A Weak Form Implementation of Nonlinear Axisymmetric Shell Equations With Examples

2019 ◽  
Vol 86 (12) ◽  
Author(s):  
Matteo Pezzulla ◽  
Pedro M. Reis
Keyword(s):  
Weak Form ◽  
Surface Shape ◽  
Energy Potential ◽  
Spherical Shells ◽  
Axisymmetric Shell ◽  
Axisymmetric Shells ◽  
Shell Equations ◽  
Thickness Variations ◽  
Nonlinear Deformations ◽  
Dynamic Community

Abstract We present a weak form implementation of the nonlinear axisymmetric shell equations. This implementation is suitable to study the nonlinear deformations of axisymmetric shells, with the capability of considering a general mid-surface shape, non-homogeneous (axisymmetric) mechanical properties and thickness variations. Moreover, given that the weak balance equations are arrived to naturally, any external load that can be expressed in terms of an energy potential can, therefore, be easily included and modeled. We validate our approach with existing results from the literature, in a variety of settings, including buckling of imperfect spherical shells, indentation of spherical and ellipsoidal shells, and geometry-induced rigidity (GIR) of pressurized ellipsoidal shells. Whereas the fundamental basis of our approach is classic and well established, from a methodological view point, we hope that this brief note will be of both technical and pedagogical value to the growing and dynamic community that is revisiting these canonical but still challenging class of problems in shell mechanics.

Small-Scale Yielding at the Tip of a Through-Crack in a Shell

1980 ◽  
Vol 102 (2) ◽  
pp. 167-173 ◽  
Author(s):  
J. D. Achenbach ◽  
T. Bubenik
Keyword(s):  
Nonlinear Behavior ◽  
Crack Edge ◽  
Small Scale ◽  
Small Scale Yielding ◽  
Spherical Shells ◽  
Shallow Shells ◽  
Scale Yielding ◽  
Hardening Curve ◽  
Thickness Variations ◽  
Independent Integral

Deformation theory is used to model plastic deformation at the tip of a through-crack in a thin shell. In the vicinity of the crack the shell is subjected to both stretching and bending, but stretching is assumed to dominate. Thus the stresses are tensile, but with a nonuniform distribution through the thickness, which depends on the material properties as well as on the geometry. The nonlinear near-tip fields (which are singular) have been analyzed asymptotically. Cracks in shallow shells and spherical shells have been investigated in some detail. It is shown that the angular variations are the same as for generalized plane-stress plate problems. Assuming small-scale yielding a path-independent integral, which is valid in a region close to the crack edge, is used to connect the nonlinear near-tip fields with the corresponding singular parts of the linear fields. It is shown that the nonlinear behavior significantly affects the through-the-thickness variations of the near-tip fields. The singular parts of the membrane stresses tend to become more uniform through the thickness of the shell with a flatter strain-hardening curve.

Parametric Resonance of Liquid Storage Axisymmetric Shell Under Horizontal Excitation

1991 ◽  
Vol 113 (4) ◽  
pp. 511-516 ◽  
Author(s):  
M. Takayanagi
Keyword(s):  
Cylindrical Shell ◽  
Free Vibration ◽  
Parametric Resonance ◽  
Circular Cylindrical Shell ◽  
Translational Motion ◽  
Harmonic Balance Method ◽  
Vibration Modes ◽  
Liquid Storage ◽  
Axisymmetric Shell ◽  
Axisymmetric Shells

A procedure for analyzing parametric resonance of liquid storage axisymmetric shells is proposed that is an extension of the procedure presented at PVP-89 for parametric resonance of empty axisymmetric shells with lumped weights. Free vibration modes of axisymmetric shells containing liquid are calculated considering the effect of initial stress due to static liquid pressure by using a conical shell finite element. The calculated free vibration modes are used to expand the free vibration modes of the axisymmetric shell with lumped weights and internal liquid. A type of Mathieu equation is derived considering the effects of the translational motion of the attached weight in the radial direction or the effects of the beam-type motion of the shell without lumped weight. The harmonic balance method is used to obtain the parametric resonance regions. Principal resonance of a circular cylindrical shell with an attached weight and combination resonance of a liquid storage circular cylindrical shell without attached weights are analyzed. Analytical results show good agreement with experimental results.

