scholarly journals Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets

2019 ◽  
Vol 25 ◽  
pp. 24 ◽  
Author(s):  
Virginia Agostiniani ◽  
Alessandro Lucantonio ◽  
Danka Lučić
Keyword(s):  
Curvature Tensor ◽  
Plate Theory ◽  
Finite Elasticity ◽  
Thin Sheet ◽  
Three Dimensional ◽  
Elastic Plates ◽  
Three Dimensional Model ◽  
Spontaneous Strain ◽  
Piecewise Constant ◽  
Γ Convergence

We rigorously derive a Kirchhoff plate theory, via Γ-convergence, from a three-dimensional model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables x′. At the same time, the spontaneous strain is h-close to the identity, where h is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into ℝ3, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from an x′-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.

A numerical comparison of the uniformly valid asymptotic plate equations with a 3D model: Clamped rectangular incompressible elastic plates

2021 ◽  
pp. 108128652110255
Author(s):  
Fan-Fan Wang ◽  
Hui-Hui Dai ◽  
Ivan Giorgio
Keyword(s):  
Thin Plate ◽  
Numerical Solutions ◽  
Plate Theory ◽  
Three Dimensional ◽  
Weak Form ◽  
Numerical Comparison ◽  
Elastic Plates ◽  
Three Dimensional Model ◽  
Finite Element Software ◽  
Lagrange Strain

In this paper, we derive the weak form for clamped plates composed of incompressible neo-Hookean material from the uniformly valid asymptotic plate theory. By using the finite-element software COMSOL, we study the numerical solutions of the weak form. We show the accuracy and the efficiency of the weak form by comparing the numerical results for the two-dimensional weak form and a three-dimensional model. As a basis for comparison we choose numerical values of the displacement, the second Piola–Kirchhoff stress, and the Green–Lagrange strain at the bottom. The numerical simulations are performed for three different cases of thickness–span ratios, including (1) very thin plate, (2) thin plate, and (3) moderately thick plate. The results show that the uniformly valid plate theory is a reliable and implementable plate theory for even moderately thick plates with large deformations.

scholarly journals Rigorous derivation of active plate models for thin sheets of nematic elastomers

2017 ◽  
Vol 25 (10) ◽  
pp. 1804-1830 ◽  
Author(s):  
Virginia Agostiniani ◽  
Antonio DeSimone
Keyword(s):  
Finite Elasticity ◽  
Three Dimensional ◽  
Energy Functional ◽  
Variable Degree ◽  
Energy Functionals ◽  
Nematic Order ◽  
Reduction Techniques ◽  
Nematic Elastomers ◽  
Plate Models ◽  
Γ Convergence

In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced from a three-dimensional description of the system using rigorous dimension reduction techniques, based on the theory of Γ-convergence. The two-dimensional models are non-linear plate theories, in which deviations from a characteristic target curvature tensor cost elastic energy. Moreover, the stored energy functional cannot be minimised to zero, thus revealing the presence of residual stresses, as observed in numerical simulations. Three nematic textures are considered: splay-bend and twisted orientations of the nematic director, and a uniform director perpendicular to the mid-plane of the film, with variable degree of nematic order along the thickness. These three textures realise three very different structural models: one with only one stable spontaneously bent configuration, a bistable model with two oppositely curved configurations of minimal energy, and a shell with zero stiffness to twisting.

scholarly journals A hierarchy of multilayered plate models

2020 ◽  
Author(s):  
Miguel de Benito Delgado ◽  
Bernd Schmidt
Keyword(s):  
Nonlinear Elasticity ◽  
Curvature Tensor ◽  
Elastic Constants ◽  
Three Dimensional ◽  
Spontaneous Curvature ◽  
Multilayered Plate ◽  
Von Karman ◽  
Plate Models ◽  
Γ Convergence ◽  
Von Kármán

We derive a hierarchy of plate theories for heterogeneous multilayers from three dimensional nonlinear elasticity by means of Γ-convergence. We allow for layers composed of different materials whose constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the Γ-limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von Kármán, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials.

