The convex shape of the aspis and slight "lip" around the rim allowed the Classical hoplitai to rest the shield upon the pauldron/shoulder while maintaining a grip on the shield utilizing its interior strap (which was innovated at two important intervals, the dates of which I cannot recall at the moment, but will expand upon later). which was a Carian innovation stemming from the mid-8th century BCE.

In doing so, the shield would cover the upper body of the hoplite, as well as the partially exposed spear-arm of the compatriot to their left in the classical hoplite phalanx. It should be noted, as well, that this shield (known as the Argive shield) (the name of which - I am leaning towards Argive - I am also blanking upon, but will correct later on) was an improvement made in the Peloponnese (not by the Spartans, either!)before the dawn of the Greek Golden Age. is an innovation attributed to the city-state of Argos.

It is a marked improvement over the Boeotian or Thracian - or even pelta that lends its name to the peltastai or skirmishing forces of ancient Greece - shields that would commonly be utilized by early Greek spearmen, and was first seen in use by Dorian tribes in the Peloponnese nearing the end of the Greek Dark Age.

The classical hoplite that is so well known to us, and most often portrayed - large, rounded bronze shield, bronze cuirass, greaves, and plumed helm - was really the Greek innovation on a Mesopotamian idea. The wonderous bronze cuirass being a Euboan design from the 8th century BCE, and that most famous helm being of Corinthian design, emerging at roughly the same time as the Argive shield: early 7th century BCE.

The above stated, back to the actual use of the shield!

Once the phalanx had been formed by both armies, the hoplites would march toward one another, spears bristling and shields snuggly tucked onto their shoulders, until they had reached the point of impact. The last few metres between the opposing phalanxes would often be marked by an increase in the speed of the combatants, as they hurl themselves forward behind their shield, grieves, and helm - a Greek encased in bronze, Hades-bent on rolling over the opponent! - and pull a manuever of smashing their shield into the line of the opposing phalanx. The weight behind such a heavily armoured soldier would be difficult to withstand, individually, but the phalanxes often stood at roughly 8 shields deep (and one should note that the Greeks would discuss how 'deep' their formation was by way of shields).

The frontmen would meet, smashing into one another, and the men behind them would press their shoulders deeper into their own shields, each rank pressing against the back of the man ahead of him, to leverage their weight against that of the enemy phalanx. This force, called osthimos (meaning massive push or forward momentum), was what gave the shield its advantage over the others.

Previous shields lacked the construction to allow the hoplite to put their full weight into the press forward, which resulted in less sturdy formations. By allowing the hoplite to focus all their energies and force forward, the Argive (again, I will correct once I review my sources later) gave the phalanx a new depth of power.

Land Battles in 5th Century B.C. Greece: A History and Analysis of 173 Engagements - Fred Eugene Ray JR.

The Wars of the Ancient Greeks - Victor David Hanson

Edit: I've crossed out the parts of the original comment regarding the review of my stated sources and added pertinent information. The shield was, in fact, the Argive. I maintain my stance on osthimos.