Ricochet of a 0.25 ACP bullet

[RetiredUSNChief]

This is an interesting question.

Rosenberg and Dekel (1989) proposed a mathematical model that can be used to predict the minimum angle necessary to induce ricochet of rods (think of a bullet as a very short ''rod'') that treated the area of contact between the rod and the surface being struck as a plastic 'hinge' where the rod—or bullet in this case—would deflect at a critical angle. The equation is—

View attachment 1185032

—where β (minimum angle of incidence at which deflection/ricochet would occur) is a function of the density of the bullet (ρ), the impact velocity of the bullet (V), the penetration velocity of the bullet's nose at the target interface (U), and the dynamic resistance (Rt) of the laminated automotive glass windshield. In the case of a .25 ACP bullet weighing 50 grains striking a laminated automotive glass windshield (with a compressive yield strength of σ = 300 MPa) at a velocity of 720 fps, the bullet would ricochet at an angle of target (the windshield) inclination of just 3° (2.997°) making deflection likely in almost any situation except where the bullet strikes the windshield dead on at 'normal' incidence (0°).

What's the value of "U" (penetration velocity) and does that change for a given projectile? How is this determined? Seems to me that if you kept all other factors the same, "U" would vary based on the angle the bullet impacted the glass.

 

[481]

What's the value of "U" (penetration velocity) and does that change for a given projectile? How is this determined? Seems to me that if you kept all other factors the same, "U" would vary based on the angle the bullet impacted the glass.

Apologies, Chief.

In an attempt (probably misguided ) to keep my post from becoming too technical (not everyone here has a degree in Mechanical Engineering), I omitted certain computations that would have distracted from the gist of the topic. I didn't want to get to far ''into the weeds'' on this, but since you asked...

To answer your question, in the case being discussed, U would be approximately 487 feet per second for the hypothetical 25 ACP 50-grain bullet striking laminated automotive glass at 720 feet per second.

'U' is the projectile-target interface velocity at the bottom of the crater during an impact into a ductile (or brittle) material measured at the tip of the projectile as it advances (and erodes) through the target. It is generally taken to be constant in steady-state penetration events (there are iterative mathematical treatments for the very small perturbations in U, but they are generally not worth the time unless one is interested in the very small variations that do occur). (Anderson & Orphal, 2008, Rosenberg & Dekel 1994 & 2016, Christman & Gehring, 1966, Alekseevskii, 1966, and Tate, 1967)

The equation for U is dependent upon the square root of the ratio of the target's density (ρt) to the projectile's density (ρp) and is not affected by any other material property except for compressibility—



—which is ignored in the hydrodynamic theory that is used to model these events; The Alekseevskii-Tate modified Bernoulli equation or AT model—

.

In the ricochet model of Rosenberg et al. (1989) the asymmetric force (F) from Tate’s model, is assumed to act only upon the mass of the rod which is engaged by and in contact with the surface of the target. This alteration was made in order to account for the plastic hinge that forms as the rod bends during rod ricochet. The length of the rod (or the bullet, in this case) does not appear in this model since it concentrates on the plastic hinge, and the asymmetric force (F) at that point, rather than upon the rod’s (or the bullet's) moment of inertia.

 

[RetiredUSNChief]

OK, a little bit of background so you'll understand why I ask the questions I do.

I'm an engineer, but not a mechanical engineer. I work as a nuclear engineer in a shipyard and have two additional years as an operator of naval nuclear propulsion plants. My degree is in EET and my time in the Navy was spent as a Reactor Operator.

I'm familiar with aspects of Fracture Mechanics as related to the development of certain temperature/pressure operating curves that must be operated within, so I understand brittle fracture, ductile failure, and in general the various tests utilized to generate these curves and how they're adjusted for the design lifetime of a reactor pressure vessel.

This is a practical understanding of Fracture Mechanics as specifically applied to my field, not a degreed-in-the-field understanding of a Mechanical Engineer, mind you.


SO...

"'U' is the projectile-target interface velocity at the bottom of the crater during an impact into a ductile (or brittle) material measured at the tip of the projectile as it advances (and erodes) through the target."

I understand that this is a property determined as an object is actually advancing through a material. Which makes sense because it's a "penetration" velocity. If the projectile was too slow, then it wouldn't penetrate, therefore... And I'll make a basic assumption that this velocity is simply determined experimentally.

But what materials was this equation developed for? This value is dependent upon the densities of the target and the projectile exclusively, with the exception of compressibility according to your explanation. Materials with a low fracture toughness would have a different U value than materials with a high fracture toughness, for a given density. HOWEVER...some materials react differently even if they are of similar density to others based on other mechanical properties.

