A review of continuum mechanics models for size-dependent analysis of beams and plates
Source abstract
Abstract
This paper presents a comprehensive review on the development of higher-order continuum models for capturing size effects in small-scale structures. The review mainly focus on the size-dependent beam, plate and shell models developed based on the nonlocal elasticity theory, modified couple stress theory and strain gradient theory due to their common use in predicting the global behaviour of small-scale structures. In each higher-order continuum theory, various size-dependent models based on the classical theory, first-order shear deformation theory and higher-order shear deformation theory were reviewed and discussed. In addition, the development of finite element solutions for size-dependent analysis of beams and plates was also highlighted. Finally a summary and recommendations for future research are presented. It is hoped that this review paper will provide current knowledge on the development of higher-order continuum models and inspire further applications of these models in predicting the behaviour of micro- and nano-structures.
Source: openaire · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Nonlocal and gradient elasticity in micro/nano structures · Composite Structure Analysis and Optimization · Mechanical and Optical Resonators
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (100+)
Show source-provided references
1. A microbend test method for measuring the plasticity length scale10.1016/S1359-6454(98)00153-0 ↗
2. Mechanical properties of thin films10.1007/BF02666659 ↗
3. Size dependent hardness of silver single crystals10.1557/JMR.1995.0853 ↗
4. Strain gradient plasticity: theory and experiment10.1016/0956-7151(94)90502-9 ↗
5. Strain gradient plasticity effect in indentation hardness of polymers10.1557/JMR.1999.0554 ↗
6. At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola10.1177/1081286513509811 ↗
7. Some cases of unrecognized transmission of scientific knowledge: From antiquity to Gabrio Piola's peridynamics and generalized continuum theories
8. Cosserat · 1909
9. Elastic materials with couple-stresses10.1007/BF00253945 ↗
10. Effects of couple-stresses in linear elasticity10.1007/BF00253946 ↗
11. Couple stresses in the theory of elasticity, I and II
12. Couple stress based strain gradient theory for elasticity10.1016/S0020-7683(02)00152-X ↗
13. Micro-structure in linear elasticity10.1007/BF00248490 ↗
14. Second gradient of strain and surface-tension in linear elasticity10.1016/0020-7683(65)90006-5 ↗
15. Experiments and theory in strain gradient elasticity10.1016/S0022-5096(03)00053-X ↗
16. Simple microfluids10.1016/0020-7225(64)90005-9 ↗
17. Linear theory of micropolar elasticity
18. Micropolar fluids with stretch10.1016/0020-7225(69)90026-3 ↗
19. Strain rate tensors and constitutive equations of inelastic micropolar materials10.1016/j.ijplas.2014.05.009 ↗
20. Micromorphic prism element10.1177/1081286516637115 ↗
21. Micromorphic first-order shear deformable plate element10.1007/s11012-015-0325-7 ↗
22. A nonclassical finite element approach for the nonlinear analysis of micropolar plates
23. Determining material constants in micromorphic theory through phonon dispersion relations10.1016/S0020-7225(02)00321-X ↗
24. Eringen · 1999
25. Eringen · 2001
26. Elasticity theory of materials with long range cohesive forces10.1016/0020-7683(67)90049-2 ↗
27. Linear theory of nonlocal elasticity and dispersion of plane waves10.1016/0020-7225(72)90050-X ↗
28. Nonlocal polar elastic continua10.1016/0020-7225(72)90070-5 ↗
29. On nonlocal elasticity10.1016/0020-7225(72)90039-0 ↗
30. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves10.1063/1.332803 ↗
31. On the gradient approach – relation to Eringen’s nonlocal theory10.1016/j.ijengsci.2011.03.016 ↗
32. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation10.1016/j.jmps.2015.02.001 ↗
33. Ultrasonic wave characteristics of nanorods via nonlocal strain gradient models10.1063/1.3345869 ↗
34. Helical microtubules of graphitic carbon10.1038/354056a0 ↗
35. Geometrically nonlinear free vibration and instability of fluid-conveying nanoscale pipes including surface stress effects10.1007/s10404-015-1669-y ↗
36. Wave characteristics of nanotubes conveying fluid based on the non-classical Timoshenko beam model incorporating surface energies10.1007/s13369-016-2132-4 ↗
37. Size-dependent thermo-mechanical vibration and instability of conveying fluid functionally graded nanoshells based on Mindlin's strain gradient theory10.1016/j.tws.2016.04.009 ↗
38. Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory10.1007/s10404-015-1577-1 ↗
39. Surface stress effect on the vibration and instability of nanoscale pipes conveying fluid based on a size-dependent Timoshenko beam model10.1007/s10409-015-0435-4 ↗
40. Size-dependent nonlinear vibration and instability of embedded fluid-conveying SWBNNTs in thermal environment10.1016/j.physe.2014.04.004 ↗
41. Vibration analysis of fluid-conveying nanotubes with consideration of surface effects10.1016/j.physe.2010.08.026 ↗
42. A simple higher-order theory for laminated composite plates10.1115/1.3167719 ↗
43. A review of theories for the modeling and analysis of functionally graded plates and shells10.1016/j.compstruct.2015.03.010 ↗
44. Calibration of nonlocal scale effect parameter for bending single-layered graphene sheet under molecular dynamics10.1142/S1793292012500336 ↗
45. Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain10.1016/j.physe.2010.03.028 ↗
46. Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams10.1063/1.4820565 ↗
47. Obtaining Eringen’s length scale coefficient for vibrating nonlocal beams via continualization method10.1016/j.jsv.2014.05.002 ↗
