pith. sign in

arxiv: 2607.01070 · v1 · pith:EKWASDLZnew · submitted 2026-07-01 · 🧮 math-ph · cond-mat.mtrl-sci· math.MP

Scattering, Trapping and Cloaking-Type Effects of Plane Waves by Point Scatterers in Strain Gradient Elasticity

E. Alevras , Th. Zisis , P. A. Gourgiotis This is my paper

Pith reviewed 2026-07-02 04:11 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.mtrl-scimath.MP
keywords strain gradient elasticityplane wave scatteringpoint scattererstrapping modescloaking effectsdispersion regimesGreen's tensorresonance identification
0
0 comments X

The pith

The scattering response of plane waves by point constraints in strain gradient elasticity is governed by the ratio of microinertial and energetic length scales, producing trapping resonances in anomalous dispersion and cloaking-type behavio

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how plane waves interact with clusters of rigid point constraints in an infinite strain gradient elastic medium. A closed-form dynamic Green's tensor is derived that stays bounded at the source, unlike the classical version. This allows modeling the multiple-scattering problem as a finite algebraic system for the reaction forces at each pin. Resonances are found from minima in the determinant of the Green matrix, and the overall behavior splits into two regimes based on the ratio of the two strain gradient lengths. In one regime sharp resonances create localized trapping, while in the other the pins cause only weak perturbations resembling cloaking.

Core claim

The response of time-harmonic P and SV waves to clusters of rigid point constraints is governed primarily by the ratio of the microinertial and energetic strain gradient lengths. In the anomalous dispersion regime, sharp resonances produce strong displacement localization, including perimeter-localized trapping modes in dense circular arrays. In the normal dispersion regime, these resonances are strongly attenuated and the pins behave as weak scatterers, producing a cloaking-type response in which the incident field is only weakly perturbed. The Green's tensor remains bounded at the source, enabling point constraints through superposition of fundamental solutions, and a frequency-domain proc

What carries the argument

The bounded dynamic Green's tensor for plane-strain strain gradient elasticity, which permits direct superposition to model rigid point constraints and reduces the scattering problem to an algebraic system for pin reaction amplitudes.

Load-bearing premise

The strain gradient Green's tensor remains bounded at the source, which allows point constraints to be introduced directly through superposition of fundamental solutions.

What would settle it

A direct computation or measurement showing whether the displacement and its gradients remain finite at the location of a point constraint in a dynamic strain gradient elastic field, as opposed to developing a singularity.

Figures

Figures reproduced from arXiv: 2607.01070 by E. Alevras, P. A. Gourgiotis, Th. Zisis.

