Suppose a spring with stiffness $k$, is strained by constant forces on each end.
In a frame where the strained spring moves at a constant velocity, what's the total relativistic energy of the moving strained string as $k\to\infty$?
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The energy is the same no matter what the strain. The stiffness affects the compression of the spring, not the energy. In other words the higher k is the higher the energy density it has in terms of (potential energy)/(strain).