Efficient formulation for dynamics of corotational 2D beams

Le, Thanh-Nam; Battini, Jean‐Marc; Hjiaj, Mohammed

doi:10.1007/s00466-011-0585-6

Cited by 89 publications

(56 citation statements)

References 13 publications

(10 reference statements)

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“…The linear damping model is a leading-order approximation to the complex dissipative nature of elastomers. The bistable potential V is numerically computed by nonlinear finite-element simulations of a quasistatically deforming, corotational, linear elastic beam in 2D (36). We validated the numerical force-displacement curves by comparison with the experimental data shown in Fig.…”

Section: Numerical Resultsmentioning

confidence: 97%

Stable propagation of mechanical signals in soft media using stored elastic energy

Raney

Nadkarni

Daraio

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

292

199

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Soft structures with rationally designed architectures capable of large, nonlinear deformation present opportunities for unprecedented, highly tunable devices and machines. However, the highly dissipative nature of soft materials intrinsically limits or prevents certain functions, such as the propagation of mechanical signals.Here we present an architected soft system composed of elastomeric bistable beam elements connected by elastomeric linear springs. The dissipative nature of the polymer readily damps linear waves, preventing propagation of any mechanical signal beyond a short distance, as expected. However, the unique architecture of the system enables propagation of stable, nonlinear solitary transition waves with constant, controllable velocity and pulse geometry over arbitrary distances. Because the high damping of the material removes all other linear, small-amplitude excitations, the desired pulse propagates with high fidelity and controllability. This phenomenon can be used to control signals, as demonstrated by the design of soft mechanical diodes and logic gates.soft | mechanical signal | stable propagation | instability S oft, highly deformable materials have enabled the design of new classes of tunable and responsive systems and devices, including bioinspired soft robots (1, 2), self-regulating microfluidics (3), adaptive optics (4), reusable energy-absorbing systems (5, 6), structures with highly programmable responses (7), and morphological computing paradigms (8). However, their highly deformable and dissipative nature also poses unique challenges. Although it has been demonstrated that the nonlinear response of soft structures can be exploited to design machines capable of performing surprisingly sophisticated functions on actuation (1, 2, 9), their high intrinsic dissipation has prevented the design of completely soft machines. Sensing and control functionalities, which require transmission of a signal over a distance, still typically rely on the integration of stiff electronic components within the soft material (10, 11), introducing interfaces that are often a source of mechanical failure.The design of soft control and sensing systems (and, consequently, completely soft machines) requires the ability to propagate a stable signal without distortion through soft media. There are two limiting factors intrinsic to materials that work against this: dispersion (signal distortion due to frequency-dependent phase velocity) and dissipation (loss of energy over time as the wave propagates through the medium). Dispersion can be controlled or eliminated through nonlinear effects produced via the control of structure in the medium (12). For example, periodic systems based on Hertzian contact (13-15), tensegrity structures (16), rigid bars and linkages (17), and bistable elastic elements (18) can behave as nondispersive media, with the nonlinearity of their local mechanical response canceling out the tendency for the signal to disperse at sufficiently large amplitudes. However, dissipation is still an ov...

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Section: Numerical Resultsmentioning

confidence: 97%

Stable propagation of mechanical signals in soft media using stored elastic energy

Raney

Nadkarni

Daraio

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

292

199

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“…The second purpose is to show that in the absence of applied external loads, the proposed algorithm conserves the linear and angular momenta. The results obtained with the present energy-momentum formulation are compared to the ones obtained with the corotational formulation proposed by Le et al [3]. In this previous formulation, the alpha method is used to solve the equations of motion.…”

Section: Numerical Examplesmentioning

confidence: 86%

“…The corotational method is an attractive approach to derive non-linear finite beam elements [1,2,3,4,5,6,7,8] . The idea is to decompose the large motion of the element into rigid body and pure deformational parts through the use of a local system which continuously rotates and translates with the element.…”

Section: Introductionmentioning

confidence: 99%

“…However, such an approach assumes that the inplane local displacements are zero, which is not accurate. For this reason, Le et al [3] used the Interpolation Interdependent Element formulation [9], and hence cubic functions, to derive both the elastic and inertia terms. They show that this formulation is more efficient than using constant mass matrices.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Energy-Momentum Method for Nonlinear Dynamic of 2d Corotational Beams

Chhang¹,

Battini²,

Sansour³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. This paper presents an energy-momentum method for nonlinear dynamics of 2D Bernoulli corotational beams. It is shown that the time stepping algorithm conserves energy, linear momentum and angular momentum. To be consistent in the corotational approach, cubic interpolations of Bernoulli element are employed to derive both inertia and elastic terms. The shallow arch strain definition is used to get an element which produce accurate results for less number of elements. To avoid membrane locking, we use a constant and average value of the axial strains. In addition, the energy-momentum method is used to preserve the conserving properties, which is able to maintain the stability and accuracy in a non-dissipative system for a long period. The midpoint velocities of kinematic fields and strains are used to tackle any non-linear form of strain displacement relations. Finally, two examples including large overall displacement are presented to illustrate the stability and efficiency of the proposed algorithms.5496

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“…Updating the mass matrix as a function of the displacements of the structure is part of future work [23].…”

Section: Governing Equations In Dynamicsmentioning

confidence: 99%

Computational Assessment of the Progressive Collapse of a Five Storey Structure Considering Two Different Building Codes

Oliveira¹,

Berke

Silveira

et al. 2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Efficient formulation for dynamics of corotational 2D beams

Cited by 89 publications

References 13 publications

Stable propagation of mechanical signals in soft media using stored elastic energy

Stable propagation of mechanical signals in soft media using stored elastic energy

Energy-Momentum Method for Nonlinear Dynamic of 2d Corotational Beams

Computational Assessment of the Progressive Collapse of a Five Storey Structure Considering Two Different Building Codes

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