Inextensible filament ($|{\mathbf{r}}_s|=1$)

An inextensible filament keeps the length of every material segment fixed. With material arc length $s$, its centerline obeys $|{\mathbf{r}}_s|=1$ and has unit tangent vector $\mathbf{t}={\mathbf{r}}_s$. A filament tension acts as a Lagrange multiplier for this constraint.

The filament bending modulus multiplies squared curvature of a space curve in the bending energy, $E_b=(B/2)\int {\kappa }^{2}ds$. It has dimensions force times length squared. In the small-slope Monge representation, the leading elastic force per unit length is $-B{y}_{xxxx}$.

Filament tension is the axial force enforcing local inextensibility. Its force density is ${\partial }_s(T{\mathbf{r}}_s)$. A pre-existing constant tension contributes $T{y}_{xx}$ to small-slope transverse dynamics; in an unloaded, initially straight filament, tension induced by transverse motion begins at second order in amplitude.