Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity

Balobanov, Viacheslav; Niiranen, Jarkko

doi:10.1016/j.cma.2018.04.028

Cited by 76 publications

(51 citation statements)

References 42 publications

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“…We remark that we have not observed any locking phenomenon in the presented codes. It could be interesting to compare our physically based discretisation with that presented in Balobanov and Niiranen, which has the same interesting property. Remarkable are the results presented in Yaghoubi et al We will accept the challenge to produce simulations based on Hencky‐type discretisation, which can reproduce the dynamical effects which cited papers allows to treat in the dynamical case.…”

Section: Closing Remarks and Future Challengesmentioning

confidence: 92%

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Turco

Dell’Isola²,

Misra

2019

Num Anal Meth Geomechanics

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Summary We formulate a discrete Lagrangian model for a set of interacting grains, which is purely elastic. The considered degrees of freedom for each grain include placement of barycenter and rotation. Further, we limit the study to the case of planar systems. A representative grain radius is introduced to express the deformation energy to be associated to relative displacements and rotations of interacting grains. We distinguish inter‐grains elongation/compression energy from inter‐grains shear and rotations energies, and we consider an exact finite kinematics in which grain rotations are independent of grain displacements. The equilibrium configurations of the grain assembly are calculated by minimization of deformation energy for selected imposed displacements and rotations at the boundaries. Behaviours of grain assemblies arranged in regular patterns, without and with defects, and similar mechanical properties are simulated. The values of shear, rotation, and compression elastic moduli are varied to investigate the shapes and thicknesses of the layers where deformation energy, relative displacement, and rotations are concentrated. It is found that these concentration bands are close to the boundaries and in correspondence of grain voids. The obtained results question the possibility of introducing a first gradient continuum models for granular media and justify the development of both numerical and theoretical methods for including frictional, plasticity, and damage phenomena in the proposed model.

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Section: Closing Remarks and Future Challengesmentioning

confidence: 92%

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Turco

Dell’Isola²,

Misra

2019

Num Anal Meth Geomechanics

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“…It is well known that, in the classical Timoshenko formulations in terms of deflection and rotation, shear energy due to shear strain (i.e., the difference between rotation and gradient of deflection) leads to the socalled numerical shear locking phenomena in standard finite element methods. Traditionally, shear locking has been avoided by introducing reduced numerical integration for the shear term, whereas recently the problem has been overcome by reformulations based on change of variables, for both Timoshenko beams and Reissner-Mindlin plates [44,45]. The present three-variable formulation (2), with shear strain of the form ϕ − θ, should not be prone to numerical locking either.…”

Section: Kinematics and Deformation Energymentioning

confidence: 99%

Large deformations of Timoshenko and Euler beams under distributed load

Corte

Battista

Dell’Isola

et al. 2019

Z. Angew. Math. Phys.

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In this paper, the general equilibrium equations for a geometrically nonlinear version of the Timoshenko beam are derived from the energy functional. The particular case in which the shear and extensional stiffnesses are infinite, which correspond to the inextensible Euler beam model, is studied under a uniformly distributed load. All the global and local minimizers of the variational problem are characterized, and the relative monotonicity and regularity properties are established. MathematicsSubject Classification. 74B20, 34B15, 49J45. Keywords. Nonlinear elasticity, Timoshenko beam, Euler beam, Stability of solutions of nonlinear ODEs.the case of concentrated load. Afterward, some numerical results for the inextensible Euler beam under distributed load were published in [37,38]. The paper is organized as follows: in Sect. 2 a nonlinear version of the extensible Timoshenko beam model is introduced and the problem of a clamped-free beam is formulated. Euler-Lagrange equations are formally derived. In Sect. 3, some numerical results concerning curled equilibrium configurations are shown. These results motivate the analytical study of the properties of the equilibrium solutions done in Sect. 4, in which the Euler-Lagrange equation is studied in the particular case of an infinite shear stiffness, which leads to the nonlinear Euler beam.

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“…However, most of these contributions focus on model derivations and analytical solutions for academic benchmark problems without enlarging the domain of practical applications, including numerical methods enabling the analysis of complex systems beyond the simplest benchmarks. Some exceptions can be found in [25,26,27], and this contribution is aimed to serve as another point of view in this direction.…”

Section: Introductionmentioning

confidence: 99%

Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: Applications to sandwich beams and auxetics

Khakalo

Balobanov

Niiranen

2018

International Journal of Engineering Science

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155

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The present work is devoted to the modelling of strongly size-dependent bending, buckling and vibration phenomena of 2D triangular lattices with the aid of a simplified first strain gradient elasticity continuum theory. As a start, the corresponding generalized Bernoulli-Euler and Timoshenko sandwich beam models are derived. The effective elastic moduli corresponding to the classical theory of elasticity are defined by means of a computational homogenization technique. The two additional length scale parameters involved in the models, in turn, are validated by matching the lattice response in benchmark problems for static bending and free vibrations calibrating the strain energy and inertia gradient parameters, respectively. It is demonstrated as well that the higher-order material parameters do not depend on the problem type, boundary conditions or the specific beam formulation. From the application point of view, it is first shown that the bending rigidity, critical buckling load and eigenfrequencies strongly depend on the lattice microstructure and these dependencies are captured by the generalized Bernoulli-Euler beam model. The relevance of the Timoshenko beam model is then addressed in the context of thick beams and sandwich beams. Applications to auxetic strut lattices demonstrate a significant increase in the stiffness of the metamaterial combined with a clear decrease in mass. Furthermore, with the introduced generalized beam finite elements, essential savings in the computational costs in computational structural analysis can be achieved. For engineering applications of architectured materials or structures with a microstructure utilizing triangular lattices, generalized mechanical properties are finally provided in a form of a design table for a wide range of mass densities.

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Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity

Cited by 76 publications

References 42 publications

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Large deformations of Timoshenko and Euler beams under distributed load

Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: Applications to sandwich beams and auxetics

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