Second strain gradient elasticity of nano-objects

Cordero, Nicolás; Forest, Samuel; Busso, Esteban P.

doi:10.1016/j.jmps.2015.07.012

Cited by 137 publications

(116 citation statements)

References 71 publications

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“…Therefore, it can be thought of as a simple prototype for providing fruitful clues in view of the development of new and more refined mathematical models of continua. The interested readers will find many insight, looking at the current research trend on generalized continua and their applications, for example in [8,9] where modeling of bones is addressed; in [6,7,18,36,40,64,90] for gradient models; in [29,71] for 2nd gradient models; in [47,60,62] for multi-constituent materials and in [26] for some hints on non-local problems.…”

Section: Resultsmentioning

confidence: 99%

An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams

2016

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The theoretical results relevant to the vibration modes of Timoshenko beams are here used as benchmarks for assessing the correctness of the numerical values provided by several finite element models, based on either the traditional Lagrangian interpolation or on the recently developed isogeometric approach. Comparison of results is performed on both spectrum error (in terms of the detected natural frequencies) and on the l2 relative error (in terms of the computed eigenmodes): this double check allows detecting for each finite element model, and for a discretization based on the same number of degrees‐of‐freedom, N, the frequency threshold above which some prescribed accuracy level is lost, and results become more and more unreliable. Hence a quantitative way of measuring the finite element performance in modeling a Timoshenko beam is proposed. The use of Fast Fourier Transform is finally employed, for a selected set of vibration modes, to explain the reasons of the accuracy decay, mostly linked to a poor separation of the natural frequencies in the spectrum, which is responsible of some aliasing of modes.

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Section: Resultsmentioning

confidence: 99%

An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams

2016

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“…Let us note that one of the sources of generalized continua and models of metamaterials is the homogenization of heterogeneous materials, see, e.g., [18,54,61] and reference therein. Homogenization may lead also to strain gradient models [14,17]. While the ideas underlying the definition of pantographic microstructures have been exploited up to now only in the context of purely mechanical phenomena, it is expected that when introducing multi-physics effects (as the piezoelectric coupling phenomena exploited as explained in [23,35]) the designed meta materials could have even more interesting features.…”

Section: Introductionmentioning

confidence: 99%

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Eremeyev

Dell’Isola

Boutin

et al. 2017

J Elast

129

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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.

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“…Another application field concerns the research trend on generalized continua and their applications such as, for example, reported in for gradient models; in for second gradient models; in for multi‐constituent materials, in for some hints on non‐local problems, in for pantographic structure and in for an asymptotic approach.…”

Section: Discussionmentioning

confidence: 99%

Elastoplastic analysis of pressure‐sensitive materials by an effective three‐dimensional mixed finite element

Bilotta

Turco

2016

Z Angew Math Mech

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This work is focused on the analysis of three‐dimensional bodies whose mechanical behavior can be modeled by an elastoplastic pressure‐sensitive material description. To this end the yield condition is assessed with respect to a general quadratic function capable to represent both standard surfaces, such as Drucker–Prager surface, or more generic surfaces. For the sake of the efficiency and robustness of the numerical analysis of general‐shape solids, an effective mixed tetrahedral finite element is used to perform a step‐by‐step analysis obtaining the complete equilibrium path and the collapse load. Some numerical results regarding technical problems, such as masonry walls and footing problems, show the effectiveness of the mechanical model and of the numerical strategy.

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Second strain gradient elasticity of nano-objects

Cited by 137 publications

References 71 publications

An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams

An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Elastoplastic analysis of pressure‐sensitive materials by an effective three‐dimensional mixed finite element

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