Dynamic problems for metamaterials: Review of existing models and ideas for further research

In this paper, the relaxed micromorphic model proposed in Ghiba et al. (Math Mech Solids, 2013), Neff et al. (Contin Mech Thermodyn, 2013) has been used to study wave propagation in unbounded continua with microstructure. By studying dispersion relations for the considered relaxed medium, we are able to disclose precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersion relations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium. We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called Cosserat couple modulus μ
 c
 , which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parameter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around a given threshold drastically changes the observed response of the material with respect to wave propagation. We finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosserat and second gradient ones. The potential fields of application of the proposed relaxed model are manifold, above all for what concerns the conception of new engineering materials to be used for vibration control and stealth technology.

Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps

We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$\begin{aligned} F\mapsto W_{\mathrm{eH}}(F):=\widehat{W}_{\mathrm{eH}}(U):=\left \{ \begin{array}{l@{\quad}l} \frac{\mu}{k} e^{k\|\operatorname {dev}_n\log{U}\|^2}+\frac{\kappa}{{2 {\widehat {k}}}} e^{\widehat{k}[\operatorname {tr}(\log U)]^2}&\text{if}\ \det F>0,\\ +\infty&\text{if}\ \det F\leq0, \end{array} \right . \end{aligned}$$ based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the infinitesimal shear modulus, $\kappa=\frac{2\mu+3\lambda}{3}>0$
 is the infinitesimal bulk modulus with λ the first Lame constant, $k,\widehat{k}$
 are additional dimensionless material parameters, F=∇φ is the gradient of deformation, $U=\sqrt{F^{T} F}$
 is the right stretch tensor and is the n-dimensional deviatoric part of the strain tensor logU. For small elastic strains, W
 eH approximates the classical quadratic Hencky strain energy $$\begin{aligned} F\mapsto W_{\mathrm{H}}(F):=\widehat{W}_{\mathrm{H}}(U)&:={\mu}\| \operatorname {dev}_n\log U\|^2+\frac{\kappa}{2}\bigl[\operatorname{tr}( \log U)\bigr]^2, \end{aligned}$$ which is not everywhere rank-one convex. In plane elastostatics, i.e., n=2, we prove the everywhere rank-one convexity of the proposed family W
 eH, for $k\geq\frac{1}{4}$
 and $\widehat{k}\geq\frac{1}{8}$
 . Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for n=2,3 and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family W
 eH is not preserved in dimension n=3 and that the energies $$\begin{aligned} F\mapsto\frac{\mu}{k}e^{k\|\log U\|^2},\qquad F\mapsto\frac{\mu }{k}e^{\frac{k}{\mu} \bigl(\mu\|\operatorname {dev}_n\log U\|^2+\frac{\kappa}{2}[\operatorname {tr}(\log U)]^2 \bigr)}, \quad F \in\mathrm{GL}^+(n), n\in \mathbb {N}, n\geq2 \end{aligned}$$ are also not rank-one convex.

https://www.math.uaic.ro/~ghiba/lucrari/NeffGhibaLankeit-Arxiv-Hencky-Part-I.pdf

The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity

In this paper the relaxed micromorphic model proposed in [Patrizio Neff, Ionel-Dumitrel Ghiba, Angela Madeo, Luca Placidi, Giuseppe Rosi. A unifying perspective: the relaxed linear micromorphic continuum, submitted, 2013, arXiv:1308.3219; and Ionel-Dumitrel Ghiba, Patrizio Neff, Angela Madeo, Luca Placidi, Giuseppe Rosi. The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics, submitted, 2013, arXiv:1308.3762] has been used to study wave propagation in unbounded continua with microstructure. By studying dispersion relations for the considered relaxed medium, we are able to disclose precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersion relations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium. We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called Cosserat couple modulus $\mu_{c}$, which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parameter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around a given threshold drastically changes the observed response of the material with respect to wave propagation. We finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosserat and second gradient ones. The potential fields of application of the proposed relaxed model are manifold, above all for what concerns the conception of new engineering materials to be used for vibration control and stealth technology.

