I read that medieval European use of plate armor was motivated by the emergence of crossbows, which easily penetrated mail, but would glance off of plate armor. Since Chinese warfare was dominated by crossbows, bows and cavalry, why didn't Chinese heavy cavalry use plate armor instead of lamellar?
According to Burt Hall's "Weapons and warfare in Rennaissance Europe," and Alan Williams's "The Knight and the Blast furnace," the use of plate armor in Europe had nothing to do with the crossbow. We know this because the the crossbow was in western Europe at least as early as the 6th century AD (and potentially even earlier), and the Arbalest (a very powerful type of crossbow utilizing steel arms) was first invented in the 12th century, but plate armor didn't really start to take off in Europe until the late 14th century (full plate didn't really exist until the 15th century).
The real reason plate armor became so popular in Europe was because of advances in manufacturing, specifically advances in the use of the blast furnace and the water powered trip hammer. These tools made it much easier to manufacture plate armor. According to Hall, a shirt of chainmail and a breastplate had similar material costs, but the chainmail took 3 months to make and the breastplate only took 3 days.
Sadly this does not answer your question why the chinese didn't have plate armor. To my knowledge, China had both the blast furnace and the water powered trip hammer as least as long as Europe, but I don't know if it existed in a form that would have been useful for making plate armor (trip hammers were often used for hulling grain and breaking up ore).
Edit: Corrected an error pointed out by r/Theoroshia
I am unqualified to speak as to the specifics of why the Chinese failed to adopt plate armor. But I would like to, if it's alright, clear up a few misconceptions that seem to have informed your question. That plate emerged due to the need to counter crossbows is a rather pat explanation, and ignores the most important reason behind the emergence of plate armor: that, due to technological advances and the development of the armor maker's craft, it became possible to manufacture large amounts of shaped steel plate relatively easily.
Yes, plate is more effective protection than mail, but the development of pre-gunpowder weapons and armor does not readily lend itself to a Cold War-style arms race explanation. People who made weapons sought to make them better; people who made armor did likewise, but as the process could take hundreds of years, progress was not anything like rapid.
The art of mail seems to date to around 400 BCE, and was probably developed in central Europe by the La Tene culture, often called the Celts. The basic mail shirt, short-sleeved and thigh length, seems to have been around since, at least, the 3rd century, and was a nearly universal armor in Europe and the Mediterranean world until perhaps 1100 CE. It was probably worn with padding, but the paucity of surviving textiles from the period makes this impossible to answer; there are Roman references to a subarmalis, and there are references from the Crusades to Europeans wearing felt padding that completely defeated Muslim bows. Following this, more extensive mail protections (along with helmets and weapons) became rapidly more common, consisting of long sleeves, mittens, and leggings.
This rapid change is not technological; it's not any more difficult to add more rings to a design; but it is both more expensive to make and more tiring to wear. The regrowth of the European economy starting around 1100 may explain the former: that the warrior classes could afford to pay for better protection. As for the latter, leg armor is especially fatigueing to wear for infantrymen, as every step is taken with a weighted appendage. This change is perhaps due to the increasingly prominent role of heavy cavalry in much of Europe during this period.
It would not be until the late 13th century that crude plate protections began to appear, and they would not become anything like full suits until the late 14th. The first proper plate was not torso armor, as one might expect a counter-measure against crossbow bolts to be, but joint armor - elbow and knee guards. These are areas that mail failed to adequately protect, and so, sensibly, were rapidly supplemented. It is true that an item referred to as a "coat of plates" appeared almost simultaneously with the development of joint protections, but this is not generally what people refer to when they speak of plate (it was a series of small iron plates sewn or riveted inside a textile garment). In due course, general leg and arm armor became regularly seen, with torso protection not becoming common until around 1350.
The crossbow, meanwhile, seems to have entered European war as early as the 10th or 11th century. It was certainly in common use by the 12th. It is difficult to say for certain what the properties of these early crossbows were, because none, to my knowledge, survive from the period. It is likely they were both less sophisticated and powerful than the well-developed examples from the 14th-16th centuries, and probably was not so murderously effective as is supposed, but still dangerous to armored men in a way that a short self bow was not. Further, it was not a peasant's weapon. It was a complex bit of machinery that took skill and training to not only use, but maintain during the rigors of campaign and siege. The best testament to this is the fact that mercenary crossbowmen remained in heavy demand for so much of the medieval period, only really disappearing with the development of effective early firearms.
Thus we see that mail was the primary armor for more than two hundred years - one could stretch it to close to 400 - after the introduction of the crossbow into medieval war, and that plate protection was only gradually adopted over a period of more than a century.
Mass armies would be prohibitively expensive to equipt w/ plate armor You're talking about farmer conscripts that were considered expendable.
Their most dangerous opponents were the central Asian archer horsemen who in fact did wear plate & mail armor. These same opponents were numerically smaller than the Chinese foot soldier armies they fought, so equipting was again not as expensive for a large army.
Now, wearing all this heavy armor suited a horseman; it's quite heavy but the horse is actually doing all the work.
Whereas if your foot soldiers wore plate armor they would tire quicker and wouldn't be as mobile. Consider that the premier heavy infantry force of the time, The Romans, didn't equipt their forces w/ plate, only mail.
Also consider:
Interesting thing about mounted archers & the range of their compound bows: Effective range was kinda close, I'm gonna say 70-130 yards. Still an effective stand-off range. But as to the fatality rate, something small like 3 in 100 arrows shot would actually be fatal. So typically a barrage of arrows against a foot soldier army was like in the MILLIONS.
Parthian victory at Carhae against the Romans involved camel ammo caravans and is estimated to have supplied at least 1 million arrows. It also helped that the Romans went against common sense and even still the Parthian victory took several days.
Firstly, as the other people have said, crossbows did not motivate plate armor in Europe. So that bit of the OP's post is flawed. Secondly, think for a second about what lamellar is. Lamellar is armor in which metal plates are strung together. If you hypothetically somehow welded together all of the plates in a lamellar piece of armour, then you now have a piece of plate armor. So when protecting against bolts and arrows, wouldn't you expect lamellar and plate armor to perform about the same?
I can see why plate is better than mail when it comes to protecting against bolts: a bolt could push it's way through the links of mail and force it open from the inside. But lamellar isn't mail. Lamellar is just a bunch of plates.
From my understanding, (at least during the time from the Song Dynasty through most of the Ming), the Chinese army was largely conscripted by the central government and not based around local/regional nobility. The plate armor with which we are familiar was not generally available to the whole army in Europe, but rather the wealthier knights, squires, etc. Chinese warfare was very different. Large battles between peasants in open areas. Numbers were more important than weaponry in many cases. Furthermore, the funding for warfare was often based on living off of the land, rather than rations provided by the central government. This led to larger problems of worn out soldiers who couldn't really transport their own supplies (let alone heavy armor).
As an interesting example of how this mindset backfired, when the Mongols initially invaded the Jin Dynasty, one of their tactics was to chase the army on horseback until they ran out of energy (since the army had so few horses and many Jin soldiers simply deserted). Genghis Khan had even commented that he believed the army he had defeated might have been a trick because they were so underarmed (comparatively to the Mongols with silk armor, horses, and light bows) and because many Jin soldiers were suffering from malnutrition due to surviving off of only white rice for the entire campaign.
Officers and nobility still had access to some plate armor and chain mail, but these aren't the images we think of when we think of Chinese warfare.
I don't have the exact citation, but I could find it later and post an update.