This article will give a brief overview of the types of armor plate used on tanks during WW2, and examine some of the different ways it could fail when struck by an armor piercing shell. Different types of projectiles can cause the same plate to fail in different ways, so we'll also need to go into detail about that. Then we'll go over the importance of thickness/diameter ratios, how they influence the effect of armor sloping, and how shell normalisation actually works. Finally, we'll look at manufacturing defects that can compromise the quality of an armor plate. The field of armor and ballistics is complicated, but not so much that a basic understanding can't be gleaned.
Types of tank armor
Generally, there are two different types of armor that were used on tanks and AFVs during WW2. The first type of steel armor was heated until the point it was red hot, and was then forged into the required shape. This is rolled homogeneous armor. The second type of steel armor was heated until the point it melted, and was then poured into the required shape. This is cast armor. Both types of armor had to be heat treated and tempered until they gained the desired metallurgical propertys, such as tensile strength, hardness, etc. The alloying elements introduced into the steel -such as manganese, tungsten, nickel, chrome, vanadium- would influence its final form. Rolled homogeneous armor (RHA) had a uniform level of hardness throughout the plate, usually about 250-350 BHN, depending on how thick it is. Face hardened armor (FHA) had two different hardness levels in the plate: The surface was about 450-650 BHN, while the rest was about 250-350 BHN.
FHA is useful against projectiles with a diameter smaller than the plate thickness, I.E., an undermatching shell. It also has the ability to shatter uncapped projectiles and defeat them. However, FHA has the disadvantage that it is more prone to cracking, which gives it lower multi hit protection than RHA. Moreover, FHA plates offers inferior protection to an equivalent RHA plate when hit by capped shells. Another type of tank armor is the so called high-hardness armor. This is similar to FHA, except the hardening extends through the entire depth of the plate, and not just the surface of it. It has a uniformly high level of hardness throughout the plate. For the most part, only the Soviet Union used this type of armor during the war. It had the same strengths and weakness' as FHA did, but taken to the extreme. High hardness armor was only useful against undermatching or uncapped shells. Interestingly enough, cast armor could also be heat treated to achieve the same levels of hardness.
Modes of armor failure
What are some of the different ways that armor plate can fail? The first thing to keep in mind is the difference between penetration vs perforation. When a shell perforates, it passes completely through the armor plate and flys into the vehicle itself. When a shell merely penetrates, however, it does not pass through the plate: It produces a crater and knocks off pieces from the back side of the plate. This effect is known as scabbing or spalling. This is a less serious mode of failure.
In instances when a shell actually perforates the armor, you have to distinguish between ductile and brittle modes of failure. This is absolutely essential. Brittle failures are more dangerous than ductile failures. We'll detail some of the different categorys below. Please note, however, that the type of projectile which strikes an armor plate will strongly influence the manner in which it fails. A shell with an armor piercing cap will usually cause ductile failures in the plate, while uncapped shells will usually cause brittle failures in the plate.
-For ductile failures, there are three different categorys. Plugging, ductile hole growth, and petaling. Ductile hole growth is when the plate (at the impact site) is holed from front to back. Petaling is when the back of the plate folds out. Plugging is when the back of the plate is ejected into the vehicle, along with the projectile.
-For brittle failures, there are two different categorys: Brittle fractures, and brittle plugging. Brittle fractures is when the plate (at the impact site) is shattered from front to back. Brittle plugging is when the back of the plate is shattered and ejected into the vehicle, along with the projectile.
Note: With ductile types of plugging, the shell is still intact and exhibits little or no change of shape. With brittle plugging, the shell is not intact and exhibits a major change of shape. The size of the plugs knocked off the backside of the armor are also larger.
Modes of armor failure
Types of armor piercing shells
There are a few different types of projectile designs, which have different advantages and disadvantages. There is the standard armor piercing (AP) shell. And the armor piercing capped (APC) shell. This is an AP projectile fitted with a cap that prevents it from shattering. There is also armor piercing ballistic capped (APBC). This is an AP shell fitted with an aerodynamic windscreen to improve its ballistic coefficient, but does not prevent it from shattering. Then there is an AP shell fitted with both types of caps (APCBC). This not only enhances its ballistic coefficient but also prevents it from shattering.
Note that most of these designs have a tiny cavity at their base, which is filled with explosive. This works to burst the shell behind the armor plate, increasing the damage it does. The other type of shell is armor piercing high explosive (APHE). This is a regular AP shell, but with a larger explosive filling. Then there is armor piercing composite rigid (APCR). These types of projectiles have a dense tungsten core and an aluminum body. They have excellent penetration, but poor aerodynamics.
As a general rule, projectiles without an armor piercing cap tend to defeat armor through brittle fractures and brittle plugging. When they do succeed in punching through an armor plate, they leave jagged holes wider than the caliber of the shell. Projectiles with an armor piercing cap tend to defeat armor through ductile hole growth, petaling, or plugging. When they punch through an armor plate, they leave neat holes that are the same width as the shells caliber. Keep in mind that these distinctions are not absolute, and there are instances when capped and uncapped shells can defeat armor in different ways than this.