Figure 1
Figure 1. Figure 1: Configuration of pins embedded in an infinite elastic microstructured medium subjected to an incident harmonic plane-wave, under plane-strain conditions. and y-axes, respectively. For a purely incident P-wave, BS = 0, and the particle motion is parallel to the direction of propagation. For a purely incident SV-wave, BP = 0, and the particle motion is perpendicular to the direction of propagation while rema… view at source ↗
Figure 2
Figure 2. Figure 2: (a) − (c) detected resonant frequencies, (d) − (f) propagating (non-localized) mode Ω = 39.63393, and (g) − (i) trapping (localized) mode Ω = 44.41294, for a circular configuration, subjected to an incident SV-wave of unit amplitude. In all cases, H = 0, ν = 0.25, ψ = 0◦ , Np = 12. Figs. 2a-c illustrate the detected local minima for a circular arrangement of 12 pins, subjected to unit amplitude SV waves, f… view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of the non-dimensional displacement components and the magnitude of the non-dimensional displacement vector, for a circular configuration, subjected to an incident (a) − (c) P and (d) − (f) SV-wave of unit amplitude. In all cases, H = 0, ν = 0.25, ψ = 0◦ , Ω = 22.22847, Np = 12. In Figs. 3a–c, a P-wave propagates along the positive X-axis with particle motion polarized in the X-direction; here… view at source ↗
Figure 4
Figure 4. Figure 4: Variation of the normalized logarithmic determinant Γ/| min(Γ)| and detected trapping (localized) modes for circular pinned configurations in the low-frequency range 0 < Ω < 200: (a) H = 0, (b) H = 0.1, (c) H = 1, and (d) H = 10. In all cases, ν = 0.25. Ω 𝑎 Ω 𝑏 Ω 𝑐 Ω 𝑑 Γ min Γ Γ min Γ Γ min Γ Γ min Γ [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation of the normalized logarithmic determinant Γ/| min(Γ)| and detected trapping (localized) modes for circular pinned configurations in the high-frequency range 1000 < Ω < 1200: (a) H = 0, (b) H = 0.1, (c) H = 1, and (d) H = 10. In all cases, ν = 0.25. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Distribution of the non-dimensional displacement components and the magnitude of the non-dimensional displacement vector at two nearby trapping (localized) modes, for a circular configuration, subjected to an incident P-wave of unit amplitude. (a)−(c) Ω = 30.28867 and (d)−(f) Ω = 26.86327. In all cases, ν = 0.25, ψ = 0◦ , Np = 12 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Distribution of the non-dimensional displacement components and of the magnitude of the non-dimensional displacement vector at two nearby high-frequency trapping modes for circular configurations with different numbers of pins: (a)–(c) Np = 48, Ω = 1073.76825695; (d)–(f) Np = 96, Ω = 1074.099918. In all cases, H = 0, ν = 0.25, ψ = 0◦ , and the incident wave is SV. (Figs. 8e,f), depending on the displacemen… view at source ↗
Figure 8
Figure 8. Figure 8: Distribution of the non-dimensional displacement components and the magnitude of the non-dimensional displacement vector for a circular configuration, subjected to an incident (a) − (c) P-wave and (d) − (f) SV-wave of unit amplitude. In all cases, H = 1, ν = 0.25, ψ = 0◦ , Ω = 8.07281, Np = 12. 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑈𝑋 𝑈𝑋 𝑈𝑌 𝑈𝑌 𝑈 𝑈 𝑌 𝑋 𝑋 𝑋 𝑌 𝑌 𝑌 𝑌 𝑋 𝑋 𝑋 𝑌 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of the non-dimensional displacement components and the magnitude of the non-dimensional displacement vector for a circular configuration, subjected to an incident (a) − (c) P-wave and (d) − (f) SV-wave of unit amplitude. In all cases, H = 10, ν = 0.25, ψ = 0◦ , Ω = 100, Np = 12. This notion is employed at a conceptual level and should be distinguished from transformation-based elastodynamic cl… view at source ↗
Figure 10
Figure 10. Figure 10: Variation of the normalized determinant of the Green’s matrix Γ/| min(Γ)| for a circular configuration, for representative values of Poisson’s ratio, H = 0, Np = 24 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Distribution of the magnitude of the non-dimensional displacement vector for a circular configuration, shown for different Poisson’s ratio values: (a) Ω = 118.03322, SV incident wave; (b) Ω = 175.03035, P incident wave; (c) Ω = 139.83396, P incident wave; (d) Ω = 28.11431, P incident wave. In all cases, H = 0 and ψ = 0◦ , Np = 24. matrix through the parameter β and modifies the coupling between the prescr… view at source ↗
Figure 12
Figure 12. Figure 12: Distribution of the magnitude of the non-dimensional displacement vector for an equilateral non-convex polygon, subjected to an incident P-wave, shown for different incidence angle values. In all cases, H = 0, ν = 0.25, Ω = 13.520056, Np = 48. 5.5 The effect of multiple pin configurations We finally investigate the response of compound pin arrangements formed by placing multiple circular configurations in… view at source ↗
Figure 13
Figure 13. Figure 13: Distribution of the magnitude of the non-dimensional displacement vector for multiple circular configu￾rations subjected to an incident P-wave at different resonant frequency values. In all cases, H = 0, ν = 0.25, ψ = 0◦ . This behavior is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Distribution of the magnitude of the non-dimensional displacement vector for multiple circular config￾urations subjected to an incident P-wave at close resonant frequencies: (a) 2 offsets, (b) 3 offsets, (c) 4 offsets, each with 12 pins. In all cases, H = 0, ν = 0.25, ψ = 0◦ , and offset spacing = 0.2. This amplification may be interpreted as a consequence of the modified propagation paths cre￾ated by the… view at source ↗
Figure 15
Figure 15. Figure 15: Distribution of the magnitude of the non-dimensional displacement vector for multiple circular configu￾rations subjected to an incident P-wave at resonant frequencies: (a) 2 offsets, (b) 3 offsets, (c) 4 offsets, each with 24 pins. In all cases, H = 0, ν = 0.25, ψ = 0◦ , and offset spacing = 0.3. a superposition of fundamental solutions. This provides the basis for extending point-scatterer methods, origi… view at source ↗
read the original abstract