Non-newtonian fluids come to mind, for example. And glass is a pretty weird substance all by itself. It's an amorphous solid (basically a liquid, but acts like a solid because of its uniquely rigid structure). Its properties can be varied radically by composition, treatment, and manufacture. And then there's laminated glass, such as American automotive windshields.

So glass, tempered or not, does not react to penetration like metals. It may be similar, but it's not going to be the same. Glass will fracture/crush at the point of impact and those glass fragments will have a significant impact on bullet path as a result, because those millions (billions? trillions?) of microscopic glass fragments will interact with the bullet differently than for a metal-on-metal interface. Especially when the point of impact between glass and projectile is anything other than 90 degrees.

Glass may, of course, cause bullet ricochets. That's obvious. But there's some question as to the nature of those ricochets as compared to other materials that a bullet may impact, because apparently the effect of glass on bullet deflection is not as intuitive as one would assume when penetration is involved...definitely when total penetration is involved, and possibly when partial penetration is involved. If no penetration at all is involved, it's a simple angle of reflection = angle of incidence.


This is such a cool subject!

 

[Palladan44]

OK, a little bit of background so you'll understand why I ask the questions I do.

I'm an engineer, but not a mechanical engineer. I work as a nuclear engineer in a shipyard and have two additional years as an operator of naval nuclear propulsion plants. My degree is in EET and my time in the Navy was spent as a Reactor Operator.

I'm familiar with aspects of Fracture Mechanics as related to the development of certain temperature/pressure operating curves that must be operated within, so I understand brittle fracture, ductile failure, and in general the various tests utilized to generate these curves and how they're adjusted for the design lifetime of a reactor pressure vessel.

This is a practical understanding of Fracture Mechanics as specifically applied to my field, not a degreed-in-the-field understanding of a Mechanical Engineer, mind you.


SO...

"'U' is the projectile-target interface velocity at the bottom of the crater during an impact into a ductile (or brittle) material measured at the tip of the projectile as it advances (and erodes) through the target."

I understand that this is a property determined as an object is actually advancing through a material. Which makes sense because it's a "penetration" velocity. If the projectile was too slow, then it wouldn't penetrate, therefore... And I'll make a basic assumption that this velocity is simply determined experimentally.

But what materials was this equation developed for? This value is dependent upon the densities of the target and the projectile exclusively, with the exception of compressibility according to your explanation. Materials with a low fracture toughness would have a different U value than materials with a high fracture toughness, for a given density. HOWEVER...some materials react differently even if they are of similar density to others based on other mechanical properties.

Non-newtonian fluids come to mind, for example. And glass is a pretty weird substance all by itself. It's an amorphous solid (basically a liquid, but acts like a solid because of its uniquely rigid structure). Its properties can be varied radically by composition, treatment, and manufacture. And then there's laminated glass, such as American automotive windshields.

So glass, tempered or not, does not react to penetration like metals. It may be similar, but it's not going to be the same. Glass will fracture/crush at the point of impact and those glass fragments will have a significant impact on bullet path as a result, because those millions (billions? trillions?) of microscopic glass fragments will interact with the bullet differently than for a metal-on-metal interface. Especially when the point of impact between glass and projectile is anything other than 90 degrees.

Glass may, of course, cause bullet ricochets. That's obvious. But there's some question as to the nature of those ricochets as compared to other materials that a bullet may impact, because apparently the effect of glass on bullet deflection is not as intuitive as one would assume when penetration is involved...definitely when total penetration is involved, and possibly when partial penetration is involved. If no penetration at all is involved, it's a simple angle of reflection = angle of incidence.


This is such a cool subject!

My ballistic experience entailed shooting everything and everything as a kid to "see what would happen" in what I felt at the time was in a "safe" manner. Looking back, much of it was errant from a safety standpoint.
Looking at how bullets skip off the ground, waters surface, or even service pistol calibers careening off of sheetrock....it's always plausible.

Something to be said about a softer medium allowing the material to redirect the bullet over the course of several inches to several feet to change the direction of the bullet. Glass can't really do this, it has about .28" of contact and it has to be deflected or break. All that kinetic energy in a smaller area.

 

[RetiredUSNChief]

My ballistic experience entailed shooting everything and everything as a kid to "see what would happen" in what I felt at the time was in a "safe" manner. Looking back, much of it was errant from a safety standpoint.
Looking at how bullets skip off the ground, waters surface, or even service pistol calibers careening off of sheetrock....it's always plausible.

Something to be said about a softer medium allowing the material to redirect the bullet over the course of several inches to several feet to change the direction of the bullet. Glass can't really do this, it has about .28" of contact and it has to be deflected or break. All that kinetic energy in a smaller area.

Check out the Box 'O Truth website for their two Buick 'O Truth videos which specifically tested bullet deflection from both sides of the windshield of their test car.

