Applied Mathematics and Mechanics (English Edition) ›› 2026, Vol. 47 ›› Issue (5): 1065-1084.doi: https://doi.org/10.1007/s10483-026-3378-8
收稿日期:2025-11-13
修回日期:2026-02-22
发布日期:2026-05-06
Xueru WANG, Pengxin WANG, Junjie CHEN, Chuanyang HUANG, Kai LI†(
)
Received:2025-11-13
Revised:2026-02-22
Published:2026-05-06
Contact:
Kai LI
E-mail:kli@ahjzu.edu.cn
Supported by:中图分类号:
. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(5): 1065-1084.
Xueru WANG, Pengxin WANG, Junjie CHEN, Chuanyang HUANG, Kai LI. Light-powered self-propelling boat via self-rotating liquid crystal elastomer rod[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(5): 1065-1084.
"
| Parameter | Definition | Value | Unit |
|---|---|---|---|
| τ | Thermal relaxation time | 0.3 | s |
| T0 | Ambient temperature | 300 | K |
| m | Total mass of LCE rod and blade | 0.015 | kg |
| g | Gravitational acceleration | 9.81 | m/s2 |
| R | Cross-sectional radius of LCE rod | 0–0.004 | m |
| p | Photothermal power | 0–100 | K/s |
| α | Thermal expansion coefficient | 0–1×10-3 | K-1 |
| β1 | Damping coefficient of blade | 0–2.5 | Kg·m·s-1 |
| β2 | Damping coefficient of boat | 0–2.5 | Kg·m·s-1 |
| Ls | Support span | 0–0.4 | m |
| d | Length of blade | 0.02 | m |
| [34] | ZHAO, J., ZHANG, Z. R., SUN, X. D., ZUO, W., and LI, K. Multi-modal self-sustained motions of a silicone oil paper disc on a surface driven by hot steam. Chaos, Solitons & Fractals, 191, 115898 (2025) |
| [35] | ZHAO, H. Y., CHEN, Z. K., LI, J. H., LUO, Y. W., PENG, Z. K., MAO, G. Y., XIAO, R., and MAO, J. Spatiotemporally programmed dielectric liquid crystal elastomer: electro-reversible 3D morphing via inverse 4D printing. Science Advances, 11, eaeb2289 (2025) |
| [36] | FAN, K. Q., WANG, C. Y., CHEN, C. G., ZHANG, Y., WANG, P. H., and WANG, F. A pendulum-plucked rotor for efficient exploitation of ultralow-frequency mechanical energy. Renewable Energy, 179, 339–350 (2021) |
| [37] | LUO, A. X., ZHANG, Y. L., DAI, X. T., WANG, Y. F., XU, W. H., LU, Y., WANG, M., FAN, K. Q., and WANG, F. An inertial rotary energy harvester for vibrations at ultra-low frequency with high energy conversion efficiency. Applied Energy, 279, 115762 (2020) |
| [38] | XU, P. B., ZHU, H. W., ZHOU, K., REN, X. L., and ZHOU, L. Self-sustained chaotic warning indicator system based on liquid crystal elastomer. Chaos, Solitons & Fractals, 206, 117984 (2026) |
| [39] | WANG, X. C., LI, L. L., ZHAO, J., and DAI, Y. T. Scallop-inspired self-oscillating actuator via liquid crystal elastomer fibers. Chaos, Solitons & Fractals, 202, 117620 (2026) |
| [40] | GE, D. L. and YANG, Q. C. Light-powered self-sustained jumper via splay-aligned liquid crystal elastomer. Chaos, Solitons & Fractals, 206, 117918 (2026) |
| [41] | SUN, X. D. and ZHAO, J. Photomechanical self-oscillation of a bifilar pendulum with liquid crystal elastomeric fiber. Chaos, Solitons & Fractals, 205, 117838 (2026) |
| [42] | ZHOU, L., CHEN, H. M., LI, Z. H., CHEN, X. H., GAO, T., and GE, D. L. Self-oscillation of a photo-actuated dual-beam system under linear optical excitation. Chaos, Solitons & Fractals, 204, 117740 (2026) |
| [43] | XU, P. B., REN, X. L., ZHU, H. W., and YU, Y. Development of a self-sustained chaotic stirrer with electrothermal response liquid crystal elastomers. Chaos, Solitons & Fractals, 203, 117656 (2026) |
