[en] This study proposes a numerical method for calculating the stress fields in nano-scale multi-phase/composite materials, where the classical continuum theory is inadequate due to the small-scale effects, including intermolecular spaces. The method focuses on weakly nonlocal and inhomogeneous materials and involves post-processing the local stresses obtained using a conventional finite element approach, applying the classical continuum theory to calculate the nonlocal stresses. The capabilities of this method are demonstrated through some numerical examples, namely, a two-dimensional case with a circular inclusion and some commonly used scenarios to model nanocomposites. Representative volume elements of various nanocomposites, including epoxy-based materials reinforced with fumed silica, silica (Nanopox F700), and rubber (Albipox 1000) are subjected to uniaxial tensile deformation combined with periodic boundary conditions. The local and nonlocal stress fields are computed through numerical simulations and after post-processing are compared with each other. The results acquired through the nonlocal theory exhibit a softening effect, resulting in reduced stress concentration and less of a discontinuous behaviour. This research contributes to the literature by proposing an efficient and standardised numerical method for analysing the small-scale stress distribution in small-scale multi-phase materials, providing a method for more accurate design in the nano-scale regime. This proposed method is also easy to implement in standard finite element software that employs classical continuum theory.
Disciplines :
Mechanical engineering
Author, co-author :
Tüfekci, Mertol; Department of Mechanical Engineering, Imperial College London, London, United Kingdom ; Centre for Engineering Research, University of Hertfordshire, Hatfield, United Kingdom
Dear, John P.; Department of Mechanical Engineering, Imperial College London, London, United Kingdom
Salles, Loïc ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Mechanical aspects of turbomachinery and aerospace propulsion
Language :
English
Title :
A finite element based approach for nonlocal stress analysis for multi-phase materials and composites
Publication date :
2024
Journal title :
Engineering with Computers
ISSN :
0177-0667
eISSN :
1435-5663
Publisher :
Springer Science and Business Media Deutschland GmbH
Mertol T\u00FCfekci would like to acknowledge the support of Scientific and Technological Research Council of Turkey (TUBITAK), (fund B\u0130DEB 2213 2016/2) that makes this research possible. The authors would also like to acknowledge computational resources and support provided by the Imperial College Research Computing Service ( http://doi.org/10.14469/hpc/2232 ). For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) license to any Author Accepted Manuscript version.
K.L. Ekinci Electromechanical transducers at the nanoscale: actuation and sensing of motion in nanoelectromechanical systems (NEMS) Small 2005 1 8–9 786 797 10.1002/smll.200500077
H. Koc E. Tufekci A novel approach of bending behavior of carbon nanotubes by combining the effects of higher-order boundary conditions and coupling through doublet mechanics Mech Adv Mater Struct 2023 10.1080/15376494.2023.2263767
J. Peddieson G.R. Buchanan R.P. McNitt Application of nonlocal continuum models to nanotechnology Int J Eng Sci 2003 41 3–5 305 312 10.1016/S0020-7225(02)00210-0
M. Di Paola G. Failla A. Pirrotta A. Sofi M. Zingales The mechanically based non-local elasticity: an overview of main results and future challenges Philos Trans R Soc A Math Phys Eng Sci 2013 10.1098/rsta.2012.0433
R. Rafiee R.M. Moghadam On the modeling of carbon nanotubes: a critical review Compos B Eng 2014 56 435 449 10.1016/j.compositesb.2013.08.037
