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Patrizio Neff

Publications and source records attributed to Patrizio Neff.

At least 73 records · Page 4Linked to original sources
arXiv2022

Modeling a labyrinthine acoustic metamaterial through an inertia-augmented relaxed micromorphic approach

We present an inertia-augmented relaxed micromorphic model that enriches the relaxed micromorphic model previously introduced by the authors via a term $\text{Curl}\dot{P}$ in the kinetic energy density. This enriched model allows us to obtain a good overall fitting of the dispersion curves while introducing the new possibility of describing modes with negative group velocity that are known to trigger negative refraction effects. The inertia-augmented model also allows for more freedom on the values of the asymptotes corresponding to the cut-offs. In the previous version of the relaxed micromorphic model, the asymptote of one curve (pressure or shear) is always bounded by the cut-off of the following curve of the same type. This constraint does not hold anymore in the enhanced version of the model. While the obtained curves' fitting is of good quality overall, a perfect quantitative agreement must still be reached for very small wavelengths that are close to the size of the unit cell.

physics.app-ph↗
arXiv2022

On H1, H(curl) and H(sym Curl) finite elements for matrix-valued Curl problems

In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of $[\mathit{H}^1]^{3\times 3}$, $[\mathit{H}(\mathrm{curl})]^3$ and $\mathit{H}(\mathrm{sym}\mathrm{Curl})$ for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely $[\mathit{H}(\mathrm{curl})]^3$ in place of the classical $[\mathit{H}^1]^{3\times 3}$. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then $\mathit{H}(\mathrm{sym}\mathrm{Curl})$. The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory.

math.NA↗
arXiv2022

Primal and mixed finite element formulations for the relaxed micromorphic model

The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are $[\mathit{H}^1]^3$ for the displacement and $[\mathit{H}(\mathrm{curl})]^3$ for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart-Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length $L_\mathrm{c}$ and the correlation to the classical Cauchy continuum theory.

math.NA↗
arXiv2022

Existence and uniqueness of Rayleigh waves in isotropic elastic Cosserat materials and algorithmic aspects

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, a new algorithm which determines the amplitudes and the wave speed in the theory of isotropic elastic Cosserat materials is described. Moreover, the method allows to prove the existence and uniqueness of a subsonic solution of the secular equation, a problem which remains unsolved in almost all generalised linear theories of elastic materials. Since the results are suitable to be used for numerical implementations, we propose two numerical algorithms which are viable for any elastic material. Explicit numerical calculations are made for alumunium-epoxy in the context of the Cosserat model. Since the novel form of the secular equation for isotropic elastic material has not been explicitly derived elsewhere, we establish it in this paper, too.

math.AP↗
arXiv2022

Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture

Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function $W$. We will demonstrate these methods using the planar isotropic rank-one convex function \[ W_{\rm magic}^+(F)=\frac{λ_{\rm max}}{λ_{\rm min}}-\log\frac{λ_{\rm max}}{λ_{\rm min}}+\log\det F=\frac{λ_{\rm max}}{λ_{\rm min}}+2\logλ_{\rm min}\,, \] where $λ_{\rm max}\geqλ_{\rm min}$ are the singular values of $F$, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies $W:\operatorname{GL}^+(2)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split of the form \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F) \] with a concave volumetric part. This example is therefore of particular interest with regard to Morrey's open question whether or not rank-one convexity implies quasiconvexity in the planar case.

math.AP↗
arXiv2021

The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

We consider the classical Mindlin-Eringen linear micromorphic model with a new strictly weaker set of displacement boundary conditions. The new consistent coupling condition aims at minimizing spurious influences from arbitrary boundary prescription for the additional microdistortion field P. In effect, P is now only required to match the tangential derivative of the classical displacement u which is known at the Dirichlet-part of the boundary. We derive the full boundary condition, in adding the missing Neumann condition on the Dirichlet-part. We show existence and uniqueness of the static problem for this weaker boundary condition. These results are based on new coercive inequalities for incompatible tensor fields with prescribed tangential part. Finally, we show that compared to classical Dirichlet conditions on u and P, the new boundary condition modifies the interpretation of the constitutive parameters.

