It seems that this discussion starts from a fundamental confusion between the CRSS which is a [material/slip system type/temperature] property, and the resolved shear stress, which is nothing but the projection of the applied stress onto the different possible slip systems (therefore depends on crystal orientation relative to straining).

Slip occurs on a given system when the resolved shear stress on THIS system reaches the CRSS of this particular system.

The CRSS is a combination of the Peierls-Nabarro stress plus all the possible hardening mechanisms that are present (work-hardening, solid solution, precipitates). It is independent of the applied stress or crystal orientation.

In some materials, the CRSS might be different for the two opposite slip directions on a given plane, an effect called twinning/anti-twinning direction, linked to the core structure of dislocations.

Hope this helps.

For a given applied normal stress along a given crystallographic direction, every possible slip system has its own CRSS that can be calculated according to Schmid's law.

AFAIK, compression does not initiate slip, tension does.

The stress to initiate permanent deformation of single crystal is dependent on the CRSS of the most favorably oriented slip system, that can be determined from schmid's relation. As the deformation is further applied the initial slip system rotates out and other slip systems with larger CRSS are activated. Thus more than one slip system is activated as deformation proceeds from yield to max stress. CRSS is not dependent on direction loading. Hence it is same in the tensile and compressive directions. However the resistance to dislocation differs in the forward and reverse directions and hence the stress to initiate deformation in tensile and compressive directions maybe different (eg. Bauschinger effect)

The "Crticial Resolved Shear Stress" is a material property dependent on temperature. This property, the CRSS, is there for every possible slip system, and is independent of the applied stress. The CRSS gives the minimum value of resolved shear stress on the slip system, which will activate slip on that system. As the stress on the sample is increased, the slip system on which the resolved shear stress crosses the CRSS first will undergo slip before others. This system is called the primary slip system.

As explained by others above, the crystals tend to rotate under applied stress, so as to maintain geometrical continuity of the sample. This rotation occurs so that the slip direction moves towards the tensile direction (in tension) and the slip plane normal moves towards the compressive axis (in compression). This rotation causes the resolved shear stress on secondary slip systems to increase, and when its value increases beyond the CRSS of any of the scondary slip systems, that system gets activated. So one might have multiple slip occuring simulatenously or in tandem, depending on the CRSS of the different slip systems and the orientation of the crystal.

Not sure I agree with absolutely everything written in the answers given so far, but these are complex topics, that don't necessarily lend themselves to the format of short, punchy answers, especially because most answers in fact have to come with additional statements as to when they are valid, to be strictly correct. (otherwise one gets another expert pointing out, 'ah well that does not apply when X occurs'). So mine too will no doubt raise objections, if one moves away from the general case to specific cases (the old adage holds, 'all generalisations are untrue', paraphrasing Mark Twain).

Of course compression (not just tension) can cause slip as stated by Gupta.

Schmid's law only really applies when one considers single crystals, in the polycrystalline case then Taylor's law applies, and in general a minimum of 5 different slip systems need to operate independently for plastic deformation to occur. (This is one of those answers/statements that would require qualification, as noted above, but this would make the answer even longer. I will not bring in Sach's theory, and upper and lower bounding of the deformation process between Taylor's and Sach's theories, but they are somewhat relevant.)

So although you can work out on which slip system the highest CRSS is to be found in a given grain, it cannot operate in splendid isolation in a polycrystalline body, i.e. it is constrained by the surrounding grains. However, it's possible to work out the 5 systems which will operate (although quite often more than one set of 5 systems is possible, the so called Taylor ambiguity), using the Taylor theory or the equivalent Bishop-Hill criterion.

The procedure is very nicely explained in 'Deformation Geometry for Material Scientists' by C.N. Reid, pub. Pergamon Press, 1973.  

A lot of the early work done on the deformation of single crystals etc. is to be found in summary form in the book, 'The Plastic Deformation of Metals' by RWK Honeycombe, pub. Edward Arnold, 1968. This should be consulted as a primer on the topic.

If you look on Professor Tony Rollet's webpages you will also find some original articles by Schmid and Boas on this topic.

