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Two- and three-dimensional size-dependent couple stress response using a displacement-based variational method. (English) Zbl 07362918
Summary: Based upon the principle of minimum total potential energy for consistent couple stress theory, a Ritz variational approach is developed using tensor product B-splines for both two- and three-dimensional elastostatic analysis. The underlying theory is size-dependent due to incorporation of the energy conjugate skew-symmetric couple-stress and mean curvature tensors, in addition to the force-stress and strain conjugate pair of classical theory. The use of B-splines as the basis functions can assure the required \(C^1\) continuity of the displacement field, while also permitting higher order representations. Both displacements and rotations are essential boundary conditions in this theory. Displacements can be enforced in the usual way, but rotations require special treatment to maintain symmetry and positive definiteness of the stiffness matrix for well-posed elastostatic problems. Several computational examples are considered to validate the formulation, illustrate convergence characteristics, and investigate mechanical behavior under consistent couple stress theory. All previous numerical analysis of consistent couple stress theory has been limited to plane strain problems. Thus, the extension here to three-dimensions is of great importance, especially because the behavior in several cases shows significant deviation from the plane strain solutions. Consequently, the variational formulations and computational methodology presented in this paper can play a critical role in assessing the predictive capability of consistent couple stress theory and in understanding size-dependent elastic response.
MSC:
74-XX Mechanics of deformable solids
Software:
Matlab
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[1] Ainsworth, M., Essential boundary conditions and multi-point constraints in finite element analysis, Comput. Methods Appl. Mech. Eng., 190, 6323-6339 (2001) · Zbl 0992.65128
[2] 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)
[3] Cosserat, E.; Cosserat, F., Théorie des Corps Déformables (Theory of Deformable Bodies) (1909), A. Hermann et Fils: A. Hermann et Fils Paris · JFM 40.0862.02
[4] Cottrell, J. A.; Hughes, T. J.R.; Reali, A., Studies of refinement and continuity in isogeometric structural analysis, Comput. Methods Appl. Mech. Eng., 196, 4160-4183 (2007) · Zbl 1173.74407
[5] Courant, R., Variational methods for the solution of problems of equilibrium and vibrations, Bull. Am. Math. Soc., 49, 1-23 (1943) · Zbl 0810.65100
[6] 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
[7] Deng, G.; Dargush, G. F., Mixed Lagrangian formulation for size-dependent couple stress elastodynamic response, Acta Mech., 227, 3451-3473 (2016) · Zbl 1394.74004
[8] 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)
[9] Ebrahimi, F.; Dabbagh, A., Wave Propagation Analysis of Smart Nanostructures (2020), CRC Press: CRC Press Boca Raton, FL
[10] Eringen, A. C., Nonlinear theory of simple micro-elastic solids, Int. J. Eng. Sci., 2, 189-203 (1964) · Zbl 0138.21202
[11] Eringen, A. C., Linear theory of micropolar elasticity, J. Math. Mech., 15, 909-923 (1966) · Zbl 0145.21302
[12] Eringen, A. C., Theory of micropolar elasticity, (Liebowitz, H., Fracture 2 (1968), Academic Press: Academic Press New York), 662-729
[13] Fischer, P.; Klassen, M.; Mergheim, J.; Steinmann, P.; Müller, R., Isogeometric analysis of 2D gradient elasticity, Comput. Mech., 47, 325-334 (2011) · Zbl 1398.74329
[14] Hadjesfandiari, A. R.; Dargush, G. F., Couple stress theory for solids, Int. J. Solid Struct., 48, 2496-2510 (2011)
[15] Hadjesfandiari, A. R.; Dargush, G. F., Boundary element formulation for plane problems in couple stress elasticity, Int. J. Numer. Methods Eng., 89, 618-636 (2012) · Zbl 1242.74185
