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Davit Harutyunyan

Publications and source records attributed to Davit Harutyunyan.

35 records · Page 2Linked to original sources
arXiv2017

Towards characterization of all $3\times3$ extremal quasiconvex quadratic forms

Given a $d\times d$ quasiconvex quadratic form, $d\geq 3,$ we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero, then the form itself is an extremal quasiconvex quadratic form, i.e. it loses its quasiconvexity whenever a convex quadratic form is subtracted from it. In the special case $d=3,$ we slightly weaken the condition, namely we prove, that if the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square, then the form itself is an extremal quadratic form. In the case $d=3$ we also prove, that if the determinant of the acoustic tensor of the form is identically zero, then the form is either an extremal or polyconvex. Also, if the determinant of the acoustic tensor of the form is a perfect square, then the form is either extremal, polycovex, or is a sum of a rank-one form and an extremal, whose acoustic tensor determinant is identically zero. Here we use the notion of extremality introduced by Milton in [\ref{bib:Mil3}]

math.AG↗
arXiv2016

When the Cauchy inequality becomes a formula

In this note we revisit the classical geometric-arithmetic mean inequality and find a formula for the difference of the arithmetic and the geometric means of given $n\in\mathbb N$ nonnegative numbers $x_1,x_2,\dots,x_n$. The formula yields new stronger versions of the geometric-arithmetic mean inequality. We also find a second version of a strong geometric-arithmetic mean inequality and show that all inequalities are optimal in some sense. Anther striking novelty is, that the equality in all new inequalities holds not only in the case when all $n$ numbers are equal, but also in other cases.

math.CA↗
arXiv2016

On ideal dynamic climbing ropes

We consider the rope climber fall problem in two different settings. The simplest formulation of the problem is when the climber falls from a given altitude and is attached to one end of the rope while the other end of the rope is attached to the rock at a given height. The problem is then finding the properties of the rope for which the peak force felt by the climber during the fall is minimal. The second problem of our consideration is again minimizing the same quantity in the presence of a carabiner. We will call such ropes \textit{mathematically ideal.} Given the height of the carabiner, the initial height and the mass of the climber, the length of the unstretched rope, and the distance between the belayer and the carabineer, we find the optimal (in the sense of minimized the peak force to a given elongation) dynamic rope in the framework of nonlinear elasticity. Wires of shape memory materials have some of the desired features of the tension-strain relation of a mathematically ideal dynamic rope, namely a plateau in the tension over a range of strains. With a suitable hysteresis loop, they also absorb essentially all the energy from the fall, thus making them an ideal rope in this sense too.

math.OC↗
arXiv2016

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors $\BC_1>0$ and $\BC_2=0$ comprising the set $U=\{\BC_1,\BC_2\}$ and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor $\BC_2=0$ corresponds to a material which is void. (For technical reasons $\BC_2$ is actually taken to be nonzero and we take the limit $\BC_2\to 0$). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.

cond-mat.mtrl-sci↗
arXiv2016

Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with positive definite elasticity tensors $\BC_1$ and $\BC_2=\Gd\BC_0$ comprising the set $U=\{\BC_1,\Gd\BC_0\}$ and mixed in proportions $f$ and $1-f$ is partly characterized in the limit $\Gd\to\infty$. The material with tensor $\BC_2$ corresponds to a material which (for technical reasons) is almost rigid in the limit $\Gd\to \infty$. The paper, and the underlying microgeometries, have many aspects in common with the companion paper "On the possible effective elasticity tensors of 2-dimensional printed materials". The chief difference is that one has a different algebraic problem to solve: determining the subspaces of stress fields for which the thin walled structures can be rigid, rather than determining, as in the companion paper, the subspaces of strain fields for which the thin walled structure is compliant. Recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is almost rigid to $6-p\leq 5$ independent applied stresses, yet is compliant to any strain in the orthogonal space. Thus the walls, by themselves, can support stress with almost no deformation. The region outside the walls contains "Avellaneda material" that is a hierarchical laminate which minimizes an appropriate sum of elastic energies.

cond-mat.mtrl-sci↗
arXiv2016

High frequency homogenization for travelling waves in periodic media

We consider high frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schr{ö}dinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wave vector $\Bk$ and frequency $ω_1$ plus a modulated Bloch carrier wave having crystal wave vector $\Bm$ and frequency $ω_2$. We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless $ω_1=ω_2$ and $(\Bk-\Bm)\odotΛ\in 2π\mathbb Z^d,$ where $Λ=(λ_1λ_2\dotsλ_d)$ is the periodicity cell of the medium and for any two vectors $a=(a_1,a_2,\dots,a_d), b=(b_1,b_2,\dots,b_d)\in\mathbb R^d,$ the product $a\odot b$ is defined to be the vector $(a_1b_1,a_2b_2,\dots,a_db_d).$ This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigor as that of Allaire and coworkers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue.

math.AP↗
arXiv2016

Sharp weighted Korn and Korn-like inequalities and an application to washers

In this paper we prove asymptotically sharp weighted "first-and-a-half" $2D$ Korn and Korn-like inequalities with a singular weight occurring from Cartesian to cylindrical change of variables. We prove some Hardy and the so-called "harmonic function gradient separation" inequalities with the same singular weight. Then we apply the obtained $2D$ inequalities to prove similar inequalities for washers with thickness $h$ subject to vanishing Dirichlet boundary conditions on the inner and outer thin faces of the washer. A washer can be regarded in two ways: As the limit case of a conical shell when the slope goes to zero, or as a very short hollow cylinder. While the optimal Korn constant in the first Korn inequality for a conical shell with thickness $h$ and with a positive slope scales like $h^{1.5}$ e.g. [10], the optimal Korn constant in the first Korn inequality for a washer scales like $h^2$ and depends only on the outer radius of the washer as we show in the present work. The Korn constant in the first and a half inequality scales like $h$ and depends only on $h.$ The optimal Korn constant is realized by a Kirchoff Ansatz. This results can be applied to calculate the critical buckling load of a washer under in plane loads, e.g. [1].

