Would Move Semantic in C++ 11 be executed if there is no rvalue parameter with function signature?

To make program run more efficiently, move semantic is introduced since C++ 11, for example:

void fun(const string& str); //1st
void fun(string&& str); //2ed since c++ 11

If I use it like fun("tmpStr");, the 2ed function signatured with rvalue would be used,and it is more efficient than the 1st function.

But the problem is that if I need ONLY 1 function signature to handle paramters with both lvalue and rvalue, what shoud I do?

1. if the 1st one is kept, it is not efficient with rvalue;
2. if the 2ed one is kept, it is efficient with rvalue, but I have to fun(std::move(lvalue)) to make it possible with lvalue, which I think the added codes std::move looks like redundant---moreover, this makes the status of lvalue undefined after this function.

With the thought above, I wonder if Move Semantic in C++ 11 would be executed if there is no rvalue parameter with function signature like the 1st one, even just in release(optimized) mode? if the answer is not , then what is the reason behind it?