Stress-driven nonlocal integral model with discontinuity for size-dependent buckling and bending of cracked nanobeams using Laplace transform

References

• Allgöwer, E. L., K. Glashoff, and H.-O. Peitgen. 1981. Numerical Solution of Nonlinear Equations. Bremen: Springer.
   Google Scholar
• Ansari, R., M. Faraji Oskouie, M. Roghani, and H. Rouhi. 2021. “Nonlinear Analysis of Laminated FG-GPLRC Beams Resting on an Elastic Foundation Based on the Two-Phase Stress-Driven Nonlocal Model.” Acta Mechanica 232 (6): 2183–2199. https://doi.org/10.1007/s00707-021-02935-4
   Web of Science ®Google Scholar
• Ansari, R., R. Gholami, and H. Rouhi. 2012a. “Vibration Analysis of Single-Walled Carbon Nanotubes Using Different Gradient Elasticity Theories.” Composites Part B: Engineering 43 (8): 2985–2989. https://doi.org/10.1016/j.compositesb.2012.05.049
   Web of Science ®Google Scholar
• Ansari, R., R. Gholami, and H. Rouhi. 2013. “Various Gradient Elasticity Theories in Predicting Vibrational Response of Single-Walled Carbon Nanotubes with Arbitrary Boundary Conditions.” Journal of Vibration and Control 19 (5): 708–719. https://doi.org/10.1177/1077546312439223
   Web of Science ®Google Scholar
• Ansari, R., M. F. Oskouie, and R. Gholami. 2016. “Size-Dependent Geometrically Nonlinear Free Vibration Analysis of Fractional Viscoelastic Nanobeams Based on the Nonlocal Elasticity Theory.” Physica E: Low-Dimensional Systems and Nanostructures 75: 266–271. https://doi.org/10.1016/j.physe.2015.09.022
   Web of Science ®Google Scholar
• Ansari, R., H. Ramezannezhad, and R. Gholami. 2012b. “Nonlocal Beam Theory for Nonlinear Vibrations of Embedded Multiwalled Carbon Nanotubes in Thermal Environment.” Nonlinear Dynamics 67 (3): 2241–2254. https://doi.org/10.1007/s11071-011-0142-z
   Web of Science ®Google Scholar
• Apuzzo, A., R. Barretta, R. Luciano, F. M. de Sciarra, and R. Penna. 2017. “Free Vibrations of Bernoulli-Euler Nano-Beams by the Stress-Driven Nonlocal Integral Model.” Composites Part B: Engineering 123: 105–111. https://doi.org/10.1016/j.compositesb.2017.03.057
   Web of Science ®Google Scholar
• Asano, K., H.-C. Tang, C.-Y. Chen, T. Nagoshi, T.-F. M. Chang, D. Yamane, K. Machida, K. Masu, and M. Sone. 2017. “Micro-Bending Testing of Electrodeposited Gold for Applications as Movable Components in MEMS Devices.” Microelectronic Engineering 180: 15–19. https://doi.org/10.1016/j.mee.2017.05.044
   Web of Science ®Google Scholar
• Barretta, R., M. Čanađija, R. Luciano, and F. M. de Sciarra. 2018a. “Stress-Driven Modeling of Nonlocal Thermoelastic Behavior of Nanobeams.” International Journal of Engineering Science 126: 53–67. https://doi.org/10.1016/j.ijengsci.2018.02.012
   Web of Science ®Google Scholar
• Barretta, R., S. A. Faghidian, R. Luciano, C. M. Medaglia, and R. Penna. 2018b. “Free Vibrations of FG Elastic Timoshenko Nano-Beams by Strain Gradient and Stress-Driven Nonlocal Models.” Composites Part B: Engineering 154: 20–32. https://doi.org/10.1016/j.compositesb.2018.07.036
   Web of Science ®Google Scholar
• Benvenuti, E., and A. Simone. 2013. “One-Dimensional Nonlocal and Gradient Elasticity: Closed-Form Solution and Size Effect.” Mechanics Research Communications 48: 46–51. https://doi.org/10.1016/j.mechrescom.2012.12.001
   Web of Science ®Google Scholar
• Bian, P. L., H. Qing, and T. T. Yu. 2022. “A New Finite Element Method Framework for Axially Functionally-Graded Nanobeam with Stress-Driven Two-Phase Nonlocal Integral Model.” Composite Structures 295: 115769. https://doi.org/10.1016/j.compstruct.2022.115769