Wave scattering by localized constraints in microstructured solids is strongly influenced by the interplay of material length scales, dispersion and geometry. This work investigates plane-strain scattering of time-harmonic P and SV waves by clusters of rigid point constraints embedded in an infinite strain gradient elastic medium. A closed-form dynamic Green's tensor is derived for the plane-strain problem. Unlike the classical elastodynamic Green's tensor, the strain gradient Green's tensor remains bounded at the source, enabling point constraints to be introduced directly through superposition of fundamental solutions. The multiple-scattering problem is reduced to a finite-dimensional algebraic system for the pin reaction amplitudes. A frequency-domain procedure is developed to identify resonance-like amplification and trapping. Candidate resonant frequencies are associated with local minima of the Green matrix determinant, while higher-order curvature criteria distinguish trapping-dominated resonances from non-localized scattering responses. The results show that the response is governed primarily by the ratio of the microinertial and energetic strain gradient lengths. In the anomalous dispersion regime, sharp resonances produce strong displacement localization, including perimeter-localized trapping modes in dense circular arrays. In the normal dispersion regime, these resonances are strongly attenuated and the pins behave as weak scatterers, producing a cloaking-type response in which the incident field is only weakly perturbed. The influence of Poisson's ratio, incidence angle and compound pin configurations is also examined, demonstrating how intrinsic material lengths and geometric arrangement can be used to tune scattering, trapping and wave-screening mechanisms in microstructured elastic media.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Central claim rests on standard linear strain gradient elasticity constitutive relations and the boundedness of the derived Green's tensor; no free parameters are explicitly fitted in the abstract, and no new entities are postulated.

free parameters (2)
  • microinertial strain gradient length
    Material length scale parameter controlling dispersion, treated as input rather than fitted.
  • energetic strain gradient length
    Material length scale parameter controlling dispersion, treated as input rather than fitted.
axioms (2)
  • domain assumption Linear strain gradient elasticity constitutive assumptions hold for the infinite medium.
    Foundational modeling choice invoked for the plane-strain problem.
  • domain assumption Time-harmonic plane P and SV waves propagate in the infinite domain.
    Setup assumption for the scattering configuration.

pith-pipeline@v0.9.1-grok · 5816 in / 1272 out tokens · 159975 ms · 2026-07-02T04:11:48.975046+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

85 extracted references

  1. [1]

    R. V. Craster, S. Guenneau, Acoustic metamaterials: negative refraction, imaging, lensing and cloaking, Vol. 166, Springer Science & Business Media, 2012

    2012

  2. [2]

    Kadic, T

    M. Kadic, T. B¨ uckmann, R. Schittny, M. Wegener, Metamaterials beyond electromagnetism, Reports on Progress in Physics 76 (12) (2013) 126501

    2013

  3. [3]

    Romeo, M

    F. Romeo, M. Ruzzene, Wave propagation in linear and nonlinear periodic media: analysis and applications, Vol. 540, Springer Science & Business Media, 2013

    2013

  4. [4]

    Madeo, Generalized continuum mechanics and engineering applications, Elsevier, 2015

    A. Madeo, Generalized continuum mechanics and engineering applications, Elsevier, 2015

    2015

  5. [5]

    G. Rosi, N. Auffray, Anisotropic and dispersive wave propagation within strain-gradient frame- work, Wave Motion 63 (2016) 120–134

    2016

  6. [6]

    Aggelis, S

    D. Aggelis, S. Tsinopoulos, D. Polyzos, An iterative effective medium approximation (IEMA) for wave dispersion and attenuation predictions in particulate composites, suspensions and emulsions, The Journal of the Acoustical Society of America 116 (6) (2004) 3443–3452. 26

    2004

  7. [7]

    Maurel, V

    A. Maurel, V. Pagneux, D. Boyer, F. Lund, Propagation of elastic waves through polycrys- tals: the effects of scattering from dislocation arrays, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462 (2073) (2006) 2607–2623

    2073

  8. [8]

    M. G. Vavva, V. C. Protopappas, L. N. Gergidis, A. Charalambopoulos, D. I. Fotiadis, D. Poly- zos, Velocity dispersion of guided waves propagating in a free gradient elastic plate: Application to cortical bone, The Journal of the Acoustical Society of America 125 (5) (2009) 3414–3427