One would assume, for example, that a bullet shot through a windshield towards a target inside the vehicle would result in a bullet deflection in an upward direction. That is not what happened, however. The bullet actually deflected DOWN upon penetration through the windshield to the target.

And the opposite happened when shooting through a windshield towards a target outside the vehicle. The bullet actually deflected in UP upon penetration through the windshield to the target.

The links were posted earlier in this thread. They're pretty cool to watch.

 

[Palladan44]

Check out the Box 'O Truth website for their two Buick 'O Truth videos which specifically tested bullet deflection from both sides of the windshield of their test car.

One would assume, for example, that a bullet shot through a windshield towards a target inside the vehicle would result in a bullet deflection in an upward direction. That is not what happened, however. The bullet actually deflected DOWN upon penetration through the windshield to the target.

And the opposite happened when shooting through a windshield towards a target outside the vehicle. The bullet actually deflected in UP upon penetration through the windshield to the target.

The links were posted earlier in this thread. They're pretty cool to watch.

I'm aware of this, my post #13.

 

[481]

OK, a little bit of background so you'll understand why I ask the questions I do.

I'm an engineer, but not a mechanical engineer. I work as a nuclear engineer in a shipyard and have two additional years as an operator of naval nuclear propulsion plants. My degree is in EET and my time in the Navy was spent as a Reactor Operator.

I'm familiar with aspects of Fracture Mechanics as related to the development of certain temperature/pressure operating curves that must be operated within, so I understand brittle fracture, ductile failure, and in general the various tests utilized to generate these curves and how they're adjusted for the design lifetime of a reactor pressure vessel.

This is a practical understanding of Fracture Mechanics as specifically applied to my field, not a degreed-in-the-field understanding of a Mechanical Engineer, mind you.


SO...

"'U' is the projectile-target interface velocity at the bottom of the crater during an impact into a ductile (or brittle) material measured at the tip of the projectile as it advances (and erodes) through the target."

I understand that this is a property determined as an object is actually advancing through a material. Which makes sense because it's a "penetration" velocity. If the projectile was too slow, then it wouldn't penetrate, therefore... And I'll make a basic assumption that this velocity is simply determined experimentally.

But what materials was this equation developed for? This value is dependent upon the densities of the target and the projectile exclusively, with the exception of compressibility according to your explanation. Materials with a low fracture toughness would have a different U value than materials with a high fracture toughness, for a given density. HOWEVER...some materials react differently even if they are of similar density to others based on other mechanical properties.

Non-newtonian fluids come to mind, for example. And glass is a pretty weird substance all by itself. It's an amorphous solid (basically a liquid, but acts like a solid because of its uniquely rigid structure). Its properties can be varied radically by composition, treatment, and manufacture. And then there's laminated glass, such as American automotive windshields.

So glass, tempered or not, does not react to penetration like metals. It may be similar, but it's not going to be the same. Glass will fracture/crush at the point of impact and those glass fragments will have a significant impact on bullet path as a result, because those millions (billions? trillions?) of microscopic glass fragments will interact with the bullet differently than for a metal-on-metal interface. Especially when the point of impact between glass and projectile is anything other than 90 degrees.

Glass may, of course, cause bullet ricochets. That's obvious. But there's some question as to the nature of those ricochets as compared to other materials that a bullet may impact, because apparently the effect of glass on bullet deflection is not as intuitive as one would assume when penetration is involved...definitely when total penetration is involved, and possibly when partial penetration is involved. If no penetration at all is involved, it's a simple angle of reflection = angle of incidence.


This is such a cool subject!

I've got a BSME, so topics like this are also of great interest to me.

Cutting to the chase, the answer to your question, ''But what materials was this equation developed for?'', is straight-forward.

Originally, the A-T model (and equations like the ricochet equations derived of the A-T model), a modified Bernoulli equation was developed independently by Alekseevskii (1966) and Tate (1967) in order to model and predict effects of the hydro-dynamic pressure interface and deceleration of high-aspect L/D KE-penetrators into ductile targets (metallic armor systems) and was inspired—at least in part—by the work of Pack et al ''Penetration by High-Velocity (‘Munroe ’) Jets: I & ll'' Armament Research Establishment, Fort Halstead, Kent (1950). The ricochet equation has seen significant examination and seems to be sufficient in its representation of the phenomena.

Rosenberg Z, Ashuach Y, Dekel E (2007) More on the ricochet of eroding long rods—validating the analytical model with 3D simulations. Int J Impact Eng 34:942–957

Over the last 57+ years, the A-T model (and its derivative expressions) has been proven to be valid for use not only in ductile solids, but also in crystalline, polymeric, brittle, porous, granular, and geological mediums. Here are just a few examples of the research supporting those findings...