   Web of Science ®Google Scholar
• Caporale, A., H. Darban, and R. Luciano. 2022. “Exact Closed-Form Solutions for Nonlocal Beams with Loading Discontinuities.” Mechanics of Advanced Materials and Structures 29 (5): 694–704. https://doi.org/10.1080/15376494.2020.1787565
   Web of Science ®Google Scholar
• Caporale, A., R. Luciano, D. Scorza, and S. Vantadori. 2023. “Local-Nonlocal Stress-Driven Model for Multi-Cracked Nanobeams.” International Journal of Solids and Structures 273: 112230. https://doi.org/10.1016/j.ijsolstr.2023.112230
   Web of Science ®Google Scholar
• Challamel, N., and C. M. Wang. 2008. “The Small Length Scale Effect for a Non-Local Cantilever Beam: A Paradox Solved.” Nanotechnology 19 (34): 345703. https://doi.org/10.1088/0957-4484/19/34/345703
   PubMed Web of Science ®Google Scholar
• Darban, H., F. Fabbrocino, L. Feo, and R. Luciano. 2021. “Size-Dependent Buckling Analysis of Nanobeams Resting on Two-Parameter Elastic Foundation through Stress-Driven Nonlocal Elasticity Model.” Mechanics of Advanced Materials and Structures 28 (23): 2408–2416. https://doi.org/10.1080/15376494.2020.1739357
   Web of Science ®Google Scholar
• Darban, H., F. Fabbrocino, and R. Luciano. 2020. “Size-Dependent Linear Elastic Fracture of Nanobeams.” International Journal of Engineering Science 157: 103381. https://doi.org/10.1016/j.ijengsci.2020.103381
   Web of Science ®Google Scholar
• Darban, H., R. Luciano, and M. Basista. 2022. “Free Transverse Vibrations of Nanobeams with Multiple Cracks.” International Journal of Engineering Science 177: 103703. https://doi.org/10.1016/j.ijengsci.2022.103703
   Web of Science ®Google Scholar
• Eringen, A. C. 1972. “Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves.” International Journal of Engineering Science 10 (5): 425–435. https://doi.org/10.1016/0020-7225(72)90050-x
   Web of Science ®Google Scholar
• Eringen, A. C. 1983. “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves.” Journal of Applied Physics 54 (9): 4703–4710. https://doi.org/10.1063/1.332803
   Web of Science ®Google Scholar
• Eringen, A. C. 1987. “Theory of Nonlocal Elasticity and Some Applications.” Res Mechanica 21 (4): 313–342.
   Google Scholar
• Eringen, A. C., and D. G. B. Edelen. 1972. “On Nonlocal Elasticity.” International Journal of Engineering Science 10 (3): 233–248. https://doi.org/10.1016/0020-7225(72)90039-0
   Web of Science ®Google Scholar
• Fakher, M., and S. Hosseini-Hashemi. 2022. “Vibration of Two-Phase Local/Nonlocal Timoshenko Nanobeams with an Efficient Shear-Locking-Free Finite-Element Model and Exact Solution.” Engineering with Computers 38 (1): 231–245. https://doi.org/10.1007/s00366-020-01058-z
   Web of Science ®Google Scholar
• Farajpour, M. R., A. R. Shahidi, and A. Farajpour. 2019. “Elastic Waves in Fluid-Conveying Carbon Nanotubes under Magneto-Hygro-Mechanical Loads via a Two-Phase Local/Nonlocal Mixture Model.” Materials Research Express 6 (8): 0850a8. https://doi.org/10.1088/2053-1591/ab2396
   Web of Science ®Google Scholar
• Feizi, S., S. Javadiyan, C. M. Cooksley, G. Shaghayegh, A. J. Psaltis, P. J. Wormald, and S. Vreugde. 2021. “Green Synthesized Colloidal Silver is Devoid of Toxic Effects on Primary Human Nasal Epithelial Cells in Vitro.” Food and Chemical Toxicology: An International Journal Published for the British Industrial Biological Research Association 157: 112606. https://doi.org/10.1016/j.fct.2021.112606
