A Justification of the Timoshenko Beam Model through $\boldsymbolΓ$-Convergence

We validate the Timoshenko beam model as an approximation of the linear-elasticity model of a three-dimensional beam-like body. Our validation is achieved within the framework of $\Gamma$-convergence theory, in two steps: firstly, we construct a suitable sequence of energy functionals; secondly, we show that this sequence $\Gamma$-converges to a functional representing the energy of a Timoshenko beam.

Similar Publications

This paper deals with the three-dimensional narrow escape problem in dendritic spine shaped domain, which is composed of a relatively big head and a thin neck. The narrow escape problem is to compute the mean first passage time of Brownian particles traveling from inside the head to the end of the neck. The original model is to solve a mixed Dirichlet-Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. Read More

The partition function of the Penner matrix model for both positive and negative values of the coupling constant can be explicitly written in terms of the Barnes G function. In this paper we show that for negative values of the coupling constant this partition function can also be represented as the product of an holomorphic matrix integral by a nontrivial oscillatory function of n. We show that the planar limit of the free energy with 't Hooft sequences does not exist. Read More

We report on the occurrence of the focus-focus type of monodromy in an integrable version of the Dicke model. Classical orbits forming a pinched torus represent extreme realizations of the dynamical superradiance phenomenon. Quantum signatures of monodromy appear in lattices of expectation values of various quantities in the Hamiltonian eigenstates and are related to an excited-state quantum phase transition. Read More

We consider an exclusion process with long jumps in the box $\Lambda_N=\{1, \ldots,N-1\}$, for $N \ge 2$, in contact with infinitely extended reservoirs on its left and on its right. The jump rate is described by a transition probability $p(\cdot)$ which is symmetric, with infinite support but with finite variance. The reservoirs add or remove particles with rate proportional to $\kappa N^{-\theta}$, where $\kappa>0$ and $\theta \in \mathbb{R}$. Read More

We relate the counting of honeycomb dimer configurations on the cylinder to the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We show that these dimer configurations yield the quantum Kostka numbers of the small quantum cohomology ring of the Grassmannian, i.e. Read More

We consider some complex-valued solutions of the Navier-Stokes equations in $R^{3}$ for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the "fluid" remains quiet. Read More

Several families of one-point interactions are derived from the system consisting of two and three $\delta$-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of $\delta$-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter $\varepsilon \to 0$, so that the intensity of each $\delta$-potential is $c_j =a_j \varepsilon^{1-\mu}$, $a_j \in {\mathbb{R}}$, $j=1,2,3$, the width of each layer $l =\varepsilon$ and the distance between the layers $r = c\varepsilon^\tau$, $c >0$. Read More

We define a family of quantum invariants of closed oriented $3$-manifolds using spherical multi-fusion categories. The state sum nature of this invariant leads directly to $(2+1)$-dimensional topological quantum field theories ($\text{TQFT}$s), which generalize the Turaev-Viro-Barrett-Westbury ($\text{TVBW}$) $\text{TQFT}$s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the $\text{TVBW}$ approach in that here the labels live not only on $1$-simplices but also on $0$-simplices. Read More

We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We find an explicit expression of the joint probability density function as a bi-orthogonal ensemble and it is shown that this ensemble reduces asymptotically to the Hermite Muttalib-Borodin model. Explicit expression for the bi-orthogonal functions as well as the correlation kernel are provided. Read More