Unrestricted second-order tensor virial equations for linear oscillations of magnetic configurations

1979 ◽  
Vol 60 (2) ◽  
pp. 401-421 ◽  
Author(s):  
M. Goossens
Keyword(s):  
Second Order ◽  
Order Tensor

scholarly journals Integrity basis for a second-order and a fourth-order tensor

1982 ◽  
Vol 5 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Josef Betten
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Tensor Function ◽  
Order Tensor ◽  
Isotropic Tensor ◽  
Anisotropic Properties ◽  
Isotropic Tensor Function ◽  
Fourth Order Tensor

In this paper a scalar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order greater than four, as shown in the paper, for instance, for a tensor of order six.

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

Core Lines in 3D Second-Order Tensor Fields

2018 ◽  
Vol 37 (3) ◽  
pp. 327-337 ◽  
Author(s):  
T. Oster ◽  
C. Rössl ◽  
H. Theisel
Keyword(s):  
Second Order ◽  
Tensor Fields ◽  
Order Tensor

On the Green function of the Lord–Shulman thermoelasticity equations

2019 ◽  
Vol 220 (1) ◽  
pp. 393-403 ◽  
Author(s):  
Zhi-Wei Wang ◽  
Li-Yun Fu ◽  
Jia Wei ◽  
Wanting Hou ◽  
Jing Ba ◽  
...  
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Parabolic Type ◽  
Second Order ◽  
P Wave ◽  
Thermal Mode ◽  
S Wave ◽  
Order Tensor ◽  
P Waves ◽  
Thermal Conductivities

SUMMARY Thermoelasticity extends the classical elastic theory by coupling the fields of particle displacement and temperature. The classical theory of thermoelasticity, based on a parabolic-type heat-conduction equation, is characteristic of an unphysical behaviour of thermoelastic waves with discontinuities and infinite velocities as a function of frequency. A better physical system of equations incorporates a relaxation term into the heat equation; the equations predict three propagation modes, namely, a fast P wave (E wave), a slow thermal P wave (T wave), and a shear wave (S wave). We formulate a second-order tensor Green's function based on the Fourier transform of the thermodynamic equations. It is the displacement–temperature solution to a point (elastic or heat) source. The snapshots, obtained with the derived second-order tensor Green's function, show that the elastic and thermal P modes are dispersive and lossy, which is confirmed by a plane-wave analysis. These modes have similar characteristics of the fast and slow P waves of poroelasticity. Particularly, the thermal mode is diffusive at low thermal conductivities and becomes wave-like for high thermal conductivities.

scholarly journals Iterative solution of quadratic tensor equations for mutual polarisation

2002 ◽  
Vol 32 (5) ◽  
pp. 301-312 ◽  
Author(s):  
Wynand S. Verwoerd
Keyword(s):  
Dipole Moment ◽  
Linear Approximation ◽  
Iterative Solution ◽  
Second Order ◽  
Local Fields ◽  
Bulk Materials ◽  
Tensor Equation ◽  
Order Tensor ◽  
Intrinsic Values ◽  
Induced Dipole

To describe mutual polarisation in bulk materials containing high polarisability molecules, local fields beyond the linear approximation need to be included. A second order tensor equation is formulated, and it describes this in the case of crystalline or at least locally ordered materials such as an idealised polymer. It is shown that this equation is solved by a set of recursion equations that relate the induced dipole moment, linear polarisability, and first hyperpolarisability in the material to the intrinsic values of the same properties of isolated molecules. From these, macroscopic susceptibility tensors up to second order can be calculated for the material.

scholarly journals Multisensor Fused Fault Diagnosis for Rotation Machinery Based on Supervised Second-Order Tensor Locality Preserving Projection and Weighted k-Nearest Neighbor Classifier under Assembled Matrix Distance Metric

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Fen Wei ◽  
Gang Wang ◽  
Bingyin Ren ◽  
Jianghua Ge ◽  
Yaping Wang
Keyword(s):  
Fault Diagnosis ◽  
Nearest Neighbor ◽  
Second Order ◽  
Distance Metric ◽  
Order Tensor ◽  
Locality Preserving Projection ◽  
Nearest Neighbor Classifier ◽  
Locality Preserving ◽  
Diagnosis Method ◽  
Neighbor Classifier

In order to sufficiently capture the useful fault-related information available in the multiple vibration sensors used in rotation machinery, while concurrently avoiding the introduction of the limitation of dimensionality, a new fault diagnosis method for rotation machinery based on supervised second-order tensor locality preserving projection (SSTLPP) and weighted k-nearest neighbor classifier (WKNNC) with an assembled matrix distance metric (AMDM) is presented. Second-order tensor representation of multisensor fused conditional features is employed to replace the prevailing vector description of features from a single sensor. Then, an SSTLPP algorithm under AMDM (SSTLPP-AMDM) is presented to realize dimensional reduction of original high-dimensional feature tensor. Compared with classical second-order tensor locality preserving projection (STLPP), the SSTLPP-AMDM algorithm not only considers both local neighbor information and class label information but also replaces the existing Frobenius distance measure with AMDM for construction of the similarity weighting matrix. Finally, the obtained low-dimensional feature tensor is input into WKNNC with AMDM to implement the fault diagnosis of the rotation machinery. A fault diagnosis experiment is performed for a gearbox which demonstrates that the second-order tensor formed multisensor fused fault data has good results for multisensor fusion fault diagnosis and the formulated fault diagnosis method can effectively improve diagnostic accuracy.

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