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February  2014, 8(1): 247-257. doi: 10.3934/ipi.2014.8.247

The linearized problem of magneto-photoelasticity

1. 

Sobolev Institute of Mathematics and Novosibirsk State University, 4 Koptyug Avenue, Novosibirsk, 630090, Russian Federation

Received  April 2013 Revised  September 2013 Published  March 2014

The equations of magneto-photoelasticity are derived for a nonhomogeneous background isotropic medium and for a variable gyration vector. They coincide with Aben's equations in the case of a homogeneous background medium and of a constant gyration vector. We obtain an explicit linearized formula for the fundamental solution under the assumption that variable coefficients of equations are sufficiently small. Then we consider the inverse problem of recovering the variable coefficients from the results of polarization measurements known for several values of the gyration vector. We demonstrate that the data can be easily transformed to a family of Fourier coefficients of the unknown function if the modulus of the gyration vector is agreed with the ray length.
Citation: Vladimir Sharafutdinov. The linearized problem of magneto-photoelasticity. Inverse Problems and Imaging, 2014, 8 (1) : 247-257. doi: 10.3934/ipi.2014.8.247
References:
[1]

H. Aben, Magnetophotoelasticity - photoelasticity in a magnetic field, Experimental Mechanics, 10 (1970), 97-105.

[2]

H. Aben, Integrated Photoelasticity, McGraw Hill, New York - London 1979.

[3]

G. P. Clarc, H. W. McKenzie and P. Stanley, The magnetophotoelastic analysis of residual stresses in thermally toughened glass, Proc. of the Royal Society of London A, 455 (1999), 1149-1173.

[4]

S. Gibson, G. W. Jewel and R. A. Tomlinson, Full-field pulsed magnetophototlasticity, J. of Strain Analysis for Engineering Design, 41 (2006), 161-182.

[5]

Yu. Kravtsov, "Quasi-isotropic'' approximation of geometric optics, Dokl. Acad. Nauk SSSR, 183 (1968), 74-76.

[6]

L. Landau and E. Lifshitz, Theoretical Physics. VIII Electrodynamics of Continuous Media, (Russian) 2nd edition, Nauka, Moscow, 1982.

[7]

H. Poincaré, Théory Mathématique de la Lumiére. Reprint of the 1889 and 1892 Originals, Éditions Jacques, Sceaux, 1995.

[8]

F. E. Puro, Magnetophotoelasticity as parametric tensor field tomography, Inverse Problems, 14 (1998), 1315-1330. Éditions Jacques, Sceaux, (1995).

[9]

V. A. Sharafutdinov, The method of integral photoelasticity in the case of weak optical anisotropy, Eesti NSV Tead. Acad. Toimetised Fuus.-Mat., 38 (1989), 379-389.

[10]

V. Sharafutdinov, Integral Geometry of Tensor Field, VSP, Utrecht, the Netherlands, 1994. doi: 10.1515/9783110900095.

show all references

References:
[1]

H. Aben, Magnetophotoelasticity - photoelasticity in a magnetic field, Experimental Mechanics, 10 (1970), 97-105.

[2]

H. Aben, Integrated Photoelasticity, McGraw Hill, New York - London 1979.

[3]

G. P. Clarc, H. W. McKenzie and P. Stanley, The magnetophotoelastic analysis of residual stresses in thermally toughened glass, Proc. of the Royal Society of London A, 455 (1999), 1149-1173.

[4]

S. Gibson, G. W. Jewel and R. A. Tomlinson, Full-field pulsed magnetophototlasticity, J. of Strain Analysis for Engineering Design, 41 (2006), 161-182.

[5]

Yu. Kravtsov, "Quasi-isotropic'' approximation of geometric optics, Dokl. Acad. Nauk SSSR, 183 (1968), 74-76.

[6]

L. Landau and E. Lifshitz, Theoretical Physics. VIII Electrodynamics of Continuous Media, (Russian) 2nd edition, Nauka, Moscow, 1982.

[7]

H. Poincaré, Théory Mathématique de la Lumiére. Reprint of the 1889 and 1892 Originals, Éditions Jacques, Sceaux, 1995.

[8]

F. E. Puro, Magnetophotoelasticity as parametric tensor field tomography, Inverse Problems, 14 (1998), 1315-1330. Éditions Jacques, Sceaux, (1995).

[9]

V. A. Sharafutdinov, The method of integral photoelasticity in the case of weak optical anisotropy, Eesti NSV Tead. Acad. Toimetised Fuus.-Mat., 38 (1989), 379-389.

[10]

V. Sharafutdinov, Integral Geometry of Tensor Field, VSP, Utrecht, the Netherlands, 1994. doi: 10.1515/9783110900095.

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