Journal articles: 'Gradient of elasticity'

1

Askes, Harm, and Miguel A. Gutiérrez. "Implicit gradient elasticity." International Journal for Numerical Methods in Engineering 67, no. 3 (2006): 400–416. http://dx.doi.org/10.1002/nme.1640.

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Tarasov, Vasily E., and Elias C. Aifantis. "Toward fractional gradient elasticity." Journal of the Mechanical Behavior of Materials 23, no. 1-2 (May 1, 2014): 41–46. http://dx.doi.org/10.1515/jmbm-2014-0006.

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AbstractThe use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The second involves the Riesz fractional derivative in three dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, in a Riesz-type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.

3

Lurie, Sergey A., Alexander L. Kalamkarov, Yury O. Solyaev, and Alexander V. Volkov. "Dilatation gradient elasticity theory." European Journal of Mechanics - A/Solids 88 (July 2021): 104258. http://dx.doi.org/10.1016/j.euromechsol.2021.104258.

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4

Lazar, Markus. "On gradient field theories: gradient magnetostatics and gradient elasticity." Philosophical Magazine 94, no. 25 (July 11, 2014): 2840–74. http://dx.doi.org/10.1080/14786435.2014.935512.

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Gutkin, M. Yu, and E. C. Aifantis. "Edge dislocation in gradient elasticity." Scripta Materialia 36, no. 1 (January 1997): 129–35. http://dx.doi.org/10.1016/s1359-6462(96)00352-1.

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Lazar, Markus, and Gérard A. Maugin. "Dislocations in gradient elasticity revisited." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2075 (June 6, 2006): 3465–80. http://dx.doi.org/10.1098/rspa.2006.1699.

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In this paper, we consider dislocations in the framework of first as well as second gradient theory of elasticity. Using the Fourier transform, rigorous analytical solutions of the two-dimensional bi-Helmholtz and Helmholtz equations are derived in closed form for the displacement, elastic distortion, plastic distortion and dislocation density of screw and edge dislocations. In our framework, it was not necessary to use boundary conditions to fix constants of the solutions. The discontinuous parts of the displacement and plastic distortion are expressed in terms of two-dimensional as well as one-dimensional Fourier-type integrals. All other fields can be written in terms of modified Bessel functions.

7

Hwang, K. C., T. F. Cuo, Y. Huang, and J. Y. Chen. "Fracture in strain gradient elasticity." Metals and Materials 4, no. 4 (July 1998): 593–600. http://dx.doi.org/10.1007/bf03026364.

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Gutkin, M. Yu, and E. C. Aifantis. "Screw dislocation in gradient elasticity." Scripta Materialia 35, no. 11 (December 1996): 1353–58. http://dx.doi.org/10.1016/1359-6462(96)00295-3.

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Giannakopoulos, Antonios E., Stylianos Petridis, and Dimitrios S. Sophianopoulos. "Dipolar gradient elasticity of cables." International Journal of Solids and Structures 49, no. 10 (May 2012): 1259–65. http://dx.doi.org/10.1016/j.ijsolstr.2012.02.008.

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Zervos, A. "Finite elements for elasticity with microstructure and gradient elasticity." International Journal for Numerical Methods in Engineering 73, no. 4 (2008): 564–95. http://dx.doi.org/10.1002/nme.2093.

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11

Mousavi, S. Mahmoud, Juha Paavola, and Djebar Baroudi. "Distributed non-singular dislocation technique for cracks in strain gradient elasticity." Journal of the Mechanical Behavior of Materials 23, no. 3-4 (August 1, 2014): 47–58. http://dx.doi.org/10.1515/jmbm-2014-0007.

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AbstractThe mode III fracture analysis of a cracked graded plane in the framework of classical, first strain gradient, and second strain gradient elasticity is presented in this paper. Solutions to the problem of screw dislocation in graded materials are available in the literature. These solutions include various frameworks such as classical elasticity, and the first strain and second strain gradient elasticity theories. One of the applications of dislocations is the analysis of a cracked medium through distributed dislocation technique. In this article, this technique is used for the mode III fracture analysis of a graded medium in classical elasticity, which results in a system of Cauchy singular integral equations for multiple interacting cracks. Furthermore, the technique is modified for gradient elasticity. Owing to the regularization of the classical singularity, a system of non-singular integral equations is obtained in gradient elasticity. A plane with one crack is studied, and the stress distribution in classical elasticity is compared with those in gradient elasticity theories. The effects of the internal lengths, introduced in gradient elasticity theories, are investigated. Additionally, a plane with two cracks is studied to elaborate the interactions of multiple cracks in both the classical and gradient theories.

