Atomic structure.
Imagine you have two boxes. One box is just a simple, one layer sheet of card paper (like the stuff they use to make playing cards). That box is gonna suck because card paper is very simple, so just about anyone can break it.
Now have a different box made of cardboard. Most cardboard boxes actually have very complex, layered structures, much makes them more durable.
Another would be fibers. If you have a sheet of fibers that are all going the same two ways, up and across (^this way, and >that way), it’s a very simple pattern, so it’s easier to break. But if you have multiple layers, all crossing over each other, almost like they’re tangled, they’ll fare much better.
The inherent strength of a material comes from the atomic or molecular bonds holding the atoms together. The strength of pure aluminum, for example, comes from the strength of an aluminum atom bonded to another aluminum atom. The strength of corrundum (or Al2O3) comes from the strength of the ionic bonds between aluminum and oxygen. The ionic bonds in this example take more energy to break, and lead to a higher strength. Theoretically, at least – that’s not the whole story.
You can calculate the theoretical strength of materials by measuring the energy of the bonds, but real materials never reach that theoretical limit. Most of the time, they get nowhere close. This is because real materials have flaws, both microscopic and macroscopic, that act as weak points that eventually cause the entire piece to break. Microscopic flaws are things like impurities and imperfections in the crystal structure (the way the atoms are arranged). Macroscopic flaws are things like cracks or pores.
Finally, engineered materials are rarely homogenous at the atomic or microscopic level (like a pure metal is). Most have complicated structures, almost like a bunch of materials all mixed together at the microscopic level, to achieve properties that pure substances can’t. Sometimes this is for additional strength (from higher energy atomic bonds), but often it’s for other reasons (better properties at different temperatures, resisting harsh environments, or resisting cracking). It gets very complex very quickly. This is why production of materials is an important science.
Note: This discussion skips important concepts like yielding (permanently deforming, like metals often do) and toughness, which are just as important as strength.
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