Scale-size dependent multi-continuum homogenization of complex bodies
Author:
Grigor Nika
Journal:
Quart. Appl. Math. 83 (2025), 361-388
MSC (2020):
Primary 74Q05, 35B27, 35J58, 35Q74
DOI:
https://doi.org/10.1090/qam/1696
Published electronically:
June 7, 2024
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Abstract: We derive effective equations of a periodically heterogeneous Cosserat material encompassing intrinsic lengths modelling scale-size effects. The resultant homogenized material supports internal body torques and leads to an asymmetric effective stress providing a connection to the theory of odd elasticity. Furthermore, a link to the classical Cauchy stress is given. Moreover, the corresponding local problem exhibits asymmetry as well, due to the micropolar couple modulus inherited from the original microscopic Cosserat problem. We validate our results by conducting numerical simulations using the finite element method on circularly perforated square and rectangular unit cells, highlighting the impact, of not only volume fraction but also of internal body torques on effective coefficients. Additionally, we numerically quantify the “amount” that the body can torque internally.
References
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- M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annu. Rev. Condens. Matter Phys. 14 (2023), 471–510.
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- P. Germain, The method of virtual power in continuum mechanics. Part 2: Microstructure, SIAM J. Appl. Math. 25 (1973), no. 3, 556–575.
- Elena Grekova and Pavel Zhilin, Basic equations of Kelvin’s medium and analogy with ferromagnets, J. Elasticity 64 (2001), no. 1, 29–70 (2002). MR 1902263, DOI 10.1023/A:1014828612841
- Giuseppe Grioli, Elasticità asimmetrica, Ann. Mat. Pura Appl. (4) 50 (1960), 389–417 (Italian). MR 115342, DOI 10.1007/BF02414525
- C. S. Ha, M. E. Plesha, and R. S. Lakes, Chiral three-dimensional lattices with tunable Poisson’s ratio, Smart Mater. Struct. 25 (2016), 6 pp.
- Soroosh Hassanpour and Glenn R. Heppler, Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations, Math. Mech. Solids 22 (2017), no. 2, 224–242. MR 3605054, DOI 10.1177/1081286515581183
- F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640, DOI 10.1515/jnum-2012-0013
- H. Joumaa and M. Ostoja-Starzewski, Stress and couple-stress invariance in non-centrosymmetric micropolar planar elasticity, Proc. R. Soc. A: Math. Phys. Eng. Sci. 467 (2011), no. 2134, 2896–2911.
- W. T. Koiter, Couple-stresses in the theory of elasticity. I, II, Nederl. Akad. Wetensch. Proc. Ser. B 67 (1964), 17–29; 67 (1964) 30–44. MR 163469
- R. S. Lakes, Size effects and micromechanics of porous solids, J. Mat. Sci. 18 (1983), 2572–2581.
- R. S. Lakes, Strongly Cosserat elastic lattice and foam materials for enhanced toughness, Cell. Polym. 12 (1993), 17–30.
- Gérard A. Maugin and Andrei V. Metrikine (eds.), Mechanics of generalized continua, Advances in Mechanics and Mathematics, vol. 21, Springer, New York, 2010. One hundred years after the Cosserats; Selected papers from the EUROMECH Colloquium 510 held in Paris, May 13–16, 2009. MR 2605626, DOI 10.1007/978-1-4419-5695-8
- G. A. Maugin, The method of virtual power in continuum mechanics: application to coupled fields, Acta Mech. 35 (1980), no. 1-2, 1–70 (English, with German summary). MR 563729, DOI 10.1007/BF01190057
- Chiang C. Mei and Bogdan Vernescu, Homogenization methods for multiscale mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. MR 2777986, DOI 10.1142/7427
- R. D. Mindlin, Micro-structure in linear elasticity, Arch. Rational Mech. Anal. 16 (1964), 51–78. MR 160356, DOI 10.1007/BF00248490
- R. D. Mindlin, On the equations of elastic materials with micro-structure, Int. J. Solids Structures 1 (1965), no. 1, 73–78.
- R. D. Mindlin and N. N. Eshel, On first strain-gradient theories in linear elasticity, Int. J. Solids Structures 4 (1968), no. 1, 109–124.
- Danial Molavitabrizi, Sergei Khakalo, Rhodel Bengtsson, and S. Mahmoud Mousavi, Second-order homogenization of 3-D lattice materials towards strain gradient media: numerical modelling and experimental verification, Contin. Mech. Thermodyn. 35 (2023), no. 6, 2255–2274. MR 4656853, DOI 10.1007/s00161-023-01246-4
- Miguel Angel Moreno-Mateos, Mokarram Hossain, Paul Steinmann, and Daniel Garcia-Gonzalez, Hard magnetics in ultra-soft magnetorheological elastomers enhance fracture toughness and delay crack propagation, J. Mech. Phys. Solids 173 (2023), Paper No. 105232, 19. MR 4546081, DOI 10.1016/j.jmps.2023.105232
- G. Nika, Cosserat continuum modelling of chiral scale-size effects and their influence on effective constitutive laws, Forces Mechanics 9 (2022), 100140.
