It depends on how the next hop routers are connected to your router.
Theoretically, if the interface of the router leading to the "next hop router" goes down, then that route will be removed from the routing table. And if there is a default route then it will be used to forward all the traffic.
IF your next hop router is connected on lan and reachable via the ethernet interface in your router, when the next hop router goes down, your router will not be aware about it. It will still try to ARP for the ip address next hop router and retrying. The default route will not take effect in this scenario.
If your next hop router is connected via a point to point serial interface, then when the next hop router goes down, the serial interface in your router goes down. Hence all routes associated with that serial interface will be removed from the routing table. In this case, the traffic will be forward via the default route.
Generic answer would be "yes, it will use default route". More detailed answer would require knowing whether your static route was configured with both next-hop and interface or only with next-hop address.
To prevent traffic from following default route when next-hop dies, typical solution is install second route for that specific destination but with higher administrative distance and point it to Null0 interface. Higher administrative distance will ensure that this route is not used as long as normal route is alive, and pointing to Null0 will cause packets for that destination to be dropped if primary route goes down.
Here is an example:
ip route 10.0.1.0 255.255.255.0 192.168.1.2
ip route 10.0.1.0 255.255.255.0 Null0 250
As long as 192.168.1.2 is reachable (=there is route to reach it, not necessarily ping'able), traffic for 10.0.1.0/24 will be sent towards it. If 192.168.1.2 becomes unreachable, first route will be removed and second route being installed will cause packets for 10.0.1.0/24 to be dropped.
If next-hop route is on serial or ethernet point-to-point link, next-hop will disappear if link goes down, so no problem. However if IP link is done via switching infrastructure, then next-hop going down will not cause link on other routers to go down, so first route will not be removed (though net-result is still the same - traffic is not sent via default route).
We are pleased to announce availability of Beta software for 16.6.3.
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