Actually, the stiffness matrix is not a matrix, but the fourth-order tensor, presented with the Voigt notation. We just present it as a matrix. There are two ways of rotation transform of VTI stiffness tensor to the global frame.

Method 1. You compose the fourth-order tensor from this matrix, rotate it according to the rotation rules for the fourth-order tensor, and then you can again present it as 6x6 matrix.

Method 2. This approach makes it possible to rotate the matrix directly, without transforming it to the fourth-order tensor. This method is explained in the paper: Bond, W., Jan. 1943, The mathematics of the physical properties of crystals, The Bell System Technical Journal, 1-72. A suitable modification of the same method is given in the book: Slawinski, M., 2015, Waves and Rays in Elastic Continua, Third Edition, World Scientific Publishing Co, ISBN 978-981-4641-75-3.

Note that in the local frame the VTI stiffness matrix has many zeros, but after rotation to the global frame it becomes fully populated. Good luck

You can find how to rotate the system in the book:

Acoustic Fields and Waves in solids. Volume 1. Author: B. A. Auld

The transformation can be found in Chapter 3: Elastic properties of solids.

Actually, the stiffness matrix is not a matrix, but the fourth-order tensor, presented with the Voigt notation. We just present it as a matrix. There are two ways of rotation transform of VTI stiffness tensor to the global frame.

Method 1. You compose the fourth-order tensor from this matrix, rotate it according to the rotation rules for the fourth-order tensor, and then you can again present it as 6x6 matrix.

Method 2. This approach makes it possible to rotate the matrix directly, without transforming it to the fourth-order tensor. This method is explained in the paper: Bond, W., Jan. 1943, The mathematics of the physical properties of crystals, The Bell System Technical Journal, 1-72. A suitable modification of the same method is given in the book: Slawinski, M., 2015, Waves and Rays in Elastic Continua, Third Edition, World Scientific Publishing Co, ISBN 978-981-4641-75-3.

Note that in the local frame the VTI stiffness matrix has many zeros, but after rotation to the global frame it becomes fully populated. Good luck

hi..you have to use directional cosine matrix to find the stiffness matrix of a rotated section. For example the stiffness matrix of an un-rotated section is known and stiffness matrix of the rotated section can be estimated by simply multiplying stiffness matrix of unrotated with directional cosine matrix. The directional cosine matrix can be obtained with respect to 3 directional coordinates. I suggest you to read the text books i.e First course in the finite element method' by Daryl L. Logan, 4th edition, 2007, published by Thomson Canada Limited..ok all the best..

Diego Salam Claro Igor Ravve Karthigeyan . S thank you very much for the references! I have tried the methods you recommended and successfully rotated the stiffness tensor. One that sparked my curiosity then is when I observed the result of rotation along x-axis with angle 45 degrees, some of the stiffness elements become negative. I just wonder if negative is okay and reasonable? Then I'm also trying to find out how to rotate a bedding plane of anisotropy with given azimuth and dip, since geologic information is often in azimuth and dip. I look forward more for your suggestions on this

To verify whether your rotation is correct, compute the invariants. The stiffness tensor has 21 independent components due to 18 invariants and three orientation angles. Six of these 18 invariants are the eigenvalues of 6x6 matrix. Compute them before and after rotation. They are the same for vertical and tilted transverse isotropy

Igor Ravve Exactly as you predicted, Igor. I found the matrix fully populated. This is a very good suggestion, thank you very much, I will check the invariants

The process is described in Appendix 2A of Grechka and Heigl (Microseismic monitoring, 2019).

Thank you very much for the valuable reference Vladimir Grechka

As per today, 10/7/2020, I posted a Python code about this in GitHub, here: https://gist.github.com/yohanesnuwara/787f53c4e2e135531c650de619eb5c4f

Opened to any feedback. As suggested by Igor Ravve , next investigation will be measuring the tensor invariants, which potentially need a kind of collaborative work. Whenever you're interested, just poke me.

Thanks guys for the valuable answers, really appreciate it!

Thank you, Ignatius Yohanes Nuwara. To summarize, the eigenvector corresponding to distinct eigenvalue of contraction of a general TTI stiffness matrix recovers the direction of its symmetry axis. Complement of this direction by any two orthogonal vectors is a rotation matrix back to VTI.

Colleagues, a small comment. A second order tensor has eigenvectors.

