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an interesting puzzle: multiscale mechanics

Henry Tan's picture

an interesting puzzle for fun:

Lame’s classical solution for an elastic 2D plate, with a hole of radius a and uniform tensile stress applied at the far field, gives a stress concentration factor (SCF) of two at the edge of the hole. This SCF=2 is independent of the hole radius.

Consider what happened to this concentration factor if the radius a approaches infinitely small. The SCF is independent of a, so it remains equal to two even when the hole disappears.

This is inconsistent with what one would expect physically, namely that the limit a->0 should be the same as when the plate is whole without a hole and has no stress concentration.



Henry Tan's picture

This puzzle can be used for teaching multiscale modeling.

Luming Shen's picture


It's a very interesting point.

I guess when the radius of the hole a is close to the lattice constant, Lame’s classical solution does not hold any more. Certainly the SCF is one when the hole disappears.


Henry Tan's picture

therefore, you see, length scale comes very naturally into the size-independent continuum result at the nanoscale.

Zhigang Suo's picture

When the hole is large, the linear elastic theory applies, and the stress concentration factor is 3.  When the hole is small, the linear elastic theory breaks down, and the stress concentration factor approaches 1, i.e., the hole does not concentrate stress.

The question is how small the hole needs to be.  To answer this question, you don't need abandon continuum theory, but you need a length scale.  The linear elastic theory has no length scale.  You can introduce a length scale in many ways.  For example, you can prescribe a cohesive zone near the hole. 

The length scale is specific to materials.  The cohesive zone model gives you a formula to estimate the length scale.  For a brittle solid, the fracture process involves breaking a plane of atomic bonds, so that the length scale is of atomic dimension, about 1 nm.  For a fiber-reinforced composite, the fracture process involves pulling out fibers from the matrix, and the length scale can be macroscopic.  Consequently, in a composite a hole as large as, say, 1cm in diameter may still not concentrate stress. 

The following papers carried out this calculation for holes and notches.  The papers also reviewed some of the history of the question posted by Henry.  

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).

G. Bao and Z. Suo, "Remarks on crack-bridging concepts," Applied Mechanics Review. 45, 355-366 (1992).

Rui Huang's picture


I agree with you on that the stress concentration factor should approach 1 for a small hole. The 2D elasticity solution no longer holds. In your papers, you introduced a cohesive zone for localized inelastic deformation. This however changes the geometry from a circular hole to a bridged crack. Can we actually define/determine a size-dependent stress concentration factor without introducing any cracks? A length scale may be introduced by considering the surface energy or surface stress at the hole or by a cohesive law (pressure-expansion for the hole, similar to traction-separation for a cohesive crack). 


Konstantin Volokh's picture

Hi Guys,

 I consider your puzzle in my blog "Griffith controversy" where you can find a paper devoted to it.


Henry Tan's picture

Indeed, the softening hyperelasticity model you proposed solved the puzzle quite nicely.

Rui Huang's picture

The Griffith approach in Kosta's paper does solve part of the puzzle regarding the failure criterion, just as it did for a sharp crack in an linear elastic solid. On the other hand, the stress fields for both the spherical and cylindrical voids appear to have the same stress concentration factors, 3/2 and 2, respectively. I don't see how the proposed softening hyperelasticity solves this controversy, since the stress field should be homogeneous as the void radius goes to zero.

Well, I think this controversy might be resolved by re-considering the boundary conditions. Simply putting, the tractions at the surface of the void are no longer zero as the hole size becomes extremely small (i.e., comparable to the range of van der waals force or even interatomic interactions). A length scale must be introduced here, and the solution will become size dependent. Two limiting cases are obvious: for large voids, the classical elasticity solutions are valid; for a hole radius of interatomic spacing, the stress is uniform.


Konstantin Volokh's picture

Hi Rui,

I think the idea that "the stress field should be homogeneous as the void radius goes to zero" is wrong. The stress field is inhomogeneous unless the cavity entirely disappears.


Rui Huang's picture

"the radius goes to zero" means that the cavity the continuum sense of course. 


Mike Ciavarella's picture

See below for an account....

Konstantin Volokh's picture

No it is not a smooth process. The cavity exists or not! Only two possibilities.

Rui Huang's picture

In crystals, individual vacancies aggregate to form microvoids, and microvoids aggregares to form big holes. From discrete to continuum, a smooth transition makes sense to me.


Konstantin Volokh's picture

Well, you are getting to the scales where the classical length-independent continuum mechanics may not be applicable. This is far from my criticism of Griffith.

