Yoshimura buckling

In 1941, crease patterns in cylindrical shells were first studied by Theodore von Kármán and Hsue-Shen Tsien at the California Institute of Technology, and was later independently studied by Yoshimaru Yoshimura in a 1951 Japanese paper, with an English version published in 1955. Isolation of Japan during and after World War II made Yoshimura unaware of the earlier work. == Mathematical derivation ==

Compatibility condition The compatibility condition of the buckling pattern is given by: \frac{1}{2} \left(2 \frac{\partial^2 F}{\partial x \partial \theta} - \frac{\partial^2 E}{\partial \theta^2} - \frac{\partial^2 G}{\partial x^2} \right) = LN - M^2 = KH H^2 = EG - F^2 where E, F, G and L, M, N represent the first and second fundamental forms of the deflection surface, respectively. K represents the Gaussian curvature, which is expressed as: K = \frac{1}{R_1 R_2} where R_1 and R_2 are the principal radii of curvature of the cylinder. H is expressed as: H = 2\left ( k^4+1\right ) where k is the length of the buckle in the circumferential direction divided by the length of the buckle in the axial direction. • On the undeformed initial surface, the Gaussian curvature of the cylinder is 0, satisfying the compatibility condition. • On the deformed surface, it is observed that the surface is nearly developable. Consequently, the Gaussian curvature of the cylinder is close to 0. Since the left side of the first compatibility condition is already small, the compatibility condition is satisfied. Buckling load prediction In classical shell theory, the asymptotic formula to predict the critical buckling load \sigma_{cr} in cylindrical shells is expressed as: \sigma_{cr} = \frac{ Eh }{\sqrt { ( 3 - \nu^2 ) } } where h = \frac{t}{R} represents the ratio of the cylinder wall thickness to the radius, and E and \nu represent the Young's modulus and Poisson ratio, respectively. after Dutch engineer Warner T. Koiter, who derived it in 1945, but was first derived by R. Lorenz in 1911. Experimental results have shown that this classical formula frequently overestimates the buckling load by a factor of 4 to 5. Conditions for equilibrium Under a Cartesian coordinate system, the equilibrium conditions for a cylinder under axial compression can be expressed as: D \nabla^4 \omega = 2h \left[\frac{\partial^2 \phi}{\partial y^2} \frac{\partial^2 \omega}{\partial x^2}-2\frac{\partial^2 \phi}{\partial x^2 \partial y^2} \frac{\partial^2 \omega}{\partial x \partial y}+\frac{\partial^2 \phi}{\partial x^2}\left( \frac{\partial^2 \omega}{\partial y^2}+\frac{1}{r}\right) \right] \nabla^4 \phi = E \left[\left(\frac{\partial^2 \omega}{\partial y^2}\right)^2-\frac{\partial^2 \omega}{\partial x^2} \frac{\partial^2 \omega}{\partial^2 y^2} \right]-\frac{E}{r} \frac{\partial^2 \omega}{\partial x^2} where E and D are the Young's modulus and flexural rigidity, respectively. \phi is derived from the second equation, and \omega can be expressed as: \omega = A \cos \lambda (y+mx) \cos \lambda (y-mx) + B [ \cos \lambda (y+mx)+ \cos \lambda (y-mx) ]+C with \lambda, m, A, B, C as the parameters. This carefully selected method == Characteristics ==

Folding pattern The Yoshimura folding pattern is composed of isosceles triangles that share a single edge at the base, forming repeated rhombuses, as seen in the Schwarz lantern crease pattern. The Yoshimura creasing pattern is related to both the Kresling and Hexagonal folds, and can be framed as a special case of the Miura fold. Unlike the Miura fold which is rigidly deformable, both the Yoshimura and Kresling patterns require panel deformation to be folded to a compact state. Local buckling Cylindrical shells under axial compression have been observed to exhibit local buckling, provided that they are comparatively long. Consequently, lengthwise along the cylinder, the buckling occurs at over 1.5 times the lobe's axial wavelength. This phenomenon can be further explained as a loss of total elastic energy. Considering a cylinder with fixed ends under Euler's critical load, the elastic energy decrease of the unbuckled region will overpower the increase in elastic energy of the buckled region when local buckling occurs. This results in a loss of total elastic energy. With respect to the asymptotic formula \sigma_{cr} = \frac{Eh}{\sqrt{( 3- \nu ^2 )}}from classical shell theory, where h = \frac{t}{R} is the shell's dimensionless thickness, the buckling load approximately scales in two different ways: • \sigma_{cr} \sim h^{3/2} for imperfections in shape. • \sigma_{cr} \sim h^{5/4} for imperfections in load. Developable surface As the thickness of the cylindrical shell decreases, the buckled surface becomes approximately developable. The surface is consequently most developable when the thickness of the shell approaches 0, as it behaves like an ideal membrane. == Applications ==

Yoshimura buckling and its related origami patterns' possible applications have been researched, but their use in engineering remains limited. Current Yoshimura origami designs lack an overarching mathematical theory between the two dimensional (2-D) creases, and three-dimensional (3-D) forms. Currently, engineering attempts to develop a deployable cylindrical structure with Yoshimura folding have only been made for membrane structures, such as soft pneumatic actuators. Current applications of the Yoshimura buckling have focused on structures built with soft membranes, such as the casing around pneumatic actuators or protective housing of robotic joints. Yoshimura pattern's compliability also makes it applicable in reconfigurable soft robots (RSRs) made from synthesized materials due to its 1800% elongation ratio under paper fabrication. == See also ==