Smartphone and TRIM

From a forensics standpoint TRIM is basically a moot point on most smartphones given most implement encryption with a hardware component preventing full dumps of the physical raw data anyway. Even if you get root access, you're limited to the file system for most scenarios on iOS and Android. You may have some success with very early Android.

encryption is an additional level of abstraction that is introduced but does not change the way the hw works.
my attenction is on the fact that the active TRIM means that simply by turning on the smartphone the TRIM works by deleting the blocks that the operating system passes on each change on filesystem.
Admitting to being able to make a physical acquisition of the smartphone the changes that the phone undergoes are not only related to the start of the operating system and the apps but also to the TRIM which in fact goes to prevent the reconstruction of the deleted data.
I am lookgin for sceientific o technical paper that confirm the implementation and use of TRIM on Android and on iPhone

TRIM can prevent you from recovering deleted data. But, as mcman said, it doesn't matter in most cases because encryption will prevent recovery of deleted data anyway.

I am lookgin for sceientific o technical paper that confirm the implementation and use of TRIM on Android and on iPhone

That sounds like a fairly simple literature search. You should not have any problems with that. If you have problems, most research library should be able to help you – most have beginner's courses on that particular topic.

I'm assuming that you have done this already, but have not come up with anything. If you've done a good job (which presupposes the basic training already mentioned), that should be a fairly strong indication that there aren't any paper on the topic.

In such case, you may stop there. Otherwise …

For implementation, you need to go the memory device itself, which is what implements TRIM. The specification is available, probably in the ATA standard or equivalent, but if it is implemented on a particular device, and how it actually implemented, is something you will have to research. The manufacturer's device specification as well as any hardware reference manuals or developer's kit documentation may help, as may any tech support forum the manufacturer provides. (It's usually not a major problem to get an account you may need to explain what you're doing to a support rep.) But this is also where you're likely to run into deliberately unspecified functionality if so, you'll have to fall back on direct HW research work if you really need the information.

For Android, you have substantial source code access for file systems covered by Android source code you should be able to find how different versions of Android handles TRIM. However, I believe that phone manufacturers are free to do their own thing – so you also need to check if a particular platform and release does do their own thing – in which case, you're fairly clearly on your own. For other file systems not supported by Android, it's the same thing … you'll have to research those separately.

You may get additional help and info in Android development forums.

Your question seems to be too broad as it is formulated. If you need details, you need to narrow it down quite a bit.

Thanks so much Athulin for your explain
I have been googling for some days looking for some tech paper that confirm that apple now is using TRIM on last version of iOS but the only thing I have found is that starting from AFS the iOS support TRIM, so I can deduce Apple is using TRIM on the last version, but nothing official from apple