Mechanics of Defects in Plates and Shells
- Sumit Basu
- 6 days ago
- 7 min read
Updated: 4 days ago
Saptarshi Paul is a PhD student at IIT Kanpur who worries about dislocations and disclinations on curved surfaces. These are defects that arise because of mismatches in rotation and translation. On curved surfaces, curvature interacts with the defects. And this has deeper technological implications.
Inspiration: Defect-mediated Growth of Cell Wall in Bacteria
All living things grow. A bud on a plant can grow into a flower, leaf, or shoot. An embryo in a mother’s womb grows into a baby with limbs, a head, and other functioning body parts. An amoeba can grow and divide into numerous daughter cells. The examples are inexhaustible. The question we want to address is this: how do we model that growth? How do we understand why things grow into the shape, and the function they do? These are very complex phenomena, with many moving parts, and many things happening simultaneously. We need a starting point, an idealized, simpler scenario, which would be easier to understand, and then things could be built up from there.
One such starting point was provided by [1] and [2], which attempted to study growth in rod-shaped bacteria by modeling them as interacting dislocations on the surface of a cylinder.
The 2012 paper describes the growth mechanism of cell walls in rod shaped bacteria as proteins associated with cell wall extension moving in circles oriented approximately along the cell circumference of the cylinder, viewing them as dislocations in the partially ordered protein- glucose scaffold (peptidoglycan) structure.
Of particular interest to us is the second paper above, in which the authors discuss, theoretically and computationally, the interaction of dislocations on a cylindrical surface. They write that ”the Laplacian on a cylinder is equivalent to that in an infinite two-dimensional flat space together with the periodicity requirement”, which implies that the physics of the problem of a single dislocation on a cylindrical surface is the same as that of the dislocation kept in an infinite 2D flat space, with the added constraint of the variables (kinetic and kinematic) repeating themselves with a period of 2πR in the direction in which the cylinder is curved (R being the radius of the cylinder).
We were not as convinced with this idea of theirs. Their final analysis might eventually be correct, but modeling a cylindrical surface so simplistically did not sit right with us. The curvature and topology of the surface are not well discussed. A cylindrical shell is a multiply-connected surface, which brings in more depth to the description of defects than is mentioned in the paper. What is a multiply-connected surface? It is a topological notion. It is the existence of
closed curves in the body that do not shrink to a point via continuous deformation without going out of the domain. In Fig. 1, the curve C1 is reducible to a point within the domain Ω, but C2 has to be cut to shrink it to a point within the domain. Hence, C1 is a ”reducible” curve, whereas C2 is irreducible.
Why is this important? This is important because the reducible and irreducible curves are key to a complete
description of defects pertaining to a cylindrical shell. The reducible curves describe defects in the bulk, and the irreducible ones describe defects in the through-hole. Examples of defects (Dislocations and Disclinations) in the hole can be seen in Fig. 2.
As we were not convinced of the theoretical underpinnings of Nelson’s work, we decided to go back from dynamics of defects in a cylindrical shell to establishing a simple, linear theory which describes the statics of a general shell with defects. It was important to us to have firm ground on, first of all, how to describe defects in a curved geometry, and secondly, how its curvature and topology of the surface would influence the effect of defects on such surfaces.
Theory, and Computations
Figure 3 shows an edge dislocation of Burgers vector b in a plate. It is essentially an extra line of atoms pushed into the ordered crystalline microstructure of the plate. In an ideal scenario, we would have liked to take into account
how every atom in that plate (including the inserted ones) behaved, but that is impossible. This is where the study of continuum mechanics proves useful. A continuum theory uses rigorous mathematical tools to transform variables pertaining to atoms in a discrete atomic lattice theory into fields (temperature, displacement, stress, etc.), which are functions of position. This enables us to homogenize the behavior of individual atoms into a small number of variables that accurately capture the macroscopic behavior of the whole system. A good place to read about it is in ”Continuum Theory of Defects” by Ekkehart Kroner (1980).
The continuum theory that we have developed for a shell allows us to describe what kind of forces are generated within it upon the introduction of a defect. Addition of the defect forces its neighborhood to adjust, or deform to accommodate the defect without losing the shell’s integrity. This is what we term elastic behavior, in which the shell recovers its original form when the defect is removed. The interaction of the shell with the defect, or any loading or other stimulus in general, is called ”stress”, which is a notion carefully derived from Euler’s Laws of Motion.