Its also important to note that the nose shape of the projectile can also have an influence on how it interacts with the target plate. Blunt nosed projectiles have an increased tendency towards plugging (of both the brittle and ductile variety). Sharp nosed projectiles have an increased tendency towards petaling and ductile hole growth.  The difference in nose shape also influences their effect on unsloped vs sloped armor. Blunt nosed projectiles are superior against sloped armor plates, whereas sharp nosed projectiles are superior against unsloped armor plates.
Types of armor piercing shells
T/D ratio and sloping
Failure mechanisms for armor are largely dictated by the T/D ratio, where T is the armor thickness and D is projectile diameter. (Simply divide the former by the latter) The T/D ratio dictates that the more the thickness of the armor plate exceeds the diameter of the projectile, the harder it is for the incoming shell to achieve a penetration. The converse is also true of course, and the more the diameter of the projectile exceeds the thickness of the plate, the easier it is for the shell to achieve penetration. To put it simply, larger shells have an advantage over relatively thinner plates. Whats more interesting is that T/D ratio can also determine whether or not the degree of sloping in an armor plate is effective or not.
One way to demonstrate this is by looking at the hull front of a Sherman tank. Early models had a 51mm thick glacis sloped at 56 degrees, for an LOS thickness of 91mm. Later models had a 63mm thick glacis sloped at 47 degrees, for an LOS thickness of 92mm. Even though there was practically no difference in either plates line of sight (LOS) thickness, the later model of Sherman actually offered superior resistance. Because of the more favorable T/D ratio, they had a higher chance of withstanding hits from 75mm APCBC projectiles. The 51mm thick glacis was equal to 98mm of RHA, while the 63mm thick glacis was equal to 118mm of RHA. (Assuming there were no flaws in the armor itself) 
Another important phenomenon to understand is shell normalisation. There is a popular misconception that when a shell hits an inclined armor plate, it will turn by a few degrees before digging into the armor, thus reducing the acute angle it has to negotiate. Such that a 50 degree slope becomes a more manageable 45 degree slope, or some other such reduction. In fact, this is not what actually happens. Normalisation only occurs after the shell has dug into the armor, not before. So when a projectile hits an inclined armor plate, the net result is that it ends up making an S-shaped hole. The difference in the angle of the entrance and exit hole is what leads people to believe that 'normalisation' has taken place.
This is not how normalisation works!
The last factor we'll examine are manufacturing defects that occur at the mill where armor plates are heat treated and tempered at. These mechanical flaws in the structure of the armor will reduce its resistance to impacting projectiles, sometimes by a small amount, sometimes by a severe amount. There are a number of different factors that can lead to substandard armor plate being produced. Early in the war, the U.S. experienced problems with manufacturing defects. The Sherman tank made extensive use of cast armor, not only in the turret but in the hull as well. These armor castings frequently suffered from hot tears and shrinkage cracks.  RHA plates were not without problems either, and were sometimes found to be riddled with stringers and laminations. These problems stemmed mainly from the huge expansion of the U.S. tank industry, which went from producing over 300 tanks in 1940, to over 23,000 tanks in 1942. The sheer quantity of armor plate required meant a decline in quality control.
Later in the war, the Germans also had their fare share of trouble with manufacturing defects. By early 1944, they were suffering from a shortage of alloying elements like nickel, tungsten, molybdenum, and manganese. This forced them to reduce the quantitys of these critical alloys, or to find substitute alloys. In order to ensure that their armor plates did not suffer from brittleness or flaws, a different method of heat treatment also had to be utilised. The Germans eventually settled on the so-called interrupted quench process.  This required great precision in the tempering of the plate, and if the timing was not within a certain margin, the plate would be mechanically flawed. It was not always possible to detect armor with defects. The steel mills were unable to achieve consistent quality control, and were unintentionally churning out many flawed plates.
The Soviet Union experienced some notable problems with manufacturing defects, as well. These issues weren't as ubiquitous as those plaguing the Sherman tanks, but they still cropped up relatively often. The Americans noted their presence in a number of different metallurgical reports. Some of these difficultys can be linked to the fact that much of the USSRs industry had to be hastily evacuated to the Urals in 1941, in order to avoid being captured by the invading German army. The rest can be linked to the fact that Russia was technologically less advanced than the other great powers. Armor castings (especially on the KV-1 tank) tended to have issues with hot tears and shrinkage cracks. RHA plates were sometimes not adequately cross-rolled, and were incompletely quench hardened.  This resulted in uneven hardness levels in an armor plate, an undesirable feature. The high hardness armor of the T-34 was noted for its tendency to create spall.
 World War II Ballistics: Armor and Gunnery, by Robert D. Livingston. (Page 15 and 16)
 World War II Ballistics: Armor and Gunnery, by Robert D. Livingston. (Page 28)
 World War II Ballistics: Armor and Gunnery, by Robert D. Livingston. (Page 6 and 7)
 World War II Ballistics: Armor and Gunnery, by Robert D. Livingston. (Page 8 and 9)
 Metallurgical Examination of Armor and Weld Joint Samples from Russian Medium Tank T-34 and Heavy Tank KV-1. (Page 1 and 9)