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Calculate bimaterial H-matrix for orthotropic material.

Submitted by BrianBak on

Hi

I have a question regarding calculating the B-matrix which is used to create the bimaterial H-matrix in the article: "Singularities, Interfaces and Cracks in Dissimilar Anisotropic Media" by Zhigang Suo, 1989. 

It seems that the roots with the positive imaginary parts in equation (2.3), \mu_1, \mu_2 and \mu_3 has to be sorted in a certain way for the B-matrix to be hermitian. I have noticed this for specific values of isotropic materials, transverse isotropic materials oriented with the "fiber" in the x_1 direction and for transverse isotropic materials rotated around the x_3 axis, respectively. It is not mentioned in the article that they should be sorted. Does anyone know if the roots has to be sorted and could you please explain why and the system they should be sorted after? 

I be glad to send Maple sheets of my calculations if anyone care to see them. 

Thanks in advance

Regards Brian Bak

Mechanical Engineering Student at Aalborg Universit, Denmark

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BrianBak

Hi Zhigang

Thank you for your reply.

For an example with an orthotropic material rotated about the 3-axis the
compliance matrix is on the form:

      | x   x   x   0   0   x |

      | x   x   x   0   0   x |

S = | x   x   x   0   0   x |

      | 0   0  0    x   x   0 |

      | 0   0  0    x   x   0 |

      | x   x   x   0   0   x |

      where x indicates a value (not the same)

Then in equation (2.4) l_3 = 0 because all the compliance components
figuring in the equation are zero. This implies that \eta_1, \eta_2 and
\eta_3 in equation (2.7) are equal to zero. Then the L-matrix in
equation (2.5) has the form:

       |-\mu_1   -\mu_2    0 |

 L = |     1            1        0 |

       |     0            0        -1|

 My question is how to choose the values for \mu_1 and \mu_2 from the
three available roots (those with positive imaginary part). The A-matrix
in equation (2.6) is also influenced by the choice of \mu_1, \mu_2 and
\mu_3. With the material properties in my example there is only one
choice of the roots which in the end makes the B-matrix in equation
(2.12) a hermitian matrix. 

Regards Brian Bak

Mon, 05/03/2010 - 07:30 Permalink
Zhigang Suo

Dear Brian:  Thank you for additional input.  I'm still not sure what causes the problem.  I'll think more.  I collected proofs of several general properties of these matrices in the following paper.  Hope it is of use to you.  See Appendix A in

Z. Suo, C.-M. Kuo, D.M. Barnett and J.R. Willis, "Fracture mechanics for piezoelectric ceramics," J. Mech. Phys. Solids 40, 739-765 (1992).

Mon, 05/03/2010 - 11:22 Permalink
BrianBak

Dear Zhigang



Thank you for your reply.



 I was not able to find out how they should be sorted. In the examples I have calculated it seems that there are two ways of sorting the roots. If \mu = [a, b, c] makes the B-matrix hermetian then \mu = [b, a, c] also makes the B-matrix hermetian.



I have to correct what wrote in my first post. I have rotated the material about the y-axis and not the z-axis. This means that the form of the compliance is as the following instead:



      | x   x   x   0   x   0 |

      | x   x   x   0   x   0 |

S = | x   x   x   0   x   0 |

      | 0   0  0    x   0   x |

      | x   x   x   0   x   0 |

      | 0   0  0    x   0   x |



My observations has to do with this and not the rotation about the z-axis.



I have noticed that for some material combinations the sign of the displacements changes. In an example with two almost identical orthotropic materials the sign of the relative y-displacement of equation (6.18) is negative for pure mode I with K_2 = K_3 = 0. The predicted displacement seems to be correct if I change the sign. Is it correct that the sign can change and if so is then only the sign that is incorrect?



Thanks in advance



Regards Brian Bak

Wed, 05/05/2010 - 18:58 Permalink