    2009

  9. [9]

    Papacharalampopoulos, M

    A. Papacharalampopoulos, M. G. Vavva, V. C. Protopappas, D. I. Fotiadis, D. Polyzos, A numerical study on the propagation of Rayleigh and guided waves in cortical bone according to Mindlin’s Form II gradient elastic theory, The Journal of the Acoustical Society of America 130 (2) (2011) 1060–1070

    2011

  10. [10]

    Charalambopoulos, L

    A. Charalambopoulos, L. N. Gergidis, G. Kartalos, On the gradient elastic wave propagation in cylindrical waveguides with microstructure, Composites Part B: Engineering 43 (6) (2012) 2613–2627

    2012

  11. [11]

    Morini, A

    L. Morini, A. Piccolroaz, G. Mishuris, Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity, International Journal of Solids and Structures 51 (18) (2014) 3087–3100

    2014

  12. [12]

    M. B. Muhlestein, B. M. Goldsberry, A. N. Norris, M. R. Haberman, Acoustic scattering from a fluid cylinder with Willis constitutive properties, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474 (2220) (2018) 20180571

    2018

  13. [13]

    Nobili, E

    A. Nobili, E. Radi, A. Vellender, Diffraction of antiplane shear waves and stress concentration in a cracked couple stress elastic material with micro inertia, Journal of the Mechanics and Physics of Solids 124 (2019) 663–680

    2019

  14. [14]

    Nobili, E

    A. Nobili, E. Radi, C. Signorini, A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476 (2235) (2020) 20190822

    2020

  15. [15]

    Brˆ ul´ e, E

    S. Brˆ ul´ e, E. Javelaud, S. Enoch, S. Guenneau, Experiments on seismic metamaterials: molding surface waves, Physical Review Letters 112 (13) (2014) 133901

    2014

  16. [16]

    Colombi, D

    A. Colombi, D. Colquitt, P. Roux, S. Guenneau, R. V. Craster, A seismic metamaterial: The resonant metawedge, Scientific Reports 6 (1) (2016) 27717

    2016

  17. [17]

    Colombi, P

    A. Colombi, P. Roux, S. Guenneau, P. Gueguen, R. V. Craster, Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances, Scientific Reports 6 (1) (2016) 19238

    2016

  18. [18]

    Colombi, R

    A. Colombi, R. V. Craster, D. Colquitt, Y. Achaoui, S. Guenneau, P. Roux, M. Rupin, Elastic wave control beyond band-gaps: shaping the flow of waves in plates and half-spaces with subwavelength resonant rods, Frontiers in Mechanical Engineering 3 (2017) 10

    2017

  19. [19]

    J. M. De Ponti, A. Colombi, R. Ardito, F. Braghin, A. Corigliano, R. V. Craster, Graded elastic metasurface for enhanced energy harvesting, New Journal of Physics 22 (1) (2020) 013013

    2020

  20. [20]

    Evans, R

    D. Evans, R. Porter, Penetration of flexural waves through a periodically constrained thin elastic plate in vacuo and floating on water, Journal of Engineering Mathematics 58 (2007) 317–337

    2007

  21. [21]

    J. N. Reddy, Theory and analysis of elastic plates and shells, CRC press, 2006

    2006

  22. [22]

    M. J. Smith, R. McPhedran, C. G. Poulton, M. H. Meylan, Negative refraction and dispersion phenomena in platonic clusters, Waves in Random and Complex Media 22 (4) (2012) 435–458. 27

    2012

  23. [23]

    Haslinger, N

    S. Haslinger, N. Movchan, A. Movchan, R. McPhedran, Transmission, trapping and filtering of waves in periodically constrained elastic plates, Proceedings of the Royal Society A: Math- ematical, Physical and Engineering Sciences 468 (2137) (2012) 76–93

    2012

  24. [24]

    Haslinger, I

    S. Haslinger, I. Jones, N. Movchan, A. Movchan, Localization in semi-infinite herringbone waveguides, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sci- ences 474 (2211) (2018) 20170590

    2018

  25. [25]

    L´ azaro, M

    M. L´ azaro, M. Mart´ ı-Sabat´ e, R. V. Craster, V. Romero-Garc´ ıa, Weak scattering formulation for flexural waves in thin elastic plates with point-like resonators, Wave Motion (2026) 103702

    2026

  26. [26]

