# Value of Mechanical Constants in Real-World Applications

I've recently been reading through introductory texts on the mechanical properties of materials such as this. Most start with a description of various material constants such as Young's modulus, Poisson's ratio and a zoo of other properties, and try to determine these properties through a study of dislocations and interatomic interactions.

However, when I read about the very specific test conditions for these kinds of properties, it's not immediately obvious how well these kinds of coefficients and measurements translate to the real world.

For example, I can find coefficients describing hardness, toughness, etc. from laboratory experiments under specific experimental conditions, but it's not obvious that these coefficients can give a more than qualitative description description of how well my nail can hold together a desk, how well my steel truss can hold up a roof, etc., which are ultimately the questions I might be interested in. These coefficients seem to describe very specific geometries, and it seems that I can only get answers about more general geometries using FEA, leaving these kinds of measurements as useful for 1) qualitative descriptions and 2) feeding parameters into FEA.

I think some of the confusion stems from the seemingly arbitrary nature of all these parameters. It appears that someone designed a test and came up with a new parameter every time they were interested in a behavior under different conditions, and there's no minimal degrees-of-freedom type argument that prevents me from coming up with my own test and modulus.

Put another way, what is the argument that the mechanical properties covered by the usual texts (Young's, Poisson's, hardness, toughness, etc.) can give me a complete mechanical description of a system under arbitrary geometries and loads?

Thinking about the parameter space more abstractly, each test can be represented as a material-dependent function that accepts a given material geometry and load geometry and maps it onto strain (throwing out hysteresis and time dependent phenomena for simplification). If we represent our inputs in R^n and our outputs in, say, R^m, what is the minimum number of moduli and constants needed to define this function? What's the argument I won't open up a textbook and find that it defines a new and non-redundant mechanical parameter.

And, if such a well defined relationship exists, where might I find out more about how to go from the parameters derived from well defined experiments (stretching a cylinder, squeezing a flat plate, etc.) and more general material and stress geometries?

Generally, the first-order model parameters you discuss (specifically elastic constants) relate the complete stress state of an infinitesimal cube of a homogeneous material to its complete strain state. There are 21 such elastic constants forming a symmetric rank four tensor. Left-multiplying the elastic tensor against the symmetric rank two strain tensor (or its inverse against the stress tensor) gives the stress tensor (or strain tensor). As with all physical models in every field the model of elastic material behavior is wrong (e.g. during plastic deformation), but useful under certain assumptions. Notably, those of linear elasticity, continuum homogeneity, and uniform standard environmental conditions. The many tests you write of are useful for determining those 21 elastic constants by experiment.

The model is still resonably useful when those assumptions are close to true but not quite perfect. The model can also be extended to include the effects of variable temperature, etc. Given resonably accurate values, a computer can be fed those values for use in deformation calculations in FEA simulation software which may be used to estimate deformations in arbitrary geometric domains subject to nearly arbitrary boundary conditions. The results of the simulations can be used to guide design decisions in real world products.

Additionally, other mathematical models may require other physical values such as elastic limits, deviation from linearity, strain rate effects, temperature effects, plastic deformation models, etc. Models have also been developed which correlate physical properties with microstructure, and microstructure with processing history and with material composition. This allows for variation of the physical properties over the geometric domain based on differential processing parameters, e.g. corners solidifying faster than hotspots in a casting, resulting in more accurate simulations.

Without physical parameters and fundamental models none of this would be possible.

Put another way, what is the argument that the mechanical properties covered by the usual texts (Young's, Poisson's, hardness, toughness, etc.) can give me a complete mechanical description of a system under arbitrary geometries and loads?

There is no such argument. All models are wrong. Some models are useful. In other words, there is absolutely nothing that can give you a complete mechanical description of any system. But the usual properties are quite useful in making predictions under many (but not all) practical situations.

They don't give a complete description, hence why there's no accurate conversion between the various hardness scales.

Hardness and toughness have arbitrary and specific test conditions because they represent a combination of more fundamental properties that we either can't measure or perhaps don't have a model of. They're useful for specifying materials that have consistent performance to each other in typical applications, not for putting into any theoretical model.

Young's modulus and Poisson's ratio are less empirical because we have a model that can use them to make predictions (linear elasticity). That gives a complete description of how a material will behave as long as it stays linearly elastic. In that regime, there's no need for hardness or toughness because those describe non-elastic behavior.

Since you're willing to neglect time dependence, and I guess any kind of history dependence, then I think you're limiting yourself to simple linear elasticity which is well defined with its 21 constants. You don't need to consider hardness and toughness because they describe permanent deformation as well as load rate dependence. Large elastic strains (like in elastomers) can also have history dependence if stretching it makes it temporarily stiffer, for example. I'm not sure if there's room for a nonlinear elasticity model that excludes any time dependence. But if there is, then I guess that's what you're looking for.

It is because materials science is fundamentally a empiric science. The models we have are adequate at best and can not be built exactly from first principles. Too many things going on at the same time.

As an example of this I know of a big engine manufacturer that has engines made out of material X on the market, has had for decades and they work well. But they can not model this, if they model it with best of knowledge they get that the engine should last for a fraction (Weeks or days as opposed to multiple decades) of what they actually do.

So by all means if you can make accurate prediction with your own model go for it. Just do not expect people to use your stuff unless you can actually show a consistent history of being right.

• If this is the case, why should we be interested in these moduli and material coefficients at all? Sep 17, 2016 at 17:15
• @Dragonsheep because they predict how steel would work in this case accurately. Unfortunately it does not hold true to all materials. Sep 17, 2016 at 18:46