Existence of Minimizers in Nonlinear Elastostatics of Micromorphic Solids

Neff, Patrizio

doi:10.1007/978-94-007-2739-7_783

Cited by 9 publications

(14 citation statements)

References 35 publications

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“…Suitably generalizing what done before, the equilibrium problem for the considered constrained micromorphic continuum subjected to a Bias Extension Test can be formulated as Find (u * , ϕ * ) ∈ Q × D such that P int (u * , ϕ * , δu, δϕ) + P ext (u * , ϕ * , δu, δϕ) = 0 for any (δu, δϕ) ∈ T u * × T ϕ * , where now P int , P ext , Q, D, T u and T ϕ are given by (19), (20), (21) and (22) respectively. The mathematical question of well-posedness of the geometrically nonlinear micromorphic approach has been discussed in [60,61,62]-the extension of these results to the anisotropic setting is straightforward. Some attendant results in the large strain and small strain setting, including an efficient numerical treatment and further modeling and well-posedness results can be found in [39,48,63,64,65,66].…”

Section: Space Of Configurations For the Bias Extension Testmentioning

confidence: 99%

Continuum and discrete models for unbalanced woven fabrics

Madeo

Barbagallo

d’Agostino

et al. 2016

International Journal of Solids and Structures

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The classical models used for describing the mechanical behavior of woven fabrics do not fully account for the whole set of phenomena that occur during the testing of such materials. This lack of precision is mainly due to the absence of energy terms related to the microstructural properties of the fabric and, in particular, to the bending stiffness of the yarns. The importance of the bending stiffness on the overall mechanical behavior of woven reinforcements, if already essential for the complete description of balanced fabrics, becomes even more important in the case of unbalanced ones. In this paper it is shown that the unbalance in the bending stiffnesses of the warp and weft yarns produces macroscopic effects that are extremely visible: we mention, for example, the asymmetric S-shape of a woven interlock subjected to a Bias Extension Test (BET).We propose to introduce a constrained micromorphic model and, simultaneously, a discrete model that are both able to account for i) the angle variation between warp and weft tows, ii) the unbalance in the bending stiffness of the yarns and iii) the relative slipping of the tows.The introduced constrained micromorphic model is rigorously framed in the spirit of the Principle of Virtual Work for the study of the equilibrium of continuum bodies. A suitable constraint is introduced in such micromorphic model by means of Lagrange multipliers in the strain energy density and the resulting constrained model is seen to tend to a particular second gradient one. The main advantage of using such constrained micromorphic model is that the kinematical and traction boundary conditions that can be imposed on some sub-portions of the boundary of the considered body take a natural and unique meaning.The discrete model is set up by opportunely interconnecting Euler-Bernoulli beams with different bending stiffnesses in the two directions by means of rotational and translational elastic springs. The main advantage of such discrete model is that the slipping of the tows is described in a rather realistic way. Suitable numerical simulations are presented for both the continuum and the discrete models and a comparison between the simulations and the experimental results is made showing a definitely good agreement.

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Section: Space Of Configurations For the Bias Extension Testmentioning

confidence: 99%

Continuum and discrete models for unbalanced woven fabrics

Madeo

Barbagallo

d’Agostino

et al. 2016

International Journal of Solids and Structures

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“…existence result for the geometrically nonlinear static case has been obtained in [34], which includes a previous result for the nonlinear Cosserat model [62]. For more details about existence results for micromorphic models at finite deformations, we refer the reader to [40,63,65,66]. Further existence results are supplied in [14,15,50,51].…”

Section: Introductionmentioning

confidence: 95%

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

Barbagallo

Madeo

d’Agostino

et al. 2017

International Journal of Solids and Structures

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106

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In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micromorphic model, also introducing their second order counterpart by using a Voigt-type vector notation. In strong contrast with the usual micromorphic theories, in our relaxed micromorphic model only classical elasticity-tensors with at most 21 independent components are studied together with rotational coupling tensors with at most 6 independent components. We show that in the limit case Lc → 0 (which corresponds to considering very large specimens of a microstructured metamaterial) the meso-and microcoefficients of the relaxed model can be put in direct relation with the macroscopic stiffness of the medium via a fundamental homogenization formula. We also show that a similar homogenization formula is not possible in the case of the standard Mindlin-Eringen-format of the anisotropic micromorphic model. Our results allow us to forecast the successful short term application of the relaxed micromorphic model to the characterization of anisotropic mechanical metamaterials.

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“…In case of type (c) of Fig. 6, we set v χ = 0 (i.e., 3χ 1 − χ 3 = 0) to illustrate that v elas = v 4 and v rot = v 3 and that the lines of v elas and v rot play the role of asymptotic lines. In this case, the matrix M of (3.14) becomes diagonal and the system looks similar to (3.15).…”

Section: Properties Of Solutionsmentioning

confidence: 99%

Soliton solutions in geometrically nonlinear Cosserat micropolar elasticity with large deformations

2019

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We study the fully nonlinear dynamical Cosserat micropolar elasticity problem in three dimensions with various energy functionals dependent on the microrotation R and the deformation gradient tensor F . We derive a set of coupled nonlinear equations of motion from first principles by varying the complete energy functional. We obtain a double sine-Gordon equation and construct soliton solutions. We show how the solutions can determine the overall deformational behaviours and discuss the relations between wave numbers and wave velocities thereby identifying parameter values where the soliton solution does not exist.

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Existence of Minimizers in Nonlinear Elastostatics of Micromorphic Solids

Cited by 9 publications

References 35 publications

Continuum and discrete models for unbalanced woven fabrics

Continuum and discrete models for unbalanced woven fabrics

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

Soliton solutions in geometrically nonlinear Cosserat micropolar elasticity with large deformations

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