×

Variational formulations and isogeometric analysis for the dynamics of anisotropic gradient-elastic Euler-Bernoulli and shear-deformable beams. (English) Zbl 1406.74409

Summary: A strain and velocity gradient framework is formulated for centrosymmetric anisotropic Euler-Bernoulli and third-order shear-deformable (TSD) beam models, reducible to Timoshenko beams. The governing equations and boundary conditions are obtained by using variational approach. The strain energy is generalized to include strain gradients and the tensor of anisotropic static length scale parameters. The kinetic energy includes velocity gradients and a tensor of anisotropic length scale parameters and hence the static and kinetic quantities of centrosymmetric anisotropic materials are distinguished in micro- and macroscales. Furthermore, the external work is written in the corresponding general form. Free vibration of simply supported centrosymmetric anisotropic TSD beams is studied by using analytical solution as well as an isogeometric numerical method verified with respect to convergence.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
49Q12 Sensitivity analysis for optimization problems on manifolds
65D07 Numerical computation using splines
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74E10 Anisotropy in solid mechanics

Software:

GeoPDEs
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Admal, N.; Marian, J.; Po, G., The atomistic representation of first strain-gradient elastic tensors, J. Mech. Phys. Solids, 99, 93-115, (2017) · Zbl 1444.74007
[2] Akgöz, B.; Civalek, Ö., A size-dependent shear-deformation beam model based on strain gradient elasticity theory, Int. J. Eng. Sci., 70, 1-14, (2013) · Zbl 1423.74452
[3] Ansari, R.; Gholami, R.; Sahmani, S., Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Compos. Struct., 94, 221-228, (2011) · Zbl 1293.74155
[4] Ansari, R.; Shojaei, M. F.; Gholami, R., Size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differentional quadrature method, Compos. Struct., 136, 669-683, (2016)
[5] Auffray, N.; Dirrenberger, J.; Rossi, G., A complete description of bi-dimensional anisotropic strain-gradient elasticity, Int. J. Solids Struct., 195-206, (2015)
[6] Auffray, N.; Quang, H. L.; He, Q., Matrix representations for 3d strain-gradient elasticity, J. Mech. Phys. Solids, 61, 1202-1223, (2013) · Zbl 1260.74012
[7] Bickford, W., A consistent higher order beam theory, (Developments in Theoretical and Applied Mechanics, (1982)), 137-150
[8] Chen, C.; Ma, M.; Liu, J.; Zheng, Q.; Xu, Z., Viscous damping of nanobeam resonators: humidity, thermal noise and a paddling effect, J. Appl. Phys., 110, (2011)
[9] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric analysis toward integration of CAD and FEA, (2009), Wiley · Zbl 1378.65009
[10] Eringen, A., Nonlocal polar elastic continua, Int. J. Eng. Sci., 10, 1-16, (1972) · Zbl 0229.73006
[11] Eringen, A., On differential equations of nonlocal elasticity and solutions of screw dislocation and waves, J. Appl. Phys., 54, 4703-4710, (1983)
[12] de Falco, C.; Reali, A.; Vázquez, R., Geopdes: a research tool for isogeometric analysis of pdes, Adv. Eng. Softw., 42, 1020-1034, (2011) · Zbl 1246.35010
[13] Gitman, I. M.; Askes, H.; Kuhl, E.; Aifantis, E., Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity, Int. J. Solids Struct., 47, 1099-1107, (2010) · Zbl 1193.74093
[14] Jing, G. Y.; Duan, H. L.; Sun, X. M.; Zhang, Z. S.; Xu, J.; Li, Y. D.; Wang, J. X.; Yu, D. P., Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy, Phys. Rev. B, 73, 235409, (2006)
[15] Kacem, N.; Hentz, S.; Pinto, D.; Reig, B.; Nguyen, V., Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors, Nanotechnology, 20, 275501, (2009)
[16] Khakalo, S.; Balobanov, V.; Niiranen, J., Modelling size-dependent bending, buckling and vibrations of 2d triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics, (2017), In review · Zbl 1423.74343
[17] Khodabakhshi, P.; Reddy, J., A unified beam theory with strain gradient effect and the von karman nonlinearity, ZAMM J. Appl. Math. Z. für Angew. Math. Mech., 97, 70-91, (2017)
[18] Kong, S.; Zhou, S.; Nie, Z.; Wang, K., Static and dynamic analysis of micro beams based on strain gradient elasticity theory, Int. J. Eng. Sci., 47, 487-498, (2009) · Zbl 1213.74190
[19] Lam, D.; Yang, F.; Chong, A.; Wang, J.; Tong, P., Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids, 51, 1477-1508, (2003) · Zbl 1077.74517
[20] Lazar, M.; Po, G., The non-singular Green tensor of Mindlin’s anisotropic gradient elasticity with separable weak non-locality, Phys. Lett. A, 379, 1538-1543, (2015) · Zbl 1370.74033
[21] Lazar, M.; Po, G., The non-singular Green tensor of gradient anisotropic elasticity of Helmholtz type, Eur. J. Mechanics-A/Solids, 50, 152-162, (2015) · Zbl 1406.74085
