122(2):548–578
Giraldo-Londono O, Paulino GH (2020) A unified approach for topology optimization with local stress constraints considering various failure criteria: von Mises, Drucker-Prager, Tresca, Mohr-Coulomb, Bresler-Pister and Willam-Warnke. Proc R Soc A 476(2238):20190861
Giraldo-Londono O, Aguiló MA, Paulino GH (2021) Local stress constraints in topology optimization of structures subjected to arbitrary dynamic loads: a stress aggregation-free approach. Struct Multidiscip Optim 64:3287–3309
da Silva G, Beck AT (2018) Reliability-based topology optimization of continuum structures subject to local stress constraints. Struct Multidiscip Optim 57:2339–2355
da Silva G, Beck AT, Sigmund O (2019) Stress-constrained topology optimization considering uniform manufacturing uncertainties. Comput Methods Appl Mech Eng 344:512–537
da Silva G, Aage N, Beck AT, Sigmund O (2021) Local versus global stress constraint strategies in topology optimization: a comparative study. Int J Numer Methods Eng 122(21):6003–6036
Da D, Chen W (2023) Simple strategy toward tailoring fracture properties of brittle architected materials. Int J Numer Methods Eng 124(2):334–357
Guo X, Zhang WS, Wang MY, Wei P (2011) Stress-related topology optimization via level set approach. Comput Methods Appl Mech Eng 200(47–48):3439–3452
Zhang W, Li D, Zhou J, Du Z, Li B, Guo X (2018) A moving morphable void (MMV)-based explicit approach for topology optimization considering stress constraints. Comput Methods Appl Mech Eng 334:381–413
Bruggi M, Venini P (2008) A mixed FEM approach to stress-constrained topology optimization. Int J Numer Methods Eng 73(12):1693–1714
Sharma A, Maute K (2018) Stress-based topology optimization using spatial gradient stabilized XFEM. Struct Multidiscip Optim 57:17–38
Yang D, Liu H, Zhang W, Li S (2018) Stress-constrained topology optimization based on maximum stress measures. Comput Struct 198:23–39
Granlund G, Wallin M, Tortorelli D, Watts S (2023) Stress-constrained topology optimization of structures subjected to nonproportional loading. Int J Numer Methods Eng 124(12):2818–2836
Ogawa S, Yamada T (2023) Stress constraint topology optimization of coupled thermo-mechanical problems using the temperature dependence of allowable stress. Comput Struct 281:107006
Bruggi M, Duysinx P (2012) Topology optimization for minimum weight with compliance and stress constraints. Struct Multidiscip Optim 46:369–384
Suresh K, Takalloozadeh M (2013) Stress-constrained topology optimization: a topological level-set approach. Struct Multidiscip Optim 48:295–309
Boissier M, Deaton JD, Beran N, Vermaak PA (2021) Elastoplastic topology optimization of cyclically loaded structures via direct methods for shakedown. Struct Multidiscip Optim 64:189–217
Fan Z, Xia L, Lai W, Xia Q, Shi T (2019) Evolutionary topology optimization of continuum structures with stress constraints. Struct Multidiscip Optim 59:647–658
Chen A, Cai K, Zhao Z-L, Zhou Y, Xia L, Xie YM (2021) Controlling the maximum first principal stress in topology optimization. Struct Multidiscip Optim 63:327–339
Ho-Nguyen-Tan T, Kim H-G (2022) Level set-based topology optimization for compliance and stress minimization of shell structures using trimmed quadrilateral shell meshes. Comput Struct 259:106695
Da D, Chan YC, Wang I, Chen W (2022) Data-driven and topological design of structural metamaterials for fracture resistance. Extreme Mech Lett 50:101528
Challis VJ, Roberts AP, Wilkins AH (2008) Fracture resistance via topology optimization. Struct Multidiscip Optim 36:263–271
Jansen M, Lombaert G, Schevenels M, Sigmund O (2014) Topology optimization of fail-safe structures using a simplified local damage model. Struct Multidiscip Optim 49:657–666