M.G. Geers R. De Borst W.A. Brekelmans R.H. Peerlings On the use of local strain fields for the determination of the intrinsic length scale Journal De Physique. IV: JP 1998 8 8 167 174 10.1051/jp4:1998821
Z.P. Bažant G. Pijaudier-Cabot Measurement of characteristic length of nonlocal continuum J Eng Mech 1989 10.1061/(asce)0733-9399(1989)115:4(755)
J.P. Salvetat G.A.D. Briggs J.M. Bonard R.R. Bacsa A.J. Kulik T. Stöckli N.A. Burnham L. Forró Elastic and shear moduli of single-walled carbon nanotube ropes Phys Rev Lett 1999 82 5 944 947 10.1103/PhysRevLett.82.944
J. Burghardt R. Brannon J. Guilkey A nonlocal plasticity formulation for the material point method Comput Methods Appl Mech Eng 2012 225–228 55 64 2917496 10.1016/j.cma.2012.03.007
H. Chen C. Meng Y. Liu Modeling elasticity of cubic crystals using a novel nonlocal lattice particle method Comput Mech 2022 69 5 1131 1146 4410047 10.1007/s00466-021-02133-y
Y. Huang A.J. Kinloch Modelling of the toughening mechanisms in rubber-modified epoxy polymers—part I finite element analysis studies J Mater Sci 1992 27 10 2753 2762 10.1007/BF00540702
D. Carolan H.M. Chong A. Ivankovic A.J. Kinloch A.C. Taylor Co-continuous polymer systems: a numerical investigation Comput Mater Sci 2015 98 24 33 10.1016/j.commatsci.2014.10.039
Y. Zhang H. Ren Implicit implementation of the nonlocal operator method: an open source code 2023 London Springer 185 219 10.1007/s00366-021-01537-x
Y.K. Khdir T. Kanit F. Zaïri M. Naït-Abdelaziz Computational homogenization of elastic–plastic composites Int J Solids Struct 2013 50 18 2829 2835 10.1016/j.ijsolstr.2013.03.019
A.R. Melro P.P. Camanho F.M. Andrade Pires S.T. Pinho Micromechanical analysis of polymer composites reinforced by unidirectional fibres: part II-micromechanical analyses Int J Solids Struct 2013 50 11–12 1906 1915 10.1016/j.ijsolstr.2013.02.007
J. Fish Q. Yu Computational mechanics of fatigue and life predictions for composite materials and structures Comput Methods Appl Mech Eng 2002 191 43 4827 4849 10.1016/S0045-7825(02)00401-2
E. Oterkus C. Diyaroglu D. De Meo G. Allegri Fracture modes, damage tolerance and failure mitigation in marine composites 2016 Amsterdam Elsevier Ltd. 79 102 10.1016/B978-1-78242-250-1.00004-1
T. Okabe M. Nishikawa H. Toyoshima A periodic unit-cell simulation of fiber arrangement dependence on the transverse tensile failure in unidirectional carbon fiber reinforced composites Int J Solids Struct 2011 48 20 2948 2959 10.1016/j.ijsolstr.2011.06.012
V.A. Buryachenko On the thermo-elastostatics of heterogeneous materials: I. General integral equation Acta Mech 2010 213 3–4 359 374 10.1007/s00707-010-0282-0 arXiv:0912.4162
V.A. Buryachenko Some general representations in thermoperistatics of random structure composites Int J Multiscale Comput Eng 2014 12 4 331 350 10.1615/IntJMultCompEng.2014010354
V.A. Buryachenko Effective properties of thermoperistatic random structure composites: some background principles Math Mech Solids 2017 22 6 1366 1386 3659619 10.1177/1081286516632581
H. Dong Computationally efficient higher-order three-scale method for nonlocal gradient elasticity problems of heterogeneous structures with multiple spatial scales Appl Math Model 2022 109 426 454 4422314 10.1016/j.apm.2022.05.010
A.C. Eringen On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves J Appl Phys 1983 54 9 4703 4710 10.1063/1.332803 arXiv:0021.8979
H.M. Numanoğlu H. Ersoy B. Akgöz Ö. Civalek A new eigenvalue problem solver for thermo-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method Math Methods Appl Sci 2022 45 2592 2614 4395616 10.1002/mma.7942
Ö. Civalek B. Uzun M. Yaylı B. Akgöz Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method Eur Phys J Plus 2020 10.1140/epjp/s13360-020-00385-w
C. Ömer Civalek B. Uzun M. Özgür Yaylı An effective analytical method for buckling solutions of a restrained fgm nonlocal beam Comput Appl Math 2022 4377276 10.1007/s40314-022-01761-1
