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# Trouble with linear elastic theory of strength

A body is subject to a load. What is the magnitude of the load that will cause the body to fracture? Let us begin with a body made of a glass, which deforms elastically by small strains. A procedure you have been taught before probably goes as follows. You first determine the maximum stress in the body. You then determine the strength of the material. The body is supposed to fracture when the maximum stress in the body reaches the strength of the material.

I’ll first review this procedure, so that you and I agree exactly what this procedure is. I’ll then explain why this procedure is difficult to apply in practice

These notes are part of a course on fracture mechanics.

Attachment | Size |
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1913 Inglis Stress in a plate due to the presence of cracks and sharp corners.pdf | 2.67 MB |

Elasticity 2016 02 02.pdf | 460.12 KB |

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## Comments

## Swain-Timoshenko confrontation

an interesting story, making me re-think about the theory of linear elasticity.

RH

## Re: Swain-Timoshenko confrontation

These files found on the Internet made Swain look really bad. However, this article gives a more historical perspective, in which Swain came out OK. I still have not found a description of the real technical issue of the Swain-Timoshenko confrontation. If anyone has any information, please leave a comment. Thanks.

## A reference on Swain-Timoshenko confrontation

I have just located a technical description of this confrontation:

V.V. Meleshko, Selected topics in the history of the two-dimensional biharmonic problem. Appl. Mech. Rev. 56, 33-85 (2003).

The "confrontation" is described on pp. 51-52.

## Come on guys

Come on guys. It is unfair to criticize a theory for not being able to do what it is not intended to do. Elasticity is a beautiful theory and subject that has enabled the construction of airplanes, bridges, skyscrapers, and other structures that have improved the human condition. Of course it cannot predict the fracture of a solid, or pick up the potential effects of size of a hole; this is because it is limited to determination of the deformation (and associated strains and stresses) of a structure when the material behaves in a way that can be approximated by the theory. We use elasticity's excellent predictions together with phenomenologically based (sometimes ad hoc) criteria for fracture/etc. (maximum stress criteria, small scale yielding for fracture initiation, etc.).

Rui, I do not know what you meant by "re-thinking" about the theory. It is a very good theory if applied within its limits of applicability.

John Dundurs (my favorite mechanics teacher from whom I had the privilige to learn about elasticity) once said at a mechanics dinner that he did not mind dying, because he knew that he could do elasticity in heaven. This is the correct spirit. The incorrect spirit is that of ex-smokers that are the biggest critics of smokers, or ex-patriots that are the biggest critics of their homeland.

All this of course in jest. Warm regards, Roberto

## Re: Come on guys

Dear Roberto: What a good line, "ex-patriots that are the biggest critics of their homeland".

It's just that some of us love a theory so dearly that we cannot stop talking about its pecularities. Even its failings amuse us.

## Roberto, Although you

Roberto,

Although you jest, you bring up the excellent point that we should use the classic elastic theory within its domain of applicability. You point out that the classic theory needs to be augmented with phenomenology, and call out the inherent limitation of classic strain.

But why not consider an elastic theory where the kinematics used is general and so discontinuous deformations are not a problem? Certainly any such notion of

straincannot be understood in a classic sense. The resulting elastic equations are then well-posed at all points, including discontinuities. Hence generalize beyond the domain of applicability of the classic elastic theory.Rich

## It is welcomed

Rich:

yes, theories that eliminate the problems associated with the linearized theory of elasticity should be pursued. But the main issue that needs to be addressed is the constitutive law, which as it stands cannot deal with material failure. In other words, as it stands "the pain experienced by the material" is the left hand side of the equation (a combination of the stresses, for example that is determined by solving the elastostatics BVP), and we set this pain equal to "the maximum pain that can be tolerated by the material" which is the right hand side. The pie in the sky is a theory that is self-contained and has both within it.

Regards

## Reliable constitutive relations

Roberto,

Summarizing your view espoused in your blog entry, although an elastic theory, not necessarily linear, making minimal assumption on the regularity deformation is of value, the real issue is that of constitutive relations. In particular, reliable relations demand accurate nucleation and propagation criteria. And so at a broad level there are two approaches

Which approach renders the development of

reliable constitutive relations for determining material faiure? This is the real question, as I believe you suggest.The second approach renders modeling consistency, e.g., the same equations can be applied everywhere; given a nucleation criterion for a discontinuity, the nonclassical elastic equations evolve the material including where it's softening and has failed. The same cannot be said of the classical approach because the model varies across the material.

Which approach is more aligned with the data from experiments? It's interesting to point out that DIC (digital image correlation) purports to accurately determine the displacement of the material. Classical strain is computed in a postprocessing step, and based on my modest understanding, complicates validating classical constitutive relations. This suggests that the nonclassical elastic equations as I've outlined above may be easier to validate because classical strain has been avoided, e.g. the ''middleman'' of classical strain has been removed.

In summary, any elastic theory, classical or nonclassical, is an idealization. The most useful idealization is probably the most pragmatic one; make minimal assumptions, provides reliable constitutive relations for determining material faiure, and is well aligned with experiments. I would also add that a further useful attribute of an elastic theory is how ameniable it is to rigorous mathematical analysis, e.g. can we show that the resulting equations are well-posed under minimal assumptions on the regularity of the deformation.