   PubMed Web of Science ®Google Scholar
• Fernández-Sáez, J., and C. Navarro. 2002. “Fundamental Frequency of Cracked Beams in Bending Vibrations: An Analytical Approach.” Journal of Sound and Vibration 256 (1): 17–31. https://doi.org/10.1006/jsvi.2001.4197
   Web of Science ®Google Scholar
• Fernández-Sáez, J., L. Rubio, and C. Navarro. 1999. “Approximate Calculation of the Fundamental Frequency for Bending Vibrations of Cracked Beams.” Journal of Sound and Vibration 225 (2): 345–352. https://doi.org/10.1006/jsvi.1999.2251
   Web of Science ®Google Scholar
• Fernández-Sáez, J., R. Zaera, J. A. Loya, and J. N. Reddy. 2016. “Bending of Euler-Bernoulli Beams Using Eringen’s Integral Formulation: A Paradox Resolved.” International Journal of Engineering Science 99: 107–116. https://doi.org/10.1016/j.ijengsci.2015.10.013
   Web of Science ®Google Scholar
• Gayen, D., R. Tiwari, and D. Chakraborty. 2019. “Static and Dynamic Analyses of Cracked Functionally Graded Structural Components: A Review.” Composites Part B: Engineering 173: 106982. https://doi.org/10.1016/j.compositesb.2019.106982
   Web of Science ®Google Scholar
• Günay, M. G. 2023a. “Buckling Analysis of Thin-Walled Beams by Two-Phase Local-Nonlocal Integral Model.” Iranian Journal of Science and Technology, Transactions of Mechanical Engineering 47 (2): 765–777. https://doi.org/10.1007/s40997-022-00546-5
   Web of Science ®Google Scholar
• Günay, M. G. 2023b. “Free Vibration Analysis of Thin-Walled Beams Using Two-Phase Local-Nonlocal Constitutive Model.” Journal of Vibration and Acoustics-Transactions of the Asme 145 (3): 031009. https://doi.org/10.1115/1.4056908
   Web of Science ®Google Scholar
• He, Y., H. Qing, and C.-F. Gao. 2020. “Theoretical Analysis of Free Vibration of Microbeams under Different Boundary Conditions Using Stress-Driven Nonlocal Integral Model.” International Journal of Structural Stability and Dynamics 20 (03): 2050040. https://doi.org/10.1142/S0219455420500406
   Google Scholar
• Jiang, P., H. Qing, and C. Gao. 2020. “Theoretical Analysis on Elastic Buckling of Nanobeams Based on Stress-Driven Nonlocal Integral Model.” Applied Mathematics and Mechanics 41 (2): 207–232. https://doi.org/10.1007/s10483-020-2569-6
   Google Scholar
• Karami, B., and M. H. Ghayesh. 2023. “Vibration Characteristics of Sandwich Microshells with Porous Functionally Graded Face Sheets.” International Journal of Engineering Science 189: 103884. https://doi.org/10.1016/j.ijengsci.2023.103884
   Web of Science ®Google Scholar
• Karami, B., M. H. Ghayesh, and N. Fantuzzi. 2024. “Quasi-3D Free and Forced Vibrations of Poroelastic Microplates in the Framework of Modified Couple Stress Theory.” Composite Structures 330: 117840. https://doi.org/10.1016/j.compstruct.2023.117840
   Web of Science ®Google Scholar
• Karami, B., and M. Janghorban. 2023. “Numerical Study on the Static Bending and Forced Vibration of Triclinic Plate with Arbitrary Boundary Conditions.” Archives of Civil and Mechanical Engineering 23 (4): 228. https://doi.org/10.1007/s43452-023-00728-1
   Web of Science ®Google Scholar
• Karami, B., M. Janghorban, and H. Fahham. 2022a. “Forced Vibration Analysis of Anisotropic Curved Panels via a Quasi-3D Model in Orthogonal Curvilinear Coordinate.” Thin-Walled Structures 175: 109254. https://doi.org/10.1016/j.tws.2022.109254
   Web of Science ®Google Scholar
