scholarly journals Closed-Form Solution for Circular Membranes under In-Plane Radial Stretching or Compressing and Out-of-Plane Gas Pressure Loading

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1238
Author(s):  
Bin-Bin Shi ◽  
Jun-Yi Sun ◽  
Ting-Kai Huang ◽  
Xiao-Ting He
Keyword(s):  
Closed Form ◽  
Plane Stress ◽  
Large Deflection ◽  
Closed Form Solution ◽  
Form Solution ◽  
Gas Pressure ◽  
Circular Membrane ◽  
Pressure Loading ◽  
Out Of Plane ◽  
Large Deflection Problem

The large deflection phenomenon of an initially flat circular membrane under out-of-plane gas pressure loading is usually involved in many technical applications, such as the pressure blister or bulge tests, where a uniform in-plane stress is often present in the initially flat circular membrane before deflection. However, there is still a lack of an effective closed-form solution for the large deflection problem with initial uniform in-plane stress. In this study, the problem is formulated and is solved analytically. The initial uniform in-plane stress is first modelled by stretching or compressing an initially flat, stress-free circular membrane radially in the plane in which the initially flat circular membrane is located, and based on this, the boundary conditions, under which the large deflection problem of an initially flat circular membrane under in-plane radial stretching or compressing and out-of-plane gas pressure loading can be solved, are determined. Therefore, the closed-form solution presented in this paper can be applied to the case where the initially flat circular membrane may, or may not, have a uniform in-plane stress before deflection, and the in-plane stress can be either tensile or compressive. The numerical example conducted shows that the closed-form solution presented has satisfactory convergence.

scholarly journals A Closed-Form Solution for the Boundary Value Problem of Gas Pressurized Circular Membranes in Contact with Frictionless Rigid Plates

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1017
Author(s):  
Dong Mei ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Xiao-Ting He
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Gas Pressure ◽  
Circular Membrane ◽  
Rigid Plate ◽  
Pressure Loading ◽  
Numerical Example

In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such that it comes into contact with a frictionless rigid plate, resulting in a restriction on the maximum deflection of the deflected circular membrane. The power series method was employed to solve the boundary value problem of the resulting nonlinear differential equation, and a closed-form solution of the problem addressed here was presented. The difference between the axisymmetric deformation caused by gas pressure loading and that caused by gravity loading was investigated. In order to compare the presented solution applying to gas pressure loading with the existing solution applying to gravity loading, a numerical example was conducted. The result of the conducted numerical example shows that the two solutions agree basically closely for membranes lightly loaded and diverge as the external loads intensify.

scholarly journals A New Solution to Well-Known Hencky Problem: Improvement of In-Plane Equilibrium Equation

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 653
Author(s):  
Xue Li ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Shou-Zhen Li ◽  
Xiao-Ting He
Keyword(s):  
Closed Form ◽  
Equilibrium Equation ◽  
Circumferential Stress ◽  
Closed Form Solution ◽  
Form Solution ◽  
Circular Membrane ◽  
Equilibrium Equations ◽  
Power Series Method ◽  
Out Of Plane ◽  
Plane Equilibrium

In this paper, the well-known Hencky problem—that is, the problem of axisymmetric deformation of a peripherally fixed and initially flat circular membrane subjected to transverse uniformly distributed loads—is re-solved by simultaneously considering the improvement of the out-of-plane and in-plane equilibrium equations. In which, the so-called small rotation angle assumption of the membrane is given up when establishing the out-of-plane equilibrium equation, and the in-plane equilibrium equation is, for the first time, improved by considering the effect of the deflection on the equilibrium between the radial and circumferential stress. Furthermore, the resulting nonlinear differential equation is successfully solved by using the power series method, and a new closed-form solution of the problem is finally presented. The conducted numerical example indicates that the closed-form solution presented here has a higher computational accuracy in comparison with the existing solutions of the well-known Hencky problem, especially when the deflection of the membrane is relatively large.