12

Eremeyev, V. A., and F. dell’Isola. "Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity." Lobachevskii Journal of Mathematics 41, no. 10 (October 2020): 1992–98. http://dx.doi.org/10.1134/s1995080220100078.

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13

Polizzotto, Castrenze. "Stress gradient versus strain gradient constitutive models within elasticity." International Journal of Solids and Structures 51, no. 9 (May 2014): 1809–18. http://dx.doi.org/10.1016/j.ijsolstr.2014.01.021.

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14

Efremidis,, G. T., and E. C. Aifantis,. "The Coefficient of Geostatic Stress: Gradient Elasticity vs. Classical Elasticity." Journal of the Mechanical Behavior of Materials 18, no. 1 (February 2007): 43–54. http://dx.doi.org/10.1515/jmbm.2007.18.1.43.

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15

Tarasov, Vasily E. "Fractional Gradient Elasticity from Spatial Dispersion Law." ISRN Condensed Matter Physics 2014 (April 3, 2014): 1–13. http://dx.doi.org/10.1155/2014/794097.

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Nonlocal elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak nonlocality) and the integral nonlocal models (strong nonlocality). This paper focuses on the fractional generalization of gradient elasticity that allows us to describe a weak nonlocality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We demonstrate how the continuum limit transforms the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in Riesz's form. A weak nonlocality of power-law type in the nonlocal elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The continuum equations with derivatives of noninteger orders, which are obtained from the lattice model, can be considered as a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.

16

Lazar, Markus. "Incompatible strain gradient elasticity of Mindlin type: screw and edge dislocations." Acta Mechanica 232, no. 9 (June 28, 2021): 3471–94. http://dx.doi.org/10.1007/s00707-021-02999-2.

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AbstractThe fundamental problem of dislocations in incompatible isotropic strain gradient elasticity theory of Mindlin type, unsolved for more than half a century, is solved in this work. Incompatible strain gradient elasticity of Mindlin type is the generalization of Mindlin’s compatible strain gradient elasticity including plastic fields providing in this way a proper eigenstrain framework for the study of defects like dislocations. Exact analytical solutions for the displacement fields, elastic distortions, Cauchy stresses, plastic distortions and dislocation densities of screw and edge dislocations are derived. For the numerical analysis of the dislocation fields, elastic constants and gradient elastic constants have been used taken from ab initio DFT calculations. The displacement, elastic distortion, plastic distortion and Cauchy stress fields of screw and edge dislocations are non-singular, finite, and smooth. The dislocation fields of a screw dislocation depend on one characteristic length, whereas the dislocation fields of an edge dislocation depend on up to three characteristic lengths. For a screw dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type agree with the corresponding ones in simplified incompatible strain gradient elasticity. In the case of an edge dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type are depicted more realistic than the corresponding ones in simplified incompatible strain gradient elasticity. Among others, the Cauchy stress of an edge dislocation obtained in incompatible isotropic strain gradient elasticity of Mindlin type looks more physical in the dislocation core region than the Cauchy stress obtained in simplified incompatible strain gradient elasticity and is in good agreement with the stress fields of an edge dislocation computed in atomistic simulations. Moreover, it is shown that the shape of the dislocation core of an edge dislocation has a more realistic asymmetric form due to its inherent asymmetry in incompatible isotropic strain gradient elasticity of Mindlin type than the dislocation core possessing a cylindrical symmetry in simplified incompatible strain gradient elasticity. It is revealed that the considered theory with the incorporation of three characteristic lengths offers a more realistic description of an edge dislocation than the simplified incompatible strain gradient elasticity with only one characteristic length.

17

Tarasov, Vasily E. "General lattice model of gradient elasticity." Modern Physics Letters B 28, no. 07 (March 13, 2014): 1450054. http://dx.doi.org/10.1142/s0217984914500547.

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In this paper, new lattice model for the gradient elasticity is suggested. This lattice model gives a microstructural basis for second-order strain-gradient elasticity of continuum that is described by the linear elastic constitutive relation with the negative sign in front of the gradient. Moreover, the suggested lattice model allows us to have a unified description of gradient models with positive and negative signs of the strain gradient terms. Possible generalizations of this model for the high-order gradient elasticity and three-dimensional case are also suggested.

