Stress Intensity vs Stress Concentration? 1

[jpaero]

I have been trying to sort of distinguish between stress concentration factors and stress intensity factors. I have managed to write a bit.

Stress concentration factor tells you by what factor the local stress is higher than the far field/reference stress.
It is purely a geometry driven parameter (not dependent on the magnitude of applied stress although might depend on the kind of loading (axial, bi-axial, shear etc), allowable stress or anything else ..just pure geometry). Based on Kt alone you cannot say much about the health of the structure in any quantitative manner. You can have a Kt=10 and still be OK cause your max. applied stress is low enough to allow for such high Kts. (Yes, it will not be a great design, but you will survive).

Also, concept of Kt itself is null and void if you have a crack where the crack tip has sharp notch with near zero radius. You will get a theoretical Kt of infinity at that crack tip based on which you will predict that your local stress is infinite any non-zero applied stress and that your structure will fail. But experience showed that just because you predicted infinite local stress, the structures in practice didn't fail. And that's where the concept of stress intensity factor(SIF) comes in.

Stress intensity factor takes into account three things.1) the applied stress 2) geometry 3) crack length. If your applied stress goes up, your SIF goes up. You make the geometry mess, SIF goes up. You increase the crack length, SIF goes up. SIF provides you with a way of quantifying the state of stress around the crack. If you do not have a crack, your stress intensity factor is 0 but your stress concentration factor will still be the same old. The stress intensity factor sort of provides you with a single number which sort of wraps everything you are concerned into one. If two different structures have the same SIF, (although they may have different applied stresses, different geometries and different crack lengths), those two structures have the same stress field under the given conditions of stress, geometry and crack length. Now having calculated the SIF, you can compare it with the critical stress intensity factor (Kcrit) to determine if your structure is OK or not. This Kcrit is the SIF at which you have a runaway crack and is akin to the traditional material allowable.

The figure attached is from a book by Jaap Schijve which essentially says that when the cracks are microscopic, (as in when there are "NO" cracks, the concept of Kt is applicable) but once a crack is formed the concept of Kt cease to be a meaningful one and you essentially transition to the stress intensity concept.


Is that a fair summary?

 

[3DDave]

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[jpaero]

I'm still learning the ways with the forms. I've tried to upload it again.

 

[rb1957]

Usually stress intensity and stress concentration are different things, although there is some overlap.

Fatigue analysis is different to crack growth.

Fatigue analysis can be accepted by the authorities as a substitute for the threshold life ... in that the threshold can be considered the safe life of the part (threshold is time before a detectable crack ... it was initially now the definitions blur initial crack growth and detectable crack growth).

Stress intensity for a very short crack (like a 0.005" typical imperfection) can be modelled as the stress concentration (as the Kt or as the stress field around the notch).

But fundamentally, stress concentration is due to the geometry of the notch but stress intensity is due to the crack.

another day in paradise, or is paradise one day closer ?

 

[LiftDivergence]

One of the fundamental differences is that the stress intensity factor is part of a criterion for determining a thermodynamically irreversible change in the media. Remember that the stress intensity factor approach can be interpreted as an energy balance approach. If the externally added or internally released energy is greater than the amount of stored and dissipated energies, a new free surface is created in the media. The stress intensity factor is associated with the rupture of cohesive bonds in the material. It can indeed be used to determine a stress field in the vicinity of a stress singularity, but it fundamentally differs from the stress concentration factor in it's application. You may think of the fracture toughness as the rate of change of dissipative energy with respect to the fracture area.

The reason why we say that the SIF concept is applicable only to macroscopic regimes is that it is derived using continuum mechanics theory where the media is assumed to be a homogeneous continuum. In reality this is not the case microstructurally, but a comprehensive theory of fracture from smallest scale to largest scale is not really possible since phenomena on a scale below ~0.01 cm like dislocation movements and slip bands are heavily dependent on the imperfections and exact arrangement of the microstructure.