| [44] | FANG, X., LOU, J., WANG, J., CHUANG, K. C., WU, H. M., and HUANG, Z. L. A self-excited bistable oscillator with a light-powered liquid crystal elastomer. International Journal of Mechanical Sciences, 271, 109124 (2024) |
| [45] | CHEN, J. J., YE, Q., HU, Z. J., WANG, X. R., and ZHOU, X. G. Jellyfish-inspired non-reciprocal light-fueled self-propelling liquid crystal elastomer-based fluidic saucer. Chaos, Solitons & Fractals, 206, 117889 (2026) |
| [46] | LI, K., WU, H. Y., LIU, Y. F., DAI, Y. T., and YU, Y. Self-swaying of oblique bending cantilevers under steady illumination. European Journal of Mechanics A/Solids, 103, 105187 (2024) |
| [47] | WANG, X. C., DAI, Y. T., and ZHAO, J. Light-driven self-swing of a liquid crystal elastomer fiber-based composite pendulum in magnetic field. Chaos, Solitons & Fractals, 202, 117570 (2026) |
| [48] | WU, H. Y., GE, D. L., QIU, Y. L., LI, K., and XU, P. B. Mechanics of light-fueled bidirectional self-rolling in a liquid crystal elastomer rod on a track. Chaos, Solitons & Fractals, 191, 115901 (2025) |
| [49] | SUN, X., ZHOU, K., and XU, P. B. Chaotic self-beating of left ventricle modeled by liquid crystal elastomer. Thin-Walled Structures, 205, 112540 (2024) |
| [50] | DENG, Z. X., LI, K., PRIIMAGI, A., and ZENG, H. Light-steerable locomotion using zero-elastic-energy modes. Nature Materials, 23, 1728–1735 (2024) |
| [51] | LI, K., XU, P. S., and ZHOU, L. Light-fueled self-rotation of a liquid crystal elastomer rod enabled by lateral constraint. Theoretical and Applied Mechanics Letters, 15, 100569 (2025) |
| [52] | DAI, L., XU, J. W., and XIAO, R. Modeling the stimulus-responsive behaviors of fiber-reinforced soft materials. International Journal of Applied Mechanics, 16, 2450041 (2024) |
| [53] | NIE, Z. Z., WANG, M., HUANG, S., LIU, Z. Y., and YANG, H. Multimodal self-sustainable autonomous locomotions of light-driven Seifert ribbon actuators based on liquid crystal elastomers. Angewandte Chemie International Edition, 62, e202304081 (2023) |
| [54] | QIU, Y. L., DAI, Y. T., and LI, K. Multimodal self-operation of a liquid crystal elastomer spring-linkage mechanism under constant light. International Journal of Solids and Structures, 302, 112998 (2024) |
| [55] | CHEN, J. J., QIU, Y. L., ZHOU, X. G., GE, D. L., and LI, K. Impatiens-inspired self-sustained photo-ejector with liquid crystal elastomer shell. Thin-Walled Structures, 219, 114312 (2025) |
| [1] | KRUSE, K. and JÜLICHER, F. Oscillations in cell biology. Current Opinion in Cell Biology, 17, 20–26 (2005) |
| [2] | LIAO, W. and YANG, Z. Q. The integration of sensing and actuating based on a simple design fiber actuator towards intelligent soft robots. Advanced Materials Technologies, 7, 2101260 (2022) |
| [3] | GE, D. L., BAO, W., LI, K., and LIANG, H. Y. Self-oscillation-driven locomotion in a liquid crystal elastomer-based robot under constant illumination. Communications in Nonlinear Science and Numerical Simulation, 145, 108706 (2025) |
| [4] | BAI, C. P., KANG, J. T., and WANG, Y. Q. Light-induced motion of three-dimensional pendulum with liquid crystal elastomeric fiber. International Journal of Mechanical Sciences, 266, 108911 (2024) |
| [5] | LIU, J. Q., XU, L. C., JI, Q. X., CHANG, L. F., HU, Y., PENG, Q. Y., and HE, X. D. A mxene-based light-driven actuator and motor with self-sustained oscillation for versatile applications. Advanced Functional Materials, 34, 2310955 (2024) |