scholarly journals A Closed-Form Solution without Small-Rotation-Angle Assumption for Circular Membranes under Gas Pressure Loading

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2269
Author(s):  
Xiao-Ting He ◽  
Xue Li ◽  
Bin-Bin Shi ◽  
Jun-Yi Sun
Keyword(s):  
Closed Form ◽  
Rotation Angle ◽  
Closed Form Solution ◽  
Form Solution ◽  
Gas Pressure ◽  
Pressure Loading ◽  
Large Rotation ◽  
Closed Form Solutions ◽  
Film Substrate ◽  
Small Rotation

The closed-form solution of circular membranes subjected to gas pressure loading plays an extremely important role in technical applications such as characterization of mechanical properties for freestanding thin films or thin-film/substrate systems based on pressured bulge or blister tests. However, the only two relevant closed-form solutions available in the literature are suitable only for the case where the rotation angle of membrane is relatively small, because they are derived with the small-rotation-angle assumption of membrane, that is, the rotation angle θ of membrane is assumed to be small so that “sinθ = 1/(1 + 1/tan2θ)1/2” can be approximated by “sinθ = tanθ”. Therefore, the two closed-form solutions with small-rotation-angle assumption cannot meet the requirements of these technical applications. Such a bottleneck to these technical applications is solved in this study, and a new and more refined closed-form solution without small-rotation-angle assumption is given in power series form, which is derived with “sinθ = 1/(1 + 1/tan2θ)1/2”, rather than “sinθ = tanθ”, thus being suitable for the case where the rotation angle of membrane is relatively large. This closed-form solution without small-rotation-angle assumption can naturally satisfy the remaining unused boundary condition, and numerically shows satisfactory convergence, agrees well with the closed-form solution with small-rotation-angle assumption for lightly loaded membranes with small rotation angles, and diverges distinctly for heavily loaded membranes with large rotation angles. The confirmatory experiment conducted shows that the closed-form solution without small-rotation-angle assumption is reliable and has a satisfactory calculation accuracy in comparison with the closed-form solution with small-rotation-angle assumption, particularly for heavily loaded membranes with large rotation angles.

scholarly journals A Revisit of the Boundary Value Problem for Föppl–Hencky Membranes: Improvement of Geometric Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 631 ◽  
Author(s):  
Yong-Sheng Lian ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Xiao-Ting He ◽  
Zhou-Lian Zheng
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Circular Membrane ◽  
Computational Accuracy ◽  
Power Series Method ◽  
Membrane Problem ◽  
Geometric Equation

In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption of the membrane was given up. In particular, a more effective geometric equation was, for the first time, established to replace the classic one, and finally the resulting new boundary value problem due to the improvement of geometric equation was successfully solved by the power series method. The conducted numerical example indicates that the closed-form solution presented in this study has higher computational accuracy in comparison with the existing solutions of the well-known Föppl–Hencky membrane problem. In addition, some important issues were discussed, such as the difference between membrane problems and thin plate problems, reasonable approximation or assumption during establishing geometric equations, and the contribution of reducing approximations or relaxing assumptions to the improvement of the computational accuracy and applicability of a solution. Finally, some opinions on the follow-up work for the well-known Föppl–Hencky membrane were presented.

scholarly journals Axisymmetric Large Deflection Elastic Analysis of Hollow Annular Membranes under Transverse Uniform Loading

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1770
Author(s):  
Jun-Yi Sun ◽  
Qi Zhang ◽  
Xue Li ◽  
Xiao-Ting He
Keyword(s):  
Closed Form ◽  
Circular Plate ◽  
Large Deflection ◽  
Closed Form Solution ◽  
Circular Ring ◽  
Form Solution ◽  
Elastic Response ◽  
Geometrically Nonlinear ◽  
Closed Form Solutions ◽  
Annular Membrane