On texture and deformation of single crystals, (Schmid's law, and lattice rotation), Professor F.J. Humphreys (formerly at the Uni of Manchester, UK)  used to have some nice DOS programs that illustrated how in this simple case,  (defm. of single crystal), slip would commence on the slip system with the highest CRSS, but then the rotation accompanying such slip would mean that another slip system takes over the mantle of having the highest CRSS, and thus slip switches to that one. The program showed how such switching of slip systems leads to certain texture 'endpoints' in the unit (stereographic) triangle, and how they differed for tension and compression. On this one should also consult chapter 5, 'Geometrical Models for Polycrystal Deformation and Texture Prediction', in 'Large Strain Work Hardening and Textures', by J.G. Sevillano, P. Van Houtte, and E. Aernoudt, Progress in Materials Science, 1980, vol. 25. That has relevance for the polycrystalline case also.

Now down to answering the questions originally posed, let's separate FCC and BCC cases, (read Honeycombe's book if you want to find out about the HCP/CPH case). In the BCC crystal there are no close packed planes, only close packed directions, i.e. , so various investigators have observed slip on planes of the following types, {110}, {112} and {123}, with {110} being preferred at low and moderate temperatures (these planes all contain the  close packed direction). Thus slip generally takes place on the planes of closest packing. You will have to read up/round yourself for the subtelties involved, i.e. when these alternatives operate and why.

FCC has both close packed planes [111} and close packed directions , these are colloquially called good slip systems CF those in BCC, and under  most circumstances, whatever the orientation of the crystal, slip will take place on them, i.e. on the one most favourably oriented (reminder single crystal case). However, you will find in the literature cases which exceptions to this, and I attach a paper which attempts to model circumstances under which this might occur. You will have to read round to see if you can find a case of this for single crystal deformation.

So the answer would appear to be yes, there are circumstances when these so called 'primary' slip systems have a higher CRSS than the alternatives, and yes they, (the alternatives), will have their own CRSS.  The actual deformation will follow the path of least resistance, whatever that is.

Is CRSS different under tensile or compressive loading ?  Gupta mentions the Bauschinger effect, but that only applies to cyclical patterns of loading (again there will be exceptions I imagine), but generally on 'virgin' material, (let's say well annealed), the slip initiation stress, (CRSS), on the most favourably oriented system will be the same in tension and compression, what happens on continued deformation and stress reversal is much more complicated. It's obvious because the shear stress which drives dislocation glide will be the same on the chosen plane for the same absolute level of applied tensile/compressive stress.

See Prof. Harry Bhadeshia's/Cambridge Uni pages for some nice (shortcut) rules/laws to work out operative systems.

If this is a topic that you get into, (deformation and texture), it will quickly become apparent to you that local constraints, e.g. grain orientations, and external constraints, e.g. exactly how any deformation is applied, play a big part in exactly how deformation and thus texture development proceeds. In fact it gets a lot more complicated than the commonly depicted situations used for the benefit of students, and the most advanced models still struggle to incorporate the detail of the different factors invloved in texture development, but it's work in progress.

I'm sure its possible to pick many holes in what I stated, but it's a reasonable basis to start from, before getting overloaded with the ifs, buts exceptions.

Regards,

John

Dear Adyita,

in case of a material reinforced in some selected directions, you should choose an anisotropic  failure criterion,  shows different strenght to slip in different directions. Obviously the criterion should be expressed by a continuous function, as I made in the nice work about reinforced concrete columns confined by FRP, that I attach to this response.

The concrete core was characterized by an Intrinsic Curve or Schleicher-like

elastic-plastic behavior endowed with associate flow rule, and the FRP by an elastic-brittle TsaiHill anisotropic yield criterion.

Please, let me know if this paper is useful to your purposes. If you need more suggestions, please do not hesitate tu contact me sending by e-mail some more information about your problem (draft or writings..) to [email protected] .

Regards,

Luciano Nunziante.

Ouch, now Luciano has taken you into deep waters, far away from 'easy' metals.

This might help you if you are interested in anisotropic flow modelling

As a primer on Luciano's slant on the topic you could try 'Mechanical Metallurgy' by GE. Dieter, 'Engineering Plasticity' by W. Johnson and P.B. Mellor starting on the isotropic plastic flow case, then going on to the anisotropic case that you need to cope with such materials.

If you are going to apply continuum mechanics to this situation, then the available reading is a bit sparse when compared with that for the isotropic case. There is a textbook by S.C. Cowin, 'Continuum Mechanics of Anisotropic Materials', devoted to this topic, no idea how helpful it is.

As a primer for the maths etc. involved in these continuum mechanics approaches, Kelly has some online articles here:- http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/

as does Prof Allan F. Bower at Brown Uni here

http://solidmechanics.org/

This used to be in his online notes for the Brown Uni courses EN175, and EN222, but those courses are now given by a new instructor at Brown.