Geological materials such as soil, rock, and granular mediums like sand—
Forrestal, M.J. Penetration into dry porous rock. Int. J. Solids Struct. 1986, 22, 1485–1500.
Forrestal, M.J.; Luk, V.K. Penetration into Soil Targets. Int. J. Impact Eng. 1992, 12, 427–444.
Khan, M.A. Mechanics of projectile penetration into non-cohesive soil targets. Int. J. Civ. Eng. 2015, 13, 28–39.
Frew, D.J.; Forrestal, M.J.; Hanchak, S.J. Penetration Experiments with Limestone Targets and Ogive-Nose Steel Projectiles. J. Appl. Mech. 2000, 67, 841–845.
Savvateev, A.F.; Budin, A.V.; Kolikov, V.A.; Rutberg, P.G. High-Seed Penetration into Sand. Int. J. Impact Eng. 2001, 26, 675–681.

Ceramics, glasses, crystalline solids—
Clayton JD. Penetration resistance of armor ceramics: Dimensional analysis and property correlations. Int J Impact Eng 2015; 85:124–31.
Clayton JD. Dimensional Analysis and Extended Hydrodynamic Theory Applied to Long-Rod Penetration of Ceramics Defence Technology 2016; 12(4): 334–342
Orphal DL, Franzen RR. Penetration of confined silicon carbide targets by tungsten long rods at impact velocities from 1.5 to 4.6 km/s. Int J Impact Eng 1997;19:1–13.
Orphal DL, Franzen RR, Charters AC, Menna TL, Piekutowski AJ. Penetration of confined boron carbide targets by tungsten long rods at impact velocities from 1.5 to 5.0 km/s. Int J Impact Eng 1997;19: 15–29.
Orphal DL, Franzen RR, Piekutowski AJ, Forrestal MJ. Penetration of confined aluminum nitride targets by tungsten long rods at 1.5–4.5 km/s. Int J Impact Eng 1996; 18:355–68.
Bless SJ, Subramanian R, Normandia M, Campos J (1999) Reverse impact results from yawed long rods perforating oblique plates. In: Proceedings of the 18th International symposium on ballistics, San Antonio, Nov 1999, pp 693–701.
Bless SJ, Rosenberg Z, Yoon B (1987) Hypervelocity penetration of ceramics. Int J Impact Eng 5:165–171.
Rosenberg Z, Tsaliah J (1990) Applying Tate’s model for the interaction of long-rod projectiles with ceramic targets. Int J Impact Eng 9:247–251
Walker JD (2003) Analytically modeling hypervelocity penetration of thick ceramic targets. Int J Impact Eng 29:747–755

Polymers—
Z. Rosenberg, Z. Surujon, Y. Yeshurun, Y. Ashuach and E. Dekel, " Ricochet of 0.3" AP projectile from inclined polymeric plates", (2005) , Int. J. Impact Engineering ,Vol. 31, pp 221–233

Structural materials—
Forrestal, M.J.; Tzou, D.Y. Spherical cavity-expansion penetration model for concrete targets. Int. J. Solids Struct. 1997, 34, 4127–4146.
Forrestal MJ, Frew DJ, Hickerson JP, Rohwer TA (2003) Penetration of concrete targets with deceleration time measurements. Int J Impact Eng 28:479–497

Dynamic strength (Rt) of these materials can be modeled using either SCE (spherical cavity expansion) or CCE (cylindrical cavity expansion) equations which are also derivative of the A-T modeling approach—
Luk, V.K., Amos, D.E. (1991). Dynamic cylindrical cavity expansion of compressible strain hardening materials. J. of Applied Mechanics, 58(2): 334-340.
Luk, V.K., Forrestal, M.J., Amos, D.E. (1991). Dynamic spherical cavity expansion of strain hardening materials. J. of Applied Mechanics, 58(1): 1-6.
Masri, R., Durban, D. (2006). Dynamic cylindrical cavity expansion in an incompressible elastoplastic medium. Acta Mechanica, 181(1-2): 105-123.
Masri, R., Durban, D. (2005). Dynamic spherical cavity expansion in an elastoplastic compressible mises solid. J. of Applied Mechanics, 72(6): 887-898.
Forrestal, M.J., Luk, V.K. (1988). Dynamic spherical cavity expansion in a compressible elastic plastic solid. J. of Applied Mechanics, 55(2): 275-279.
Warren, T.L. (1999). The effect of strain rate on the dynamic expansion of cylindrical cavities. J. of Applied Mechanics, 66(3): 818-821.

 

[RetiredUSNChief]

I've got a BSME, so topics like this are also of great interest to me.

Cutting to the chase, the answer to your question, ''But what materials was this equation developed for?'', is straight-forward.