    Alevras, T

    E. Alevras, T. Zisis, P. A. Gourgiotis, Scattering of antiplane SH waves by fractal structures in strain gradient elasticity, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 482 (2331) (2026) 20250640

    2026

  27. [27]

    Gavardinas, A

    I. Gavardinas, A. Giannakopoulos, T. Zisis, A von Karman plate analogue for solving anti-plane problems in couple stress and dipolar gradient elasticity, International Journal of Solids and Structures 148 (2018) 169–180

    2018

  28. [28]

    Madeo, P

    A. Madeo, P. Neff, I.-D. Ghiba, L. Placidi, G. Rosi, Wave propagation in relaxed micromor- phic continua: modeling metamaterials with frequency band-gaps, Continuum Mechanics and Thermodynamics 27 (4) (2015) 551–570

    2015

  29. [29]

    Greenleaf, M

    A. Greenleaf, M. Lassas, G. Uhlmann, On nonuniqueness for Calder´ on’s inverse problem, Math- ematical Research Letters 10 (5) (2003) 685–693

    2003

  30. [30]

    Greenleaf, M

    A. Greenleaf, M. Lassas, G. Uhlmann, Anisotropic conductivities that cannot be detected by EIT, Physiological Measurement 24 (2) (2003) 413–419

    2003

  31. [31]

    J. B. Pendry, D. Schurig, D. R. Smith, Controlling electromagnetic fields, Science 312 (5781) (2006) 1780–1782

    2006

  32. [32]

    Leonhardt, Optical conformal mapping, Science 312 (5781) (2006) 1777–1780

    U. Leonhardt, Optical conformal mapping, Science 312 (5781) (2006) 1777–1780

    2006

  33. [33]

    G. W. Milton, M. Briane, J. R. Willis, On cloaking for elasticity and physical equations with a transformation invariant form, New Journal of Physics 8 (10) (2006) 248

    2006

  34. [34]

    Colquitt, M

    D. Colquitt, M. Brun, M. Gei, A. Movchan, N. V. Movchan, I. S. Jones, Transformation elastodynamics and cloaking for flexural waves, Journal of the Mechanics and Physics of Solids 72 (2014) 131–143

    2014

  35. [35]

    M. Brun, D. Colquitt, I. Jones, A. Movchan, N. Movchan, Transformation cloaking and radial approximations for flexural waves in elastic plates, New Journal of Physics 16 (9) (2014) 093020

    2014

  36. [36]

    P. A. Gourgiotis, D. Bigoni, The dynamics of folding instability in a constrained Cosserat medium, Philosophical Transactions of the Royal Society A: Mathematical, Physical and En- gineering Sciences 375 (2093) (2017) 20160159

    2093

  37. [37]

    Bigoni, P

    D. Bigoni, P. A. Gourgiotis, Folding and faulting of an elastic continuum, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472 (2187) (2016) 20160018

    2016

  38. [38]

    R. A. Toupin, Theories of elasticity with couple-stress, Archive for Rational Mechanics and Analysis 17 (2) (1964) 85–112

    1964

  39. [39]

    A. C. Eringen, Mechanics of micromorphic continua, in: Mechanics of Generalized Continua: Proceedings of the IUTAM-Symposium on the Generalized Cosserat Continuum and the Con- tinuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart (Germany) 1967, Springer, 1968, pp. 18–35. 28

    1967

  40. [40]

    A. C. Eringen, Microcontinuum field theories: I. Foundations and solids, Springer Science & Business Media, 2012

    2012

  41. [41]

    Neff, I.-D

    P. Neff, I.-D. Ghiba, A. Madeo, L. Placidi, G. Rosi, A unifying perspective: the relaxed linear micromorphic continuum, Continuum Mechanics and Thermodynamics 26 (5) (2014) 639–681

    2014

  42. [42]

    Delfani, H

    M. Delfani, H. Shodja, An enhanced continuum modeling of the ideal strength and the angle of twist in tensile behavior of single-walled carbon nanotubes, Journal of Applied Physics 114 (5) (2013) 053521

    2013

  43. [43]

    Delfani, M

    M. Delfani, M. Latifi Shahandashti, Elastic field of a spherical inclusion with non-uniform eigen- fields in second strain gradient elasticity, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473 (2205) (2017) 20170254

    2017

  44. [44]

    Majdoub, P

    M. Majdoub, P. Sharma, T. Cagin, Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect, Physical Review B—Condensed Matter and Materials Physics 77 (12) (2008) 125424

