Density of an object in Relativity

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If the density of an object increases in special relativity (as mass increases and volume decreases), wouldn't we see strange phenomena as skyrockets exploding (because some materials can't sustain to much pressure)??
 
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It's only the "relativistic mass", which is similar to velocity in that it depends on your choice of reference frame, that increases...in the object's own frame its relativistic mass does not increase, and its "rest mass" (which is what physicists usually mean by 'mass') is not dependent on your choice of frame. So, the rocket won't explode or show any other odd behaviors which would be noticed by someone at rest relative to it, regardless of its velocity relative to us.
 
So the mass (and furthermore the density) can only be measured in a non relative frame of reference (as that of an accelerating one) otherwise you would never know the real mass of yours, hence the density.
 

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