Describe the “American Way of War” from World War I to the War on Terror. How does it evolve, and how do American views of the military and military mission change over this period?

The American Way of War, especially in conflicts since the turn of the twentieth century, is not necessarily one of strategy and surrender but one of decimation. The idea is to stamp out all opposition and win a decisive victory (to paraphrase Ender's Game, "not just to win this battle, but to go ahead and win every other battle"). In prior conflict, such as the Revolutionary and Civil Wars, it was not necessary, possible, or beneficial to completely eliminate the opponent.
In recent years, however, the conflict has changed. Think, for example, of the bombing of Hiroshima and Nagasaki. The idea was to decimate the national morale of Japan and claim a decisive victory in one fell swoop. Against terror groups in modern-day conflicts, the objective is to completely eliminate or capture all opposing agents and, by doing so, prevent anyone from coming back to wage a subsequent battle.
It is brutal, and it is not like historical conflicts. However, its supporters argue that this method of battle—the "American Way of War"—is one of decisive and enduring victory, as opposed to one which, in all likelihood, will result in a resurgence of the enemy.