How do we describe defects? They are geometric objects, and are therefore described kinematically. Their description stems from the notion of ”compatibility”. Suppose that we have an unstrained body, which is given a strain to put it into a different configuration. Compatibility is the condition in which the given strain has an accompanying displacement, which maps every point in the original configuration to the current one. If such a displacement does not exist, it means that some material has been added or removed from the original to obtain the current configuration, which we term as ”defects”. Hence, defects are sources of ”incompatibilities” in strain, and, therefore, sources of stress if the body is governed by a constitutive law between the stress and strain.
These notions have been developed in literature over the years primarily for three-dimensional solids. There are works which reduce such 3D theories to one- and two-dimensional ”slender” structures like rods, plates, and shells. However, we had not come across a theory which homogenizes the 3D theory to a thin shell, incorporating the effect of curvature and topology and their interaction with the defect distribution. This has been our little contribution to the field.
The governing equations we obtain via our theory for any instance of shells are complicated partial differential equations, which are analytically very difficult to solve. Hence, we had to take the help of numerical techniques to solve them. The technique that we used was the Finite Element Method, where the governing equations are converted to a ”weak-form”, i.e., into an integral form with lower order of derivatives, and applied over small discretizations (or elements) of the shell surface. The variables to be solved for are approximated in terms of polynomials (shape functions), and the intent is that the correct solution is obtained asymptotically as the element size is decreased, or a higher degree of polynomial is chosen, or both. The methodology, of course, needs to be tested against available analytical solutions for simpler cases, so as to check whether the correct solution is indeed obtained.
There are many software packages available on the market that do FEM computations, such as ANSYS and ABAQUS. However, as our formulation is setup in a different manner (in terms of ”Stress Function”) and it deals with singularities which are used to describe discrete defects, we have had to write our own codes from scratch. An example problem of a plate with a hole, with a defect in the said hole, is described in the next section, which has been solved using our
code and validated against its analytical solution. Available FEM solvers also don’t deal with ”global” defects (defects within the hole).
Example Problem: Disclination in the Hole of an Annular Disc
Figure 4 shows an example problem from the book Elasticity, by J. R. Barber, in which a thin wedge of material is inserted at the cut AB of an annular disc. In our formulation, this is a wedge disclination (rotational defect) in the hole of the disc, since the tip of the wedge lies in the hole. This problem can be solved analytically, and has also been solved with our code. The analytical and computational solutions for stress components are presented in Fig. 5. Our FEM methodology is also validated in Fig. 6, where we see that the error between analytical and computational values decreases with a finer mesh, i.e., decreasing element-size.
I would like to pose a question to interested readers at this point, something that troubled us for a long time, which we have just figured out at the time of writing: Suppose that the tip of the wedge in Fig. 4 is shifted off the center, such that it still lies within the hole. Would the response of the disc change? The disclination charge within the hole is still the same as before, as the tip is still within the hole and the wedge angle has not changed. Food for thought.
Conclusion
We started from the question of growth in biological matter, and ended with solving a problem of the response of an annular disc to an inserted wedge of some material. We intend to build back up from this basic analysis, to the following questions:
The above case was for a flat surface, but with multiply-connected topology. A full cylindrical shell, the premise for our original inquiry, is a multiply-connected, as well as curved surface. How does that affect the presence of isolated defects? We have also drawn up codes for spherical and hyperboloid surfaces.
How do two or more defects interact on such geometries?
Nelson et al. talk about growth via motion of dislocations in their papers. What effect does a particular geometry have on the dynamics of defects that are interacting with each other and moving on the surface?
[1] Amir, Ariel, and David R. Nelson. ”Dislocation-mediated growth of bacterial cell walls.” Proceedings of the National Academy of Sciences 109, no. 25 (2012): 9833-9838,
[2] Amir, Ariel, Jayson Paulose, and David R. Nelson. ”Theory of interacting dislocations on cylinders.” Physical Review E 87, no. 4 (2013): 042314.
Figure 7: An Edge Dislocation.
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