Rui Huang's picture

No, this is not about Griffith. But the puzzle cannot be fully resolved without considering the discrete nature down below in the scale. Yet, a slight modification can extend the continuum theory much further, narrowing the gap between the continuum world and the quantum world.


Henry Tan's picture

Fundamentally, a buk material is a collection of atoms with movements controlled by the atomic potential (suppose that we do not go further into electron scale).

The continuum concept is an assumption.

Other terms, like displacement, strain, stress, constitutive law, stress concentration, stress intensity factors, surface energy, etc., are all based on this assumption.

Geometries, such as void (e.g., spherical), crack, surface, etc. are also based on the continuum concept.

Rui Huang's picture

Yes. The question is: how far can we go with the continuum assumption? The answer would be problem-dependent, rather than a definite scale. 


This particular problem has been dealt with by Mindlin (1962) in terms of a reduced Cosserat couple stress theory.  That theory can be regarded like an extension of the continuum assumption which allows bringing in length scales into the problem in order to handle size dependent response. Such size dependency should be expected below some characteristic dimension (like the hole radius in this case) of the specimen in question. 


PhD Computational Mechanics


Applied Mechanics Group

EAFIT University


Henry Tan's picture

To teach myself

The Cosserat theory of elasticity, also known as micropolar elasticity, incorporates a local rotation of points as well as the translation assumed in classical elasticity; and a couple stress (a torque per unit area) as well as the force stress (force per unit area). The force stress is referred to simply as 'stress' in classical elasticity in which there is no other kind of stress.

The idea of a couple stress can be traced to Voigt during the early development of the theory of elasticity.

More recently, theories incorporating couple stresses were developed using the full capabilities of modern continuum mechanics.

Early theoretical work was done by the Cosserat brothers, by Mindlin (, and by Nowacki. Eringen incorporated micro-inertia and renamed Cosserat elasticity micropolar elasticity.

Henry Tan's picture

To teach myself:

In the isotropic Cosserat solid there are six elastic constants, in contrast to the classical elastic solid in which there are two, and the uniconstant material in which there is one.

Cosserat or micropolar elasticity has the following consequences in isotropic materials.

(i) A size-effect is predicted in the torsion of circular cylinders and of square section bars of Cosserat elastic materials. Slender cylinders appear more stiff than expected classically. A similar size effect is also predicted in the bending of plates and of beams. No size effect is predicted in tension.

(ii) The stress concentration factor for a circular hole, is smaller than the classical value, and small holes exhibit less stress concentration than larger ones.

(iii) The wave speed of plane shear waves and dilatational waves in an unbounded Cosserat elastic medium is independent of frequency as in the classical case. The speed of shear waves depends on frequency in a Cosserat solid. A new kind of wave associated with the micro- rotation is predicted to occur in Cosserat solids.

(iv) The range in Poisson's ratio is from -1 to +0.5, the same as in the classical case.

Amir Naeiji's picture
yahoo messenger ID: amirnaeiji



Would you please send me the refrence of this writing especially " (iii) A new kind of wave associated with the micro- rotation is predicted to occur in Cosserat solids."


Rui Huang's picture

Thanks for pointing out Mindlin's work! I thought someone must have done it. I am glad it was Mindlin, my academic grandpa.

Although I have not read all his works, I believe Mindlin's approach is essentially a strain-gradient elasticity theory, one of many ways to introduce a length scale in elasticity. This approach has been much elaborated recently (an example), but unfortunately not my favorite.


Pradeep Sharma's picture


Unfortunately, Mindlin's approach (in particular, the strain gradient theories) can only provide reasonable answers in the class of materials represented by polymers, metallic glass and composites (under certain restrictions). For single element materials (including metals, ceramics etc.), it seems, the strain gradient constants calculated by us (from atomistics) are so small that classical local elasticity remains (surprisingly) valid down to a 1-2 lattice parameters. On the other hand, surface energy effects are fairly appreciable at that size. This conclusion however changes when coupling with electromagnetic fields is invoked

Regarding the stress concentration of a hole, another phenomenological way to look at the problem (perhaps equivalent to a other views stated in this section) is to invoke surface energy effects. The stress concenetration is then (very roughly!): Actual SCF=classical SCF-tauo/(R*app_stress). At some critical size (which in my simple equation is simply a function of size of the hole and surface tension), the effective stress concentration will be cancelled out.