You can see how the stress state is influenced by macroscopic details which lead to stress concentrations which therefore become likely points of nucleation. But addressing the process on the nucleation scale is challenging. For that, we need a model of the slip bands resulting in intrusions and extrusion. Therefore, we generally let peak stresses (stress concentrations) inform where we want to examine behavior of macroflaws, and we treat these flaws from a simpler standpoint of homogeneity which allows us to used LEFM. As I've said in other posts, remember that the continuum model is semi-empirical because some constants are experimentally derived.

A stress concentration factor is actually a "manufactured" quantity, like for example, the stress ratio, R. The stress state in the part is only dependent on the geometry, material, and applied load. Kt is just a ratio of the stresses at two different points in the material. Remember too, that for purposes of fatigue, we don't work with Kt, but rather Kf, which is dependent on the notch sensitivity.

You can get a further sense of the difference in these two quantities by examining how they are mathematically derived.

For stresses in the vicinity of singularities, i.e. cracks, the start of "modern" work is Griffith, Irwin, Barenblatt. Foundation for some early work is partly found in Westergaard.

If you're familiar with continuum mechanics or stress function methods from school, original formulations were an extension of that...rather than an Airy stress function, which most people have heard of, practical usage mostly involves Westergaard's stress function. Although the basic premise is the same... an Airy stress function is a way to solve the equilibrium equations of elasticity, which include the stress-strain relations and the strain-displacement relations. Airy assumes the body forces are small relative to the applied forces, and solve only for the stress-strain relations (ignoring strain-displacement). But to do this, you need to make sure the compatibility equation is satisfied (you always need equilibrium and compatibility somehow). Usually this is written as the biharmonic equation, thus any valid stress function is one which also satisfies the biharmonic equation.

Westergaard is based on Muskhelishvili's method but simplified so that it only applies to cracks along a straight segment of an axis. The "issue" with Muskhelishvili and Westergaard is that they both leave the stress solutions in complex (real + imaginary) coordinates. It wasn't really until Irwin that the approach was made more practical.

Note: Westergaard's original formulas were later shown to only be applicable to periodic cracks specifically in biaxial tension, and has sense been rectified.

There are many mathematical approaches to solving the stress field equations in the vicinity of a crack tip. For example, the problem of a central crack in a finite width plate has been studied extensively and refined. There is a general solution by Irwin, a "tangent formula". To date, probably the most accurate closed form solution is by Isida, basically a truncated polynomial series. But Feddersen has also given an accurate and more compact secant formula.

In general, the main difference is that SIF solutions are developed to be used in LEFM engineering methods. There is an assumption of a macroscopic defect in the material which produces a stress singularity. The solution for the surrounding stress field is based on this assumption, and the boundary conditions.

For a stress concentration, stress fields are generally still found analytical using the equations of elasticity, but there is no stress singularity. There is a stress riser, which is normally a geometric design feature, which results in a preak stress, but this is not the same as the assumption of a crack tip.

Actually if you look in Tada/Paris/Irwin, "The Stress Analysis of Cracks Handbook", there is an entire subsection on applicability of LEFM stress field solutions to stress concentration factors for very slender notches where they refer to Savin, Neuber, and Peterson.

As far as readability I generally prefer Murakami or Sih, but one great thing about Tada/Paris/Irwin is that for all of their solutions they cite the original author on the same page, and list the method used to obtain the solution, and the accuracy. (For example, the problem I reference above, is included in Part III... it refers to Isida 1965).


Keep em' Flying
//Fight Corrosion!

 

[Lariliss]

Hello, the summary is fair enough.
Making it clear in several lines:

Stress concentration factor accounts for the presence of geometric discontinuities in a structure (e. g. Notches, fillets or holes). It depends on the geometry.
Stress intensity factor is dependent on geometry and load. And can be determined experimentally or calculated for a certain crack. It is dependent on cracks and other material and geometric defects.