scholarly journals Stability analysis of a rotationally restrained microbar embedded in an elastic matrix using strain gradient elasticity

2019 ◽  
Vol 6 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Mustafa Özgür Yayli
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Scale Parameter ◽  
Gradient Elasticity ◽  
Buckling Load ◽  
Elastic Matrix ◽  
Small Scale ◽  
Strain Gradient Elasticity ◽  
Critical Buckling Load ◽  
Fourier Sine Series

AbstractThe buckling of rotationally restrained microbars embedded in an elastic matrix is studied within the framework of strain gradient elasticity theory. The elastic matrix is modeled in this study as Winkler’s one-parameter elastic matrix. Fourier sine series with a Fourier coefficient is used for describing the deflection of the microbar. An eigenvalue problem is obtained for buckling modes with the aid of implementing Stokes’ transformation to force boundary conditions. This mathematical model bridges the gap between rigid and the restrained boundary conditions. The influences of rotational restraints, small scale parameter and surrounding elastic matrix on the critical buckling load are discussed and compared with those available in the literature. It is concluded from analytical results that the critical buckling load of microbar is dependent upon rotational restraints, surrounding elastic matrix and the material scale parameter. Similarly, the dependencies of the critical buckling load on material scale parameter, surrounding elastic medium and rotational restraints are significant.

Nonlinear Vibration Analysis of Microscale Functionally Graded Timoshenko Beams using the Most General form of Strain Gradient Elasticity

2013 ◽  
Vol 30 (2) ◽  
pp. 161-172 ◽  
Author(s):  
R. Ansari ◽  
M. Faghih Shojaei ◽  
V. Mohammadi ◽  
R. Gholami ◽  
H. Rouhi
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Vibration ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Functionally Graded ◽  
Small Scale ◽  
Length Scale ◽  
Beam Model ◽  
Strain Gradient Elasticity ◽  
Governing Equations

ABSTRACTBased on the Timoshenko beam model, the nonlinear vibration of microbeams made of functionally graded (FG) materials is investigated under different boundary conditions. To consider small scale effects, the model is developed based on the most general form of strain gradient elasticity. The nonlinear governing equations and boundary conditions are derived via Hamilton's principle and then discretized using the generalized differential quadrature technique. A pseudo-Galerkin approach is used to reduce the set of discretized governing equations into a time-varying set of ordinary differential equations of Duffing-type. The harmonic balance method in conjunction with the Newton-Raphson method is also applied so as to solve the problem in time domain. The effects of boundary conditions, length scale parameters, material gradient index and geometrical parameters are studied. It is found that the importance of the small length scale is affected by the type of boundary conditions and vibration mode. Also, it is revealed that the classical theory tends to underestimate the vibration amplitude and linear frequency of FG microbeams.

Strain Gradient Finite Element Analysis on the Vibration of Double-Layered Graphene Sheets

2016 ◽  
Vol 13 (03) ◽  
pp. 1650011 ◽  
Author(s):  
Wei Xu ◽  
Lifeng Wang ◽  
Jingnong Jiang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Scale Effect ◽  
Strain Gradient ◽  
Virtual Work ◽  
Gradient Elasticity ◽  
Kirchhoff Plate ◽  
Small Scale ◽  
Graphene Sheets ◽  
Simply Supported

A nonlocal Kirchhoff plate model with the van der Waals (vdW) interactions taken into consideration is developed to study the vibration of double-layered graphene sheets (DLGS). The dynamic equations of multi-layered Kirchhoff plate are derived based on strain gradient elasticity. An explicit formula is derived to predict the natural frequency of the DLGS with all edges simply supported. Then a 4-node 24-degree of freedom (DOF) Kirchhoff plate element is developed to discretize the higher order partial differential equations with the small scale effect taken into consideration by the theory of virtual work. It can be directly used to predict the scale effect on the vibrational DLGS with different boundary conditions. A good agreement between finite element method (FEM) results and theoretical natural frequencies of the vibration simply supported double-layered graphene sheet (DLGS) validates the reliability of the FEM. Finally, this new FEM is used to investigate the effect of vdW coefficients, sizes, nonlocal parameters, vibration mode and boundary conditions on the vibration behaviors of DLGS.

Prediction of the Critical Energy Release Rate of Nanostructured Solids Using the Laplacian Version of the Strain Gradient Elasticity Theory

2018 ◽  
Vol 774 ◽  
pp. 447-452 ◽  
Author(s):  
Michal Kotoul ◽  
Petr Skalka ◽  
Tomáš Profant ◽  
Martin Friák ◽  
Petr Řehák ◽  
...  
Keyword(s):  
Dft Calculations ◽  
Strain Gradient ◽  
Scale Parameter ◽  
Gradient Elasticity ◽  
Length Scale ◽  
Strain Gradient Elasticity ◽  
Material Length Scale Parameter ◽  
Gradient Elasticity Theory ◽  
Material Length Scale ◽  
Material Length