| [6] | XU, P. B., ZHANG, J. H., ZHOU, K., WANG, M. H., and HU, W. F. Self-sustaining chaotic characteristics of liquid crystal elastomer pendulum under different linear temperature fields. Chaos, Solitons & Fractals, 205, 117812 (2026) |
| [7] | GE, D. L., BAO, W., CHEN, H. M., and LI, K. A liquid crystal elastomer-based generator using light-powered self-oscillations. Chaos, Solitons & Fractals, 199, 116690 (2025) |
| [8] | QIU, Y. L. and LI, K. Light-powered self-striking liquid crystal elastomer hammers inspired by mantis shrimp. Communications in Nonlinear Science and Numerical Simulation, 146, 108802 (2025) |
| [9] | LI, Z. H., XIA, D., KANG, S. Z., LI, Y., and LI, T. A comparative study of multi-tentacled underwater robot with different self-steering behaviors: maneuvering and cruising modes. Physics of Fluids, 36, 115118 (2024) |
| [10] | LI, Z. H., XIA, D., LEI, M., and YAN, H. Numerical investigation on self-propelled hydrodynamics of squid-like multiple tentacles with synergistic expansion. Ocean Engineering, 287, 115808 (2023) |
| [11] | GE, D. L., WEI, S. S., LIANG, Y. S., and LI, K. Snap-through self-peeling of liquid crystal elastomer bilayers under constant light. Thin-Walled Structures, 219, 114260 (2026) |
| [12] | YU, Y., QUAN, Y. Y., LI, T. Y., HUANG, C. Y., MA, H. H., and LI, K. Self-synchronization of a two-span liquid crystal elastomer continuous beam above a hot platform. Chaos, Solitons & Fractals, 202, 117528 (2026) |
| [13] | HE, Q. G., YIN, R., HUA, Y. C., JIAO, W. J., MO, C. Y., SHU, H., and RANEY, J. R. A modular strategy for distributed, embodied control of electronics-free soft robots. Science Advances, 9, eade 9247 (2023) |
| [14] | XU, P. B., ZHOU, K., SUN, X., and LI, K. Self-sustainable chaotic dynamics of a liquid crystal elastomer pendulum in radial linear temperature fields. Communications in Nonlinear Science and Numerical Simulation, 152, 109338 (2026) |
| [15] | LI, J. J., GUO, W. J., ZHAO, W. Q., ZHU, Y. T., BAI, J., XIA, Z. G., ZHOU, X., and LIU, Z. F. Recent advances in flexible self-oscillating actuators. eScience, 4, 100250 (2024) |
| [16] | LIAO, B., ZANG, H. B., CHEN, M. Y., WANG, Y. J., LANG, X., ZHU, N. N., YANG, Z., and YI, Y. Soft rod-climbing robot inspired by winding locomotion of snake. Soft Robotics, 7, 500–511 (2020) |
| [17] | HUANG, C. Y., YANG, F., LI, K., DAI, Y. T., and YU, Y. Modeling and analysis of self-sustaining oscillation behavior of liquid crystal elastomer fiber/baffle system under stable full-field illumination. Chaos, Solitons & Fractals, 194, 116259 (2025) |
| [18] | ZHAO, D. and LIU, Y. A prototype for light-electric harvester based on light sensitive liquid crystal elastomer cantilever. Energy, 198, 117351 (2020) |
| [19] | YU, Y., HUANG, C. Y., LIU, C. Y., WANG, Z. J., MA, H. H., and XU, P. B. Heat-driven self-wobbling of a liquid crystal elastomer double-wheel linkage. Thin-Walled Structures, 217, 113829 (2025) |
| [20] | LI, K., ZHAO, C. F., QIU, Y. L., and DAI, Y. T. Light-powered self-rolling of a liquid crystal elastomer-based dicycle. Applied Mathematics and Mechanics (English Edition), 46(2), 253–268 (2025) https://doi.org/10.1007/s10483-025-3221-8 |
