  In a recent paper, Bona and Smith define the notion of \textit{strong
avoidance}, in which a permutation and its square both avoid a given pattern.
In this paper, we generalize this idea to what we call \textit{chain
avoidance}. We say that a permutation avoids a chain of patterns $(\tau_1 :
\tau_2: \cdots : \tau_k)$ if the $i$-th power of the permutation avoids the
pattern $\tau_i$. We enumerate the set of permutations $\pi$ which avoid the
chain $(213, 312 : \tau)$, i.e.,~unimodal permutations whose square avoids
$\tau$, for $\tau \in \S_3$ and use this to find a lower bound on the number of
permutations that avoid the chain $(312: \tau)$ for $\tau \in \S_3$. We finish
the paper by discussing permutations that avoid longer chains.
