Torsional rigidity D vs torsional rigidity factor k vs torsion constant J

Question

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.

Answer 1

Different texts use different notation.

$K$ is sometimes called $J$, torsional stiffness constant, units of $L^4$.

$G$ is shear modulus of rigidity which is sometimes assumed to be 40% of $E$.

$GJ$ or $JK$ is called section modulus of torsion rigidity as $EI$ is in a beam bending moment.

The frequently used equations can be found in Roark's Formulas for Stress and Strain, 7th edition, chapter 10, pp 404.

I am using my phone app, but if you need more detail I responded on my computer using $\LaTeX$.

The equations for torsion in a rectangular solid section:

$$\theta =\frac { TL }{ KG }$$

$${ \tau }_{ max }=\frac { 3T }{ 8a{ b }^{ 2 } } \left[ 1+0.6095\left(\frac { b }{ a }\right) + 0.8865{ \left(\frac { b }{ a } \right) }^{ 2 }-1.8023{ \left(\frac { b }{ a } \right) }^{ 3 }+0.9100{ \left(\frac { b }{ a } \right) }^{ 4 } \right]$$

$$K=a{ b }^{ 3 }\left[ \frac { 16 }{ 3 } -3.36\frac { b }{ a } \left(1-\frac { { b }^{ 4 } }{ { 12a }^{ 4 } } \right) \right] \text{ for } a\ge b $$

$\theta = \text{angle of rotation [rad]} \\ \tau = \text{torsion stress} \\ K = \text{torsional stiffness constant}$