What was the point of horse armour?

Note that, in the link you provide, the thickness cited for the horse's barding in the accepted answer is mostly similar to the thicknesses measured on the knight's armor, which also ranges from 1.2-1.6mm. Given the weight/protection tradeoffs that was probably as thick as it could be made.

There are other reasons for barding as well, such as charging pikemen or engaging in close combat with infantry/other knights.

Just because some of the battles of French knights vs English longbowmen were a mess (and note that Agincourt, which gets lumped in with Crecy in that regard, was a mostly dismounted affair), doesn't mean that cavalry wasn't used successfully in many other combat situations and that there wasn't ample empirical evidence justifying protecting those very costly and highly trained combat horses.

In fact, cavalry vs foot archers didn't usually resolve to the latter's advantage. The success of Medieval heavy cavalry vs other types of combattants often depended on whether the commanding general knew what he was doing and whether he could keep control of his often undisciplined knights. Neither Agincourt, nor Crecy, were indicative of particularly effective French tactics or leadership, and while the longbowmen certainly shone, they were also operating in a quite favorable tactical context.

It would be just as risky to draw too many conclusions re. modern tanks by referring only to the Russian armor's dismal performance in the first battle of Grozny.

The part you specifically draw attention to as being thinnest at 0.4mm in that link was the ‘crinet’ which is the articulated piece on the horse’s upper neck. If you look at the illustration you will see that the panels are lapped, such that the effective thickness doubles to 0.8mm. It still sounds hellish thin, but not absolutely to airy thinness beat.

Question: What was the point of horse armour?

Short Answer
The practice of giving body armour to a horse was called barding. It was more widely used and to great effect in antiquity long before the middle ages in the time of Alexander. During the late Middle ages when such tactics as using mixed troops were being re-introduced to European warfare, heavy barding began to make a comeback in response to such weapons as the long bow. This reintroduction was cut short by the introduction of gunpowder in the early 16th century. Thus heavy barding during the middle ages had a short lived popularity and existing examples of it during this period are extremely limited. Due to this it's hard to judge how effective it would have been or could have eventually become against archers.

Detailed Answer
Full or heavy barding was more prevalent in antiquity than in the middle ages, Alexander the Great who used many specialized units including multiple heavy and light cavalry and infantry units,

Macedonian Army Units employed by Alexander the Great
Heavy Cavalry Unites

• Companion cavalry ( in which Alexander often rode)...
• Thessalian cavalry

light Cavalry units.

• Prodromoi/Sarissophoroi
• Paeonian cavalry
• Thracian cavalry
• Horse archers

Light Infantry

• javelin
• archers
• slings

Alexander the great used these specialized troops to gain advantage (mismatch) when employed against different kinds of enemy troops arrayed against him. Europe in the early middle ages was a step backwards with respect to military tactics. Specialized troops were less prevalent early on. Where Alexander had many kinds of cavalry, balistic and infantry; middle ages European armies were not as diverse and often consisted of just one type of unit.

Take the battle of Hastings 1066. The Normands had knights, archers and infantry an early exception to the rule but still not at Alexanders level of diversity. The Saxons mostly had only infantry. The Saxon's rode horses to battles but always fought on foot. The Bayeux Tapestry only has only one depiction of a Saxon archer while it exhibits many examples of the more than 1000 Normand archers on the field at Hastings, suggesting Harold didn't have many archers.

Weapons of 1066
Knights
Knights, armoured warriors fighting on horseback with lance, sword and shield, were the Norman 'secret weapon' at the Battle of Hastings. Nothing like them had been seen in England before. For though the Saxons rode to battle or on journeys, they always fought on foot.

There were between 1,000 and 2,000 knights in the Norman army, and it was these who mainly won William's victory.

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Archers
Many Norman archers are shown in the Bayeux Tapestry, and it's estimated that there were over 1,000 of them in William's army. They played an important part in the battle, especially after William ordered them to shoot high, firing their arrows onto the heads of the Saxons behind their shield-wall.

