Andrew Smith is a 3rd-degree BJJ black belt based out of Richmond, VA (Revolution BJJ). He runs BJJ Path, a video tutorial website.
The armbar from the closed guard is a true staple of Brazilian Jiu Jitsu training. As a result, so are the typical follow up attacks. If the person pulls their arm free, you can switch to a triangle. If the angle changes sufficiently, you can switch to an omoplata. While these are incredibly effective moves, all of your training partners will see them coming at some point. A nice addition here is the "armbar to armbar" combination, or the "side switch armbar." This is just a fancy way of putting the idea that you are switching to the other arm to armbar, but there are some subtle, sneaky steps involved.
Often times, even though your partner knows it's not the greatest idea in the world, they're going to pull their arm completely free whenever you go for an armbar, especially in no-gi training or competition. When this happens, sometimes a transition to a triangle or omoplata either isn't possible or isn't in your best interest at the time. Switching sides is a great possibility. As they pull free from the traditional armbar from guard, assuming you're going for your partner's left arm, slide your right shin across your partner's neck, creating a frame. The simplest way to think about it is to point your right knee toward the ground. Use this frame to push off and get the armbar on the other side.
Dealing With the Hand
When your partner defends the armbar, you may have to contend with them hiding their hand behind your knee. Here, as you're sliding your shin across their face, you'll also want to pull their hand through as you begin to turn. Because their grip is predicated on blocking the leg you're trying to throw over their head, you can take advantage of the fact that you're taking this "platform" away from them, sneaking your focus over to the other arm while grabbing the wrist you'll need. If your partner continues trying to pass your guard (a very likely next step for them once they get the arm free; people often build entire guard passing games off of failed submission attempts, and the armbar is no exception), you should be able to finish the armbar from your back. If they hesitate here, fearing the trap you've set, you may need to do a belly-down armbar instead.
With the Gi
While the basic mechanics of the position are the same for gi or no-gi, with the gi, you can add an element of sleeve control.
Here, Jarrett sets up a really nice lapel grip armbar from guard. Never mind all that fanciness, though, as the armbar is essentially the same. However, this particular set up allows your partner to defend by hiding the wrist of the arm you're trying to "break," so this will be a fairly common scenario from this exact set up. This time, as your partner begins to hide their arm to stack, you will need to lift your hips up so that you will be able to pivot into the correct position. Notice that, while this is a similar scenario as those described before, it's fundamentally different because your partner isn't simply ripping their arm free, but rather trying to drive their weight forward on top of you. Act accordingly.
Per usual, there are a great many practical applications for this technique, as demonstrated by the subtle differences in the way the technique is taught here. You ultimately must understand the fundamental principles at play so that you can then apply your own reasoning to determine which specific steps are necessary, and so that you can further do your own troubleshooting. Remember that these are specific examples with the ultimate goal of you understanding the overall concepts well enough for you to be able to come up with your own variations. As always, let me know if these techniques work well for you!