The Nonlinear Theory of Elastic Shells

Libai, A.; Simmonds, J. G.

doi:10.1017/cbo9780511574511

Cited by 289 publications

(310 citation statements)

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“…The potential energy functional for films made of orthotropic material is obtained in a similar manner to the isotropic case [18]. We assume that the principal directions of the material orthotropy are being aligned with the x and y coordinate axes.…”

Section: A Model Of Thin Orthotropic Plates Under Finite In-plane Strmentioning

confidence: 99%

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Disappearance of stretch-induced wrinkles of thin sheets: A study of orthotropic films

Sipos

Fehér

2016

International Journal of Solids and Structures

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Abstract. A recent paper (Healey et al., J. Nonlin. Sci., 2013, 23, 777-805.) predicted the disappearance of the stretch-induced wrinkled pattern of thin, clamped, elastic sheets by numerical simulation of the Föppl-von Kármán equations extended to the finite in-plane strain regime. It has also been revealed that for some aspect ratios of the rectangular domain wrinkles do not occur at all regardless of the applied extension. To verify these predictions we carried out experiments on thin (20 µm thick adhesive covered), previously prestressed elastomer sheets with different aspect ratios under displacement controlled pull tests. On one hand the the adjustment of the material properties during prestressing is highly advantageous as in targeted strain regime the film becomes substantially linearly elastic (which is far not the case without prestress). On the other hand a significant, non-ignorable orthotropy develops during this first extension. To enable quantitative comparisons we abandoned the assumption about material isotropy inherent in the original model and derived the governing equations for an orthotropic medium. In this way we found good agreement between numerical simulations and experimental data.Analysis of the negativity of the second Piola-Kirchhoff stress tensor revealed that the critical stretch for a bifurcation point at which the wrinkles disappear must be finite for any aspect ratio. On the contrary there is no such a bound for the aspect ratio as a bifurcation parameter. Physically this manifests as complicated wrinkled patterns with more than one highly wrinkled zones on the surface in case of elongated rectangles. These arrangements have been found both numerically and experimentally. These findings also support the new, finite strain model, since the Föppl-von Kármán equations based on infinitesimal strains do not exhibit such a behavior.

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Section: A Model Of Thin Orthotropic Plates Under Finite In-plane Strmentioning

confidence: 99%

“…The potential energy of the enitre domain follows by substituting (18) and (19) into eq.(11). After rescaling (i.e.…”

Section: A Model Of Thin Orthotropic Plates Under Finite In-plane Strmentioning

confidence: 99%

Disappearance of stretch-induced wrinkles of thin sheets: A study of orthotropic films

Sipos

Fehér

2016

International Journal of Solids and Structures

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“…This displacement is then known as the buckling displacement and the corresponding loading parameter is the critical load. This case implies that ∆W is a minimum at v c , motivating the Trefftz criterion for buckling, which states that the buckling load can be found by searching for a stationary value of ∆W (6). In fact, since we consider arbitrarily small virtual displacements, the Trefftz criterion is usually applied to the quadratic terms of ∆W , so that we search for stationary values of ∆W 2 .…”

Section: Conditions For Change In Stabilitymentioning

confidence: 99%

“…Thus we replace the middle-surface strain tensor for state I, defined γ (I) αβ , by its linearised counterpart θ (I) αβ , which is defined by 6) where the notation v α | β denotes covariant differentiation and b αβ is the second fundamental tensor of the surface. Likewise, we replace the tensor of changes of curvature ρ (I) αβ by its linearised counterpartρ αβ = w| αβ .…”

Section: Conditions For Change In Stabilitymentioning

confidence: 99%

Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity

Jones

Chapman

Allwright

2008

The Quarterly Journal of Mechanics and Applied Mathematics

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SummaryThe problem of a thin spherical linearly-elastic shell, perfectly bonded to an infinite linearly-elastic medium is considered. A constant axisymmetric stress field is applied at infinity in the matrix, and the displacement and stress fields in the shell and matrix are evaluated by means of harmonic potential functions. In order to examine the stability of this solution, the buckling problem of a shell which experiences this deformation is considered. Using Koiter's nonlinear shallow shell theory, restricting buckling patterns to those which are axisymmetric, and using the Rayleigh-Ritz method by expanding the buckling patterns in an infinite series of Legendre functions, an eigenvalue problem for the coefficients in the infinite series is determined. This system is truncated and solved numerically in order to analyse the behaviour of the shell as it undergoes buckling, and to identify the critical buckling stress in two cases -namely where the shell is hollow and the stress at infinity is either uniaxial or radial.

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“…The corresponding 2D stretch and bending vectors ε α , κ α defined by u, ψ each had three independent non-vanishing components in any vector base. This approach was developed in a number of papers and summarized in the books by Libai and Simmonds [13], Chróścielewski et al [14], and Eremeyev and Zubov [15]. Such non-linear resultant six-field shell model could be consistently linearized for small translations, rotations and 2D strain measures.…”

Section: Introductionmentioning

confidence: 99%

The resultant linear six‐field theory of elastic shells: What it brings to the classical linear shell models?

Pietraszkiewicz

2015

Z Angew Math Mech

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Key words Theory of shells, linear six-field model, static-geometric analogy, complex shell relations, Cesáro type formulas, gradients of strain measures.Basic relations of the resultant linear six-field theory of shells are established by consistent linearization of the resultant 2D non-linear theory of shells. As compared with the classical linear shell models of Kirchhoff-Love and TimoshenkoReissner type, the six-field linear shell model contains the drilling rotation as an independent kinematic variable as well as two surface drilling couples with two work-conjugate surface drilling bending measures are present in description of the shell stress-strain state. Among new results obtained here within the six-field linear theory of elastic shells there are: 1) formulation of the extended static-geometric analogy; 2) derivation of complex BVP for complex independent variables; 3) description of deformation of the shell boundary element; 4) the Cesáro type formulas and expressions for the vectors of stress functions along the shell boundary contour; 5) discussion on explicit appearance of gradients of 2D stress and strain measures in the resultant stress working.

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The Nonlinear Theory of Elastic Shells

Cited by 289 publications

References 0 publications

Disappearance of stretch-induced wrinkles of thin sheets: A study of orthotropic films

Disappearance of stretch-induced wrinkles of thin sheets: A study of orthotropic films

Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity

The resultant linear six‐field theory of elastic shells: What it brings to the classical linear shell models?

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