• Karami, B., M. Janghorban, and H. Fahham. 2022b. “On the Stress Analysis of Anisotropic Curved Panels.” International Journal of Engineering Science 172: 103625. https://doi.org/10.1016/j.ijengsci.2022.103625
   Web of Science ®Google Scholar
• Ke, L.-L., J. Yang, S. Kitipornchai, and Y. Xiang. 2009. “Flexural Vibration and Elastic Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials.” Mechanics of Advanced Materials and Structures 16 (6): 488–502. https://doi.org/10.1080/15376490902781175
   Web of Science ®Google Scholar
• Khaniki, H. B. 2018. “Vibration Analysis of Rotating Nanobeam Systems Using Eringen’s Two-Phase Local/Nonlocal Model.” Physica E: Low-Dimensional Systems and Nanostructures 99: 310–319. https://doi.org/10.1016/j.physe.2018.02.008
   Web of Science ®Google Scholar
• Khaniki, H. B., and M. H. Ghayesh. 2023. “Airy Stress Based Nonlinear Forced Vibrations and Internal Resonances of Nonlocal Strain Gradient Nanoplates.” Thin-Walled Structures 192: 111147. https://doi.org/10.1016/j.tws.2023.111147
   Web of Science ®Google Scholar
• Khaniki, H. B., S. Hosseini-Hashemi, and H. B. Khaniki. 2018. “Dynamic Analysis of Nano-Beams Embedded in a Varying Nonlinear Elastic Environment Using Eringen’s Two-Phase Local/Nonlocal Model.” The European Physical Journal Plus 133 (7): 283. https://doi.org/10.1140/epjp/i2018-12128-5
   Web of Science ®Google Scholar
• Kitipornchai, S., L. L. Ke, J. Yang, and Y. Xiang. 2009. “Nonlinear Vibration of Edge Cracked Functionally Graded Timoshenko Beams.” Journal of Sound and Vibration 324 (3-5): 962–982. https://doi.org/10.1016/j.jsv.2009.02.023
   Web of Science ®Google Scholar
• Li, C., L. Q. Yao, W. Q. Chen, and S. Li. 2015. “Comments on Nonlocal Effects in Nano-Cantilever Beams.” International Journal of Engineering Science 87: 47–57. https://doi.org/10.1016/j.ijengsci.2014.11.006
   Web of Science ®Google Scholar
• Loya, J., J. López-Puente, R. Zaera, and J. Fernández-Sáez. 2009. “Free Transverse Vibrations of Cracked Nanobeams Using a Nonlocal Elasticity Model.” Journal of Applied Physics 105 (4): 044309. https://doi.org/10.1063/1.3068370
   Web of Science ®Google Scholar
• Loya, J. A., L. Rubio, and J. Fernández-Sáez. 2006. “Natural Frequencies for Bending Vibrations of Timoshenko Cracked Beams.” Journal of Sound and Vibration 290 (3-5): 640–653. https://doi.org/10.1016/j.jsv.2005.04.005
   Web of Science ®Google Scholar
• Mahmoudpour, E., S. H. Hosseini-Hashemi, and S. A. Faghidian. 2018. “Nonlinear Vibration Analysis of FG Nano-Beams Resting on Elastic Foundation in Thermal Environment Using Stress-Driven Nonlocal Integral Model.” Applied Mathematical Modelling 57: 302–315. https://doi.org/10.1016/j.apm.2018.01.021
   Web of Science ®Google Scholar
• Meng, L., D. Zou, H. Lai, Z. Guo, X. He, Z. Xie, and C. Gao. 2018. “Semi-Analytic Solution of Eringen’s Two-Phase Local/Nonlocal Model for Euler-Bernoulli Beam with Axial Force.” Applied Mathematics and Mechanics 39 (12): 1805–1824. https://doi.org/10.1007/s10483-018-2395-9
   Google Scholar
• Norouzzadeh, A., and R. Ansari. 2017. “Finite Element Analysis of Nano-Scale Timoshenko Beams Using the Integral Model of Nonlocal Elasticity.” Physica E: Low-Dimensional Systems and Nanostructures 88: 194–200. https://doi.org/10.1016/j.physe.2017.01.006
   Web of Science ®Google Scholar
• Okamura, H., K. Watanabe, and T. Takano. 1973. Applications of the compliance concept in fracture mechanics, Progress in flaw growth and fracture toughness testing. ASTM special technical publications, Philadelphia, Pa., 423–438.
