Large bending deformations of a cylindrical membrane with internal pressure

Kolesnikov, A. A.; Зубов, Л. М.

doi:10.1002/zamm.200800182

Cited by 12 publications

(11 citation statements)

References 16 publications

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“…Thus the equilibrium Equation can be rewritten

\frac{d L^{11}}{d s} + L^{11} false(2 Γ_{11}^{1} + Γ_{21}^{2} false) + L^{22} {normalΓ}_{22}^{1} = - ξ^{1},

0 = - ξ^{2},

L^{11} B_{11} + L^{22} B_{22} = - ξ .

…”

Section: Pure Bending Of Pressurized Curved Tubementioning

confidence: 99%

“…In undeformed state its middle surface is given by the equations

\begin{matrix} true \\ right boldr = x_{1} (s) i_{1} + x_{2} (s) i_{2} + t i_{3}, s \in [0; S], t \in [0; T] . \end{matrix}

The natural basis

r_{α}

and the components

g_{α β}

of the metric tensor have the following form

\begin{matrix} true \\ right r_{1} = x_{1}^{'} i_{1} + x_{2}^{'} i_{2}, r_{2} = i_{3}, \\ right g_{11} = {x_{1}^{'}}^{2} + {x_{2}^{'}}^{2}, g_{12} = g_{21} = 0, g_{22} = 1 . \end{matrix}

Replacing by , the problem of pure bending of the inflated straight tube can reduce to Equations , , , (or , , ) with the same boundary and end conditions. The bending of cylindrical membrane was investigated in [] using the tension‐field theory, and in [] using the ordinary membrane theory.…”

Section: Pure Bending Of Pressurized Curved Tubementioning

confidence: 99%

“…Replacing (12) by (34), the problem of pure bending of the inflated straight tube can reduce to Equations (16), (17), (27), (28) (or (16), (30), (31)) with the same boundary and end conditions. The bending of cylindrical membrane was investigated in [14,15] using the tension-field theory, and in [21] using the ordinary membrane theory.…”

Section: Pure Bending Of Pressurized Straight Tubementioning

confidence: 99%

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Bending of inflated curved hyperelastic tubes

2019

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We discuss large flexure of an inflated curved thin‐walled tube within the framework of the nonlinear theory of elastic membranes. Wrinkling of the membrane is taken into account by using a relaxed strain energy density derived from a strain energy function of incompressible hyperelastic material. We consider curved tubes with elliptical cross‐section and analyze the influence of a cross‐section shape and initial curvature of the tube at its deformation. Dependence of the bending moment on the tube curvature is obtained and discussed. In particular, the comparison with the pure bending of a cylindrical tube same material is made. We study the fact that the curving of the curved tube under in‐plane bending might be realized in two directions. Depending on the flexure direction the behavior of a curved tube can be similar or significantly different from the behavior of a cylindrical tube.

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“…Thus the equilibrium Equation can be rewritten

\frac{d L^{11}}{d s} + L^{11} false(2 Γ_{11}^{1} + Γ_{21}^{2} false) + L^{22} {normalΓ}_{22}^{1} = - ξ^{1},

0 = - ξ^{2},

L^{11} B_{11} + L^{22} B_{22} = - ξ .

…”

Section: Pure Bending Of Pressurized Curved Tubementioning

confidence: 99%

“…In undeformed state its middle surface is given by the equations

\begin{matrix} true \\ right boldr = x_{1} (s) i_{1} + x_{2} (s) i_{2} + t i_{3}, s \in [0; S], t \in [0; T] . \end{matrix}

The natural basis

r_{α}

and the components

g_{α β}

of the metric tensor have the following form

\begin{matrix} true \\ right r_{1} = x_{1}^{'} i_{1} + x_{2}^{'} i_{2}, r_{2} = i_{3}, \\ right g_{11} = {x_{1}^{'}}^{2} + {x_{2}^{'}}^{2}, g_{12} = g_{21} = 0, g_{22} = 1 . \end{matrix}

Section: Pure Bending Of Pressurized Curved Tubementioning

confidence: 99%

Section: Pure Bending Of Pressurized Straight Tubementioning

confidence: 99%

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Bending of inflated curved hyperelastic tubes

2019

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“…In particular, the constitutive equations of the membrane made of a neo Hookean [7] material are written as [8] where h is the shell thickness, and μ is the shear mod ulus.…”

Section: Set Of Equations Of Nonlinear Staticsmentioning

confidence: 99%

Large deformations of elastic shells with distributed dislocations

Зубов

2012

Dokl. Phys.

