arXiv subjects
Peter Lewintan
Publications and source records attributed to Peter Lewintan.
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.
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.
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 $$ \|A\|_{L^p(Ω,\mathfrak{so}(3))}\le c\, \|\operatorname{Curl} A\|_{L^p(Ω,\mathbb{R}^{3\times3})} \quad \text{with } A \times ν= 0 \ \Leftrightarrow \ A=0 \quad \text{on $\partial Ω$}. $$
A constrained Cosserat shell model up to order $O(h^5)$: Modelling, existence of minimizers, relations to classical shell models and scaling invariance of the bending tensor
We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$. We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the $O(h^5)$ and $O(h^3)$ case and we draw some connections to existing models and classical shell strain measures. Notably, the role of the appearing new bending tensor is highlighted and investigated with respect to an invariance condition of Acharya [Int. J. Solids and Struct., 2000] which will be further strengthened.
The isotropic Cosserat shell model including terms up to $O(h^5)$. Part II: Existence of minimizers
We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including $O(h^5)$ terms. The energy allows us to show the coercivity for terms up to order $O(h^5)$ and the convexity of the energy. Secondly, we consider only that part of the energy including $O(h^3)$ terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including $O(h^5)$ the conditions on the thickness $h$ are those considered in the modelling process and they are independent of the constitutive parameter, in the $O(h^3)$-case the coercivity is proven on some more restrictive conditions under the thickness $h$.
The isotropic Cosserat shell model including terms up to $O(h^5)$. Part I: Derivation in matrix notation
We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$. The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensional sets of equations in variational form. We give an explicit form of the curvature energy using the orthogonal Cartan-decomposition of the wryness tensor. Moreover, we consider the matrix representation of all tensors in the derivation of the variational formulation, because this is convenient when the problem of existence is considered, and it is also preferential for numerical simulations. The step by step construction allows us to give a transparent approximation of the three-dimensional parental problem. The resulting 6-parameter isotropic shell model combines membrane, bending and curvature effects at the same time. The Cosserat shell model naturally includes a frame of orthogonal directors, the last of which does not necessarily coincide with the normal of the surface. This rotation-field is coupled to the shell-deformation and augments the well-known Reissner-Mindlin kinematics (one independent director) with so-called in-plane drill rotations, the inclusion of which is a decisive for subsequent numerical treatment and existence proofs. As a major novelty, we determine the constitutive coefficients of the Cosserat shell model in dependence on the geometry of the shell which are otherwise difficult to guess.
On $α$-minimizing hypercones
In this paper we considerably extend the class of known $α$-minimizing hypercones using sub-calibration methods. Indeed, the improvement of previous results follows from a careful analysis of special cubic and quartic polynomials.
On Bernstein-type theorems
We summarize results concerning the Bernstein property of differential equations.
Geometric Calculus of the Gauss Map
In this paper we connect classical differential geometry with the concepts from geometric calculus. Moreover, we introduce and analyze a more general Laplacian for multivector-valued functions on manifolds. This allows us to formulate a higher codimensional analog of Jacobi`s field equation.
The "Wrong Minimal Surface Equation" does not have the Bernstein property
A celebrated result of S. Bernstein states that every solution of the minimal surface equation over the entire plane has to be an affine linear function. Since the paper of Bernstein appeared in 1927, many different proofs and generalizations of this beautiful theorem were given, namely to higher dimensions and to more general equations, for a careful account we refer to the paper by Simo and to the monograph by Dierkes-Hildebrandt-Tromba. In his paper Simon posed the question whether the equation \begin{equation} (1+{u_x}^2)u_{xx}+2u_x u_y u_{xy}+ (1+{u_y}^2)u_{yy} = 0 \label{toll} \end{equation} has the Bernstein property i.e. whether every $C^2$-solution defined over the entire plane necessarily has to be affine. We here show by a very simple argument that this is not the case.