Searcharxiv⌕ Search

arXiv subjects

Ionel-Dumitrel Ghiba

Publications and source records attributed to Ionel-Dumitrel Ghiba.

At least 55 records · Page 3Linked to original sources
arXiv2017

A polyconvex extension of the logarithmic Hencky strain energy

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function $W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ which is equal to the classical Hencky strain energy \[ W_{\mathrm{H}}(F) = μ\,\lVert\operatorname{dev}_n\log U\rVert^2+\fracκ{2}\,[\operatorname{tr}(\log U)]^2 = μ\,\lVert\log U\rVert^2+\fracΛ{2}\,[\operatorname{tr}(\log U)]^2 \] in a neighborhood of the identity matrix; here, $\operatorname{GL^+}(n)$ denotes the set of $n\times n$-matrices with positive determinant, $F\in\operatorname{GL^+}(n)$ denotes the deformation gradient, $U=\sqrt{F^TF}$ is the corresponding stretch tensor, $\log U$ is the principal matrix logarithm of $U$, $\operatorname{tr}$ is the trace operator, $\lVert X\rVert$ is the Frobenius matrix norm and $\operatorname{dev}_n X$ is the deviatoric part of $X\in\mathbb{R}^{n\times n}$. The extension can also be chosen to be coercive, in which case Ball's classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions $W_{\mathrm{VL}}$ in the so-called Valanis-Landel form \[ W_{\mathrm{VL}}(F) = \sum_{i=1}^n w(λ_i) \] with $w\colon(0,\infty)\to\mathbb{R}$, where $λ_1,\dotsc,λ_n$ denote the singular values of $F$.

math.AP↗
arXiv2017

A non-ellipticity result, or the impossible taming of the logarithmic strain measure

The logarithmic strain measures $\lVert\log U\rVert^2$, where $\log U$ is the principal matrix logarithm of the stretch tensor $U=\sqrt{F^TF}$ corresponding to the deformation gradient $F$ and $\lVert\,.\,\rVert$ denotes the Frobenius matrix norm, arises naturally via the geodesic distance of $F$ to the special orthogonal group $\operatorname{SO}(n)$. This purely geometric characterization of this strain measure suggests that a viable constitutive law of nonlinear elasticity may be derived from an elastic energy potential which depends solely on this intrinsic property of the deformation, i.e. that an energy function $W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ of the form \begin{equation} W(F)=Ψ(\lVert\log U\rVert^2) \tag{1} \end{equation} with a suitable function $Ψ\colon[0,\infty)\to\mathbb{R}$ should be used to describe finite elastic deformations. However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function $Ψ$ such that $W$ of the form (1) is Legendre-Hadamard elliptic. Similarly, we consider the related isochoric strain measure $\lVert\operatorname{dev}_n\log U\rVert^2$, where $\operatorname{dev}_n \log U$ is the deviatoric part of $\log U$. Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case $n=2$, we show that for $n\geq3$, no strictly monotone function $Ψ\colon[0,\infty)\to\mathbb{R}$ exists such that $F\mapsto Ψ(\lVert\operatorname{dev}_n\log U\rVert^2)$ is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form $F\mapsto Ψ(\lVert\operatorname{dev}_n\log U\rVert^2) + W_{\mathrm{vol}}(\det F)$ cannot be rank-one convex for any function $W_{\mathrm{vol}}\colon(0,\infty)\to\mathbb{R}$ if $Ψ$ is strictly monotone.

math.CA↗
arXiv2017

The modified indeterminate couple stress model: Why Yang et al.'s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless

We show that the reasoning in favor of a symmetric couple stress tensor in Yang et al.'s introduction of the modified couple stress theory contains a gap, but we present a reasonable physical hypothesis, implying that the couple stress tensor is traceless and may be symmetric anyway. To this aim, the origin of couple stress is discussed on the basis of certain properties of the total stress itself. In contrast to classical continuum mechanics, the balance of linear momentum and the balance of angular momentum are formulated at an infinitesimal cube considering the total stress as linear and quadratic approximation of a spatial Taylor series expansion.

math.AP↗
arXiv2017

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

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 $L_c\rightarrow 0$ (which corresponds to considering very large specimens of a microstructured metamaterial the meso- and micro-coefficients 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.

