• Structural Engineering and Mechanics
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• v.87 no.3
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• pp.283-296
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• 2023

Free-vibration and buckling analyses of plate problems are investigated with the aid of the strain gradient notation finite element method (SGN-FEM). As SGN-FEM employs physically interpretable polynomials in developing finite elements, parasitic shear sources, which are the cause of shear locking, can be precisely identified and subsequently eliminated. This allows two mutually complementary objectives to be defined in this work, namely, evaluate the efficiency of free-vibration and buckling results provided by corrected models, and study the severity of parasitic shear effects on plate models performance. Parasitic shear are flexural terms erroneously present in shear strain polynomials. It is reviewed here that six parasitic shear terms arise during the formulation of the four-node Mindlin plate element. Two parasitic shear terms have been identified in the in-plane shear strain polynomial while other two have been identified in each of the transverse shear strain polynomials. The element is corrected a-priori, i.e., during development, by simply removing the spurious terms from the shear strain polynomials. The computational implementation of the element in its two versions, namely, containing the parasitic shear terms (PS) and corrected for parasitic shear (SG), allows for assessments of the accuracy of results and of the deleterious effects of parasitic shear in free vibration and buckling analyses. This assessment of the parasitic shear effects is a novelty of this work. Validation of the SG model is done comparing its results with analytical results and results provided by other numerical procedures. Analyses are performed for square plates with different thickness-to-length ratios and boundary conditions. Results for thin plates provided by the PS model do not converge to the correct solutions, which indicates that parasitic shear must be eliminated. That is, analysts should not rely on refinement alone. For thick plates, PS model results can be considered acceptable as deleterious effects are really critical in thin plates. On the other hand, results provided by the SG model converge well for both thin and thick plates. The effectiveness of the SG model is established via high-accuracy results obtained in several examples. It is concluded that corrected SGN-FEM models are efficient alternatives for free-vibration and buckling analysis of Mindlin plate problems, and that precise elimination of parasitic shear is a requirement for sound analyses.

• Structural Engineering and Mechanics
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• v.28 no.5
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• pp.603-633
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• 2008

A four-node plate finite element for the analysis of laminated composites which is developed using strain gradient notation is presented. The element is based on a first-order shear deformation theory and on the equivalent lamina assumption. Strains and stresses can be calculated at different points through the thickness of the plate. They are averaged values due to the equivalent lamina assumption. A shear correction factor is used as the transverse shear strain is taken to be constant over the plate thickness while its actual variation is parabolic. Strain gradient notation, which is physically interpretable, allows for the detailed a-priori analysis of the finite element model. The polynomial expansions are inspected and spurious terms responsible for modeling errors are identified in the shear strains polynomial expansions. The element is corrected by simply removing the spurious terms from the shear strains expansions. The element is implemented into a FORTRAN finite element code in two versions; namely, with and without spurious terms. Results are compared to show the effects of the spurious terms on the solutions. It is also shown that a refined mesh composed of corrected elements provides solutions which approximate very well the analytical solutions, validating the procedure.

• Structural Engineering and Mechanics
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• v.40 no.3
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• pp.393-410
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• 2011

The eight-node 3D solid element is one of the most extensively used elements in computational mechanics. This is due to its simple shape and easy of discretization. However, due to the parasitic shear locking, it should not be used to simulate the behaviour of structural members in bending dominant conditions. Previous researches have indicated that the introduction of incompatible mode into the displacement field of the solid element could significantly reduce the shear locking phenomenon. In this study, an incompatible mode eight-node solid element, which considers both geometric and material nonlinearities, is developed for modelling of structural members at elevated temperatures. An algorithm is developed to extend the state determination procedure at ambient temperature to elevated temperatures overcoming initially converged stress locking when the external load is kept constant. Numerical studies show that this incompatible element is superior in terms of convergence, mesh insensitivity and reducing shear locking. It is also showed that the solid element model developed in this paper can be used to model structural behaviour at both ambient and elevated temperatures.

