# zbMATH — the first resource for mathematics

Mixed displacement and couple stress finite element method for anisotropic centrosymmetric materials. (English) Zbl 07305824
Summary: The classical theory of elasticity is an idealized model of a continuum, which works well for many engineering applications. However, with careful experiments one finds that it may fail in describing behavior in fatigue, at small scales and in structures having high stress concentration factors. Many size-dependent theories have been developed to capture these effects, one of which is the consistent couple stress theory. In this theory, couple stress $$\mu_{ij}$$ is present in addition to force stress $$\sigma_{ij}$$ and its tensor form is shown to have skew symmetry. The mean curvature $$\kappa_{ij}$$, which is defined as the skew-symmetric part of the gradient of rotations, is the correct energy conjugate of the couple stress. This mean curvature $$\kappa_{ij}$$ and strain $$e_{ij}$$ together contribute to the elastic energy. The scope of this paper is to extend the work to study anisotropic materials and present a corresponding finite element method. A fully displacement based finite element method for couple stress elasticity requires $$C^1$$ continuity. To avoid this, a mixed formulation is presented with primary variables of displacements $$u_i$$ and couple stress $$\mu_i$$ vectors, both of which require only $$C^0$$ continuity. Centrosymmetric classes of materials are considered here for which force stress and strain are decoupled from couple stress and mean curvature in the constitutive relations. Details regarding the numerical implementation are discussed and the effect of couple stress elasticity on anisotropic materials is examined through several computational examples.
##### MSC:
 74-XX Mechanics of deformable solids
Full Text:
##### References:
 [1] Apostolakis, G.; Dargush, G. F., Mixed Lagrangian formulation for linear thermoelastic response of structures, J. Eng. Mech., 138, 508-518 (2011) [2] Apostolakis, G.; Dargush, G. F., Variational methods in irreversible thermoelasticity: theoretical developments and minimum principles for the discrete form, Acta Mech., 224, 2065-2088 (2013) · Zbl 1398.74062 [3] Chakravarty, S.; Hadjesfandiari, A. R.; Dargush, G. F., A penalty-based finite element framework for couple stress elasticity, Finite Elem. Anal. Des., 130, 65-79 (2017) [4] Chen, S.; Wang, T., Strain gradient theory with couple stress for crystalline solids, Eur. J. Mech. Solid., 20, 739-756 (2001) · Zbl 1055.74012 [5] Chen, S.; Wang, T., Finite element solutions for plane strain mode i crack with strain gradient effects, Int. J. Solid Struct., 39, 1241-1257 (2002) · Zbl 1090.74683 [6] Cosserat, E.; Cosserat, F., Théorie des corps déformables (Theory of Deformable Bodies) (1909), A. Hermann et fils · JFM 40.0862.02 [7] Darrall, B. T.; Dargush, G. F.; Hadjesfandiari, A. R., Finite element Lagrange multiplier formulation for size-dependent skew-symmetric couple-stress planar elasticity, Acta Mech., 225, 195-212 (2014) · Zbl 1401.74269 [8] Dehkordi, S. F.; Beni, Y. T., Electro-mechanical free vibration of single-walled piezoelectric/flexoelectric nano cones using consistent couple stress theory, Int. J. Mech. Sci., 128, 125-139 (2017) [9] Deng, G.; Dargush, G. F., Mixed Lagrangian formulation for size-dependent couple stress elastodynamic and natural frequency analyses, Int. J. Numer. Methods Eng., 109, 809-836 (2017) [10] Eringen, A. C., Theory of micropolar elasticity, (Microcontinuum Field Theories (1999), Springer), 101-248 [11] Ghosh, S.; Liu, Y., Voronoi cell finite element model based on micropolar theory of thermoelasticity for heterogeneous materials, Int. J. Numer. Methods Eng., 38, 1361-1398 (1995) · Zbl 0823.73066 [12] Hadjesfandiari, A.; Dargush, G., Boundary element formulation for plane problems in couple stress elasticity, Int. J. Numer. Methods Eng., 89, 618-636 (2012) · Zbl 1242.74185 [13] Hadjesfandiari, A. R.; Dargush, G. F., Couple stress theory for solids, Int. J. Solid Struct., 48, 2496-2510 (2011) [14] Hu, H. C., On some variational principles in the theory of elasticity and the theory of plasticity, Sci. Sin., 4, 33-54 (1955) · Zbl 0066.17903 [15] Huang, F. Y.; Yan, B. H.; Yan, J. L.; Yang, D. U., Bending analysis of micropolar elastic beam using a 3-d finite element method, Int. J. Eng. Sci., 38, 275-286 (2000) · Zbl 1210.74166 [16] Koiter, W., Couple stresses in the theory of elasticity, Proc. Koninklijke Nederl. Akademie Wetenschappen, 67 (1964) · Zbl 0119.39504 [17] Lata, P.; Kaur, H., Axisymmetric deformation in transversely isotropic thermoelastic medium using new modified couple stress theory, Coupled Syst. Mech., 8, 501-522 (2019) [18] Lata, P.; Kaur, H., Deformation in transversely isotropic thermoelastic medium using new modified couple stress theory in frequency domain, Geomech. Eng., 19, 369 (2019) [19] Lavan, O., Dynamic analysis of gap closing and contact in the mixed Lagrangian framework: toward progressive collapse prediction, J. Eng. Mech., 136, 979-986 (2010) [20] Lavan, O.; Sivaselvan, M.; Reinhorn, A.; Dargush, G., Progressive collapse analysis through strength degradation and fracture in the mixed Lagrangian formulation, Earthq. Eng. Struct. Dynam., 38, 1483-1504 (2009) [21] Lazar, M.; Maugin, G. A.; Aifantis, E. C., On dislocations in a special class of generalized elasticity, Phys. Status Solidi, 242, 2365-2390 (2005) [22] Li, A.; Zhou, S.; Zhou, S.; Wang, B., Size-dependent analysis of a three-layer microbeam including electromechanical coupling, Compos. Struct., 116, 120-127 (2014) [23] Li, L.; Xie, S., Finite element method for linear micropolar elasticity and numerical study of some scale effects phenomena in mems, Int. J. Mech. Sci., 46, 1571-1587 (2004) · Zbl 1098.74054 [24] Ma, H.; Gao, X. L.; Reddy, J., A microstructure-dependent timoshenko beam model based on a modified couple stress theory, J. Mech. Phys. Solid., 56, 3379-3391 (2008) · Zbl 1171.74367 [25] Mindlin, R.; Eshel, N., On first strain-gradient theories in linear elasticity, Int. J. Solid Struct., 4, 109-124 (1968) · Zbl 0166.20601 [26] Mindlin, R.; Tiersten, H., Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal., 11, 415-448 (1962) · Zbl 0112.38906 [27] Mindlin, R. D., Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16, 51-78 (1964) · Zbl 0119.40302 [28] Mohammadi, K.; Mahinzare, M.; Rajabpour, A.; Ghadiri, M., Comparison of modeling a conical nanotube resting on the winkler elastic foundation based on the modified couple stress theory and molecular dynamics simulation, The Eur. Phys. J. Plus, 132, 115 (2017) [29] Nowacki, W., Theory of Asymmetric Elasticity (1986), Pergamon Press, Headington Hill Hall: Pergamon Press, Headington Hill Hall Oxford OX 3 0 BW, UK, 1986 · Zbl 0604.73020 [30] Nye, J. F., Physical Properties of Crystals: Their Representation by Tensors and Matrices (1985), Oxford university press · Zbl 0079.22601 [31] Patel, B. N.; Pandit, D.; Srinivasan, S. M., A simplified moment-curvature based approach for large deflection analysis of micro-beams using the consistent couple stress theory, Eur. J. Mech. Solid., 66, 45-54 (2017) · Zbl 1406.74401 [32] Providas, E.; Kattis, M., Finite element method in plane cosserat elasticity, Comput. Struct., 80, 2059-2069 (2002) [33] Reddy, J., Microstructure-dependent couple stress theories of functionally graded beams, J. Mech. Phys. Solid., 59, 2382-2399 (2011) · Zbl 1270.74114 [34] Reissner, E., On a variational theorem in elasticity, Stud. Appl. Math., 29, 90-95 (1950) · Zbl 0039.40502 [35] Riahi, A.; Curran, J. H., Full 3d finite element cosserat formulation with application in layered structures, Appl. Math. Model., 33, 3450-3464 (2009) · Zbl 1205.74112 [36] Romanoff, J.; Reddy, J., Experimental validation of the modified couple stress timoshenko beam theory for web-core sandwich panels, Compos. Struct., 111, 130-137 (2014) [37] Sachio, N.; Benedict, R.; Lakes, R., Finite element method for orthotropic micropolar elasticity, Int. J. Eng. Sci., 22, 319-330 (1984) · Zbl 0536.73007 [38] Sharbati, E.; Naghdabadi, R., Computational aspects of the cosserat finite element analysis of localization phenomena, Comput. Mater. Sci., 38, 303-315 (2006) [39] Simmons, G.; Wang, H., Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook. Mass (1971), MIT Press [40] Sivaselvan, M. V.; Lavan, O.; Dargush, G. F.; Kurino, H.; Hyodo, Y.; Fukuda, R.; Sato, K.; Apostolakis, G.; Reinhorn, A. M., Numerical collapse simulation of large-scale structural systems using an optimization-based algorithm, Earthq. Eng. Struct. Dynam., 38, 655-677 (2009) [41] Sivaselvan, M. V.; Reinhorn, A. M., Lagrangian approach to structural collapse simulation, J. Eng. Mech., 132, 795-805 (2006) [42] Subramaniam, C.; Mondal, P. K., Effect of couple stresses on the rheology and dynamics of linear maxwell viscoelastic fluids, Phys. Fluids, 32, Article 013108 pp. (2020) [43] Tan, Z. Q.; Chen, Y. C., Size-dependent electro-thermo-mechanical analysis of multilayer cantilever microactuators by joule heating using the modified couple stress theory, Compos. B Eng., 161, 183-189 (2019) [44] Toupin, R. A., Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., 11, 385-414 (1962) · Zbl 0112.16805 [45] Voigt, W., Theoretische Studien über die Elasticitätsverhältnisse der Krystalle(Theoretical studies on the elasticity relationships of crystals), (Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen (1887), Dieterichsche Verlags-Buchhandlung) [46] Washizu, K., Variational Methods in Elasticity and Plasticity (1975), Pergamon press · Zbl 0164.26001 [47] Wei, Y., A new finite element method for strain gradient theories and applications to fracture analyses, Eur. J. Mech. Solid., 25, 897-913 (2006) · Zbl 1105.74044 [48] Wood, R., Finite element analysis of plane couple-stress problems using first order stress functions, Int. J. Numer. Methods Eng., 26, 489-509 (1988) · Zbl 0629.73054 [49] Yang, F.; Chong, A.; Lam, D. C.C.; Tong, P., Couple stress based strain gradient theory for elasticity, Int. J. Solid Struct., 39, 2731-2743 (2002) · Zbl 1037.74006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.