math-ph↗
arXiv2016

A panorama of dispersion curves for the weighted isotropic relaxed micromorphic model

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 Cartan-Lie decomposition of the tensors $P_{,t}$ and Curl $P$ in their dev, dev sym, skew and spheric part. It is shown that the split of the tensor $P_{,t}$ in the micro-inertia provide an independent control of the cut-offs of the optic benches. This is crucial for the future calibration of the relaxed micromorphic model on real band-gap metamaterials. Even if the physical interest of the introduction of the split of the tensor Curl $P$ is less evident than in the previous case, we discuss in detail which is its effect on the dispersion curves. Finally, we also provide a complete parametric study involving all the constitutive parameters of the introduced model, so giving rise to an exhaustive panorama of dispersion curves for the relaxed micromorphic model.

math-ph↗
arXiv2016

A review on wave propagation modeling in band-gap metamaterials via enriched continuum models

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 the microstructure 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.

math-ph↗
arXiv2016

Reflection and transmission of elastic waves at interfaces embedded in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model

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 different types of micro-macro connections which are intrinsically compatible with the governing bulk equations. To illustrate the interest of the many introduced jump conditions, we focus on the case of an interface between a classical Cauchy continuum on one side and a relaxed micromorphic one on the other side. As expected, we find a complete reflection in the frequency intervals for which band-gaps are known to occur in the relaxed micromorphic continuum and precise microstructure-related reflective patterns are identified. We repeat a similar study for analogous connections between a classical Cauchy continuum and a Mindlin's micromorphic one and we show that the reflective properties of the considered interfaces are drastically modified due to the fact that band-gaps are not allowed in standard Mindlin's micromorphic media. The present work opens the way towards the possibility of conceiving complex metastructures in which band-gap metamaterials and classical materials are coupled together to produce structures with completely new and unorthodox properties with respect to wave propagation, transmission and reflection. Last, but not least, indirect measurements of the material coefficients of the relaxed micromorphic model based upon real experiments of reflection and transmission in band-gap metamaterials are uncovered by the present work which makes them finally realizable in the short term.

math-ph↗
arXiv2016

Real wave propagation in the isotropic relaxed micromorphic model

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 linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, while it does in isotropic linear elasticity.

math-ph↗
arXiv2016

A new view on boundary conditions in the Grioli-Koiter-Mindlin-Toupin indeterminate couple stress model

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.

math-ph↗
arXiv2015

Loss of ellipticity for non-coaxial plastic deformations in additive logarithmic finite strain plasticity

In this paper we consider the additive logarithmic finite strain plasticity formulation from the view point of loss of ellipticity in elastic unloading. We prove that even if an elastic energy $F\mapsto W(F)=\hat{W}(\log U)$ defined in terms of logarithmic strain $\log U$, where $U=\sqrt{F^T\, F}$, is everywhere rank-one convex as a function of $F$, the new function $F\mapsto \widetilde{W}(F)=\hat{W}(\log U-\log U_p)$ need not remain rank-one convex at some given plastic stretch $U_p$ (viz. $E_p^{\log}:=\log U_p$). This is in complete contrast to multiplicative plasticity in which $F\mapsto W(F\, F_p^{-1})$ remains rank-one convex at every plastic distortion $F_p$ if $F\mapsto W(F)$ is rank-one convex. We show this disturbing feature with the help of a recently considered family of exponentiated Hencky energies.

math.CA↗
arXiv2015

An ellipticity domain for the distortional Hencky-logarithmic strain energy

We describe ellipticity domains for the isochoric elastic energy $ F\mapsto \|{\rm dev}_n\log U\|^2=\bigg\|\log \frac{\sqrt{F^TF}}{(\det F)^{1/n}}\bigg\|^2 =\frac{1}{4}\,\bigg\|\log \frac{C}{({\rm det} C)^{1/n}}\bigg\|^2 $ for $n=2,3$, where $C=F^TF$ for $F\in {\rm GL}^+(n)$. Here, ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the logarithmic strain tensor $\log U$. For $n=2$ we identify the maximal ellipticity domain, while for $n=3$ we show that the energy is Legendre-Hadamard elliptic in the set $\mathcal{E}_3\bigg(W_{_{\rm H}}^{\rm iso}, {\rm LH}, U, \frac{2}{3}\bigg)\,:=\,\bigg\{U\in{\rm PSym}(3) \;\Big|\, \|{\rm dev}_3\log U\|^2\leq \frac{2}{3}\bigg\}$, 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 $ W_{_{\rm H}}(F)=μ\,\|{\rm dev}_3\log U\|^2+ \fracκ{2}\,[{\rm tr} (\log U)]^2 $, $U=\sqrt{F^TF}$ with $μ>0$ and $κ>\frac{2}{3}\, μ$, previously obtained by Bruhns et al.

