Global Continuation in Second-Gradient Nonlinear Elasticity

Mareno, A.; Healey, Timothy J.

doi:10.1137/050626065

Cited by 24 publications

(21 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Therefore the results by Healey and Chambon [15,36,44] could not be applied directly and there was the appearance of a lack of the coercivity of considered energy. Indeed the second gradient deformation energy for pantographic sheets is not coercive if one considers the standard Sobolev Space, whose norm involves all second order derivatives.…”

Section: Discussionmentioning

confidence: 99%

“…Moreover also the results for second gradient continua proven by Healey et al [36,44] are not applicable here, as the energy of pantographic sheets is not coercive with respect to the highest order derivatives. Before framing the problem in the appropriate energy space we present here some heuristic preliminary considerations.…”

Section: Heuristicsmentioning

confidence: 91%

“…with positive constants c 1 and c 2 , see [4] for the general framework, and [44] for the strain gradient elasticity. So, it is clear that here the equilibrium equations are not strongly elliptic.…”

Section: On Ellipticity Of Equilibrium Equationsmentioning

confidence: 99%

See 2 more Smart Citations

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Eremeyev

Dell’Isola

Boutin

et al. 2017

J Elast

129

View full text Add to dashboard Cite

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 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. 39:2255Engl. 39: -2259Engl. 39: , 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 E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (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.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Heuristicsmentioning

confidence: 91%

See 1 more Smart Citation

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Eremeyev

Dell’Isola

Boutin

et al. 2017

J Elast

129

View full text Add to dashboard Cite

show abstract

“…[11,16,17,26,28], and references therein. But to the best of our knowledge the only works with a rigorous global analysis are [11] for forced phase transitions in one-dimensional shape memory models, and [16] with results on global continuation in nonlinear three dimensional elasticity.…”

Section: Introductionmentioning

confidence: 99%

Local Bifurcation Analysis of a Second Gradient Model for Deformations of a Rectangular Slab

Montes-Pizarro

Negrón-Marrero

2006

J Elasticity

View full text Add to dashboard Cite

In this paper we carry out a derivation of the equilibrium equations of nonlinear elasticity with an added second-gradient term proportional to a small parameter ε > 0. These equations are given by a fourth order semilinear system of pdes. We discuss different types of possible boundary conditions for these equations. We then specialize the equations to a rectangular slab and study the linearized problem about a homogenous deformation. We show that these equations admit solutions representable as Fourier series in one of the independent variables. Furthermore, we obtain the characteristic equation for the eigenvalues (possible bifurcation points) for the linear problem and derive asymptotic representations for this equation for small ε. We used these expressions to show that in the limit as ε → 0 the characteristic equation for ε > 0 converges uniformly (in certain regions of the parameter space) to the corresponding characteristic equation for ε = 0. When the base material (ε = 0) is that of a Blatz-Ko type, we get conditions for the existence of eigenvalues of the linear problem with ε > 0 and small. Our numerical results in this case indicate that the number of bifurcation points is finite when ε > 0 and that this number monotonically increases as ε → 0. For the problem with ε > 0 we get conditions for the existence of local branches of non-trivial solutions.

show abstract

“…In particular, when the model is linear in the higher-gradient term, the complementing condition is always satisfied, cf. [19]. We plan to pursue these questions, in the context of global bifurcation problems, in future work.…”

Section: Discussionmentioning

confidence: 97%

Global Bifurcation in Nonlinear Elasticity with an Application to Barrelling States of Cylindrical Columns

Healey

Montes-Pizarro

The Rational Spirit in Modern Continuum Mechanics

Self Cite

View full text Add to dashboard Cite

Abstract. We present rigorous local and global bifurcation results for a concrete example from 3-dimensional nonlinear elastostatics -the problem of barrelling of compressed cylindrical columns. We use standard tools of bifurcation theory for the local analysis, already producing results that are rare in our field. For the global part we employ the generalized degree designed by Healey and Simpson to overcome the specific difficulties of 3-dimensional nonlinear elasticity. Ours are the first global bifurcation results for a problem from 3-dimensional nonlinear elastostatics not governed by ordinary differential equations. Moreover, our approach to the barrelling problem provides a paradigm for the solution of a large class of problems in nonlinear elastostatics concerning bifurcation from a homogeneously deformed state. (2000): 74B20, 74G25, 37G99. Mathematics Subject Classifications

show abstract

Global Continuation in Second-Gradient Nonlinear Elasticity

Cited by 24 publications

References 19 publications

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Local Bifurcation Analysis of a Second Gradient Model for Deformations of a Rectangular Slab

Global Bifurcation in Nonlinear Elasticity with an Application to Barrelling States of Cylindrical Columns

Contact Info

Product

Resources

About