Why doesn't the Young's modulus change when an alloy is strengthened? There is one thing about Young's modulus that I find unexpected and confusing.
When certain solid materials, pure metal, steel or an alloy of a certain composition, gets strengthened by cold working or by heat treating, the Young's modulus stays exactly the same as before even though the yield strength of that material gets doubled, and the elongation gets reduced by an order of magnitude.
Take maraging steel 350 for example.
Annealed yield strength = 830 MPa ... Annealed elongation = 18% ... Annealed Young's modulus = 190 GPa
Aged yield strength = 2300 MPa ... Aged elongation = 4% ... Aged Young modulus = 190 GPa
This seems crazy to me. Strength triples and elongation is reduced to less than a quarter, yet the Young's modulus doesn't change one bit? I don't understand.
If the definition of Young's modulus is the ratio between stress and strain, when steel after aging gets 300% stronger, and that strength is achieved at 20% elongation, how could the Young's modulus possibly not get massively changed too?
 A: Here’s a diagram relating pre and post aging performance:

pre vs post aging performance
For that material, you can see how yield point elongation decreases while the slope of the linear part (modulus) doesn’t change much. 
A: It's important to distinguish between two very different regimes when considering the stress-strain behavior of metals: (1) The elastic regime and (2) the plastically deforming regime. When relatively small stresses are applied to metals, they tend to behave elastically. If you apply a stress it bends a little, and if you then remove the stress it goes back to its original position. Elastic properties of metals depend on the elemental composition of the metal, but they tend to be insensitive to the microstructural details of the metal. Things like dislocations (produced by work hardening a metal) or fine precipitates (which can be produced by age hardening the metal) don't affect the elastic properties of metals like the Young's modulus much since they tend to be a relatively small volume fraction of the overall volume of the metal.
So the question is really why do these microstructural details become so important when the metal starts to plastically strain? When the stress on a metal becomes large enough to plastically stain it, we enter into a very different regime in which the material is undergoing large deformation which is enabled by the movement of dislocations through it. When this starts to happen, then all those little microstructural details in the metal such as grain boundaries, pre-existing dislocations, and fine precipitates become very important because they all act to block the smooth flow of dislocations through the metal. As a result, more stress has to be applied to the metal in order to overcome the dislocation barriers and make the metal plastically flow. That's why work-hardening and precipitation hardening (i.e., "age hardening") are so effective at increasing the yield strength, which is a measure of the stress required in order to make the metal plastically deform. 
Bottom Line:
Elastic Properties (e.g., Young's modulus, Bulk modulus, Poisson's ratio) depend on the elemental composition of a metal but are insensitive to the microstructural details of a metal.
Plastic Properties (e.g., Yield strength, tensile strength, elongation at maximum yield) are sensitive to the microstructural details of a metal (e.g., grain boundaries, pre-existing dislocation bundles, fine precipitates) because these microstructural features can block the movement of dislocations through a metal which enable the metal to plastically deform.
A: It's a common misconception that work hardening a metal increases its Young's Modulus.  It looks simply like, "Well, steel has a higher Young's Modulus than does brass and steel is harder than brass, so I guess if you make the hardness of brass up closer to that of steel, you will also make its Young's Modulus up closer to that of steel." 
The misconception arises because we are talking about two different mechanisms within the metal.  One is elasticity and the other is plasticity. 
Young's Modulus is a feature of elasticity, and elasticity is possible only for small enough stresses.  Such sufficiently small stresses involve only the intermolecular forces within the material.  You can visualize them as tiny springs.  That implies that only the chemical nature of the material participates in the observed elastic property. 
Once you stress the material so much as to cause plastic deformation, you involve other forces than just the intermolecular forces.  Every metal has internal features that tend to resist deformation, such as grain boundaries, inclusions, and dislocations. With plastic movement, these features get "locked in" more than they were just before deformation, and subsequent deformation escalates the resistance to relative movement that these features provide.  That explains the hardening, why it's harder to cause more deformation. 
But the intermolecular state is not changed appreciably by the deformation.  That's probably because the total volume of the material occupied by the grain boundaries and other defects is very small compared to the total volume of the material.  Thus, if the material is only slightly stressed again after the deformation, those tiny springs behave just as they did before deformation. 
A: Remember that to get the hardness, you measure with a hardness tester. (eg see Vickers, Brinell). That means, put simply, you push a ball or pyramid into the surface and measure the size of the dent. So it's easy to imagine that hardness is a measure of plastic deformation.
When you're measuring the Young's Modulus, you stretch (usually, but it can be compress) your specimen elastically, so when you let go, there's NO deformation, it goes back to what it was. You  know how much you stretched it and the force that that created, and you know the cross sectional area of the specimen, so you work out stress/strain.
Using different words from the previous poster, when you  elastically deform something, you're stretching the elastic bonds between the atoms, which ping back when you let go. When you plastically deform something, all the faults in the crystal structure move around to accommodate the movement. Like ripples in a carpet - you can move the ripple to get the whole carpet moved a little, but when the ripple's reached the wardrobe, or simply the edge of the carpet, you have to put a lot more effort in to move the carpet. That's work hardening.
The work hardenability depends on many things. You can easily imagine the size of the crystals (called grains) makes a difference. Fine precipitated hard particles distributed through the crystal also make a difference, because they stop the ripples moving.