Originally, the A-T model (and equations like the ricochet equations derived of the A-T model), a modified Bernoulli equation was developed independently by Alekseevskii (1966) and Tate (1967) in order to model and predict effects of the hydro-dynamic pressure interface and deceleration of high-aspect L/D KE-penetrators into ductile targets (metallic armor systems) and was inspired—at least in part—by the work of Pack et al ''Penetration by High-Velocity (‘Munroe ’) Jets: I & ll'' Armament Research Establishment, Fort Halstead, Kent (1950). The ricochet equation has seen significant examination and seems to be sufficient in its representation of the phenomena.

Rosenberg Z, Ashuach Y, Dekel E (2007) More on the ricochet of eroding long rods—validating the analytical model with 3D simulations. Int J Impact Eng 34:942–957

Over the last 57+ years, the A-T model (and its derivative expressions) has been proven to be valid for use not only in ductile solids, but also in crystalline, polymeric, brittle, porous, granular, and granular mediums. Here are just a few examples of the research supporting those findings...

Geological materials such as soil, rock, and granular mediums like sand—
Forrestal, M.J. Penetration into dry porous rock. Int. J. Solids Struct. 1986, 22, 1485–1500.
Forrestal, M.J.; Luk, V.K. Penetration into Soil Targets. Int. J. Impact Eng. 1992, 12, 427–444.
Khan, M.A. Mechanics of projectile penetration into non-cohesive soil targets. Int. J. Civ. Eng. 2015, 13, 28–39.
Frew, D.J.; Forrestal, M.J.; Hanchak, S.J. Penetration Experiments with Limestone Targets and Ogive-Nose Steel Projectiles. J. Appl. Mech. 2000, 67, 841–845.
Savvateev, A.F.; Budin, A.V.; Kolikov, V.A.; Rutberg, P.G. High-Seed Penetration into Sand. Int. J. Impact Eng. 2001, 26, 675–681.

Ceramics, glasses, crystalline solids—
Clayton JD. Penetration resistance of armor ceramics: Dimensional analysis and property correlations. Int J Impact Eng 2015; 85:124–31.
Clayton JD. Dimensional Analysis and Extended Hydrodynamic Theory Applied to Long-Rod Penetration of Ceramics Defence Technology 2016; 12(4): 334–342
Orphal DL, Franzen RR. Penetration of confined silicon carbide targets by tungsten long rods at impact velocities from 1.5 to 4.6 km/s. Int J Impact Eng 1997;19:1–13.
Orphal DL, Franzen RR, Charters AC, Menna TL, Piekutowski AJ. Penetration of confined boron carbide targets by tungsten long rods at impact velocities from 1.5 to 5.0 km/s. Int J Impact Eng 1997;19: 15–29.
Orphal DL, Franzen RR, Piekutowski AJ, Forrestal MJ. Penetration of confined aluminum nitride targets by tungsten long rods at 1.5–4.5 km/s. Int J Impact Eng 1996; 18:355–68.
Bless SJ, Subramanian R, Normandia M, Campos J (1999) Reverse impact results from yawed long rods perforating oblique plates. In: Proceedings of the 18th International symposium on ballistics, San Antonio, Nov 1999, pp 693–701.
Bless SJ, Rosenberg Z, Yoon B (1987) Hypervelocity penetration of ceramics. Int J Impact Eng 5:165–171.
Rosenberg Z, Tsaliah J (1990) Applying Tate’s model for the interaction of long-rod projectiles with ceramic targets. Int J Impact Eng 9:247–251
Walker JD (2003) Analytically modeling hypervelocity penetration of thick ceramic targets. Int J Impact Eng 29:747–755

Polymers—
Z. Rosenberg, Z. Surujon, Y. Yeshurun, Y. Ashuach and E. Dekel, " Ricochet of 0.3" AP projectile from inclined polymeric plates", (2005) , Int. J. Impact Engineering ,Vol. 31, pp 221–233

Structural materials—
Forrestal, M.J.; Tzou, D.Y. Spherical cavity-expansion penetration model for concrete targets. Int. J. Solids Struct. 1997, 34, 4127–4146.
Forrestal MJ, Frew DJ, Hickerson JP, Rohwer TA (2003) Penetration of concrete targets with deceleration time measurements. Int J Impact Eng 28:479–497

Dynamic strength (Rt) of these materials can be modeled using either SCE (spherical cavity expansion) or CCE (cylindrical cavity expansion) equations which are also derivative of the A-T modeling approach—
Luk, V.K., Amos, D.E. (1991). Dynamic cylindrical cavity expansion of compressible strain hardening materials. J. of Applied Mechanics, 58(2): 334-340.
Luk, V.K., Forrestal, M.J., Amos, D.E. (1991). Dynamic spherical cavity expansion of strain hardening materials. J. of Applied Mechanics, 58(1): 1-6.
Masri, R., Durban, D. (2006). Dynamic cylindrical cavity expansion in an incompressible elastoplastic medium. Acta Mechanica, 181(1-2): 105-123.
Masri, R., Durban, D. (2005). Dynamic spherical cavity expansion in an elastoplastic compressible mises solid. J. of Applied Mechanics, 72(6): 887-898.
Forrestal, M.J., Luk, V.K. (1988). Dynamic spherical cavity expansion in a compressible elastic plastic solid. J. of Applied Mechanics, 55(2): 275-279.
Warren, T.L. (1999). The effect of strain rate on the dynamic expansion of cylindrical cavities. J. of Applied Mechanics, 66(3): 818-821.

