• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.649-660
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• 2020

This paper proves a new regularity criterion for solutions to the Cauchy problem of the 3D Boussinesq equations via one directional derivative of the horizontal component of the velocity field (i.e., (∂_iu₁; ∂_ju₂; 0) where i, j ∈ {1, 2, 3}) in the framework of the anisotropic Lebesgue spaces. More precisely, for 0 < T < ∞, if $$\large{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_o}^T}({\HUGE\left\|{\small{\parallel}{\partial}_iu_1(t){\parallel}_{L^{\alpha}_{x_i}}}\right\|}{\small^{\gamma}_{L^{\beta}_{x_{\hat{i}}x_{\bar{i}}}}+}{\HUGE\left\|{\small{\parallel}{\partial}_iu_2(t){\parallel}_{L^{\alpha}_{x_j}}}\right\|}{\small^{\gamma}_{L^{\beta}_{x_{\hat{i}}x_{\bar{i}}}}})dt<{{\infty}},$$ where ${\frac{2}{{\gamma}}}+{\frac{1}{{\alpha}}}+{\frac{2}{{\beta}}}=m{\in}[1,{\frac{3}{2}})$ and ${\frac{3}{m}}{\leq}{\alpha}{\leq}{\beta}<{\frac{1}{m-1}}$, then the corresponding solution (u, θ) to the 3D Boussinesq equations is regular on [0, T]. Here, (i, ${\hat{i}}$, ${\tilde{i}}$) and (j, ${\hat{j}}$, ${\tilde{j}}$) belong to the permutation group on the set 𝕊₃ := {1, 2, 3}. This result reveals that the horizontal component of the velocity field plays a dominant role in regularity theory of the Boussinesq equations.

• Interaction and multiscale mechanics
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• v.4 no.2
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• pp.123-137
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• 2011

We review a series of crack problems arising in the general deformations of a linearly elastic solid (Mode-I, Mode-II and Mode-III crack) and, perhaps more significantly, when the contribution of surface effects are taken into account. The surface mechanics are incorporated using the continuum based surface/interface model of Gurtin and Murdoch. We show that the deformations of an elastic solid containing a single crack can be decoupled into in-plane (Mode-I and Mode-II crack) and anti-plane (Mode-III crack) parts, even when the surface mechanics is introduced. In particular, it is shown that, in contrast to classical fracture mechanics (where surface effects are neglected), the incorporation of surface elasticity leads to the more accurate description of a finite stress at the crack tip. In addition, the corresponding stress fields exhibit strong dependency on the size of crack.

• Transactions of the Korean Society of Mechanical Engineers
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• v.15 no.6
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• pp.1792-1798
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• 1991

In this paper the path independent $J_{k}$(k=1, 2) integrals are evaluated in a rectilinear anisotropic body for two dimensional case. The relationship among elastic constants are examined. Using those relationship the expression of $J_{2}$ Integral in terms of $K_{I}$ is found to be very simple.e.e.

• Journal of Mechanical Science and Technology
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• v.15 no.10
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• pp.1386-1397
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• 2001

In this paper, the plane elasticity equations are used to investigate the in-plane normal (mode I) and shear (mode II) behavior of a crack perpendicular to and terminating at the interface in bonded media with a graded interfacial zone. The interfacial Bone is treated as a nonhomogeneous interlayer with the continuously varying elastic modulus between the two dissimilar, homogeneous semi-infinite constituents. For each of the individual loading modes, based on the Fourier integral transform technique, a singular integral equation with a Cauchy kernel is derived in a separate but parallel manner. In the numerical results, the values of corresponding modes of stress intensity factors are illustrated for various combinations of material and geometric parameters of the bonded media in conjunction with the effect of the material nonhomogeneity within the graded interfacial zone.

• Journal of Aerospace System Engineering
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• v.1 no.1
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• pp.25-35
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• 2007

In this paper, the interlaminar crack problems of a fiber metal laminate (FML) under generalized plane deformation are studied using the theory of anisotropic elasticity. The crack is considered to be embedded in the matrix interlaminar region (including adhesive zone and resin rich zone) of the FML. Based on Fourier integral transformation and the stress matrix formulation, the current mixed boundary value problem is reduced to solving a system of Cauchy-type singular integral equations of the 1st kind. Within the theory of linear fracture mechanics, the stress intensity factors are defined on terms of the solutions of integral equations and numerical results are obtained for in-plane normal (mode I) crack surface loading. The effects of location and length of crack in the 3/2 and 2/1 ARALL, GLARE or CARE type FML's on the stress intensity factors are illustrated.