D. Albas Şeref H. Ersoy B. Akgöz C. Ömer Civalek Dynamic analysis of a fiber-reinforced composite beam under a moving load by the ritz method Mathematics 2021 10.3390/math9091048
Ç. Demir Çiğdem Ö. Civalek Ömer On the analysis of microbeams Int J Eng Sci 2017 121 14 33 3715723 10.1016/j.ijengsci.2017.08.016
B. Akgöz Ö. Civalek Ömer Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment Compos B Eng 2018 150 68 77 10.1016/j.compositesb.2018.05.049
B. Akgöz C. Ömer Civalek Buckling analysis of functionally graded tapered microbeams via Rayleigh–Ritz method Mathematics 2022 10.3390/math10234429
M. Aydogdu Axial vibration of the nanorods with the nonlocal continuum rod model Phys E 2009 41 5 861 864 10.1016/j.physe.2009.01.007
A.F. Russillo G. Failla R. Barretta F. Marotti de Sciarra On the dynamics of 3D nonlocal solids Int J Eng Sci 2022 180 10.1016/j.ijengsci.2022.103742
E. Tufekci S.A. Aya A nonlocal beam model for out-of-plane static analysis of circular nanobeams Mech Res Commun 2016 76 11 23 10.1016/j.mechrescom.2016.06.002
E. Tufekci S.A. Aya O. Oldac A unified formulation for static behavior of nonlocal curved beams Struct Eng Mech 2016 59 3 475 502 10.12989/sem.2016.59.3.475
S.A. Aya E. Tufekci Modeling and analysis of out-of-plane behavior of curved nanobeams based on nonlocal elasticity Compos B Eng 2017 119 184 195 10.1016/j.compositesb.2017.03.050
E. Tufekci S.A. Aya Nonlocal continuum modeling of curved nanostructures 2018 Amsterdam Elsevier Inc. 101 158 10.1016/B978-0-323-48061-1.00003-8
C.M. Twinkle J. Pitchaimani A semi-analytical nonlocal elasticity model for static stability and vibration behaviour of agglomerated CNTs reinforced nano cylindrical panel under non-uniform edge loads Appl Math Model 2022 103 68 90 4336443 10.1016/j.apm.2021.10.027
Tufekci M, Rendu Q, Yuan J, Dear JP, Salles L, Cherednichenko AV (2020) Stress and modal analysis of a rotating blade and the effects of nonlocality. American Society of Mechanical Engineers, pp 1–12. https://doi.org/10.1115/GT2020-14821. https://asmedigitalcollection.asme.org/GT/proceedings/GT2020/84225/Virtual, Online/1095287
A.A. Pisano A. Sofi P. Fuschi Nonlocal integral elasticity: 2D finite element based solutions Int J Solids Struct 2009 46 21 3836 3849 10.1016/j.ijsolstr.2009.07.009
A.A. Pisano A. Sofi P. Fuschi Finite element solutions for nonhomogeneous nonlocal elastic problems Mech Res Commun 2009 36 7 755 761 10.1016/j.mechrescom.2009.06.003
T.H. Nguyen T.Q. Bui S. Hirose Smoothing gradient damage model with evolving anisotropic nonlocal interactions tailored to low-order finite elements Comput Methods Appl Mech Eng 2018 328 498 541 3726135 10.1016/j.cma.2017.09.019
Sidorov V, Shitikova M, Badina E, Detina E (2023) Review of nonlocal-in-time damping models in the dynamics of structures. Axioms 12(7). https://doi.org/10.3390/axioms12070676. https://www.mdpi.com/2075-1680/12/7/676
K. Bertoldi D. Bigoni W.J. Drugan Structural interfaces in linear elasticity. Part I: nonlocality and gradient approximations J Mech Phys Solids 2007 55 1 1 34 2284279 10.1016/j.jmps.2006.06.004
M. Javanbakht S. Mirzakhani M. Silani Local vs. nonlocal integral elasticity-based phase field models including surface tension and simulations of single and two variant martensitic transformations and twinning Eng Comput 2023 39 1 489 503 10.1007/s00366-021-01598-y
I. Monetto W.J. Drugan A micromechanics-based nonlocal constitutive equation for elastic composites containing randomly oriented spheroidal heterogeneities J Mech Phys Solids 2004 52 2 359 393 2033977 10.1016/S0022-5096(03)00103-0
W.J. Drugan Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials J Mech Phys Solids 2003 51 9 1745 1772 1994167 10.1016/S0022-5096(03)00049-8
W.J. Drugan J.R. Willis A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites J Mech Phys Solids 1996 44 4 497 524 1381396 10.1016/0022-5096(96)00007-5
