Elastic instability

External-pressure failure of shells can result from overstress at one extreme or n om elastic instability at the other or at some intermediate loading. The code provides the solution for most shells by using a number of charts. One chart is used for cylinders where the shell diameter-to-thickness ratio and the length-to-diameter ratio are the variables. The rest of the charts depic t curves relating the geometry of cyhnders and spheres to allowable stress by cui ves which are determined from the modulus of elasticity, tangent modulus, and yield strength at temperatures for various materials or classes of materials. The text of this subsection explains how the allowable stress is determined from the charts for cylinders, spheres, and hemispherical, ellipsoidal, torispherical, and conical heads. 

I. Durand, K. Kassner, C. Misbah, H. Muller-Krumbhaar. Strong coupling between diffusive and elastic instabilities in directional solidification. Phys Rev Lett 76 10X1, 1996 I. Cantat, K. Kassner, C. Misbah, H. Muller-Krumbhaar, Directional solidification under stress. Phys Rev E 58 6011, 1998. 

The theories of elastic and viscoelastic materials can be obtained as particular cases of the theory of materials with memory. This theory enables the description of many important mechanical phenomena, such as elastic instability and phenomena accompanying wave propagation. The applicability of the methods of the third approach is, on the other hand, limited to linear problems. It does not seem likely that further generalization to nonlinear problems is possible within the framework of the assumptions of this approach. The results obtained concern problems of linear viscoelasticity. 

The classic example of failure due to elastic instability is the buckling of tall thin columns (struts), which is described in any elementary text on the Strength of Materials . 

Two types of process vessel are likely to be subjected to external pressure those operated under vacuum, where the maximum pressure will be 1 bar (atm) and jacketed vessels, where the inner vessel will be under the jacket pressure. For jacketed vessels, the maximum pressure difference should be taken as the full jacket pressure, as a situation may arise in which the pressure in the inner vessel is lost. Thin-walled vessels subject to external pressure are liable to failure through elastic instability (buckling) and it is this mode of failure that determines the wall thickness required. 

Zhang Y, Matsumoto EA, Peter A, Lin PC, Kamien RD, Yang S (2008) One-step nanoscale assembly of complex structures via harnessing of elastic instability. Nano Lett 8 1192-1196... 

During extrusion of polymer melts with high throughputs, the elastic melt properties can also lead to elastic instabilities which can result in surface distortions of the extrudate. One example are wavy distortions also described as sharkskin. Depending on the polymer, this can also lead to helical extrudate structures (stick-slip effect) or to very irregular extrudate structures (melt fracture) at even higher throughput rates [10]. 

K.P. Chen, Elastic instability of the interface in Couette flow of viscoelastic liquids, J. Non-Newtonian Fluid Mech., 40 (1991) 261-267. 

The Prandtl Tomlinson (PT) model [17] (usually only referred to as the Tomlinson model) is the simplest model that allows for elastic instability and hence for pinning between two incommensurate solids. In its original version (see Fig. 7), atoms in the upper wall are coupled harmonically to their ideal... 

The first discussion of the effect of thermal fluctuations on friction forces in the Prandtl Tomlinson model was given by Prandtl in 1928 [18]. He considered a mass point attached to a single spring in a situation where the spring fei in Fig. 7 was compliant enough to exhibit elastic instabilities, but yet sufficiently strong to allow at most two mechanically stable positions see also Fig. 8, in which this scenario is shown. Prandtl argued that at finite temperatures, the atom... 

For Vo below the second threshold denoted by Vq, the kinetic friction is zero in the limit of quasi-static sliding that is, for sliding velocity v Q. That is, for Vo < Vq" the kinetic friction behaves like a viscous drag. For Vo > the dynamics is determined by the Prandtl Tomlinson-like mechanism of elastic instability, which leads to a finite kinetic friction. The threshold amplitude Vq increases with k and is always larger than zero. Therefore, in the commensurate case, vanishing kinetic friction does not imply vanishing static friction just like in the PT model. The FKT model for Vj, < Vo < is an example of a dry-friction system that behaves dynamically like a viscous fluid under shear even though the static friction is not zero. 

Sprensen et al. [63] also examined the effect of incommensurability. The tip was made incommensurate by rotating it about the axis perpendicular to the substrate by an angle 0. The amount of friction and wear depended sensitively on the size of the contact, the load, and 0. The friction between large slabs exhibited the behavior expected for incommensurate surfaces There was no wear, and the kinetic friction was zero within computational accuracy. The friction on small tips was also zero until a threshold load was exceeded. Then elastic instabilities were observed leading to a finite friction. Even larger loads lead to wear like that found for commensurate surfaces. 

It might yet be possible to predict trends from Eq. (52). Persson and Tosatti [27] argue that decreases quickly with increasing normal pressure. This has important consequences for tectonic motion for instance, small earthquakes typically do not occur close to the earth s surface. Sokoloff [122] concluded recently that even for his small value of the contribution to kinetic friction due to elastic instabilities that result from a competition between surface roughness and elastic interactions would lead to rather small friction coefficients on the order of 10 provided that no contamination layer or other local friction mechanisms were present in the contacts. 

Y. L. Joo and E. S. G. Shaqfeh, Observations of purely elastic instabilities in the Taylor-Dean flow of a Boger fluid, J. Fluid Mech. 262, 21-1A (1994). 

P.E. Arratia, C.C. Thomas, J. Diorio, and J.P. Gollub. Elastic instabilities of polymer solutions in cross-channel flow. Phys. Rev. Lett., 96 144502, 2006. 

In the case of vessels subject to vacuum, the shell buckling could occur because of elastic instability. A critical pressure p can be approximated by ... 

Two fundamentally different types of failure may occur in vessels operated under vacuum (as opposed to high pressure). The problem in vacuum operation is the elastic stability of the vessel shell when it is under an external pressure loading. In general, elastic instability is a problem that must be considered in all structures having limited rigidity when subjected to bending, torsion, compression or a combination of these loadings. In failure by elastic instability, the structure buckles or collapses like an evacuated thin-shelled vessel. 

Elastic deformation—Elastic instability or elastic buckling, vessel geometry, and stiffness as well as properties of materials are protection against buckling. 

Tall towers or columns should be checked for dynamic response. If the vessel is above the critical line in Figure 3-9, Rm/t ratio is above 200 or the h/D ratio is above 15, then dynamic stability (elastic instability) should be investigated. See Procedure 4-8, Vibration of Tall Towers and Stacks, for additional information. 

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