• Nuclear Engineering and Technology
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• v.50 no.7
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• pp.1120-1130
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• 2018

An analysis of a pyroprocessing safeguards methodology employing the Pu-to-$^{244}Cm$ ratio is presented. The analysis includes characterization of representative used nuclear fuel assemblies with respect to computed nuclide composition. The nuclide composition data computationally generated is appropriately reformatted to correspond with the material conditions after each step in the head-end stage of pyroprocessing. Uncertainty in the Pu-to-$^{244}Cm$ ratio is evaluated using the Geary-Hinkley transformation method. This is because the Pu-to-$^{244}Cm$ ratio is a Cauchy distribution since it is the ratio of two normally distributed random variables. The calculated uncertainty of the Pu-to-$^{244}Cm$ ratio is propagated through the mass flow stream in the pyroprocessing steps. Finally, the probability of Type-I error for the plutonium Material Unaccounted For (MUF) is evaluated by the hypothesis testing method as a function of the sizes of powder particles and granules, which are dominant parameters to determine the sample size. The results show the probability of Type-I error is occasionally greater than 5%. However, increasing granule sample sizes could surmount the weakness of material accounting because of the non-uniformity of nuclide composition.

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.1071-1084
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• 2000

We show that a weak solution of the Cauchy problem for he nonlinear Schrodinger equation, {i∂(sub)t u + ∂$^2$(sub)x u = f(u,u), t∈(-T,T), x∈R, u(0,x) = ø(x).} in the negative solbolev space H(sup)s has a smoothing effect up to real analyticity if the initial data only have a single point singularity such as the Dirac delta measure. It is shown that for H(sup)s (R)(s>-3/4) data satisfying the condition (※Equations, See Full-text) the solution is analytic in both space and time variable. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [18] and previous work by Kato-Ogawa [12]. We give an improved new argument in the regularity argument.

• Journal of Ocean Engineering and Technology
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• v.16 no.3
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• pp.34-39
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• 2002

본 논문에서는 수면파(Surface wave)와 수중파 (Internal wave)간의 동적 상관관계에 관하여 수행된 연구결과를 정리하였다. 표면파의 비선형 문제는 파의 경사매개변수를 2차원으로 가정하여 해석하였으며, Cauchy 문제는 불균일 조류상의 균일 수면중력파에 대하여 해석하였다. 또한, 파의 경사, 주기의 범위(Frequency range) 그리고 자유표면하의 조류의 분포들간의 조화에 대한 연구가 수행되었으며 해류 및 이동파와 연계되어 수중파의 최전 후방에 형성될 수 있는 정적 파형 (Steady wave pattern)이 수면파형에 포함되었다.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.18 no.1
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• pp.143-149
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• 1993

An approach is presented for efficient numerical integration of the Sormnerfeld type integrals pertinent to the microstrip surface Green's function arising in the problem of an electric current point source on an infinite planar grounded dielectric substrate. This approach, valid for both lossless and lossy dielectric substrates, is based on the deformation of the integration contour via a coordinate transformation and Cauchy's residue theory, and identifies clearly the effects of surface waves. I ts useful application is in a rigorous moment method analysis of micros trip antenna arrays and microstrip guided wave structures. The efficiency and the usefulness of the present approach are emphasized through some numerical calculations of the impedance matrix elements with associated CPU times.

• International Journal of Ocean System Engineering
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• v.1 no.1
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• pp.28-31
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• 2011

An attempt was made to compute the free surface deformation due to the impact of a water droplet. The Cauchy Poisson, i.e. the initial value problem, was solved with the kinematic and dynamic free surface boundary conditions linearized. The zero order Hankel transformation and Laplace transform were applied to the related equations. The initial condition for the free surface profile was derived from a captured video image. The effect of the surface tension was not significant with the water mass used in this investigation. The computed and observed free surface deformations were compared.