18

Eremeyev, Victor A., Antonio Cazzani, and Francesco dell’Isola. "On nonlinear dilatational strain gradient elasticity." Continuum Mechanics and Thermodynamics 33, no. 4 (March 8, 2021): 1429–63. http://dx.doi.org/10.1007/s00161-021-00993-6.

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AbstractWe call nonlinear dilatational strain gradient elasticity the theory in which the specific class of dilatational second gradient continua is considered: those whose deformation energy depends, in an objective way, on the gradient of placement and on the gradient of the determinant of the gradient of placement. It is an interesting particular case of complete Toupin–Mindlin nonlinear strain gradient elasticity: indeed, in it, the only second gradient effects are due to the inhomogeneous dilatation state of the considered deformable body. The dilatational second gradient continua are strictly related to other generalized models with scalar (one-dimensional) microstructure as those considered in poroelasticity. They could be also regarded to be the result of a kind of “solidification” of the strain gradient fluids known as Korteweg or Cahn–Hilliard fluids. Using the variational approach we derive, for dilatational second gradient continua the Euler–Lagrange equilibrium conditions in both Lagrangian and Eulerian descriptions. In particular, we show that the considered continua can support contact forces concentrated on edges but also on surface curves in the faces of piecewise orientable contact surfaces. The conditions characterizing the possible externally applicable double forces and curve forces are found and examined in detail. As a result of linearization the case of small deformations is also presented. The peculiarities of the model is illustrated through axial deformations of a thick-walled elastic tube and the propagation of dilatational waves.

19

Gusev, Andrei A., and Sergey A. Lurie. "Symmetry conditions in strain gradient elasticity." Mathematics and Mechanics of Solids 22, no. 4 (September 21, 2015): 683–91. http://dx.doi.org/10.1177/1081286515606960.

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We study the variational significance of the “order-of-differentiation” symmetry condition of strain gradient elasticity. This symmetry condition stems from the fact that in strain gradient elasticity, one can interchange the order of differentiation in the components of the second displacement gradient tensor. We demonstrate that this symmetry condition is essential for the validity of free variational formulations commonly employed for deriving the field equations of strain gradient elasticity. We show that relying on this additional symmetry condition, one can restrict consideration to strain gradient constitutive equations with a considerably reduced number of independent material coefficients. We explicitly derive a symmetry unified theory of isotropic strain gradient elasticity with only two independent strain gradient material coefficients. The presented theory has simple stability criteria and its factorized displacement form equations of equilibrium allow for expedient identification of the fundamental solutions operative in specific theoretical and application studies.

20

Volkov-Bogorodskii, D. B., and S. A. Lurie. "Eshelby integral formulas in gradient elasticity." Mechanics of Solids 45, no. 4 (August 2010): 648–56. http://dx.doi.org/10.3103/s0025654410040138.

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21

Tsagrakis, Ioannis, Igor S. Yasnikov, and Elias C. Aifantis. "Gradient elasticity for disclinated micro crystals." Mechanics Research Communications 93 (October 2018): 159–62. http://dx.doi.org/10.1016/j.mechrescom.2017.11.007.

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22

Konstantopoulos, Iason, and Elias Aifantis. "Gradient elasticity applied to a crack." Journal of the Mechanical Behavior of Materials 22, no. 5-6 (December 1, 2013): 193–201. http://dx.doi.org/10.1515/jmbm-2013-0026.

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AbstractThe aim of this paper is to investigate the stress and the displacement field of a crack within a robust version of gradient elasticity, focusing at the standard Mode I, II, III problems. Special treatment is attributed to the crack configuration near its tip, deriving the gradient elasticity results that are analogous to the classical asymptotical solutions near the crack tip.

23

Sun, Bohua, and E. C. Aifantis. "Gradient Elasticity Formulations for Micro/Nanoshells." Journal of Nanomaterials 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/846370.

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The focus of this paper is on illustrating how to extend the second author’s gradient theory of elasticity to shells. Three formulations are presented based on the implicit gradient elasticity constitutive relation1 -ld2∇2σij=Cijkl(1-ls2∇2)εkland its two approximations1+ls2∇2-ld2∇2σij=Cijklεklandσij=Cijkl(1+ld2∇2-ls2∇2)εkl.

24

Gutkin, M. Yu, and E. C. Aifantis. "Dislocations and Disclinations in Gradient Elasticity." physica status solidi (b) 214, no. 2 (August 1999): 245–84. http://dx.doi.org/10.1002/(sici)1521-3951(199908)214:2<245::aid-pssb245>3.0.co;2-p.