That's awesome!

Now for the $64,000 question (I just dated myself, there), what's the layman's working interpretation of all this as applied to a .25 caliber bullet fired into a windshield from the back seat of a vehicle (such as that in post #8)?

The presumption, based on the OP's information, is that the bullet actually ricocheted and did not fully penetrate. But we know that penetrating bullets are deflected counter intuitively by glass windshields, at least as demonstrated by a couple of actual videos on the subject.

 

[481]

That's awesome!

Now for the $64,000 question (I just dated myself, there), what's the layman's working interpretation of all this as applied to a .25 caliber bullet fired into a windshield from the back seat of a vehicle (such as that in post #8)?

The presumption, based on the OP's information, is that the bullet actually ricocheted and did not fully penetrate. But we know that penetrating bullets are deflected counter intuitively by glass windshields, at least as demonstrated by a couple of actual videos on the subject.

In reference to this specific set of conditions—

Thanks to all of you. I should have clarified that I am talking about a windshield with a fairly steep angle, similar to the one in the photo.

View attachment 1183858

—the ''short answer'' is that the OP's presumption is correct that the 25 ACP bullet would ricochet.

The ''longer answer'' is that given the angle and internal concavity of the windshield that getting the nearly perpendicular angle of incidence (< 3°) required to avoid ricochet of the 25 ACP 50-grain bullet at 720 fps under those conditions is going to be very difficult to achieve. However...there is a solution to this problem albeit an impractical one.

Consider that the Rosenberg & Dekel ricochet equation is also highly dependent upon the impact velocity of the bullet...



Once pushed past its Vc (critical velocity) of ≈1,105 fps which is another A-T model equation, a 50-grain 25 ACP bullet would have a critical deflection angle of ≈59°. Of course, getting a 25 ACP 50-grain bullet to move at more than 1,105 fps from the limited case volume, propellant mass, and typically short barrel lengths (< 3 inches) of pistols that fire the 25 ACP is a 'big ask' and one that I don't have an answer to.

Of course, the 25 ACP is a miserable choice for self-defense. In every gang-related shooting that I've ever seen with it, it's been a huge failure; the local thugs went to 9x19 years ago having figured that out for themselves.

Using the US Army BRL Bio-Physics Division Provisional Personnel Incapacitation model's ΔE15 parameter as the basis to establish an expected time to incapacitation (or T(I/H)) model, the paltry .25 ACP 50-grain FMJ at 720 fps fired into COM has a one-shot T(I/H) of 40.1 seconds which leaves a lot of time for someone intent on doing all sorts of evil things to us plenty of time to do so before their time is up. The two-shot COM T(I/H) for the 25 ACP at 6.3 seconds is not very impressive either.

More desirable JHPs like the Speer 9mm 124-grain Gold Dot +P JHP at 1,220 fps which expands to 0.60''—



—and retains 100% of its pre-impact mass has a significantly better one-shot COM T(I/H) of 13.25 seconds and two shots reduces its COM T(I/H) to 3.6 seconds.

 

[481]

This is such a cool subject!

I agree! It is!

Well, now that you've gone an' got me all revved up on the topic , I'd like to address one of the most challenging aspects of penetration mechanics—the accurate determination of Rt.; the dynamic resistance of the target material.

On page 148 of Terminal Ballistics (2020), Rosenberg and Dekel state that (they refer to Rt as σr) with a proper definition of projectile length (the A-T model is sensitive to projectile length) that the ballistic limit (Vbl) of any projectile/target pair can be determined without firing a single shot using this rather basic Poncelet modification of the A-T model—



—if a correct model of Rt can be constructed—



The problem is that modeling Rt accurately is a very complex task because it requires the use of cavity expansion theory to determine the amount of energy that is necessary for a projectile to open a cavity in target material from zero radius. The accurate computation of Rt had been debated for decades without much resolution until Anderson, CE (1994) realized that Rt is not solely a property of the target material but rather one that is velocity-dependent. That is to say, that as impact velocity changes so too does Rt. This fact carries with it some profound and rather surprising implications the most fascinating of which is that as impact velocity approaches zero, Rt approaches infinity.

