# Ionel-Dumitrel GhibaUniversitatea Alexandru Ioan Cuza | UAIC · Department of Mathematics

Ionel-Dumitrel Ghiba

Associate Professor

## About

125

Publications

21,891

Reads

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1,829

Citations

Introduction

APPOINTMENTS: Associate Professor at the Faculty of Mathematics of the Alexandru Ioan Cuza University of Iasi, Romania. Scientific Researcher III at Octav Mayer Institute of Mathematics of the Romanian Academy, Iasi, Romania.

Additional affiliations

February 2018 - present

February 2018 - present

October 2013 - October 2016

Education

October 2007 - October 2010

**Faculty of Mathematics, Alexandru Ioan Cuza University of Iasi**

Field of study

- Mathematics

October 2005 - June 2007

**Faculty of Mathematics, Alexandru Ioan Cuza University of Iasi**

Field of study

- Mathematics

October 2001 - June 2005

**Faculty of Mathematics, Alexandru Ioan Cuza University of Iasi**

Field of study

- Mathematics

## Publications

Publications (125)

We study convexity properties of energy functions in plane nonlinear elasticity of incompressible materials and show that rank-one convexity of an objective and isotropic elastic energy $W$ on the special linear group $\mathrm{SL}(2)$ implies the polyconvexity of $W$.

We consider conformally invariant energies W on the group GL^+(2) of 2x2-matrices with positive determinant, i.e., W: GL^+(2) -> IR such that
W(A F B) = W(F) for all A, B in {aR in GL^+(2) | a in IR+, R \in SO(2)},
where SO(2) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasico...

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W : SL(2) → R with W (RF) = W (FR) = W (F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work b...

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$. The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensional sets...

For the recently introduced isotropic relaxed micromorphic generalized continuum model, we show that under the assumption of positive definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than positive-definiteness of the energy. Connections to isotropic...

Concerning a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating higher order effects, we prove the coercivity of the proposed strain energy density for shells. This result is useful to show the existence of minimizers for the energy functional using the direct methods of the calculus of variations. Then, we linea...

We consider Morrey’s open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies \(W{:}{\text {GL}}^+(n)\ri...

In this paper we derive the linear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$ as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of K...

Using $\Gamma$-convergence arguments, we construct a nonlinear membrane-like Cosserat shell model on a curvy reference configuration starting from a geometrically nonlinear, physically linear three-dimensional isotropic Cosserat model. Even if the theory is of order $O(h)$ in the shell thickness $h$, by comparison to the membrane shell models propo...

In this paper we do a comparative presentation of the linear isotropic Cosserat elastic model from two perspectives: the classical Mindlin-Eringen-Nowacki description in terms of a microrotation vector and a new formulation in terms of a skew-symmetric matrix and a curvature energy in dislocation form. We provide the reader with an alternative repr...

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-on...

We discuss the propagation of surface waves in an isotropic half space modelled with the linear Cosserat theory of isotropic elastic materials. To this aim we use a method based on the algebraic analysis of the surface impedance matrix and on the algebraic Riccati equation, and which is independent of the common Stroh formalism. Due to this method,...

We consider the regularity question of solutions for the dynamic initial‐boundary value problem for the linear relaxed micromorphic model. This generalized continuum model couples a wave‐type equation for the displacement with a generalized Maxwell‐type wave equation for the microdistortion. Naturally, solutions are found in H1 for the displacement...

We derive analytical solutions for the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua. These solutions may help in the identification of material parameters of generalized continua which are able to disclose size effects.

We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order \(O(h^{5})\) in the shell thickness \(h\). We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the \(O(h^{5})\) and \(O(h^{3})\) case and we draw some con...

We derive analytical solutions for the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua. These solutions may help in the identification of material parameters of generalized continua which are able to disclose size-effects.

We solve the St. Venant torsion problem for an infinite cylindrical rod whose behaviour is described by a family of isotropic generalized continua, including the relaxed micromorphic and classical micromorphic model. The results can be used to determine the material parameters of these models. Special attention is given to the possible nonphysical...

We consider Morrey's open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies W: GL+(n) → R with an addi...

We discuss the propagation of surface waves in an isotropic half space modelled with the linear Cosserat theory of isotropic elastic materials. To this aim we use a method based on the algebraic analysis of the surface impedance matrix and on the algebraic Riccati equation, and which is independent of the common Stroh formalism. Due to this method,...

We solve the St.Venant torsion problem for an infinite cylindrical rod whose behaviour is described by a family of isotropic generalized continua, including the relaxed micromorphic and classical micromorphic model. The results can be used to determine the material parameters of these models. Special attention is given to the possible nonphysical s...

We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce...

In this paper, we notice a property of the extension operator from the space of tangential traces of H(curl; Ω) in the context of the linear relaxed micromorphic model, a theory that is recently used to describe the behavior of some metamaterials showing unorthodox behaviors with respect to elastic wave propagation. We show that the new property is...

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order \(O(h^{5})\) in the shell thickness \(h\). The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensiona...

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solut...

We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$. We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the $O(h^5)$ and $O(h^3)$ case and we draw some connections to ex...

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group GL+(2) of invertible 2x2-matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity...

It is well known that a twice-differentiable real-valued function W:GL^+(n)->R on the group GL^+(n) of invertible nxn-matrices with positive determinant is rank-one convex if and only if it is Legendre-Hadamard elliptic. Many energy functions arising from interesting applications in isotropic nonlinear elasticity, however, are not necessarily twice...

