• Computers and Concrete
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• 제15권4호
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• pp.607-642
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• 2015

The stress-strain relationship of concrete in flexure is one of the essential parameters in assessing the flexural strength and ductility of reinforced concrete (RC) columns. An overview of previous research studies revealed that the presence of strain gradient would affect the maximum concrete stress developed in flexure. However, no quantitative model was available to evaluate the strain gradient effect on concrete under flexure. Previously, the authors have conducted experimental studies to investigate the strain gradient effect on maximum concrete stress and respective strain and developed two strain-gradient-dependent factors k3 and ko for modifying the flexural concrete stress-strain curve. As a continued study, the authors herein will extend the investigation of strain gradient effects on flexural strength and ductility of RC columns to concrete strength up to 100 MPa by employing the strain-gradient-dependent concrete stress-strain curve using nonlinear moment-curvature analysis. It was evident from the results that both the flexural strength and ductility of RC columns are improved under strain gradient effect. Lastly, for practical engineering design purpose, a new equivalent rectangular concrete stress block incorporating the combined effects of strain gradient and concrete strength was proposed and validated. Design formulas and charts have also been presented for flexural strength and ductility of RC columns.

• Journal of Mechanical Science and Technology
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• 제20권5호
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• pp.621-633
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• 2006

To reflect the size effect of material $(1\sim15{\mu}m)$ during plastic deformation of polycrystalline copper, a constitutive equation which includes the strain gradient plasticity theory and intrinsic material length model is coupled with the finite element analysis and applied to plane strain deformation problem. The method of least square has been used to calculate the strain gradient at each element during deformation and the effect of distributed force on the strain gradient is investigated as well. It shows when material size is less than the intrinsic material length $(1.54{\mu}m)$, its deformation behavior is quite different compared with that computed from the conventional plasticity. The generation of strain gradient is greatly suppressed, but it appears again as the material size increases. Results also reveal that the strain gradient leads to deformation hardening. The distributed force plays a role to amplify the strain gradient distribution.

• 소성∙가공
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• 제15권8호
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• pp.544-549
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• 2006

In this work, we have employed the strain gradient plasticity theory to investigate the effect of material size on the deformation behavior in metal forming process. Flow stress is expressed in terms of strain, strain gradient (spatial derivative of strain) and intrinsic material length. The least square method coupled with strain gradient plasticity was used to calculate the components of strain gradient at each element of material. For demonstrating the size effect, the proposed approach has been applied to plane compression process and micro rolling process. Results show when the characteristic length of the material comes to the intrinsic material length, the effect of strain gradient is noteworthy. For the microcompression, the additional work hardening at higher strain gradient regions results in uniform distribution of strain. In the case of micro-rolling, the strain gradient is remarkable at the exit section where the actual reduction of the rolling finishes and subsequently strong work hardening take places at the section. This results in a considerable increase in rolling force. Rolling force with the strain gradient plasticity considered in analysis increases by 20% compared to that with conventional plasticity theory.

• Multiscale and Multiphysics Mechanics
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• 제1권1호
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• pp.15-33
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• 2016

In this paper, an asymptotic method is employed to formulate nano- or micro-beams based on strain gradient elasticity. Although a basic theory for the strain gradient elasticity has been well established in literature, a systematic approach is relatively rare because of its complexity and ambiguity of higher-order elasticity coefficients. In order to systematically identify the strain gradient effect, an asymptotic approach is adopted by introducing the small parameter which represents the beam geometric slenderness and/or the internal atomistic characteristic. The approach allows us to systematically split the two-dimensional strain gradient elasticity into the microscopic one-dimensional through-the-thickness analysis and the macroscopic one-dimensional beam analysis. The first-order beam problem turns out to be different from the classical elasticity in terms of the bending stiffness, which comes from the through-the-thickness strain gradient effect. This subsequently affects the second-order transverse shear stress in which the surface shear stress exists. It is demonstrated that a careful derivation of a first strain gradient elasticity embraces "Gurtin-Murdoch traction" as the surface effect of a one-dimensional Euler-Bernoulli-like beam model.

• Structural Engineering and Mechanics
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• 제48권3호
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• pp.415-434
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• 2013

This work is concerned with transverse vibrations of axially traveling nanobeams including strain gradient and thermal effects. The strain gradient elasticity theory and the temperature field are taken into consideration. A new higher-order differential equation of motion is derived from the variational principle and the corresponding higher-order non-classical boundary conditions including simple, clamped, cantilevered supports and their higher-order "offspring" are established. Effects of strain gradient nanoscale parameter, temperature change, shape parameter and axial traction on the natural frequencies are presented and discussed through some numerical examples. It is concluded that the factors mentioned above significantly influence the dynamic behaviors of an axially traveling nanobeam. In particular, the strain gradient effect tends to induce higher vibration frequencies as compared to an axially traveling macro beams based on the classical vibration theory without strain gradient effect.