Shobeiri V (2015) The topology optimization design for cracked structures. Eng Anal Bound Elem 58:26–38
Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229–256
Gu GX, Dimas L, Qin Z, Buehler MJ (2016) Optimization of composite fracture properties: method, validation, and applications. J Appl Mech 83(7):071006
Gu GX, Wettermark S, Buehler MJ (2017) Algorithm-driven design of fracture resistant composite materials realized through additive manufacturing. Addit Manuf 17:47–54
Kang Z, Liu P, Li M (2017) Topology optimization considering fracture mechanics behaviors at specified locations. Struct Multidiscip Optim 55:1847–1864
Banh TT, Lee D (2018) Multi-material topology optimization design for continuum structures with crack patterns. Compos Struct 186:193–209
Banh TT, Luu NG, Lee D (2021) A non-homogeneous multi-material topology optimization approach for functionally graded structures with cracks. Compos Struct 273:114230
Hu J, Yao S, Gan N, Xiong Y, Chen X (2019) Fracture strength topology optimization of structural specific position using a bi-directional evolutionary structural optimization method. Eng Optim 52(4):583–602
Klarbring A, Torstenfelt B, Edlund U, Schmidt P, Simonsson K, Ansell H (2018) Minimizing crack energy release rate by topology optimization. Struct Multidiscip Optim 58:1695–1703
Marbœuf A, Budinger M, Pommier-Budinger V, Palanque V, Bennani L (2022) Improving mechanical ice protection systems with topology optimization. Struct Multidiscip Optim 65(5):147
Silling AS (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209
Kefal A, Sohouli A, Oterkus E, Yildiz M, Suleman A (2019) Topology optimization of cracked structures using peridynamics. Continuum Mech Thermodyn 31:1645–1672
Sohouli A, Kefal A, Abdelhamid A, Yildiz M, Suleman A (2020) Continuous density-based topology optimization of cracked structures using peridynamics. Struct Multidiscip Optim 62:2375–2389
Lahe Motlagh P, Kefal A (2021) Comparative study of peridynamics and finite element method for practical modeling of cracks in topology optimization. Symmetry 13(8):1407
Kendibilir A, Kefal A, Sohouli A, Yildiz M, Koc B, Suleman A (2022) Peridynamics topology optimization of three-dimensional structures with surface cracks for additive manufacturing. Comput Methods Appl Mech Eng 401:115665
Chen Z, Long K, Wen P, Nouman S (2020) Fatigue-resistance topology optimization of continuum structure by penalizing the cumulative fatigue damage. Adv Eng Softw 150:102924
Hou J, Zhu J, Wang J, Zhang W (2018) Topology optimization of the multi-fasteners jointed structure considering fatigue constraints. Int J Simul Multidiscip Des Optim 9:A4
Lemaitre J, Chaboche J (1994) Mechanics of solid materials. Cambridge University Press, Cambridge
Lemaitre J, Desmorat R (2006) Engineering damage mechanics: ductile, creep, fatigue and brittle failures. Springer, Berlin
Mrzyglod M, Zielinski AP (2007) Parametric structural optimization with respect to the multiaxial high-cycle fatigue criterion. Struct Multidiscip Optim 33:161–171
Sines G (1959) Behavior of metals under complex static and alternating stresses. Met Fatigue 1:145–169
Crossland B (1956) Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel. In: Proceedings of the international conference on fatigue of metals, vol 138. Institution of Mechanical Engineers, London, p 12
Van K, Griveau B (1989) On a new multiaxial fatigue limit criterion—theory and application. Biaxial and multiaxial fatigue (A 90-16739 05-39). Mechanical Engineering Publications, London, pp 479–496
Mrzygłód M (2010) Two-stage optimization method with fatigue constraints for thin-walled structures. J Theor Appl Mech 48(3):567–578
