A strain gradient generalized continuum approach for modelling elastic scale effects

Sansour, Carlo; Skatulla, Sebastian

doi:10.1016/j.cma.2008.12.031

Cited by 34 publications

(12 citation statements)

References 27 publications

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“…Certain authors have used meshless methods for the implementation of gradient elasticity, by which C 1 -continuity is easily accomodated (Askes and Aifantis, 2002;Tang et al, 2003;Sansour and Skatulla, 2009). Boundary element implementations for gradient elasticity have also been developed, in particular by Tsepoura et al (2002), Polyzos et al (2003), Tsepoura et al (2003), Polyzos et al (2005), Karlis et al (2007).…”

Section: Numerical Implementationsmentioning

confidence: 97%

Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results

Askes

Aifantis

2011

International Journal of Solids and Structures

774

503

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a b s t r a c tIn this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity.

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Section: Numerical Implementationsmentioning

confidence: 97%

Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results

Askes

Aifantis

2011

International Journal of Solids and Structures

774

503

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“…is the axial distance between the rings of carbon atoms when the carbon nanotube is modeled as a strain-gradient material. [37][38][39][40][41] Since nanotubes are thin-walled structures, it can be assumed that only membrane stresses of constant magnitude need to be taken into account in the derivation of governing equations. The differential equations governing the longitudinal and radial vibrations of single-walled nanotubes are given in Ref.…”

Section: Governing Equations Based On Strain-gradient Theorymentioning

confidence: 99%

“…[14][15][16] with the strain-gradient model developed in Refs. [40,41] to study the effect of initial strains on wave propagation in multi-walled nanotubes. A review of the modeling techniques for carbon nanotubes undergoing vibrations and a discussion of related subjects are given in Ref.…”

Section: Introductionmentioning

confidence: 99%

Variational Principles for Vibrating Carbon Nanotubes Modeled as Cylindrical Shells Based on Strain Gradient Nonlocal Theory

Adali¹

2011

Jnl of Comp & Theo Nano

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Variational principles are derived for single and double-walled carbon nanotubes undergoing longitudinal and radial vibrations and the corresponding Hamilton's principles are obtained. Derivations are based on the strain gradient theory of cylindrical shells which provide a continuum model for carbon nanotubes. Strain gradient theory employed in the present study is a nonlocal elastic theory and as such it is capable of taking small scale effects into account. The variational formulation also enables to obtain the natural and geometric boundary conditions for the problem which lead to boundary conditions coupled in the longitudinal and radial directions. The method of variational formulation uses the techniques of calculus of variations and the semi-inverse method of deriving the variational integrals for problems governed by a system of differential equations. Variational formulations provide the basis for a number of approximate and numerical methods of solutions.

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“…Neff [13] investigated geometrically exact generalized continua of Cosserat continuum. Sansour and Skatulla [14][15] discussed some aspects of non-linear formulation of the Cosserat continuum by using a modified variational principle and a first strain gradient formulation is derived, which features a generalized deformation description. Cao et al [16] and Liu et al [17] developed a Cosserat rod element and modified Cosserat rod element modelling strategy to model slender structures in the certain dynamical system.…”

Section: Introductionmentioning

confidence: 99%

Variational principle for a special Cosserat rod

Liu

Wang

2009

Appl. Math. Mech.-Engl. Ed.

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A strain gradient generalized continuum approach for modelling elastic scale effects

Cited by 34 publications

References 27 publications

Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results

Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results

Variational Principles for Vibrating Carbon Nanotubes Modeled as Cylindrical Shells Based on Strain Gradient Nonlocal Theory

Variational principle for a special Cosserat rod

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