The concept of plane stress and plane strain are quite fundamental in nature. All stresses and strains are three dimensional in nature (mathematically expressed as a stress or strain tensor). However, there are many special (and practical) cases where some of the many components of this tensor are zero leading to 2-D version of stresses and strains. Some of these conditions are plane-stress, plane strain, and axisymmetric. Lets consider the plane stress situation. This means that all the non-zero components of stresses are confined within the plane and all the stress components on the surface that is accessible (i.e the planes that can be touched) are zero. So if the x and y axes point in the traditional x and y directions within the plane and z points perpendicular to the plane then sigmazz is zero and sigma zy (or tau zy) and sigma zx( or tau zx) are zero. This is because no load is acting on the z plane (it is open to the atmosphere). This approximation is usually true for thin surfaces. So, plane stress condition works well for thin plates.

Consider the opposite situation, a thick plate. A structure that is like a dam that is locked between two cliffs: a very think plate. In this case the plate has no place to move in the z direction but sure enough there will be a lot of stress buildup. So the strain in the z direction and the shear strains in the z plane will be zero but the stresses will not be zero. I hope this is helpful.

At the root of any answer to this question has to be, because it makes the maths easier! I don't mean that in a trivialising way, so I'll explain: as Shuvra points out, all real-world stress and strain states are three-dimensional. In many cases, however, we can make assumptions that greatly simplify - or make tractable - the analysis, without greatly affecting the result.

In the case of a thin film loaded biaxially, we can assume that there will be no normal or shear stresses in the direction perpendicular to the film surface - that is, that the film is in plane stress. This must be the case at the surface itself, because no such stress can be sustained at a free surface. If the film is thin enough, we can assume that its whole thickness behaves in this way. It's an approximation, but a pretty good one, and two-dimensional stress states are much easier to analyse. Note that there will, in general, be a strain in the perpendicular direction - in the case of biaxial tension, this will be a Poisson contraction.

As we make the film thicker, this approximation begins to break down: stresses in the perpendicular direction appear, and it is no longer possible to assume plane stress. Under certain conditions, we may be able to assume that there are no strains in one direction - plane strain. This requires that the material be constrained and unable to change its size in one direction. The most common example of this is rolling, where friction between the film and the rollers constrains the film in the transverse direction. In this case, strains are confined to the plane containing the film normal and the rolling direction.

I think plane stress is still followed in thick plates but considering shear deformation

Certainly for dam type of structures where the thickness direction is much more unlike thick plates plane strain is used.

Thanks for the answers

I do not want to add further mathematics and take a class on stress and strain tensors in 2D and 3D space. Shuvra and Philip have given you sufficient explanation. If I have to put it simple, in plane stress condition, all the z components of stresses are assumed to vanish (2D version as described by Shuvra). The condition would be satisfied if you load a "thin"plate. Note here the strain in thickness and transverse directions are NOT zero because of poisson effect.

In plain strain condition, all the strain components in z direction is assumed to be zero. The situation can be represented by a "thick" plate.

Therefore these are anologies drawn to explain the simplification of mathematical equations.

If you think of a cracked body (part through crack) loaded in tension. The the crack front in the interior will have plain strain (as the mid section is surrounded by sufficient volume of material thus making it anologous to thick section) whereas the crack front at the surface will have plane stress. The plastic zone will also be plane stress to plain strain as you go from the surface to interior. If the plate is sufficiently thick, the componentsof plane stress are assumed to be insignificant compared to the components of plain strain ( as majority of the crack front would experience 3D state of stress) and the whole body can be approximated to be loaded in plain strain.

Hope that clarifies your doubt.

@Abhinav, above explanations are good enough to understand the problem. What I want to add is the physical behavior which led to plane strain and plane stress assumption. In plane strain case for a given load the strain in the thickness direction is negligible because more material is available in the thickness direction which will resist any deformation in that direction (ie, deformation due to poisson effect). So Strain in the thickness direction for a thick plate can be assumed as zero which is actually from the physical behavior of thick plates. In case of thin plates the material in thickness direction is not sufficient to resist the compression due to poisson effect and the deformation in that direction can not be neglected compared with the dimension in the thickness direction ( which is thin). So there is a deformation which is due to the inability of the material to resist the compressive stress, and the stress at a point is the resistance to applied traction which is negligible in this case so an approximation of zero stress in thickness direction is really works in thin plates and it is known as plane stress condition.

deep understanding of this question needs fundamental of principal stress.

