Hookean Spring
Hookean Spring is a constitutive model for simulating linear elastic springs connecting two particles in 3D space.
#12 Hookean Spring
For a spring element connecting two particles at positions \(\mathbf{x}_0\) and \(\mathbf{x}_1\), we define:
Distance Vector:
\[
\mathbf{d} = \mathbf{x}_1 - \mathbf{x}_0
\]
Current Length:
\[
L = \|\mathbf{d}\|_2
\]
Strain:
\[
\epsilon = \frac{L - L_0}{L_0}
\]
where \(L_0\) is the rest length of the spring.
Strain Energy Density
The strain energy of the Hookean spring is given by:
\[
E = \frac{\kappa}{2} \epsilon^2 = \frac{\kappa}{2} \left(\frac{L - L_0}{L_0}\right)^2
\]
Substituting the expressions for \(L\):
\[
E = \frac{\kappa}{2} \left(\frac{\|\mathbf{d}\|_2 - L_0}{L_0}\right)^2
\]
where:
-
\(\kappa\) is the spring constant (stiffness parameter)
-
\(L_0\) is the rest length of the spring
-
\(\mathbf{d}\) is the current displacement vector between the two particles