Sherif K, Witteveen W, Puchner K, Irschik H (2010) Efficient topology optimization of large dynamic finite element systems using fatigue. AIAA J 48(7):1339–1347
Choi WS, Park GJ (1999) Transformation of dynamic loads into equivalent static loads based on modal analysis. Int J Numer Methods Eng 46(1):29–43
Jeong SH, Choi DH, Yoon GH (2015) Fatigue and static failure considerations using a topology optimization method. Appl Math Model 39(3–4):1137–1162
Lee JW, Yoon GH, Jeong SH (2015) Topology optimization considering fatigue life in the frequency domain. Comput Math Appl 70(8):1852–1877
Jeong SH, Lee JW, Yoon GH, Choi DH (2018) Topology optimization considering the fatigue constraint of variable amplitude load based on the equivalent static load approach. Appl Math Model 56:626–647
Holmberg E, Torstenfelt B, Klarbring A (2014) Fatigue constrained topology optimization. Struct Multidiscip Optim 50:207–219
Oest J, Lund E (2017) Topology optimization with finite-life fatigue constraints. Struct Multidiscip Optim 56:1045–1059
Suresh S, Lindström SB, Thore C-J, Torstenfelt B, Klarbring A (2020) Topology optimization using a continuous-time high-cycle fatigue model. Struct Multidiscip Optim 61:1011–1025
Suresh S, Lindström SB, Thore CJ, Klarbring A (2022) Acceleration of continuous-time, high-cycle fatigue constrained problems in topology optimization. Eur J Mech A Solids 96:104723
Zhang S, Le C, Gain AL, Norato JA (2019) Fatigue-based topology optimization with non-proportional loads. Comput Methods Appl Mech Eng 345:805–825
Zhao L, Xu B, Han Y, Xue J, Rong J (2020) Structural topological optimization with dynamic fatigue constraints subject to dynamic random loads. Eng Struct 205:110089
Nabaki K, Shen J, Huang X (2019) Evolutionary topology optimization of continuum structures considering fatigue failure. Mater Des 166:107586
Teng X, Wang C, Jiang X, Chen X (2023) Structural topology optimization with local finite-life fatigue constraints. Mathematics 11(5):1220
Collet M, Bruggi M, Duysinx P (2017) Topology optimization for minimum weight with compliance and simplified nominal stress constraints for fatigue resistance. Struct Multidiscip Optim 55:839–855
Chen Y, Wang Q, Wang C, Gong P, Shi Y, Yu Y, Liu Z (2021) Topology optimization design and experimental research of a 3D-printed metal aerospace bracket considering fatigue performance. Appl Sci 11(15):6671
Demir S, Kurt M, Kotil T (2022) Fatigue damage-based topology optimization of helicopter tail rotor pitch arm. J Aerosp Eng 35(5):04022073
Niutta CB, Tridello A, Barletta G, Gallo N, Baroni A, Berto F, Paolino DS (2022) Defect-driven topology optimization for fatigue design of additive manufacturing structures: application on a real industrial aerospace component. Eng Fail Anal 142:106737
Al-Ali MA, Al-Ali MA, Takezawa A, Kitamura M (2017) Topology optimization and fatigue analysis of temporomandibular joint prosthesis. World J Mech 7(12):323–339
Olesen AM, Hermansen SM, Lund E (2021) Simultaneous optimization of topology and print orientation for transversely isotropic fatigue. Struct Multidiscip Optim 64:1041–1062
Suresh S, Lindström SB, Thore CJ, Klarbring A (2021) Topology optimization for transversely isotropic materials with high-cycle fatigue as a constraint. Struct Multidiscip Optim 63:161–172
Trudel E, ElSayed MSA (2022) Penalization techniques for fatigue-based topology optimizations of structures with embedded functionally graded lattice materials. Int J Numer Methods Eng 123(9):1991–2011
Dagkolu A, Gokdag I, Yilmaz O (2021) Design and additive manufacturing of a fatigue-critical aerospace part using topology optimization and L-PBF process. Procedia Manuf 54:238–243