Does the above represent an accurate elaboration of your blog entry?

rich

## Stress concentration factor

Dear Zhigang:

It is true that stress concentration factor for a circular hole in an infinite plate is 3 in classical linear elasticity but this is not just an artifact of linear elasticity. If you consider a "Cosserat" body, the simplest possible one (which has a characteristic length as a material property), and use a linearized theory, you would see that stress concentration factor depends on the ratio of the material characteristic length and the hole radius. When the hole radius becomes smaller and smaller the calculated stress concentration factor deviates from 3 (would be smaller than 3 for small holes). I remember seeing this in the following paper, though don't remember the details.

Mindlin, R. D., "Effect of couple stresses on stress concentrations", Experimental Mechanics, 3, 1-7, 1963.

Regards,

Arash

## Re: Stress concentration factor

Dear Arash: The notes are for the first lecture of a course on fracture mechanics, which started on Tuesday this week. I'd like to return to the issue of stress concentration later in the class within the context of the cohesive-zone model. In the cohesive-zone model, which introduces a length scale, a small hole will not cause any stress concentration. In this sense, the stress concentration factor of 3 is an "artifact" of linear elasticity. See Fig. 8 in

Z. Suo, S. Ho and X. Gong, "Notch ductile-to-brittle transition due to localized inelastic band," ASME

J. Engng. Mater. Tech.115,319-326 (1993).## Re: Re: Stress concentration factor

Dear Zhigang:

I agree that introducing a cohesive zone introduces a length scale and exactly because of this length scale the stress concentration factor will not be "3" anymore (maybe there are other ways of introducing a length scale other than a cohesive zone or couple stresses). The question that I pose now is the following: In the same infinite plate with a circular hole under far-field tensile stresses, in the framework of nonlinear elasticity (given some strain energy function), what is the stress concentration factor? I don't know the answer but my guess is that the calculated stress concentration factor may depend on the applied load (of course the circular hole will deform and the stress concentration factor will change with this change in shape). However, because of the lack of a characteristic length even in classical nonlinear elasticity, what the theory predicts would not be affected by the size of the hole (as long as the hole is circular). In this sense, I still think stress concentration factor being "3" is not an artifact of linear elasticity; it is a consequence of the lack of any characteristic length in both classical linear and nonlinear elasticity theories.

Of course, I would very much like to know what you think and continue the discussion.

Regards,

Arash

## Stress concentration in nonlinear solids

Dear Arash: I agree that in nonlinear elasticity, which also lacks a length scale, the stress concentraion will be independent of the size of the hole. For example, an analytical solution exists for neo-Hookean materials with full nonlinear kinematics.

A.N. Gent and D.A. Tompkins (1969) Nucleation and growth of gas bubbles in elastomers, Journal of Applied Physics 6, 2520-2525.

## Kinematic assumptions of classical linear elasticity

Zhigang,

Interesting note. A key assumption of classic linear elasticity is that strain is a well-defined quantity. Unfortunately, the classsic notion of strain is undefined where the displacement field is not differentiable (let alone discontinuous). Hence the kinematic assumptions of classical linear elasticity are a source (maybe the source) of the problem because material failure often manifests itself in non-differentiable motion. The classic solution is to infer where the non-differentiable motion occurs. But this is a challenge, probably impossible for many complex materials.

Question: At a purely mechanical level, what is it about molecular mechanics that allows it to succeed where classical linear elasticity struggles mightly? Better constitutuve relations? If so, what are the constitutive assumptions of molecular mechanics? What are the kinematic assumptions? How are the latter mapped to the force (or work if you like) associated with a colleciton of particles (or the class of consitutive relations)?

--rich

## Re: Kinematic assumptions of classical linear elasticity

Dear Rich: In so far as fracture is concerned, what is obviously wrong with linear elasticity includes

As they say, all models are wrong, but some are useful. Tomorrow in

the class, I'll desceibe how Griffith used lineasr elasticity to obtain

useful results. Later in the class I'll talk about how Irwin formulated linear

elastic fracture mechanics even for things very nonlinear, such as

metals. In these classic examples, special cares are taken to use linear

elasticity where it is correct. In particular, the crack-tip elastic

field is never really used to calculate stress at the crack tip.

I'll post notes as I go along at the website of the course on fracture mechanics.

## Clarification

Zhigang,

Look forward to reading your notes but let me prod you for some clarification.

classicallinear elasticity contains no length scale. There are linear elastic theories that do contain a length scale.Thanks.

rich

## About linear elasticity in fracture mechanics

Dear Rich: I'm preparing for the lecture this afternoon. Quick responses to your questions.

## Linear elasticity w/o strain

Zhigang,

As you explain, strain gradients or other high-order terms can be used to augment classical strain so introducing a length-scale. Unfortunately, such kinematics assumes even more differentiability than simply using classical strain. How can this be viable for fracture mechanics (without assuming that you can predict where the deformation discontinuities will appear)?

Why not consider a linear elastic theory that uses the deformation itself (not an approximation of the deformation as in classical strain)? Such linear elastic equations are then well posed even at discontinuities of the deformation.

Rich

## Re: Linear elasticity w/o strain

Dear Rich: The cohesive-zone model is perhaps what you are looking for. The model is a hybrid of linear elasticity and a nonlinear traction-displacement relation. If you are unfamiliar with this model, here is an introduction to the basic ideas and results:

G. Bao and Z. Suo, "Remarks on crack-bridging concepts,"

Applied Mechanics Review45, 355-366 (1992).## discussion when i updated the notes for the 2014 class

Additional discussion on this lecture when the notes were updated for the 2014 class.

## Inglis Paper

The figures in the Inglis papers are not included in the link. Could you please upload them as well.

Rajesh

## Re: Inglis Paper

Sorry, I don't have the figures of the paper. Please post a note if anyone find the figures.