Symmetry-breaking mechanism in viscoelastic recovery time of curved nanobeams under time-varying loads

doi:10.1007/s10483-026-3389-7

摘要/Abstract

Abstract:

The dissipative characteristics of deformation recovery in curved nanobeams are crucial for micro/nano-device reliability. In situ experiments reveal the dependence of recovery characteristics on loading direction, but existing theories fail to elaborate this mechanism, and efficient prediction methods are scarce. This study proposes an elastic-viscoelastic core-shell model under Kirchhoff’s small deformation hypothesis, accounting for surface-inner dissipation and geometric differences. Using the time-domain differential method, we convert the integral model into an equivalent differential model and derive an analytical solution for the deformation recovery. Combined with finite element (FE) analysis, we study the load/geometric effects on the viscoelastic recovery of the nano-circular-arc. The results agree well with the experimental results and the FE simulations. They show that laminated structures and surface dissipation endow the recovery process with two intrinsic characteristic time scales and two stages, including an instantaneous jump and long-term evolution, in which the synergy and competition between bending and axial deformation cause the dependence of recovery behavior on the loading direction and symmetry-breaking phenomena. This study clarifies experimental mechanisms and provides a new dissipation control approach.

Key words: nanowire, viscoelastic recovery, core-shell model, symmetry-breaking, time-varying load

中图分类号:

O345

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(6): 1263-1278.

Xiuquan WANG, Nenghui ZHANG, Hanlin LIU, Jiawei LING, Qiqi LI. Symmetry-breaking mechanism in viscoelastic recovery time of curved nanobeams under time-varying loads[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(6): 1263-1278.