It should be note that for a free surface (such as the wall of any plates) the vertical stress (in the lack of external body forces) will be zero and the surface will be the principal plan. For case 1 of your question, a plates, the vertical stress act on the surface will be zero. If the thickness of the plate is very low (thin plates), therefore the area between two surface will be free of component of stress that is perpendicular to the plate surface. This situation equal to plane stress.

There is a bit more to it. In addition to vanishing stresses acting on z-plane (in plane stress approximation) or strains (in plane strain) there is a condition that stress strain and displacement do not change with z.

Another addition is about what is thin and what is thick. 'Thin' means the thickness is much smaller than the size of the problem domain. 'Thick' means the size in z directions is much larger than the domain size. Therefore thick plates may not be qualified for the plane strain approximation.

I am a bit uncertain about the statement 'On the application of tensile load in the presence of a crack in both thin and thick plates, the tensile stresses act in different directions at the crack tip. ' Could you please elaborate on what you mean? In plane stress and plane strain both magnitudes and directions of in-plane stresses coincide.

It can be prooved experimentally very easily measuring (e.g. by DIC) the z-displacement at the crack tip for thin or thick specimens of the same material. You would find similar results because the amount of material free to deform in that direction is limited to the proximity of the surface even for thick specimens.

I agree with Arcady. How much thick is thick enough (or thin is thin enough) to define a plain strain / plane stress condition is difficult. Practically, you will never have a "pure"plain strain/plane stress condition.

I am also not sure what Abhinav mean by " on application of ....different directions at the crack tip"....perhaps he is referring to the triaxial state of stress ahead of the crack tip...

You consider a thin plate (thickness = e) in direction 3 and you proceed to a rescaling in this third direction y_3 = x_3/e. After that you rewrite the equilibrium equation. You obtain terms at order -1 in e. These terms have to be equal to zero. These terms are \sigma_i3,3. Integrating and applying the limit conditions in stress in the third direction (i.e. 0) you obtain \sigma_i3 =0.

For which direction does you consider the plane strain hypothesis ?

Do you study plates in a 3D space or in a 2D space (a section) ?

Consider that my answer (below) is for plate models (i.e. 2D plates immersed in a 3D space, described with at least 5 unhnowns : u,v and w displacement, and for example, two rotations theta_x and theta_y)

Thin plates => small load for big deflection => big s11, s22, s12, s13, s23

but small load => small s33 => (generalized) plane stresses

Thick plates => high loads for small deflection => small s11, s22, s12, s13, s23,

but high load => "high" s33 that can no longer be neglected but I am not sure that plane strains will be a good choice in this case (3D solutions shows that e33 is not null)

The difference between plane strain energy and 3D energy is often very small, and remains acceptable even for very small length-to-thichness ratios of 4 !

It can be different if materials with very different Young's modulus are used together (sandwiches, for example)

Alexandre, The plane strain and plane stress approximations presume the absence of bending.

The plane stress has only s11, s22, s12=s21, as we are in the frame of the classical theory. (It would be different in the Cosserat mechanics.) Since the surfaces normal to z-axis are free, s31=s32=s33=0. Due to the moment equilibrium s13=s31=0, s23=s32=0. So, only 3 unknown stresses. Strains: only 3 unknowns, e11, e12, e22. e33 is not zero but can be calculated from e11, e22 using the Poisson's ratio.

The plane strain, the same 6 unknowns, e33=e13=e23=0 but s33#0. That is why these two approximations are 2D and in elasticity they can be reduced one to another by a proper replacement of the moduli

Thank you very much to all of you to clarify my doubt. A fruitful discussion is going on.

Arcady, as rightly understood by Sivaprasad I am talking about the triaxiality state of stress at the crack tip. I just wanted to explain my question well.

Most of the times, after a mathematic model is developed to describe a phenomenon, it is usually simplifyed to reduce complex analysis of a well-know system under specific circunstances (e.g. study of metals), as result some variables could be neglected and the final outcoming offers good predictions. For that reason it is strongly recommended to understand that considerations before applying the model.

@ Arcady :

Plane stress (or generalized plane stress) hypothesis is used in pretty all "Equivalent Single Layer" plate models (with bending) I have encountered, like the Love-Kirchhoff, Mindlin-Reissner, Reddy's models and many other.

http://en.wikipedia.org/wiki/Plate_theory

But I understand now that the question of Abhinav does not concern plate models in the sense I am used to, i.e. which first goal is to predict the bending produced by out-of-plane loads, but concerns the behavior of more or less thick 2D structures (which, I admit, can be assimilated to plates restricted to in-plane (or membrane) loading).