A JUSTIFICATION OF THE REISSNER–MINDLIN PLATE THEORY THROUGH VARIATIONAL CONVERGENCE

2007 ◽  
Vol 05 (02) ◽  
pp. 165-182 ◽  
Author(s):  
ROBERTO PARONI ◽  
PAOLO PODIO-GUIDUGLI ◽  
GIUSEPPE TOMASSETTI
Keyword(s):  
Plate Theory ◽  
Scaling Law ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Energy Functional ◽  
Mindlin Plate ◽  
Variational Convergence ◽  
Mindlin Plate Theory ◽  
Γ Convergence ◽  
Three Dimensional Elasticity

We provide a justification of the Reissner–Mindlin plate theory, using linear three-dimensional elasticity as framework and Γ-convergence as technical tool. Essential to our developments is the selection of a transversely isotropic material class whose stored energy depends on (first and) second gradients of the displacement field. Our choices of a candidate Γ-limit and a scaling law of the basic energy functional in terms of a thinness parameter are guided by mechanical and formal arguments that our variational convergence theorem is meant to validate mathematically.

Wave scattering by multiple rows of circular ice floes

2009 ◽  
Vol 639 ◽  
pp. 213-238 ◽  
Author(s):  
L. G. BENNETTS ◽  
V. A. SQUIRE
Keyword(s):  
Finite Number ◽  
Wave Scattering ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Linear Solution ◽  
Ice Floes ◽  
The Individual

A three-dimensional model of ocean-wave scattering in the marginal ice zone is constructed using linear theory under time-harmonic conditions. Individual floes are represented by circular elastic plates and are permitted to have a physically realistic draught. These floes are arranged into a finite number of parallel rows, and each row possesses an infinite number of identical floes that are evenly spaced. The floe properties may differ between rows, and the spacing between the rows is arbitrary.The vertical dependence of the solution is expanded in a finite number of modes, and through the use of a variational principle, a finite set of two-dimensional equations is generated from which the full-linear solution may be retrieved to any desired accuracy. By dictating the periodicity in each row to be identical, the scattering properties of the individual rows are combined using transfer matrices that take account of interactions between both propagating and evanescent waves.Numerical results are presented that investigate the differences between using the three-dimensional model and using a two-dimensional model in which the rows are replaced with strips of ice. Furthermore, Bragg resonance is identified when the rows are identical and equi-spaced, and its reduction when the inhomogeneities, that are accommodated by the model, are introduced is shown.

Three-Dimensional Model for Virtual Surgical Simulation, during Complex Skull Base Surgery and Aneurysm Surgery

Skull Base ◽  
2008 ◽  
Vol 18 (S 01) ◽  
Author(s):  
Akio Morita ◽  
Toshikazu Kimura ◽  
Shigeo Sora ◽  
Kengo Nishimura ◽  
Hisayuki Sugiyama ◽  
...  
Keyword(s):  
Skull Base ◽  
Surgical Simulation ◽  
Skull Base Surgery ◽  
Three Dimensional ◽  
Aneurysm Surgery ◽  
Dimensional Model ◽  
Three Dimensional Model

Digitalization system of ancient architecture decoration art based on neural network and image features

2020 ◽  
pp. 1-12
Author(s):  
Wu Xin ◽  
Qiu Daping
Keyword(s):  
Neural Network ◽  
Construction Industry ◽  
Three Dimensional ◽  
Performance Testing ◽  
Image Features ◽  
Three Dimensional Model ◽  
Performance Effect ◽  
Data Process ◽  
And Performance ◽  
Construction Mode

The inheritance and innovation of ancient architecture decoration art is an important way for the development of the construction industry. The data process of traditional ancient architecture decoration art is relatively backward, which leads to the obvious distortion of the digitalization of ancient architecture decoration art. In order to improve the digital effect of ancient architecture decoration art, based on neural network, this paper combines the image features to construct a neural network-based ancient architecture decoration art data system model, and graphically expresses the static construction mode and dynamic construction process of the architecture group. Based on this, three-dimensional model reconstruction and scene simulation experiments of architecture groups are realized. In order to verify the performance effect of the system proposed in this paper, it is verified through simulation and performance testing, and data visualization is performed through statistical methods. The result of the study shows that the digitalization effect of the ancient architecture decoration art proposed in this paper is good.

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