- Grigor Nika, On a hierarchy of effective models for the biomechanics of human compact bone tissue, IMA J. Appl. Math. 88 (2023), no. 2, 282–307. MR 4597179, DOI 10.1093/imamat/hxad011
- Grigor Nika, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Ric. Mat. 73 (2024), no. 1, 381–406. MR 4715431, DOI 10.1007/s11587-021-00610-3
- W. Noll and B. Coleman, The thermodynamics of elastic materials with heat conduction and viscosity, in W. Noll, editor, The Foundations of Mechanics and Thermodynamics: Selected Papers, Springer, 1974, pp. 145–156.
- W. Nowacki, The theory of micropolar elasticity, Springer, 1972.
- W. Nowacki, Theory of asymmetric elasticity, Pergamon Press, Oxford; PWN—Polish Scientific Publishers, Warsaw, 1986. Translated from the Polish by H. Zorski. MR 894254
- Martin Ostoja-Starzewski, Microstructural randomness and scaling in mechanics of materials, CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2008. MR 2341287
- H. C. Park and R. S. Lakes, Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent, J. Biomech. 19 (1986), 385–397.
- R. Rodríguez-Ramos, V. Yanes, Y. Espinosa-Almeyda, J. A. Otero, F. J. Sabina, C. F. Sánchez-Valdés, and F. Lebon, Micro–macro asymptotic approach applied to heterogeneous elastic micropolar media. Analysis of some examples, Int. J. Solids Structures 239-240 (2022), 111444.
- Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345
- C. Scheibner, A. Souslov, D. Banerjee, P. Surówka, W. T. M. Irvine, and V. Vitelli, Odd elasticity, Nat. Phys. 16 (2020), no. 4, 475–480.
- R. A. Toupin, Elastic materials with couple-stresses, Arch. Rational Mech. Anal. 11 (1962), 385–414. MR 144512, DOI 10.1007/BF00253945
- R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech. Anal. 17 (1964), 85–112. MR 169425, DOI 10.1007/BF00253050
- N. Triantafyllidis and S. Bardenhagen, The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models, J. Mech. Phys. Solids 44 (1996), no. 11, 1891–1928. MR 1420125, DOI 10.1016/0022-5096(96)00047-6
- Ruike Zhao, Yoonho Kim, Shawn A. Chester, Pradeep Sharma, and Xuanhe Zhao, Mechanics of hard-magnetic soft materials, J. Mech. Phys. Solids 124 (2019), 244–263. MR 3872586, DOI 10.1016/j.jmps.2018.10.008
References
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- Nicolas Auffray, Saad El Ouafa, Giuseppe Rosi, and Boris Desmorat, Anisotropic structure of two-dimensional linear Cosserat elasticity, Math. Mech. Complex Syst. 10 (2022), no. 4, 321–356. MR 4580724, DOI 10.2140/memocs.2022.10.321
- N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Mathematics and its Applications (Soviet Series), vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials; Translated from the Russian by D. Leĭtes. MR 1112788, DOI 10.1007/978-94-009-2247-1
- A. Bensoussan, J.-L. Lions, and G. Papanicolaou. Asymptotic analysis for periodic structures, AMS Chelsea Publishing, Providence, RI, 1978.
- S. Bytner and B. Gambin, Homogenization of Cosserat continuum, Archiwum Mechaniki Stosowanej 38 (1986), no. 3, 289–299.
- Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
- C. Ciorănescu and P. Donato, An introduction to homogenization, Oxford University Press, Oxford, UK, 2000.
- Georges Griso, Estimation d’erreur et éclatement en homogénéisation périodique, C. R. Math. Acad. Sci. Paris 335 (2002), no. 4, 333–336 (French, with English and French summaries). MR 1931511, DOI 10.1016/S1631-073X(02)02477-9
- D. Cioranescu, A. Damlamian, and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal. 40 (2008), no. 4, 1585–1620. MR 2466168, DOI 10.1137/080713148
- Doina Cioranescu, Alain Damlamian, and Georges Griso, The periodic unfolding method, Series in Contemporary Mathematics, vol. 3, Springer, Singapore, 2018. Theory and applications to partial differential problems. MR 3839345, DOI 10.1007/978-981-13-3032-2
- E. Cosserat and F. Cosserat, Théorie des Corps Déformables, Librairie Scientifique A. Hermann et Fils, 6, Rue de la Sorbonne, 1909.