What Anderson proposed is that one must first solve this transcendental equation for 'α' which is the extent of the plastic zone to the cavity radius:



Once 'α' has been solved for, it can be used to compute Rt using the target material's elastic (E), bulk (K), or shear (G) moduli which I have solved for in the following illustration:



Note: 'm' accounts for the decrease in U across the plastic zone using the slope of intact yield strength-pressure curve where m = 0.75 for ductile materials (metals) and 1.00 brittle materials (ceramic, crystalline materials, glass, etc.).

So, while the 25 ACP 50-grain bullet striking a typical laminated glass windshield at 720 fps would encounter an Rt of 732 MPa, Rt would decrease above the critical velocity (where 'U' = 0) of 1,105 fps to 431 MPa allowing for the 50-grain 25 ACP bullet to defeat the windshield and pass through the laminated glass.

Even more surprising is that at very low projectile impact velocities, Rt approaches infinity!

For example, imagine taking that very same .25-caliber 50-grain bullet and, holding it nose-forward in your fingers, thrusting it forwards as fast as you possibly can (the speed of an average punch is about 20 fps) into the windshield's surface. Besides experiencing bruised—and possibly broken—fingers, the bullet being thrust into the laminated glass windshield at 20 fps would encounter a computed Rt of 3,240 MPa—an increase in Rt of 443%! At even lower speeds, say 5 fps, Rt increases to 4,211 MPa; an increase in computed Rt of 575%. In both cases (5 and 20 fps), the laminated glass behaves as if it has a much, much greater Rt than at 720 fps or 1,105 fps and the laminated glass remains undamaged, although maybe a bit smudged from our hands making contact with it.

This stuff is definitely very cool! I agree!

 

[481]

Hello:

I have read a few opinions about the power, or lack of power, of weapons that fire .25 ACP ammunition. But I would like to know your opinion in this scenario:

Inside a car. If someone fires a .25 ACP bullet from the back seat to the front windshield:

1) Is there a reasonable probability that the bullet will ricochet down the windshield, without going through it, causing only a few cracks and bouncing inside the car twice more? Or is it almost certain the bullet to go through the glass?

2) If ricochet is reasonably likely, would we expect some deformation in the bullet or is it more likely that it appears almost intact?

I would like to know your opinions, and please, be lenient with my little knowledge about firearms.

To answer your questions;

1.) More likely than not a .25 ACP bullet will ricochet due to the typical curvature that exists in most automotive windshields. Damage in the form of cracks, comminution, and bulging of the windshield at the impact site is the most probable outcome. Residual velocity of the bullet will be much lower after bullet rebound/ricochet so it is unlikely that the bullet will rattle around the car's interior as portrayed in cartoons and movies.

2.) Deformation of the bullet will be dependent upon the compressive yield strength of the alloys making up the bullet jacket and core, but it is not unreasonable to expect that the bullet's nose will at least partially flatten against the windshield at impact.

 

[RetiredUSNChief]

I agree! It is!

Well, now that you've gone an' got me all revved up on the topic , I'd like to address one of the most challenging aspects of penetration mechanics—the accurate determination of Rt.; the dynamic resistance of the target material.

On page 148 of Terminal Ballistics (2020), Rosenberg and Dekel state that (they refer to Rt as σr) with a proper definition of projectile length (the A-T model is sensitive to projectile length) that the ballistic limit (Vbl) of any projectile/target pair can be determined without firing a single shot using this rather basic Poncelet modification of the A-T model—

View attachment 1185472

—if a correct model of Rt can be constructed—

View attachment 1185463

The problem is that modeling Rt accurately is a very complex task because it requires the use of cavity expansion theory to determine the amount of energy that is necessary for a projectile to open a cavity in target material from zero radius. The accurate computation of Rt had been debated for decades without much resolution until Anderson, CE (1994) realized that Rt is not solely a property of the target material but rather one that is velocity-dependent. That is to say, that as impact velocity changes so too does Rt. This fact carries with it some profound and rather surprising implications the most fascinating of which is that as impact velocity approaches zero, Rt approaches infinity.

What Anderson proposed is that one must first solve this transcendental equation for 'α' which is the extent of the plastic zone to the cavity radius:

View attachment 1185467

Once 'α' has been solved for, it can be used to compute Rt using the target material's elastic (E), bulk (K), or shear (G) moduli which I have solved for in the following illustration:

View attachment 1185469

Note: 'm' accounts for the decrease in U across the plastic zone using the slope of intact yield strength-pressure curve where m = 0.75 for ductile materials (metals) and 1.00 brittle materials (ceramic, crystalline materials, glass, etc.).