   Google Scholar
• Oskouie, M. F., R. Ansari, and H. Rouhi. 2018. “Bending of Euler-Bernoulli Nanobeams Based on the Strain-Driven and Stress-Driven Nonlocal Integral Models: A Numerical Approach.” Acta Mechanica Sinica 34 (5): 871–882. https://doi.org/10.1007/s10409-018-0757-0
   Web of Science ®Google Scholar
• Palmeri, A., and A. Cicirello. 2011. “Physically-Based Dirac’s Delta Functions in the Static Analysis of Multi-Cracked Euler-Bernoulli and Timoshenko Beams.” International Journal of Solids and Structures 48 (14-15): 2184–2195. https://doi.org/10.1016/j.ijsolstr.2011.03.024
   Web of Science ®Google Scholar
• Penna, R. 2023. “Bending Analysis of Functionally Graded Nanobeams Based on Stress-Driven Nonlocal Model Incorporating Surface Energy Effects.” International Journal of Engineering Science 189: 103887. https://doi.org/10.1016/j.ijengsci.2023.103887
   Web of Science ®Google Scholar
• Qing, H. 2022. “Well-Posedness of Two-Phase Local/Nonlocal Integral Polar Models for Consistent Axisymmetric Bending of Circular Microplates.” Applied Mathematics and Mechanics 43 (5): 637–652. https://doi.org/10.1007/s10483-022-2843-9
   Google Scholar
• Qing, H. 2023. “Theoretical Fracture Analysis of Double Cantilever Microbeam Using Two-Phase Local/Nonlocal Integral Models with Discontinuity.” Mechanics Based Design of Structures and Machines: 1–16. https://doi.org/10.1080/15397734.2023.2207622
   Web of Science ®Google Scholar
• Qing, H., and Y. Tang. 2023. “Size-Dependent Fracture Analysis of Centrally-Cracked Nanobeam Using Stress-Driven Two-Phase Local/Nonlocal Integral Model with Discontinuity and Symmetrical Conditions.” Engineering Fracture Mechanics 282: 109193. https://doi.org/10.1016/j.engfracmech.2023.109193
   Web of Science ®Google Scholar
• Qing, H., and L. Wei. 2022. “Linear and Nonlinear Free Vibration Analysis of Functionally Graded Porous Nanobeam Using Stress-Driven Nonlocal Integral Model.” Communications in Nonlinear Science and Numerical Simulation 109: 106300. https://doi.org/10.1016/j.cnsns.2022.106300
   Web of Science ®Google Scholar
• Ren, Y., and H. Qing. 2021b. “On the Consistency of Two-Phase Local/Nonlocal Piezoelectric Integral Model.” Applied Mathematics and Mechanics 42 (11): 1581–1598. https://doi.org/10.1007/s10483-021-2785-7
   Google Scholar
• Ren, Y., and H. Qing. 2022. “Bending and Buckling Analysis of Functionally Graded Timoshenko Nanobeam Using Two-Phase Local/Nonlocal Piezoelectric Integral Model.” Composite Structures 300: 116129. https://doi.org/10.1016/j.compstruct.2022.116129
   Web of Science ®Google Scholar
• Ren, Y.-M., and H. Qing. 2021a. “Bending and Buckling Analysis of Functionally Graded Euler–Bernoulli Beam Using Stress-Driven Nonlocal Integral Model with Bi-Helmholtz Kernel.” International Journal of Applied Mechanics 13 (04): 2150041. https://doi.org/10.1142/S1758825121500411
   Google Scholar
• Romano, G., and R. Barretta. 2017. “Stress-Driven versus Strain-Driven Nonlocal Integral Model for Elastic Nano-Beams.” Composites Part B: Engineering 114: 184–188. https://doi.org/10.1016/j.compositesb.2017.01.008
   Web of Science ®Google Scholar
• Romano, G., R. Barretta, M. Diaco, and F. M. de Sciarra. 2017. “Constitutive Boundary Conditions and Paradoxes in Nonlocal Elastic Nanobeams.” International Journal of Mechanical Sciences 121: 151–156. https://doi.org/10.1016/j.ijmecsci.2016.10.036
   Web of Science ®Google Scholar
• Scorza, D., R. Luciano, and S. Vantadori. 2022. “Fracture Behaviour of Nanobeams through Two-Phase Local/Nonlocal Stress-Driven Model.” Composite Structures 280: 114957. https://doi.org/10.1016/j.compstruct.2021.114957
   Web of Science ®Google Scholar
• Shahsavari, D., and B. Karami. 2022. “Assessment of Reuss, Tamura, and LRVE Models for Vibration Analysis of Functionally Graded Nanoplates.” Archives of Civil and Mechanical Engineering 22 (2): 92.https://doi.org/10.1007/s43452-022-00409-5
   Web of Science ®Google Scholar
• Singh, T., S. Shukla, P. Kumar, V. Wahla, and V. K. Bajpai. 2017. “Application of Nanotechnology in Food Science: Perception and Overview.” Frontiers in Microbiology 8: 1501. https://doi.org/10.3389/fmicb.2017.01501
   PubMed Web of Science ®Google Scholar
• Stan, G., C. V. Ciobanu, P. M. Parthangal, and R. F. Cook. 2007. “Diameter-Dependent Radial and Tangential Elastic Moduli of ZnO Nanowires.” Nano Letters 7 (12): 3691–3697. doi:. https://doi.org/10.1021/nl071986e
   Web of Science ®Google Scholar
• Tada, H., P. C. Paris, and G. R. Irwin. 2000. Stress Analysis of Cracks Handbook, 3rd ed. New York: ASME Press.
