From Pressure Vessel Design: The Direct Route
The following theorems from [41,32,4] are useful in the setting up of the load case specification lists, see Section 3.3.1, and they are also useful in pre-checks in the context of the Progressive Plastic Deformation Design Check, in pre-checks whether detailed checks or detailed investigations are required. The theorems are proven, and therefore valid, for shakedown to linear-elastic behaviour.
Theorem 1
Each cyclic action for which the corresponding elastic stress field is enveloped by the envelope of the elastic stress field of a shakedown action is also a shakedown action.
Theorem 2
A cyclic action can be a shakedown action only if for each point of the model a time-invariant stress 
can be found such that the superposition of this stress with the cyclic elastic stress due to the cyclic action is always compatible with the yield condition. The time-invariant stress may be different for each point of the structure.
In cases in which, in each point of the model and during the whole action cycle, the yield stress is invariant, this means that it must be possible to shift all of the cyclic elastic stress trajectories (in the stress-space) due to the cyclic action translatorially such that they are enveloped by the (invariant) yield surface. The translatorial shift may be different for each point of the model.
In the more general cases in which, in each point of the model and during the cycle, the yield stress may vary (cyclically), e.g. due to temperature changes, this means...
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