Liu YJ, Ren DC, Li SJ, Wang H, Zhang LC, Sercombe TB (2020) Enhanced fatigue characteristics of a topology-optimized porous titanium structure produced by selective laser melting. Addit Manuf 32:101060
Barbier T, Shakour E, Sigmund O, Lombaert G, Schevenels M (2022) Topology optimization of damage-resistant structures with a predefined load-bearing capacity. Int J Numer Methods Eng 123(4):1114–1145
Maute K, Schwarz S, Ramm E (1998) Adaptive topology optimization of elastoplastic structures. Struct Optim 15:81–91
Schwarz S, Maute K, Ramm E (2001) Topology and shape optimization for elastoplastic structural response. Comput Methods Appl Mech Eng 190(15–17):2135–2155
Kato J, Hoshiba H, Takase S, Terada K, Kyoya T (2015) Analytical sensitivity in topology optimization for elastoplastic composites. Struct Multidiscip Optim 52:507–526
Nakshatrala PB, Tortorelli DA (2015) Topology optimization for effective energy propagation in rate-independent elastoplastic material systems. Comput Methods Appl Mech Eng 295:305–326
Wallin M, Jönsson V, Wingren E (2016) Topology optimization based on finite strain plasticity. Struct Multidiscip Optim 54:783–793
Maury A, Allaire G, Jouve F (2018) Elasto-plastic shape optimization using the level set method. SIAM J Control Optim 56(1):556–581
Jia J, Da D, Hu J, Yin S (2021) Crashworthiness design of periodic cellular structures using topology optimization. Compos Struct 271:114164
Achtziger W, Bendsøe MP (1995) Design for maximal flexibility as a simple computational model of damage. Struct Optim 10:258–268
Bendsøe MP, Díaz AR (1998) A method for treating damage related criteria in optimal topology design of continuum structures. Struct Optim 16:108–115
Amir O (2013) A topology optimization procedure for reinforced concrete structures. Comput Struct 114:46–58
Amir O, Sigmund O (2013) Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization. Struct Multidiscip Optim 47:157–174
James KA, Waisman H (2014) Failure mitigation in optimal topology design using a coupled nonlinear continuum damage model. Comput Methods Appl Mech Eng 268:614–631
Noël L, Duysinx P, Maute K (2017) Level set topology optimization considering damage. Struct Multidiscip Optim 56(4):737–753
Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree HPJ (1996) Gradient-enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39(39):3391–3403
Alberdi R, Khandelwal K (2017) Topology optimization of pressure dependent elastoplastic energy absorbing structures with material damage constraints. Finite Elem Anal Des 133:42–61
Li L, Khandelwal K (2017) Design of fracture resistant energy absorbing structures using elastoplastic topology optimization. Struct Multidiscip Optim 56:1447–1475
Li L, Zhang G, Khandelwal K (2017) Topology optimization of energy absorbing structures with maximum damage constraint. Int J Numer Methods Eng 112(7):737–775
Li L, Zhang G, Khandelwal K (2018) Failure resistant topology optimization of structures using nonlocal elastoplastic-damage model. Struct Multidiscip Optim 58:1589–1618
Zhang G, Khandelwal K (2022) Gurson–Tvergaard–Needleman model guided fracture-resistant structural designs under finite deformations. Int J Numer Meth Eng 123(14):3344–3388
Zhang Z, Chen J, Li E, Li W, Swain M, Li Q (2016) Topological design of all-ceramic dental bridges for enhancing fracture resistance. Int J Numer Methods Biomed Eng 32(6):e02749
Ambati M, Gerasimov T, de Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383–405
Wu JY, Nguyen VP, Nguyen CT, Sutula D, Bordas S, Sinaie S (2018) Phase field modeling of fracture. In: Advances in applied mechanics: multi-scale theory and computation, vol 52. Elsevier, Amsterdam
Da D, Yvonnet J, Xia L, Li G (2018) Topology optimization of particle–matrix composites for optimal fracture resistance taking into account interfacial damage. Int J Numer Methods Eng 115(5):604–626