scholarly journals A Shell Description of Beams Incorporating Transverse Thickness Strain Energies for Receding Contact

2020 ◽  
Vol 87 (5) ◽  
Author(s):  
Adam R. Brink ◽  
Allen T. Mathis ◽  
D. Dane Quinn
Keyword(s):  
Shell Theory ◽  
Weak Form ◽  
Momentum Balance ◽  
Rigid Surface ◽  
Thickness Strain ◽  
Cambridge University ◽  
Receding Contact ◽  
Order Representation ◽  
Shell Equations ◽  
Geometrically Exact Shell

Abstract The geometrically exact nonlinear deflection of a beamshell is considered here as an extension of the formulation derived by Libai and Simmonds (1998, The Nonlinear Theory of Elastic Shells, Cambridge University Press, Cambridge, UK) to include deformation through the thickness of the beam, as might arise from transverse squeezing loads. In particular, this effect can lead to receding contact for a uniform beamshell resting on a smooth, flat, rigid surface; traditional shell theory cannot adequately such behavior. The formulation is developed from the weak form of the local equations for linear momentum balance, weighted by an appropriate tensor. Different choices for this tensor lead to both the traditional shell equations corresponding to linear and angular momentum balance, as well as the additional higher-order representation for the squeezing deformation. In addition, conjugate strains for the shell forces are derived from the deformation power, as presented by Libai and Simmonds. Finally, the predictions from this approach are compared against predictions from the finite element code abaqus for a uniform beam subject to transverse applied loads. The current geometrically exact shell model correctly predicts the transverse shell force through the thickness of the beamshell and is able to describe problems that admit receding contact.

Large Deflection Stability of Spherical Shells With Ring Loads

1986 ◽  
Vol 53 (4) ◽  
pp. 897-901 ◽  
Author(s):  
J. Cagan ◽  
L. A. Taber
Keyword(s):  
Large Deflection ◽  
Local Buckling ◽  
Point Load ◽  
Spherical Shells ◽  
Peak Loads ◽  
Ring Radius ◽  
Nonlinear Shell ◽  
Shell Equations ◽  
Transition Type ◽  
Good Agreement

Large deflections of shallow and deep spherical shells under ring loads are studied. The axisymmetric problem is solved through a Newton-Raphson technique on discretized nonlinear shell equations. Comparison of computed load-deflection curves to experimental data from both thick and thin shells generally shows good agreement in peak loads and the type of instability. For a point load, the load increases monotonically with deflection; as the ring radius increases, transition-type (snap-through) and then local buckling occurs. In addition, the pre- and post-buckled mechanical behaviors of the shell are examined.

Elliptical discontinuities in spherical shells

1967 ◽  
Vol 2 (1) ◽  
pp. 34-42 ◽  
Author(s):  
F A Leckie ◽  
D J Payne ◽  
R K Penny
Keyword(s):  
Pressure Vessels ◽  
Mathieu Functions ◽  
Flat Plates ◽  
Spherical Shells ◽  
Simple Method ◽  
Factors Affecting ◽  
Concentration Factors ◽  
Shell Equations ◽  
Stress Concentration Factors ◽  
Theoretical Results

In Part 1 shallow-shell equations expressed in elliptic co-ordinates have been solved in terms of Mathieu functions. Boundary conditions for the rigid insert and for the unreinforced hole are discussed in some detail. Results for an unreinforced opening are compared with experiment and satisfactory agreement is obtained. In Part 2 a parametric study has been made of the factors affecting the stresses at rigid inclusions and at unreinforced holes of elliptical shape in spherical shells. This work makes possible a simple method of representing the results in terms of the stress-concentration factors in flat plates, and a single geometrical variable. The results are verified by experiment. On the basis of this study predictions of the stresses at the intersection of non-radial nozzles in pressure vessels have been made. The theoretical results are compared with experimental evidence already available and reasonable agreement is found.

