Orthogonality and closure relations in Stroh formalism

Dear iMechanica members,

I deal with exteded Stroh formalism for piezoelectric media. I have problems with orthogonalization of Stroh´s matrices A, B, respectively how to fulfil the ortogonality relations (see attachments). I tried to "rape" it and made the left matrix inverse of the right matrix (then the product is an unit matrix) and I became very similar singularity expomemt as in the paper

(Hirai, H., Chiba, M., Abe, M., Ikeda, T., Miyazaki, N. Stress intensity factoranalysisofaninterfacialcornerbetweenpiezoelectricbimaterialsusingthe H-integralmethod. Eng. Fract. Mech., vol. 82, pp. 60-72, 2012.). 

But! the characteristic equation is not zero in [0,1]. Then the exponent does not converge to 0.5 for notch angle->180° and exhibits instabilities near 180 (attachment Hirai).

Have there anybody more experiences with Stroh formalism? (I have no doubts :-)) I tried also to use the explicit expression and it does not work at all.

In addition, I multiplied the left and right matrices (in attachment) and the resulting matrix was not unit, but made of scaling coefficients, which I multiplied the BA AB matrix elements. And I became the same resutls as in the case above.

Thank you

Regards

Miroslav