So, while the 25 ACP 50-grain bullet striking a typical laminated glass windshield at 720 fps would encounter an Rt of 732 MPa, Rt would decrease above the critical velocity (where 'U' = 0) of 1,105 fps to 431 MPa allowing for the 50-grain 25 ACP bullet to defeat the windshield and pass through the laminated glass.

Even more surprising is that at very low projectile impact velocities, Rt approaches infinity!

For example, imagine taking that very same .25-caliber 50-grain bullet and, holding it nose-forward in your fingers, thrusting it forwards as fast as you possibly can (the speed of an average punch is about 20 fps) into the windshield's surface. Besides experiencing bruised—and possibly broken—fingers, the bullet being thrust into the laminated glass windshield at 20 fps would encounter a computed Rt of 3,240 MPa—an increase in Rt of 443%! At even lower speeds, say 5 fps, Rt increases to 4,211 MPa; an increase in computed Rt of 575%. In both cases (5 and 20 fps), the laminated glass behaves as if it has a much, much greater Rt than at 720 fps or 1,105 fps and the laminated glass remains undamaged, although maybe a bit smudged from our hands making contact with it.

This stuff is definitely very cool! I agree!

OK, I've got some of this down to the "duh" level. But the film layer over my eyes is crystalizing over much of the rest!

Like "as impact velocity approaches zero, Rt approaches infinity". That's a "duh" because as velocity drops to zero, the dynamic resistance to penetration of virtually ANY material approaches infinity. A non-moving object cannot penetrate anything by definition.

But isn't any material's elastic (E), bulk (K), or shear (G) moduli something determined experimentally? And aren't these material properties based upon, and unique to, any given alloy and alloy treatment, which affects fracture toughness?

Let's take the cast and rolled steel used for the manufacturing of AR500. I believe it's safe to say that the dynamic resistance of this steel prior to the specific heat treatment required of the finished AR500 product would be significantly different than after the heat treatment.

Which would mean there's got to be a wealth of experimental data gleened from a huge variety of materials to draw upon in order to do the calculations you've cited.

We have swerved so far off my baseline understanding of fracture mechanics that it isn't even funny! (Well...except that it is!)

 

[481]

OK, I've got some of this down to the "duh" level. But the film layer over my eyes is crystalizing over much of the rest!

Like "as impact velocity approaches zero, Rt approaches infinity". That's a "duh" because as velocity drops to zero, the dynamic resistance to penetration of virtually ANY material approaches infinity. A non-moving object cannot penetrate anything by definition.

But isn't any material's elastic (E), bulk (K), or shear (G) moduli something determined experimentally? And aren't these material properties based upon, and unique to, any given alloy and alloy treatment, which affects fracture toughness?

Let's take the cast and rolled steel used for the manufacturing of AR500. I believe it's safe to say that the dynamic resistance of this steel prior to the specific heat treatment required of the finished AR500 product would be significantly different than after the heat treatment.

Which would mean there's got to be a wealth of experimental data gleened from a huge variety of materials to draw upon in order to do the calculations you've cited.

We have swerved so far off my baseline understanding of fracture mechanics that it isn't even funny! (Well...except that it is!)

Well, of course, a projectile with zero velocity is incapable of penetrating anything. My remarks were restricted to addressing the computation of non-zero velocity impacts using the equations under discussion; if zero velocity was being discussed, I'd have been talking about dividing by zero and we both know how that turns out.

Elastic (E), bulk (K), or shear (G) moduli are often determined experimentally. Manufacturers will test batches of alloys if requested not to mention that there are lots of online sources like matweb and AZoM that provide searchable databases of material properties of various materials with a lot of it for very specific materials much like the military does for RHA under MIL-12560K. Of course, elastic (E), bulk (K), or shear (G) moduli are related to Kc, but that material property is rarely discussed in the field we're talking about with the exception of the work that I cited earlier in 'Clayton JD. Dimensional Analysis and Extended Hydrodynamic Theory Applied to Long-Rod Penetration of Ceramics Defence Technology 2016; 12(4): 334–342' which deals specifically with the failure mechanisms and microstructure-controlled and density-controlled penetration resistance (Rt) of ceramic armors. I think that given your professional experience in fracture mechanics that you might find that technical paper to be very interesting.

 

[RetiredUSNChief]

I'm geeky enough to put myself through the read!

However, I doubt I'll be geeky enough to learn enough to train on the concepts.

 

[481]

I'm geeky enough to put myself through the read!

However, I doubt I'll be geeky enough to learn enough to train on the concepts.

Don't sell yourself short. The Clayton paper is written such that even "old dogs'' (like you and I) can learn the ''new trick''. Sure, it's technical, but Clayton's style translates well even to the lay audience who will probably "read past" the math anyway.

Expanding our knowledge is good medicine for the brain and why geeks like us stay sharp for so long.