• Journal of the Microelectronics and Packaging Society
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• v.16 no.3
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• pp.67-73
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• 2009

Compared to the chip-bonding process utilizing solder bumps, flip chip process using Cu pillar bumps can accomplish fine-pitch interconnection without compromising stand-off height. Cu pillar bump technology is one of the most promising chip-mounting process for RF packages where large gap between a chip and a substrate is required in order to suppress the parasitic capacitance. In this study, Cu pillar bumps and Sn bumps were electroplated on a chip and a substrate, respectively, and were flip-chip bonded together. Contact resistance and chip shear force of the Cu pillar bump joints were measured with variation of the electroplated Sn-bump height. With increasing the Sn-bump height from 5 ${\mu}m$ to 30 ${\mu}m$, the contact resistance was improved from 31.7 $m{\Omega}$ to 13.8 $m{\Omega}$ and the chip shear force increased from 3.8 N to 6.8 N. On the contrary, the aspect ratio of the Cu pillar bump joint decreased from 1.3 to 0.9. Based on the variation behaviors of the contact resistance, the chip shear force, and the aspect ratio, the optimum height of the electroplated Sn bump could be thought as 20 ${\mu}m$.

• The Journal of the Petrological Society of Korea
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• v.21 no.2
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• pp.129-150
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• 2012

The tectonic evolution of the Central Ogcheon Belt has been newly analyzed in this paper from the detailed geological maps by lithofacies classification, the development processes of geological structures, microstructures, and the time-relationship between deformation and metamorphism in the Ogcheon, Cheongsan, Mungyeong Buunnyeong, Busan areas, Korea and the fossil and radiometric age data of the Ogcheon Supergroup(OSG). The 1st tectonic phase($D^*$) is marked by the rifting of the original Gyeonggi Massif into North Gyeonggi Massif(present Gyeonggi Massif) and South Gyeonggi Massif (Bakdallyeong and Busan gneiss complexes). The Joseon Supergroup(JSG) and the lower unit(quartzose psammitic, pelitic, calcareous and basic rocks) of OSG were deposited in the Ogcheon rift basin during Early Paleozoic time, and the Pyeongan Supergroup(PSG) and its upper unit(conglomerate and pelitic rocks and acidic rocks) appeared in Late Paleozoic time. The 2nd tectonic phase(Ogcheon-Cheongsan phase/Songnim orogeny: D1), which occurred during Late Permian-Middle Triassic age, is characterized by the closing of Ogcheon rift basin(= the coupling of the North and South Gyeonggi Massifs) in the earlier phase(Ogcheon subphase: D1a), and by the coupling of South China block(Gyeonggi Massif and Ogcheon Zone) and North China block(Yeongnam Massif and Taebaksan Zone) in the later phase(Cheongsan subphase: D1b). At the earlier stage of D1a occurred the M1 medium-pressure type metamorphism of OSG related to the growth of coarse biotites, garnets, staurolites. At its later stage, the medium-pressure type metamorphic rocks were exhumed as some nappes with SE-vergence, and the giant-scale sheath fold, regional foliation, stretching lineation were formed in the OSG. At the D1b subphase which occurs under (N)NE-(S)SW compression, the thrusts with NNE- or/and SSW-vergence were formed in the front and rear parts of couple, and the NNE-trending Cheongsan shear zone of dextral strike-slip and the NNE-trending upright folds of the JSG and PSG were also formed in its flank part, and Daedong basin was built in Korean Peninsula. After that, Daedong Group(DG) of the Late Triassic-Early Jurassic was deposited. The 3rd tectonic phase(Honam phase/Daebo orogeny: D2) occurred by the transpression tectonics of NNE-trending Honam dextral strike-slip shearing in Early~Late Jurassic time, and formed the asymmetric crenulated fold in the OSG and the NNE-trending recumbent folds in the JSG and PSG and the thrust faults with ESE-vergence in which pre-Late Triassic Supergroups override DG. The M2 contact metamorphism of andalusite-sillimanite type by the intrusion of Daebo granitoids occurred at the D2 intertectonic phase of Middle Jurassic age. The 4th tectonic phase(Cheongmari phase: D3) occurred under the N-S compression at Early Cretaceous time, and formed the pull-apart Cretaceous sedimentary basins accompanying the NNE-trending sinistral strike-slip shearing. The M3 retrograde metamorphism of OSG associated with the crystallization of chlorite porphyroblasts mainly occurred after the D2. After the D3, the sinistral displacement(Geumgang phase: D4) occurred along the Geumgang fault accompanied with the giant-scale Geumgang drag fold with its parasitic kink folds in the Ogcheon area. These folds are intruded by acidic dykes of Late Cretaceous age.