A General Solution for Two-Dimensional Stress Distributions in Thin Films

2004 ◽  
Vol 71 (5) ◽  
pp. 691-696 ◽  
Author(s):  
R. Krishnamurthy ◽  
D. J. Srolovitz
Keyword(s):  
Thin Film ◽  
Fourier Coefficients ◽  
Fourier Transforms ◽  
Fourier Integral ◽  
Two Dimensional ◽  
Stress Distributions ◽  
Closed Form Solutions ◽  
Thick Substrate ◽  
Film Substrate ◽  
Free Strain

We present closed-form solutions for stresses in a thin film resulting from a purely dilatational stress-free strain that can vary arbitrarily within the film. The solutions are specific to a two-dimensional thin film on a thick substrate geometry and are presented for both a welded and a perfectly slipping film/substrate interface. Variation of the stress-free strain through the thickness of the film is considered to be either arbitrary or represented by a Fourier integral, and solutions are presented in the form of a Fourier series in the lateral dimension x. The Fourier coefficients can be calculated rapidly using Fast Fourier Transforms. The method is applied to determine the stresses in the film and substrate for three cases: (a) where the stress-free strain is a sinusoidal modulation in x, (b) where the stress-free strain varies only through the thickness, and (c) where a rectangular inclusion is embedded within the film, and the calculated stresses match accurately with the exact solutions for these cases.

Delamination-Based Measurement and Prediction of the Adhesion Energy of Thin Film/Substrate Interfaces

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
Liangliang Zhu ◽  
Xi Chen
Keyword(s):  
Thin Film ◽  
Flexible Electronics ◽  
Current Method ◽  
Adhesion Energy ◽  
Two Dimensional ◽  
Soft Substrate ◽  
Interface Adhesion ◽  
Compliant Substrates ◽  
And Performance ◽  
Film Substrate

With the rapid emerging of two-dimensional (2D) micro/nanomaterials and their applications in flexible electronics and microfabrication, adhesion between thin film and varying substrates is of great significance for fabrication and performance of micro devices and for the understanding of the buckle delamination mechanics. However, the adhesion energy remains to be difficult to be measured, especially for compliant substrates. We propose a simple methodology to deduce the adhesion energy between a thin film and soft substrate based on the successive or simultaneous emergence of wrinkles and delamination. The new metrology does not explicitly require the knowledge of the Young's modulus, Poisson's ratio, and thickness of the 2D material, the accurate measurement of which could be a challenge in many cases. Therefore, the uncertainty of the results of the current method is notably reduced. Besides, for cases where the delamination width is close to the critical wrinkle wavelength of the thin film/substrate system, the procedure can be further simplified. The simple and experimentally easy methodology developed here is promising for determining/estimating the interface adhesion energy of a variety of thin film/soft substrate systems.

Contribution of substrate to converse piezoelectric response of constrained thin films

2004 ◽  
Vol 19 (10) ◽  
pp. 2853-2858 ◽  
Author(s):  
Lang Chen ◽  
J-H. Li ◽  
J. Slutsker ◽  
J. Ouyang ◽  
A.L. Roytburd
Keyword(s):  
Thin Film ◽  
Thin Films ◽  
Finite Element ◽  
Piezoelectric Response ◽  
Element Calculation ◽  
Theoretical Estimation ◽  
Various Boundary Conditions ◽  
Total Displacement ◽  
Thick Substrate ◽  
Film Substrate

The converse piezoelectric response of a thin film constrained by a substrate is analyzed in different geometries under various boundary conditions. We considerthe effects of elastic deformation of the substrate on the total displacement of thefilm surface induced by the electric field. The change of film thickness and the bending curvature of a film/substrate couple are calculated. For a thin film island clamped on a large thick substrate, the theoretical estimation of the piezoresponse, including a local bending in the vicinity of the island/substrate interface, is in agreement with the finite element calculation.

Some Problems of a Rigid Elliptical Disk-Inclusion Bonded Inside a Transversely Isotropic Space: Part I

1999 ◽  
Vol 66 (3) ◽  
pp. 612-620 ◽  
Author(s):  
M. Rahman
Keyword(s):  
Fourier Transforms ◽  
Failure Load ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Two Dimensional ◽  
Closed Form Solutions ◽  
Isotropic Space ◽  
Uniform Stress Field ◽  
Perfect Contact ◽  
Dimensional Integral

The paper addresses the problem of contact of an elliptical inclusion in the form of a thin disk, bonded in the interior of a transversely isotropic space. The inclusion is assumed to be absolutely rigid and in perfect contact with the medium. Three different cases of loading are considered, namely, (a) the inclusion is loaded in its plane by a shearing force, whose line of action passes through the center of the disk; (b) the inclusion is rotated by a torque whose axis is perpendicular to the plane of the inclusion; (c) the medium is under uniform stress field at infinity in a plane parallel to the plane of the inclusion. In the first part of the article, the problems corresponding to all three cases are reduced, in a unified manner, to a system of coupled two-dimensional integral equations by using the theory of two-dimensional Fourier transforms. In the second part, closed-form solutions for these equations are obtained by using Dyson’s theorem and Willis’ generalization of Galin’s theorem. Explicit expressions for the stress intensity factors near the edge of the inclusion are extracted from these solutions. Numerical results are plotted illustrating how these coefficients vary with transverse isotropy and the parametric angle of the ellipse. The results can be used to determine the critical failure load and angle of crack initiation for solids containing elliptical inclusions.

