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Patrizio Neff

Publications and source records attributed to Patrizio Neff.

At least 127 records · Page 7Linked to original sources
arXiv2017

The exponentiated Hencky energy: Anisotropic extension and case studies

In this paper we propose an anisotropic extension of the isotropic exponentiated Hencky energy, based on logarithmic strain invariants. Unlike other elastic formulations, the isotropic exponentiated Hencky elastic energy has been derived solely on differential geometric grounds, involving the geodesic distance of the deformation gradient F to the group of rotations. We formally extend this approach towards anisotropy by defining additional anisotropic logarithmic strain invariants with the help of suitable structural tensors and consider our findings for selected case studies.

cs.CE↗
arXiv2017

Geometrically nonlinear Cosserat elasticity in the plane: applications to chirality

Modelling two-dimensional chiral materials is a challenging problem in continuum mechanics because three-dimensional theories reduced to isotropic two-dimensional problems become non-chiral. Various approaches have been suggested to overcome this problem. We propose a new approach to this problem by formulating an intrinsically two-dimensional model which does not require references to a higher dimensional one. We are able to model planar chiral materials starting from a geometrically non-linear Cosserat type elasticity theory. Our results are in agreement with previously derived equations of motion but can contain additional terms due to our non-linear approach. Plane wave solutions are briefly discussed within this model.

math-ph↗
arXiv2017

Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: new perspectives towards meta-structural design

In the present paper, the material parameters of the isotropic relaxed micromorphic model derived for a specific metamaterial in a previous contribution are used to model its transmission properties. Specifically, the reflection and transmission coefficients at an interface between a homogeneous solid and the chosen metamaterial are analyzed by using both the relaxed micromorphic model and a direct FEM implementation of the detailed microstructure. The obtained results show an excellent agreement between the transmission spectra derived via our enriched continuum model and those issued by the direct FEM simulation. Such excellent agreement validates the indirect measure of the material parameters and opens the way towards an efficient meta-structural design.

cond-mat.mtrl-sci↗
arXiv2017

The relaxed-polar mechanism of locally optimal Cosserat rotations for an idealized nanoindentation and comparison with 3D-EBSD experiments

The rotation ${\rm polar}(F) \in {\rm SO}(3)$ arises as the unique orthogonal factor of the right polar decomposition $F = {\rm polar}(F) \cdot U$ of a given invertible matrix $F \in {\rm GL}^+(3)$. In the context of nonlinear elasticity Grioli (1940) discovered a geometric variational characterization of ${\rm polar}(F)$ as a unique energy-minimizing rotation. In preceding works, we have analyzed a generalization of Grioli's variational approach with weights (material parameters) $μ> 0$ and $μ_c \geq 0$ (Grioli: $μ= μ_c$). The energy subject to minimization coincides with the Cosserat shear-stretch contribution arising in any geometrically nonlinear, isotropic and quadratic Cosserat continuum model formulated in the deformation gradient field $F := \nablaφ: Ω\to {\rm GL}^+(3)$ and the microrotation field $R: Ω\to {\rm SO}(3)$. The corresponding set of non-classical energy-minimizing rotations $$ {\rm rpolar}^\pm_{μ,μ_c}(F) := \substack{{\rm argmin}\\ R\,\in\,{\rm SO(3)}} \Big\{ W_{μ, μ_c}(R\,;F) := μ\, || {\rm sym}(R^TF - 1)||^2 + μ_c\, ||{\rm skew}(R^TF - 1)||^2 \Big\} $$ represents a new relaxed-polar mechanism. Our goal is to motivate this mechanism by presenting it in a relevant setting. To this end, we explicitly construct a deformation mapping $φ_{\rm nano}$ which models an idealized nanoindentation and compare the corresponding optimal rotation patterns ${\rm rpolar}^\pm_{1,0}(F_{\rm nano})$ with experimentally obtained 3D-EBSD measurements of the disorientation angle of lattice rotations due to a nanoindentation in solid copper. We observe that the non-classical relaxed-polar mechanism can produce interesting counter-rotations. A possible link between Cosserat theory and finite multiplicative plasticity theory on small scales is also explored.

math-ph↗
arXiv2017

A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by \[ W_\mathrm{eH}(\boldsymbol{F}) = \dfracμ{k}\, e^{\displaystyle k \left\lVert\mbox{dev}_n \log\boldsymbol{U}\right\rVert^2} + \dfracκ{2 \hat{k}}\, e^{\displaystyle \hat{k} [\mbox{tr} (\log\boldsymbol{U})]^2 }\,, \] where $μ>0$ is the (infinitesimal) shear modulus, $κ>0$ is the (infinitesimal) bulk modulus, $k$ and $\hat{k}$ are additional dimensionless material parameters, $\boldsymbol{U}=\sqrt{\boldsymbol{F}^T\boldsymbol{F}}$ and $\boldsymbol{V}=\sqrt{\boldsymbol{F}\boldsymbol{F}^T}$ are the right and left stretch tensor corresponding to the deformation gradient $\boldsymbol{F}$, $\log$ denotes the principal matrix logarithm on the set of positive definite symmetric matrices, $\mbox{dev}_n \boldsymbol{X} = \boldsymbol{X}-\frac{\mbox{tr} \boldsymbol{X}}{n}\boldsymbol{1}$ and $\lVert \boldsymbol{X} \rVert = \sqrt{\mbox{tr}\boldsymbol{X}^T\boldsymbol{X}}$ are the deviatoric part and the Frobenius matrix norm of an $n\times n$-matrix $\boldsymbol{X}$, respectively, and $\mbox{tr}$ denotes the trace operator. To do so, the equivalent different forms of the constitutive equation are recast in terms of the principal logarithmic stretches by use of the spectral decomposition together with the undergoing properties. We show the capability of our approach with a number of relevant examples, including the challenging "eversion of elastic tubes" problem.

