Theory
Theory
Let's consider a vertical plane defined by a phi rotation around the X3 axis.
![](images/00.bmp)
The unit vector is defined:
![](images/01.bmp)
Let's consider also the following convention for the stiffness tensor and the symmetric condition:
![](images/02.bmp)
We can also use that convention for lambda. Defining l, m and n as follows and expanding for lambda 1 to 6 we have:
![](images/03.bmp)
![](images/04.bmp)
From the Christofell's equation we have:
![](images/05.bmp)
And the determinant must be zero:
![](images/06.bmp)
Defining A as follows, expanding the determinant we arrive to the following polynomial in A:
![](images/07.bmp)
Now let's define the polynomial coefficients:
![](images/08.bmp)
In order to find the roots, we will find first the location in A of the maximum and minimum values of the polynomial, by taking the derivative:
![](images/09.bmp)
The three roots
![](images/10.bmp)
of the original third grade polynomial will be located one at the left of A1, one in between of A1 and A2, and one at the right of A2.
![](images/11.bmp)
Now, we can express the following:
![](images/13.bmp)
and considering that alfa is a unit vector:
![](images/12.bmp)
We arrive to the following results for alfa:
![](images/14.bmp)
wn which beta1, beta2 and DB are defined:
![](images/16.bmp)
![](images/15.bmp)