Sorry for the mix-up.

Plain straincondition means, that the strain in thickness direction of the probe, starting from the surface, tends to zero. The reason is the constraint condition for this strain direction in thick plates. In thin plates the strain in thickness direction is not constrained, because of the free defomability of the plate surfaces, which are in little distance to each other.

The constraint condition for strains in thickness direction plays a role for all strongly non uniform stress states in plates (for example notch effects or cracks).

I have a question.what is the condition of plane strain in rolling without any crack(or any defect)?how can calculate the critical dimensions of sheet plate for this condition?

I believe several people have answered this question. However, Saranath came very close to what I believe is the right answer. Let us take the example of plane strain. As someone else has pointed out, due to the action of loads (and therefore, stresses), there will be deformations in all three directions (using the cartesian coordinate system). If we assume the plane is the xy plane and z is the thickness direction, in the case of plane strain due to the large thickness, the strain in the z direction is essentially zero, using the classical definition of strain = change in length divided by original length. It follows that in the case of plane strain there has to be a stress in the z direction (or the thickness direction).

In the case of plane stress, one uses thin plates and here is why: As has been pointed out earlier, there is a deformation in the z direction (or the thickness direction). But this deformation divided by the original length is not negligible, so there does exist a strain in the z direction. However, there is no stress in the z direction for a plane stress case. Hope this answers your original question, Abhinav.

Dear Abhinav,

Correct me if I'm wrong, but I think that you were looking for a conceptual answer. I will try my best to explain why you have to consider plain stress in thin plates and plane strain in thick ones. It's easier for me to develop the latter case, so here it comes:

Let's consider that we are analyzing an infinitesimal element in the middle of a thick plate with a crack under tensile loading applied in the "vertical" direction, or Z-Axis (classic Mode I of fracture, right?). That element will try to conserve it's volume when deformed (volume constancy), it means that while it gets elongated in the Z direction, it will shrink in the X and Y direction. So far, so good. The thing is that this element has neighbours, and they are all trying to strecht at the same time. This is the so-called "constraint effect". The neighbouring elements excerts a resistance to be deformed, imposing tensile stress in the X and Y direction, it's like the Newton's action-and-reaction principle ;)

So, if you apply some yield criteria, like Tresca or Von Mises, you will find that you reach the yield point at lower stresses, because now you have 3 tensile components acting on this elemental cube (Sz + Sx + Sy = Y).

That's why the plastic zone is smaller in the center of thick plates, meaning that the material will absorb less energy before fracture. This derivates in the "size effect", so problematic in some fracture toughness asessments.

I hope I was helpful, let me know if my response was useful!

Also, If I made any mistake, please feel free to point them out.

Kind regards from Argentina!

When thin plate/sheet/plane is subjected to loading, stresses induced normal to the plate is negligible as it is non material direction and shear stresses corresponding to normal direction of plate becomes zero.But all normal and shear strains are present.-------------------------Plane stress.

Try the following reference:

https://www.researchgate.net/publication/27255824_Fundamental_solutions_for_the_generalised_plane_strain_theory

Article Fundamental solutions for the generalised plane strain theory

Further thoughts on this question:

Firstly plane strain and stress conditions apply to the plate-membrane part of the plate behaviour and not the plate-bending behaviour. So the question therefore relates only to the former type of behaviour.

One can explore the plane-stress, plane-strain affect by taking a simple plate-membrane problem and modelling it with plane-stress, plane-strain and three-dimensional solid elements where you monitor the structural response on the plane at the centre of the thickness. The thickness can be changed to mimic ‘thin’ and ‘thick’ plates.

If this is done you will find:

1) Good correlation between plane-stress and three-dimensional models when the plate is thin (thickness is much less than the diameter of the plate). You will also find good correlation at the free ends of the three-dimensional model for thick plates since there is a local plane-stress situation being ‘imposed’ – there is a transition length where the out-of-plane stress transition to zero at the free ends of the plate.

2) Good correlation between plane-strain and three-dimensional models will only occur when the out-of-plane displacements are constrained in the three-dimensional model. If this displacement is not constrained then the response at the centre of a three-dimensional model even with significant thickness (much greater than the diameter of the plate) will not correlate with either of the planar models.

There is also the so-called generalised plane-strain condition where out-of-plane strains conform to the displacements of a pair of initially parallel rigid planes, i.e. it is defined by three parameters corresponding to the out-of-plane relative translation and in-plane relative rotations of the two planes. This feature extends the applicability of plane-strain models to applications where the out-of-plane displacements are known to behave in such a manner, e.g. when there is a linear temperature variation across the plate.