The anticipated use of a hollow linearly elastic annular membrane for designing elastic shells has provided an impetus for this paper to investigate the large deflection geometrically nonlinear phenomena of such a hollow linearly elastic annular membrane under transverse uniform loads. The so-called hollow annular membranes differ from the traditional annular membranes available in the literature only in that the former has the inner edge attached to a movable but weightless rigid concentric circular ring while the latter has the inner edge attached to a movable but weightless rigid concentric circular plate. The hollow annular membranes remove the transverse uniform loads distributed on “circular plate” due to the use of “circular ring” and result in a reduction in elastic response. In this paper, the large deflection geometrically nonlinear problem of an initially flat, peripherally fixed, linearly elastic, transversely uniformly loaded hollow annular membrane is formulated, the problem formulated is solved by using power series method, and its closed-form solution is presented for the first time. The convergence and effectiveness of the closed-form solution presented are investigated numerically. A comparison between closed-form solutions for hollow and traditional annular membranes under the same conditions is conducted, to reveal the difference in elastic response, as well as the influence of different closed-form solutions on the anticipated use for designing elastic shells.

Large Deflection Behavior of Composite Laminated Plates under End-Shortening and Normal Pressure Loading Using a Semi-Energy Finite Strip Method

2011 ◽  
Vol 471-472 ◽  
pp. 432-437
Author(s):  
Hamid Reza Ovesy ◽  
Mohammad Homayoun Sadr-Lahidjani ◽  
Mohammad Hajikazemi ◽  
Hassan Assaee
Keyword(s):  
Large Deflection ◽  
Closed Form Solution ◽  
Normal Pressure ◽  
Form Solution ◽  
Laminated Plates ◽  
Pressure Loading ◽  
Finite Strip ◽  
Composite Laminated Plates ◽  
Strip Method ◽  
Finite Strip Method

In this paper, the application of previously the semi energy finite strip method (FSM) for the non-linear post-buckling analysis of rectangular anti-symmetric laminates is extended to include the effects of normal pressure loading in addition to the progressive end-shortening. One of the main advantages of the semi-energy FSM is that it is based on the closed form solution of von Kármán’s compatibility equation. The developed finite strip method is applied to analyze the large deflection behavior of anti-symmetric angle ply composite laminated plates with simply supported boundary conditions at its loaded ends. To validate the results, they are compared with those obtained from finite element method (FEM) of analysis.

Thermal Induced Stresses in Bridge-Wire Initiator Glass-to-Metal Seals

2006 ◽  
Vol 129 (3) ◽  
pp. 300-306 ◽  
Author(s):  
Luke M. Thompson ◽  
Michael R. Maughan ◽  
Karl K. Rink ◽  
Donald M. Blackketter ◽  
Robert R. Stephens
Keyword(s):  
Closed Form ◽  
Plane Stress ◽  
Thermal Stresses ◽  
Closed Form Solution ◽  
Form Solution ◽  
Manufacturing Processes ◽  
Material Degradation ◽  
Element Analysis ◽  
Parametric Plane ◽  
Lumped Models

Cracks have been observed in the insulating glass of bridge-wire initiators that may allow moisture to penetrate the assembly, potentially leading to the corrosion and degradation of the bridge wire and the pyrotechnic material. Degradation of the pyrotechnic or the bridge wire may result in initiator failure or diminished performance. The goal of this research is to determine if the manufacturing processes could produce thermal stresses great enough to crack the glass. A parametric plane stress closed-form solution was used to determine the effects of changing material properties and dimensions of the initiator, and to determine potential stresses within the initiator from two different manufacturing scenarios. To verify and expand the plane stress closed-form solution, a two-dimensional axisymmetric finite element analysis was performed. To reproduce the two manufacturing scenarios, lumped models and models that included the effects of cooling the initiator were used. Both models showed that if the manufacturing process involves pouring molten glass into the initiator, the potential for cracking exists. Furthermore, if the surface of the initiator cools faster than the center, cracking is more likely.

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