scholarly journals Note on sufficient symmetry conditions for isotropy of the elastic moduli tensor

1991 ◽  
Vol 6 (5) ◽  
pp. 1114-1118 ◽  
Author(s):  
M.S. Dresselhaus ◽  
G. Dresselhaus
Keyword(s):  
Elastic Properties ◽  
Direct Product ◽  
Elastic Moduli ◽  
Strain Tensor ◽  
Basis Functions ◽  
Irreducible Representations ◽  
Symmetry Groups ◽  
Symmetric Representation ◽  
Theoretical Methods ◽  
Independent Parameters

Group theoretical methods are used to obtain the form of the elastic moduli matrices and the number of independent parameters for various symmetries. Particular attention is given to symmetry groups for which 3D and 2D isotropy is found for the stress-strain tensor relation. The number of independent parameters is given by the number of times the fully symmetric representation is contained in the direct product of the irreducible representations for two symmetrical second rank tensors. The basis functions for the lower symmetry groups are found from the compatibility relations and are explicitly related to the elastic moduli. These types of symmetry arguments should be generally useful in treating the elastic properties of solids and composites.

scholarly journals Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 548
Author(s):  
Thomas M. Mellor ◽  
Sergei N. Yurchenko ◽  
Per Jensen
Keyword(s):  
Symmetry Group ◽  
Group Algebra ◽  
Basis Functions ◽  
Basis Set ◽  
Irreducible Representations ◽  
Molecular Axis ◽  
Symmetry Groups ◽  
Molecular Symmetry ◽  
Linear Molecules ◽  
Group D

Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the a (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the 3N−6 approach, of the form ∣ν1,ν2,ν3ℓ3;J,k,m⟩, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with k=ℓ3, which are the projections, in units of ℏ, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the C2v(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition ℓ3=k is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group D∞h(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to ℓ and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group D∞h(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.

The symmetric group and the method of diatomics in molecules: an application to small lithium clusters

1973 ◽  
Vol 333 (1592) ◽  
pp. 69-87 ◽  
Keyword(s):  
Alkali Metal ◽  
Symmetric Group ◽  
Metal Clusters ◽  
Basis Functions ◽  
Basis Set ◽  
Irreducible Representations ◽  
Secular Equations ◽  
Alkali Metal Clusters ◽  
Lithium Clusters

A new ‘most economical’ algorithm for the construction of diatomics in molecules secular equations is described. The method does not require the basis functions to be written down explicitly, since overlap may be factored out of the equations entirely. The theory is presented in detail for the particular case of homogeneous alkali metal clusters. A knowledge of the irreducible representations of the symmetric group for the Jahn-Serber basis set is necessary. The irreducible representations are derived by a genealogical procedure. Some preliminary calculations are presented for the molecules Li 3 through Li 6 , Li + 3 and Li + 4 . The lithium clusters are found to be stable with respect to all possible dissociations, and the i.ps of Li 3 and Li 4 are in agreement with the trends for the species Na 3 , Na 4 , K 3 , K 4 , etc., whose i.ps have been measured experimentally.

scholarly journals Irreducible representations of E theory

2019 ◽  
Vol 34 (24) ◽  
pp. 1950133 ◽  
Author(s):  
Peter West
Keyword(s):  
Irreducible Representation ◽  
Direct Product ◽  
Simple Form ◽  
Light Cone ◽  
Degrees Of Freedom ◽  
Irreducible Representations ◽  
Vector Representation ◽  
Maximal Supergravity ◽  
Duality Relations ◽  
Cartan Involution

We construct the [Formula: see text] theory analogue of the particles that transform under the Poincaré group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of [Formula: see text] with its vector representation. We show that one such irreducible representation has only the degrees of freedom of 11-dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in [Formula: see text] theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of [Formula: see text]. We also argue that our familiar space–times have to be extended by additional coordinates when considering extended objects such as branes.

Elastic properties and phonon conductivities of Ti3Al(C0.5,N0.5)2 and Ti2Al(C0.5,N0.5) solid solutions

2008 ◽  
Vol 23 (6) ◽  
pp. 1517-1521 ◽  
Author(s):  
M. Radovic ◽  
A. Ganguly ◽  
M.W. Barsoum
Keyword(s):  
Elastic Properties ◽  
Solid Solutions ◽  
Room Temperature ◽  
Elastic Moduli ◽  
Unit Cell ◽  
Young’S Moduli ◽  
Temperature Shear ◽  
End Members ◽  
Cell Volumes ◽  
Thermal Conductivities

Herein we compare the lattice parameters, room temperature shear and Young’s moduli, and phonon thermal conductivities of Ti2AlC0.5N0.5 and Ti3Al(C0.5, N0.5)2 solid solutions with those of their end members, namely Ti2AlC, Ti2AlN, Ti3AlC2, and Ti4AlN2.9. In general, the replacement of C by N decreases the unit cell volumes and increases the elastic moduli and phonon thermal conductivities. The increase in the latter two properties, however, is sensitive to the concentrations of defects, most likely vacancies on one or more of the sublattices.

Automated Symmetry Exploitation in Engineering Analysis

2004 ◽  
Author(s):  
Krishnan Suresh
Keyword(s):  
Irreducible Representations ◽  
Point Symmetry ◽  
Symmetry Groups ◽  
Engineering Analysis ◽  
Symmetry Detection ◽  
Important Concept ◽  
Improve Accuracy ◽  
Speed Up ◽  
Boundary Mapping ◽  
Cell Construction

It is well known that one can exploit symmetry to speed-up engineering analysis and improve accuracy, at the same time. Not surprisingly, most CAE systems have standard ‘provisions’ for exploiting symmetry. However, these provisions are inadequate in that they needlessly burden the design engineer with time consuming and error-prone tasks of symmetry detection, symmetry cell construction and reformulation. In this paper, we propose and discuss an automated methodology for symmetry exploitation. First, we briefly review the theory of point symmetry groups that symmetry exploitation rests on. We then address symmetry detection and ‘symmetry cell’ construction. We then address an important concept of boundary mapping of symmetry cells, and relate it to the irreducible representations of point symmetry groups. By formalizing these concepts, we show how automated symmetry exploitation can be achieved, and discuss an implementation of the proposed work within the FEMLAB CAE environment.

Elastic Properties of Artificially Layered Thin Films

1987 ◽  
Vol 103 ◽  
Author(s):  
Robert C. Cammarata
Keyword(s):  
Thin Films ◽  
Mechanical Properties ◽  
Composite Materials ◽  
Elastic Properties ◽  
Elastic Moduli ◽  
Metallic Thin Films ◽  
Theoretical Understanding ◽  
Composition Modulation

ABSTRACTEnhancements in the elastic moduli by factors of two or more in compositionally modulated metallic thin films have been observed for a certain range of composition modulation wavelengths. The experimental and theoretical understanding of this phenomenon, known as the supermodulus effect, is reviewed. Also, the mechanical properties of other artificially layered and composite materials are discussed and compared with the behavior of metallic superlattice thin films.

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