Returning to your original question then it is not a matter of convention but a matter of fact that for linear-elastic plate problems, where by definition the thickness is small compared to the other dimensions, plane-stress should be used. For a sharp crack the stress at the crack tip is infinite under any load and so an elastic-plastic analysis might need to be considered. The development of plasticity around the crack tip will be three-dimensional in nature and there will be variation in the out-of-plane direction. This may, presumably, only be captured accurately with an appropriate three-dimensional model.

Some years ago I did the above comparisons and when I find the time to tidy them up will post them here.

Hi,

It turned out to be a very interesting discussion. I have a question if someone could answer. If I have to model a square plate ( 100 mm x 100 mm ) with a thickness of 4 mm in 2D (plane-stress as the plate drops in thin plate criteria); and I instead of ignoring the thickness of plate, put the width of the plate in z-direction (meaning that processing length is in x-direction and thickness is in y-direction), would that still count as a plane stress problem ? I mean I read a lot of comments saying that in plane stress condition thickness is normally in z-direction and we ignore the stress in that direction and thats why for thin plates we use plane stress constraint. So, I thought to share this question here with you all.

Moreover, , to answer the question of this original thread, I would say first the person who asked the question needs to know we refer these terms plane-stress and plane-strain only for 2D ! When you are deforming a material you will have to study the stress tensor ( a 9x9 matrix for isotropic materials or most metals) which is having all three dimensions (x, y and z). To make problems simple we assume 2D conditions (a 2 x 2 matrix for stress tensor, without z direction) and thats when we decide that if the problem is a plane-stress problem or plane-strain problem. A plane-stress problem would have to be assumed if you have a plate that is falling in a thin plate criteria ( to my understanding, thickness of the plate divided by the processed length of the plate - this ratio if is less than 0.05, a plate is regarded as a thin plate). In my case above, 4/100 is 0.04 so it would be a thin plate. For thin plates we assume that stress is not considered along one of the direction (normally the direction of the thickness) or it is uniformly distributed; as the thickness is so small that in the direction of thickness stresses would not change much and hence we can ignore the stresses in one direction and work the problem as a 2D problem. To this we call as plane-stress condition meaning we have a stress constrained in one direction, for a problem when solved in 2D.

Plane-strain however, would be the case if the plate is thick. For plates which have thickness to processing length ratio greater than 0.05 (thick plates) we cannot assume the stress in the direction of the thickness as zero or not changing. The thickness now is well defined and cannot be ignored and so we have to consider the stresses in that direction. But then how to make the problem back to 2D now ? For this we assume lets keep the strains in that direction equal to zero or just ignore those or say they are not changing and are uniform so delta.epsilon is zero. This means we are now putting a constrain in terms of strain along the direction of the thickness. You must be wondering why making one zero and other not and why strain in thick plates and stress in thin plates are kept as a constraints. Well dont worry about that just think of hooke's law, stress = strain x youngs modulus ( S=e*E). If you would make stresses zero ( plane stress - thin plates) in say z-direction, even though you are assuming strains in that direction but they would also become zero because of the stresses that are zero ( S = 0 ---> 0/E = e ----> e = 0). So ultimately you will end up in 2D conditions. Same goes for plane strain. This would hold at least for elastic limit for plastic limit you could find the constitutive laws and equations.

Thank you and good luck,

Plane stress and plane strain are 2D approximations of real 3D objects. For this to happen the dependence of the z coordinate (in Angus' notations) should be ignored. This can be done in 2 cases.

1. Plane stress - very thin plate such that the stress variation over thickness can be ignored. If the plate is not very thin than all variables should be averaged over thickness. On top of that sigma_z=sigma_zx=sigma_zy=0. Subsequently strain e_zx=e_zy=0, but e_z is not zero. As a rule of thumb the thickness should be at least 3 times smaller (in elasticity 3 is infinity).

2. Plane strain - infinitely thick (long) body and no dependence of z. (If the body is not infinitely long there will be dependence of z near the ends). In plane strain e_z=e_zx=e_zy=0, igma_zx=sigma_zy=0, but sigma_z is not zero (can be calculated from sigma_x and sigma_y. As a rule of thumb the thickness should be at least 3 times higher than other dimensions. Approximation is only valid far from the ends,

Both approximations give exactly the same equations, but with different moduli: there are simple rules to recalculate.

Source - any textbook on elasticity.

Caution. In plasticity the situation is more complex and needs a separate consideration.