Dead loading

In an elastic boundary value problem, a dead load is a prescribed force independent of the incremental displacement. A dead reference traction has $\delta t^0=0$, even when the current face changes orientation. This differs from a follower load, whose direction follows the moving body, and from the structural-engineering use of the same phrase for permanent self-weight. Under pure dead tractions, a constant incremental translation remains a zero-frequency mode unless a displacement condition fixes it.

For $\Phi =\mu [{\sum }_i{v}_i^2/2+{\sum }_i{v}_i^{-1}]$ and positive stretch product one, the energy simplifies to $\mu S^2/2$, where $S={\sum }_i{v}_i$. Equal nominal tensile loading adds potential $-\tau S$. The homogeneous diagonal equilibrium equations reduce to $(\mu S-\tau )(v_i-v_j)=0$. Thus the unit triple exists for every load, while the other equilibria form the displayed sum-product family. Since $S\ge 3$, only the unit triple exists for $\tau \le 3\mu$. Above this load, a continuous family is obtained by choosing $v_1=t$ near one and taking $v_2,v_3$ as the two positive roots with sum $\tau /\mu -t$ and product $1/t$.

A dead-load traction is prescribed force per unit reference surface area, with its vector held fixed as the body deforms. Its boundary condition is $T_{R}N=t_R$. A follower load instead depends on the current deformation, for example through a current normal or current area.