Dependence of the Depolarization Tensor on the Displacement Gradient

1971 ◽  
Vol 49 (17) ◽  
pp. 2284-2286
Author(s):  
A. Redlack ◽  
J. Grindlay
Keyword(s):  
Second Order ◽  
Displacement Gradient ◽  
Derivatives Of ◽  
Special Case

Expressions are given for the first- and second-order derivatives of the depolarization tensor with respect to the displacement gradient. The results are applied to the special case of the sphere.

scholarly journals Constructing Carrollian CFTs

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Nishant Gupta ◽  
Nemani V. Suryanarayana
Keyword(s):  
Scalar Fields ◽  
Higher Dimensions ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Relativistic Limit ◽  
Conformal Field ◽  
Local Symmetries ◽  
Derivatives Of ◽  
Special Case

Abstract We construct classical theories for scalar fields in arbitrary Carroll spacetimes that are invariant under Carrollian diffeomorphisms and Weyl transformations. When the local symmetries are gauge fixed these theories become Carrollian conformal field theories. We show that generically there are at least two types of such theories: one in which only time derivatives of the fields appear and the other in which both space and time derivatives appear. A classification of such scalar field theories in three (and higher) dimensions up to two derivative order is provided. We show that only a special case of our theories arises in the ultra-relativistic limit of a covariant parent theory.

scholarly journals Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

2021 ◽  
Author(s):  
V. Calisti ◽  
A. Lebée ◽  
A. A. Novotny ◽  
J. Sokolowski
Keyword(s):  
Circular Inclusion ◽  
Elasticity Tensor ◽  
Second Order ◽  
Topological Derivative ◽  
Microstructural Changes ◽  
Topological Derivatives ◽  
Spatial Dimensions ◽  
Elasticity Tensors ◽  
Derivatives Of ◽  
Optimization Of Microstructures

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

scholarly journals On the derivatives of the principal invariants of a second-order tensor

1986 ◽  
Vol 16 (2) ◽  
pp. 221-224 ◽  
Author(s):  
Donald E. Carlson ◽  
Anne Hoger
Keyword(s):  
Second Order ◽  
Order Tensor ◽  
Derivatives Of

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

On second-order directional derivatives of value functions

Optimization ◽  
2013 ◽  
Vol 64 (2) ◽  
pp. 389-407 ◽  
Author(s):  
L. Minchenko ◽  
A. Tarakanov
Keyword(s):  
Second Order ◽  
Directional Derivatives ◽  
Value Functions ◽  
Derivatives Of

DFT STUDY ON SECOND-ORDER NONLINEAR OPTICAL PROPERTIES OF THE DERIVATIVES OF DISUBSTITUTED SEVEN-VERTEX COBALTACARBORANE METALLOCENYL

2010 ◽  
Vol 09 (01) ◽  
pp. 219-231 ◽  
Author(s):  
XIAODONG LIU ◽  
YONGQING QIU ◽  
SHILING SUN ◽  
CHUNGUANG LIU ◽  
ZHONGMIN SU
Keyword(s):  
Nonlinear Optical ◽  
Membered Ring ◽  
Second Order ◽  
Meta Position ◽  
Nlo Response ◽  
B3lyp Method ◽  
Acceptor Group ◽  
Calculation Results ◽  
Optimum Model ◽  
Derivatives Of

DFT B3LYP method was employed to calculate the second-order nonlinear optical (NLO) responses of the derivatives of disubstituted seven-vertex cobaltacarborane metallocenyl. The results show that cobaltacarborane metallocenyl plays a pushing/pulling role and a bridge role to transfer electron in these molecules. The five-membered ring of cyclopentadiene is more beneficial to increase second-order NLO response than the five-membered ring composed of two C atoms and three B atoms in cobaltacarborane. Moreover, the second-order NLO response is more powerful when one substituent containing electron donor group and one substituent containing electron acceptor group are located at meta position. Accordingly, among the nine models, model c2 is the optimum model with largest value of βtot. The calculation results also show that cobaltacarborane metallocenyl and ferrocene parts play the same roles to increase second-order NLO response. Thus, cobaltacarborane metallocenyl can be a promising second-order NLO material.

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