A Method for Rotation of Spherical Shells by Any Law with a Small Center Beating.

1994 ◽  
Vol 372 ◽  
Author(s):  
A. V. Veselov ◽  
A. G. Golubinskij ◽  
A. V. Zaharov ◽  
A. V. Malkov
Keyword(s):  
High Speed ◽  
High Accuracy ◽  
Spherical Shells ◽  
Shell Surface ◽  
Accuracy Of Measurements ◽  
Thickness Variations ◽  
Automated Installation ◽  
Wide Access

AbstractTo provide control of the whole surface the shell should be rotated. But the shell center beating within 1 μm gives thickness variations from 90 to 1200 Å. For high accuracy of measurements it is necessary to decrease the shell center beating upon rotation down to 0.1–0.2 μm. This problem can be solved using the proposed method for rotation of shells.Several units for rotation of shells have been developed and investigated. Those laws of rotation have been selected which allow to inspect the whole surface of the sell.Obvious advantages of the proposed method are the following: ease of installation and removal of the shell in the unit, automated installation of the shell in the desired position, high speed of rotation, small beating of the shell center during its rotation, wide access to the shell surface to be controlled, any required law of the shell rotation.

Effects of Vibration on the Preparation of Ultrathin Sections

1973 ◽  
Vol 31 ◽  
pp. 358-359
Author(s):  
Joseph M. Blum ◽  
Edward P. Gargiulo ◽  
J. R. Sawers
Keyword(s):  
Cutting Speed ◽  
Ground Level ◽  
Environmental Vibration ◽  
Block Face ◽  
Ultrathin Sections ◽  
Embedding Medium ◽  
Thickness Variations ◽  
Building Walls ◽  
Cutting Direction ◽  
Diamond Knife

It is now well-known that chatter (Figure 1) is caused by vibration between the microtome arm and the diamond knife. It is usually observed as a cyclical variation in “optical” density of an electron micrograph due to sample thickness variations perpendicular to the cutting direction. This vibration might be induced by using too large a block face, too large a clearance angle, excessive cutting speed, non-uniform embedding medium or microtome vibration. Another prominent cause is environmental vibration caused by inadequate building construction. Microtomes should be installed on firm, solid floors. The best floors are thick, ground-level concrete pads poured over a sand bed and isolated from the building walls. Even when these precautions are followed, we recommend an additional isolation pad placed on the top of a sturdy table.

Various technic for the measurement of step thickness on crystal surfaces

1978 ◽  
Vol 36 (1) ◽  
pp. 438-439
Author(s):  
C. Boulesteix ◽  
C. Colliex ◽  
C. Mory ◽  
B. Pardo ◽  
D. Renard
Keyword(s):  
Critical Thickness ◽  
Symmetric Case ◽  
Transmitted Intensity ◽  
Crystal Surfaces ◽  
Bright Field ◽  
Excited Wave ◽  
Highly Sensitive ◽  
Contrast Mechanisms ◽  
Step Thickness ◽  
Thickness Variations

Contrast mechanisms, which are responsible of the various types of image formation, are generally thickness dependant. In the following, two imaging modes in the 100 kV CTEM are described : they are highly sensitive to thickness variations and can be used for quantitative estimations of step heights.Detailed calculations (1) of the bright-field intensity have been carried out in the 3 (or 2N+l)-beam symmetric case. They show that in given conditions, the two important symmetric Bloch waves interfere most strongly at a critical thickness for which they have equal emergent amplitudes (the more excited wave at the entrance surface is also the more absorbed). The transmitted intensity I for a Nd2O3 specimen has been calculated as a function of thickness t. The capacity of the method to detect a step and measure its height can be more clearly deduced from a plot of dl/Idt as shown in fig. 1.

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