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A nonlinear theory of elastic shells of the Kirch hoff-Love type with continuously distributed disloca tions is proposed. For the principal unknowns in the set of solving equations, we accepted the components of the distortion tensor for a deforming surface. The set of equations includes the incompatibility equations in which the density of dislocations is assumed as a set function of coordinates on the surface and the equilib rium equations with given external load density. The problems of torsion and bending of a circular cylindri cal shell with dislocations were considered. These problems are reduced either to ordinary differential equations or to algebraic equations. Without external loads, we found several exact solutions on isometric deformations of shells with dislocations. It is shown that the cylindrical shell with dislocations can be twisted or bent with no resistance, i.e., without the occurrence of stresses. We also investigated the infla tion of a closed elastic sphere and the buckling of a flat membrane with dislocations.Defects of the dislocation type play a significant role in the mechanical behavior of surface crystals, nanotubes, nanofilms, and other two dimensional physical systems [1]. The distributions of dislocations can be used for modeling various other defects in the indicated systems. DENSITY OF DISLOCATIONS IN SHELLSLet σ be the elastic shell surface considered as a two dimensional material continuum in the reference configuration, i.e., in an undeformed state. We desig nate the radius vector of a point on σ as r and the Gaussian coordinates as q α (α = 1, 2). We introduce the vectors r α of the principal basis and the vectors of the mutual basis r β , as well as the gradient operator on σ:(1)Here, n is the unit vector of the normal to the surface σ. We designate the radius vector of the deformed surface Σ referred to the same curvilinear coordinates q α as R and the vectors of the principal basis, the mutual basis, and the unit normal to Σ as R α , R β , and N, respectively. Assuming that the surface distortion tensor [2, 3] F = gradR = r α ⊗ R α is a single valued and continuously differentiable function of coordinates q α , we consider the problem of determining the field of shell displace ments u = R -r from the field of the tensor F set in the multiply connected region. The vector field u(r) is determined in this case, generally speaking, ambigu ously, which assumes the presence of translational dis locations in the shell, each of which is characterized by the Burgers vector. Assuming that the number of dislo cations on a limited portion of the shell is high and fol lowing the method of [4], we pass from the discrete set of dislocations to their continuous distribution. Let us determine the dislocation density α as the vector field, the integral of which over an arbitrary region on σ is equal to the summary Burgers vector for all disloca tions contained in this region. The indicated defini tion results in the following equation with respect to the distortion:(2)Here, E is the three...

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“…The large deflection phenomena of membrane problem usually give rise to nonlinear differential equations [8][9][10][11]. These nonlinear equations generally present serious analytical difficulties when applied to boundaryvalue problems.…”

Section: Introductionmentioning

confidence: 99%

Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions

Wang

2018

Mathematical Problems in Engineering

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The problem of axisymmetric deformation of a peripherally fixed and uniformly loaded circular membrane under deflection restrictions (by a frictionless horizontal rigid plate) was analytically solved, where the assumption of constant membrane stress adopted in the existing work was given up, and a closed-form solution of this problem was presented for the first time. The numerical analysis shows that the closed-form solution presented here has higher calculation accuracy than the existing approximate solution.

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Large bending deformations of a cylindrical membrane with internal pressure

Cited by 12 publications

References 16 publications

Bending of inflated curved hyperelastic tubes

Bending of inflated curved hyperelastic tubes

Large deformations of elastic shells with distributed dislocations

Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions

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