    2008

  45. [45]

    Ahmadpoor, P

    F. Ahmadpoor, P. Sharma, Flexoelectricity in two-dimensional crystalline and biological mem- branes, Nanoscale 7 (40) (2015) 16555–16570

    2015

  46. [46]

    Giannakopoulos, C

    A. Giannakopoulos, C. Knisovitis, A. Charalambopoulos, T. Zisis, A. J. Rosakis, Hyperbolicity, Mach lines, and super-shear mode III steady-state fracture in magneto-flexoelectric materials, part I: methodology, Journal of Applied Mechanics 90 (12) (2023) 121009

    2023

  47. [47]

    Giannakopoulos, C

    A. Giannakopoulos, C. Knisovitis, T. Zisis, A. J. Rosakis, Hyperbolicity, mach lines, and super-shear Mode III steady-state fracture in magneto-flexoelectric materials, Part II: crack- tip asymptotics, Journal of Applied Mechanics 90 (12) (2023) 121010

    2023

  48. [48]

    Knisovitis, A

    C. Knisovitis, A. Giannakopoulos, A. J. Rosakis, Anti-plane Yoffe-type crack in flexoelectric material, Engineering Fracture Mechanics 311 (2024) 110551

    2024

  49. [49]

    Giannakopoulos, C

    A. Giannakopoulos, C. Knisovitis, A. J. Rosakis, Failure induced by a dynamic anti-plane slip pulse in flexoelectric materials, Mechanics of Materials 214 (2025) 105587

    2025

  50. [50]

    Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11 (1) (1962) 385–414

    R. Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11 (1) (1962) 385–414

    1962

  51. [51]

    R. D. Mindlin, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis 16 (1964) 51–78

    1964

  52. [52]

    Jaunzemis, Continuum mechanics, MacMillan, New York, 1967

    W. Jaunzemis, Continuum mechanics, MacMillan, New York, 1967

    1967

  53. [53]

    Mindlin, N

    R. Mindlin, N. Eshel, On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures 4 (1968) 109–124

    1968

  54. [54]

    J. L. Bleustein, A note on the boundary conditions of Toupin’s strain-gradient theory, Inter- national Journal of Solids and Structures 3 (6) (1967) 1053–1057

    1967

  55. [55]

    Georgiadis, C

    H. Georgiadis, C. Grentzelou, Energy theorems and the J-integral in dipolar gradient elasticity, International Journal of Solids and Structures 43 (18-19) (2006) 5690–5712

    2006

  56. [56]

    Georgiadis, E

    H. Georgiadis, E. Velgaki, High-frequency Rayleigh waves in materials with micro-structure and couple-stress effects, International Journal of Solids and Structures 40 (10) (2003) 2501–2520

    2003

  57. [57]

    Lazar, G

    M. Lazar, G. A. Maugin, Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity, International Journal of Engineering Science 43 (13-14) (2005) 1157–1184. 29

    2005

  58. [58]

    G. A. Maugin, Nonlinear waves in elastic crystals, Oxford University Press, 1999

    1999

  59. [59]

    Brock, Plane waves with dispersion and decay: surface reflection for thermoelastic solids with thermal relaxation, Journal of Thermal Stresses 34 (7) (2011) 687–701

    L. Brock, Plane waves with dispersion and decay: surface reflection for thermoelastic solids with thermal relaxation, Journal of Thermal Stresses 34 (7) (2011) 687–701

    2011

  60. [60]

    R. D. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11 (1) (1962) 415–448

    1962

  61. [61]

    P. A. Gourgiotis, D. Bigoni, Stress channelling in extreme couple-stress materials Part I: Strong ellipticity, wave propagation, ellipticity, and discontinuity relations, Journal of the Mechanics and Physics of Solids 88 (2016) 150–168

    2016

  62. [62]

    Achenbach, Wave propagation in elastic solids, Elsevier, 2012

    J. Achenbach, Wave propagation in elastic solids, Elsevier, 2012

    2012

  63. [63]

    P. A. Gourgiotis, H. Georgiadis, I. Neocleous, On the reflection of waves in half-spaces of microstructured materials governed by dipolar gradient elasticity, Wave Motion 50 (3) (2013) 437–455

    2013

  64. [64]