Q. Tong S. Li Multiscale coupling of molecular dynamics and peridynamics J Mech Phys Solids 2016 95 169 187 3541868 10.1016/j.jmps.2016.05.032 arXiv:0701029v1 [arXiv:physics]
G. Lu J. Chen A new nonlocal macro–meso-scale consistent damage model for crack modeling of quasi-brittle materials Comput Methods Appl Mech Eng 2020 362 4061800 10.1016/j.cma.2019.112802
A. Candaş E. Oterkus C.E. İmrak Peridynamic simulation of dynamic fracture in functionally graded materials subjected to impact load Eng Comput 2023 39 1 253 267 10.1007/s00366-021-01540-2
M. Paggi P. Wriggers A nonlocal cohesive zone model for finite thickness interfaces—part II: FE implementation and application to polycrystalline materials Comput Mater Sci 2011 50 5 1634 1643 10.1016/j.commatsci.2010.12.021
D. Srivastava C. Wei K. Cho Nanomechanics of carbon nanotubes and composites Appl Mech Rev 2003 56 2 215 229 10.1115/1.1538625
F.M. de Sciarra P. Russo Experimental characterization, predictive mechanical and thermal modeling of nanostructures and their polymer composites 2018 Cambridge Elsevier 10.1016/C2016-0-00081-5
M.O. Steinhauser S. Hiermaier A review of computational methods in materials science: examples from shock-wave and polymer physics Int J Mol Sci 2009 10 12 5135 5216 10.3390/ijms10125135
G. Shahin E.B. Herbold S.A. Hall R.C. Hurley Quantifying the hierarchy of structural and mechanical length scales in granular systems Extrem Mech Lett 2022 10.1016/j.eml.2021.101590
J. Llorca C. González J.M. Molina-Aldareguía J. Segurado R. Seltzer F. Sket M. Rodríguez S. Sádaba R. Muñoz L.P. Canal Multiscale modeling of composite materials: a roadmap towards virtual testing Adv Mater 2011 23 44 5130 5147 10.1002/adma.201101683
A.C. Eringen E.S. Suhubi Nonlinear theory of simple micro-elastic solids-I Int J Eng Sci 1964 2 2 189 203 169423 10.1016/0020-7225(64)90004-7
A.C. Eringen Theory of nonlocal elasticity and some applications Res Mech 1987 21 4 313 342
Eringen A, Wegner J (2003) Nonlocal Continuum field theories, vol 56. Springer, pp B20–B22. https://doi.org/10.1115/1.1553434. http://appliedmechanicsreviews.asmedigitalcollection.asme.org/article.aspx?articleid=1397591
A.C. Eringen Nonlocal continuum mechanics based on distributions Int J Eng Sci 2006 44 3–4 141 147 2226653 10.1016/j.ijengsci.2005.11.002
A.C. Eringen Linear theory of micropolar viscoelasticity Int J Eng Sci 1967 5 2 191 204 10.1016/0020-7225(67)90004-3
M. Tuna M. Kirca Exact solution of Eringen’s nonlocal integral model for vibration and buckling of Euler–Bernoulli beam Int J Eng Sci 2016 107 54 67 10.1016/j.ijengsci.2016.07.004
R. Barretta F. Fabbrocino R. Luciano F. Marotti de Sciarra Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams Phys E Low Dimens Syst Nanostruct 2018 97 13 30 10.1016/j.physe.2017.09.026
M. Tuna L. Leonetti P. Trovalusci M. Kirca ‘Explicit’ and ‘implicit’ non-local continuous descriptions for a plate with circular inclusion in tension Meccanica 2020 55 927 944 4082395 10.1007/s11012-019-01091-3
M. Tüfekci B. Özkal C. Maharaj H. Liu J.P. Dear L. Salles Strain-rate-dependent mechanics and impact performance of epoxy-based nanocomposites Compos Sci Technol 2023 233 10.1016/j.compscitech.2022.109870
A. Pontefisso M. Zappalorto M. Quaresimin An efficient rve formulation for the analysis of the elastic properties of spherical nanoparticle reinforced polymers Comput Mater Sci 2015 96 319 326 10.1016/j.commatsci.2014.09.030
I.V. Singh A.S. Shedbale B.K. Mishra Material property evaluation of particle reinforced composites using finite element approach J Compos Mater 2016 50 2757 2771 10.1177/0021998315612539
R.V. Pucha J. Worthy Representative volume element-based design and analysis tools for composite materials with nanofillers J Compos Mater 2014 48 2117 2129 10.1177/0021998313494916
