How to transform the stiffness tensor of a rhombohedral crystal from crystallographic frame of reference to laboratory fame of reference. For crystal structures having orthogonal crystallographic axes (like tetragonal or orthorhombic), one can simply use the transformation of axes system (like Euler's angles based transformation matrix) to simply rotate the tensor property to achieve the goal. But I am not sure how to achieve the same with crystal with non orthogonal axes like rhombohedral/trigonal crystals with axes making angles of 60 degrees with each other.

Alexus Deleon

Alexus Deleon

Answered question

2022-09-24

How to transform the stiffness tensor of a rhombohedral crystal from crystallographic frame of reference to laboratory fame of reference
For crystal structures having orthogonal crystallographic axes (like tetragonal or orthorhombic), one can simply use the transformation of axes system (like Euler's angles based transformation matrix) to simply rotate the tensor property to achieve the goal. But I am not sure how to achieve the same with crystal with non orthogonal axes like rhombohedral/trigonal crystals with axes making angles of 60 degrees with each other.

Answer & Explanation

Marley Stone

Marley Stone

Beginner2022-09-25Added 13 answers

Let's say your crystal basis is: a , b , c . These vectors are linearly independent, but need not be orthogonal or normalized.
Let's say your lab frame basis is: x ^ , y ^ , z ^ . These vectors are orthogonal and normalized.
In general any vector V can be expressed in the lab-frame
V = V 1 x ^ + V 2 y ^ + V 3 z ^ = V i e ^ i
Note the notation, V i is the i-th component of the vector in the lab-frame, and e ^ i = 1 , 2 , 3 = x ^ , y ^ , z ^ . The repeated index implies summation, i.e. V i e ^ i = V 1 e ^ 2 + V 1 e ^ 2 + V 3 e ^ 3
The same vector can be written in the crystal basis:
V = V ¯ i u i
Where V ¯ i is the i-th component in crystal frame and u i = 1 , 2 , 3 = a , b , c
Since both crystal frame basis set and the lab-frame basis set are linearly independent there will exist an invertible matrix Λ j i , such that:
V i = Λ j i V ¯ j
I.e. it converts the vectors between the two frames. For example, if you were to define
x ^ = ( 1 0 0 ) , y ^ = ( 0 1 0 ) , z ^ = ( 0 0 1 )
and
a = ( a x a y a z ) , b = ( b x b y b z ) , c = ( c x c y c z )
Then the Λ j i would be in the i-th row and j-th column of matrix:
( a x b x c x a y b y c y a z b z c z )
Now we can talk about the stiffness tensor. Firstly, 6 × 6 makes no sense in 3d space. Stiffness tensor is a rank- 4 tensor with exchange symmetry in the first two and the last two indices. It does mean that it has 36 components
So, define this stiffness tensor like that:
σ i j = C i j k l ϵ k l , C i j k l = C j i k l = C i j l k
Where C is the stiffness tensor, ϵ is elsticity tensor and σ is the stress.
Assuming that, defined the tensor correctly, you can follow the standard conversion rules for the tensors. If the components of the stress tensor are:
C ¯ i j k l in crystal basis,
the componentents in the lab frame are:
C i j k l = ( Λ 1 ) i i ( Λ 1 ) j j ( Λ 1 ) k k ( Λ 1 ) l l C ¯ i j k l

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Relativity

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?