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Madeo A.,INSA Lyon | Madeo A.,Institut Universitaire de France | Neff P.,University of Duisburg - Essen | Barbagallo G.,INSA Lyon | And 4 more authors.
Advanced Structured Materials | Year: 2017

In the present contribution we show that the relaxed micromorphic model is the only non-local continuum model which is able to account for the description of band-gaps in metamaterials for which the kinetic energy accounts separately for micro and macro-motions without considering a micro-macro coupling. Moreover, we show that when adding a gradient inertia term which indeed allows for the description of the coupling of the vibrations of themicrostructure to the macroscopic motion of the unit cell, other enriched continuum models of the micromorphic type may allow the description of the onset of band-gaps. Nevertheless, the relaxed micromorphic model proves to be yet the most effective enriched continuum model which is able to describe multiple band-gaps in non-local metamaterials. © Springer Nature Singapore Pte Ltd. 2017.

Neff P.,University of Duisburg - Essen | Ghiba I.D.,University of Duisburg - Essen | Ghiba I.D.,Al. I. Cuza University | Ghiba I.D.,Octav Mayer Institute of Mathematics of the Romanian Academy | And 4 more authors.
Quarterly Journal of Mechanics and Applied Mathematics | Year: 2015

We consider the equilibrium problem in the relaxed linear model of micromorphic elastic materials. The basic kinematical fields of this extended continuum model are the displacement u ε R3 and the non-symmetric micro-distortion density tensor P ε R3×3. In this relaxed theory, a symmetric force-stress tensor arises despite the presence of microstructure and the curvature contribution depends solely on the micro-dislocation tensor Curl P. However, the relaxed model is able to fully describe rotations of the microstructure and to predict non-polar size-effects. In contrast to classical linear micromorphic models, we allow the usual elasticity tensors to become positive-semidefinite. We prove that, nevertheless, the equilibrium problem has a unique weak solution in a suitable Hilbert space. The mathematical framework also settles the question of which boundary conditions to take for the micro-distortion. Similarities and differences between linear micromorphic elasticity and dislocation gauge theory are discussed and pointed out. © The Author, 2015.

Neff P.,University of Duisburg - Essen | Munch I.,Karlsruhe Institute of Technology | Ghiba I.-D.,University of Duisburg - Essen | Ghiba I.-D.,Al. I. Cuza University | And 3 more authors.
International Journal of Solids and Structures | Year: 2016

Madeo A.,INSA Lyon | Madeo A.,International Center Mand Mathematics and Mechanics of Complex Systems | Ghiba I.-D.,University of Duisburg - Essen | Ghiba I.-D.,Al. I. Cuza University | And 4 more authors.
European Journal of Mechanics, A/Solids | Year: 2016

In this paper we consider the Grioli-Koiter-Mindlin-Toupin linear isotropic indeterminate couple stress model. Our main aim is to show that, up to now, the boundary conditions have not been completely understood for this model. As it turns out, and to our own surprise, restricting the well known boundary conditions stemming from the strain gradient or second gradient models to the particular case of the indeterminate couple stress model, does not always reduce to the Grioli-Koiter-Mindlin-Toupin set of accepted boundary conditions. We present, therefore, a proof of the fact that when specific "mixed" kinematical and traction boundary conditions are assigned on the boundary, no "a priori" equivalence can be established between Mindlin's and our approach. © 2016 Elsevier Masson SAS. All rights reserved.

Ghiba I.-D.,University of Duisburg - Essen | Ghiba I.-D.,Al. I. Cuza University | Ghiba I.-D.,Octav Mayer Institute of Mathematics of the Romanian Academy | Neff P.,University of Duisburg - Essen | Martin R.J.,University of Duisburg - Essen
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2015

We describe ellipticity domains for the isochoric elastic energy F →||devn logU||2 = ||log √ FTF (det F)1/n||2 = 1/4||log C (detC)1/n || 2 for n=2, 3, where C=FTF for F ε GL+(n). Here, devn logU =logU - (1/n) tr(logU) 1 is the deviatoric part of the logarithmic strain tensor logU. For n=2, we identify the maximal ellipticity domain, whereas for n=3, we show that the energy is Legendre- Hadamard (LH) elliptic in the set E3(Wiso H , LH,U, 23 ) := U PSym(3)| dev3 logU2 ≤ 23 , which is similar to the von Mises-Huber-Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy WH(F)=μ dev3 logU2 + (k/2)[tr(logU)]2, U = √ FTF with μ>0 andk > 23 μ, previously obtained by Bruhns et al.

