Finite displacements of annular elastic membranes

Grabmüller, Hans; Weinitschke, H. J.

doi:10.1007/bf00043581

Cited by 20 publications

(18 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The starting point for the work on elastic membranes [4] was a representation theorem, expressing the solution of the finite deflection annular membrane problem explicitly in terms of a solution of the circular membrane problem [7]. Here, it is assumed that either the radial stress or or the radial displacement u is precribed at the outer edge, while the inner edge is assumed free of traction, that is, 0r(£) = O.…”

Section: Nonlinear Membrane Problemsmentioning

confidence: 99%

Boundary Layer Analysis for Some Problems of Singular Domain Perturbation

Weinitschke

1988

Trends in Applications of Mathematics to Mechanics

View full text Add to dashboard Cite

This article is concerned with analyzing the effect of small circular holes in a plane domain on the solution of certain nonlinear boundary value problems. Apart from its intrinsic mathematical interest. this problem has various applications. Examples include the stress concentration in thin elastic membranes and shells at the edge of a small hole. and the buckling of plates and shells. which may be of the bifurcation or snapping type. It is usually much simpler to calculate stresses and critical loads for a shell without hole. Therefore. it is of interest to assess asymptotically the effect of the singular domain perturbation on the stresses and on the critical loads. Another application relates to tubular chemical reactors. where internal cooling pipes of small diameter are inserted into the reactor vessel.The study of this type of problems by asymptotic methods was initiated by LAGERSTROM and CASTEN [1]. A model for incompressible flow at low Reynolds number is given by the boundary value problem n-l , In the past few years. investigation of this type of problems has been continued by WAN, LANGE. and this writer [4,5.6], with a view towards new applications to non-linear problems. whose solutions depend on an additional parameter ~. Of interest are not only regular solution branches. but also critical points at ~

show abstract

Section: Nonlinear Membrane Problemsmentioning

confidence: 99%

Boundary Layer Analysis for Some Problems of Singular Domain Perturbation

Weinitschke

1988

Trends in Applications of Mathematics to Mechanics

View full text Add to dashboard Cite

show abstract

“…In particular, the radial stress o, and the radial displacement u are given, in terms of y(x), by on= cly(x), u = c2x(xy'(x) + (1 -v)y(x)), with certain constants c~ > 0. We refer the reader to [4] and [5] for more details and for an account of earlier work on this problem. Physical reasoning suggests to prescribe or or u at the inner and outer edges.…”

Section: ) G(x T (T) Dr P--g-- Po= Max Lp(t) Imentioning

confidence: 99%

“…

The study of axisymmetric deformations of annular membranes under normal surface loads within the framework of the FSppl-Hencky small finite-deflection theory is continued after progress in this field has been made in the recent work of Grabmtiller and Weinitschke [5]. When a radial displacement is applied at the inner edge and a radial tension or a displacement at the outer edge, the mathematical question of uniqueness of tensile solutions of the resulting nonlinear boundary value problems has not been settled yet.

…”

mentioning

confidence: 99%

Nonlinear boundary value problems for the annular membrane: A note on uniqueness of positive solutions

Grabmüller

Novak

1987

J Elasticity

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is only in the 0(e2) terms that the solution /x, gx is needed (as c3, c4 enter into the calculation of B2, C2). In the case of a membrane, a rigorous proof for lim e_0SN(e) = 2 has been obtained in [7]. We have not attempted to extend the method of [7] to the present problem.…”

mentioning

confidence: 99%

Asymptotic solutions for finite deformation of thin shells of revolution with a small circular hole

Weinitschke¹,

Lange²

1987

Quart. Appl. Math.

Self Cite

View full text Add to dashboard Cite

Abstract. The method of matched asymptotic expansions is used to describe the finite deformation of thin shells of revolution with a small circular hole at the apex. The loading is assumed to be a rotationally symmetric, smoothly varying normal pressure. The mathematical problem is of singular perturbation type characterized by a boundary layer region at the inner edge of the small hole. The analytical results are compared with numerical approximations, and formulas for the stress concentration factors at the hole are presented.1. Introduction. In the field of linear and nonlinear elasticity of thin structures, many important problems whose solutions are of considerable complexity can be analyzed by boundary layer methods. This involves a small parameter 8, which is usually related to the ratio of the shell thickness h to a shell length L such as the radius of a spherical shell. In the present paper, we discuss nonlinear shell problems where the small parameter e is the ratio of the radius of a small hole at the apex of a shell of revolution to the radius of the

show abstract

Finite displacements of annular elastic membranes

Cited by 20 publications

References 12 publications

Boundary Layer Analysis for Some Problems of Singular Domain Perturbation

Boundary Layer Analysis for Some Problems of Singular Domain Perturbation

Nonlinear boundary value problems for the annular membrane: A note on uniqueness of positive solutions

Asymptotic solutions for finite deformation of thin shells of revolution with a small circular hole

Contact Info

Product

Resources

About