Only one Anglo-Saxon archer is shown in the Tapestry, symbolising that Harold's army included very few bowmen. Archers were poor men, and it's possible they couldn't afford horses to help them keep up with Harold's rapid move to the battlefield.

As such even Williams army which was vastly more diverse than Harold's, didn't have what we would call heavy cavalry. Knight's at Hastings rode small horses called destriers prized for their mobility and agility and were lightly barded. They didn't need to protect themselves from archers, because knights themselves were the innovation.

It wasn't until late middle ages when knight armor improved that the necessity for horse armor became apparent. The longbow actually drove the practice of heavy barding. Arrows killed knights they improved the knights armor.. Then Archers learned to target the horses and heavy infantry would kill the now imobalized horseless knights; so they tried to improve the horse's armor.

barding
During the Late Middle Ages as armour protection for knights became more effective, their mounts became targets. This vulnerability was exploited by the Scots at the Battle of Bannockburn in the 14th century, when horses were killed by the infantry, and for the English at the Battle of Crécy in the same century where longbowmen shot horses and the then dismounted French knights were killed by heavy infantry. Barding developed as a response to such events.

So the battle of Agincourt (1415) where the English with an army mostly made up of longbowmen (7000) defeated a larger French conventional army with a large contingent of knights; are the kinds of defeats which drove full Barding. The practice was short lived however; as gunpowder was introduced onto the battlefield of Europe in 1503 at the Battle of Cerignola and that would shortly negate all traditional forms of armor.

So while the reason to armour your horse in battle was to protect him. Your observation that existing examples of full barding during middle ages Europe would not have accomplished this task can probable be attributed to:

1. The time period in question. Early middle ages knights did not bard their horses as knights were shock troops which relied on speed and mobility. Heavy barding to protect against arrows arose only in the late middle ages.

2. There are very few examples of full barding which exist today from this period so we don't have a representative sample of them to choose from.
   Rare surviving period examples of barding- medieval armor for horses

3. Examples of full barding that do exist may have been used more for tournaments or parade than during actual battles. Full bard armor was used on the parade and tournament grounds for a longer period of time than it was used on the battlefield in the middle ages and into the 16th century. Medieval Military Technology, Second Edition

4. Full Barding was a late innovation in the middle ages and short lived as gun powder was introduced and quickly made all conventional armor obsolete. This further explains why we have so few examples of it today.

The long bow was a particularly effective weapon against armored cavalry, and the French were surprised by this fact.

The (relatively thin) armor that you mentioned had earlier provided the horse some protection against "spears," particularly those wielded by enemy infantry. Although Swiss "pikes" (about 50 years into the future from the end of the Hundred Year's War), were also very effective against armored cavalry.

Basically, every weapons system becomes obsolete at some point, after having served a purpose earlier. The point of horse armor was "earlier."

Knights didn't just face longbows. There were also swords, pikes, maces etc. on the battlefield and good plate armor also protected against firearms. Two knights fighting on the battlefield - they're trying to hit each other but sometimes the horses get hit instead.. There's an article called Armour which says a lot about this. For example,

The horse was afforded protection from lances and infantry weapons by steel plate barding. This gave the horse protection and enhanced the visual impression of a mounted knight. Late in the era, elaborate barding was used in parade armour.

Also the article says horse armor was not as heavy as one might think.

In fact, even the heaviest tournament armour (for knights) weighed little more than 90 pounds (41 kg), and field (war) armour 40 to 70 pounds (18 to 32 kg); barding, or horse armour, more common in tournaments than war, rarely weighed more than 70 pounds (32 kg).

Some research shown in this article How Much Weight Can Horses Comfortably Carry? found that horses can carry up to 29% of their weight so for a 1,200 pound horse, the armor is not such a big deal. Also, they didn't always have metal. Leather could also be effective, and it's much lighter. Armor didn't always offer full protection but it could easily mean the difference between a wound that would heal and one that was fatal.