Russ JB, Waisman H (2019) Topology optimization for brittle fracture resistance. Comput Methods Appl Mech Eng 347:238–263
Russ JB, Waisman H (2020) A novel topology optimization formulation for enhancing fracture resistance with a single quasi-brittle material. Int J Numer Methods Eng 121(13):2827–2856
Wu C, Fang J, Zhou S, Zhang Z, Sun G, Steven GP, Li Q (2021) A path-dependent level set topology optimization with fracture criterion. Comput Struct 249:106515
Desai J, Allaire G, Jouve F (2022) Topology optimization of structures undergoing brittle fracture. J Comput Phys 458:111048
Jia Y, Lopez-Pamies O, Zhang XS (2023) Controlling the fracture response of structures via topology optimization: from delaying fracture nucleation to maximizing toughness. J Mech Phys Solids 173:105227
Li P, Yvonnet J, Wu Y (2022) Improved fracture resistance of 3D-printed elastoplastic structures with respect to their topology and orientation of deposited layers. Int J Mech Sci 220:107147
Noii N, Jahangiry HA, Waisman H (2023) Level-set topology optimization for ductile and brittle fracture resistance using the phase-field method. Comput Methods Appl Mech Eng 409:115963
Li P, Wu Y, Yvonnet J (2021) A SIMP-phase field topology optimization framework to maximize quasi-brittle fracture resistance of 2D and 3D composites. Theoret Appl Fract Mech 114:102919
Wu Y, Yvonnet J, Li P, He ZC (2022) Topology optimization for enhanced dynamic fracture resistance of structures. Comput Methods Appl Mech Eng 394:114846
Verhoosel CV, de Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96(1):43–62
Nguyen TT, Yvonnet J, Zhu QZ, Bornert M, Chateau C (2016) A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography. Comput Methods Appl Mech Eng 312:567–595
Wu C, Fang J, Zhou S, Zhang Z, Sun G, Steven GP, Li Q (2020) Level-set topology optimization for maximizing fracture resistance of brittle materials using phase-field fracture model. Int J Numer Methods Eng 121(13):2929–2945
Da D, Yvonnet J (2020) Topology optimization for maximizing the fracture resistance of periodic quasi-brittle composites structures. Materials 13(15):3279
Da D (2019) Topology optimization design of heterogeneous materials and structures. Wiley, New York
Da D, Qian X (2020) Fracture resistance design through biomimicry and topology optimization. Extreme Mech Lett 40:100890
Liu F, Li T, Jia Z, Wang L (2020) Combination of stiffness, strength, and toughness in 3D printed interlocking nacre-like composites. Extreme Mech Lett 35:100621
Da D (2022) Model reduction on 3D fracture resistance design. J Comput Phys 463:111274
Da D, Shu X (To be submitted) Enhancing fracture resistance in architected bi-material structures: design optimization, experimental validation, and mechanical analysis
Singh S, Pflug L, Stingl M (2021) Material optimization to enhance delamination resistance of composite structures using viscous regularization. Comput Methods Appl Mech Eng 382:113881
Singh S, Pflug L, Mergheim J, Stingl M (2023) Robust design optimization for enhancing delamination resistance of composites. Int J Numer Methods Eng 124(6):1381–1404
Singh S, Pflug L, Mergheim J, Stingl M (2023) On optimization of heterogeneous materials for enhanced resistance to bulk fracture. Forces Mech 2:100200
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding authors state that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Yvonnet, J., Da, D. Topology Optimization to Fracture Resistance: A Review and Recent Developments. Arch Computat Methods Eng 31, 2295–2315 (2024). https://doi.org/10.1007/s11831-023-10044-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11831-023-10044-9