iMechanica - an interesting puzzle: multiscale mechanics - Comments

Mindlin Centennial

Henry Tan — Sun, 29 Apr 2007 22:07:46 -0400

Mindlin Centennial
http://imechanica.org/node/309

Micropolar material constants

Henry Tan — Sun, 29 Apr 2007 20:18:36 -0400

A link to discussions on Micropolar material constants
http://imechanica.org/node/1298

microstructure theory of elasticity

Henry Tan — Fri, 23 Mar 2007 19:05:11 -0400

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.

An introduction to the Cosserat theory of elasticity

Henry Tan — Fri, 23 Mar 2007 16:27:34 -0400

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

Henry Tan — Fri, 23 Mar 2007 15:56:44 -0400

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.

No, I just wrote it down

Pradeep Sharma — Thu, 15 Mar 2007 17:11:20 -0400

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.

Pradeep, how you get the equation?

Henry Tan — Thu, 15 Mar 2007 16:41:43 -0400

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?

I agree that the real result

Pradeep Sharma — Thu, 15 Mar 2007 13:39:25 -0400

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.

on strain-gradient elasticity

Rui Huang — Thu, 15 Mar 2007 12:33:01 -0400

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.

stress around a small hole

Rui Huang — Thu, 15 Mar 2007 12:19:39 -0400

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).

move to Education forum

Henry Tan — Thu, 15 Mar 2007 09:17:47 -0400

The discussions may benifit many curious students.

I moved this blog entry to Education forum.

further on size effects

Pradeep Sharma — Thu, 15 Mar 2007 08:52:39 -0400

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.

Stress concentration around a hole and length scale

Zhigang Suo — Thu, 15 Mar 2007 08:06:55 -0400

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).

behavior of granular material: solid-fluid transition

Henry Tan — Thu, 15 Mar 2007 02:17:39 -0400

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.

To flow or to fracture? That is the question

Zhigang Suo — Thu, 15 Mar 2007 00:44:06 -0400

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.

an interesting puzzle: multiscale mechanics

Henry Tan — Fri, 02 Mar 2007 16:06:51 -0500

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.