Neff P.,University of Duisburg - Essen | Ghiba I.-D.,University of Duisburg - Essen | Ghiba I.-D.,Al. I. Cuza University | Ghiba I.-D.,Octav Mayer Institute of Mathematics of the Romanian Academy
Continuum Mechanics and Thermodynamics | Year: 2015

We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric–isochoric decoupled strain energies(Formula presented.)based on the Hencky-logarithmic (true, natural) strain tensor (Formula presented.) . Here, (Formula presented.) is the infinitesimal shear modulus, (Formula presented.) is the infinitesimal bulk modulus with λ the first Lamé constant, (Formula presented.) are additional dimensionless material parameters, (Formula presented.) is the gradient of deformation, (Formula presented.) is the right stretch tensor, and devn(Formula presented.) is the deviatoric part of the strain tensor (Formula presented.) . Based on the multiplicative decomposition (Formula presented.) , we couple these energies with some isotropic elasto-plastic flow rules (Formula presented.) defined in the plastic distortion Fp, where (Formula presented.) is the subdifferential of the indicator function (Formula presented.) of the convex elastic domain (Formula presented.) in the mixed-variant (Formula presented.) -stress space, (Formula presented.) , and (Formula presented.) represents the isochoric part of the energy. While (Formula presented.) may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of (Formula presented.) on the one hand and the elastic domain in (Formula presented.) -stress space on the other hand are closely related. Thus, the new formulation remains elliptic in elastic unloading at any given plastic predeformation. In addition, in this domain, the true stress–true strain relation remains monotone, as observed in experiments. © 2015 Springer-Verlag Berlin Heidelberg

Neff P.,University of Duisburg - Essen | Ghiba I.-D.,University of Duisburg - Essen | Ghiba I.-D.,Al. I. Cuza University | Ghiba I.-D.,Octav Mayer Institute of Mathematics of the Romanian Academy
Lecture Notes in Applied and Computational Mechanics | Year: 2016

We discuss in detail existing isotropic elasto-plastic models based on 6-dimensional flow rules for the positive definite plastic metric tensor Cp = FT p Fp and highlight their properties and interconnections. We show that seemingly different models are equivalent in the isotropic case. © Springer International Publishing Switzerland 2016.

Neff P.,University of Duisburg - Essen | Lankeit J.,University of Paderborn | Ghiba I.-D.,University of Duisburg - Essen | Ghiba I.-D.,Al. I. Cuza University | And 3 more authors.
Zeitschrift fur Angewandte Mathematik und Physik | Year: 2015

We consider a family of isotropic volumetric–isochoric decoupled strain energies (Formula Presented.) based on the Hencky-logarithmic (true, natural) strain tensor log U, where μ > 0 is the infinitesimal shear modulus, κ=2μ+3λ3>0 is the infinitesimal bulk modulus with $${\lambda}$$λ the first Lamé constant, (Formula Presented.) are dimensionless parameters, F=∇φ is the gradient of deformation, (Formula Presented.) is the right stretch tensor and (Formula Presented.) is the deviatoric part (the projection onto the traceless tensors) of the strain tensor log U. For small elastic strains, the energies reduce to first order to the classical quadratic Hencky energy (Formula Presented.) which is known to be not rank-one convex. The main result in this paper is that in plane elastostatics the energies of the family (Formula Presented.) are polyconvex for (Formula Presented.) extending a previous finding on its rank-one convexity. Our method uses a judicious application of Steigmann’s polyconvexity criteria based on the representation of the energy in terms of the principal invariants of the stretch tensor U. These energies also satisfy suitable growth and coercivity conditions. We formulate the equilibrium equations, and we prove the existence of minimizers by the direct methods of the calculus of variations. © 2015, Springer Basel.

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