Yoshimura Crush Patterns

One of the signature moves of the John Belushi character in the movie Animal House is Belushi crushing an aluminum can against his forehead.  The shape of the crushed can presents an interesting problem in material science, which has a nice mathematical component. For perfectly symmetrically crushed cans, the shapes are known as Yoshimura Crush Patterns. 

I first learned of Yoshimura crush patterns some years ago, when my friends David Wright and Lisa Mantini visited my family and gave us a set of tumblers that were made in the distinctive shape.  Two of the tumblers are in the photo at the start of this post.

The history of the crush patterns (also called “crease patterns”) demonstrates the interplay between theory and experiment, as described below.  It also demonstrates how dissemination of scientific information has changed over the last eighty years.  Yoshimaru Yoshimura studied the patterns that bear his name in a paper published in Japan in 1951.  An article in English was published in 1955.  The patterns were analyzed even earlier, though, by Theodore von Kármán and Hsue-Shen Tsien in 1941.  Yoshimura, however, was unaware of their work due to the war.  Over the decades, Mathematical Reviews has worked to provide information about published research to researchers everywhere.  This was especially important during the Cold War.  Today, of course, the internet provides many ways for people to discover research done by researchers halfway around the world: via MathSciNet, the arXiv, or simply via a Google search.  It seems that the greatest impediment to keeping up on the literature now is the immense quantity of what is published.

Both studies were motivated by the gap between experiment and theory.  As von Kármán and Tsien write, they had discussed in two previous papers “the inadequacy of the classical theory of thin shells in explaining the buckling phenomenon of cylindrical and spherical shells. It was shown that not only the calculated buckling load is 3 to 5 times higher than that found by experiments, but the observed wave pattern of the buckled shell is also different from that predicted.”  Their 1941 paper begins with the buckling patterns observed in experiment and provides an analysis that explains them.  The starting point is the observation that the bending energy is related to the curvature of the surface, as previous investigators, such as L.H. Donnell, had shown.  The analysis by von Kármán and Tsien seeks to minimize the energy computed using curvature terms arising only via the radial displacement of the shell.  The resulting PDEs are nonlinear, which they solve by considering products of cosine functions and the method of undetermined coefficients.   In their conclusion, they point out limitations of their approach.  “However, due to the complexity of the problem, the results given in this paper can be only considered as a rough approximation and most of the statements made are qualitative rather than quantitative. To put the new theory on a solid footing, a more accurate solution of the differential equations of equilibrium is necessary.”  Von Kármán included his work with Tsien in his Gibbs Lecture on the topic “the engineer grapples with nonlinear problems”, delivered at the December 1939 meeting of the American Mathematical Society in Columbus, Ohio.  The lecture was published in the Bulletin of the AMS.

Yoshimura’s investigation also acknowledges the gap between experiment and theory.  As compared to von Kármán and Tsien, though, he takes a broader view, and makes intrinsic use of the geometry of the buckling.  He writes, “The state which may be actually realized after buckling must be determined by minimizing the energy, not only with respect to the magnitude of deflection, but also to the aspect ratio and the circumferential number of buckling waves. The actual buckling load will be given by a comparison of energy levels before and after buckling and the energy barrier to be jumped over in buckling. Based on such a concept, the general buckling and the local buckling of a cylindrical shell are considered to be quite different phenomena from the energy viewpoint, though they are equivalent with respect to the load.”  An important observation by Yoshimura is that while both a flat sheet and a cylindrical shell are geometrically flat (zero Gauss curvature), the cylindrical shell has nearby deformations that are also developable surfaces, i.e., deformations that also have zero Gauss curvature.  These are his eponymous crush patterns.

Crumpled paper in the news

Crumpling paper has been in the news recently.  Siobhan Roberts has published two articles about crumpling paper in the New York Times, one in 2018 and the other in March 2021.  The first article describes the doctoral work of Omer Gottesman, along with Jovana Andrejevic, Chris H. Rycroft, and  Shmuel M. Rubinstein, who studied the growth of crease length in repeatedly crumpled sheets of paper.  The primary finding is that, despite complications and variations, the total length of the creases grows logarithmically.  The second article describes work by Jovana Andrejevic, Lisa M. Lee, Shmuel M. Rubinstein, and Chris H. Rycroft on a two-dimensional aspect of the work: analysis of the facets that develop upon repeated crumpling.  Their work relates paper crumpling to fragmentation distributions, including work by Kolmogorov from around the time the von Kármán and Tsien were doing their work on buckling of thin cylinders.