math.NA↗
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

Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices

We consider the problem to determine the optimal rotations $R \in {\rm SO}(n)$ which minimize $$W: {\rm SO}(n) \to \mathbb{R}^+_0,\quad W(R\,;D) := ||{\rm sym}(RD - 1)||^2$$ for a given diagonal matrix $D := {\rm diag}(d_1, ..., d_n) \in \mathbb{R}^{n \times n}$. The function $W$ subject to minimization is the reduced form of the Cosserat shear-stretch energy, which, in its general form, is a contribution in any geometrically nonlinear, isotropic and quadratic Cosserat micropolar (extended) continuum model. We characterize the critical points of the energy $W(R\,;D)$, determine the global minimizers and the global minimum. This proves the correctness of previously obtained formulae for the optimal Cosserat rotations in dimensions two and three. The key to the proof is a characterization of the entire set of (possibly non-symmetric) real matrix square roots of (possibly non-positive definite) real symmetric matrices which does not seem to be known in the literature.

math-ph↗
arXiv2017

Grioli's Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations

Let $F \in {\rm GL}^+(3)$ and consider the right polar decomposition $F = R_p(F)\cdot U$ into an orthogonal factor $R_p(F) \in {\rm SO}(3)$ and a symmetric, positive definite factor $U(F) = \sqrt{F^TF} \in {\rm Psym}(3)$. In 1940 Giuseppe Grioli proved that $$ {\rm argmin}_{R \in {\rm SO}(3)} ||R^TF - 1{||}^2 \quad=\quad \{\,R_p(F)\,\} \quad=\quad {\rm argmin}_{R \in {\rm SO}(3)} ||F - R{||}^2\;. $$ This variational characterization of the orthogonal factor $R_p(F) \in {\rm SO}(n)$ holds in any dimension $n \geq 2$ (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations $$ {\rm rpolar}_{μ,μ_c}(F) \;:=\, {\rm argmin}_{R \in {\rm SO}(n)} \left\lbrace μ\, ||{\rm sym}(R^TF - 1){||}^2 \;+\, μ_c\, ||{\rm skew}(R^TF - 1){||}^2 \right\rbrace $$ for given weights $μ> 0$ and $μ_c \geq 0$. We identify a classical parameter range $μ_c \geq μ> 0$ for which Grioli's Theorem is recovered and a non-classical parameter range $μ> μ_c \geq 0$ giving rise to a new type of globally energy-minimizing rotations which can substantially deviate from $R_p(F)$. In mechanics, the weighted energy subject to minimization appears as the shear-stretch contribution in any geometrically nonlinear, quadratic, and isotropic Cosserat theory.

math-ph↗
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

Hyperelastic bodies under homogeneous Cauchy stress induced by three-dimensional non-homogeneous deformations

In isotropic finite elasticity, unlike in the linear elastic theory, a homogeneous Cauchy stress may be induced by non-homogeneous strains. To illustrate this, we identify compatible non-homogeneous three-dimensional deformations producing a homogeneous Cauchy stress on a cuboid geometry, and provide an example of an isotropic hyperelastic material, which is not rank-one convex, and for which the homogeneous stress and the associated non-homogeneous strains on a domain similar to those analysed are given explicitly.

math-ph↗
arXiv2016

Geometry of logarithmic strain measures in solid mechanics

We consider the two logarithmic strain measures\[ω_{\rm iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and }\quad ω_{\rm vol}=|\mathrm{tr}(\log U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,\]which are isotropic invariants of the Hencky strain tensor $\log U$, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group $\mathrm{GL}(n)$. Here, $F$ is the deformation gradient, $U=\sqrt{F^TF}$ is the right Biot-stretch tensor, $\log$ denotes the principal matrix logarithm, $\|.\|$ is the Frobenius matrix norm, $\mathrm{tr}$ is the trace operator and $\mathrm{dev}_n X$ is the $n$-dimensional deviator of $X\in\mathbb{R}^{n\times n}$. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor $\varepsilon=\mathrm{sym}\nabla u$, which is the symmetric part of the displacement gradient $\nabla u$, and reveals a close geometric relation between the classical quadratic isotropic energy potential \[μ\,\|\mathrm{dev}_n\mathrm{sym}\nabla u\|^2+\fracκ{2}\,[\mathrm{tr}(\mathrm{sym}\nabla u)]^2=μ\,\|\mathrm{dev}_n\varepsilon\|^2+\fracκ{2}\,[\mathrm{tr}(\varepsilon)]^2\]in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy\[μ\,\|\mathrm{dev}_n\log U\|^2+\fracκ{2}\,[\mathrm{tr}(\log U)]^2=μ\,ω_{\rm iso}^2+\frac\kappa2\,ω_{\rm vol}^2\,,\]where $μ$ is the shear modulus and $κ$ denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor $R$, where $F=R\,U$ is the polar decomposition of $F$. We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity.

math.DG↗
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

Modeling real phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia

In this paper the relaxed micromorphic continuum model with weighted free and gradient micro-inertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the band-gap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model, which are independent of frequency, are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic structure as the basic building block and which preserve its ability of "stopping" elastic wave propagation at the scale of the structure.

physics.class-ph↗
arXiv2016

Existence result for a dislocation based model of single crystal gradient plasticity with isotropic or linear kinematic hardening

We consider a dislocation-based rate-independent model of single crystal gradient plasticity with isotropic or linear kinematic hardening. The model is weakly formulated through the so-called primal form of the flow rule as a variational inequality for which a result of existence and uniqueness is obtained using the functional analytical framework developed by Han-Reddy.

math.AP↗
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↗