scholarly journals Ruga-formation instabilities of a graded stiffness boundary layer in a neo-Hookean solid

2014 ◽  
Vol 470 (2168) ◽  
pp. 20140218 ◽  
Author(s):  
Mazen Diab ◽  
Kyung-Suk Kim
Keyword(s):  
Boundary Layer ◽  
Half Space ◽  
Compressive Strain ◽  
Three Dimensional ◽  
Decay Length ◽  
Element Analysis ◽  
Dimensional Parameter ◽  
First Order ◽  
Homogeneous Half Space ◽  
Lateral Plane

We present an analysis of ruga-formation instabilities arising in a graded stiffness boundary layer of a neo-Hookean half space, caused by lateral plane-strain compression. In this study, we represent the boundary layer by a stiffness distribution exponentially decaying from a surface value Q 0 to a bulk value Q B with a decay length of 1/ a . Then, the normalized perturbation wavenumber, k ¯ = k / a , and the compressive strain, ε , control formation of a wrinkle pattern and its evolution towards crease or fold patterns for every stiffness ratio η = Q B / Q 0 . Our first-order instability analysis reveals that the boundary layer exhibits self-selectivity of the critical wavenumber for nearly the entire range of 0< η <1, except for the slab ( η =0) and homogeneous half-space ( η =1) limits. Our second-order analysis supplemented by finite-element analysis further uncovers various instability-order-dependent bifurcations, from stable wrinkling of the first order to creasing of the infinite-order cascade instability, which construct diverse ruga phases in the three-dimensional parameter space of ( ε , k ¯ , η ) . Competition among film-buckling, local film-crease and global substrate-crease modes of energy release produces diverse ruga-phase domains. Our analysis also reveals the subcritical crease states of the homogeneous half space. Our results are, then, compared with the behaviour of equivalent bilayer systems for thin-film applications.

Three-dimensional analytical model for the dynamic interaction of twin tunnels in a homogeneous half-space

2019 ◽  
Vol 230 (3) ◽  
pp. 1159-1179 ◽  
Author(s):  
Chao He ◽  
Shunhua Zhou ◽  
Peijun Guo ◽  
Quanmei Gong
Keyword(s):  
Half Space ◽  
Analytical Model ◽  
Three Dimensional ◽  
Dynamic Interaction ◽  
Homogeneous Half Space ◽  
Twin Tunnels

Computation of Green’s tensor integrals for three‐dimensional electromagnetic problems using fast Hankel transforms

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1754-1759 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Hankel Transform ◽  
Electromagnetic Modeling ◽  
Electromagnetic Problems ◽  
Hankel Transforms ◽  
Homogeneous Half Space ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
The Many

A new method is presented that rapidly evaluates the many Green’s tensor integrals encountered in three‐dimensional electromagnetic modeling using an integral equation. Application of a fast Hankel transform (FHT) algorithm (Anderson, 1982) is the basis for the new solution, where efficient and accurate computation of Hankel transforms are obtained by related and lagged convolutions (linear digital filtering). The FHT algorithm is briefly reviewed and compared to earlier convolution algorithms written by the author. The homogeneous and layered half‐space cases for the Green’s tensor integrals are presented in a form so that the FHT can be easily applied in practice. Computer timing runs comparing the FHT to conventional direct convolution methods are discussed, where the FHT’s performance was about 6 times faster for a homogeneous half‐space, and about 108 times faster for a five‐layer half‐space. Subsequent interpolation after the FHT is called is required to compute specific values of the tensor integrals at selected transform arguments; however, due to the relatively small lagged convolution interval used (same as the digital filter’s), a simple and fast interpolation is sufficient (e.g., by cubic splines).

scholarly journals Mathematical model of falling of a viscous jet onto a moving surface