References

1. Andrejevic, J., Lee, L.M., Rubinstein, S.M. et al. A model for the fragmentation kinetics of crumpled thin sheets. Nat Commun 12, 1470 (2021). https://doi.org/10.1038/s41467-021-21625-2.
2. Donnell, L. H., Stability of Thin-Walled Tubes Under Torsion, N.A.C.A. Technical Report No. 479, 1934.
3. Gottesman, O., Andrejevic, J., Rycroft, C.H. et al. A state variable for crumpled thin sheets. Commun Phys 1, 70 (2018). https://doi.org/10.1038/s42005-018-0072-x.
4. von Kármán, Theodore; Tsien, Hsue-Shen, The buckling of thin cylindrical shells under axial compression. J. Aeronaut. Sci. 8 (1941), 303–312.
5. von Kármán, Theodore, The engineer grapples with non-linear problems.
   Bull. Amer. Math. Soc. 46 (1940), 615–683. MR0003131
6. Kolmogoroff, A. N., Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C. R. (Doklady) Acad. Sci. URSS (N. S.) 31, (1941). 99–101. MR0004415
7. Yoshimura, Y.: Theory of Thin Shells with Finite Deformation. Rep. Inst. Sci. and Tech., Tokyo Univ., 2, 1948, P. 167; 3, 1949,P. 19.
8. Yoshimura, Y.: Local Buckling of Circular Cylindrical Shells and Scale Effects. Proc. of the 1st Japan National Congress for Appl. Mech., 1951.
9. Yoshimura, Yoshimaru: On the mechanism of buckling of a circular cylindrical shell under axial compression. Technical Memorandum 1390. National Advisory Committee for Aeronautics, July 1955.
   Note: this paper had been available from the NASA site: https://ntrs.nasa.gov/citations/19930093840. At the time of this writing (18 July 2021), I can no longer find it on the NASA server.  A copy is available from the University of North Texas: https://digital.library.unt.edu/ark:/67531/metadc62872/m1/1/.

Reviews

MR0006926
von Kármán, Theodore; Tsien, Hsue-Shen
The buckling of thin cylindrical shells under axial compression.
J. Aeronaut. Sci. 8 (1941), 303–312.
73.2X

This paper is devoted to the solution of the problem stated in the title under the same general assumptions as were made in an earlier paper by the same authors on buckling of the spherical shell [J. Aeronaut. Sci. 7, 43–50 (1939); MR0003177]. [See also a paper by v. Kármán, Dunn and Tsien [J. Aeronaut. Sci. 7, 276–289 (1940); MR0003178] and a paper by K. Friedrichs [Theodore von Kármán Anniversary Volume, California Institute of Technology, Pasadena, 1941, pp. 258–272; MR0004599].] The essentially new idea in this as in the earlier papers is that it is possible to explain why thin shells buckle at a much lower pressure than that predicted by the linear theory of buckling by considering the effect of certain nonlinear terms; even the quantitative results of the nonlinear theory for the spherical shell were found to be in quite good accord with experiment. The cylindrical shell is a much more difficult case than that of the spherical shell (because of a lack of symmetry in the buckled state) so that the authors restrict themselves in the main to qualitative rather than quantitative comparison with experiment. Solutions are obtained by an energy method.

Reviewed by J. J. Stoker

MR0003131 (2,167d) Reviewed
von Kármán, Theodore
The engineer grapples with non-linear problems.
Bull. Amer. Math. Soc. 46 (1940), 615–683.
71.0X

von Kármán appeals to pure mathematicians for cooperation with engineers who are struggling with a great variety of mathematical problems. He summarizes the intention of his Gibbs lecture as follows: “An attempt is made to show the application of analytical methods available for the solution of certain nonlinear problems in which the engineer is interested. Some gaps are shown and frontiers indicated beyond which the safe guidance of the mathematical analysis is for the time being lacking.” After contrasting linear with nonlinear problems von Kármán states: “in most nonlinear problems physical reasoning is not sufficient or fully convincing, so that in these cases the questions of existence and uniqueness represent a real challenge to the mathematician.” The whole field of nonlinear mathematical engineering problems is then discussed in a rather detailed survey.