Another article called Horse Armor in Europe mentions jousting, and that armor was used in tournaments to protect the horse and it was also decorative. It could protect against glancing blows and help prevent or reduce injuries. Maybe based on movies people think knights always went to battle with lots of armor on them and their horses but was not the case. The article 'Armour' mentioned before says

Tournament armour is always heavier, clumsier and more protective than combat armour. Combat armour is a compromise between protection and mobility, while tournament armour stresses protection on cost of mobility.

So horses had armor because it offered some protection. The armor wasn't always enough but it was better than no armor which offered no protection.

I would contend that we tend to overestimate the effectiveness of bows vs armour, and that the armour would likely prevent at least some percentage of the damage to the mount.

If we look at the wiki article on Barding we find the following (emphasis mine):

During the Late Middle Ages as armour protection for knights became more effective, their mounts became targets. This vulnerability was exploited by the Scots at the Battle of Bannockburn in the 14th century, when horses were killed by the infantry, and for the English at the Battle of Crécy in the same century where longbowmen shot horses and the then dismounted French knights were killed by heavy infantry. Barding developed as a response to such events.

So why would you develop and build something, unless it worked?

I have watched a video, ARROWS vs ARMOUR - Medieval Myth Busting , showing a bit of experimental archaeology which investigates the issue of bows vs armour. Though I would normally cringe to use a YouTube video as a source, I feel this one is extremely well done, has measured the physics involved, goes to lengths to try to simulate period weapons and armours, and Tobias Capwell, curator of Arms and Armour at the Wallace Collection in London is present to oversee the process.

This above-mentioned video, though it is concerning human armour and not horse barding, does use energies and armour thicknesses which fall into the range you mention, so the physics should be similar. The archer is shooting 160lb, 80g arrows, which are developing 123J of energy at 10m, and down to 109J (about 80 ft-lbs) at the testing range of 25m. Similar to the (unsourced) energy from the OP's question.

The armour being shot at, in order to simulate examined museum specimens, varies from 1.5mm to 2.5mm in thickness. Again similar range to the numbers quoted in the question.

Information we can draw from the video is that arrows which hit head on may dent, but not penetrate (very reminiscent as to how modern body armor works). Most arrows break, and any that hit slightly obliquely, or in the more curved regions, ricochet away. This deflection may be of primary importance with a horse, keeping raking shots which might tear at the animals flanks from doing any damage, and deflecting away at least a percentage of the shots finding a target.

Note that one of only three complete sets of equestrian armour is found in the same above mentioned Wallace Collection, a beautiful set of German armor, dated to about 1480:

We can draw some information from the Wallace Collection exhibit as well. In looking at the information from the exhibit we find weights listed concerning the armor:

27.161kg (man's armour); 30.07kg (horse's armour); 10.17kg (mail)

(About 150lbs) From this we can see that the plate mail barding was expected to be worn in conjunction with chainmail protection as well. The chainmail can deal with slashing damage when engaged close-up. We can also observe the plate portion of the barding was mainly applied in curving sweeps along the breast, the top of the neck, and the upper region of the horses hind quarters. These are exactly the regions you might expect to receive fire from, during either an overhead barrage of arrows, or directly from the front during a charge. One more note concerning the Wallace Collection armour, comes from the description on this page where it defines this armor as

‘field’ armour (i.e. armour for war)

This set of armor was field armour, it was made for war, not for parades or tournaments.

So in conclusion, if we can compare the bit of experimental archaeology can be applied to your barding question, it would seem that barding would provide at least some protection from archers. The curved metal surfaces would deflect many indirect strikes, decreasing the amount of damage possibly taken by the horse. The physics tested out in the video indicate the possibility that even direct shots might not have been able to penetrate, further reducing the chance of the horse being brought down. Every arrow deflected or stopped increased your odds off reaching the battle. Statistically it increased the number of knights delivered to the fight.

That was the point of horse armour. Increased survivability.