Here's some more relevant material on this general area.

http://imechanica.org/node/1551

Dr Scott Hollister also has some useful online materials here:-

http://www.umich.edu/~bme456/

On the algorithms for the more standard cases, the go to reference was often, 'Computational Inelasticity' by J.C. Simo and T.J.R. Hughes, vol. 7 Interdiciplinary Applied Mathematics, Eds. J.E. Marsden, L. Sirovich and S. Wiggins.

It's quite a heavy gooing road to get up to speed, but it all depends upon your background/starting point. Luciano's field is more at the Everest side of things, rather than the foothills where this discussion started.

It wouldn't surprise me at all if in the composite/anisotropic  work is quite difficult to obtain highly accurate and verifiable models for different loading paths/situations, that's still the case for isotropic cases, so I can not imagine just how fierce the debates are raging on the most appropriate approach for anistropic bodies. Huge safety factors on application no doubt, and many adjustable fitting parameters in such models ?

Indeed, compression can cause slip, otherwise there would be no plastic deformation during compression unless there was twinning! Clearly, that is not the case, and we see slip bands form on compression samples.

Moreover, the CRSS can indeed be different in tension and compression, e.g. in alpha/beta Ti alloys, see Neeraj et al, Phil Mag. 2005 (V 81, No. 2-3, pp. - 279-295).

Aditya,

Oh dear, first composites, now HCP/CPH systems and twinning, well you might as well be aware of some of the possible exceptions, I suppose, just in case your material is Ti, Zn, Zr based, superalloy, directional material etc., or some composite.

As I warned you above, when you move away from the general case into specific examples, then there will be exceptions (one can be left wondering when do the general principles apply ?), here it's worth mentioning that one class of metals where this is more common are the CPH/HCP metals (which I purposely avoided, along with composites/anisotropic/directional cases etc. etc.), these have an inherently less isotropic nature than BCC and FCC based alloys, and this is reflected in their slip systems. I've not seen the full paper that Dhriti cites for assymetry of the CRSS, but French workers, J.S. Lecomte et al, and Fundenberger et al cite Pyramidial slip and Lecomte interaction between the dislocations and oxygen atoms.

"The asymmetry of the yield stress in the transverse direction between tension and compression is associated with the movement of < c+a> dislocations and is due to the effect ofthe non-centrosymmetry of the {1101} glide."  (how's your French ? Lecomte paper attached)

The abstract of the paper that Dhriti cited states the following

"Titanium alloys are known to creep at room temperature at stresses below the macroscopic yield stress. In this study a two-phase α/β alloy, Ti-6Al-2Sn-4Zr-2Mo (Ti-6242), and a single phase binary alloy, Ti-6Al, were observed to exhibit a dramatic asymmetry between tension and compression during room temperature creep. It was observed that the aero-engine alloy, Ti-6242, accumulated about five to six times higher creep strains in tension than in compression. A much smaller asymmetry was observed for the 0.2% yield strength such that it is higher in compression. It is shown that the present observations cannot be explained by the activation of predominantly slip. Further, CRSS measurements from single colony crystals of Ti-6242Si oriented for basal -type slip showed that there is a significant asymmetry between tension and compression, with the CRSS observed to be higher in compression. Studies on the single-phase α alloy Ti-6Al with low oxygen content suggest that the interstitial content could also play a significant role in determining the behaviour of these alloys. The single phase alloy also exhibited tension-compression asymmetry in creep, although it was observed to be less pronounced compared to the two phase alloy. HRTEM results suggest that the core structure of -type dislocations is altered by the addition of alloying elements, especially oxygen, and that the tension-compression asymmetry is primarily due to the behaviour of -type dislocations."

In passing Dhriti mentioned twinning, now in the context of your question if we are going to complicate things for you, twinning can cause asymmetric plastic deformation behaviour, that is both when acting 'alone' as the primary deformation mechanism, and it has been postulated that it also causes CRSS asymmetry when it acts when the stress is applied in one direction and slip when the stress is applied in the opposite direction (e.g. creep of superalloys).

[However, the whole concept of a CRSS for twinning is still under debate as discussed by Bo Qin in his master's thesis http://www.msm.cam.ac.uk/phase-trans/2007/QinBo_Thesis.pdf, chapter 3, section 3.3.