J.D. Fidelus E. Wiesel F.H. Gojny K. Schulte H.D. Wagner Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites Compos A Appl Sci Manuf 2005 36 1555 1561 10.1016/j.compositesa.2005.02.006
I.M. Gitman H. Askes L.J. Sluys Representative volume: existence and size determination Eng Fract Mech 2007 74 2518 2534 10.1016/j.engfracmech.2006.12.021
G. Catalanotti On the generation of rve-based models of composites reinforced with long fibres or spherical particles Compos Struct 2016 138 84 95 10.1016/j.compstruct.2015.11.039
F.L. Jin X. Li S.J. Park Synthesis and application of epoxy resins: a review J Ind Eng Chem 2015 29 1 11 10.1016/j.jiec.2015.03.026
Z. Yang H. Peng W. Wang T. Liu Crystallization behavior of poly(ϵ-caprolactone)/layered double hydroxide nanocomposites J Appl Polym Sci 2010 116 5 2658 2667 10.1002/app
L.T. Zhuravlev The surface chemistry of amorphous silica. Zhuravlev model Colloids Surf A 2000 173 1–3 1 38 10.1016/S0927-7757(00)00556-2
S. Ghosh A. Kumar V. Sundararaghavan A.M. Waas Non-local modeling of epoxy using an atomistically-informed kernel Int J Solids Struct 2013 50 19 2837 2845 10.1016/j.ijsolstr.2013.04.025
K. Karapiperis M. Ortiz J.E. Andrade Data-Driven nonlocal mechanics: discovering the internal length scales of materials Comput Methods Appl Mech Eng 2021 386 4303351 10.1016/j.cma.2021.114039
G. Kuvyrkin I. Savelyeva A. Sokolov Solution of two-dimensional problems of nonlocal elasticity theory by the finite element method IOP Conf Ser Mater Sci Eng 2021 1191 10.1088/1757-899x/1191/1/012014
S.H. Moghtaderi A. Jedi A.K. Ariffin A review on nonlocal theories in fatigue assessment of solids Materials 2023 10.3390/ma16020831
M. Huang Z. Li Size effects on stress concentration induced by a prolate ellipsoidal particle and void nucleation mechanism Int J Plast 2005 21 8 1568 1590 10.1016/j.ijplas.2004.07.006
A.C. Eringen Vistas of nonlocal continuum physics Int J Eng Sci 1992 30 10 1551 1565 1187113 10.1016/0020-7225(92)90165-D
A. Cemal Eringen B.S. Kim Stress concentration at the tip of crack Mech Res Commun 1974 1 4 233 237 10.1016/0093-6413(74)90070-6
M. Mehdi A.R. Bhagat G.R. Selokar Evaluation of effective elastic moduli using micromechanics IOP Conf Ser Mater Sci Eng 2018 455 1 90 10.1088/1757-899X/455/1/012116
J.J. Luo I.M. Daniel Characterization and modeling of mechanical behavior of polymer/clay nanocomposites Compos Sci Technol 2003 63 11 1607 1616 10.1016/S0266-3538(03)00060-5
A. Bisoi M. Tüfekci V. Öztekin E. Denimal Goy L. Salles Experimental investigation of mechanical properties of additively manufactured fibre-reinforced composite structures for robotic applications Appl Compo Mater 2023 10.1007/s10443-023-10179-9
J. Summerscales N.P. Dissanayake A.S. Virk W. Hall A review of bast fibres and their composites. Part 1—fibres as reinforcements Compos A Appl Sci Manuf 2010 41 10 1329 1335 10.1016/j.compositesa.2010.06.001
J. Summerscales N. Dissanayake A. Virk W. Hall A review of bast fibres and their composites. Part 2—composites Compos A Appl Sci Manuf 2010 41 10 1336 1344 10.1016/j.compositesa.2010.05.020
J. Summerscales A. Virk W. Hall A review of bast fibres and their composites: part 3—modelling Compos A Appl Sci Manuf 2013 44 1 132 139 10.1016/j.compositesa.2012.08.018
A. Daliri J. Zhang C.H. Wang Hybrid polymer composites for high strain rate applications 2016 Amsterdam Elsevier Ltd 121 163 10.1016/B978-1-78242-325-6.00006-2
L. Shan C.Y. Tan X. Shen S. Ramesh M.S. Zarei R. Kolahchi M.H. Hajmohammad The effects of nano-additives on the mechanical, impact, vibration, and buckling/post-buckling properties of composites: a review J Market Res 2023 24 7570 7598 10.1016/j.jmrt.2023.04.267
J. Liu X. Huang K. Zhao Z. Zhu X. Zhu L. An Effect of reinforcement particle size on quasistatic and dynamic mechanical properties of Al-Al2O3 composites J Alloy Compd 2019 797 1367 1371 10.1016/j.jallcom.2019.05.080