• Structural Engineering and Mechanics
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• 제66권6호
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• pp.693-701
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• 2018

This paper develops a nonlocal strain gradient plate model for vibration analysis of graphene sheets under thermal environments. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton's principle. Differential quadrature method (DQM) is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as temperature rise, nonlocal parameter, length scale parameter, elastic foundation and aspect ratio on vibration characteristics a graphene sheets are studied. It is seen that vibration frequencies and critical buckling temperatures become larger and smaller with increase of strain gradient and nonlocal parameter, respectively.

• Advances in nano research
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• 제12권5호
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• pp.441-455
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• 2022

This study explores the linear and nonlinear solutions of sigmoid functionally graded material (S-FGM) nanoplate with porous effects. A size-dependent numerical solution is established using the strain gradient theory and isogeometric finite element formulation. The nonlinear nonlocal strain gradient is developed based on the Reissner-Mindlin plate theory and the Von-Karman strain assumption. The sigmoid function is utilized to modify the classical functionally graded material to ensure the constituent volume distribution. Two different patterns of porosity distribution are investigated, viz. pattern A and pattern B, in which the porosities are symmetric and asymmetric varied across the plate's thickness, respectively. The nonlinear finite element governing equations are established for bending analysis of S-FGM nanoplates, and the Newton-Raphson iteration technique is derived from the nonlinear responses. The isogeometric finite element method is the most suitable numerical method because it can satisfy a higher-order derivative requirement of the nonlocal strain gradient theory. Several numerical results are presented to investigate the influences of porosity distributions, power indexes, aspect ratios, nonlocal and strain gradient parameters on the porous S-FGM nanoplate's linear and nonlinear bending responses.

• Advances in nano research
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• 제13권2호
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• pp.147-164
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• 2022

This article investigates the longitudinal vibration of a nanorod embedded in viscoelastic medium according to the nonlocal strain gradient theory. Viscoelastic medium is considered based on Kelvin-Voigt model. Governing partial differential equation is derived based on longitudinal equilibrium and analytical solution is obtained by adopting harmonic motion solution for the nanorod. Modal frequencies and corresponding damping ratios are presented to demonstrate the influences of nonlocal parameter, material length scale, elastic and damping parameters of the viscoelastic medium. It is observed that material length scale parameter is very influential on modal frequencies especially at lower values of nonlocal parameter whereas increase in length scale parameter has less effect at higher values of nonlocal parameter when the medium is purely elastic. Elastic stiffness and damping coefficient of the medium have considerable impacts on modal frequencies and damping ratios, and the highest impact of these parameters on frequency and damping ratio is seen in the first mode. Results calculated based on strain gradient theory are quite different from those calculated based on classical elasticity theory. Hence, nonlocal strain gradient theory including length scale parameter can be used to get more accurate estimations of frequency response of nanorods embedded in viscoelastic medium.

• Steel and Composite Structures
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• 제47권6호
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• pp.795-811
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• 2023

When studying the resonance problem of nanoplates, the existing papers do not consider the influences of geometric nonlinearity and initial geometric imperfection, so this paper is to fill this gap. In this paper, based on the nonlocal strain gradient theory (NSGT), the nonlinear resonances of functionally graded (FG) nanoplates with initial geometric imperfection under different boundary conditions are established. In order to consider the small size effect of plates, nonlocal parameters and strain gradient parameters are introduced to expand the assumptions of the first-order shear deformation theory. Subsequently, the equations of motion are derived using the Euler-Lagrange principle and solved with the help of perturbation method. In addition, the effects of initial geometrical imperfection, functionally graded index, strain gradient parameter, nonlocal parameter and porosity on the nonlinear forced vibration behavior of nanoplates under different boundary conditions are discussed.

• 한국추진공학회:학술대회논문집
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• 한국추진공학회 2011년도 제37회 추계학술대회논문집
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• pp.528-535
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• 2011

본 논문에서는 변형율 독립 탄-소성 구성방정식을 이용, 전단 변형 하에서의 국부적 소성변형 집중현상이 분석되었다. 또한 변형량 기울기 (strain gradient) 항이 포함된 비구역적 (non-local) 구성방정식이 유도되었으며 이는 다시 이중후방응력 경화 모델로 표현되었다. 더욱이 본 모델은 연속체 파손역학과 조합되었다. 국부적 변형집중 현상은 수치해석을 통해 분석되었으며 변형량 기울기 항이 구성방정식에 포함될 때 본 항의 크기가 증가할수록 전단 밴드의 크기는 감소하는 것으로 밝혀졌다.