   Google Scholar
• Tang, Y., and H. Qing. 2023. “Bi-Helmholtz Kernel Based Stress-Driven Nonlocal Integral Model with Discontinuity for Size-Dependent Fracture Analysis of Edge-Cracked Nanobeam.” Mechanics of Advanced Materials and Structures: 1–11. https://doi.org/10.1080/15376494.2023.2214922
   Web of Science ®Google Scholar
• Thrall, J. H. 2004. “Nanotechnology and Medicine.” Radiology 230 (2): 315–318. https://doi.org/10.1148/radiol.2302031698
   PubMed Web of Science ®Google Scholar
• Torabi, K., and J. N. Dastgerdi. 2012. “An Analytical Method for Free Vibration Analysis of Timoshenko Beam Theory Applied to Cracked Nanobeams Using a Nonlocal Elasticity Model.” Thin Solid Films. 520 (21): 6595–6602. https://doi.org/10.1016/j.tsf.2012.06.063
   Web of Science ®Google Scholar
• Tuna, M., and M. Kirca. 2016. “Exact Solution of Eringen’s Nonlocal Integral Model for Bending of Euler-Bernoulli and Timoshenko Beams.” International Journal of Engineering Science 105: 80–92. https://doi.org/10.1016/j.ijengsci.2016.05.001
   Web of Science ®Google Scholar
• Tuna, M., M. Kirca, and P. Trovalusci. 2019. “Deformation of Atomic Models and Their Equivalent Continuum Counterparts Using Eringen’s Two-Phase Local/Nonlocal Model.” Mechanics Research Communications 97: 26–32. https://doi.org/10.1016/j.mechrescom.2019.04.004
   Web of Science ®Google Scholar
• Vaccaro, M. S., R. Barretta, F. Marotti de Sciarra, and J. N. Reddy. 2022. “Nonlocal Integral Elasticity for Third-Order Small-Scale Beams.” Acta Mechanica 233 (6): 2393–2403. https://doi.org/10.1007/s00707-022-03210-w
   Web of Science ®Google Scholar
• Vaccaro, M. S., F. Marotti de Sciarra, and R. Barretta. 2021. “On the Regularity of Curvature Fields in Stress-Driven Nonlocal Elastic Beams.” Acta Mechanica 232 (7): 2595–2603. https://doi.org/10.1007/s00707-021-02967-w
   Web of Science ®Google Scholar
• Varadan, V. K., A. S. Pillai, D. Mukherji, M. Dwivedi, and L. Chen. 2010. Nanoscience and nanotechnology in engineering. Singapore: World Scientific Publishing Company.