- Alain Damlamian, An elementary introduction to periodic unfolding, Multi scale problems and asymptotic analysis, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 24, Gakkōtosho, Tokyo, 2006, pp. 119–136. MR 2233174
- A. Cemal Eringen, Microcontinuum field theories. I. Foundations and solids, Springer-Verlag, New York, 1999. MR 1720520, DOI 10.1007/978-1-4612-0555-5
- A. Cemal Eringen, Linear theory of micropolar elasticity, J. Math. Mech. 15 (1966), 909–923. MR 198744
- A. Cemal Eringen and E. S. Suhubi, Nonlinear theory of simple micro-elastic solids. I, Internat. J. Engrg. Sci. 2 (1964), 189–203 (English, with French, German, Italian and Russian summaries). MR 169423, DOI 10.1016/0020-7225(64)90004-7
- E. S. Suhubi and A. Cemal Eringen, Nonlinear theory of micro-elastic solids. II, Internat. J. Engrg. Sci. 2 (1964), 389–404 (English, with French, German, Italian and Russian summaries). MR 170513, DOI 10.1016/0020-7225(64)90017-5
- S. Forest, Milieux continus généralisés et matériaux hétérogènes, Presses des MINES, 2006.
- Samuel Forest, Francis Pradel, and Karam Sab, Asymptotic analysis of heterogeneous Cosserat media, Internat. J. Solids Structures 38 (2001), no. 26-27, 4585–4608. MR 1837063, DOI 10.1016/S0020-7683(00)00295-X
- S. Forest and K. Sab, Cosserat overall modeling of heterogeneous materials, Mech. Res. Comm. 25 (1998), no. 4, 449–454. MR 1635846, DOI 10.1016/S0093-6413(98)00059-7
- S. Forest and K. Sab, Estimating the overall properties of heterogeneous Cosserat materials, Modelling Simul. Mater. Sci. Eng. 7 (1999), no. 5, 829–840.
- M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annu. Rev. Condens. Matter Phys. 14 (2023), 471–510.
- D. Garcia-Gonzalez, Magneto-visco-hyperelasticity for hard-magnetic soft materials: theory and numerical applications, Smart Mater. Struct. 28 (2019), no. 8, 085020.
- P. Germain, La méthode des puissances virtuelles en mécanique des milieux continus. I. Théorie du second gradient, J. Mécanique 12 (1973), 235–274 (French, with English summary). MR 423935
- P. Germain, The method of virtual power in continuum mechanics. Part 2: Microstructure, SIAM J. Appl. Math. 25 (1973), no. 3, 556–575.
- Elena Grekova and Pavel Zhilin, Basic equations of Kelvin’s medium and analogy with ferromagnets, J. Elasticity 64 (2001), no. 1, 29–70 (2002). MR 1902263, DOI 10.1023/A:1014828612841
- Giuseppe Grioli, Elasticità asimmetrica, Ann. Mat. Pura Appl. (4) 50 (1960), 389–417 (Italian). MR 115342, DOI 10.1007/BF02414525
- C. S. Ha, M. E. Plesha, and R. S. Lakes, Chiral three-dimensional lattices with tunable Poisson’s ratio, Smart Mater. Struct. 25 (2016), 6 pp.
- Soroosh Hassanpour and Glenn R. Heppler, Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations, Math. Mech. Solids 22 (2017), no. 2, 224–242. MR 3605054, DOI 10.1177/1081286515581183
- F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640, DOI 10.1515/jnum-2012-0013
- H. Joumaa and M. Ostoja-Starzewski, Stress and couple-stress invariance in non-centrosymmetric micropolar planar elasticity, Proc. R. Soc. A: Math. Phys. Eng. Sci. 467 (2011), no. 2134, 2896–2911.
- W. T. Koiter, Couple-stresses in the theory of elasticity. I, II, Nederl. Akad. Wetensch. Proc. Ser. B 67 (1964), 17–29; 67 (1964) 30–44. MR 163469
- R. S. Lakes, Size effects and micromechanics of porous solids, J. Mat. Sci. 18 (1983), 2572–2581.
- R. S. Lakes, Strongly Cosserat elastic lattice and foam materials for enhanced toughness, Cell. Polym. 12 (1993), 17–30.
- Gérard A. Maugin and Andrei V. Metrikine (eds.), Mechanics of generalized continua, Advances in Mechanics and Mathematics, vol. 21, Springer, New York, 2010. One hundred years after the Cosserats; Selected papers from the EUROMECH Colloquium 510 held in Paris, May 13–16, 2009. MR 2605626, DOI 10.1007/978-1-4419-5695-8
- G. A. Maugin, The method of virtual power in continuum mechanics: application to coupled fields, Acta Mech. 35 (1980), no. 1-2, 1–70 (English, with German summary). MR 563729, DOI 10.1007/BF01190057
- Chiang C. Mei and Bogdan Vernescu, Homogenization methods for multiscale mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. MR 2777986, DOI 10.1142/7427
- R. D. Mindlin, Micro-structure in linear elasticity, Arch. Rational Mech. Anal. 16 (1964), 51–78. MR 160356, DOI 10.1007/BF00248490
- R. D. Mindlin, On the equations of elastic materials with micro-structure, Int. J. Solids Structures 1 (1965), no. 1, 73–78.