A fourth order tensor has second-order eigentensors.

We can present the fourth-order stiffness tensor as a 6x6 matrix, it has six eigenvalues and six eigenvectors of length 6.

However, each vector of length 6 is actually a second-order tensor (3x3 matrix). Due to symmetry of this matrix, there are only six distinct components (and these six components can be presented as a vector - but this is not a true vector).

In the TTI case, there is a double eigenvalue, another double eigenvalue and two distinct simple eigenvalues. We present the eigenvectors corresponding to the two simple eigenvalues as two 3x3 matrices. Each of these matrices, in turn, has three eigenvectors of length 3, which may be arranged as another matrices. For the two distinct simple eigenvalues, these two 3x3 matrices )consisting of eigenvectors of length 3) are identical up to the order of columns. Each of them represents a rotation matrix between the tilted (global) and the crystal (local) frames.

In the global frame, any tilted symmetry (TI, orthorhombic, monoclinic, and) represents a fully populated 6x6 matrix with no zeros. However, for all types symmetries, the eigenvalue analysis can reveal a rotation matrix to the crystal frame. The 21 independent stiffness components are defined by 18 elastic invariants and three orientation angles.

In 1856, Lord Kelvin published a paper:

Thomson, W., 1856, Elements of a mathematical theory of elasticity, Part I: On stresses and strains, Philosophical Transactions of the Royal Society of London, 146, 481–498. The second part of this study was never published. The continuation of the Kelvin's paper appeared more than 150 years later,

Helbig, K., 2013, Review paper: what Kelvin might have written about elasticity, Geophysical Prospecting 61 (1), 1–20.

Both papers are accessible in the internet.

This discussion has some interesting points. I would like to add some hints and a warning. Stress and strain are 2nd order tensors which are transformed twice with a proper orthogonal rotation matrix. The stiffness is a 4th order tensor which is transformed four times.

The matrix representation looses the tensor properties. There is no rotation in 6 dimensional space. But a 6x6 proper orthogonal matrix still does the transformation. There is 1 transformation of stress and strain matrix. The transformation matrix now contains products of sine and cosine to represent the 2 transformations of the 2nd order tensors. The stiffness matrix needs 2 transformations so that sine and cosine appear up to 4th power.

You may find the transformation matrix in several sources such as

T.C.T. Ting: Anisotropic Elasticity. Theory and Applications. Oxford Univ Press 1996, page 55.

Warning: Use the transformation matrix with some care because the transformation matrix depends on the ordering of shear in the matrix presentation of stress and strain and not all authors use the same order.

Dear Ignatius,

Take a look at the supplementary data in Article Compressive failure of fiber composites containing stress co...

Good luck,

Vedad.

This 6x6 matrix comes from the fourth order tensor. So why not rotate this fourth order tensor first and then redefine the rotated one into 6x6.

It is simpler to rotate the 6x6 stiffness matrix, not the tensor. It is suitable to write the equations in matrix form than in component-wise form

Hello Ignatius Yohanes Nuwara, I just found your question yesterday and I would like to share my thoughts. You can find it in the attached pdf. But I still need to emphasize that the transition matrix depends on the ordering of shear, as mentioned by professor Staat Manfred Staat. My convention is xy, xz, yz. I have also verified the result with that generated from direct transformation of the 4-th order tensor.

However, as for the tensor invariants, I found the six eigenvalues are not the same for VTI and TTI Igor Ravve . I guess this is because we are doing a congruent transform, not similarity transform. The congruent transform can only guarantee the inertia of the matrix is not changed. I would like to hear your opinions. Thanks!

Ignatius Yohanes Nuwara I found in your code, you use:

Y_inv = np.linalg.inv(Y) S_dot = np.dot(Y, S) S_trans = np.dot(S_dot, Y_inv)

This is not correct. It should be the transpose of Y, not the inverse of Y, because Y is not orthogonal.

Dear Dr. Qi Zhang , please specify the parameters of the transverse isotropy that you used in your numerical example, like axial compressional and shear velocities, Thomsen delta, epsilon and gamma, and two orientation angles: zenith and azimuth of the medium axis of symmetry (given in the global frame), and I will compute the eigenvalues in the global frame and in the crystal frame. I am pretty sure that they will be identical because these are inherent properties of this medium, independent of its spatial orientation. Thank you.