Strengthen damage added to an attack is non-elemental, I believe. That is, when using a Fire attack against a Wood virus, only the original chip damage is doubled.

Does Strengthen damage get modified by other things in the attack? i.e. can it Break, Impact, Spread, Splash, Blast, Wide Hit, and Backstab?
Further confusing the issue is Twi saying this:

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Kazuhiro: no, I was under the impression that Strenghten can never be elemental
Kazuhiro: you can't Strengthen a sigattack
Kazuhiro: and if you Strengthen an elemental chip, the extra damage isn't elemental
Soundman: Imbue Null: 20 (40 if off-elemental) [Allows you to add an element to a Null Element attack. The element must be declared when you register this ability, and can only target a Null Element attack. You cannot use this ability on a Rapid Buster Attack (known as a Buster Shot), but it may be used on any Null-Element offensive or defensive ability. You cannot imbue multiple elements on a single attack. It costs 20 if you are imbuing your element, and costs 40 if it is any other element. You cannot select Null as a choice.]
Soundman: Theoretically
Soundman: You could imbue the non-elemental damage

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Strengthen: Add 1:1 Damage [This ability allows you to add damage to a single Chip or Charged Buster attack. The damage does not stack over multiple hits or multiple targets and does not duplicate itself for multiple targets, or multiple hits, but the damage is splittable. This ability cannot be used to imbue Effects, only damage.]

Let's see...
Break: I think Strengthen adds to the total damage, and I think it gets modified by Break.
Impact: See above.
Spread, Splash, Blast, Wide: Does not stack over multi-hits. You CAN, however, Strengthen the Spread'd/Splash'd/whatever shot. I've done it before.
Backstab: It's basically just where you put the attack. I think Strengthen adds to it.

Also, I think Strengthen damage takes the element of the attack it affects.