- R. D. Mindlin and N. N. Eshel, On first strain-gradient theories in linear elasticity, Int. J. Solids Structures 4 (1968), no. 1, 109–124.
- Danial Molavitabrizi, Sergei Khakalo, Rhodel Bengtsson, and S. Mahmoud Mousavi, Second-order homogenization of 3-D lattice materials towards strain gradient media: numerical modelling and experimental verification, Contin. Mech. Thermodyn. 35 (2023), no. 6, 2255–2274. MR 4656853, DOI 10.1007/s00161-023-01246-4
- Miguel Angel Moreno-Mateos, Mokarram Hossain, Paul Steinmann, and Daniel Garcia-Gonzalez, Hard magnetics in ultra-soft magnetorheological elastomers enhance fracture toughness and delay crack propagation, J. Mech. Phys. Solids 173 (2023), Paper No. 105232, 19. MR 4546081, DOI 10.1016/j.jmps.2023.105232
- G. Nika, Cosserat continuum modelling of chiral scale-size effects and their influence on effective constitutive laws, Forces Mechanics 9 (2022), 100140.
- Grigor Nika, On a hierarchy of effective models for the biomechanics of human compact bone tissue, IMA J. Appl. Math. 88 (2023), no. 2, 282–307. MR 4597179, DOI 10.1093/imamat/hxad011
- Grigor Nika, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Ric. Mat. 73 (2024), no. 1, 381–406. MR 4715431, DOI 10.1007/s11587-021-00610-3
- W. Noll and B. Coleman, The thermodynamics of elastic materials with heat conduction and viscosity, in W. Noll, editor, The Foundations of Mechanics and Thermodynamics: Selected Papers, Springer, 1974, pp. 145–156.
- W. Nowacki, The theory of micropolar elasticity, Springer, 1972.
- W. Nowacki, Theory of asymmetric elasticity, Pergamon Press, Oxford; PWN—Polish Scientific Publishers, Warsaw, 1986. Translated from the Polish by H. Zorski. MR 894254
- Martin Ostoja-Starzewski, Microstructural randomness and scaling in mechanics of materials, CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2008. MR 2341287
- H. C. Park and R. S. Lakes, Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent, J. Biomech. 19 (1986), 385–397.
- R. Rodríguez-Ramos, V. Yanes, Y. Espinosa-Almeyda, J. A. Otero, F. J. Sabina, C. F. Sánchez-Valdés, and F. Lebon, Micro–macro asymptotic approach applied to heterogeneous elastic micropolar media. Analysis of some examples, Int. J. Solids Structures 239-240 (2022), 111444.
- Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345
- C. Scheibner, A. Souslov, D. Banerjee, P. Surówka, W. T. M. Irvine, and V. Vitelli, Odd elasticity, Nat. Phys. 16 (2020), no. 4, 475–480.
- R. A. Toupin, Elastic materials with couple-stresses, Arch. Rational Mech. Anal. 11 (1962), 385–414. MR 144512, DOI 10.1007/BF00253945
- R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech. Anal. 17 (1964), 85–112. MR 169425, DOI 10.1007/BF00253050
- N. Triantafyllidis and S. Bardenhagen, The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models, J. Mech. Phys. Solids 44 (1996), no. 11, 1891–1928. MR 1420125, DOI 10.1016/0022-5096(96)00047-6
- Ruike Zhao, Yoonho Kim, Shawn A. Chester, Pradeep Sharma, and Xuanhe Zhao, Mechanics of hard-magnetic soft materials, J. Mech. Phys. Solids 124 (2019), 244–263. MR 3872586, DOI 10.1016/j.jmps.2018.10.008
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Additional Information
Grigor Nika
Affiliation:
Department of Mathematics & Computer Science, Karlstad University, Universitetsgatan 2, 65188 Karlstad, Sweden
MR Author ID:
1127661
ORCID:
0000-0002-4403-6908
Email:
grigor.nika@kau.se
Keywords:
Multi-continuum homogenization,
Cosserat continuum,
scale-size effects,
odd elasticity
Received by editor(s):
January 7, 2024
Received by editor(s) in revised form:
April 17, 2024
Published electronically:
June 7, 2024
Additional Notes:
The author was financially supported by the Knowledge Foundation (project nr. KK 2020-0152).
Article copyright:
© Copyright 2024
Brown University
