×

zbMATH — the first resource for mathematics

Mixed Lagrangian formulation for size-dependent couple stress elastodynamic response. (English) Zbl 1394.74004
Summary: In this paper, a mixed Lagrangian multiplier formulation is developed for consistent size-dependent couple stress elastodynamic response. This extends previous work on quasistatic couple stress response that also requires the rotation to be one half of the curl of displacement. A Lagrange multiplier, which turns out to be related to the skew-symmetric part of the force stress, is used to constrain the relation between rotation and displacement, so that rotation becomes an independent variable and \(\mathrm{C}^{1}\) continuity is avoided in the weak form. The finite element method is applied first to obtain the matrix form of the Lagrangian formulation in space, and then, discrete variational time integration is introduced to produce the discrete action. Afterward, the variation of discrete action is applied to derive the governing equations in algebraic form, which are used, along with the initial condition relations, to obtain the solutions at finite time steps. Finally, the problems of uniaxial deformation of a square plate, transverse deformation of a cantilever, and uniform traction on a square plate with a hole are investigated under this formulation, and the results are compared to existing methods where possible.

MSC:
74A10 Stress
Software:
ABAQUS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Voigt, W.: Theoretische Studien über die Elastizitätsverhältnisse der Kristalle (Theoretical Studies on the Elasticity Relationships of Crystals). In: Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen, vol. 34 (1887) · Zbl 0193.07601
[2] Cosserat, E., Cosserat, F.: Théorie des corps déformables (Theory of Deformable Bodies). A. Hermann et Fils, Paris (1909) · JFM 40.0862.02
[3] Mindlin, RD, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16, 51-78, (1964) · Zbl 0119.40302
[4] Eringen, AC; Liebowitz, H (ed.), Theory of micropolar elasticity, No. 2, 662-729, (1968), New York
[5] Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon Press, Oxford (1986) · Zbl 0604.73020
[6] Chen, S; Wang, T, Strain gradient theory with couple stress for crystalline solids, Eur. J. Mech. A Solids, 20, 739-756, (2001) · Zbl 1055.74012
[7] Toupin, RA, Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., 11, 385-414, (1962) · Zbl 0112.16805
[8] Mindlin, RD; Tiersten, HF, Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal., 11, 415-448, (1962) · Zbl 0112.38906
[9] Koiter, W.T.: Couple stresses in the theory of elasticity, I and II. In: Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. Series B. Physical Sciences, vol. 67, pp. 17-44 (1964) · Zbl 0124.17405
[10] Hadjesfandiari, AR; Dargush, GF, Couple stress theory for solids, Int. J. Solids Struct., 48, 2496-2510, (2011)
[11] 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 A.R.hadjesfandiari and G.F. dargush), Int. J. Solids Struct., 81, 233-243, (2016)
[12] Neff, P., Ghiba, I.-D., Madeo, A., Münch, I.: Correct traction boundary conditions in the indeterminate couple stress model. arXiv:1504.00448 [math-ph] · Zbl 1398.74062
[13] 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. arXiv:1505.00995 [math-ph] · Zbl 1406.74035
[14] Hadjesfandiari, A.R., Dargush, G.F.: Evolution of generalized couple-stress continuum theories: a critical analysis. arXiv:1501.03112 [physics.gen-ph] · Zbl 1242.74185
[15] Hadjesfandiari, A.R., Dargush, G.F.: Foundations of consistent couple stress theory. arXiv:1509.06299 [physics.gen-ph] · Zbl 1242.74185
[16] Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1970) · Zbl 0257.70001
[17] Fung, Y.C.: Foundations of Solid Mechanics. Prentice-Hall, N.J. (1965)
[18] Reissner, E, On a variational theorem in elasticity, J. Math. Phys., 29, 90-95, (1950) · Zbl 0039.40502
[19] Hu, HC, On some variational principles in the theory of elasticity and the theory of plasticity, Sci. Sin., 4, 33-54, (1955) · Zbl 0066.17903
[20] Washizu, K.: On the variational principles of elasticity and plasticity. In: Technical Report 25-18 MIT, Aeroelastic and Structures Research Laboratory, Cambridge, MA (1955) · Zbl 0064.37703
[21] Washizu, K.: Variational Methods in Elasticity and Plasticity. Pergamon, New York (1968) · Zbl 0164.26001
[22] Sivaselvan, M.V., Reinhorn, A.M.: Nonlinear Structural Analysis Towards Collapse Simulation: A Dynamical Systems Approach. Rep. MCEER-05-0004, MCEER/SUNY at Buffalo, Buffalo, NY (2004)
[23] Sivaselvan, MV; Reinhorn, AM, Lagrangian approach to structural collapse simulation, J. Eng. Mech., 132, 795-805, (2006)
[24] Sivaselvan, MV; Lavan, O; Dargush, GF; Kurino, H; Hyodo, Y; Fukuda, R; Sato, K; Apostolakis, G; Reinhorn, AM, Numerical collapse simulation of large-scale structural systems using an optimization-based algorithm, Earthq. Eng. Struct. Dyn., 38, 655-677, (2009)
[25] Lavan, O; Sivaselvan, MV; Reinhorn, AM; Dargush, GF, Progressive collapse analysis through strength degradation and fracture in the mixed Lagrangian formulation, Earthq. Eng. Struct. Dyn., 38, 1483-1504, (2009)
[26] 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)
[27] Apostolakis, G; Dargush, GF, Mixed Lagrangian formulation for linear thermoelastic response of structures, J. Eng. Mech., 138, 508-518, (2012)
[28] Apostolakis, G; Dargush, GF, Mixed variational principles for dynamic response of thermoelastic and poroelastic continua, Int. J. Solids Struct., 50, 642-650, (2013)
[29] Apostolakis, G; Dargush, GF, Variational methods in irreversible thermoelasticity: theoretical developments and minimum principles for the discrete form, Acta Mech., 224, 2065-2088, (2013) · Zbl 1398.74062
[30] Cadzow, JA, Discrete calculus of variations, Int. J. Control, 11, 393-407, (1970) · Zbl 0193.07601
[31] Darrall, BT; Dargush, GF; Hadjesfandiari, AR, Finite element Lagrange multiplier formulation for size-dependent skew-symmetric couple-stress planar elasticity, Acta. Mech., 225, 195-212, (2014) · Zbl 1401.74269
[32] Darrall, BT; Hadjesfandiari, AR; Dargush, GF, Size-dependent piezoelectricity: a 2D finite element formulation for electric field-Mean curvature coupling in dielectrics, Eur. J. Mech. A Solids, 49, 308-320, (2015) · Zbl 1406.74213
[33] Park, SK; Gao, X-L, Variational formulation of a modified couple stress theory and its application to a simple shear problem, Z. Angew. Math. Phys., 59, 904-917, (2008) · Zbl 1157.74014
[34] Wang, X.X., Wu, H.A.: The Basis of Computational Mechanics. University of Science and Technology of China Press, Hefei (2009)
[35] Simulia: Abaqus 6.12 documentation. Dassault Systèmes Simulia Corp. (2012) · Zbl 1406.74213
[36] Hadjesfandiari, AR; Dargush, GF, Boundary element formulation for plane problems in couple stress elasticity, Int. J. Num. Methods Eng., 89, 618-636, (2012) · Zbl 1242.74185
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.