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Archive of Applied Mechanics
Erschienen in: Archive of Applied Mechanics 5/2021

28.01.2021 | Original

Analytical solutions of the simple shear problem for micromorphic models and other generalized continua

verfasst von: Gianluca Rizzi, Geralf Hütter, Angela Madeo, Patrizio Neff

Erschienen in: Archive of Applied Mechanics | Ausgabe 5/2021

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Abstract

To draw conclusions as regards the stability and modelling limits of the investigated continuum, we consider a family of infinitesimal isotropic generalized continuum models (Mindlin–Eringen micromorphic, relaxed micromorphic continuum, Cosserat, micropolar, microstretch, microstrain, microvoid, indeterminate couple stress, second gradient elasticity, etc.) and solve analytically the simple shear problem of an infinite stripe. A qualitative measure characterizing the different generalized continuum moduli is given by the shear stiffness \(\mu ^{*}\). This stiffness is in general length-scale dependent. Interesting limit cases are highlighted, which allow to interpret some of the appearing material parameter of the investigated continua.
Vorheriger Artikel Influence of contact interface friction of bolted disk joint on motion stability of rotor-bearing system
Nächster Artikel Large deformation behavior of functionally graded porous curved beams in thermal environment

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Fußnoten
1
The following energy expression has been used in [41]: \(W \left( \varvec{\nabla u}, \varvec{P}, \varvec{\nabla P}\right) = \mu \left\Vert \hbox {sym} \, \varvec{\nabla u} \right\Vert ^2 + \lambda /2 \, \hbox {tr}^2 \left( \varvec{\nabla u} \right) + \alpha \, \mu \left\Vert \varvec{\nabla u} - \varvec{P} \right\Vert ^2 + \alpha \, \lambda /2 \, \hbox {tr}^2 \left( \varvec{\nabla u} - \varvec{P} \right) +\mu \, L_{c}^2/2 \, \left\Vert \varvec{\nabla P} \right\Vert ^2 \). This formulations is not reconcilable with the relaxed micromorphic model even if we neglect the curvature part.
 
