Abstract
The virtual element method has been developed over the last decade and applied to problems in solid mechanics. Different formulations have been used regarding the order of ansatz, stabilization of the method and applied to a wide range of problems including elastic and inelastic materials and fracturing processes. This paper is concerned with formulations of virtual elements for higher gradient elastic theories of solids using the possibility, inherent in virtual element methods, of formulating C1-continuous ansatz functions in a simple and efficient way.
摘要
虚拟单元法在过去十年中得到了发展, 并应用于固体力学中的问题. 目前已使用了不同的公式来解释该方法的顺序和稳定性, 并应用到包括弹性和非弹性材料以及压裂过程的多种问题中. 本文利用虚拟单元法固有的可能性, 以简单有效的方式, 建立C1连续的弹性函数, 研究固体高梯度弹性理论的虚拟单元公式.
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References
Z. P. Bažant, Size effect, Int. J. Solids Struct. 37, 69 (2000).
N. A. Fleck, G. M. Muller, M. F. Ashby, and J. W. Hutchinson, Strain gradient plasticity: Theory and experiment, Acta Metall. Mater. 42, 475 (1994).
H. T. Zhu, H. M. Zbib, and E. C. Aifantis, Strain gradients and continuum modeling of size effect in metal matrix composites, Acta Mech. 121, 165 (1997).
H. Gao, Mechanism-based strain gradient plasticity? I. Theory, J. Mech. Phys. Solids 47, 1239 (1999).
E. Cosserat, and F. Cosserat, Sur la théorie de l’élasticité. Premier mémoire, Annales de la Faculté des sciences de Toulouse: Mathématiques, 10, I1 (1896).
E. Hellinger, Die allgemeinen Ansätze der Mechanik der Kontinua, in: F. Klein and C. Müller, eds. Mechanik (Springer, 1907), pp. 601–694.
R. D. Mindlin, and H. F. Tiersten, Effects of couple-stresses in linear elasticity, Arch. Rational Mech. Anal. 11, 415 (1962).
W. Koiter, Couple-stress in the theory of elasticity I, in: Proceedings of the Royal Netherlands Academy of Arts and Sciences (B) (North Holland Pub, 1964a), pp. 17–29.
W. Koiter, Couple-stress in the theory of elasticity II, in: Proceedings of the Royal Netherlands Academy of Arts and Sciences (B) (North Holland Pub, 1964b), pp. 30–44.
R. A. Toupin, Elastic materials with couple-stresses, Arch. Rational Mech. Anal. 11, 385 (1962).
R. A. Toupin, Theory of elasticity with couple-stress, Arch. Rational Mech. Anal. 17, 85 (1984).
A. C. Eringen, and E. Suhubi, Nonlinear theory of simple micro-elastic solids, Int. J. Eng. Sci. 2, 189 (1964).
R. D. Mindlin, and N. N. Eshel, On first strain-gradient theories in linear elasticity, Int. J. Solids Struct. 4, 109 (1968).
S. B. Altan, and E. C. Aifantis, On the structure of the mode iii crack-tip in gradient elasticity, Scr. Metall. Mater. 26, 319 (1992).
E. C. Aifantis, Strain gradient interpretation of size effects, Int. J. Fract. 95, 299 (1999).
D. C. C. Lam, F. Yang, A. C. M. Chong, J. Wang, and P. Tong, Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids 51, 1477 (2003).
E. C. Aifantis, On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity, Int. J. Eng. Sci. 47, 1089 (2009).
A. Bertram, Finite gradient elasticity and plasticity: A constitutive mechanical framework, Continuum Mech. Thermodyn. 27, 1039 (2015).
A. Beheshti, Generalization of strain-gradient theory to finite elastic deformation for isotropic materials, Continuum Mech. Thermodyn. 29, 493 (2017).
J. Y. Shu, W. E. King, and N. A. Fleck, Finite elements for materials with strain gradient effects, Int. J. Numer. Meth. Eng. 44, 373 (1999).
E. Amanatidou, and N. Aravas, Mixed finite element formulations of strain-gradient elasticity problems, Comput. Methods Appl. Mech. Eng. 191, 1723 (2002).
A. Zervos, S. A. Papanicolopulos, and I. Vardoulakis, Two finite-element discretizations for gradient elasticity, J. Eng. Mech. 135, 203 (2009).
J. C. Reiher, I. Giorgio, and A. Bertram, Finite-element analysis of polyhedra under point and line forces in second-strain gradient elasticity, J. Eng. Mech. 143, 04016112 (2017).
G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Eng. 191, 3669 (2002).
T. Lesičar, Z. Tonković, and J. Sorić, Two-scale computational approach using strain gradient theory at microlevel, Int. J. Mech. Sci. 126, 67 (2017).
S. Papargyri-Beskou, K. G. Tsepoura, D. Polyzos, and D. E. Beskos, Bending and stability analysis of gradient elastic beams, Int. J. Solids Struct. 40, 385 (2003).
N. Challamel, and C. M. Wang, The small length scale effect for a nonlocal cantilever beam: A paradox solved, Nanotechnology 19, 345703 (2008).