math.CA↗
arXiv2015

Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case

We show that in the two-dimensional case, every objective, isotropic and isochoric energy function which is rank-one convex on $\mathrm{GL}^+(2)$ is already polyconvex on $\mathrm{GL}^+(2)$. Thus we negatively answer Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasiconvexity. Our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.

math.AP↗
arXiv2015

On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of A.R. Hadjesfandiari and G.F. Dargush

In a series of papers which are either published [A.R. Hadjesfandiari and G.F. Dargush, Couple stress theory for solids, Int. J. Solids Struct. 48, 2496-2510, 2011; A.R. Hadjesfandiari and G.F. Dargush, Fundamental solutions for isotropic size-dependent couple stress elasticity, Int. J. Solids Struct. 50, 1253-1265, 2013] or available as preprints Hadjesfandiari and Dargush have reconsidered the linear indeterminate couple stress model. They are postulating a certain physically plausible split in the virtual work principle. Based on this postulate they claim that the second-order couple stress tensor must always be skew-symmetric. Since they use an incomplete set of boundary conditions in their virtual work principle their statement contains unrecoverable errors. This is shown by specifying their development to the isotropic case. However, their choice of constitutive parameters is mathematically possible and still yields a well-posed boundary value problem.

math-ph↗
arXiv2015

A variant of the linear isotropic indeterminate couple stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and complete traction boundary conditions

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 problem. Relations with other gradient elastic theories and the possibility to switch from a {4th order} (gradient elastic) problem to a 2nd order micromorphic model are also discussed with a view of obtaining symmetric force-stress tensors. It is shown that the indeterminate couple stress model can be written entirely with symmetric force-stress and symmetric couple-stress. The difference of the alternative models rests in specifying traction boundary conditions of either rotational type or strain type. If rotational type boundary conditions are used in the partial integration, the classical anti-symmetric nonlocal force stress tensor formulation is obtained. Otherwise, the difference in both formulations is only a divergence--free second order stress field such that the field equations are the same, but the traction boundary conditions are different. For these results we employ a novel integrability condition, connecting the infinitesimal continuum rotation and the infinitesimal continuum strain. Moreover, we provide the complete, consistent traction boundary conditions for both models.

math-ph↗
arXiv2015

Correct traction boundary conditions in the indeterminate couple stress model

In this paper we consider the Grioli-Koiter-Mindlin-Toupin indeterminatecouple stress model. The main aim is to show that the traction boundary conditions were not yet completely deduced. As it turns out, and to our own surprise, restricting the boundary condition framework from the strain gradient models to the couple stress model does not reduce to Mindlin's set of accepted boundary conditions. We present therefore, for the first time the complete, consistent set of traction boundary conditions.

math-ph↗
arXiv2015

The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity

In this paper we improve the result about the polyconvexity of the energies from the family of isotropic volumetric-isochoric decoupled strain exponentiated Hencky energies defined in the first part of this series, i.e. $$ W_{_{\rm eH}}(F)= \left\{\begin{array}{lll} \fracμ{k}\,e^{k\,\|{\rm dev}_n\log U\|^2}+\fracκ{2\,\widehat{k}}\,e^{\widehat{k}\,[(\log {\rm det} U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0\,, \end{array}\right. $$ where $F=\nabla φ$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U}$ is the deviatoric part of the strain tensor $\log U$. The main result in this paper is that in plane elastostatics, i.e. for $n=2$, the energies of this family are polyconvex for $k\geq \frac{1}{4}$, $\widehat{k}\geq \frac{1}{8}$, extending a previous result which proves polyconvexity for $k\geq \frac{1}{3}$, $\widehat{k}\geq \frac{1}{8}$. This leads immediately to an extension of the existence result.

math.CA↗