The aim of the paper is quantify the material length scale parameter of the simplified form of the strain gradient elasticity theory (SGET) using first principles density-functional theory (DFT). The single material length scale parameterlis extracted from phonon-dispersions generated by DFT calculations and, for comparison, by adjusting the analytical SGET solution for the displacement field near the screw dislocation with the DFT calculations of this field. The obtained results are further used in the SGET modeling of cracked nanopanel formed by the single tungsten crystal where due to size effects and nonlocal material point interactions the classical fracture mechanics breaks down.

scholarly journals Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

2017 ◽  
Vol 24 (1) ◽  
pp. 312-335 ◽  
Author(s):  
J. Niiranen ◽  
V. Balobanov ◽  
J. Kiendl ◽  
SB Hosseini
Keyword(s):  
Boundary Conditions ◽  
Elasticity Theory ◽  
Strain Gradient ◽  
Size Effects ◽  
Gradient Elasticity ◽  
Stability And Convergence ◽  
Strain Gradient Elasticity ◽  
Strain Gradient Elasticity Theory ◽  
Gradient Elasticity Theory ◽  
Variational Formulations

As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s strain gradient elasticity theory of form II. For both models, this leads to sixth-order boundary value problems with new types of boundary conditions that are given additional attributes singly and doubly, referring to a physically relevant distinguishing feature between free and prescribed curvature, respectively. Second, the variational formulations are analyzed with rigorous mathematical tools: the existence and uniqueness of weak solutions are established by proving continuity and ellipticity of the associated symmetric bilinear forms. This guarantees optimal convergence for conforming Galerkin discretization methods. Third, the variational analysis is extended to cover two other generalized beam models: another modification of the strain gradient elasticity theory and a modified version of the couple stress theory. A model comparison reveals essential differences and similarities in the physicality of these four closely related beam models: they demonstrate essentially two different kinds of parameter-dependent stiffening behavior, where one of these kinds (possessed by three models out of four) provides results in a very good agreement with the size effects of experimental tests. Finally, numerical results for isogeometric Galerkin discretizations with B-splines confirm the theoretical stability and convergence results. Influences of the gradient and thickness parameters connected to size effects, boundary layers and dispersion relations are studied thoroughly with a series of benchmark problems for statics and free vibrations. The size-dependency of the effective Young’s modulus is demonstrated for an auxetic cellular metamaterial ruled by bending-dominated deformation of cell struts.

scholarly journals Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions

2021 ◽  
Author(s):  
Lidiia Nazarenko ◽  
Rainer Glüge ◽  
Holm Altenbach
Keyword(s):  
Quadratic Form ◽  
Potential Energy ◽  
Boundary Conditions ◽  
Uniqueness Theorem ◽  
Strain Gradient ◽  
Mixed Boundary ◽  
Displacement Vector ◽  
Gradient Elasticity ◽  
Strain Gradient Elasticity ◽  
Second Gradient

AbstractThe equilibrium equations and the traction boundary conditions are evaluated on the basis of the condition of the stationarity of the Lagrangian for coupled strain gradient elasticity. The quadratic form of strain energy can be written as a function of the strain and the second gradient of displacement and contains a fourth-, a fifth- and a sixth-order stiffness tensor $${\mathbb {C}}_4$$ C 4 , $${\mathbb {C}}_5$$ C 5 and $${\mathbb {C}}_6$$ C 6 , respectively. Assuming invariance under rigid body motions the balance of linear and angular momentum is obtained. The uniqueness theorem (Kirchhoff) for the mixed boundary value problem is proved for the case of the coupled linear strain gradient elasticity (novel). To this end, the total potential energy is altered to be presented as an uncoupled quadratic form of the strain and the modified second gradient of displacement vector. Such a transformation leads to a decoupling of the equation of the potential energy density. The uniqueness of the solution is proved in the standard manner by considering the difference between two solutions.

Extended Hill’s lemma for non-Cauchy continua based on the simplified strain gradient elasticity theory

2016 ◽  
Vol 01 (03n04) ◽  
pp. 1640004 ◽  
Author(s):  
X.-L. Gao
Keyword(s):  
Boundary Conditions ◽  
Elasticity Theory ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Strain Gradient Elasticity ◽  
Classical Counterpart ◽  
Gradient Effect ◽  
Strain Gradient Elasticity Theory ◽  
Gradient Elasticity Theory ◽  
Hill’S Lemma

Hill's lemma for the Cauchy continuum has been playing an important role in micromechanics. An extended version of Hill's lemma for non-Cauchy continua is formulated using the simplified strain gradient elasticity theory (SSGET), which contains only one material length scale parameter and can account for the microstructure-dependent strain gradient effect. As a corollary of the extended Hill's lemma, the Hill–Mandel macro-homogeneity condition for non-Cauchy continua is obtained along with the general forms of kinematically and statically admissible boundary conditions that are required for constructing an energetically equivalent homogeneous comparison material. Based on these general forms, four sets of uniform boundary conditions are identified, which are implementable in material tests and can be directly used in homogenization analyses of heterogeneous materials. It is shown that when the strain gradient effect is suppressed, the extended Hill's lemma recovers the classical Hill's lemma for the Cauchy continuum and the extended Hill–Mandel condition reduces to its classical counterpart.

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