Asymptotic–numerical methods and Pade approximants for non‐linear elastic structures

Cochelin, Bruno; Damil, Noureddine; Potier‐Ferry, Michel

doi:10.1002/nme.1620370706

Cited by 195 publications

(158 citation statements)

References 18 publications

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“…0 value in order to highlight a classical result of AMN which is an accumulation of steps at damage inception which correspond to stiff damage rate evolution. This has been already underlined in previous works dealing with other regularized problems [3,8,13]. Figure 6 where the damage rate has been plotted.…”

Section: Numerical Resultssupporting

confidence: 67%

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Solving fracture problems using an asymptotic numerical method

Daridon

Wattrisse

Chrysochoos

et al. 2011

Computers & Structures

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The present work deals with the use of asymptotic numerical methods (ANM) to manage crack onset and crack growth in the framework of Continuum Damage Mechanics (CDM). More specifically, an application of regularization techniques to a 1D cohesive model is proposed. The standard "triangle" damageable elastic model, often used in finite element codes to describe fracture of brittle materials, was chosen. Results associated with load-unload cycle showed that ANM is convenient to take numerically this specific non regular behaviour into account.Moreover, the present paper also shows that the chosen damageable interface model can be introduced in the generalized standard material formalism which unables us to define a complete energy balance associated with the damage process. In such a framework, the new damage state variable is a displacement. Finally, a 1D finite element application to a simple elastic damageable structure is shown to emphasize the potentialities of such an approach.

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Section: Numerical Resultssupporting

confidence: 67%

“…ANM has been widely applied to smooth partial differential equations such as non-linear elasticity [2,3] and Newtonian fluid mechanics [4,5]. It cannot be used directly for non smooth models such as unilateral contact, plasticity or damage, because the Taylor series exists only if the governing equations are defined by smooth functions.…”

Section: Introductionmentioning

confidence: 99%

Solving fracture problems using an asymptotic numerical method

Daridon

Wattrisse

Chrysochoos

et al. 2011

Computers & Structures

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“…The left hand side of Equation (23) is similar to Equation (22), hence is singular. The right hand side vector Q is the pseudo load vector given by Equation (24), and it is used to determine the post-buckling displacements 2 U,…”

Section: Asymptotic Post-buckling Analysis In Finite Element Formmentioning

confidence: 99%

“…To obtain these, W j is substituted into Equation (22), and the resulting expression is differentiated to become…”

Section: Sensitivity Analysis Of (Simple) Eigenvectorsmentioning

confidence: 99%

Post‐buckling optimization of composite structures using Koiter's method

Henrichsen

Lindgaard

Lund

2016

Numerical Meth Engineering

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SUMMARYThin-walled structures, when compressed, are prone to buckling. To fully utilize the capabilities of such structures, the post-buckling response should be considered and optimized in the design process. This work presents a novel method for gradient based design optimization of the post-buckling performance of structures. The post-buckling analysis is based on Koiter's asymptotic method. To perform gradient based optimization, the design sensitivities of the Koiter factors are derived and new design optimization formulations based on the Koiter factors are presented. The proposed optimization formulations are demonstrated on a composite square plate and a curved panel where the post-buckling stability is optimized. Copyright c 0000 John Wiley & Sons, Ltd. Thin-walled structures are often designed such that buckling does not occur during service. To ensure a structure does not buckle, the specified buckling load is often much greater than the design load. If a structure can be allowed to buckle during operation, thus operating in the post-buckling regime, it enables the possibility to design lighter and more efficient structures. To enable such an approach, the engineer must optimize the post-buckling response of the structure. Fiber reinforced polymers are ideally suited for such design tasks, as these materials allow a high degree of tailoring of the considered structure, and thus applied here for the post-buckling design optimization of structures. When optimizing structures, robustness of the resulting structure is of major importance, as the imperfection sensitivity can increase during the optimization process [1]. One method is to collect these into an equivalent geometric imperfection and use that to evaluate the knockdown in performance. Refs. [2,3] demonstrate robust design optimization by combining "worst" shape imperfection optimization and laminate optimization, thereby efficiently decreasing the imperfection sensitivity of laminated composite structures. A different approach is to handle all imperfections simultaneously by modeling the uncertainties using statistics, therefore quantifying * Correspondence to: Department of Mechanical and Manufacturing Engineering, Aalborg University (AAU), the imperfections arising from material, geometry, load etc., and perform robust buckling optimization based on the uncertainties, see e.g., [4] for a review of different approaches. Many textbooks describe the coupling between imperfections, buckling load factor, and post-buckling stability, see e.g., [5]. Focusing on simple i.e., distinct buckling load factors, the sensitivity towards imperfections relates to the stability of the post-buckling response. Generally speaking, a stable post-buckled structure is less sensitive towards imperfections than an unstable post-buckled structure. Because of these considerations, the post-buckling stability can be utilized to design robust structures.Post-buckling analysis of plates and shells has been subject to much research. Driven by the aerospace industry, large re...

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“…Different FE-methods were applied with respect to special purposes (global deformation analysis, stress analysis, first ply failure criteria, local stress singularities, crack analysis), see e.g. [2,8]. By admitting a priori delaminations between plies and regarding their growth, load depending disturbances of the perfect geometry of cross-sections are growing and diminish the critical load within stability analysis.…”

Section: Introductionmentioning

confidence: 99%

Theory and Computation of Multilayer Composites

Stein

Teßmer

1999

Mechanics of Composite Materials and Structures

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Abstract. An hierarchical concept for the analysis of thin-walled composite shells is presented within the finite element method for nonlinear deformation of thin-walled composite structures. For non-disturbed subdomains of the structure an effective 4-node shell element with 5 or 6 d.o.f. per node is used for the whole laminate. For the analysis of 3D-stress states a multidirector shell element along the thickness with piecewise polynomials is presented. n physical layers are approximated by N numerical layers, discretized with hierarchical trial-and test functions. The coupling of both elements is performed by special transition elements. An example shows the technique and efficiency of coupling both types of elements.

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Asymptotic–numerical methods and Pade approximants for non‐linear elastic structures

Cited by 195 publications

References 18 publications

Solving fracture problems using an asymptotic numerical method

Solving fracture problems using an asymptotic numerical method

Post‐buckling optimization of composite structures using Koiter's method

Theory and Computation of Multilayer Composites

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