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To analyze sword penetration through medieval armor, you can use principles from physics and material science. The basic idea involves calculating the force exerted by the sword and comparing it to the armor's resistance, which can be influenced by factors such as material properties, thickness, and design.

A simplified formula to estimate penetration depth (d) might look like this:

[ d = \frac{F}{A \cdot R} ]

Where:

( F ) is the force applied by the sword,
( A ) is the area of the sword's blade that is in contact with the armor,
( R ) is the resistance of the armor material (which can be determined through its hardness or yield strength).
However, this formula is quite basic and doesn't take into account many variables such as the angle of attack, the type of armor, or the dynamics of the impact. For a more accurate analysis, you might want to look into studies on ballistics and materials science.
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When a sword strikes at an angle, the dynamics of penetration change due to the angle of impact affecting both the effective force and the area of contact.

In this case, you can modify the force component acting perpendicular to the armor surface. The effective force (( F_{effective} )) can be calculated using the angle of attack (( \theta )):

[ F_{effective} = F \cdot \cos(\theta) ]

Where:

( F ) is the total force applied by the sword,
( \theta ) is the angle of impact (in this case, 45°).
Thus, the penetration depth formula becomes:

[ d = \frac{F \cdot \cos(45°)}{A \cdot R} ]

Since ( \cos(45°) ) is approximately ( 0.707 ), the formula simplifies to:

[ d = \frac{0.707 \cdot F}{A \cdot R} ]

This adjustment accounts for the reduced effective force due to the angle of impact. Keep in mind that this is still a simplified model and real-world results can vary based on numerous factors, including the specific materials involved and the dynamics of the strike.

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