math.AP↗
arXiv2021

Lagrange and $H(\operatorname{curl},{\cal B})$ based Finite Element formulations for the relaxed micromorphic model

Modeling the unusual mechanical properties of metamaterials is a challenging topic for the mechanics community and enriched continuum theories are promising computational tools for such materials. The so-called relaxed micromorphic model has shown many advantages in this field. In this contribution, we present the significant aspects related to the relaxed micromorphic model realization with the finite element method. The variational problem is derived and different FEM-formulations for the two-dimensional case are presented. These are a nodal standard formulation $H^1({\cal B}) \times H^1({\cal B})$ and a nodal-edge formulation $H^1({\cal B}) \times H(\operatorname{curl}, {\cal B})$, where the latter employs the Nédélec space. However, the implementation of higher-order Nédélec elements is not trivial and requires some technicalities which are demonstrated. We discuss the convergence behavior of Lagrange-type and tangential-conforming finite element discretizations. Moreover, we analyze the characteristic length effect on the different components of the model and reveal how the size-effect property is captured via this characteristic length.

math.NA↗
arXiv2021

Polyconvex anisotropic hyperelasticity with neural networks

In the present work, two machine learning based constitutive models for finite deformations are proposed. Using input convex neural networks, the models are hyperelastic, anisotropic and fulfill the polyconvexity condition, which implies ellipticity and thus ensures material stability. The first constitutive model is based on a set of polyconvex, anisotropic and objective invariants. The second approach is formulated in terms of the deformation gradient, its cofactor and determinant, uses group symmetrization to fulfill the material symmetry condition, and data augmentation to fulfill objectivity approximately. The extension of the dataset for the data augmentation approach is based on mechanical considerations and does not require additional experimental or simulation data. The models are calibrated with highly challenging simulation data of cubic lattice metamaterials, including finite deformations and lattice instabilities. A moderate amount of calibration data is used, based on deformations which are commonly applied in experimental investigations. While the invariant-based model shows drawbacks for several deformation modes, the model based on the deformation gradient alone is able to reproduce and predict the effective material behavior very well and exhibits excellent generalization capabilities. In addition, the models are calibrated with transversely isotropic data, generated with an analytical polyconvex potential. For this case, both models show excellent results, demonstrating the straightforward applicability of the polyconvex neural network constitutive models to other symmetry groups.

cond-mat.mtrl-sci↗
arXiv2021

Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach

In this paper, a coherent boundary value problem to model metamaterials' behavior based on the relaxed micromorphic model is established. This boundary value problem includes well-posed boundary conditions, thus disclosing the possibility of exploring the scattering patterns of finite-size metamaterials' specimens. Thanks to the simplified model's structure (few frequency- and angle-independent parameters), we are able to unveil the scattering metamaterial's response for a wide range of frequencies and angles of propagation of the incident wave. These results are an important stepping stone towards the conception of more complex large-scale meta-structures that can control elastic waves and recover energy.

physics.app-ph↗
arXiv2021

$L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

For $1 0$ such that \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev} \operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] holds for all tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$, i.e., for all $P\in W^{1,\,p}(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$ with vanishing tangential trace $ P\times ν=0 $ on $ \partialΩ$ where $ν$ denotes the outward unit normal vector field to $\partialΩ$ and $\operatorname{dev} P := P -\frac13 \operatorname{tr}(P)\,\mathbb{1}_3$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}+\|{\operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev}\operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] for tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $Γ\subseteq \partialΩ$ of the boundary.

math.AP↗
arXiv2021

On in-plane drill rotations for Cosserat surfaces

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a $C^2$-smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary, is the identity mapping.

