Strengthening materials

Materials such as ceramics, glasses and metals could be strengthened by adding extra chemical element that could form connections with as many atoms as possible. In case of chemical bonds that's with around 4 atoms and such elements are in middle of periodic table away from left and right edge.

Electrostatic attraction is other force that hold many materials together. Even hydrogen bond in water is example of electrostatic attraction and it helps with increasing melting and boiling temperatures.

 Electronegativity shows how strongly some element attracts electrons. Substances with higher electronegativity usually get negative charge and the ones with smaller values get more positive charge in at least ionic bonds. That's partly because less electronegative elements have larger atoms and their electron are further away from positive charge in nucleus making these electrons easy to remove (giving atom positive charge) while in top right corner of periodic table atoms are smallest and positive charge of nucleus is closer to all its electrons so the strong attraction can give it extra electron(s).

Soft metals have very weak structure. For example sodium, lithium and other group 1 elements only have 1 electron for chemical bond so they could only form strong connection with 1 other atom and that makes these metals very soft and fragile. This lack of connection with other atoms may explain why these metals have low melting temperature (lithium melts at 180 C and sodium at 97 C).

One early example of material strengthening was in production of steel. Iron is fragile with too low or too high carbon % but if carbon forms 0,2-2,1% of steel by weight then this mixture can get extra strength. Usually at first carbon was burned out by pumping oxygen into melted iron mix to burn it away as CO2. After that people add required amount of carbon into molten metal in atmosphere that can't burn carbon like argon, helium and other noble gases.
These solutions may be common in production of other materials because often different elements get mixed in liquid hot environment where different ingredients melt and evaporate at different temperatures. 
Carbon has more negative charge (more electronegative) than iron and that somewhat large difference in electronegativity can give steel strong electrostatic pull. Although oxygen and fluorine have higher electronegativity they have much less free electrons for forming bonds. 

Iron has cubic body centered crystal structure that has 9 iron atoms in crystal unit. In case of 2% carbon it would mean about 10% of atoms are carbon and that makes about 1 carbon atom for each such cubical unit in iron crystal although iron could also have structure shown in middle.

One way to appreciate the strength of steel is to see it supporting high buildings. Above clip (climbing starts at around 50 seconds into the video) shows a steel antenna that is above 500 meters high and supports climber with increasingly minimalistic safety features. Much of the climbing is done on metal (probably steel) ladder sticks that are about as thin as human finger and attached to tower from only one side and they are still enough to support human with a 10 kg bag.

In case of clay and porcelain diversity of ingredients can also help with durability.
Bone china is one more historic type of porcelain that Europeans learned to produce. One recipe for it is to mix kaolinite (25%), feldspar (25%) and bone ash (50%).  Kaolinite (Al2S2O5(OH)4) is basically clean clay. Feldspar has also aluminium, silicon, oxygen and hydrogen but in addition sodium, calcium and potassium. Bone ash is left after burning bones and that composes mostly of phosphate groups and calcium (CaHPO4).
Special thing about bone ash is that both phosphate and calcium have double charges. Calcium has +2 charge and phosphate group has -2 charge making electrostatic with them extra strong and porcelain was partly valued because it resisted drops to floor better than clay.

Elements found in clay can also be used to produce very strong and scratch resistant transparent glass (porcelain is also somewhat transparent). One version of this strong glass has brand name Gorilla Glass and it used on newer more expensive phones and tablets. It is made from aluminum, silicon and oxygen like clay with additional potassium and sodium. After glass gets made from Al, Si, O and Na it gets strengthened by putting it in molten potassium salt at 400 C. Smaller sodium ions leave glass and larger potassium moves into glass that expanded during heating. After cooling glass constricts again and forces atoms tighter together but now atom are more densely packed due to larger potassium with larger differences in electronegativity between elements (and stronger electrostatic attraction) than before.

Example of Gorilla Glass in use.