A theorem on potential of isotropic symmetric second-order tensor function

1983 ◽  
Vol 4 (1) ◽  
pp. 93-96
Author(s):  
Cheng Yuan-sheng
Keyword(s):  
Second Order ◽  
Tensor Function ◽  
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.

Two-dimensional tensor function representation for all kinds of material symmetry

1993 ◽  
Vol 443 (1917) ◽  
pp. 127-138 ◽  
Keyword(s):  
Finite Number ◽  
Recent Work ◽  
Dimensional Space ◽  
Second Order ◽  
Irreducible Representations ◽  
Material Symmetry ◽  
Two Dimensional ◽  
Tensor Function ◽  
Order Tensor ◽  
Function Representation

All kinds of physically possible material symmetry in two-dimensional space were investigated in a recent work of Q. -S. Zheng and J. P. Boehler. In this paper, we establish the complete and irreducible representations with respect to every kind of material symmetry for scalar-, vector-, and second-order tensor-valued functions in two-dimensional space of any finite number of vectors and second-order tensors. These representations allow general invariant forms of physical and constitutive laws of anisotropic materials to be developed in plane problems.

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.

Sign in / Sign up

Export Citation Format

Share Document