Courant’s minimax variational principle is considered in application to the six-parameter theory of prestressed shells. The equations of a prestressed micropolar shell are deduced in detail. Courant’s principle is used to study the dependence of the least and higher eigenfrequencies on shell parameters and boundary conditions. Cases involving boundary reinforcements and shell junctions are also treated.
LibaiASimmondsJG. The nonlinear theory of elastic shells. Cambridge: Cambridge University Press, 1998.
5.
ChróścielewskiJMakowskiJPietraszkiewiczW. Statics and dynamics of multifolded shells. Nonlinear theory and finite element method (in Polish). Warsaw: Wydawnictwo IPPT PAN, 2004.
6.
EremeyevVAZubovLM. Mechanics of elastic shells (in Russian). Moscow: Nauka, 2008.
7.
EremeyevVALebedevLPAltenbachH. Foundations of micropolar mechanics. Heidelberg: Springer, 2013.
8.
KonopińskaVPietraszkiewiczW. Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells. Int J Solids Struct2007; 44: 352–369.
9.
PietraszkiewiczWKonopińskaV. On unique kinematics for the branching shells. Int J Solids Struct2011; 48: 2238–2244.
10.
PietraszkiewiczWKonopińskaV. Singular curves in the resultant thermomechanics of shells. Int J Eng Sci2014; 80: 21–31.
11.
AkayAXuZCarcaterraAKoçIM. Experiments on vibration absorption using energy sinks. J Acoust Soc Am2005; 118: 3043–3049.
12.
CarcaterraAAkayABernardiniC. Trapping of vibration energy into a set of resonators: Theory and application to aerospace structures. Mech Syst Signal Process2012; 26: 1–14.
13.
CarcaterraAAkayA. Vibration damping device. Patent application 13/910,752, USA, 2014.
14.
KoçIMCarcaterraAXuZAkayA. Energy sinks: vibration absorption by an optimal set of undamped oscillators. J Acoust Soc Am2005; 118: 3031–3042.
15.
MauriniCdell’IsolaFDel VescovoD. Comparison of piezoelectronic networks acting as distributed vibration absorbers. Mech Syst Signal Process2004; 18: 1243–1271.
VidoliSdell’IsolaF. Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks. Eur J Mech A: Solids2001; 20: 435–456.
18.
EremeyevVA. (2005) Nonlinear micropolar shells: theory and applications. In: PietraszkiewiczWSzymczakC (ed.) Shell structure: Theory and applications. London: Taylor & Francis, 2005, 11–18.
19.
EremeyevVAIvanovaEAMorozovNF. On free oscillations of an elastic solids with ordered arrays of nano-sized objects. Continuum Mech Thermodyn2015; 27(4–5): 583–607.
20.
PietraszkiewiczWKonopińskaV. Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells. Int J Solids Struct2014; 51: 2133–2144.
21.
BîrsanMNeffP. Shells without drilling rotations: A representation theorem in the framework of the geometrically nonlinear 6-parameter resultant shell theory. Int J Eng Sci2014; 80: 32–42.
22.
EremeyevVALebedevLP. Existence theorems in the linear theory of micropolar shells. Z Angew Math Mech2011; 91: 468–476.
23.
BîrsanMNeffP. Existence theorems in the geometrically non-linear 6-parameter theory of elastic plates. J Elasticity2013; 112: 185–198.
24.
BîrsanMNeffP. Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations. Math Mech Solids2014; 19: 376–397.
25.
EremeyevVAAltenbachH. Rayleigh variational principle and vibrations of prestressed shells. In: PietraszkiewiczWGórskiJ (ed.) Shell structure: Theory and applications. London: Taylor & Francis, 2005, 285–288.
LebedevLPCloudMJEremeyevVA. Tensor analysis with applications in mechanics. New Jersey: World Scientific, 2010.
28.
PietraszkiewiczWEremeyevVA. On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. Int J Solids Struct2009; 46: 2477–2480.
29.
WiśniewskiK. Finite rotation shells. Basic equations and finite elements for Reissner kinematics (Lecture Notes on Numerical Methods in Engineering and Sciences, vol. 14). Berlin: Springer, 2010.
30.
PietraszkiewiczW. Refined resultant thermomechanics of shells. Int J Eng Sci2011; 49: 1112–1124.
31.
EremeyevVAPietraszkiewiczW. Local symmetry group in the general theory of elastic shells. J Elasticity2006; 85: 125–152.
32.
ReissnerE. On the theory of bending of elastic plates. J Math Phys1944; 23: 184–194.
33.
ReissnerE. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech1945; 12: A69–A77.
34.
MindlinRD. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. J Appl Mech1951; 18: 31–38.
35.
PietraszkiewiczW. Consistent second approximation to the elastic strain energy of a shell. Z Angew Math Mech1979; 59: 206–208.
36.
PietraszkiewiczW. Finite rotations and Langrangian description in the non-linear theory of shells. Warsaw–Poznan: Polish Scientific Publishers, 1979.
37.
ChróścielewskiJPietraszkiewiczWWitkowskiW. On shear correction factors in the non-linear theory of elastic shells. Int J Solids Struct2010; 47: 3537–3545.
38.
ChróścielewskiJWitkowskiW. On some constitutive equations for micropolar plates. Z Angew Math Mech2010; 90: 53–64.
39.
ChróścielewskiJWitkowskiW. FEM analysis of Cosserat plates and shells based on some constitutive relations. Z Angew Math Mech2011; 91: 400–412.
40.
ChróścielewskiJKrejaISabikAWitkowskiW. Modeling of composite shells in 6-parameter nonlinear theory with drilling degree of freedom. Mech Adv Mater Struct2011; 18: 403–419.
ParlettBN. The symmetric eigenvalue problem (Classics in Applied Mathematics, vol. 20). Philadelphia: SIAM, 1998.
55.
SaadY. Numerical methods for large eigenvalue problems (Classics in Applied Mathematics, vol. 66). Philadelphia: SIAM, 2011.
56.
AntmanSS. Nonlinear problems of elasticity. New York: Springer, 2005.
57.
BîrsanMAltenbachHSadowskiTEremeyevVAPietrasD. Deformation analysis of functionally graded beams by the direct approach. Composites Part B2012; 43: 1315–1328.
58.
EremeyevVALebedevLP. Existence of weak solutions in elasticity. Math Mech Solids2013; 18: 204–217.
59.
PlacidiLdell’IsolaFIaniroNSciarraG. Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena. Eur J Mech A: Solids2008; 27: 582–606.
60.
dell’IsolaFGuarascioMHutterK. A variational approach for the deformation of a saturated porous solid. a second-gradient theory extending Terzaghi’s effective stress principle. Arch Appl Mech2000; 70: 323–337.
61.
SciarraGdell’IsolaFIaniroNMadeoA. A variational deduction of second gradient poroelasticity part I: General theory. J Mech Mater Struct2008; 3: 507–526.
62.
NeffPGhibaIDMadeoAPlacidiLRosiG. A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech Thermodyn2014; 26: 639–681.
63.
Del VescovoDGiorgioI. Dynamic problems for metamaterials: Review of existing models and ideas for further research. Int J Eng Sci2014; 80: 153–172.
