ON CONTACT POWERS AND NULL LAGRANGIAN FLUXES

2005 ◽  
Author(s):  
PAOLO PODIO-GUIDUGLI ◽  
GIORGIO VERGARA CAFFARELLI
Keyword(s):  
Null Lagrangian

Improved bounds on the conductivity of composites by interpolation

1994 ◽  
Vol 444 (1921) ◽  
pp. 363-374 ◽  
Keyword(s):  
Composite Material ◽  
Lower Bounds ◽  
Approximation Method ◽  
Interpolation Method ◽  
Padé Approximation ◽  
Upper Bounds ◽  
Pade Approximation ◽  
Two Component ◽  
Null Lagrangian

Different bounds on the conductivity of a composite material may improve on each other in different conductivity régimes. If so, the question arises of how to efficiently interpolate between the bounds. In this paper I show how to do an interpolation with a two-point Padé approximation method. For bounds on two-component composites the interpolation method is shown to be, in a sense, the best possible. The method is discussed in the context of equiaxed polycrystals where the classic Hashin-Shtrikman bounds and the more recent null-lagrangian bounds, partly improve on each other. Denoting the principal conductivities of the crystallite σ 1 ≼ σ 2 ≼ σ 3 , the method gives improved lower bounds for equiaxed polycrystals which have σ 2 (0.77σ 1 + 0.23σ 3 ) ≽ σ 1 σ 3 . The method also gives improved upper bounds.

Second variation of liquid crystal energy at x /| x |

1992 ◽  
Vol 437 (1900) ◽  
pp. 475-487 ◽  
Keyword(s):  
Liquid Crystal ◽  
Elastic Constants ◽  
Second Variation ◽  
Surface Integral ◽  
Frame Indifference ◽  
The Second Variation ◽  
Null Lagrangian

A new and simplified proof of the sign of the second variation of the Oseen–Frank energy in terms of the elastic constants is given. The proof relies on the frame indifference of the energy and a new expression for the second invariant null lagrangian as a surface integral.

Rank one plus a null-Lagrangian is an inherited property of two-dimensional compliance tensors under homogenisation

1998 ◽  
Vol 128 (2) ◽  
pp. 283-299 ◽  
Author(s):  
Yury Grabovsky ◽  
Graeme W. Milton
Keyword(s):  
Shear Modulus ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Pure Crystal ◽  
Compliance Tensor ◽  
Rank One ◽  
Two Space Dimensions ◽  
Local Compliance ◽  
Null Lagrangian ◽  
Surprising Discovery

Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.

scholarly journals Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant

2019 ◽  
Vol 30 (05) ◽  
pp. 1950021
Author(s):  
Benjamin Audoux ◽  
Delphine Moussard
Keyword(s):  
Finite Type ◽  
Rational Homology ◽  
Direct Sums ◽  
Finite Type Invariants ◽  
Null Lagrangian

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined, respectively, by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of [Formula: see text]-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.

NULL LAGRANGIANS IN LINEAR ELASTICITY

1995 ◽  
Vol 05 (04) ◽  
pp. 415-427 ◽  
Author(s):  
M.R. LANCIA ◽  
G. VERGARA CAFFARELLI ◽  
P. PODIO-GUIDUGLI
Keyword(s):  
Linear Elasticity ◽  
Energy Functional ◽  
Elasticity Tensor ◽  
Theory Of Elasticity ◽  
Stored Energy ◽  
Cauchy Relations ◽  
Null Lagrangian

The concept of null Lagrangian is exploited in the context of linear elasticity. In particular, it is shown that the stored energy functional can always be split into a null Lagrangian and a remainder; the null Lagrangian vanishes if and only if the elasticity tensor obeys the Cauchy relations, and is therefore determined by only 15 independent moduli (the so-called “rari-constant” theory of elasticity).

scholarly journals Solutions of the differential inequality with a null Lagrangian: higher integrability and removability of singularities. I

2014 ◽  
Author(s):  
A.A. Egorov
Keyword(s):  
Differential Inequality ◽  
Sobolev Class ◽  
The Self ◽  
Hölder Regularity ◽  
Higher Integrability ◽  
Stability Theorems ◽  
Sobolev Exponent ◽  
Derivatives Of ◽  
Null Lagrangian ◽  
Structural Assumption

The aim of this paper is to derive the self-improving property of integrability for derivatives of solutions of the differential inequality with a null Lagrangian. More precisely, we prove that the solution of the Sobolev class with some Sobolev exponent slightly smaller than the natural one determined by the structural assumption on the involved null Lagrangian actually belongs to the Sobolev class with some Sobolev exponent slightly larger than this natural exponent. We also apply this property to improve Holder regularity and stability theorems of [19].

scholarly journals Решения дифференциального неравенства с нуль-лагранжианом: повышающаяся интегрируемость и устранимость особенностей. II

2014 ◽  
Author(s):  
A.A. Egorov
Keyword(s):  
Differential Inequality ◽  
Jacobian Matrix ◽  
Differential Forms ◽  
Null Lagrangian

The aim of this paper is to establish a result on removability of singularities for solutions of the differential inequality with a null Lagrangian. Also, we obtain integral estimates for wedge products of closed differential forms and for minors of a Jacobian matrix.

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