Young's Modulus
The Young's modulus is the ratio of stress (pounds / square inch) divided by strain (inches deformation / inch). When you apply a force to your barrel, that is stress. When the barrel moves in response to the force that is strain. This ratio defines how stiff a material is. Stiffer materials move less under the same stress. An aluminum object will move three times as much as a steel object of the same size under the same load. The modulus of iron is roughly 30*10^6, thirty million as is that of cobalt, manganese, nickel. Chromium is somewhat higher, about 45*10^6 as is vanadium. All iron alloys, tool steel, structural, stainless, inconel, all of them have approximately the same modulus, 30*10^6. This means how stiff the barrel is is independent of the alloy it is made of. The other factor that defines the stiffness of an object is it's dimensions. Generally, stiffness of a given shape is proportional to it's width, the cube of the thickness and inversely proportional to the cube of the length. For example, a 2X8 is twice as stiff as a 2X4 bent the easy way but eight time as stiff bent the hard way. A 2X4 8 feet long will bend eight times a much under the same load as a 2X4 4 feet long. An object twice as wide is twice as stiff, an object twice as thick is eight times as stiff and an object half as long is eight times stiffer. If you double the diameter of an object it is sixteen times as stiff because it has twice the width and twice the thickness.
Toughness is a different thing, it is the opposite of brittleness and is a measure of the amount of energy it take to fracture a material. This varies widely with alloy and temperature. A material may be very tough but not very strong, very strong and not very tough, tough at room temperature but brittle in the cold. The engineer specifies all materials so they are adequately tough under anticipated conditions. The Liberty Ships of WWII were made of steel that became brittle in the cold of the North Sea. They broke in half and sunk. Steels must be designed to have adequate toughness under the coldest conditions they will face or they will break, just snap.
Strength is yet another thing, it is the measure of how much stress (pounds / square inch) it takes to deform a material to the point it will not return to it's previous shape. Deformation may be elastic or plastic. An object (such as your gun barrel) deforms elastically then returns to it's previous shape. Other objects, say a beer can you crush has been deformed plastically. Stregnth varies widely with temperature and alloy. Generally when we talk about "hard" or "soft" we are talking about strength. Most structural components and all firearms components are designed so they never (hopefully) deform plastically.


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