Rui Huang's picture

I am not surprised that the strain-gradient elasticity constants are small for metals and ceramics. The classical elastcity has been working very well to very small scales, unless there is a huge strain gradient in the field. Plasticity of course is a different story.

For the stress concentration around a hole, invoking surface energy effect would lead to a size effect. The result may not be as simple as you estimated. I think either one of us can work this out. 


Pradeep Sharma's picture

I agree that the real result for surface energy effect is not as simple as the one I quoted but should qualitatively reflect the physics down to a few lattice spacings. I am reminded that in my "old" life when I used to study creep damage in materials, void nucleation criterion was often based on this sort of concept (because, again roughly speakinng, thermal fluctuations alone will close a void if the local stresses are below ~tauo/R.

Henry Tan's picture

Dear Pradeep, I very am interested in how you get the equation, SCF=classical SCF-tauo/(R*app_stress)?

Do you have any publication concerning this?

Pradeep Sharma's picture

No, I just wrote it down based on physical intuition. It can be formally derived as well. You may check one of my papers that appeared in APL a while back (2003) or the one in JAM (2004). I had set tau_o (i.e. surface tension) to zero while plotting the results but that term can be retained. Both papes have typographical errors which were subsequently corrected in published "erratta". The papers are on my website.

Henry Tan's picture

Continuum assumption is the conerstone of Fluid mechanics too. And In turbulence, there are too many different spatial and temporal scales. If study fluid from molecular scale, it would be too complicate to do. In turbulence modelling, people tried to establish different model for different scales. I don't know if there is an unified scaling law for all scales.

In solid mechanics, people study crack or fracture, and a microscopic mechanics has been built. However, in fluid mechanics, it is assumed that fluid can endure very large stress, and "fracture" would never happen! This seems strange! But everybody accepts it as a "fact". I don't know if it is a good attitude!  

Zhigang Suo's picture

Solids also flow. They creep at elevated temperatures. They deform plastically at room temperatures. Both modes of deformation share the same essential feature with the flow of a fluid: atoms change neighbors.

Some years ago, Jim Rice and others attempted to classify crystalline solids according to the following model. Consider a pre-existing, atomistically-sharp crack. Under load, the crack concentrates stress; some materials break atomic bonds, but other materials emit dislocations. The bond breakers are called brittle materials, the dislocation emitters are called ductile materials. Thus, at low temperatures, aluminum is ductile, but silicon is brittle.

Of course, a model is just a model. We should never hope to capture the wondrous nature neatly by a single model.

Even for a ductile metal, fracture may still occur. As the crack tip blunts, the strain becomes large, if there is a small void ahead of the crack, the ligament between the void the main crack will neck down, so that the crack advances.

What if the solid is perfect that there are no voids? Then the crack will keep blunting until the whole sample necks down to a point. Not so different from pulling honey (a fluid) from a jar.

One significant difference between a solid and a liquid is that a solid has a large yield stress. Consequently, at a crack tip, a large stress can be built up to activate damage processes, such as open up voids or break hard particles.

You can also build up a large stress in fluid in special geometries. For example, you can fracture a thin layer of liquid between two glass slides.

Another significant difference between a solid and a liquid is that the liquid flows quickly. Under surface energy, a crack-like flaw is simply unstable in the liquid, and will heal. But in a solid, crack-like flaws are all over the place. Atoms just move too slowly to heal them.

My favorite text on comparing fluid and solid mechanics is a 1963 book, The Mechanical Properties of Matter, by Sir Alan Cottrell. He is an extraordinary scientist and writer.

Henry Tan's picture

The behavior of granular material is interesting. It can be either solid-like or fluid-like.

The solid-fluid transition depends on the loading rate and other factors, and is not an intrinsic matertial property (as the cases for ductile or brittle materials).

Granular materials can be treated as continuum, with special constitutive models. But fundamentally, the are assembles of partcles.

Henry Tan's picture

Essentially, fluid is a assembly of flowing moleculars. 

Molecular dynamics can be applied to liquids , on the largest computers.

Molecular calculations can provide extremely useful information concerning the point at which we must abandon the usual image of a fluid as a seamless continuum, and must consider it instead as a collection of molecules.

Henry Tan's picture

These discussions remind me of the Flip Test idea from Zhigang Suo, posted two months ago.

Henry Tan's picture

The discussions may benifit many curious students. 

I moved this blog entry to Education forum.

