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We consider the rigorously derived thin shell membrane Γ-limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation m:ω⊂R2→R3 and the orthogonal microrotation tensor field R:ω⊂R2→SO(3). The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet-type energy term ∣DR∣2. We use Rivière’s regularity techniques of harmonic-map-type systems for our system which couples harmonic maps to SO(3) with a linear equation for m. The additional coupling term in the harmonic map equation is of critical integrability and can only be handled because of its special structure.

For $1< p<\\infty$ we prove an $L^{p}$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\\operatorname {Curl}; \\Omega ,\\mathbb {R}^{3\\times 3})$. More precisely, let $\\Omega \\subset \\mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \\[ \\lVert{ P }\\rVert_{L^{p}(\\Omega,\\mathbb{R}^{3\\times 3})}\\leq c\\,\\left(\\lVert{\\operatorname{dev} \\operatorname{sym} P }\\rVert_{L^{p}(\\Omega,\\mathbb{R}^{3\\times 3})} + \\lVert{ \\operatorname{dev} \\operatorname{Curl} P }\\rVert_{L^{p}(\\Omega,\\mathbb{R}^{3\\times 3})}\\right) \\] holds for all tensor fields $P\\in W^{1,p}_0(\\operatorname {Curl}; \\Omega ,\\mathbb {R}^{3\\times 3})$, i.e., for all $P\\in W^{1,p} (\\operatorname {Curl}; \\Omega ,\\mathbb {R}^{3\\times 3})$ with vanishing tangential trace $P\\times \\nu =0$ on $\\partial \\Omega$ where $\\nu$ denotes the outward unit normal vector field to $\\partial \\Omega$ and $\\operatorname {dev} P : = P -\\frac 13 \\operatorname {tr}(P) {\\cdot } {\\mathbb {1}}$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence \\begin{align*} &\\lVert{ P }\\rVert_{L^{p}(\\Omega,\\mathbb{R}^{3\\times 3})}+\\lVert{ \\operatorname{Curl} P }\\rVert_{L^{p}(\\Omega,\\mathbb{R}^{3\\times 3})}\\\ &\\quad\\leq c\\,\\left(\\lVert{P}\\rVert_{L^{p}(\\Omega,\\mathbb{R}^{3\\times 3})} + \\lVert{ \\operatorname{dev} \\operatorname{Curl} P }\\rVert_{L^{p}(\\Omega,\\mathbb{R}^{3\\times 3})}\\right) \\end{align*} for tensor fields $P\\in W^{1,p}(\\operatorname {Curl}; \\Omega ,\\mathbb {R}^{3\\times 3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\\Gamma \\subseteq \\partial \\Omega$ of the boundary.

For$1< p<\\infty$we prove an$L^{p}$-version of the generalized trace-free Korn inequality for incompatible tensor fields$P$in$W^{1,p}_0(\\operatorname {Curl}; \\Omega ,\\mathbb {R}^{3\\times 3})$. More precisely, let$\\Omega \\subset \\mathbb {R}^{3}$be a bounded Lipschitz domain. Then there exists a constant$c>0$such that \\[ P _L^p( R^3 3) c\\,(dev sym P _L^p( R^3 3) + dev Curl P _L^p( R^3 3)) \\] holds for all tensor fields$P\\in W^{1,p}_0(\\operatorname {Curl}; \\Omega ,\\mathbb {R}^{3\\times 3})$, i.e., for all$P\\in W^{1,p} (\\operatorname {Curl}; \\Omega ,\\mathbb {R}^{3\\times 3})$with vanishing tangential trace$P\\times \\nu =0$on$\\partial \\Omega$where$\\nu$denotes the outward unit normal vector field to$\\partial \\Omega$and$\\operatorname {dev} P : = P -\\frac 13 \\operatorname {tr}(P) {\\cdot } {\\mathbb {1}}$denotes the deviatoric (trace-free) part of$P$. We also show the norm equivalence align* & P _L^p( R^3 3)+ Curl P _L^p( R^3 3)\\\ & c\\,(P_L^p( R^3 3) + dev Curl P _L^p( R^3 3)) align* for tensor fields$P\\in W^{1,p}(\\operatorname {Curl}; \\Omega ,\\mathbb {R}^{3\\times 3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset$\\Gamma \\subseteq \\partial \\Omega$of the boundary.

We characterise all linear maps A : R n × n → R n × n such that, for 1 ≤ p < n , P L p ∗ ( R n ) ≤ c ( A [ P ] L p ∗ ( R n ) + Curl P L p ( R n ) ) holds for all compactly supported P ∈ C c ∞ ( R n ; R n × n ) , where Curl P displays the matrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different combinations between the ellipticities of A , the integrability p and the underlying space dimensions n , especially requiring a finer analysis in the two-dimensional situation.

In this paper we model the size-effects of metamaterial beams under bending with the aid of the relaxed micromorphic continuum. We analyze first the size-dependent bending stiffness of heterogeneous fully discretized metamaterial beams subjected to pure bending loads. Two equivalent loading schemes are introduced which lead to a constant moment along the beam length with no shear force. The relaxed micromorphic model is employed then to retrieve the size-effects. We present a procedure for the determination of the material parameters of the relaxed micromorphic model based on the fact that the model operates between two well-defined scales. These scales are given by linear elasticity with micro and macro elasticity tensors which bound the relaxed micromorphic continuum from above and below, respectively. The micro elasticity tensor is specified as the maximum possible stiffness that is exhibited by the assumed metamaterial while the macro elasticity tensor is given by standard periodic first-order homogenization. For the identification of the micro elasticity tensor, two different approaches are shown which rely on affine and non-affine Dirichlet boundary conditions of candidate unit cell variants with the possible stiffest response. The consistent coupling condition is shown to allow the model to act on the whole intended range between macro and micro elasticity tensors for both loading cases. We fit the relaxed micromorphic model against the fully resolved metamaterial solution by controlling the curvature magnitude after linking it with the specimen’s size. The obtained parameters of the relaxed micromorphic model are tested for two additional loading scenarios.

In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model. As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary ϵ > 0 , the displacement vector u belongs to W 3 2 - ϵ , 2 ( Ω ) , and the microdistortion tensor P belongs to W 1 2 - ϵ , 2 ( Ω ) while Curl P belongs to W 1 2 - ϵ , 2 ( Ω ) .

In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model. As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary ϵ>0, the displacement vector u belongs to W32-ϵ,2(Ω), and the microdistortion tensor P belongs to W12-ϵ,2(Ω) while CurlP belongs to W12-ϵ,2(Ω).

In the present paper we consider a 2D pantographic structure composed of two orthogonal families of Euler beams. Pantographic rectangular 'long' waveguides are considered in which imposed boundary displacements can induce the onset of travelling (possibly non-linear) waves. We performed numerical simulations concerning a set of dynamically interesting cases. The system undergoes large rotations, which may involve geometrical non-linearities, possibly opening a path to appealing phenomena such as the propagation of solitary waves. Boundary conditions dramatically influence the transmission of the considered waves at discontinuity surfaces. The theoretical study of this kind of objects looks critical, as the concept of pantographic 2D sheets seems to have promising possible applications in a number of fields, e.g. acoustic filters, vascular prostheses, and aeronautic/aerospace panels.

We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of$H(\\operatorname {div};\\Omega )\\cap H_0(\\operatorname {curl};\\Omega )$into$H^1(\\Omega )$.