Distributed dislocation approach for cracks in couple-stress elasticity: shear modes

Gourgiotis, P.A.; Georgiadis, H.G.

doi:10.1007/978-1-4020-6929-1_9

Cited by 9 publications

(27 citation statements)

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“…. The same ratio was also obtained by Sternberg and Muki (1967) for the mode I problem and by Gourgiotis and Georgiadis (2007) for the mode II problem in couple-stress elasticity. Of course, from the physical point of view, the case    is of no interest since the characteristic length is a small quantity.…”

Section: Reduction Of the Crack Problem To A System Of Singular Integsupporting

confidence: 71%

“…However, this is not the case in couple-stress elasticity because a discrete climb dislocation produces both normal stresses yy σ and couple-stresses yz m along the dislocation line 0  y . Therefore, it is not possible to satisfy both (41a) and (41b) It is noteworthy that in the mode II crack problem of couple-stress elasticity studied by the present authors (Gourgiotis and Georgiadis, 2007), only a distribution of glide dislocations was indeed sufficient to generate the requisite shear stress yx σ along the crack-faces. This is because a discrete glide dislocation produces neither normal stresses yy σ nor couple-stresses yz m along the crack-line 0  y…”

Section: The Corrective Solutionmentioning

confidence: 74%

“…The additional effort in dealing with the mode I case here is due to the nature of the boundary conditions that arise in couple-stress elasticity (involving rotations and couple-stresses). However, such a situation does not appear in the mode II case of couple-stress elasticity (Gourgiotis and Georgiadis, 2007).…”

Section: Introductionmentioning

confidence: 96%

“…Recently, the present authors (Gourgiotis and Georgiadis, 2007) applied the standard DDT to solve finite-length crack problems, under mode II and mode III conditions, within the framework of couple-stress elasticity. Within this framework, and having solved now the mode I (opening mode) case, a comparison between the two plane-strain crack modes (mode I and mode II) shows that mode I is mathematically more involved than mode II.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An approach based on distributed dislocations and disclinations for crack problems in couple-stress elasticity

Gourgiotis

Georgiadis

2008

International Journal of Solids and Structures

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a b s t r a c tThe technique of distributed dislocations proved to be in the past an effective approach in studying crack problems within classical elasticity. The present work is intended to extend this technique in studying crack problems within couple-stress elasticity, i.e. within a theory accounting for effects of microstructure. This extension is not an obvious one since rotations and couple-stresses are involved in the theory employed to analyze the crack problems. Here, the technique is introduced to study the case of a mode I crack. Due to the nature of the boundary conditions that arise in couple-stress elasticity, the crack is modeled by a continuous distribution of climb dislocations and constrained wedge disclinations (the concept of 'constrained wedge disclination' is first introduced in the present work). These distributions create both standard stresses and couple stresses in the body. In particular, it is shown that the mode-I case is governed by a system of coupled singular integral equations with both Cauchy-type and logarithmic kernels. The numerical solution of this system shows that a cracked solid governed by couple-stress elasticity behaves in a more rigid way (having increased stiffness) as compared to a solid governed by classical elasticity. Also, the stress level at the crack-tip region is appreciably higher than the one predicted by classical elasticity.

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Section: Reduction Of the Crack Problem To A System Of Singular Integsupporting

confidence: 71%

Section: The Corrective Solutionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An approach based on distributed dislocations and disclinations for crack problems in couple-stress elasticity

Gourgiotis

Georgiadis

2008

International Journal of Solids and Structures

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“…It is observed that the couple stress decays rapidly as the distance from the crack-tip increases. Also, it is interesting to note that m xz is bounded at the crack-tip, as in the stationary crack case (Huang et al, 1997;Gourgiotis and Georgiadis, 2007). In particular, the couple-stress m xz takes a finite negative value at the crack tip then increases to a bounded positive maximum, and diminishes monotonically to zero for X >> ℓ, thus recovering the classical solution of linear elasticity.…”

Section: Analytical Representation Of Displacements Stresses and Coumentioning

confidence: 85%

Steady-state propagation of a mode II crack in couple stress elasticity

Gourgiotis

2014

Int J Fract

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The present work deals with the problem of a semi-infinite crack steadily propagating in an elastic body subject to plane-strain shear loading. It is assumed that the mechanical response of the body is governed by the theory of couple-stress elasticity including also micro-rotational inertial effects. This theory introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying microstructure and has proved to be very effective for modeling complex microstructured materials. It is assumed that the crack propagates at a constant sub-Rayleigh speed. An exact full field solution is then obtained based on integral transforms and the Wiener-Hopf technique. Numerical results are presented illustrating the dependence of the stress intensity factor and the energy release rate upon the propagation velocity and the characteristic material lengths in couple-stress elasticity. The present analysis confirms and extends previous results within the context of couple-stress elasticity concerning stationary cracks by including inertial and micro-inertial effects.

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Singularity-free defect mechanics for polar media

Mousavi

2019

Continuum Mech. Thermodyn.

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We present singularity-free solution for cracks within polar media in which material points possess both position and orientation. The plane strain problem is addressed in this study for which the generalized continua including micropolar, nonlocal micropolar, and gradient micropolar elasticity theories are employed. For the first time, the variationally consistent boundary conditions are derived for gradient micropolar elasticity. Moreover, having reviewed the solution to line defects including glide edge dislocation and wedge disclination from the literature, the fields of a climb edge dislocation within micropolar, nonlocal micropolar and gradient micropolar elasticity are derived. This completes the collection of line defects needed for an inplane strain analysis. Afterward, as the main contribution, using both types of line defects (i.e., dislocation being displacement discontinuity and disclination being rotational discontinuity), the well-established dislocation-based fracture mechanics is systematically generalized to the dislocation-and disclination-based fracture mechanics of polar media for which we have three translations together with three rotations as degrees of freedom. Due to the application of the line defects, incompatible elasticity is employed throughout the paper. Cracks under three possible loadings including stress and couple stress components are analyzed, and the corresponding line defect densities and stress and couple stress fields are reported. The singular fields are obtained once using the micropolar elasticity, while nonlocal micropolar, and gradient micropolar elasticity theories give rise to singularity-free fracture mechanics. IntroductionGeneralized (or extended) continua include (strong) nonlocal theories as well as higher-order (or microcontinuum) and higher-grade (or weak nonlocal) extensions to the classical continuum theories. Within nonlocal elasticity, nonlocal stresses at a material point is expressed as a function of weighted values of the entire stress field. On the other hand, by increasing the order of the theory, microcontinuum field theories (e.g., micromorphic elasticity, microstretch elasticity, micropolar elasticity, and dilatation elasticity) are obtained. Further, higher-grade versions include gradient elasticity theories.A microcontinuum is a continuous collection of deformable point particles [14] for which the stress measures include force stress as well as couple stress [13,54,55]. Microcontinuum field theories address media for which the microstructure is no longer rigid (e.g., polar media). These theories are in fact a two-level continuum models where macroscopic deformation field is enhanced with the microscopic deformation. Within Communicated by Victor Eremeyev, Holm Altenbach.

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Distributed dislocation approach for cracks in couple-stress elasticity: shear modes

Cited by 9 publications

References 50 publications

An approach based on distributed dislocations and disclinations for crack problems in couple-stress elasticity

An approach based on distributed dislocations and disclinations for crack problems in couple-stress elasticity

Steady-state propagation of a mode II crack in couple stress elasticity

Singularity-free defect mechanics for polar media

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