The atomistic representation of first strain-gradient elastic tensors

Admal, Nikhil Chandra; Marian, Jaime; Po, Giacomo

doi:10.1016/j.jmps.2016.11.005

Cited by 39 publications

(58 citation statements)

References 65 publications

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“…Nonetheless, the validity of such an approximation can be verified numerically. In order to do so, let us consider the Voigt representation of the tensors D (see Admal et al, 2017). In Voigt notation, the tensor D is represented as an 18 × 18 symmetric matrix D. Its component D αβ is the strain-energy contribution associated to strain-gradient components α and β.…”

Section: The Projection Methodsmentioning

confidence: 99%

A non-singular theory of dislocations in anisotropic crystals

Lazar

Admal

et al. 2018

International Journal of Plasticity

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We develop a non-singular theory of three-dimensional dislocation loops in a particular version of Mindlin's anisotropic gradient elasticity with up to six length scale parameters. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of linearized incompatible elasticity. The non-singular version of all key equations of anisotropic dislocation theory are derived as line integrals, including the Burgers displacement equation with isolated solid angle, the Peach-Koehler stress equation, the Mura-Willis equation for the elastic distortion, and the Peach-Koehler force. The expression for the interaction energy between two dislocation loops as a double line integral is obtained directly, without the use of a stress function. It is shown that all the elastic fields are nonsingular, and that they converge to their classical counterparts a few characteristic lengths away from the dislocation core. In practice, the non-singular fields can be obtained from the classical ones by replacing the classical (singular) anisotropic Green's tensor with the non-singular anisotropic Green's tensor derived by Lazar and Po (2015b). The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Green's tensor depends on a second order symmetric tensor of length scale parameters modeling a weak non-locality, whose structure depends on the specific class of crystal symmetry. The anisotropic Helmholtz operator defined by such tensor admits a Green's function which is used as the spreading function for the Burgers vector density. As a consequence, the Burgers vector density spreads differently in different crystal structures. Two methods are proposed to determine the tensor of length scale parameters, based on independent atomistic calculations of classical and gradient elastic constants. The anisotropic non-singular theory is shown to be in good agreement with molecular statics without fitting parameters, and unlike its singular counterpart, the sign of stress components does not show reversal as the core is approached. Compared to the isotropic solution, the difference in the energy density per unit length between edge and screw dislocations is more pronounced.

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Section: The Projection Methodsmentioning

confidence: 99%

A non-singular theory of dislocations in anisotropic crystals

Lazar

Admal

et al. 2018

International Journal of Plasticity

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“…For convenience, the values of the independent elastic and gradient-elastic constants are reported in table 1. These components are used to populate the elastic tensors C and D [Admal et al (2016), Auffray et al (2013)]. The Voigt structure of the resulting tensors C and D is shown in Fig.…”

Section: A Comparison With Molecular Statics: the Kelvin Problemmentioning

confidence: 99%

“…Despite the fact that these potentials were never fitted to gradient-elastic constants, it can be observed that the analytical predictions are in good agreement with MS calculations, with a maximum error at the origin in the order of 5-30%, depending on the potential used. It should be noted that, compared to the EAM potential, the MEAM potential better compares to the analytical results, possibly as a result of artifacts in gradient-elastic constants evaluated by EAM potentials [Admal et al (2016)].…”

Section: A Comparison With Molecular Statics: the Kelvin Problemmentioning

confidence: 99%

“…In particular, Mindlin's anisotropic strain gradient elasticity has received renewed attention as a tool to solve engineering problems at the micro-and nano-scales for realistic materials [Polizzotto (2018)]. Only recently, the structure of the gradient-elastic tensor has been rationalized for different material symmetry classes [Auffray et al (2013)], and its atomistic representation and ensuing determination from interatomic potentials has become available [Admal et al (2016)].…”

Section: Introductionmentioning

confidence: 99%

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The Green tensor of Mindlin’s anisotropic first strain gradient elasticity

Admal

Lazar

2019

Mater Theory

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We derive the Green tensor of Mindlin's anisotropic first strain gradient elasticity. The Green tensor is valid for arbitrary anisotropic materials, with up to 21 elastic constants and 171 gradient elastic constants in the general case of triclinic media. In contrast to its classical counterpart, the Green tensor is non-singular at the origin, and it converges to the classical tensor a few characteristic lengths away from the origin. Therefore, the Green tensor of Mindlin's first strain gradient elasticity can be regarded as a physical regularization of the classical anisotropic Green tensor. The isotropic Green tensor and other special cases are recovered as particular instances of the general anisotropic result. The Green tensor is implemented numerically and applied to the Kelvin problem with elastic constants determined from interatomic potentials. Results are compared to molecular statics calculations carried out with the same potentials.

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“…These characterize a boundary-value problem in which the body finds itself clamped at the points of in a response dependent position with displacements ≠̂, which is indeed a problem different from the one we intended to solve. 1 The above means that the notion of nonsimple material does not completely characterize a stress gradient elastic material, which instead belongs to the class of micromorphic materials.…”

Section: Formulations Based On the Notion Of Nonsimple Materialsmentioning

confidence: 99%

A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects

Polizzotto

2018

Z Angew Math Mech

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Basing on thermodynamics principles, a stress gradient elasticity theory is presented whereby the material is modeled as a centro‐symmetric micromorphic anisotropic material. A principle of minimum complementary energy and a stationarity principle of Hellinger–Reissner type are provided, by which the boundary conditions are uniquely determined in a form consistent with continuum mechanics. For 3‐D solids, these conditions include, beside the three ordinary boundary conditions for the assigned boundary data (displacements and/or tractions), six extra boundary conditions by which a microstructure/continuum compatibility condition is enforced, generally through the vanishing of the normal derivative of the stress, ∂nbold-italicσ=0, on the whole boundary surface. A typical boundary‐value problem for isotropic materials with a single length scale parameter under static loads is discussed, showing that it is governed by the same Navier PDEs of classical elasticity, along with a set of Helmholtz PDEs as constitutive equations. Size effects are shown to arise from two distinct sources, that is, the spatial fluctuating of the body force and the double curl of the Hookean stress field. It is proved that a formulation of stress gradient elasticity theory based on the pure notion of nonsimple material (like the ones existing in the literature) leads to anomalies in the ordinary and extra boundary conditions. A comparison of the present theory with that by Eringen is also discussed. Two examples of stress gradient structural models are analytically worked out, namely, a thick‐walled cylinder and a Kirchhoff–Love circular plate. For comparison, the solutions for strain gradient material are also presented. Graphic illustrations show that in both examples stiffening size effects like with the ancillary strain gradient models are found.

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The atomistic representation of first strain-gradient elastic tensors

Cited by 39 publications

References 65 publications

A non-singular theory of dislocations in anisotropic crystals

A non-singular theory of dislocations in anisotropic crystals

The Green tensor of Mindlin’s anisotropic first strain gradient elasticity

A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects

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