Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

Erbaş, Barış; Kaplunov, Julius; Elishakoff, Isaac

doi:10.1177/10812865211023885

Cited by 17 publications

(13 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Comparison of mid-plane displacements performed using finite element software shows that for a sliding contact the tangential displacement significantly dominates over the vertical one, whereas for a fixed lower face both displacements seem to be of the same order. This is in line with initial physical insight, as well as observations of an extra longitudinal mode in Achenbach and Keshava [15], and also more recent findings reported for an elastic contact in Erbaş et al [27,28].…”

Section: Introductionsupporting

confidence: 91%

Elastodynamics of a coated half-space under a sliding contact

Bratov

Kaplunov

Lapatsin

et al. 2022

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

The paper deals with elastic wave propagating in a layer on a half-space induced by a vertical force. The focus is on the effect of a sliding contact along the interface and its comparative study with a perfect one. The effective boundary conditions substituting the presence of the layer are derived. The leading order term in these conditions corresponds to vertical inertia of the layer, whereas next order correction involves the effect of plate waves in the coating. Analysis of the associated dispersion relation confirms the existence of a Rayleigh-type wave, along with extensional and shear plate waves. An asymptotic hyperbolic-elliptic formulation for surface wave field is also presented. This includes a hyperbolic equation singularly perturbed by a pseudo-differential operator playing a role of a boundary condition for the elliptic equation governing decay over the interior. The sign of the coefficient at the pseudo-differential operator is demonstrated to be always negative, corresponding to a local maximum of the phase speed at zero wave number, and consequently to a distinct receding type of the Rayleigh-type wave quasi-front induced by an impulse load.

show abstract

Section: Introductionsupporting

confidence: 91%

Elastodynamics of a coated half-space under a sliding contact

Bratov

Kaplunov

Lapatsin

et al. 2022

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

show abstract

“…It is then tempting to model the response of the layer using the Timoshenko beam theory but still model the response of the half-space using the Winkler assumption. This possibility has indeed been analysed by Erbas et al (2022) for the dynamic case without prestress; see also Kaplunov et al (2019). Adapting equation ( 17) in Goldenveizer et al (1993) to our static case, we obtain…”

Section: Introductionmentioning

confidence: 62%

A refined model for the buckling of film/substrate bilayers

Wang¹,

Liu²,

Fu³

2022

Preprint

View full text Add to dashboard Cite

The classical reduced model for a film/substrate bilayer is one in which the film is governed by the Euler-Bernoulli beam equation and the substrate is replaced by an array of springs (the so-called Winkler foundation assumption). We derive a refined model in which the normal and shear tractions at the bottom of the film are expressed in terms of the corresponding horizontal and vertical displacements, and the response of the half-space is described by the exact theory. The self-consistency of the refined model is confirmed by showing that it yields a four-term (in the incompressible case) or six-term (in the compressible case) expansion for the critical strain that agrees with the expansion given by the exact theory.

show abstract

“…So we can say that (30) is a more general version for the leading order equation while the correction factor comes from the next order and the term with fourth order derivative of time can be trivial under the special scalings in [4] as is also pointed out by Elishakoff et al [3]. A similar procedure in works [3,4] is also applied by Erbaș et al [33] to derive the refined equation for an elastic beam resting on a Winkler foundation. Recently, Wang et al [34] studied the buckling of a film/substrate bilayer.…”

Section: Application To the Natural Vibration Of A Cantilever Rodmentioning

confidence: 89%

“…A similar procedure in works [3,4] is also applied by Erbaaè t al. [33] to derive the refined equation for an elastic beam resting on a Winkler foundation. Recently, Wang et al [34] studied the buckling of a film/substrate bilayer.…”

Section: Application To the Natural Vibration Of A Cantilever Rodmentioning

confidence: 99%

See 1 more Smart Citation

A uniform framework for the dynamic behavior of linearized anisotropic elastic rods

Chen

Pruchnicki²,

Dai

et al. 2022

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

A uniform framework for the dynamic behavior of a rod composed of a linearized anisotropic material is constructed based on an asymptotic reduction method. Taylor–Young series expansions of the displacement vector and stress tensor are adopted at the beginning. By substituting the series expansions into the three-dimensional (3D) equilibrium equations and the lateral traction condition followed by proper manipulations, four scalar rod equations are obtained for the leading-order displacement vector and the twist angle of the central line. The associated one-dimensional (1D) boundary conditions are obtained through the virtual work principle. One key feature of the present theory is the establishment of recursive relations so that most of the unknowns from the expansion can be eliminated. Also, all the remainders are retained, so the pointwise error estimate of the equations is established. One important advantage of the present theory is that no ad hoc assumption on the forms of displacement and external loading is adopted. Therefore, the rod theory provides a uniform framework to study dynamic behaviors. As an illustrative example, the natural vibration of a linearized isotropic rod with fixed-free boundary conditions is studied. The natural frequency is calculated by applying this uniform framework for the bending, torsional, and longitudinal modes. The obtained results are compared with those of the 3D simulations, Euler–Bernoulli, and Timoshenko beam theories. It turns out that the present theory is accurate enough for moderate and long rods in the bending vibration, and provides explicit results with high accuracy for either long or short rods in torsional and longitudinal vibrations.

show abstract

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

Cited by 17 publications

References 26 publications

Elastodynamics of a coated half-space under a sliding contact

Elastodynamics of a coated half-space under a sliding contact

A refined model for the buckling of film/substrate bilayers

A uniform framework for the dynamic behavior of linearized anisotropic elastic rods

Contact Info

Product

Resources

About