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Electrical Resistance and chemistry

Can some one describe or explain what happens when too much current is passed through lets say a copper wire, I am looking for an explanation to do with physics and chemistry. For instance we have ...
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1answer
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What is the approximate electrical conductivity $\sigma$ of graphene in S/m or S/cm?

I am trying to find an approximate value of the electrical conductivity $\sigma$ of graphene in units of S/m or S/cm. This table on Wikipedia gives $\sigma$ values for a variety of materials ...