If I used a via of 8 mils diameter with 20 um thickness. As per sierra tool calculator it can carry current of 1 A at 10 degree rise in temperature. What happens if the current is more than 1 A? When will the via fail? How the failure will be in PCB?
10 °C rise isn’t much for a PCB trace. Even 20 °C or 30 °C is usually fine. In some cases, voltage drop across the trace, or power dissipated in the trace, are too high for system design reasons when the rise gets high, say 30 °C or 50 °C.
From a strictly thermal standpoint, there are maximum temperatures for the dielectric being used. For common FR4, this is often 110 °C to 130 °C and is determined by the rated Tg (glass transition temperature). Above this temperature, the board becomes soft, and mechanical and electrical properties are degraded.
In the absence of detailed information, I’d keep thermal rise to 10 °C to 30 °C absolute worst case, and if pushing 30 °C, check that voltage drops aren’t an issue.
Higher temperatures are also undesirable due to thermal stress and cycling. The laminate expands much more than the copper plated onto it, causing the copper to stretch, fatigue, and eventually crack.
Reduce heat by using less current in the first place, using more vias in parallel, wider traces, more layers, etc.