Wrinkle-free solutions of circular membrane problems

Beck, Armin; Grabmüller, Hans

doi:10.1007/bf00946242

Cited by 14 publications

(15 citation statements)

References 11 publications

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“…3]), from which we obtain d = 0, and therefore jc, = x2. In general, if y < 0 is assumed, D(x) may become negative for some values of (see also [3]). Now, Eq.…”

Section: Bvp (S S) Letting D(t) := Xx(t) -X2(t)mentioning

confidence: 99%

“…The mathematical problems of existence and uniqueness of wrinkle-free solutions for curved circular membranes under variable vertical loads have been solved only recently by A. Beck and H. Grabmiiller [3], but the corresponding case of curved annular aximembranes has apparently not been investigated as yet. The present results originate from the doctoral thesis of the first author [2] with a number of considerable improvements obtained by the use of a suitable transformation of variables.…”

mentioning

confidence: 99%

“…We introduce a nonlinear operator T: E -* E and a convex set Ms s c E by setting where K{ := ||2 C(cop/k2 -p) + vp2q/{4k)\\ < 00. Using arguments as in [3,Lemma 3.1], the operator T is shown to fulfill all requirements of the Schauder fixed-point theorem (see, e.g., [20,§7,XII]) when acting on the closure of the set Ms s. Since T{E) C Ms s, there must exist at least one fixed-point x = Tx e Ms s of the required regularity of an rn-solution. Furthermore, if we let F(t) ajCp(t)/(k2(t)J7(t) + @2(0) > 0 and if y > 0 is assumed, it follows from condition (Q) that F(t, x(t))-F(t)x{t) < 0 holds for all teIQ.…”

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On finite displacements of curved annular elastic membranes without wrinkling

Beck¹,

Grabmüller²

1995

Quart. Appl. Math.

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Abstract. Axisymmetric deformations of curved annular elastic membranes subjected to vertical surface loads and radial edge loads or displacements are considered within Reissner's finite-rotation theory of thin shells of revolution, assuming linear stress-strain relations. The principal stresses in the membrane are determined by the solutions of a nonlinear second-order ODE, dependent on a geodesic variable. Analytical methods are used in order to determine the range of those boundary data for which the solutions of the differential equation are wrinkle-free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.1. Introduction. For many events, a linear theory of elastic membranes and thin shells of revolution is not applicable, for instance, if the applied load is not small enough. In these cases a nonlinear theory is required. Of particular interest are geometrically nonlinear theories where nonlinear strain-displacement relations but a linear stress-strain relation like Hooke's law are assumed.A membrane theory of this kind that is widely applied and that covers the special case of aximembranes, originates from E. Reissner's theory of thin shells of revolution

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“…3]), from which we obtain d = 0, and therefore jc, = x2. In general, if y < 0 is assumed, D(x) may become negative for some values of (see also [3]). Now, Eq.…”

Section: Bvp (S S) Letting D(t) := Xx(t) -X2(t)mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On finite displacements of curved annular elastic membranes without wrinkling

Beck¹,

Grabmüller²

1995

Quart. Appl. Math.

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“…Integral equation methods have first been applied to nonlinear membrane problems by Dickey [9] and later successfully been refined by adding concavity and monotonicity arguments. Equipped with these tools, the mathematical problems of existence and uniqueness of a stable membrane state were solved for flat circular and annular membranes not only within the small finite deflection theory [10,11,12] but also within a simplified version of the Reissner theory of finite rotations [13,14,15,16,17]. The results are summarized in [18].…”

Section: Introductionmentioning

confidence: 99%

“…Concerning the displacement boundary value problem, we employ a mapping argument which originates from R. Pirner's diploma thesis [19] and which has been improved later on by H. Grabmiiller et al [14,15,16]. The idea of this method consists of mapping the set of parameters related to a stable state of the stress problem into the parameter set associated with the displacement problem.…”

Section: Introductionmentioning

confidence: 99%

Wrinkle-free solutions in the theory of curved circular membranes

Beck

Grabmüller

1993

J Eng Math

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Rotationally symmetric deformations of a curved circular elastic membrane under a vertical surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations, the determination of the principal stresses in the membrane is shown to be equivalent to the solution of a single, second-order ODE, expressed in terms of a geodesic variable. Analytical techniques are applied in order to determine a limit curve of those boundary data, which subdivide the parameter range into complementary domains of existence and non-existence of tensile solutions. Finally, the more restricted subdomain of those boundary parameters is determined which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.

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Recent mathematical resuits in the nonlinear theory of flat and curved elastic membranes of revolution

Weinitschke

Grabmüller

1992

J Eng Math

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The present article reviews some recent developments in nonlinear elastic membrane theory with special emphasis on axisymmetric deformation of flat circular and annular membranes subjected to a vertical surface load and with prescribed radial stresses or radial displacements at the edges. The nonlinear F6ppl membrane theory of small finite deflections as well as a simplified version of Reissner's finite-rotation theory is employed, assuming linear stress-strain relations. The main analytical techniques are reported which have been applied recently in order to determine the ranges of those boundary parameters for which solutions of the relevant nonlinear boundary value problems exist, and ranges of parameters for which the principal stresses are nonnegative everywhere. Concerning plane membranes, it is shown how the mathematical theory of existence and uniqueness was nearly completed in recent works in contrast to curved membranes where references can be given to rather few results.

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Wrinkle-free solutions of circular membrane problems

Cited by 14 publications

References 11 publications

On finite displacements of curved annular elastic membranes without wrinkling

On finite displacements of curved annular elastic membranes without wrinkling

Wrinkle-free solutions in the theory of curved circular membranes

Recent mathematical resuits in the nonlinear theory of flat and curved elastic membranes of revolution

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