math.DG↗
arXiv2021

Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

Let $Ω\subset \mathbb{R}^3$ be an open and bounded set with Lipschitz boundary and outward unit normal $ν$. For $1 1 \quad \text{if $p = \frac32$.}$$ Specifically, there exists a constant $c=c(p,Ω,r)>0$ such that the inequality \[ \|P \|_{L^p}\leq c\,\left(\|\operatorname{sym} P \|_{L^p} + \|\operatorname{dev}\operatorname{sym} \operatorname{Curl} P \|_{L^{r}}\right) \] holds for all tensor fields $P\in W^{1,\,p, \, r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl})$. Here, $\operatorname{dev} X := X -\frac13 \operatorname{tr}(X)\,\mathbb{1}$ denotes the deviatoric (trace-free) part of a $3 \times 3$ matrix $X$ and the boundary condition is understood in a suitable weak sense.

math.AP↗
arXiv2021

Morrey's conjecture for the planar volumetric-isochoric split. Part I: least convex energy functions

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 $W:\operatorname{GL}^+(n)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split, i.e. \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F)\,, \] which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of one-dimensional coupled differential inequalities. We identify a number of \enquote{least} rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function \[ W_{\rm magic}^+(F)=\frac{λ_{\rm max}}{λ_{\rm min}}-\log\frac{λ_{\rm max}}{λ_{\rm min}}+\log\det F=\frac{λ_{\rm max}}{λ_{\rm min}}-2\logλ_{\rm min} \] is quasiconvex. In addition, we demonstrate that under affine boundary conditions, $W_{\rm magic}^+(F)$ allows for non-trivial inhomogeneous deformations with the same energy level as the homogeneous solution, and show a surprising connection to the work of Burkholder and Iwaniec in the field of complex analysis.

math.AP↗
arXiv2021

Analytical solutions of the simple shear problem for certain types of micromorphic continuum models -- including full derivations

To draw conclusions as regards the stability and modelling limits of the investigated continuum, we consider a family of infinitesimal isotropic generalized continuum models (Mindlin-Eringen micromorphic, relaxed micromorphic continuum, Cosserat, micropolar, microstretch, microstrain, microvoid, indeterminate couple stress, second gradient elasticity, etc.) and solve analytically the simple shear problem of an infinite stripe. A qualitative measure characterizing the different generalized continuum moduli is given by the shear stiffness $μ^{*}$. This stiffness is in general length-scale dependent. Interesting limit cases are highlighted, which allow to interpret some of the appearing material parameter of the investigated continua.

physics.class-ph↗
arXiv2021

Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

For $1 0$ such that \begin{equation*} \| P\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left( \|\operatorname{sym} P\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \| \operatorname{Curl}P \|_{L^p(Ω, \mathbb{R}^{3\times3})}\right)\end{equation*} holds for all tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$, i.e., for all $P\in W^{1,\,p}(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$ with vanishing tangential trace $ P\times ν=0 $ on $ \partialΩ$ where $ν$ denotes the outward unit normal vector field to $\partialΩ$. For compatible $P=D u$ this recovers an $L^p$-version of the classical Korn's first inequality $$ \|D u \|_{L^p(Ω,\mathbb{R}^{3\times 3})} \le c\, \|\operatorname{sym}D u\|_{L^p(Ω,\mathbb{R}^{3\times3})} \quad \text{with }D u \times ν= 0 \quad \text{on $\partial Ω$}, $$ and for skew-symmetric $P=A\in\mathfrak{so}(3)$ an $L^p$-version of the Poincaré inequality $$ \|A\|_{L^p(Ω,\mathfrak{so}(3))}\le c\, \|\operatorname{Curl} A\|_{L^p(Ω,\mathbb{R}^{3\times3})} \quad \text{with } A \times ν= 0 \ \Leftrightarrow \ A=0 \quad \text{on $\partial Ω$}. $$

math.AP↗
arXiv2021

A note on local higher regularity in the dynamic linear relaxed micromorphic model

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 micro-distortion. Naturally solutions are found in ${\rm H}^1$ for the displacement $u$ and ${\rm H}({\rm Curl})$ for the microdistortion $P$. Using energy estimates for difference quotients, we improve this regularity. We show ${\rm H}^1_{\rm loc}$-regularity for the displacement field, ${\rm H}^1_{\rm loc}$-regularity for the micro-distortion tensor $P$ and that ${\rm Curl}\,P$ is ${\rm H}^1$-regular if the data is sufficiently smooth.

math.AP↗