Spatial chaos as a governing factor for imperfection sensitivity in shell buckling

Groh, Rainer; Pirrera, Alberto

doi:10.1103/physreve.100.032205

Cited by 23 publications

(23 citation statements)

References 19 publications

(23 reference statements)

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“…Imperfection sensitive structures that usually show clustered buckling modes and clustered natural frequencies [21][22][23] have also been studied using the asymptotic expansion with multiple modes [14][15][16][17]. In such structures, small imperfections due to variations in manufacturing parameters can induce different bifurcation paths [18,19], which can be studied by Koiter's multi-modal perturbation analysis. The single-mode asymptotic expansion of Eq.…”

Section: Multi-modal Asymptotic Expansionmentioning

confidence: 99%

“…In recent years, the method has been applied in the analysis of imperfection sensitive shells [14][15][16][17]. Particularly the multi-modal formulation of Koiter's approach provides a systematic and efficient procedure to assess the nonlinear behavior of the structure in cases where several buckling modes interact, such as in structures highly optimized for buckling and imperfection-sensitive shell structures, where small imperfections due to variations in manufacturing parameters can induce different bifurcation paths [18,19], which can be studied by Koiter's perturbation analysis.…”

Section: Introductionmentioning

confidence: 99%

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Displacement-based multi-modal formulation of Koiter's method applied to cylindrical shells

Castro¹,

Jansen²

2021

Preprint

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The multi-modal formulation of Koiter's asymptotic method provides a systematic and efficient procedure to evaluate the initial post-buckling behaviour and to assess the nonlinear behavior of structures. This manuscript presents a displacement-based multi-modal formulation of Koiter's method for cylindrical shells, which are structures known for their high imperfection sensitivity and for having clustered bifurcation modes that highly interact. A third-order interpolation is used for the in-plane and out-of-plane displacements by means of the Bogner-Fox-Schmit-Castro (BFSC) element, with 4 nodes and 10 degrees-of-freedom per node, aiming at an accurate representation of the second-order fields required in the initial post-buckling analysis. The single-curvature of the shell is considered in the finite element kinematics and the study includes nonlinear kinematics from Von Kármán and Sanders. The mesh is obtained by closing the circumferentially oriented coordinate at the position where the mesh completes one revolution about the shell perimeter. The proposed formulation and implementation is verified in detail by comparing results for composite shells against established literature for multi-mode asymptotic expansions. A fast convergence of the proposed formulation is observed for linear buckling, pre-buckling state and the initial post-buckling coefficients. The developed formulation enables a close relationship between formulae and the implemented code, and is implemented using state-of-the-art collaborative software. The authors made the implemented routines in a publicly available data set with the aim to popularize Koiter's method.

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Section: Multi-modal Asymptotic Expansionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Displacement-based multi-modal formulation of Koiter's method applied to cylindrical shells

Castro¹,

Jansen²

2021

Preprint

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“…Owing to the rotational invariance of the cylinder, each unique combination of dimples can occur anywhere around the circumference, and the spatial multiplicity of these localizations implies a large set of possible trajectories to instability, with each trajectory affine to a particular imperfection signature (Groh and Pirrera, 2019b). The resulting complexity of multiple and oftentimes entangled post-buckling paths of this symmetric problem creates unique challenges for numerical methods, for the interpretation of results, and for obtaining physical insight into problem.…”

Section: Introductionmentioning

confidence: 99%

Exploring Adaptive Behavior of Non-linear Hexagonal Frameworks

et al. 2020

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Non-linear structural responses offer a rich design space that can be integrated into the development of novel (meta)-materials to enable transformative capabilities. By exploiting robust, repeatable, non-linear elasticity the (meta)-material's performance can be tailored to significantly exceed what has been demonstrated through traditional linear design paradigms. Embracing geometric non-linearity offers the designer the potential for truly bespoke large-range elastic behavior. Herein we explore the behavior of bistable elements that can be arranged into a space-filling triangular lattice, constructed in a hierarchical manner. We develop an analysis of the system that can be readily extended to more complex hierarchies, such as the hexagonal arrangement considered herein, whilst retaining physical insight. Specifically, we present the geometric restrictions of lattice elements that govern the existence of stable stress-free states and subsequently characterize the transitions between such states by searching for the minimum energetic transition pathway. Through our approach, we explore how hierarchical multi-stable systems can be analyzed, thus contributing to the development of truly bespoke adaptive (meta-)materials.

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“…In particular, the multi-modal formulation of Koiter's approach provides a systematic and efficient procedure to assess the nonlinear behavior of the structure in cases where several buckling modes interact, such as in structures highly optimized for buckling [24,25,26] and imperfectionsensitive shell structures [27,28,29,30,31,32]. In such designs, small imperfections due to variations in manufacturing parameters can induce different bifurcation paths [33,34], which can be studied by Koiter's perturbation analysis.…”

Section: Introductionmentioning

confidence: 99%

“…33 33 33 for isotropic plate with ∕ = 3; each row corresponds to one model; columns from left to right are: , ,…”

mentioning

confidence: 99%

Displacement-based formulation of Koiter’s method: application to multi-modal post-buckling finite element analysis of plates

Castro¹,

Jansen²

2020

Preprint

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Koiter’s asymptotic method enables the calculation and deep understanding of the initial post-buckling behaviour of thin-walled structures. For the single-mode asymptotic analysis, Budiansky (1974) presented a clear and general formulation for Koiter’s method, based on the expansion of the total potential energy function. The formulation from Budiansky is herein revisited and expanded for the multimodal asymptotic analysis, of primordial importance in structures with clustered bifurcation modes. Given the admittedly difficult implementation of Koiter’s method, especially for multi-modal analysis and during the evaluation of the third– and fourth–order tensors involved in Koiter’s analysis; the presented study proposes a formulation and notation with close correspondence with the implemented algorithms. The implementation is based on state-of-the-art collaborative tools: Python, NumPy and Cython. The kinematic relations are specialized using von Kármán shell kinematics, and the displacement field variables are approximated using an enhanced Bogner-Fox-Schmit (BFS) finite element, modified to reach third-order interpolation also for the in-plane displacements, using only 4 nodes per element and 10 degrees-of-freedom per node, aiming an accurate representation of the second-order fields. The formulation and implementation are verified by comparing results for isotropic and composite plates against established literature. Finally, results for multi-modal displacement fields with up to 5 modes and corresponding post-buckling factors are reported for future reference.

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Spatial chaos as a governing factor for imperfection sensitivity in shell buckling

Cited by 23 publications

References 19 publications

Displacement-based multi-modal formulation of Koiter's method applied to cylindrical shells

Displacement-based multi-modal formulation of Koiter's method applied to cylindrical shells

Exploring Adaptive Behavior of Non-linear Hexagonal Frameworks

Displacement-based formulation of Koiter’s method: application to multi-modal post-buckling finite element analysis of plates

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