    P. A. Gourgiotis, H. Georgiadis, Torsional and SH surface waves in an isotropic and homogenous elastic half-space characterized by the Toupin–Mindlin gradient theory, International Journal of Solids and Structures 62 (2015) 217–228

    2015

  65. [65]

    Zisis, X

    T. Zisis, X. Kuci, H. Georgiadis, Wave reflection and Rayleigh waves in the context of the complete Toupin–Mindlin theory of strain gradient elasticity, Journal of Mechanics of Materials and Structures 18 (4) (2023) 567–592

    2023

  66. [66]

    Georgiadis, I

    H. Georgiadis, I. Vardoulakis, E. Velgaki, Dispersive Rayleigh-wave propagation in microstruc- tured solids characterized by dipolar gradient elasticity, Journal of Elasticity 74 (2004) 17–45

    2004

  67. [67]

    Maranganti, P

    R. Maranganti, P. Sharma, A novel atomistic approach to determine strain-gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies, Journal of the Mechanics and Physics of Solids 55 (9) (2007) 1823–1852

    2007

  68. [68]

    N. D. Sharma, R. Maranganti, P. Sharma, On the possibility of piezoelectric nanocomposites without using piezoelectric materials, Journal of the Mechanics and Physics of Solids 55 (11) (2007) 2328–2350

    2007

  69. [69]

    Eason, J

    G. Eason, J. Fulton, I. N. Sneddon, The generation of waves in an infinite elastic solid by variable body forces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 248 (955) (1956) 575–607

    1956

  70. [70]

    Polyzos, K

    D. Polyzos, K. Tsepoura, S. Tsinopoulos, D. Beskos, A boundary element method for solving 2-D and 3-D static gradient elastic problems: Part I: Integral formulation, Computer Methods in Applied Mechanics and Engineering 192 (26-27) (2003) 2845–2873

    2003

  71. [71]

    P. A. Gourgiotis, T. Zisis, H. Georgiadis, On concentrated surface loads and Green’s functions in the Toupin–Mindlin theory of strain-gradient elasticity, International Journal of Solids and Structures 130 (2018) 153–171

    2018

  72. [72]

    Giannakopoulos, E

    A. Giannakopoulos, E. Amanatidou, N. Aravas, A reciprocity theorem in linear gradient elastic- ity and the corresponding Saint-Venant principle, International Journal of Solids and Structures 43 (13) (2006) 3875–3894

    2006

  73. [73]

    Srivastava, Elastic metamaterials and dynamic homogenization: a review, International Journal of Smart and Nano Materials 6 (1) (2015) 41–60

    A. Srivastava, Elastic metamaterials and dynamic homogenization: a review, International Journal of Smart and Nano Materials 6 (1) (2015) 41–60

    2015

  74. [74]

    Yavari, A

    A. Yavari, A. Golgoon, Nonlinear and linear elastodynamic transformation cloaking, Archive for Rational Mechanics and Analysis 234 (1) (2019) 211–316. 30

    2019

  75. [75]

    Sozio, A

    F. Sozio, A. Golgoon, A. Yavari, Elastodynamic transformation cloaking for non- centrosymmetric gradient solids, Zeitschrift f¨ ur angewandte Mathematik und Physik 72 (3) (2021) 123

    2021

  76. [76]

    Sozio, M

    F. Sozio, M. F. Shojaei, A. Yavari, Optimal elastostatic cloaks, Journal of the Mechanics and Physics of Solids 176 (2023) 105306

    2023

  77. [77]

    G. N. Greaves, A. L. Greer, R. S. Lakes, T. Rouxel, Poisson’s ratio and modern materials, Nature Materials 10 (11) (2011) 823–837

    2011

  78. [78]

    B¨ uckmann, M

    T. B¨ uckmann, M. Kadic, N. Stenger, M. Thiel, M. Wegener, On the feasibility of pentamode mechanical metamaterials, in: The Sixth International Congress on Advanced Electromagnetic Materials in Microwaves and Optics, 2012

    2012

  79. [79]

    Lakes, Foam structures with a negative Poisson’s ratio, Science 235 (4792) (1987) 1038–1040

    R. Lakes, Foam structures with a negative Poisson’s ratio, Science 235 (4792) (1987) 1038–1040

    1987

  80. [80]

    G. W. Milton, Composite materials with Poisson’s ratios close to -1, Journal of the Mechanics and Physics of Solids 40 (5) (1992) 1105–1137

    1992

Showing first 80 references.