Henry Tan's picture

for self-teaching (from Dr. Michail Kulesh (

In the framework of the Cosserat continuum theory the displacements of particles in the examined medium are described in terms of two variables - an ordinary displacement field and kinematically independent vector field, which is introduced to characterize small rotations of particles. Thus, in the couple-stress theory there are two independent kinematic unknown quantities, and the stress tensor and the couple-stress tensor are asymmetric. In the context of this theory the elastic behavior of isotropic linear medium is described by six elastic constants: two Lame constants and four new constants describing microstructure. In the case of quadratic-nonlinear medium the number of new constants increases to nine.
The history of microstructure theories goes back to works by W.Voigt, who was the first to introduce a model of the medium with rotational interaction of its particles for studying elastic properties of a crystal. An early effort to develop an elasticity theory with asymmetric stress tensor evidently belongs to E.Cosserat and F.Cosserat. According to the Cosserat brothers' conception, which takes into account rotational interactions of material particles the most effective approach to the problems of stress-strain state in deformable solids is to introduce in the problem formulation the couple-stresses (moment of force per unit of area) in addition to the ordinary stresses (force per unit of area).
There has been a number of works reported in the literature in which the asymmetric theory is extended to the case of thermoelasticity and large deformations. Few works presenting solutions to a number of dynamic problems are also available in the literature. This is, for example, a systematic development of the modern theory by V.I.Erofeev, who considers the problem of propagation and interaction of elastic waves in solids with microstructure. Moreover, the idea of allowing for the internal rotation vector is often used for modeling plastic deformation in materials. However, a detailed discussion of these problems is beyond the scope of this paper, which is restricted to a static state of plane bodies in the framework of the elastic Cosserat continuum theory.
The asymmetric theory of elasticity for the Cosserat continuum (especially for the pseudo-Cosserat continuum) was successfully used by many authors to construct exact analytical solutions. In the majority works the obtained solutions are analyzed and compared with the corresponding solutions of the classical elasticity theory. In this comparison, new physical constants specifying the contribution of the couple-stress components generally assume the values from the energetically admissible range. This can be explained by deficiency of information on the material constants of microstructure media, which is one of the main factors restricting further investigation of asymmetric media models.
In some works a comparison between the solutions of the asymmetric and classical theories is carried out based on the analysis of the stress concentration coefficient and its dependence on the characteristic dimension of the stress concentrator. The analysis clearly demonstrated that compared to the classical theory the coefficient of the stress concentration increases (or decreases) with characteristic dimension of the concentrator. Although this fact is of obvious interest, the use of the concentration coefficient as a measurable parameter seems to be rather problematic. Thus, for example, an attempt to measure variation of the concentration index by the photoelasticity method has failed, since the resolving power of this method is too low to apply strictly to the desired characteristic dimension of the concentrator.

Henry Tan's picture

A link to discussions on Micropolar material constants

Dear sir/madam

I want to model the fiber reinforced composite’s interface by using “COHESIVE ELEMENT” in ABAQUS.

Please help me by sending tutorial.

Many thanks.




           I  do not see any Problem in this
unless we are going to a scale in the order of intermolecular distance. In such
case the continuum assumption may not be valid and it looks as a assembly of
atoms connected through bond.

In case of a plate with a hole, the stress
concentration is 2, provided the  area which experience the higher level
of stress is in the same order of the hole radius.  In case of a hole
which is approaching a dimension infinitely small, the area which experience
the higher level of stress also comes down to a similar dimension.  And in
effect when we are looking in a macro sense, it looks as there is no stress
concentration, other than an infinitely small region.

Asst. Professor
Mechanical Engineering Department
National Institute of Technology

Mike Ciavarella's picture

I take this from Suo's notes.  I am surprised Zhigang doesn't raise this point, but he has a good few lines in his notes.

The confrontation between Timoshenko and Swain. When I was a graduate student, a large portrait of George Swain could be found on the third floor of Pierce Hall. He was a professor of Engineering at Harvard from 1881 to 1927. I was told of a confrontation between him and Timoshenko. Searching on the Internet this morning, I found quite a few entries about the confrontation, some quite inflammatory, but I did not find a clear description of the technical issue that leads to the confrontation. (Google yourself for Swain Timoshenko.) In any event, here is what I heard. The issue concerned the stress concentration factor. It was said that Swain did not believe that the stress concentration factor of a circular hole is independent of the radius of the hole. He would say, a very small hole should have negligible effect on strength.

