Plane stress and plane strain assumptions are something that we hear all the time in FEA, and in solid mechanics in general, but what actually are they? It is often assumed that everyone in the field of engineering implicitly understands what these terms mean, hence straightforward explanations can be somewhat hard to find.
Here, we describe in simple terms how plane stress and plane strain assumptions can be used to simplify 3-dimensional problems into 2-dimensional slices in order to save both model building effort and computational expense.
Why Model Things In 2D?
It’s hard to believe for new analysts that, in the early days, we (not me, I’m too young) had to stamp cards and feed them into a computer to run FEA – if you ever wondered why the different sections of an input ‘deck’ are called ‘cards’… now you know. Going back to those days, pre-processing, post-processing and computational expense were obviously huge concerns, and so mathematicians had to come up with clever ways to simplify complex 3-dimensional problems into something a little more manageable.
Although our computers have improved somewhat (OK, a lot) since then, it is still often useful to take advantage of the 2D formulations that remain. Fracture mechanics studies, for example, can benefit greatly from the use of plane stress and plane strain analysis, where we are looking to idealize the stress states in the center and at the ends of a crack tip in a laboratory specimen.
In this case, 2D refers to the degrees of freedom in the x-y plane and the lack thereof in the z plane, which is ‘out-of’ plane.
2D Formulations
Plane Strain
Plane strain assumptions can provide very good representations of real life components. Essentially, the in-plane strains are developed as in the full 3D formulation, but the out-of-plane or z-direction strains are set to zero. This condition would exist in an object that is constrained in the z-direction by rigid walls; the formulation only allows the resolution of strain ‘in (the) plane’, hence the name ‘plane strain’:
εzz = εxz = εyz = 0
In the constitutive equations derived from Hooke’s law, we simply remove the strain components (εzz, εxz, εyz) that are to be considered zero.
Plane strain assumptions tend to be a very good approximation of the behavior inside a thick component that is loaded only in one plane. The large amount of material through the thickness essentially renders through-thickness strain irrelevant (or at least negligible).
Plane stress
Plane stress, on the other hand, assumes that the three stress tensor components relating to the z-direction are zero. This is, of course, never actually the case inside of a real part, but the approximation trends towards applicability as thickness of the component approaches zero; there is not enough bounding material to maintain the through-thickness stress:
σz = σxz = σyz = 0
This is great for analyzing very thin plates that are loaded only in the plane, but it can also be applied to the surface of thicker components. In fact, the surface of a plate is the only location where true plane stress conditions can exist. It is a perfect representation of the boundary condition.
In the plane stress constitutive model, we use σz = 0 to make εz = 0 and simply calculate εzz from -v(εxx + εyy), where v is Poisson’s Ratio. However, some FEA codes deal with this differently for various reasons including accounting for thermal expansion. Check your code’s manual for more information.
Fracture Mechanics
Knowledge of plane stress and plane strain is often very useful in fracture mechanics analysis, since the constraint of the crack-tip is heavily influenced by the transition between plane strain conditions (at the center of the specimen) and plane stress conditions (at the surface). In the schematic below, we can see how the stress intensity factor is affected by the distance from the center of the crack front. At the center of the specimen, where plane strain conditions exist, constraint is high and conditions are close to small-scale yielding. At the surface, however, the lack of out-of-plane stress results in loss of crack-tip constraint and the stress intensity factor is much lower.
Which Formulation Should You Choose?
As we’ve seen in this post, 2D assumptions can be really useful when modeling certain, idealized, geometries. Here’s a quick cheat sheet to take away:
- Choose plane strain to simulate the interior of a very thick component loaded in a single plane
- Choose plane stress to simulate a very thin component when loaded in-plane only
- We can also use plane stress assumptions to estimate the surface behavior of thick plates loaded in-plane only
Final Thoughts
The aim of this post was to provide a quick hitting explanation of why we would use 2D assumptions, what plane stress and plane strain assumptions are, and which one to choose based on the geometry of interest. Although these mathematical methods were developed out of the need to simplify problems, they do still have their uses in modern engineering problems, particularly when we’re looking at fracture mechanics. It should be noted that a third option, generalized plane strain, was intentionally left out of this discussion to keep the ideas and terminology straightforward. Perhaps we’ll tackle that in another blog post soon?
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