   Google Scholar
• Wang, Y. B., X. W. Zhu, and H. H. Dai. 2016. “Exact Solutions for the Static Bending of Euler-Bernoulli Beams Using Eringen’s Two-Phase Local/Nonlocal Model.” AIP Advances 6 (8): 085114. https://doi.org/10.1063/1.4961695
   Web of Science ®Google Scholar
• Wei, D., Y. Liu, and Z. Xiang. 2012. “An Analytical Method for Free Vibration Analysis of Functionally Graded Beams with Edge Cracks.” Journal of Sound and Vibration 331 (7): 1686–1700. https://doi.org/10.1016/j.jsv.2011.11.020
   Web of Science ®Google Scholar
• Xu, C., Y. Li, M. Lu, and Z. Dai. 2022a. “Buckling Analysis of Functionally Graded Nanobeams under Non-Uniform Temperature Using Stress-Driven Nonlocal Elasticity.” Applied Mathematics and Mechanics 43 (3): 355–370. https://doi.org/10.1007/s10483-022-2828-5
   Google Scholar
• Xu, C., Y. Li, M. Lu, and Z. Dai. 2022b. “Stress-Driven Nonlocal Timoshenko Beam Model for Buckling Analysis of Carbon Nanotubes Constrained by Surface Van Der Waals Interactions.” Microsystem Technologies 28 (5): 1115–1127. https://doi.org/10.1007/s00542-022-05266-z
   Web of Science ®Google Scholar
• Yee, K., O. Z. S. Ong, M. H. Ghayesh, and M. Amabili. 2024. “Various Homogenisation Schemes for Vibration Characteristics of Axially FG Core Multilayered Microbeams with Metal Foam Face Layers Based on Third Order Shear Deformation Theory.” Applied Mathematical Modelling 125: 189–217. https://doi.org/10.1016/j.apm.2023.08.037
   Web of Science ®Google Scholar
• Yoon, H.-I., I.-S. Son, and S.-J. Ahn. 2007. “Free Vibration Analysis of Euler-Bernoulli Beam with Double Cracks.” Journal of Mechanical Science and Technology 21 (3): 476–485. https://doi.org/10.1007/BF02916309
   Web of Science ®Google Scholar
• Zhang, P., and H. Qing. 2022a. “Buckling Analysis of Curved Sandwich Microbeams Made of Functionally Graded Materials via the Stress-Driven Nonlocal Integral Model.” Mechanics of Advanced Materials and Structures 29 (9): 1211–1228. https://doi.org/10.1080/15376494.2020.1811926
   Web of Science ®Google Scholar
• Zhang, P., and H. Qing. 2022b. “Well-Posed Two-Phase Nonlocal Integral Models for Free Vibration of Nanobeams in Context with Higher-Order Refined Shear Deformation Theory.” Journal of Vibration and Control 28 (23-24): 3808–3822. https://doi.org/10.1177/10775463211039902
   Web of Science ®Google Scholar
• Zhang, J.-Q., H. Qing, and C.-F. Gao. 2020a. “Exact and Asymptotic Bending Analysis of Microbeams under Different Boundary Conditions Using Stress-Derived Nonlocal Integral Model.” Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik 100 (1): e201900148. https://doi.org/10.1002/zamm.201900148
   Web of Science ®Google Scholar
• Zhang, P., H. Qing, and C. F. Gao. 2020b. “Analytical Solutions of Static Bending of Curved Timoshenko Microbeams Using Eringen’s Two-Phase Local/Nonlocal Integral Model.” Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik 100 (7):e201900207. https://doi.org/10.1002/zamm.201900207
   PubMed Web of Science ®Google Scholar
• Zhang, P., H. Qing, and C. Gao. 2019a. “Theoretical Analysis for Static Bending of Circular Euler-Bernoulli Beam Using Local and Eringen’s Nonlocal Integral Mixed Model.” Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik 99 (8): e201800329. https://doi.org/10.1002/zamm.201800329
   Web of Science ®Google Scholar
• Zhang, P., H. Qing, and C. Gao. 2019b. “Theoretical Analysis for Static Bending of Circular Euler–Bernoulli Beam Using Local and Eringen’s Nonlocal Integral Mixed Model.” ZAMM - Journal of Applied Mathematics and Mechanics/Zeitschrift Für Angewandte Mathematik Und Mechanik 99 (8): e201800329. https://doi.org/10.1002/zamm.201800329
   Web of Science ®Google Scholar
• Zhang, P., P. Schiavone, and H. Qing. 2022a. “Stress-Driven Local/Nonlocal Mixture Model for Buckling and Free Vibration of FG Sandwich Timoshenko Beams Resting on a Nonlocal Elastic Foundation.” Composite Structures 289: 115473. https://doi.org/10.1016/j.compstruct.2022.115473
   Web of Science ®Google Scholar
• Zhang, P., P. Schiavone, and H. Qing. 2022b. “Two-Phase Local/Nonlocal Mixture Models for Buckling Analysis of Higher-Order Refined Shear Deformation Beams under Thermal Effect.” Mechanics of Advanced Materials and Structures 29 (28): 7605–7622. https://doi.org/10.1080/15376494.2021.2003489
   Web of Science ®Google Scholar
• Zhang, S., M. Wang, C. Chen, D. Shahsavari, B. Karami, and A. Tounsi. 2022c. “Wave Propagation in Carbon Nanotube-Reinforced Nanocomposite Doubly-Curved Shells Resting on a Viscoelastic Foundation.” Waves in Random and Complex Media: 1–24. https://doi.org/10.1080/17455030.2022.2058710
   Web of Science ®Google Scholar