We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce...

In this paper the relaxed micromorphic material model for anisotropic elasticity is used to describe the dynamical behavior of a band-gap metamaterial with tetragonal symmetry. Unlike other continuum models (Cauchy, Cosserat, second gradient, classical Mindlin–Eringen micromorphic etc.), the relaxed micromorphic model is endowed to capture the main...

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solut...

In this paper we use a property of the extension operator from the space of tangential traces of H(curl; Ω) in the context of the linear relaxed micromorphic model, a theory which is recently used to describe the behaviour of some metamaterials showing unorthodox behaviors with respect to elastic wave propagation. We show that the new property is i...

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytic...

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function $W\colon\operatorname{SL}(2)\to\mathbb{R}$ with $W(RF)=W(FR)=W(F)$ for all $F\in\operatorname{SL}(2)$ and all $R\in\operatorname{SO}(2)$, where $\operatorname{SL}(2)$ and $\operatorname{SO}(2)$ denote the special linear group and the spec...

We study the existence of solutions arising from the modelling of elastic materials using generalized theories of continua. In view of some evidence from physics of metamaterials, we focus our effort on two recent nonstandard relaxed micromorphic models including novel micro-inertia terms. These novel micro-inertia terms are needed to better captur...

We propose a continuum model (the relaxed micromorphic model) to describe band‐gap phenomena in metamaterials.

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function W: GL^+(n) ->R which is equal to the classical Hencky strain energy
W_H(F) = \mu |dev_n log U|^2+\kappa/2}{2} (tr(log U))^2 = \mu |log U|^2+Lambda/2 (tr(log U))^2 \] in a neighborhood of the identity matrix; here, GL^+(n) denotes th...

The logarithmic strain measures |log U|^2, where log U is the principal matrix logarithm of the stretch tensor U=\sqrt{F^TF} corresponding to the deformation gradient F and |.|$ denotes the Frobenius matrix norm, arises naturally via the geodesic distance of F to the special orthogonal group SO(n). This purely geometric characterization of this str...

This is a revised version of the talk given in Lyon. We show how the dispersion and band-gap behaviour of an anisotropic metamaterial (a periodic composite) can be described by the relaxed micromorphic model. Decisive use is made of numerical homogenization on the unit-cell level and a recently discovered micro-macro homogenization formula. Agreeme...

We show that, in the two-dimensional case, every objective, isotropic and isochoric energy function that is rank-one convex on GL ⁺ (2) is already polyconvex on GL ⁺ (2). Thus, we answer in the negative Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasi-convexity. Our methods are based on differen...

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 gradien...

Talk at the 2017 GAMM Annual Meeting in Weimar, Germany

Talk at the 2017 GAMM Annual Meeting in Weimar, Germany

We consider the weighted isotropic relaxed micromorphic model and provide an in depth investigation of the characteristic dispersion curves when the constitutive parameters of the model are varied. The weighted relaxed micromorphic model generalizes the classical relaxed micromorphic model previously introduced by the authors, since it features the...

For the recently introduced isotropic relaxed micromorphic generalized continuum model, we show a necessary and sufficient condition for real wave velocity which is weaker than positive-definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established.

The aims of this note is to present a new model based on a new representation of the curvature energy in the indeterminate couple stress model and to discuss some related choices from the literature. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

The aim of this paper is to present some results regarding the Legendre-Hadamard ellipticity and loss of ellipticity of some energies depending on the logarithmic strain tensor. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on GL⁺(2) is already polyconvex on GL⁺(2). Thus we negatively answer Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasiconvexity. Our methods are based on different represent...

We describe our result that rank-one convexity implies polyconvexity in two-dimensions for incompressible and isotropic hyperelasticity

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 gradien...

For the recently introduced isotropic relaxed micromorphic generalized continuum model, we show a necessary and sufficient condition for real wave velocity which is weaker than positive-definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established.

The relaxed micromorphic model is the only linear, isotropic, reversibly elastic,nonlocal generalized continuum model known to date able to predict complete frequency band gaps. Here, we have shown that it can be calibrated against real experiments of nontrivial wave transmission problems in phononic crystals.

In this paper we venture a new look at the linear isotropic indeterminate couple-stress model in the general framework of second-gradient elasticity and we propose a new alternative formulation which obeys Cauchy–Boltzmann’s axiom of the symmetry of the force-stress tensor. For this model we prove the existence of solutions for the equilibrium prob...

In this paper we derive, by means of a suitable least action principle, the duality jump conditions to be imposed at surfaces of discontinuity of the material properties in non-dissipative, linear-elastic, isotropic, Mindlin's and relaxed micromorphic media, respectively. The introduced theoretical framework allows the transparent set-up of differe...

In this talk we consider certain properties of nonlinear elasticity combined with plasticity and show that the additive logarithmic plasticity models may loose ellipticity in elastic unloading.

The relaxed micromorphic model is the only linear, isotropic, reversibly elastic, nonlocal generalized continuum model known to be able to predict complete frequency band gaps. It is decisive to use Curl P instead of the full micro-distortion gradient ∇ P and to take a positive Cosserat couple modulus µc > 0. A material not showing band gaps must b...

We present novel relations for the anisotropic relaxed micromorphic generalized continuum model which allow to easily interpret and determine constitutive coefficients in that theory. The simplifications and the transparency which we obtain is not satisfied for the standard Mindlin-Eringen micromorphic continuum. Our formulas also encode as a gener...