图/表 6

参考文献 33

[1]	PAN, Z. L., LEE, S. H., WANG, K., MAO, Z. Q., and LI, D. Y. Elastic stiffening induces one-dimensional phonons in thin Ta₂Se₃ nanowires. Applied Physics Letters, 120(6), 062201 (2022)
[2]	YANG, W. L., HU, Y. T., and PAN, E. N. Electronic band energy of a bent ZnO piezoelectric semiconductor nanowire. Applied Mathematics and Mechanics (English Edition), 41(6), 833–844 (2020) https://doi.org/10.1007/s10483-020-2619-7
[3]	BORDIN, A., WANG, G. Z., LIU, C. X., TEN HAAF, S. L. D., VAN LOO, N., MAZUR, G. P., XU, D., VAN DRIEL, D., ZATELLI, F., GAZIBEGOVIC, S., BADAWY, G., BAKKERS, E. P. A. M., WIMMER, M., KOUWENHOVEN, L. P., and DVIR, T. Tunable crossed Andreev reflection and elastic cotunneling in hybrid nanowires. Physical Review X, 13(3), 031031(2023)
[4]	XIANG, J. X., WANG, H. Y., ZHOU, J. Z., and LU, Y. Developments and future perspectives in nanowires mechanics. Acta Mechanica Solida Sinica, 38(2), 240–251(2025)
[5]	VLASSOV, S., BOCHAROV, D., POLYAKOV, B., VAHTRUS, M., ŠUTKA, A., ORAS, S., ZADIN, V., and KYRITSAKIS, A. Critical review on experimental and theoretical studies of elastic properties of wurtzite-structured ZnO nanowires. Nanotechnology Reviews, 12(1), 20220505 (2023)
[6]	ALDAVE, D. A., LOPEZ-POLIN, G., MORENO, C., ZAMORA, F., ARES, P., and GÓMEZHERRERO, J. All-dry deterministic transfer of thin gold nanowires for electrical connectivity. Advanced Electronic Materials, 9(7), 2300107 (2023)
[7]	HUANG, G. J., YU, H. P., WANG, X. L., NING, B. B., GAO, J., SHI, Y. Q., ZHU, Y. J., and DUAN, J. L. Highly porous and elastic aerogel based on ultralong hydroxyapatite nanowires for high-performance bone regeneration and neovascularization. Journal of Materials Chemistry B, 9(5), 1277–1287 (2021)
[8]	MARTÍN-MARTÍN, S., DELGADO, Á. V., ARENAS-GUERRERO, P., and JIMÉNEZ, M. L. Electro-orientation of Ag nanowires in viscoelastic fluids. Journal of Colloid and Interface Science, 622, 700–707 (2022)
[9]	CHEN, Y. J., BURGESS, T., AN, X. H., MAI, Y. W., TAN, H. H., ZOU, J., RINGER, S. P., JAGADISH, C., and LIAO, X. Z. Effect of a high density of stacking faults on the Young’s modulus of GaAs nanowires. Nano Letters, 16(3), 1911–1916 (2016)
[10]	WANG, X. G., CHEN, K., ZHANG, Y. Q., WAN, J. C., WARREN, O. L., OH, J., LI, J., MA, E., and SHAN, Z. W. Growth conditions control the elastic and electrical properties of ZnO nanowires. Nano Letters, 15(12), 7886–7892 (2015)
[11]	TANG, D. M., REN, C. L., WANG, M. S., WEI, X. L., KAWAMOTO, N., LIU, C., BANDO, Y., MITOME, M., FUKATA, N., and GOLBERG, D. Mechanical properties of Si nanowires as revealed by in situ transmission electron microscopy and molecular dynamics simulations. Nano Letters, 12(4), 1898–1904 (2012)
[12]	WANG, S. L., SHAN, Z. W., and HUANG, H. The mechanical properties of nanowires. Advanced Science, 4(4), 1600332 (2017)
[13]	CHEN, B., GAO, Q., WANG, Y. B., LIAO, X. Z., MAI, Y. W., TAN, H. H., ZOU, J., RINGER, S. P., and JAGADISH, C. Anelastic behavior in GaAs semiconductor nanowires. Nano Letters, 13(7), 3169–3172 (2013)
[14]	VLASSOV, S., POLYAKOV, B., DOROGIN, L. M., VAHTRUS, M., METS, M., ANTSOV, M., SAAR, R., ROMANOV, A. E., LÕHMUS, A., and LÕHMUS, R. Shape restoration effect in Ag-SiO₂ core-shell nanowires. Nano Letters, 14(9), 5201–5205 (2014)
[15]	VLASSOV, S., POLYAKOV, B., VAHTRUS, M., METS, M., ANTSOV, M., ORAS, S., TARRE, A., ARROVAL, T., LÕHMUS, R., and AARIK, J. Enhanced flexibility and electron-beam-controlled shape recovery in alumina-coated Au and Ag core-shell nanowires. Nanotechnology, 28(50), 505707 (2017)
[16]	CHEN, Z. F., DING, C. R., SUN, J. Y., SUN, Y., and WANG, C. X. Enhanced bendability and viscoelastic behavior in high-quality 2H-SiC@SiO₂ nanowires. Nanotechnology, (2025) https://doi.org/10.1088/1361-6528/ae0419