[22] Levinson, M., A new rectangular beam theory, J. Sound Vib., 74, 81-87, (1981) · Zbl 0453.73058
[23] Liebold, C.; Müller, W. H., Applications of higher-order continua to size effects in bending: theory and recent experimental results, (Altenbach, H.; Forest, S., Generalized Continua as Models for Classical and Advanced Materials, (2016), Springer), 237-260, (chapter 12)
[24] Mindlin, R., Micro-structure in linear elasticity, Archive Ration. Mech. Analysis, 16, 51-78, (1964) · Zbl 0119.40302
[25] Mindlin, R.; Tiersten, H., Effects of couple-stresses in linear elasticity, Archives Ration. Mech. Analysis, 11, 415-448, (1962) · Zbl 0112.38906
[26] Mousavi, S.; Paavola, J.; Reddy, J., Variational approach to dynamic analysis of third-order shear deformable plate within gradient elasticity, Meccanica, 50, 1537-1550, (2015) · Zbl 1329.74172
[27] Mousavi, S.; Reddy, J.; Romanoff, J., Analysis of anisotropic gradient elastic shear deformable plates, Acta Mech., 227, 3639-3656, (2016) · Zbl 1433.74075
[28] Niiranen, J.; Balobanov, V.; Kiendl, J.; Hosseini, S., Variational formulations, model comparisons and isogeometric analysis for Euler-Bernoulli micro- and nano-beam models of strain gradient elasticity, Math. Mech. Solids, (2017)
[29] Niiranen, J.; Khakalo, S.; Balobanov, V.; Niemi, A. H., Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems, Comput. Methods Appl. Mech. Eng., 308, 182-211, (2016) · Zbl 1439.74036
[30] Niiranen, J.; Niemi, A., Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates, Eur. J. Mechanics-A/Solids, 61, 164-179, (2017) · Zbl 1406.74446
[31] Nye, J., Physical properties of crystals: their representation by tensors and matrixes, (1957), Oxford University Press Oxford · Zbl 0079.22601
[32] Po, G.; Lazar, M.; Admal, N.; Ghoniem, N., A non-singular theory of dislocations in anisotropic crystals, (2017), Eprint Archive
[33] Reda, H.; Goda, I.; Ganghoffer, J.; L’Hostis, G.; Lakiss, Dynamical analysis of homogenized second gradient anisotropic media for textile composite structures and analysis of size effects, Compos. Struct., 161, 540-551, (2017)
[34] Reddy, J., A simple higher-order theory for laminated composite plates, J. Appl. Mech., 51, 745-752, (1984) · Zbl 0549.73062
[35] Rossi, G.; Auffray, N., Anisotropic and dispersive wave propagation within strain-gradient framework, Wave Motion, 63, 120-134, (2016) · Zbl 1469.74030
[36] Sahmani, S.; Aghdam, M., Nonlocal strain gradient beam model for nonlinear vibration of prebuckled and postbuckled multilayer functionally graded GPLRC nanobeams, Compos. Struct., 179, 77-88, (2017)
[37] Sahmani, S.; Ansari, R., Size-dependent buckling analysis of functionally graded third-order shear deformable microbeams including thermal environment effect, Appl. Math. Model., 37, 9499-9515, (2013) · Zbl 1427.74098
[38] Shaat, M.; Abdelkefi, A., Modeling the material structure and couple stress effects of nanocrystalline silicon beams for pull-in and bio-mass sensing applications, Int. J. Mech. Sci., 101-102, 280-291, (2015)
[39] Takamatsu, H.; Fukunaga, T.; Tanaka, Y.; Kurata, K.; Takahashi, K., Micro-beam sensor for detection of thermal conductivity of gases and liquids, Sensors Actuators A Phys., 206, 10-16, (2014)
[40] Tian, W.; Chen, Z.; Cao, Y., Analysis and test of a new MEMS micro-actuator, Microsyst. Technol., 22, 943-952, (2016)
[41] Timoshenko, S., On the correction factor for shear of the differential equation for transverse vinbrations of bars of uniform cross-section, Philos. Mag., 41, 744-746, (1921)
[42] Toupin, R., Theory of elasticity with couple stresses, Archives Ration. Mech. Analysis, 17, 85-112, (1964) · Zbl 0131.22001
[43] Voigt, W., Lehrbuch der kristallphysik, (1928), Teubner Leipzig, reprint of 1st edn · JFM 54.0929.03
[44] Wang, B.; Liu, M.; Zhao, J.; Zhou, S., A size-dependent reddy-Levinson beam model based on a strain gradient elasticity theory, Meccanica, 49, 1427-1441, (2014) · Zbl 1316.74031
[45] Xu, X. J.; Deng, Z. C., Closed-form frequency solutions for simplified strain gradient beams with higher-order inertia, Eur. J. Mech. A/Solids, 56, 59-72, (2016) · Zbl 1406.74414
[46] Yaghoubi, S. T.; Mousavi, S.; Paavola, J., Strain and velocity gradient theory for higher-order shear deformable beams, Archive Appl. Mech., 85, 877-892, (2015) · Zbl 1341.74107
[47] Yaghoubi, S. T.; Mousavi, S.; Paavola, J., Size effects on centrosymmetric anisotropic shear deformable beam structures, ZAMM J. Appl. Math. Mech. Zeitschrift für angewandte Math. und Mech., 97, 586-601, (2017)
[48] Yaghoubi, S. T.; Mousavi, S.; Paavola, J., Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity, Int. J. Solids Struct., 109, 84-92, (2017)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.