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25

Fischer, Paul, Julia Mergheim, and Paul Steinmann. "𝒞1 continuous discretization of gradient elasticity." PAMM 9, no. 1 (March 5, 2010): 435–36. http://dx.doi.org/10.1002/pamm.200910190.

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26

Lazopoulos, K. A., and A. K. Lazopoulos. "Strain gradient elasticity and stress fibers." Archive of Applied Mechanics 83, no. 9 (April 24, 2013): 1371–81. http://dx.doi.org/10.1007/s00419-013-0752-7.

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27

Ma, Hansong, Gengkai Hu, Yueguang Wei, and Lihong Liang. "Inclusion problem in second gradient elasticity." International Journal of Engineering Science 132 (November 2018): 60–78. http://dx.doi.org/10.1016/j.ijengsci.2018.07.003.

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28

Fischer, Paul, Markus Klassen, Julia Mergheim, Paul Steinmann, and Ralf Müller. "Isogeometric analysis of 2D gradient elasticity." Computational Mechanics 47, no. 3 (October 30, 2010): 325–34. http://dx.doi.org/10.1007/s00466-010-0543-8.

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29

Polizzotto, Castrenze. "Gradient elasticity and nonstandard boundary conditions." International Journal of Solids and Structures 40, no. 26 (December 2003): 7399–423. http://dx.doi.org/10.1016/j.ijsolstr.2003.06.001.

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30

Lazar, Markus, Gérard A. Maugin, and Elias C. Aifantis. "Dislocations in second strain gradient elasticity." International Journal of Solids and Structures 43, no. 6 (March 2006): 1787–817. http://dx.doi.org/10.1016/j.ijsolstr.2005.07.005.

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31

Papanicolopulos, S. A. "Chirality in isotropic linear gradient elasticity." International Journal of Solids and Structures 48, no. 5 (March 2011): 745–52. http://dx.doi.org/10.1016/j.ijsolstr.2010.11.007.

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32

Lazopoulos, K. A., and A. K. Lazopoulos. "Fractional derivatives and strain gradient elasticity." Acta Mechanica 227, no. 3 (November 14, 2015): 823–35. http://dx.doi.org/10.1007/s00707-015-1489-x.

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33

Owen, David R. "Elasticity with Gradient-Disarrangements: A Multiscale Perspective for Strain-Gradient Theories of Elasticity and of Plasticity." Journal of Elasticity 127, no. 1 (October 6, 2016): 115–50. http://dx.doi.org/10.1007/s10659-016-9599-9.

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34

XU, LIANG, and SHENGPING SHEN. "SIZE-DEPENDENT PIEZOELECTRICITY AND ELASTICITY DUE TO THE ELECTRIC FIELD-STRAIN GRADIENT COUPLING AND STRAIN GRADIENT ELASTICITY." International Journal of Applied Mechanics 05, no. 02 (June 2013): 1350015. http://dx.doi.org/10.1142/s1758825113500154.

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A size-dependent nonclassical Bernoulli–Euler beam model based on the strain gradient elasticity is proposed for piezoelectric nanowires. The governing equations and the corresponding boundary conditions are naturally derived from the variational principle. Different from the classical piezoelectric beam theory, the electric field–strain gradient coupling and the strain gradient elasticity are both taken into account. Static bending problem of a cantilever piezoelectric nanobeam is solved to illustrate the effect of strain gradient. The present model contains material length scale parameters and can capture the size dependent piezoelectricity and elasticity for nanoscale piezoelectric structures. The numerical results reveal that the deflections predicted by the present model are smaller than that by the classical beam theory and the effective electromechanical coupling coefficient is dramatic enhanced by the electric field–strain gradient coupling effect. However, the differences in both the deflections and effective EMC coefficient between the two models are very significant when the beam thickness is very small; they are diminishing with the increase of the beam thickness. This model is helpful for understanding the electromechanically coupling mechanism and in designing piezoelectric nanowires based devices.

35

Lazar, Markus, and Giacomo Po. "On Mindlin’s isotropic strain gradient elasticity: Green tensors, regularization, and operator-split." Journal of Micromechanics and Molecular Physics 03, no. 03n04 (September 2018): 1840008. http://dx.doi.org/10.1142/s2424913018400088.

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The theory of Mindlin’s isotropic strain gradient elasticity of form II is reviewed. Three-dimensional and two-dimensional Green tensors and their first and second derivatives are derived for an unbounded medium. Using an operator-split in Mindlin’s strain gradient elasticity, three-dimensional and two-dimensional regularization function tensors are computed, which are the three-dimensional and two-dimensional Green tensors of a tensorial Helmholtz equation. In addition, a length scale tensor is introduced, which is responsible for the characteristic material lengths of strain gradient elasticity. Moreover, based on the Green tensors of Mindlin’s strain gradient elasticity, point, line and double forces are studied.