Wave propagation in relaxed micromorphic continua: modelling metamaterials with frequency band-gaps

In this paper 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 2 R 3 and the non-symmetric micro-distortion density tensor P 2 R 3×3 . In this relaxed theory a symmetric forcestress tensor arises despite the presence of microstructure and the curvature contribution depends solely on the micro-dislocation tensor CurlP. 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.

https://www.math.uaic.ro/~ghiba/lucrari/Ghiba-QJMAM-Revised.pdf

The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations

We describe a principle of bounded stiffness and show that bounded stiffness in torsion and bending implies a reduction of the curvature energy in linear isotropic Cosserat models leading to the so-called conformal curvature case $${\mu\,\frac{L_c^2}{2}\,\|\,{{\rm dev \, sym}\,\nabla{\rm axl}\,\overline{A}}^2\|}$$
 where $${\overline{A} \in \mathfrak{so}(3)}$$
 is the Cosserat microrotation. Imposing bounded stiffness greatly facilitates the Cosserat parameter identification and allows a well-posed, stable determination of the one remaining length scale parameter L
 c
 and the Cosserat couple modulus μ
 c
 .

Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature

We show that symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of finite isotropic hyperelasticity with exact rotational degrees of freedom. This is contrary to claims in the literature which are valid, however, in the linear isotropic case.

/pdf/symmetric-cauchy-stresses-do-not-imply-symmetric-biot-29s73xassa.pdf

Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom

The unitary polar factor $Q=U_p$ in the polar decomposition of $Z=U_p \, H$ is the minimizer over unitary matrices $Q$ for both $\|{\rm Log}(Q^* Z)\|^2$ and its Hermitian part $\|{{\rm sym}{_{_*}}\!}({\rm Log}(Q^* Z))\|^2$ over both $\mathbb{R}$ and $\mathbb{C}$ for any given invertible matrix $Z\in\mathbb{C}^{n\times n}$ and any matrix logarithm Log, not necessarily the principal logarithm log. We prove this for the spectral matrix norm for any $n$ and for the Frobenius matrix norm for $n\leq 3$. The result shows that the unitary polar factor is the nearest orthogonal matrix to $Z$ not only in the normwise sense but also in a geodesic distance. The derivation is based on Bhatia's generalization of Bernstein's trace inequality for the matrix exponential and a new sum of squared logarithms inequality. Our result generalizes the fact for scalars that for any complex logarithm and for all $z\in\mathbb{C}{\setminus}\{0\} \min_{\vartheta\in(-\pi,\pi]}{|{\rm Log}_{\mathbb{C}}(e^{-i\vartheta} z)|}^2={|{\rm log}|...

A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms

The unitary polar factor Q = Up in the polar decomposition of Z = Up H is the minimizer over unitary matrices Q for both kLog(QZ)k 2 and its Hermitian part ksym * (Log(QZ))k 2 over both R and C for any given invertible matrix Z ∈ C n×n and any matrix logarithm Log, not necessarily the principal logarithm log. We prove this for the spectral matrix norm for any n and for the Frobenius matrix norm in two and three dimensions. The result shows that the unitary polar factor is the nearest orthogonal matrix to Z not only in the normwise sense, but also in a geodesic distance. The derivation is based on Bhatia's generalization of Bernstein's trace inequality for the matrix exponential and a new sum of squared logarithms inequality. Our result generalizes the fact for scalars that for any complex logarithm and for all z ∈ C \ {0} min ϑ2( π,π) |Log C (e iϑ z)| 2 = |log|z|| 2 , min ϑ2( π,π) |ReLog C (e iϑ z)| 2 = |log|z|| 2 .

A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm

We show that symmetric Cauchy stresses do not imply symmetric Biot strains in relaxed weak formulations of finite isotropic hyperelasticity with exact rotational degrees of freedom. This is contrary to claims in the literature. It is noted that in the linear isotropic case with skew symmetric infinitesimal rotational degrees of freedom a similar discrepancy does not arise. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Andreas Fischle

Papers

Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature

Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom

A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms

A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm

Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom.