ASSUMING PROPER SHOT PLACEMENT

I have determined, for myself, that energy doesn't really matter.  What matters is EXPANSION and PENETRATION.  

edited because I can't say bullshit in the title!

Agreed - but it is a calculation that some think they can get their headspace around.

Besides penitration and expansion into a human - is not exactly easy to test


Of the two attributes you cite - I greatly favor penitration. In the words of Mike McNett - "two holes bleed better than one"

Good luck

In Before Michael_Courtney and the "Pressure Wave of Dead."

Quoted: ASSUMING PROPER SHOT PLACEMENT I have determined, for myself, that energy doesn't really matter.  What matters is EXPANSION and PENETRATION.   edited because I can't say bullshit in the title!

It's a tripod. Get rid of one leg, and the other two are meaningless.

If you have enough "energy", then depending on the round's construction you will have that energy on hand for expansion and pentration. Having 20ft-lbs of energy ain't gonna do you shit if you "expand" without ever penetrating denim. Even on a heart shot. Conversely, if you have 900ft-lbs of energy and crap loads of penetration, but no expansion, you just poked a hole clean through with a very small wound channel.

Different rounds work for different tasks. If energy didn't matter we'd all be sporting BB guns.

Quoted: Quoted: ASSUMING PROPER SHOT PLACEMENT I have determined, for myself, that energy doesn't really matter.  What matters is EXPANSION and PENETRATION.   edited because I can't say bullshit in the title! It's a tripod. Get rid of one leg, and the other two are meaningless. If you have enough "energy", then depending on the round's construction you will have that energy on hand for expansion and pentration. Having 20ft-lbs of energy ain't gonna do you shit if you "expand" without ever penetrating denim. Even on a heart shot. Conversely, if you have 900ft-lbs of energy and crap loads of penetration, but no expansion, you just poked a hole clean through with a very small wound channel. Different rounds work for different tasks. If energy didn't matter we'd all be sporting BB guns.

Please explain how ENERGY has anything to do with either penetration or expansion? Energy is the ability to do work. It is a function of mass and velocity. It is NOT an independent factor.

Duncan MacPherson notes 29 factors in his physics models of bullet expansion and penetration. Energy is not on the list.

Quoted: Quoted: Quoted: ASSUMING PROPER SHOT PLACEMENT I have determined, for myself, that energy doesn't really matter.  What matters is EXPANSION and PENETRATION.   edited because I can't say bullshit in the title! It's a tripod. Get rid of one leg, and the other two are meaningless. If you have enough "energy", then depending on the round's construction you will have that energy on hand for expansion and pentration. Having 20ft-lbs of energy ain't gonna do you shit if you "expand" without ever penetrating denim. Even on a heart shot. Conversely, if you have 900ft-lbs of energy and crap loads of penetration, but no expansion, you just poked a hole clean through with a very small wound channel. Different rounds work for different tasks. If energy didn't matter we'd all be sporting BB guns. Please explain how ENERGY has anything to do with either penetration or expansion? Energy is the ability to do work. It is a function of mass and velocity. It is NOT an independent factor. Duncan MacPherson notes 29 factors in his physics models of bullet expansion and penetration. Energy is not on the list.

If it takes 7 psi of pressure to break human skin on a point source, then you are going to need at LEAST that much energy on a bullet to get it to penetrate. Penetration and expansion are both examples of "work" are they not?

In analyzing the effects of various rounds on various targets, you would break down each variable into a usable quantity. Duncan breaks these down into 29 catagories. And? It all comes down to energy transfer from the round to the target and the resultant effects.

Quoted: If it takes 7 psi of pressure to break human skin on a point source, then you are going to need at LEAST that much energy on a bullet to get it to penetrate. Penetration and expansion are both examples of "work" are they not? In analyzing the effects of various rounds on various targets, you would break down each variable into a usable quantity. Duncan breaks these down into 29 catagories. And? It all comes down to energy transfer from the round to the target and the resultant effects.

PSI is a pressure measure, not an energy measure. It is units of force applied over area. Energy is one-half mass times velocity squared.

You've never read MacPherson's book, have you? He has a great discussion of the energy myth in wounding.

Since velocity is square in calculating energy, lower mass higher velocity rounds will have more energy than lower velocity, but higher mass rounds (all else being equal). Lower mass-higher velocity projectiles have a well know propensity to underpenetrate. There is no free lunch.

Quoted: PSI is a pressure measure, not an energy measure. It is units of force applied over area. Energy is one-half mass times velocity squared. You've never read MacPherson's book, have you? He has a great discussion of the energy myth in wounding. Since velocity is square in calculating energy, lower mass higher velocity rounds will have more energy than lower velocity, but higher mass rounds (all else being equal). Lower mass-higher velocity projectiles have a well know propensity to underpenetrate. There is no free lunch.

No. I haven't. However, I have read others that are more concerned with rigid, elastic, and fluid impact and energy transfer dynamics. Not necessarily one geared directly towards your average shooter however. I'm coming at this discussion from the "pointy headed physics geek" side.

Energy is energy. Force is energy applied. Transfering said energy from the bullet to the target is what it's all about. Expansiom, penetration, and fragmentation are all ways to transfer the kinetic energy of the bullet to the mass of your target.

I'm from the Elmer Keith school of thought...  I like momentum.  No favoritism given to velocity.  Mass and velocity are treated equally.  (Momentum = mass * velocity)  When I compare hunting cartridges, for which there is little gel testing published, I use momentum to make crude comparisions, assuming similar bullet performance.

That being said I choose my defensive ammo based on actual gel testing, penetration plus consistent expansion.  

Kent

This energy thing is relatively meaningless unless you can compare it to real life in some fashion.  enter www.1728.com/energy.htm

A good ol' fashioned energy calculator.  

My fat ass walking at 4 mph is equal to about 150 ftlbs of energy.  My bumping into you will most likely not kill you (unless I stepped on your foot and the accompanying sudden increase in blood pressure bursts a vessel in your brain.)

Getting smacked by a linebacker at full speed will get you about 1000 ftlbs of energy.  Still not fatal.

a 500 gr arrow at 250 fps is a whole 70 ft pounds, and could be fatal.  The variable is the method in which it delivers this energy.  

Energy works with other factors, and certain implements work is certain ways to inflict damage to the intended target.  Energy itself is not fatal.  Placement is not by itself fatal.  The amount of damage is not determined by those two factors alone either.  

I would not use a 22lr or a .25acp fpr protection.  .50 bmg is not suited for that either.  

Quoted: Quoted: PSI is a pressure measure, not an energy measure. It is units of force applied over area. Energy is one-half mass times velocity squared. You've never read MacPherson's book, have you? He has a great discussion of the energy myth in wounding. Since velocity is square in calculating energy, lower mass higher velocity rounds will have more energy than lower velocity, but higher mass rounds (all else being equal). Lower mass-higher velocity projectiles have a well know propensity to underpenetrate. There is no free lunch. No. I haven't. However, I have read others that are more concerned with rigid, elastic, and fluid impact and energy transfer dynamics. Not necessarily one geared directly towards your average shooter however. I'm coming at this discussion from the "pointy headed physics geek" side. Energy is energy. Force is energy applied. Transfering said energy from the bullet to the target is what it's all about. Expansiom, penetration, and fragmentation are all ways to transfer the kinetic energy of the bullet to the mass of your target.

Um, Duncan MacPherson's whole book is about the physics of penetration and expansion. It is aimed at people who understand fluid dynamics, and thermodynamics, not at your average gun show ex-Navy SEAL. It includes dozensn of formulae to explain how bullets penetrate and expand. MacPherson dismisses energy as essentially irrelevant to both. I recommend you read his book.

Quoted: Um, Duncan MacPherson's whole book is about the physics of penetration and expansion. It is aimed at people who understand fluid dynamics, and thermodynamics, not at your average gun show ex-Navy SEAL. It includes dozensn of formulae to explain how bullets penetrate and expand. MacPherson dismisses energy as essentially irrelevant to both. I recommend you read his book.

I probably will. It will be interesting to see how he posits that mass can have velocity, over come sheer forces, or create a shockwave in a semi-elastic fluid without energy.

Quoted: Quoted: Um, Duncan MacPherson's whole book is about the physics of penetration and expansion. It is aimed at people who understand fluid dynamics, and thermodynamics, not at your average gun show ex-Navy SEAL. It includes dozensn of formulae to explain how bullets penetrate and expand. MacPherson dismisses energy as essentially irrelevant to both. I recommend you read his book. I probably will. It will be interesting to see how he posits that mass can have velocity, over come sheer forces, or create a shockwave in a semi-elastic fluid without energy.

MacPherson never claimed that such objects lack energy. He demonstrates that it is NOT the energy that matters, but things like projectile velocity, form factor, target density, target viscosity, cavitation thresholds, etc. Kinetic energy, being a derived value, does not correlate directly to either penetration or expansion. It's simply NOT that simple.

BTW, what other physics-geek articles/books on bullet penetration did you refer to earlier. I am always lookng for more information on this topic.

Quoted: MacPherson never claimed that such objects lack energy. He demonstrates that it is NOT the energy that matters, but things like projectile velocity, form factor, target density, target viscosity, cavitation thresholds, etc. Kinetic energy, being a derived value, does not correlate directly to either penetration or expansion. It's simply NOT that simple.

Duh shit it's not that simple. That was the whole point of my posts above. Over simplifying it the other way and saying "energy doesn't matter" is, as I stated, utter bullshit. Newton would kick your ass for even implying that the velocity of a mass has no kinetic energy compared to a relatively "at rest" body.

BTW, what other physics-geek articles/books on bullet penetration did you refer to earlier. I am always lookng for more information on this topic.

Bullet penetration? None. Nor did I ever claim to. Any other words you would like to atrbitue to me that I didn't actually say?

Quoted: <snip> Newton would kick your ass for even implying that the velocity of a mass has no kinetic energy compared to a relatively "at rest" body. <snip> Bullet penetration? None. Nor did I ever claim to. Any other words you would like to atrbitue to me that I didn't actually say?

Oh, pardon me. I thought we were discussing BULLETS, not general Newtonian physics. So your statement about "rigid, elastic, and fluid impact and energy transfer dynamics. Not necessarily one geared directly towards your average shooter however." had nothing to do with the actual topic of this thread. Got it.

I never said that a mass with velocity has no kinetic energy. That is you attributing to me something I never said. I stated that energy (in the case of a bullet) is the ability to do work. It is not a good measure of the effectiveness of the bullet. It is not the energy, but the work done by the energy (maximum tissue destruction with adequate penetration to reach vital organs) that matters.

A simple example: Two identical bad guys. One wearing level IV body armor, one without. Both are shot dead center in the upper chest with identical major caliber JHP loads. Both loads will have the same kinetic energy (same mass and same velocity). Assume that for the non-armor-wearing bad guy the round stops within the torso. In both cases, our targets are subject to the same kinetic energy transfer from the projectiles. Same energy transfer, VASTLY different results.  

Hence, my conclusion, like Collin's, that kinetic energy is a poor measure of projectile effectiveness against human targets. Expansion and penetration matter far more (assuming good shot placement).

Quoted: Oh, pardon me. I thought we were discussing BULLETS, not general Newtonian physics. So your statement about "rigid, elastic, and fluid impact and energy transfer dynamics. Not necessarily one geared directly towards your average shooter however." had nothing to do with the actual topic of this thread. Got it.

So you are now asserting that bullets operate under some magical principle outside of Newton's laws?

I never said that a mass with velocity has no kinetic energy. That is you attributing to me something I never said. I stated that energy (in the case of a bullet) is the ability to do work. It is not a good measure of the effectiveness of the bullet. It is not the energy, but the work done by the energy (maximum tissue destruction with adequate penetration to reach vital organs) that matters.

Nor did I say it was. Again, I refer you to the comment at the top of this thread that started it all.

A simple example: Two identical bad guys. One wearing level IV body armor, one without. Both are shot dead center in the upper chest with identical major caliber JHP loads. Both loads will have the same kinetic energy (same mass and same velocity). Assume that for the non-armor-wearing bad guy the round stops within the torso. In both cases, our targets are subject to the same kinetic energy transfer from the projectiles. Same energy transfer, VASTLY different results.

Which, again, I did in fact mention. "Different rounds work for different tasks. If energy didn't matter we'd all be sporting BB guns." Of course, you simply chose to immediately jump to the assumption that I was trying to make a "brute force is the end all and be all" type statement, which anyone with a room temp IQ could see that I wasn't.

Hence, my conclusion, like Collin's, that kinetic energy is a poor measure of projectile effectiveness against human targets. Expansion and penetration matter far more (assuming good shot placement).

I was VERY clear in my statement that all the "expansion and penetration" you could possibly want isn't going to do you much good unless you have enough ENERGY to EXPAND into or PENETRATE your target. That each variable is pretty much useless without the other forces involved. A correlary to my statements would be that any round which maximizes efficiency in ALL of the above defined variables would axiomatically be the "best" round.

You can continue to be argumentitive about this, but you aren't doing yourself any favors.

Expansion and penetration torn my hand up........."energy" blew the side wide open (the exit wound was a different hole entirely.)

I believe in the mythical "hydrostatic shock".

ETA: The round on the right "penetrated" aprox. 4 inches of flesh and bone. In that 4 inches it had enough time to "expand", and the "energy" to exit as well as blast a hole open.

you guys need some real physics instruction, more than can be provided here i'm afraid. not gunstore fat guy theories.

The units of work and energy are the same thing (force times distance) lb-ft for example. Energy is just capacity for work.

True it takes very little energy to penetrate skin. Without energy there would be no penetration. The tissue or whatever you're shooting exerts a reactive force back on the bullet. If the force of the bullet and the reactive force were the same the bullet would not penetrate and the 2 would be in static equilibrium. Even with force the bullet would not travel any distance without energy (back to force times distance), In other words without energy force is useless because there is no movement. As for expansion, what do you think expands the bullet, the small computer controlled charge inside the projectile? Not trying to be a smarta$$ but think about it. The force has to overcome to bullets desire to retain its shape. The force has to move the bullet outward. And here we have it again, a force moving somthing (force times distance).

there is also something called a coefficient of restituion involved in collisions, this affects the deformation (expansion) of the bullet and is dependent on the material being impacted (tissue vs. bone vs. kevlar) for instance.

incase you were wondering about credentials
Mechanical Engineer with a minors in physics, materials science and metallurgy

not trying to be snooty, just wanted to let you know i'm not pulling this out of my ass

Willis

Quoted: Quoted: If it takes 7 psi of pressure to break human skin on a point source, then you are going to need at LEAST that much energy on a bullet to get it to penetrate. Penetration and expansion are both examples of "work" are they not? In analyzing the effects of various rounds on various targets, you would break down each variable into a usable quantity. Duncan breaks these down into 29 catagories. And? It all comes down to energy transfer from the round to the target and the resultant effects. PSI is a pressure measure, not an energy measure. It is units of force applied over area. Energy is one-half mass times velocity squared. You've never read MacPherson's book, have you? He has a great discussion of the energy myth in wounding. Since velocity is square in calculating energy, lower mass higher velocity rounds will have more energy than lower velocity, but higher mass rounds (all else being equal). Lower mass-higher velocity projectiles have a well know propensity to underpenetrate. There is no free lunch.

That doesn't make a lot of sense since you just said that a smaller bullet with higher velocity will have more energy.  Not to mention a smaller bullet will require less force to penetrate, thereby dissipating it's energy slower.

Yes energy is important but it is utterly useless without all of the other factors there.  If you cant transfer the energu, ie the bullet doesnt expand, tumble, etc, the all of that energy is useless.  For a bullet to work properly BE PUT IN THE RIGHT SPOT, somehow transfer its energy, (expand) but to do this it has to have a high enough velocity.  Well, thanks to physics its gets its energy from its velocity.  In the end you cant have one thing w/o the other.  For example, a 5.56 round has alot of energy but it is a very small caliber.  Because of this the bullet was designed to tumble as soon as it entered the body.  Now you have a .75" long bullet going through you sideways and this transfers ALOT of energy fairly quickly.  If the bullet didnt tumble and went straight through you, well it would be about like getting stuck w/ a 5.56 mm rod because the bullet would transfer very little energy.  Yes, energy is important but a lot of thigs have to work right for it to be effective.

once again if you haven't had an education in it there's probably not enough space on this site to explain it

please don't think i'm trying to talk down to you, just trying to clarify these things

I did not disclaim the other factors or shot placement at all. I was just trying to explain the contribution of energy.

your quote "somehow transfer its energy, (expand) but to do this it has to have a high enough velocity" pavelow is true but fyi velocity is used to calculate energy (1/2 mass times velocity squared). Translation, they are directly proportional.  If velocity is zero, then energy is zero. It is kinetic energy, meaning moving energy, if something is moving it obviously has velocity.


Willis

Quoted: You've never read MacPherson's book, have you? He has a great discussion of the energy myth in wounding.

Please provide references to publications where the "energy myth in wounding" is espoused.

In fact, there are none (or very few) publications that assert that energy is an important factor in creating measurable wounding in a target.  The point most authors assert that energy is well-correlated with incapacitation.

The "energy myth in wounding" is a strawman that is easy to disprove by shooting into gelatin.  Disproving the assertion that energy is well-correlated with incapacitation (as long as penetration is within a certain range) would require studing a measure of incapacitation in living targets, and showing that this measure of incapacitation is correlated only with observed wounding and not with the energy.  Of course, every published data set that has actually used some measure of incapacitation (rather than wounding) reveals a correlation with bullet energy.

The most pervasive myth in terminal ballistics is the unproven pressuposition that only easily detectable wounding can contribute to rapid incapacitation.

Quoted: Since velocity is square in calculating energy, lower mass higher velocity rounds will have more energy than lower velocity, but higher mass rounds (all else being equal). Lower mass-higher velocity projectiles have a well know propensity to underpenetrate. There is no free lunch.

This is a accurate in many cases, but this overgeneralization is much more likely to apply to JHP bullets of conventional design.  There are a number of newer designs on the market (Barnes X, Gold Dot, Nosler Partition, Golden Sabre, etc.) where the velocity can be pushed much higher without demonstrating a propensity to underpenetrate.  

Consequently, your suggestion that increasing the energy necessarily leads to underpenetration is simply false.  For example, the 115 grain .355 Gold Dot can be pushed to 1550 FPS and still exceed the suggested minimum penetration of 12".  There is a .357 Sig load (Double Tap) that does this and produces 614 ft-lbs of energy.

The 115 grain .355 Barnes X bullet can be pushed to 1800 FPS (and perhaps much faster) and still exceed the suggested minimum penetration distance.  This would be 825 ft-lbs of energy with no penetration problems.  The 125 grain .355 Gold Dot can be pushed to 1725 FPS and penetrate ballistic gelatin to 15".  The Double Tap loading of this bullet in 9x25 does this and provides 825 ft-lbs of energy.

Comparatively high levels of energy are available in "light and fast" bullets without sacrificing penetration.  The assertion that this cannot be done is "old news" that is no longer correct with modern bullet designs.

Michael Courtney

I see Michael and the Pressure Wave of Dead has arrived...

How soon 'til we can read your study? Oh wait, you can't tell us anything except most everyone else is wrong. Because you are under a non-disclosure agreement...

Quoted: Oh wait, you can't tell us anything except most everyone else is wrong.

I've given sufficient information for others to repeat our experiments, and there are some other groups using substantially similar methodologies in other species.  I've also reported quantitative average incapacitation distances for deer, along with estimates of uncertainty determined from standard methods.

There are very few details I have left out, mainly information regarding the number of test subjects, location of the work, identities of all the collaborators, and funding/support sources.  

And I have never said that "everyone else is wrong."

There are many ballistic researchers with whom I mostly agree:

The authors of the Strasbourg Goat Tests provide a pretty good idea of how bullets work in causing incapacitation.

The Marshall and Sanow OSS numbers are a valid metric of relative handgun load effectiveness, with a typical error of about 5% (larger for loads with few data points).

Steve Fuller presented a pretty good analysis of handgun bullet effectiveness in development of the Fuller index.  How models give the wrong limiting behavior, and there are some minor technical flaws in his paper, but on the whole I agree with his analysis.

Tom Burczynski has also made many interesting and relevant contributions to the field of bullet design, effectiveness, and testing.  

I also agree with some work done by Col. Frank Chamberlain of the US Army, and there's a lot of other researchers with whom I agree on many points.

In my discussions here, I have been open in my agreement with several of the parties mentioned above.  So it is dishonest for you to say that I've taken a view that "most everyone else is wrong."

I even agree with Fackler, MacPherson, Roberts, Dodson, etc. on many points.  They have done a good job correlating bullet parameters and gelatin measurements with wounding.  MacPherson's book is nearly flawless for the ranges of bullet parameters considered, the primary flaw being leap of faith that incapacitation can only come from easily detectable wounding, and that flaw isn't really his, since he references Fackler.  