Literatur
1.
Aifantis, E.C.: The physics of plastic deformation. Int. J. Plast 3(3), 211–247 (1987)MATHCrossRef
2.
Aifantis, K.E., Willis, J.R.: The role of interfaces in enhancing the yield strength of composites and polycrystals. J. Mech. Phys. Solids 53(5), 1047–1070 (2005)MathSciNetMATHCrossRef
3.
Cosserat, E., Cosserat, F.: Théorie des Corps déformables. Hermann, Paris (1909)MATH
4.
d’Agostino, M.V., Barbagallo, G., Ghiba, I.D., Eidel, B., Neff, P., Madeo, A.: Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model. J. Elas 139, 299 (2019)MathSciNetMATHCrossRef
5.
Diebels, S., Steeb, H.: Stress and couple stress in foams. Comput. Mater. Sci. 28(3–4), 714–722 (2003)CrossRef
6.
Dunn, M., Wheel, M.: Size effect anomalies in the behaviour of loaded 3d mechanical metamaterials. Phil. Mag. 100(2), 139–156 (2020)CrossRef
7.
Eringen, A. C.: Mechanics of micromorphic continua. In: Mechanics of Generalized Continua, pp. 18–35. Springer, Berlin (1968)
8.
Forest, S.: Generalized continua from the theory to engineering applications. In: Altenbach, H., Eremeyev, V. (eds.) Micromorphic Media, vol. 541, pp. 249–300. Springer, Berlin (2013)MATH
9.
Forest, S.: Questioning size effects as predicted by strain gradient plasticity. J. Mech. Behavior Mater. 22(3–4), 101–110 (2013)CrossRef
10.
Forest, S.: Micromorphic approach to materials with internal length. In: Encyclopedia of Continuum Mechanics, pp. 1–11. Springer, Berlin, Heidelberg (2018)
11.
12.
Hadjesfandiari, A.R., Dargush, G.F.: Couple stress theory for solids. Int. J. Solids Struct. 48(18), 2496–2510 (2011)CrossRef
13.
Hütter, G.: Application of a microstrain continuum to size effects in bending and torsion of foams. Int. J. Eng. Sci. 101, 81–91 (2016)CrossRef
14.
Hütter, G.: On the micro-macro relation for the microdeformation in the homogenization towards micromorphic and micropolar continua. J. Mech. Phys. Solids 127, 62–79 (2019)MathSciNetCrossRef
15.
Hütter, G., Mühlich, U., Kuna, M.: Micromorphic homogenization of a porous medium: elastic behavior and quasi-brittle damage. Continuum Mech. Thermodyn. 27(6), 1059–1072 (2015)MathSciNetMATHCrossRef
16.
Iltchev, A., Marcadon, V., Kruch, S., Forest, S.: Computational homogenisation of periodic cellular materials: application to structural modelling. Int. J. Mech. Sci. 93, 240–255 (2015)CrossRef
17.
Jeong, J., Neff, P.: Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids 15(1), 78–95 (2010)MathSciNetMATHCrossRef
18.
Kruch, S., Forest, S.: Computation of coarse grain structures using a homogeneous equivalent medium. Le Journal de Physique IV 8(PR8), Pr8–197 (1998)
19.
Liebenstein, S., Sandfeld, S., Zaiser, M.: Size and disorder effects in elasticity of cellular structures: from discrete models to continuum representations. Int. J. Solids Struct. 146, 97–116 (2018)CrossRef
20.
Madeo, A., Ghiba, I.D., Neff, P., Münch, I.: A new view on boundary conditions in the Grioli-Koiter-Mindlin-Toupin indeterminate couple stress model. Euro. J. Mech.-A/Solids 59, 294–322 (2016)MathSciNetMATHCrossRef
21.
Mazière, M., Forest, S.: Strain gradient plasticity modeling and finite element simulation of lüders band formation and propagation. Continuum Mech. Thermodyn. 27(1–2), 83–104 (2015)MathSciNetMATHCrossRef
22.
23.
Münch, I., Neff, P., Madeo, A., Ghiba, I.D.: The modified indeterminate couple stress model: why Yang et al.’s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless. Z. Angew. Math Me. 97(12), 1524–1554 (2017)MathSciNetCrossRef
24.
Neff, P.: On material constants for micromorphic continua. In: Trends in Applications of Mathematics to Mechanics, STAMM Proceedings, Seeheim, pp. 337–348. Shaker–Verlag (2004)
25.
Neff, P.: The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. Z. Angew. Math. Me. 86(11), 892–912 (2006)MathSciNetMATHCrossRef
26.
Neff, P., Eidel, B., d’Agostino, M.V., Madeo, A.: Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization. J. Elast. 139, 269–298 (2020)MathSciNetMATHCrossRef
27.
Neff, P., Ghiba, I.D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech. Thermodyn. 26(5), 639–681 (2014)MathSciNetMATHCrossRef
28.
Neff, P., Jeong, J.: A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy. Z. Angew. Math. Me. 89(2), 107–122 (2009)MathSciNetMATHCrossRef
29.
Neff, P., Jeong, J., Fischle, A.: Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature. Acta Mech. 211(3–4), 237–249 (2010)MATHCrossRef
30.
Neff, P., Münch, I.: Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure. Continuum Mech. Thermodyn. 21(3), 195–221 (2009)MathSciNetMATHCrossRef
31.
Neff, P., Münch, I., Ghiba, I.D., Madeo, A.: On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of AR Hadjesfandiari and GF Dargush. Int. J. Solids Struct. 81, 233–243 (2016)CrossRef
32.
Nourmohammadi, N., O’Dowd, N.P., Weaver, P.M.: Effective bending modulus of thin ply fibre composites with uniform fibre spacing. Int. J. Solids Struct 196, 26 (2020)CrossRef
33.
Pham, R.D., Hütter, G.: Influence of topology and porosity on size effects in cellular materials with hexagonal structure under shear, tension and bending. arXiv preprint arXiv:​2009.​10404 (2020)
34.
Rizzi, G., Dal Corso, F., Veber, D., Bigoni, D.: Identification of second-gradient elastic materials from planar hexagonal lattices. part ii: Mechanical characteristics and model validation. Int. J. Solids Struct. 176, 19–35 (2019)CrossRef
35.
Rizzi, G., Hütter, G., Madeo, A., Neff, P.: Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua (including full derivations). arXiv preprint (2020)
36.
Rizzi, G., Hütter, G., Madeo, A., Neff, P.: Analytical solutions of the simple shear problem for certain types of micromorphic continuum models—including full derivations. arXiv preprint arXiv:​2006.​02391 (2020)
37.
Rueger, Z., Lakes, R.S.: Experimental study of elastic constants of a dense foam with weak Cosserat coupling. J. Elast. 137(1), 101–115 (2019)MathSciNetMATHCrossRef
38.
Shaat, M.: A reduced micromorphic model for multiscale materials and its applications in wave propagation. Compos. Struct. 201, 446–454 (2018)CrossRef
39.
Tekoğlu, C., Onck, P.R.: Size effects in two-dimensional voronoi foams: a comparison between generalized continua and discrete models. J. Mech. Phys. Solids 56(12), 3541–3564 (2008)MATHCrossRef
40.
Yoder, M., Thompson, L., Summers, J.: Size effects in lattice-structured cellular materials: material distribution. J. Mater. Sci. 54(18), 11858–11877 (2019)CrossRef
41.
Zhang, Z., Liu, Z., Gao, Y., Nie, J., Zhuang, Z.: Analytical and numerical investigations of two special classes of generalized continuum media. Acta Mech. Solida Sin. 24(4), 326–339 (2011)CrossRef
Metadaten
Titel
Analytical solutions of the simple shear problem for micromorphic models and other generalized continua
verfasst von
Gianluca Rizzi
Geralf Hütter
Angela Madeo
Patrizio Neff
Publikationsdatum
28.01.2021
Verlag
Springer Berlin Heidelberg
Erschienen in
Archive of Applied Mechanics / Ausgabe 5/2021
Print ISSN: 0939-1533
Elektronische ISSN: 1432-0681
DOI
https://doi.org/10.1007/s00419-021-01881-w

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