It is unimportant for us who said what. Let’s simply focus on the effect of the size of the hole on the strength of a body. Indeed, the stress concentration factor of 3 is a result of linear elasticity. The result is correct so long as the assumption of linear elasticity is correct. In particular, the body has to be linearly elastic. This assumption can be violated, for example, by metals undergoing plastic deformation. Even for a brittle solid such as a silica glass, when the hole approaches atomic dimension, linear elastic assumption breaks down. It is entirely possible that a small enough hole will not reduce the breaking stress of a body. We will return to this question later in the course. But already you can tell that you should not trust linear elasticity.

Zhigang Suo's picture

Dear Mike:  I did make this point early in this thread of discussion.

Mike Ciavarella's picture

Anyway Zhigang, I have been reading your " Remarks on crack-bridging concepts,"

You start off with saying that a brittle ceramic would indeed be "notch sensitive" by reducing its strength by a factor 3 even for a zero-radius hole (this is perhaps confusing at this stage).  Whereas a ductile material is "notch insensitive"because plasticity makes sure the  hole has only the effect of reducing the actual real width of the specimen.  You then move on to introduce cohesive laws concepts to make estimates of intermediate behaviour.  It turns out that you introduce a length scale of the order of the bridging zone size lets call it here a0 = d0 E/s0.  You then move in par.3.3 to look at notch sensitivity to find that the crack-like notch should bound the behaviour of the notch, and indeed the notch becomes brittle onlywhen a/a0>10. 

Perhaps a much simpler approach would be to consider LEFM from the beginning, avoiding to start with the more complicated concepts of cohesive laws. Hence, considering G=KIc^2/E, you find easily that the a0 above is also (except for a prefactor of the order 1) is nothing but the crossover between nominal strength s0 and K=KIc condition.  This is usually in the nanoscale and means that details about the geometry are irrelevant at this size and even the presence o the defect itself --- what Huajian Gao calls emphatically "flaw tolerance at the nano-scale".  In your Fig.14 I see both hole and crack-like start together, but they don´t start horizontal.. very likely because you have the additional effect of cohesive law building up crack resistance so that the KIc property effectively starts from zero.

Then, another limit condition could be easily put by equating the highest stress in the hole to the nominal strength.  This crosses the K=KIc line at obviously a1 = Kt^2 a0, which for a hole is 9 a0, quite close to what you say as a factor 10  to reach again the brittle behaviour.  Therefore I would draw a line in your Fig14 at 0.333 and this would cross the "crack-like" behaviour on x-axes at about 3 (consider the prefactor here makes the number different).  Some details do not convince me however since I would expect the hole curve to approach asymptotically this 0.333 asymptote, especially as you say the behaviour is brittle at a/a0>10 and indeed it seems the curve is by then already horizontal. So why this departure from the 0.333 limit?

Anyway, is this a much easier explanations for students, and for the original "puzzle"? 

Of course Swain had conceptual difficulties in the 1920´s as he did not have Fracture Mechanics.

Zhigang Suo's picture

Here are a few thoughts as I read your comments.

  1. Both the bridging model and the singular field have values.  They should both be taught to the student.  Which one is simpler depends on what the student knows.  For example, if the student has no background in singular stress field, but has some intuition about stress field, then it is easier to form a picture of the situation with the bridging model.  The love of singularity is an acquired taste.  The singular field is an idealized model that is wrong in a very essential detail:  stress is singular at the tip of the crack.  Of course, as they say, all models are wrong, but some are useful.  The student may learn to forgive this obvious error, and learn the use of the singular field.
  2. Fig. 14 of the paper does look puzzeling.  In a subsequent paper, the same problem was analyzed with more details, and was presented in Fig. 8.  This figure looks more reasonable.      
Mike Ciavarella's picture

Zhighang, I am not saying crack bridging is not useful, but you have to admit that a simple equation a1 = Kt^2 a0, where a0 is what we described earlier, to describe the transition from ductile to brittle notch, is much more immediate to write than a FEM model with an artificial crack on which a cohesive zone is implemented, which means actually a non-linear solution!   And actually it explains quite nicely how, if you had another geometry, you should expect this transition.  Notice a1 obviously goes to infinity for a crack, which means that in your notation a crack is always "ductile" ...   The problem in notation emerges since we have really two transitions here, a/a0=1 ductile to brittle, and a/a1=1 which should be brittle to ductile again, but this time not ductile in the whole section of the specimen, only at the notch edge....

In your second paper, fig.8, the asymptote continues to be a little higher than 0.333, not sure why.

