an interesting puzzle: multiscale mechanics
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.
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teaching multiscale modeling
This puzzle can be used for teaching multiscale modeling.
very interesting
Henry,
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.
Luming
length scale
therefore, you see, length scale comes very naturally into the size-independent continuum result at the nanoscale.
Stress concentration around a hole and length scale
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).
stress around a small hole
Zhigang:
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).
RH
Puzzle
Hi Guys,
I consider your puzzle in my blog "Griffith controversy" where you can find a paper devoted to it.
Kosta
softening hyperelasticity constitutive model
Indeed, the softening hyperelasticity model you proposed solved the puzzle quite nicely.
Griffith approach and stress concentration factors
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.
RH
Hi Rui
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.
-Kosta
exactly
"the radius goes to zero" means that the cavity disappears...in the continuum sense of course.
RH
No
No it is not a smooth process. The cavity exists or not! Only two possibilities.
why not smooth?
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.
RH
Well, you are getting to the
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.
from continuum to quantum
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.
RH
The continuum concept is an assumption
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.
how far can continuum go?
Yes. The question is: how far can we go with the continuum assumption? The answer would be problem-dependent, rather than a definite scale.
RH
Cosserat theory
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.
JUAN GOMEZ
PhD Computational Mechanics
Professor
Applied Mechanics Group
EAFIT University
Medellin,Colombia
An introduction to the Cosserat theory of elasticity
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 (http://imechanica.org/node/309), and by Nowacki. Eringen incorporated micro-inertia and renamed Cosserat elasticity micropolar elasticity.
consequences from Cosserat or micropolar elasticity
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.
Thanks, Mindlin!
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.
RH
further on size effects
Rui,
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.
on strain-gradient elasticity
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.
RH
I agree that the real result
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.
Pradeep, how you get the equation?
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?
No, I just wrote it down
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.
Mindlin Centennial
Mindlin Centennial
http://imechanica.org/node/309
Continuum assumption is the
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!
To flow or to fracture? That is the question
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.
behavior of granular material: solid-fluid transition
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.
Essentially, fluid is a assembly of flowing moleculars
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.
Flip Test of continuum assumption
These discussions remind me of the Flip Test idea from Zhigang Suo, posted two months ago.
move to Education forum
The discussions may benifit many curious students.
I moved this blog entry to Education forum.
microstructure theory of elasticity
for self-teaching (from Dr. Michail Kulesh (http://users.math.uni-potsdam.de/~mkulesh/rus/projectnons.html)):
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.
Micropolar material constants
A link to discussions on Micropolar material constants
http://imechanica.org/node/1298