My main disagreement with these rresearchers is that wounding is not well correlated with rapid incapacitation.  To the extent that they don't claim that it is, I am not really saying that they are wrong as much as I am saying their work is incomplete.   (Their claim is that wounding is well correlated with reliable eventual incapacitation.)

In areas where Fackler, MacPherson, Roberts, Dodson, etc. make assertions based on published data, I agree with almost every assertion they have made.  I do disagree with some of their untested hypotheses in areas for which they offer no data.  

Such is science.  Disagreeing with the opinion of some "experts" in areas in which they have offered no published data is how interesting discoveries are made.  I keep asking "where's the data correlating rapid incapacitation to wounding?"  Yet all you can do is quote tired expert opinions.  

Even if you don't want to wait for our experimental report, we're not really saying anything that can't be seen in the M&S data set or the published incapacitation times from the Strasbourg Goat Tests.   At least the points I am making are supported with published data.  All publication of our work will do is confirm and elucidate what is already present in the published data.

I invite interested readers to go to the thread at Glocktalk and read the working definition of stopping power that my team and I have put together.

http://www.glocktalk.com/showthread.php?s=&threadid=474811

(I tried to post this thread here also, but there was some trouble with the graph.)

Michael Courtney

Quoted: I see Michael and the Pressure Wave of Dead has arrived...

Do you always resort to mocking when your reasoning gets shredded?

Some trustworth ballistics viewpoint you are, unaware of the fact that modern handgun bullet designs can deliver over 800 ft-lbs of energy at over 1700 FPS without penetration problems.

Real men admit they made an overgeneralization and move on.  What kind of man change the subject and resorts to mockery?  

Men without sound reasoning behind their ideas resort to mockery and personal attacks when the flaws in their thinking are exposed.  

Michael Courtney

Energy is BS?

If energy has nothing to with it, then the 30-378 should have the same wound profile than a 30 Carbine.

If it is only the permanent cavity that warrants consideration than why don't we hunt elk with a 9mm Para given it has adequate penetration?  Hopefully nobody would, but one should be more than confident taking elk with a 35 Whelen...They have the same bullet diameter.

I know for a fact that there are those who say there is no difference between a 38 SPL and a 9mm and a 357 Mag when it comes to terminal ballistics, given they are using the same projectile; and the truth is that gelatin supports this assertion so it is a convincing argument, but I do KNOW that penetration depth, bullet expansion and measurable wound channels are only part of the equation.

How do I KNOW?

Easy.  I shot a Rockchuck with a 38 SPL and a 357 Mag loaded with the same 158 LSWC out of the same revolver.  Effect of the 38 was a hole through the animal.  Effect of the 357 was annihilation; essentially all of the organs were blown out and I basically skinned the animal.  Again both were using a non-expanding 158 Lead Semi Wad Cutter.  The only differences between the rounds were velocity and subsequent energy.


Here is a bit of homework:  Shoot a rabbit with a 22 LR, analyze wound.  Shoot another rabbit with a NON-FRAGMENTING 223 and analyze wound.  Post pics.

If energy has no effect on wounding, then a hand tossed 230 grain .45 projectile and a similar one fired from a 1911 would have equal wounding potential.  Seems unlikely.

In my opinion (which we all know what opinions are like) is this.

You need sufficient energy to penetrate the human body and properly expand the bullet. Once the bullet  has approx twice the frontal area, you need the rest of that energy to carry it through the body as far as possible tearing out as much flesh as it can.

So, assuming pistol cartridges, you want "sufficient" energy to achieve your goals. Beyond that, you want a good structure of bullet that will expand properly with clothing and make as big a hole as possible.

Since handgun rounds travel so slow anyway, there is little "temporary cavity" damage usually.

Of course, this isn't assuming going through glass, car doors, etc. etc.

So, energy is important, up to a certain point. After that, expansion size/placement/round count matters.

Rifle rounds are completely different as they are typically 2-3 times as fast and cause different damage.

I think the biggest arguments come from rounds that perform so similarly, like 9mm, .357Sig, .40S&W, .45ACP. They almost all do the same damn thing. Glass and door penetrations can be substantially different, and overall wound size can be substantially different. If you look at the "energy numbers" they're damn near equal for the overall purpose.

Well stated noaim.

Willis