48. Eringen’s length scale coefficient for buckling of nonlocal rectangular plates from microstructured beam-grid model10.1016/j.ijsolstr.2014.08.017 ↗
49. Eringen's length-scale coefficients for vibration and buckling of nonlocal rectangular plates with simply supported edges10.1061/(ASCE)EM.1943-7889.0000838 ↗
50. The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes10.1088/0957-4484/18/7/075702 ↗
51. Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: an isogeometric approach10.1007/s00339-017-0887-4 ↗
52. Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity10.1016/j.physe.2017.01.006 ↗
53. Bending of Euler-Bernoulli beams using Eringen’s integral formulation: A paradox resolved10.1016/j.ijengsci.2015.10.013 ↗
54. The small length scale effect for a non-local cantilever beam: a paradox solved10.1088/0957-4484/19/34/345703 ↗
55. Application of nonlocal continuum models to nanotechnology10.1016/S0020-7225(02)00210-0 ↗
56. Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics10.1063/1.1625437 ↗
57. Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity10.1103/PhysRevB.71.195404 ↗
58. Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models10.1016/j.physleta.2006.04.026 ↗
59. Axial vibration of the nanorods with the nonlocal continuum rod model10.1016/j.physe.2009.01.007 ↗
60. Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory10.1016/j.commatsci.2009.04.019 ↗
61. Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory10.1016/j.apm.2010.11.004 ↗
62. Free transverse vibration of an axially loaded non-prismatic single-walled carbon nanotube embedded in a two-parameter elastic medium10.1016/j.commatsci.2010.10.005 ↗
63. Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force10.1142/S0219455411004087 ↗
64. Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method10.1016/j.compstruct.2012.08.024 ↗
65. Nonlocal beam theory for nonlinear vibrations of embedded multiwalled carbon nanotubes in thermal environment10.1007/s11071-011-0142-z ↗
66. Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory10.1016/j.apm.2012.03.032 ↗
67. Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory10.1016/j.compositesb.2013.08.082 ↗
68. Non-linear vibration of nanobeams with various boundary condition based on nonlocal elasticity theory10.1016/j.compositesb.2015.05.030 ↗
69. Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods10.1016/j.commatsci.2012.04.001 ↗
70. Analytical solutions for bending of transversely or axially FG nonlocal beams10.12989/scs.2014.17.5.641 ↗
71. A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation10.1016/j.compstruct.2014.09.023 ↗
72. Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions10.1016/j.compositesb.2015.03.068 ↗
73. Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment10.1016/j.actaastro.2015.03.031 ↗
74. Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams10.1016/j.ijengsci.2016.05.005 ↗
75. Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory10.1016/j.ijengsci.2016.03.001 ↗
76. Nonlocal nonlinear free vibration of functionally graded nanobeams10.1016/j.compstruct.2013.12.006 ↗
77. Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation10.1016/j.ijnonlinmec.2015.09.013 ↗
78. Nonlinear vibration of axially functionally graded non-uniform nanobeams10.1016/j.ijengsci.2016.05.009 ↗
79. Wave propagation in carbon nanotubes via nonlocal continuum mechanics10.1063/1.2141648 ↗
80. Vibration of carbon nanotubes studied using nonlocal continuum mechanics10.1088/0964-1726/15/2/050 ↗
81. Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory10.1088/0022-3727/39/17/029 ↗
82. Vibration of nonlocal Timoshenko beams10.1088/0957-4484/18/10/105401 ↗
83. Beam bending solutions based on nonlocal Timoshenko beam theory10.1061/(ASCE)0733-9399(2008)134:6(475) ↗
84. Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures10.1016/j.physleta.2006.10.093 ↗
85. Application of nonlocal beam models for carbon nanotubes10.1016/j.ijsolstr.2006.12.034 ↗
86. Nonlocal continuum theories of beams for the analysis of carbon nanotubes10.1063/1.2833431 ↗
87. Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM10.1016/j.physe.2009.02.004 ↗
88. Thermal buckling analysis of embedded single-walled carbon nanotubes with arbitrary boundary conditions using the nonlocal timoshenko beam theory10.1080/01495739.2011.616802 ↗
89. Small-scale effect on vibration analysis of single-walled carbon nanotubes embedded in an elastic medium using nonlocal elasticity theory10.1063/1.3151703 ↗
90. Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method10.1016/j.ijengsci.2011.05.010 ↗
91. Free vibration of an embedded single-walled carbon nanotube with various boundary conditions using the RMVT-based nonlocal Timoshenko beam theory and DQ method10.1016/j.physe.2014.12.004 ↗
92. Thermal vibration analysis of carbon nanotubes embedded in two-parameter elastic foundation based on nonlocal Timoshenko's beam theory
93. The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory10.1016/j.commatsci.2011.07.021 ↗
94. Dynamic stability analysis of embedded multi-walled carbon nanotubes in thermal environment10.1016/S0894-9166(16)30007-6 ↗
95. Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon the nonlocal elasticity theory10.1016/j.compstruct.2015.02.068 ↗
96. Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory10.1016/j.compstruct.2012.10.038 ↗
97. Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory10.1016/j.ijengsci.2013.12.003 ↗
98. Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments10.1016/j.compstruct.2015.03.023 ↗
99. Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent functionally graded nanobeams10.1080/15376494.2015.1091524 ↗
100. Nonlocal theories for bending, buckling and vibration of beams10.1016/j.ijengsci.2007.04.004 ↗