For a good primer on deformation twinning, read  'Deformation Twinning' by J.W. Christian, and S. Mahajan, Prog. in Mat. Sci. vol. 39 pp 1-157, 1995. and for CRSS see section 5.1 Orientation Dependence: Is there a CRSS for Twinning, pages 78-84.]

So I end this again with the quote attributed (correctly ?) to Mark Twain (and sometimes Einstein),  " All generalisations are false (including  this one)".

So beware the exceptions, simplifications and approximations, they all tend to break down at some point, and the questions become how severe is the departure, and is a new theory/explanation required, can we verify any of the competing possibilities ? If you really pick at the studies in a critical manner, the latter is often where they are lacking.

As Dhriti mentioned, the slip can be activated by compression, this can be demonstrated in the compression of micro/nano pillars.

Is that really being debated/in question here ? It should not be, it's fundamental.

The majority of commercial operations to change the shape of metals involves compression of some form, e.g. hot and cold rolling. Leaving HCP aside and the relatively few/minor cases of deformation twinning in the BCC, (say in high silicon steels, or explosive/high rate deformation), and FCC, (say in the TWIP steels), metals, that deformation is by largely by slip/dislocation movement.  (and largely is putting it mildly, overwhelmingly might be a better term to use).

(Let's forestall another somewhat tangential input on exceptions to this, e.g. bringing in the possibility of deformation, typically at high temps.  by grain boundary sliding as in some cases of creep and superplastic flow, ala Wadsworth, Ruano and Sherby, the Stanford/Jones debate.)

Taking a simple, very highly idealised case, (pure metal, not HCP, so also no composites, no alloying/tramp elements segregating to the dislocation cores to minimise their elastic strain contribution to the free energy of the body), dislocation glide is driven by shear stress acting on the slip plane in the slip direction. Imagine you have a single crystal of a cube shape, consider orthogonal x,y,z reference system. Assume a primary slip plane is orientated exactly at 45 degrees to the z direction (and assume it has a convenient (say Edge) dislocation already existing on that plane (see D. Hull,  'Introduction to dislocations', or the Honeycombe book cited above, and if you are really keen the references that Honeycombe cites  at the end of chapter 3 i his book, Dieter too has a good section on dislocations) . Apply a principal tensile stress, to the upper and lower faces of the cube. On that 45 degree plane (assume it sort of bisects the cube) you can compute the shear stress acting. In this case it's very easy, it's half the principal tensile stress, and it's the maximum shear that body experiences under these conditions. Now draw arrows to indicate the direction of that shear stress acting on our 45 degree plane. They are in opposite directions above and below our ideal shear/slip plane. Now reverse the principal tensile stress on the outer faces of the cube, and replace it with an equal and opposite principal compressive stress. On the same 45 degree plane, the maximum shear stress has the same absolute value, but now your arrows are reversed and the dislocation moves accordingly in the opposite direction.

In the basic idealised picture the dislocation 'sees' and reacts to the imposed shear stress, it's a secondary effect that, in case one the lattice bonds are being elastically stretched in the Z direction, whilst in case two they are being compressed. Typically dislocation movement starts way below the theoretical lattice strength, as almost attained in dislocation free metal whiskers.

I should have also said let's confine all the above to the initiation stage of dislocation glide, so the load application constraint efcets, and lattice rotation don't feature enough to significantly effect the process described.  And of course there's the complication that this is in some ways the isotropic continuum picture, vs the orders of magnitude more complex discrete multiple event driven nature of such deformation in practice, and most materials are not single crystals, and are rarely if ever truly pure (even in zone refined metals).

So nothing in reality is as simple as this*, but this is the basis from which the deviations, more complicated cases and exceptions etc. etc. start.

*This is a massive understatement as anyone who has actually observed dislocations in materials will attest.

It seems that this discussion starts from a fundamental confusion between the CRSS which is a [material/slip system type/temperature] property, and the resolved shear stress, which is nothing but the projection of the applied stress onto the different possible slip systems (therefore depends on crystal orientation relative to straining).

Slip occurs on a given system when the resolved shear stress on THIS system reaches the CRSS of this particular system.

The CRSS is a combination of the Peierls-Nabarro stress plus all the possible hardening mechanisms that are present (work-hardening, solid solution, precipitates). It is independent of the applied stress or crystal orientation.

In some materials, the CRSS might be different for the two opposite slip directions on a given plane, an effect called twinning/anti-twinning direction, linked to the core structure of dislocations.

Hope this helps.