36

Lazar, Markus. "Irreducible decomposition of strain gradient tensor in isotropic strain gradient elasticity." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 96, no. 11 (May 27, 2016): 1291–305. http://dx.doi.org/10.1002/zamm.201500278.

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37

Aifantis, Elias C. "On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity." International Journal of Engineering Science 47, no. 11-12 (November 2009): 1089–99. http://dx.doi.org/10.1016/j.ijengsci.2009.07.003.

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38

Peerlings, R. H. J., and N. A. Fleck. "Computational Evaluation of Strain Gradient Elasticity Constants." International Journal for Multiscale Computational Engineering 2, no. 4 (2004): 599–620. http://dx.doi.org/10.1615/intjmultcompeng.v2.i4.60.

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39

Solyaev, Yury, and Sergey A. Lurie. "ESHELBY INTEGRAL FORMULAS IN SECOND GRADIENT ELASTICITY." Nanoscience and Technology: An International Journal 11, no. 2 (2020): 99–107. http://dx.doi.org/10.1615/nanoscitechnolintj.2020031434.

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40

Carta, Giorgio, Terry Bennett, and Harm Askes. "Determination of dynamic gradient elasticity length scales." Proceedings of the Institution of Civil Engineers - Engineering and Computational Mechanics 165, no. 1 (March 2012): 41–47. http://dx.doi.org/10.1680/eacm.2012.165.1.41.

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41

Metrikine, A. V. "On causality of the gradient elasticity models." Journal of Sound and Vibration 297, no. 3-5 (November 2006): 727–42. http://dx.doi.org/10.1016/j.jsv.2006.04.017.

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42

Lazar, Markus. "Non-singular dislocation loops in gradient elasticity." Physics Letters A 376, no. 21 (April 2012): 1757–58. http://dx.doi.org/10.1016/j.physleta.2012.04.009.

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43

Gutkin, M. Yu, and E. C. Aifantis. "Dislocations in the theory of gradient elasticity." Scripta Materialia 40, no. 5 (February 1999): 559–66. http://dx.doi.org/10.1016/s1359-6462(98)00424-2.

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44

Gutkin, M. Yu, K. N. Mikaelyan, and E. C. Aifantis. "Screw dislocation near interface in gradient elasticity." Scripta Materialia 43, no. 6 (August 2000): 477–84. http://dx.doi.org/10.1016/s1359-6462(00)00445-0.

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45

Lam, D. C. C., F. Yang, A. C. M. Chong, J. Wang, and P. Tong. "Experiments and theory in strain gradient elasticity." Journal of the Mechanics and Physics of Solids 51, no. 8 (August 2003): 1477–508. http://dx.doi.org/10.1016/s0022-5096(03)00053-x.

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46

Auffray, N., H. Le Quang, and Q. C. He. "Matrix representations for 3D strain-gradient elasticity." Journal of the Mechanics and Physics of Solids 61, no. 5 (May 2013): 1202–23. http://dx.doi.org/10.1016/j.jmps.2013.01.003.

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47

Cordero, Nicolas M., Samuel Forest, and Esteban P. Busso. "Second strain gradient elasticity of nano-objects." Journal of the Mechanics and Physics of Solids 97 (December 2016): 92–124. http://dx.doi.org/10.1016/j.jmps.2015.07.012.

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48

Papanicolopulos, S. A., A. Zervos, and I. Vardoulakis. "A three-dimensionalC1finite element for gradient elasticity." International Journal for Numerical Methods in Engineering 77, no. 10 (March 5, 2009): 1396–415. http://dx.doi.org/10.1002/nme.2449.

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49

Georgiadis, H. G., P. A. Gourgiotis, and D. S. Anagnostou. "The Boussinesq problem in dipolar gradient elasticity." Archive of Applied Mechanics 84, no. 9-11 (May 29, 2014): 1373–91. http://dx.doi.org/10.1007/s00419-014-0854-x.

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50

Anagnostou, DS, PA Gourgiotis, and HG Georgiadis. "The Cerruti problem in dipolar gradient elasticity." Mathematics and Mechanics of Solids 20, no. 9 (December 29, 2013): 1088–106. http://dx.doi.org/10.1177/1081286513514882.

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Journal articles on the topic 'Gradient of elasticity'

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