1. Nonlinear vibrations result from differential equations of the type: $$ \ddot x+\omega^2x=f(x,\dot x), $$ where $x$ represents a deflection and $\omega$ the natural frequency of the system. Periodic solutions, corresponding to Poincaré’s limit cycles, represent “self-excited” vibrations. In some limit cases there are sudden transitions between deflections of opposite sign, “relaxation vibrations” according to van der Pol. If a periodic force is applied to the system, the phenomenon of “subharmonic resonance” may result; it is important in radio-technique but also occurs in airplane vibrations; it is here treated by a perturbation method analogous to a procedure developed by Poincaré for celestial mechanics.

2. Nonlinear differential equations occur in the theory of elasticity when deflections are permitted to be large. Bending and buckling of thin rods (“elastica” problem) and plates are discussed. The differential equations for plates, derived by von Kármán in 1910, are $$ \Delta\Delta F=E(w_{xy}^2-w_{xx}w_{yy}),\quad C\Delta\Delta w=F_{yy}w_{xx}+F_{xx}w_{yy}-2F_{xy}w_{xy}, $$ where $E$ and $C$ are constants. The most interesting phenomenon is to be found in the case of very thin plates $(C\rightarrow 0)$. The solutions, which recently have been treated by asymptotic integration, will be constants except in narrow strips near the boundary where sudden changes occur. For curved arches and shells the presence of nonlinear terms provides a solution which is attained in reality even before the linear theory comes into play; in this case, therefore, the linear theory fails entirely to give account of the actual situation. A brief review is given of the rather recent general theories of nonlinear elasticity due to Murnaghan and Biot. Plastic deformations are characterized by quite a different type of nonlinear problems, namely, boundary value problems for hyperbolic differential equations of the type $$ (f_{xx}+f_{yy})^2-C(f_{xx}f_{yy}-f_{xy}{}^2)=0. $$

3. In the theory of fluid flow a variety of problems occur in which the differential equation (the potential equation) is linear while the boundary conditions are nonlinear. Many problems in this field have been thoroughly treated, for example, the flow around obstacles and through nozzles, also waves with large amplitudes; many other problems remain unsolved, for example, that of heavy jets over spill-ways or the meteorological problem of the progressing cold front; a main difficulty seems to be the determination of the proper singularity of the analytic function which represents the flow.

4. The flow of viscous fluids is governed by a nonlinear differential equation (essentially of the fourth order) for which exact solutions are known only in few instances. From an engineering point of view the limiting case that the viscosity approaches zero is more important; it had defied the powers of analysis until it was made accessible to methods of asymptotic integration through Prandtl’s ingenious boundary layer theory. There remain, however, unanswered purely mathematical questions; for example, what is the flow pattern around a submerged body?

5. The interesting feature of the nonlinear differential equations of compressible fluids is that they are elliptic or hyperbolic depending on whether the velocity is below or above that of sound. For both cases some solutions are known. For problems, however, where the equation is elliptic in one part and hyperbolic in another part (a condition encountered, for example, with aerial bombs dropped from large heights), no methods are as yet available. The invention of such methods “would be an achievement both from a practical and mathematical point of view.”

The paper is furnished with a great many instructive figures and supplemented by an extensive bibliography.

Reviewed by K. Friedrichs

MR0004415
Kolmogoroff, A. N.
Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. (German)
C. R. (Doklady) Acad. Sci. URSS (N. S.) 31, (1941). 99–101.
60.0X

It is stated that observations show that the logarithms of the sizes of particles such as mineral grains are frequently normally distributed [Rasumovski, in the same C. R. 28, 814–816 (1940)]. The purpose of the present paper is to explain this phenomenon by a plausible probabilistic scheme. Consider a random process in which the number of particles at time $t\ (=0,1,2,\cdots)$ is $N(t)$, and the number of particles of dimension not greater than $r$ is $N(r,t)$ (it is irrelevant how this “dimension” is defined). It is supposed that the probability that a particle of size $r$ splits, during $(t,t+1)$, into $n$ particles of sizes $x_1r,x_2r,\cdots,x_nr$ is independent of $t$ and $r$. Let then $Q(x)$ be the mean value of the number of particles of size not greater than $xr$ originated during $(t,t+1)$ from a particle of size $r$. It is shown that under some slight additional assumptions $N(e^x,t)/N(t)$ tends to a Gaussian distribution with mean value $mt=t\int_0^1\log ydQ(y)/Q(1)$ and variance $t\int_0^1(\log y-m)^2dQ(y)/Q(1)$.

Reviewed by W. Feller