Nontrivial extensions of a representation of the Poincar� group with mass and helicity zero by its tensorial product

1984 ◽  
Vol 8 (5) ◽  
pp. 421-433 ◽  
Author(s):  
G. Rideau
Keyword(s):  
Tensorial Product

A Micromechanical Description of Granular Material Behavior

1981 ◽  
Vol 48 (2) ◽  
pp. 339-344 ◽  
Author(s):  
J. Christoffersen ◽  
M. M. Mehrabadi ◽  
S. Nemat-Nasser
Keyword(s):  
Granular Material ◽  
Virtual Work ◽  
Contact Forces ◽  
Material Behavior ◽  
Principle Of Virtual Work ◽  
General Representation ◽  
Volume Average ◽  
Macroscopic Deformation ◽  
The Individual ◽  
Tensorial Product

Considered is a sample of cohesionless granular material, in which the individual granules are regarded rigid, and which is subjected to overall macroscopic average stresses. On the basis of the principle of virtual work, and by an examination of the manner by which adjacent granules transmit forces through their contacts, a general representation is established for the macroscopic stresses in terms of the volume average of the (tensorial) product of the contact forces and the vectors which connect the centroids of adjacent contacting granules. Then the corresponding kinematics is examined and the overall macroscopic deformation rate and spin tensors are developed in terms of the volume average of relevant microscopic kinematical variables. As an illustration of the application of the general expressions developed, two explicit macroscopic results are deduced: (1) a dilatancy equation which both qualitatively and quantitatively seems to be in accord with experimental observation, and (2) a noncoaxiality equation which seems to support the vertex plasticity model. Since the development is based on a microstructural consideration, all material coefficients entering the results have well-defined physical interpretations.

scholarly journals A Unstructured Nodal Spectral-Element Method for the Navier-Stokes Equations

2012 ◽  
Vol 12 (1) ◽  
pp. 315-336 ◽  
Author(s):  
Lizhen Chen ◽  
Jie Shen ◽  
Chuanju Xu
Keyword(s):  
Spectral Element Method ◽  
Stokes Equations ◽  
Spectral Element ◽  
Tetrahedral Mesh ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Numerical Examples ◽  
Tensorial Product ◽  
Nodal Basis ◽  
Element Method

AbstractAn unstructured nodal spectral-element method for the Navier-Stokes equations is developed in this paper. The method is based on a triangular and tetrahedral rational approximation and an easy-to-implement nodal basis which fully enjoys the tensorial product property. It allows arbitrary triangular and tetrahedral mesh, affording greater flexibility in handling complex domains while maintaining all essential features of the usual spectral-element method. The details of the implementation and some numerical examples are provided to validate the efficiency and flexibility of the proposed method.

NEW RESULTS ON DYNAMICAL DISORDER IN AMORPHOUS III–V COMPOUNDS

1993 ◽  
Vol 07 (18) ◽  
pp. 1201-1207 ◽  
Author(s):  
M. A. GRADO CAFFARO ◽  
M. GRADO CAFFARO
Keyword(s):  
Infrared Spectrum ◽  
Structural Disorder ◽  
Optical Mode ◽  
Tensorial Product ◽  
Product Of Matrices ◽  
Dynamical Disorder ◽  
Phonon Absorption

For amorphous III–V semiconductors, infrared spectrum corresponding to dynamical disorder is considered in the context of one-phonon absorption. In particular, an expression for the frequency of transversal optical mode is derived for minimum absorption. Various properties of this frequency as a function of distance are discussed. Furthermore, this frequency and minimum spectrum are determined in terms of a tensorial product of matrices. Our results are compared with experiment. Average absorption due to structural disorder is also formulated.

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