Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

被引:106
作者
Eremeyev, Victor A. [1 ,2 ,3 ]
dell'Isola, Francesco [3 ,4 ]
Boutin, Claude [5 ]
Steigmann, David [6 ]
机构
[1] GdaAsk Univ Technol, Fac Civil & Environm Engn, Ul Gabriela Narutowicza 11-12, PL-80233 Gdansk, Poland
[2] South Fed Univ, Dept Math Mech & Comp Sci, Milchakova 8a, Rostov Na Donu 344090, Russia
[3] Univ Aquila, Int Res Ctr Math & Mech Complex Syst M&MOCS, Laquila, Italy
[4] Univ Roma La Sapienza, DISG, Rome, Italy
[5] Univ Lyon, ENTPE, Lyon, France
[6] Univ Calif Berkeley, Dept Mech Engn, Berkeley, CA 94720 USA
基金
俄罗斯科学基金会;
关键词
Strain gradient elasticity; Linear pantographic sheets; Existence; Uniqueness; Anisotropic Sobolev's space; VIRTUAL POWER; BOUNDARY; MODELS; MECHANICS; ELASTICITY;
D O I
10.1007/s10659-017-9660-3
中图分类号
T [工业技术];
学科分类号
08 ;
摘要
The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771-2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103-115, 2006). The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell'Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127-162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850-858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255-2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677-678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev's space as the most suitable energy space relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in ; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.
引用
收藏
页码:175 / 196
页数:22
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