c++ size_t arithmetic with negative value

I was asked to make a copy of std::string in an assignment and I am having problem implementing the substr function. In a set of tests the teacher gave us there was one with a length equal to -1. Mine declaration of substr is:

Cadena substr(size_t start, size_t length) const;

Which I (thought) that the size_t would prevent negative values to be passed. The problem is that in the definition I check for size() < start + length (assume tam_ is the same as size()):

    if (tam_ <  start + length)
        throw std::out_of_range("Error");

In my system -1 in unsigned is 18446744073709551615, so that, for example assume start is 9 and tam_ is 10.

I expect:

10 <  9 + 18446744073709551615

So that the exception is thrown, but in reality I get

10 < 9 + (-1)

Which is false and exception isn't thrown. As the function continues it allocates a char array for size length + 1, which the system refuses due to that new[] threats the size_t as it should be, 18446744073709551615, which is so big and make the program crash.

I want to know why my expected result isn't the correct.

Thanks