I have a physics background so apologize my ignorance.
I'm confused by the constants $D$, $\kappa$, and $J$ when it comes to torsion of beams. From what I understand $\kappa$ and $J$ are the same thing, $J$ is (usually) for round beams with isotropic materials while $\kappa$ is for beams with unusual shapes and/or material properties.
What I really don't understand is how you get from $D$ to $\kappa$ though. In a paper, call it paper 1, I found, in the case of an orthotropic beam with rectangular cross section, they derive equations for torsional rigidity $D$ then mention equations such as
$$ \kappa=\frac{D}{2ba^3G^0_x} $$
where $b$ and $a$ are the height and width of the beam while $G^0_x$ is the shear modulus. Where did this equation come from? Is it a standard result? Did they derive the equation for $\kappa$ from FEM simulations?
Then in this paper (call it 2), which deals with torsion of arbitrarily shaped orthotropic composites, they talk about the torsional rigidity $GJ$, which makes me doubt my previous statement about $J$. One of the equations that they give is that the torsional rigidity factor $\beta$ (which is $\kappa$, right?) is
$$ \beta=\frac{GJ}{ab^3G_{zx}} $$
From what I can tell $GJ=D$ from paper 1 and like I said $\beta=\kappa$. So once again I wonder how $D$ (or $GJ$) and $\kappa$ (or $\beta$) are related.
I should explain that the reason why I'm looking to understand better these relationships is because I'm 3D printing springs. I'm looking to find a relationship between the shear moduli of an orthotropic material and the spring constant of a coil spring.
One method that I found is to relate the energy stored in a spring to the energy stored by a beam under torsion $\frac{1}{2}kx^2=\frac{1}{2}\kappa \theta^2$, where $k$ and $x$ are the spring constant and deflection, and $\kappa$ and $\theta$ are the torsional constant and angle of twist of the material.
The two big problems that I have are that my springs have square cross sections (because they're easier to print) and are orthotropic. Hence why I'm having so much trouble.