P. Fischer, J. Mergheim, and P. Steinmann, On the C1 continuous discretization of non-linear gradient elasticity: A comparison of NEM and FEM based on Bernstein-B�E9;zier patches, Int. J. Numer. Meth. Eng. 82, 1282 (2010).
P. Fischer, M. Klassen, J. Mergheim, P. Steinmann, and R. Müller, Isogeometric analysis of 2D gradient elasticity, Comput. Mech. 47, 325 (2011).
P. E. Fischer, C1 Continuous Methods in Computational Gradient Elasticity, Dissertation for Doctoral Degree (Friedrich-Alexander-Universitaet Erlangen-Nuernberg, Nuernberg, 2011).
L. Beirão da Veiga, F. Brezzi, and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal. 51, 794 (2013).
L. Beirão da Veiga, C. Lovadina, and D. Mora, A Virtual Element Method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Eng. 295, 327 (2015), arXiv: 1503.02042.
K. Berbatov, B. S. Lazarov, and A. P. Jivkov, A guide to the finite and virtual element methods for elasticity, Appl. Numer. Math. 169, 351 (2021).
M. L. De Bellis, P. Wriggers, and B. Hudobivnik, Serendipity virtual element formulation for nonlinear elasticity, Comput. Struct. 223, 106094 (2019).
P. Wriggers, M. L. De Bellis, and B. Hudobivnik, A Taylor-Hood type virtual element formulations for large incompressible strains, Comput. Methods Appl. Mech. Eng. 385, 114021 (2021).
F. Aldakheel, B. Hudobivnik, E. Artioli, L. Beirão da Veiga, and P. Wriggers, Curvilinear virtual elements for contact mechanics, Comput. Methods Appl. Mech. Eng. 372, 113394 (2020).
A. Hussein, B. Hudobivnik, and P. Wriggers, A combined adaptive phase field and discrete cutting method for the prediction of crack paths, Comput. Methods Appl. Mech. Eng. 372, 113329 (2020).
F. Brezzi, and L. D. Marini, Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Eng. 253, 455 (2013).
C. Chinosi, and L. D. Marini, Virtual element method for fourth order problems: L2-estimates, Comput. Math. Appl. 72, 1959 (2016).
D. Mora, and I. Velásquez, Virtual element for the buckling problem of Kirchhoff-Love plates, Comput. Methods Appl. Mech. Eng. 360, 112687 (2020), arXiv: 1905.02030.
P. Wriggers, B. Hudobivnik, and F. Aldakheel, NURBS-based geometries: A mapping approach for virtual serendipity elements, Comput. Methods Appl. Mech. Eng. 378, 113732 (2021).
F. Brezzi, and L. D. Marini, Finite elements and virtual elements on classical meshes, Vietnam J. Math. 49, 871 (2021).
P. Wriggers, On a virtual element formulation for trusses and beams, Arch. Appl. Mech. 92, 1655 (2022).
P. F. Antonietti, L. Beirão da Veiga, S. Scacchi, and M. Verani, A C1 virtual element method for the Cahn-Hilliard equation with polygonal meshes, SIAM J. Numer. Anal. 54, 34 (2016).
P. F. Antonietti, G. Manzini, and M. Verani, The fully nonconforming virtual element method for biharmonic problems, Math. Model. Methods Appl. Sci. 28, 387 (2018).
P. Wriggers, B. Hudobivnik, and O. Allix, On two simple virtual Kirchhoff-Love plate elements for isotropic and anisotropic materials, Comput. Mech. 69, 615 (2022).
J. Korelc, Automatic generation of numerical codes with introduction to AceGen 4.0 symbolc code generator, http://www.fgg.unilj.si/Symech (2000).
J. Korelc, and P. Wriggers, Automation of Finite Element Methods (Springer, Berlin, 2016).
B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66, 376 (2013).
L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, The Hitchhiker’s guide to the virtual element method, Math. Model. Methods Appl. Sci. 24, 1541 (2014).
D. Mora, G. Rivera, and I. Velásquez, A virtual element method for the vibration problem of Kirchhoff plates, ESAIM-M2AN 52, 1437 (2018).
A. M. D’Altri, S. de Miranda, L. Patruno, and E. Sacco, An enhanced VEM formulation for plane elasticity, Comput. Methods Appl. Mech. Eng. 376, 113663 (2021), arXiv: 2101.05548.
A. Chen, and N. Sukumar, Stabilization-free virtual element method for plane elasticity, arXiv: 2202.10037.
Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD, EXC 2122 (Grant No. 390833453).
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Peter Wriggers formulated the overarching research goals and aims, developed the methodology, and created the theoretical models. Peter Wriggers and Blaž Hudobivnik wrote the first draft of the manuscript. Blaž Hudobivnik did the computational work, coded the software related to the research, and performed the numerical simulations. Peter Wriggers and Blaž Hudobivnik revised and edited the final version.
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Wriggers, P., Hudobivnik, B. Virtual element formulation for gradient elasticity. Acta Mech. Sin. 39, 722306 (2023). https://doi.org/10.1007/s10409-022-22306-x
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DOI: https://doi.org/10.1007/s10409-022-22306-x