[16] Hadjesfandiari, A. R., Size-dependent piezoelectricity, Int. J. Solid Struct., 50, 2781-2791 (2013)
[17] Hadjesfandiari, A. R., Size-dependent thermoelasticity, Lat. Am. J. Solid. Struct., 11, 1679-1708 (2014)
[18] Hadjesfandiari, A. R.; Dargush, G. F., Fundamental Governing Equations of Motion in Consistent Continuum Mechanics (2018), Preprint arXiv: 1810.04514
[19] Höllig, K., Finite Element Methods with B-Splines (2003), SIAM: SIAM Philadelphia · Zbl 1020.65085
[20] Höllig, K.; Reif, U.; Wipper, J., Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal., 39, 442-462 (2001) · Zbl 0996.65119
[21] Höllig, K.; Reif, U.; Wipper, J., Multigrid methods with web-splines, Numer. Math., 91, 237-256 (2002) · Zbl 0996.65138
[22] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng., 194, 4135-4195 (2005) · Zbl 1151.74419
[23] Hughes, T. J.R.; Reali, A.; Sangalli, G., Efficient quadrature for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Eng., 199, 301-313 (2010) · Zbl 1227.65029
[24] Kagan, P.; Fischer, A.; Bar-Yoseph, P. Z., Mechanically based models: adaptive refinement for B-spline finite element, Int. J. Numer. Methods Eng., 57, 1145-1175 (2003) · Zbl 1062.74615
[25] Koiter, W. T., Couple stresses in the theory of elasticity, I and II, (Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. Series B. Physical Sciences, vol. 67 (1964)), 17-44 · Zbl 0124.17405
[26] Kolo, I.; Askes, H.; de Borst, R., Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework, Finite Elem. Anal. Des., 135, 56-67 (2017)
[27] Madeo, A.; Ghiba, I.-D.; Neff, P.; Munch, I., A new view on boundary conditions in the Grioli-Koiter-Mindlin-Toupin indeterminate couple stress model, Eur. J. Mech. Solid., 59, 294-322 (2016) · Zbl 1406.74035
[28] Malagù, M.; Benvenuti, E.; Duarte, C.; Simone, A., One-dimensional nonlocal and gradient elasticity: assessment of high order approximation schemes, Comput. Methods Appl. Mech. Eng., 275, 138-158 (2014) · Zbl 1296.74011
[29] Matlab, Release 2020b (2020), The MathWorks, Inc.: The MathWorks, Inc. Natick, Massachusetts
[30] Mindlin, R. D., Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16, 51-78 (1964) · Zbl 0119.40302
[31] Mindlin, R. D.; Tiersten, H. F., Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal., 11, 415-448 (1962) · Zbl 0112.38906
[32] Neff, P.; Munch, I.; Ghiba, I.-D.; Madeo, A., On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of A.R. Hadjesfandiari and G.F. Dargush, Int. J. Solid Struct., 81, 233-243 (2016)
[33] Nowacki, W., The linear theory of micropolar elasticity, (Nowacki, W.; Olszak, W., Micropolar Elasticity. International Centre for Mechanical Sciences (Courses and Lectures), 151 (1974), Springer: Springer Vienna) · Zbl 0314.73001
[34] Ritz, W., Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik, J. für die Reine Angewandte Math. (Crelle’s J.), 135, 1-61 (1909) · JFM 39.0449.01
[35] Rudraraju, S.; van der Ven, A.; Garikipati, K., Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains, Comput. Methods Appl. Mech. Eng., 278, 705-728 (2014) · Zbl 1423.74105
[36] Runge, C., Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten, Zeitschrift für Mathematik und Physik, 46, 224-243 (1901) · JFM 32.0272.02
[37] Rvachev, V. L.; Sheiko, T. I., R-functions in boundary value problems in mechanics, Appl. Mech. Rev., 48, 151-188 (1995)
[38] Rvachev, V. L.; Sheiko, T. I.; Shapiro, V.; Tsukanov, I., On completeness of RFM solution structures, Comput. Mech., 25, 305-316 (2000) · Zbl 1129.74348
[39] Toupin, R. A., Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., 11, 385-414 (1962) · Zbl 0112.16805
[40] Voigt, W., Theoretische Studien über die Elastizitätsverhältnisse der Kristalle, vol. 34 (1887), (Theoretical Studies on the Elasticity Relationships of Crystals). Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen
[41] Yang, F.; Chong, A. C.M.; 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
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