2007 ◽  
Vol 18 (6) ◽  
pp. 659-677 ◽  
Author(s):  
A. HLOD ◽  
A. C. T. AARTS ◽  
A. A. F. van de VEN ◽  
M. A. PELETIER
Keyword(s):  
Three Dimensional ◽  
Integral Condition ◽  
Stationary Flow ◽  
Third Order ◽  
Convex Shape ◽  
Dimensional Parameter ◽  
Moving Surface ◽  
First Order ◽  
Existence Region ◽  
Unknown Length

The stationary flow of a jet of a Newtonian fluid that is drawn by gravity onto a moving surface is analyzed. It is assumed that the jet has a convex shape and hits the moving surface tangentially. The flow is modelled by a third-order ODE on a domain of unknown length and with an additional integral condition. By solving part of the equation explicitly, the problem is reformulated as a first-order ODE with an integral constraint. The corresponding existence region in the three-dimensional parameter space is characterized in terms of an easily calculable quantity. In a qualitative sense, the results from the model are found to correspond with experimental observations.

The three-dimensional laminar boundary layer on a flat plate

1965 ◽  
Vol 22 (3) ◽  
pp. 587-598 ◽  
Author(s):  
L. Sowerby
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Three Dimensional ◽  
Leading Edge ◽  
Main Stream ◽  
Dimensional Parameter ◽  
Stress Components ◽  
Dimensional Boundary ◽  
Layer Flow ◽  
Limiting Streamlines

A series expansion is derived for the three-dimensional boundary-layer flow over a flat plate, arising from a general main-stream flow over the plate. The series involved are calculated as far as terms of order ξ2, where ξ is a non-dimensional parameter defining distance measured from the leading edge of the plate. The results are applied to an example in which the main stream arises from the disturbance of a uniform stream by a circular cylinder mounted downstream from the leading edge of the plate, the axis of the cylinder being normal to the plate. Calculations are made for shear stress components on the plate, and for the deviation of direction of the limiting streamlines from those in the main stream.

scholarly journals Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150800 ◽  
Author(s):  
R. Chebakov ◽  
J. Kaplunov ◽  
G. A. Rogerson
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Boundary Conditions ◽  
Half Space ◽  
Elastic Half Space ◽  
Interior Solution ◽  
Homogeneous Half Space ◽  
Classical Boundary ◽  
Non Local ◽  
Rayleigh Wave Speed

The dynamic response of a homogeneous half-space, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the ‘local’ problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of non-local elastic phenomena on the Rayleigh wave speed is investigated.

Elasto-Plastic Finite Element Analysis of Three-Dimensional Pure Rolling Contact Above the Shakedown Limit

1991 ◽  
Vol 58 (2) ◽  
pp. 347-353 ◽  
Author(s):  
S. M. Kulkarni ◽  
G. T. Hahn ◽  
C. A. Rubin ◽  
V. Bhargava
Keyword(s):  
Finite Element ◽  
Half Space ◽  
Kinematic Hardening ◽  
Three Dimensional ◽  
Bearing Steel ◽  
Complex Stress State ◽  
Rolling Contact ◽  
Stress Strain ◽  
Element Analysis ◽  
Shakedown Limit

This paper describes calculations for repeated, frictionless, three-dimensional rolling contact, for a relative peak pressure (po/k) of 6.0 (above the shakedown limit) for a circular contact patch. This analysis was carried out for two material responses, elastic-perfectly plastic (EPP) and elastic-linear-kinematic-hardening plastic (ELKP), using the elasto-plastic finite element model developed earlier. The ELKP material parameters are those appropriate for hardened bearing steel. Frictionless three-dimensional rolling contact is simulated by repeatedly translating a Hertzian pressure distribution across the surface of an elasto-plastic half space. The half space is represented by a finite mesh with elastic boundaries. The paper describes the complex stress state existing in the half space and the attending plasticity, as the load translates. The calculations present the distortion of the rim, the residual stress-strain distributions, stress-strain histories, and the cyclic plastic strain increments in the vicinity of the contact. Compared with the analyses at the shakedown limit, higher residual stresses and strains are observed.

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