math.AP↗
arXiv2016

Korn inequalities for shells with zero Gaussian curvature

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a "first-and-a-half Korn inequality" - a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.

math.AP↗
arXiv2014

Existence and the stability of minimizers in ferromagnetic nanowires

We study static 180 degree domain walls in infinite magnetic wires with bounded, $C^1$ and rotationally symmetric cross sections. We prove an existence of global minimizers for the energy of micromagnetics for any bounded $C^1$ cross sections. Under some asymmetry of cross sections we prove a stability result for the minimizers, namely, we show that vectors of micromagnetics having an energy close to the minimal one, must be $H^1$ close to a minimizer of the limiting energy up to a rotation and a translation.

math.AP↗
arXiv2014

Explicit examples of extremal quasiconvex quadratic forms that are not polyconvex

We prove that if the associated fourth order tensor of a quadratic form has a linear elastic cubic symmetry then it is quasiconvex if and only if it is polyconvex, i.e. a sum of convex and null-Lagrangian quadratic forms. We prove that allowing for slightly less symmetry, namely only cyclic and axis-reflection symmetry, gives rise to a class of extremal quasiconvex quadratic forms, that are not polyconvex. Non-affine boundary conditions on the potential are identified which allow one to obtain sharp bounds on the integrals of these extremal quasiconvex quadratic forms of $\nabla u$ over an arbitrary region.

math.AP↗
arXiv2014

Rigorous derivation of the formula for the buckling load in axially compressed circular cylindrical shells

The goal of this paper is to apply the recently developed theory of buckling of arbitrary slender bodies to a tractable yet non-trivial example of buckling in axially compressed circular cylindrical shells, regarded as three-dimensional hyperelastic bodies. The theory is based on a mathematically rigorous asymptotic analysis of the second variation of 3D, fully nonlinear elastic energy, as the shell's slenderness parameter goes to zero. Our main results are a rigorous proof of the classical formula for buckling load and the explicit expressions for the relative amplitudes of displacement components in single Fourier harmonics buckling modes, whose wave numbers are described by Koiter's circle. This work is also a part of an effort to quantify the sensitivity of the buckling load of axially compressed cylindrical shells to imperfections of load and shape.

math.AP↗
arXiv2014

Scaling instability of the buckling load in axially compressed circular cylindrical shells

In this paper we initiate a program of rigorous analytical investigation of the paradoxical buckling behavior of circular cylindrical shells under axial compression. This is done by the development and systematic application of general theory of "near-flip" buckling of 3D slender bodies to cylindrical shells. The theory predicts scaling instability of the buckling load due to imperfections of load. It also suggests a more dramatic scaling instability caused by shape imperfections. The experimentally determined scaling exponent 1.5 of the critical stress as a function of shell thickness appears in our analysis as the scaling of the lower bound on safe loads given by the Korn constant. While the results of this paper fall short of a definitive explanation of the buckling behavior of cylindrical shells, we believe that our approach is capable of providing reliable estimates of the buckling loads of axially compressed cylindrical shells.

math.AP↗
arXiv2014

Scaling laws and the rate of convergence in thin magnetic films

We study static 180 degree domain walls in thin infinite magnetic films. We establish the scaling of the minimal energy by Γ-convergence and the energy minimizer pro-file, which turns out to be the so called transverse wall as predicted in earlier numerical and experimental work. Surprisingly, the minimal energy decays faster than the area of the film cross section at an infinitesimal cross section diameter. We establish a rate of convergence of the rescaled energies as well.

math.AP↗
arXiv2013

Exact scaling exponents in Korn and Korn-type inequalities for cylindrical shells

Understanding asymptotics of gradient components in relation to the symmetrized gradient is im- portant for the analysis of buckling of slender structures. For circular cylindrical shells we obtain the exact scaling exponent of the Korn constant as a function of shell's thickness. Equally sharp results are obtained for individual components of the gradient in cylindrical coordinates. We also derive an ana- logue of the Kirchho? ansatz, whose most prominent feature is its singular dependence on the slenderness parameter, in marked contrast with the classical case of plates and rods.

math.AP↗
arXiv2013

New asymptotically sharp Korn and Korn-like inequalities in thin domains

It is well known that Korn inequality plays a central role in the theory of linear elasticity. In the present work we prove new asymptotically sharp Korn and Korn-like inequalities in thin curved domains with a non-constant thickness. This new results will be useful when studying the buckling of compressed shells, in particular when calculating the critical buckling load.

math.AP↗
arXiv2013

Rigorous asymptotic analysis of buckling of thin-walled cylinders under axial compression

Using rigorous constitutive linearization of second variation introduced in [6] we study weak stability of homogeneous deformation of the axially compressed circular cylindrical shell, regarded as a 3-dimensional hyperelastic body. We show that such deformation becomes weakly unstable at the citical load that coincides with value of the bifurcation load in von-Kármán-Donnel shell theory. We also show that the linear bifurcation modes described by the Koiter circle [11] minimize the second variation asymptotically. The key ingredients of our analysis are the asymptoticaly sharp estimates of the Korn constant for cylindrical shells and Korn-like inequalities on components of the deformation gradient tensor in cylindrical coordinates. The notion of buckling equivalence introduced in [6] is developed further and becomes central in this work. A link between features of this theory and sensitivity of the critical load to imprefections of load and shape is conjectured.

math.AP↗