Ardumoto motor driver shield can send 2amps current per channel.But what if we provide a battery with larger amp .Will it send/bear the excess current ? Or it will resist excess current??
it will either overhead/melt/break or if it has over current/heat protection, that will kick in.
The only way to really get more out of the Motor Shield is it replace or piggy-back the H-bridge IC.
If you were looking to get small gains you could use a heat sink on the IC, it is most limited by over heating.
First off, you draw amps - you don't provide them; you could have a 100 amp capacity battery powering your motor, but if the motor only draws one amp, that's all it's going to draw.
Now - let's say you had that same battery, and a 4 amp motor, and the h-bridge in-between (single channel mode) - on some h-bridges, you would likely smoke the bridge if it was limited to 2 amps output. However, on the L298 (which the ardumotor is based around), thermal limitations and overcurrent protection would kick in, shutting down the bridge; hence, no smoke.
Now - the Ardumotor doesn't have any heatsinks mounted on it, so I tend to wonder if you could even get near the 2 amp limit on a single channel (without a heatsink, I think the max you can pull is something like 500-750mA). So - before you even consider anything - you need to put some heatsinks on that board. You'll either have to clamp the heatsinks on, or use epoxy thermal paste or something (like what is used on some chipset/videochip heatsinks). The Ardumotor is based around the SMT version of the L298, and it doesn't appear like it has provisions for heatsink mounting, so epoxy or some mechanical clamping (which would preclude its use as a shield, of course) is needed.
Finally - the L298 has what is called a "bridge mode" where you can parallel the inputs and outputs of the chip to allow the use of the chip as a single channel, at double the current (so 2 + 2 amps = 4 amps) - you will definitely need a heatsink to do this, though (and note, since it would act as a single channel, only one motor could be controlled). It looks like you -might- be able to parallel the lines together, but you would want to consult the L298 datasheet, the schematic for the board, and the board itself - to figure out how to wire it properly for the bridge mode (the outputs definitely look "doable" - it's the inputs that seem unclear to me).
Note that there is as least one guy on the comments thread for the Ardumotor at Sparkfun asking the same question (how to use it in "bridge mode")...