No confusion here Claire about the distinction between CRSS and how to resolve stresses for planes at an angle to the principal stress. Schmid's law is based on that 'yield' effect for the onset plastic flow (CRSS), the law/postulate being confirmed for the case of magnesium crystals by Burke and Hibbard, vol. 194, Trans AIME, 1952, p. 295. (but how much of a 'law' is it really ?)

The model of Peierls and Nabarro, for the stress to move a dislocation, also explains why the preferred slips system are on close packed planes (or the planes of closest packing) and proceeds in the close packed direction.

However, if you need a  modern overview on dislocations, modelling and simulations at the mesoscale (because this is the intended area of research), then obtain the recent book, 'Dislocations, Mesoscale Simulations and Plastic Flow', by Ladislas Kubin, pub. by Oxford Uni. Press 2013. Have a look at Chapter 3. 'Lattice-controlled Plastic Flow, and if tension/compression asymmetry, particularly in BCC structures is relevant to your research, then section 3.2.3 should interest you, 'Non-Schmid effect and Peierls stresses'. One paragrah starts, 'The crictical resolved shear stresses (CRSSs) cannot be rationalized in terms of the Schmid law, (Christian, 1982, Duesbury, 1989, Duesbury and Vitek, 1998).

The isotropic continuum  picture thus comes into conflict with the true local, discrete, anisotropic nature of plastic flow, where its specific details, i.e. at the dislocation level really matter.

Attached is paper by Kim et al. where such asymmetry is observed in Mo (BCC) nanopillars,  and explanations for the behaviour are proposed.  Such work interfaces with that of the groups using atomisitic models to describe the behaviour. Also attached is a pdf of presentation on such dislocation modelling approaches by Richard LeSar of Iowa State University.

Perhaps LeSar's ppt does not cover the really fine scale atomistic modelling except briefly in passing, so I attach a few more papers, their reference lists in turn could be helpful to those with a developing interest in this area. I include one paper which actually covers the effect of the other components of the stress tensor (Vitek et al.) on dislocation movement, so you can judge the failure of the generalisation I made that it is the level of resolved shear stress which is paramount to dislocation movement, not whether the principal stresses are tensile or compressive, generating the shear stresses. Plus now you know that the sense of the stresses can be important. 

The closer you look at  most anything, the more those specifics affect the general picture. But get the general picture clear first, then look at the multitude of deviations, the material system, regimes in which they occur, their level etc. etc.. Then there's the practical tests, e.g how does defm features observed at nanopillar levels affect the performance of a hip implant ? Of course these fundamental studies may or may not be helpful, and scientific study isn't necessarily driven by practical application. (Oops getting way off topic here, so I'll stop).

BTW Dierk Raabe @ the MPI centre for Iron and Steel  Research has written a useful  overview on different modelling approaches (book), 'Computational Materials Science: The Simulation of Materials Microstructures and Properties', pub. Wiley-VCH, 1998.

Thank you all for the answers and being a part of this discussion. But I have one more doubt regarding the same issue in hand.

The problem was that, in HCP metals we observe 3 different slips (Basal, Prismatic and Pyramidal) so I was curious whether these three will have their own CRSS. To which I got a YES, every system has its own CRSS.

It is a resolved shear stress, so that means depending on the type of loading the orientation of unfavoured planes and slip of the favoured ones will take place. That also helped a lot.

But there is this last doubt I have. If we have 3 slip systems that means we have 3 different CRSS and one of it is a material property. But in most single crystal studies the researchers, out of curiosity, tend to avoid the dominant slip system by placing the crystal in a particular way that this dominant slip system is avoided, and still they get deformation and the reason is because of the other non dominant sip system getting activated.So my last question is how does CRSS become a material property ?

Hi Aditya,

The CRSS (Critical resolved shear stress) for each slip system is a material property, i.e., the CRSS for each slip system has a fixed value at a fixed temperature for a given material, and does not depend on the loading conditions in general. Not just one, but all the CRSS values for all slip systems are material properties, i.e., they are characteristics of the materials. e.g., The CRSS for the {10-11} pyramidal slip system will be different for Ti, Zr, Mg, etc. at the same temperature. Of course, as discussed by others above, there can be variations to this rule, but they are exceptions. 

 I asked the similar nature question. Can anybody wanna give some comments.

How to find the critical resolved shear stress (CRSS) at specfic slip plane using the hertzian contact equations? 

I think the CRSS is related to macroscopic yield strength. Did anyone discussed this? Thanks