Actually in a sense the problem of the singularity comes from LEFM and could be avoided!  In other words, people looked at the puzzle as described by Tan here, already quite before the advent of stress singularities (ie the 1950s with Williams and Irwin).  This was more to do with too high stress concentration, than with too small holes, but the question is the same.  I am referring here to names as Neuber, and Peterson, and others, who spoke of "process zones" mostly in fatigue, and notch sensitivity there was found there with equations depending on notch radius (a characteristic size of the notch).

David Taylor has even re-introduced these concepts (or shall we say, systematically reviewed?) in what he calls quite emphatically Theory of Critical Distances, which includes the quite trivial method to estimate the stress field using simple elastic field, at a distance L=a0/2.

Specifically for composites, the point method was suggested in 1974 by Whitney & Nuismer [1], in a quasi-isotropic glass/epoxy laminate. A lot of experimental data had been generated to demonstrate the truth of Whitney and Nuismer’s proposal. Awerbuch and Madhukar [2] have an enormous study covering over 2800 test results, Wetherhold and Mahmoud [3] also mostly polymer-matrix long-fibre materials but also including some metal-matrix composites and some materials with discontinuous fibres. Interestingly, despite the wide range of strengths and toughnesses in these materials, it was found that L fell within a narrow range of values, being almost always between 1 and 5mm, sometimes as high as 15mm.

All this is quite simple and practical engineering, I guess, and David Taylor has made even recently quite a big deal of it, especially if you read his book [4] which claims to be "A New Perspective in Fracture Mechanics"! Obviously your approach tries to look at more details, like MD people look at even more...

This approach in general requires no calculation at all, except for the corrective prefactors in the elastic solutions. Taylor here usually recurs to FEM, spoiling the beauty of the approach.  In fact, in a related transition for notch fatigue, I have rather used the prefactors for a crack replacing the notch, and these usually do not require FEM.  

You can see the diagram I have in mind for the fatigue-related counterpart of this in this paper. On fatigue limit in the presence of notches: classical vs. recent unified formulations


[1] J.M. Whitney, Nuismer, R. J. , "Stress fracture criteria for laminated composites containing stress concentrations," Journal of Composite Materials, Vol. 8, 1974, pp. 253-265.
[2] J. Awerbuch, M.S. Madhukar, Journal of Reinforced Plastics and Composites, 4 (1985) 3.
[3] R.C. Wetherhold, M.A. Mahmoud, Materials Science and Engineering, 79 (1986) pp. 55.
[4] D. Taylor, The Theory of Critical Distances: A New Perspective in Fracture Mechanics , Elsevier, Oxford, UK, (2007).
see also this or, specifically for composites, this.


Mike Ciavarella's picture

Zhigang, that Swain made a quite naive error is instructive for many reasons.  First, of the conceptual difficulties in size effects, as witnessed by this very discussion on Imechanica in 2011. Second, by the style some authoritative professors adopted in those days, even at Harvard. And third, by the fact that some controversy is often useful to make interesting progress.

You can find at pag.51 of the quite extensive review paper by my friend Kiev professor (but not from the Institute where Timoshenko was Dean!) Meleshko.

Here is a few lines:

 It all started in a talk Timoshenko and Dietz delivered at the Spring Meeting of the American Society of Mechanical Engineers, Milwaukee, May 16–21, 1925. Both this talk and the analytical Kirsch solution met a severe reaction from Swain who was at that time a Professor of Civil Engineering at Harvard University and one of the leading figures in bridge design. He had just published a textbook in which on pp 121–123 he pointed out that if the result of threefold increase of the stress on the edge of the hole is independent of the size then it will ~assuming the material to be perfectly homogeneous and elastic- be the same if the diameter of the hole be diminished to an infinitesimal size. Based upon ‘‘common sense,’’ the author took the illigitimate step of equating this infinitesimal to zero, thus abolishing the hole, with the ‘‘result’’, which he advanced as a proof of some error in the Fo¨ppl -Kirsch solution ~he reproduced, however, the main formulas ~ for stress and concluded that ‘‘it is unnecessary to give their derivation.’’ Further the author provided additional arguments based upon elementary strength of materials reasons to support his conclusion, and he noted on p 122: Perhaps in this may be found the fallacy in the theoretical demonstration, but the writer has not gone through with it. He has no time for such illusory mathematical recreations.

Selected topics in the history of the two-dimensional biharmonic problem
Appl. Mech. Rev.  -- January 2003 --  Volume 56,  Issue 1, 33 (53 pages)

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