[17]	CHENG, G. M., MIAO, C. Y., QIN, Q. Q., LI, J., XU, F., HAFTBARADARAN, H., DICKEY, E. C., GAO, H. J., and ZHU, Y. Large anelasticity and associated energy dissipation in single-crystalline nanowires. Nature Nanotechnology, 10(8), 687–691 (2015)
[18]	LI, L., CHEN, G., ZHENG, H., MENG, W. W., JIA, S. F., ZHAO, L. G., ZHAO, P. L., ZHANG, Y., HUANG, S. S., HUANG, T. L., and WANG, J. B. Room-temperature oxygen vacancy migration induced reversible phase transformation during the anelastic deformation in CuO. Nature Communications, 12, 3863 (2021)
[19]	SHENG, H. P., ZHENG, H., CAO, F., WU, S. J., LI, L., LIU, C., ZHAO, D. S., and WANG, J. B. Anelasticity of twinned CuO nanowires. Nano Research, 8(11), 3687–3693 (2015)
[20]	KUBACKOVÁ, J., SLABÝ, C., HORVATH, D., HOVAN, A., IVÁNYI, G. T., VIZSNYICZAI, G., KELEMEN, L., ŽOLDÁK, G., TOMORI, Z., and BÁNÓ, G. Assessing the viscoelasticity of photopolymer nanowires using a three-parameter solid model for bending recovery motion. Nanomaterials, 11(11), 2961 (2021)
[21]	WANG, Y. C., LIANG, B. M., XU, S. G., TIAN, L., MINOR, A. M., and SHAN, Z. W. Tunable anelasticity in amorphous Si nanowires. Nano Letters, 20(1), 449–455 (2020)
[22]	LI, Y. S., FENG, W. J., and WEN, L. Free vibration of piezoelectric semiconductor composite structure with fractional viscoelastic layer. Applied Mathematics and Mechanics (English Edition), 46(4), 683–698 (2025) https://doi.org/10.1007/s10483-025-3237-8
[23]	XU, F., QIN, Q. Q., MISHRA, A., GU, Y., and ZHU, Y. Mechanical properties of ZnO nanowires under different loading modes. Nano Research, 3(4), 271–280 (2010)
[24]	CHEN, C. Q., SHI, Y., ZHANG, Y. S., ZHU, J., and YAN, Y. J. Size dependence of Young’s modulus in ZnO nanowires. Physical Review Letters, 96(7), 075505 (2006)
[25]	JIN, Q. D. and REN, Y. R. Parametric-forced coupling resonance of core-shell nanowires with interfacial damage under weak viscoelastic boundary constraint. European Journal of Mechanics-A, 100, 105022 (2023)
[26]	RU, C. Q. Thermoelastic dissipation of nanowire resonators with surface stress. Physica E: Low-dimensional Systems and Nanostructures, 41(7), 1243–1248 (2009)
[27]	TAWFIK, S. A., OSBORNE, D. A., and SPENCER, M. J. S. Theoretical insight on the origin of anelasticity in zinc oxide nanowires. Nanoscale, 12(4), 2439–2444 (2020)
[28]	XIAO, Y., SHANG, J., KOU, L. Z., and LI, C. Surface deformation-dependent mechanical properties of bending nanowires: an ab initio core-shell model. Applied Mathematics and Mechanics (English Edition), 43(2), 219–232 (2022) https://doi.org/10.1007/s10483-022-2814-6
[29]	CHENG, G. M., YIN, S., LI, C. J., CHANG, T. H., RICHTER, G., GAO, H. J., and ZHU, Y. In-situ TEM study of dislocation interaction with twin boundary and retraction in twinned metallic nanowires. Acta Materialia, 196, 304–312 (2020)
[30]	JOSHI, S. K., SINGH, S. K., and DUBEY, S. Machine learning and molecular dynamics based models to predict the temperature dependent elastic properties of silver nanowires. International Journal for Computational Methods in Engineering Science and Mechanics, 24(5), 345–353 (2023)
[31]	LIN, K. C. and LIN, C. W. Finite deformation of 2-D laminated curved beams with variable curvatures. International Journal of Non-Linear Mechanics, 46(10), 1293–1304 (2011)
[32]	LYU, Q., LI, J. J., and ZHANG, N. H. Quasi-static and dynamical analyses of a thermoviscoelastic Timoshenko beam using the differential quadrature method. Applied Mathematics and Mechanics (English Edition), 40(4), 549–562 (2019) https://doi.org/10.1007/s10483-019-2470-8
[33]	CAI, X. S., ZHANG, N. H., and LIU, H. L. Thickness dependence of viscoelastic stress relaxation of laminated microbeams due to mismatch strain. Applied Mathematics and Mechanics (English Edition), 43(4), 467–478 (2022) https://doi.org/10.1007/s10483-022-2841-5