Microtomy of spherical Al2O3 powders

1983 ◽  
Vol 41 ◽  
pp. 74-75
Author(s):  
E.J. Jenkins ◽  
D.S. Tucker ◽  
J.J. Hren
Keyword(s):  
Thin Film ◽  
Size Range ◽  
Particle Aggregation ◽  
Ion Milling ◽  
Thin Sections ◽  
Α Phase ◽  
Convenient Means ◽  
Van Der Waals Attraction ◽  
Negative Feature ◽  
Film Substrate

The size range of mineral and ceramic particles of one to a few microns is awkward to prepare for examination by TEM. Electrons can be transmitted through smaller particles directly and larger particles can be thinned by crushing and dispersion onto a substrate or by embedding in a film followed by ion milling. Attempts at dispersion onto a thin film substrate often result in particle aggregation by van der Waals attraction. In the present work we studied 1-10 μm diameter Al2O3 spheres which were transformed from the amprphous state to the stable α phase.After the appropriate heat treatment, the spherical powders were embedded in as high a density as practicable in a hard EPON, and then microtomed into thin sections. There are several advantages to this method. Obviously, this is a rapid and convenient means to study the microstructure of serial slices. EDS, ELS, and diffraction studies are also considerably more informative. Furthermore, confidence in sampling reliability is considerably enhanced. The major negative feature is some distortion of the microstructure inherent to the microtoming operation; however, this appears to have been surprisingly small. The details of the method and some typical results follow.

Microstructural investigation of surface roughness in MBE-grown AlAs

1995 ◽  
Vol 53 ◽  
pp. 454-455
Author(s):  
R. Rajesh ◽  
R. Droopad ◽  
C. H. Kuo ◽  
R. W. Carpenter ◽  
G. N. Maracas
Keyword(s):  
Thin Film ◽  
Real Time ◽  
Growth Control ◽  
Native Oxide ◽  
Effusion Cell ◽  
Surface Oxide Layer ◽  
Microstructural Investigation ◽  
Mbe Growth ◽  
Film Substrate ◽  
And Control

Knowledge of material pseudodielectric functions at MBE growth temperatures is essential for achieving in-situ, real time growth control. This allows us to accurately monitor and control thicknesses of the layers during growth. Undesired effusion cell temperature fluctuations during growth can thus be compensated for in real-time by spectroscopic ellipsometry. The accuracy in determining pseudodielectric functions is increased if one does not require applying a structure model to correct for the presence of an unknown surface layer such as a native oxide. Performing these measurements in an MBE reactor on as-grown material gives us this advantage. Thus, a simple three phase model (vacuum/thin film/substrate) can be used to obtain thin film data without uncertainties arising from a surface oxide layer of unknown composition and temperature dependence.In this study, we obtain the pseudodielectric functions of MBE-grown AlAs from growth temperature (650°C) to room temperature (30°C). The profile of the wavelength-dependent function from the ellipsometry data indicated a rough surface after growth of 0.5 μm of AlAs at a substrate temperature of 600°C, which is typical for MBE-growth of GaAs.

Two-dimensional BN-doped ZnO thin-film deposition by a thermionic vacuum arc system

2020 ◽  
Vol 31 (9) ◽  
pp. 6948-6955
Author(s):  
Mustafa Özgür ◽  
Suat Pat ◽  
Reza Mohammadigharehbagh ◽  
Uğur Demirkol ◽  
Nihan Akkurt ◽  
...  
Keyword(s):  
Thin Film ◽  
Thin Film Deposition ◽  
Film Deposition ◽  
Vacuum Arc ◽  
Zno Thin Film ◽  
Two Dimensional ◽  
Doped Zno ◽  
Thermionic Vacuum Arc

Influence of ion beam parameters onto two-dimensional optical thin film thickness distributions deposited by ion beam sputtering

2019 ◽  
Vol 682 ◽  
pp. 109-120 ◽  
Author(s):  
Wjatscheslaw Sakiew ◽  
Stefan Schrameyer ◽  
Marco Jupé ◽  
Philippe Schwerdtner ◽  
Nick Erhart ◽  
...  
Keyword(s):  
Thin Film ◽  
Film Thickness ◽  
Ion Beam ◽  
Two Dimensional ◽  
Ion Beam Sputtering ◽  
Thin Film Thickness ◽  
Optical Thin Film ◽  
Beam Sputtering ◽  
Beam Parameters

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

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