AWS has a standard, A5.1, that explains the different levels of testing required for each "schedule" per AWS and ASME.
Unless the purchaser states additional testing is required, the testing described in the manufacturer's QC manual is adequate. That might involve welding a procedure plate and testing for the mechanical properties once a year using materials drawn from stock.
Filler metals for military applications require more stringent testing, but there are several filler metal standards used by different armed services. Each has different requirements. While the filler metal is "certified" to produce the mechanical properties required by the military standard, the materials may not be tested as frequently as required by military standards you are using.
It is best to order filler metals that have to meet military requirements directly from the manufacturer and make sure you specify what level of testing is required and what supporting documentation is required with the shipment.
One of the areas reviewed during a military audit is the purchase order used to purchase filler metals and how incoming receiving inspection handles the filler metals once it arrives at the fabricator's door step. Receiving inspection is supposed to check the information provided by the filler metal manufacturer against the purchase order and the applicable military standard. In some case, a sampling of the filler metal is to be sent out to a third party laboratory and the results compared to the information provided by the supplier if the filler metal isn't order directly from the manufacturer.
Again, each filler metal standard has it's own requirements. You, as the purchaser, have to have a copy of the appropriate standard to know what is required.
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Best regards - Al