| [21] | ZHOU, X. G., PENG, H. L., QIU, Y. L., and LI, K. Delayed oscillation of light-responsive hydrogel layers bonded to a rigid substrate under periodic illumination. Chaos, Solitons & Fractals, 202, 117445 (2026) |
| [22] | JU, X. K., KONG, J., QI, G. H., DUAN, X. Z., PENG, C., WANG, E. K., DONG, S. J., LI, J., and JIN, Y. D. Dual-function polyionic liquid piezoelectric hydrogel: combining antibacterial activity and electrical stimulation for infected wound treatment. Journal of Colloid and Interface Science, 700, 138317 (2025) |
| [23] | PAI, Y. H., XU, C., ZHU, R. Y., DING, X. Y., BAI, S. Q., LIANG, Z. Q., and CHEN, L. D. Piezoelectric-augmented thermoelectric ionogels for self-powered multimodal medical sensors. Advanced Materials, 37, 2414663 (2025) |
| [56] | WANG, Y. C., LIU, J. Q., and YANG, S. Multi-functional liquid crystal elastomer composites. Applied Physics Reviews, 9(1), 011301 (2022) |
| [57] | KIM, H., SUNDARAM, S., KANG, J. H., TANJEEM, N., EMRICK, T., and HAYWARD, R. C. Coupled oscillation and spinning of photothermal particles in Marangoni optical traps. Proceedings of the National Academy of Sciences of the United States of America, 118, 2024581118 (2021) |
| [58] | WANG, Y. C., YIN, R., JIN, L. S., LIU, M. Z., GAO, Y. C., RANEY, J., and YANG, S. 3D-printed photoresponsive liquid crystal elastomer composites for free-form actuation. Advanced Functional Materials, 33, 2210614 (2023) |
| [59] | LI, K., CHEN, Z. F., and XU, P. B. Light-propelled self-sustained swimming of a liquid crystal elastomer torus at low Reynolds number. International Journal of Mechanical Sciences, 219, 107128 (2022) |
| [60] | YU, Y., ZHOU, L., DU, C. S., ZHU, F. Y., DAI, Y. T., GE, D. L., and LI, K. Self-galloping of a liquid crystal elastomer catenary cable under a steady temperature field. Thin-Walled Structures, 202, 112071 (2024) |
| [61] | ZHAO, J., DAI, C. F., DAI, Y. T., WU, J., and LI, K. Self-oscillation of cantilevered silicone oil paper sheet system driven by steam. Thin-Walled Structures, 203, 112270 (2024) |
| [62] | GE, D. L. and LI, K. Self-oscillating buckling and postbuckling of a liquid crystal elastomer disk under steady illumination. International Journal of Mechanical Sciences, 221, 107233 (2022) |
| [63] | ZHOU, L., DU, C. S., WANG, W., and LI, K. A thermally-responsive fiber engine in a linear temperature field. International Journal of Mechanical Sciences, 225, 107391 (2022) |
| [64] | ZHU, L. R., HE, M. Q., CHEN, B. H., QIAN, J., and XIAO, R. Inflation of a polydomain nematic elastomeric membrane. Journal of the Mechanics and Physics of Solids, 198, 106075 (2025) |
| [65] | YU, Y., LIU, F. F., HUANG, C. Y., and LI, K. Self-swing of a thermo-responsive L-shaped rod partially resting on a hot plate. Thin-Walled Structures, 221, 114464 (2026) |
| [66] | DAI, L., WANG, L. Q., CHEN, B. H., XU, Z. T., WANG, Z. J., and XIAO, R. Shape memory behaviors of 3D printed liquid crystal elastomers. Soft Science, 3, 5 (2023) |
| [67] | BAI, C. P., KANG, J. T., and WANG, Y. Q. Kirigami-inspired light-responsive conical spiral actuators with large contraction ratio using liquid crystal elastomer fiber. ACS Applied Materials & Interfaces, 17, 14488–14498 (2025) |
| [68] | KANG, J. T., BAI, C. P., and WANG, Y. Q. Tunable energy absorption characteristics of 3D-printed anisotropic liquid crystal elastomers. ACS Applied Polymer Materials, 7, 14497–14507 (2025) |
| [69] | YANG, L. L., CHANG, L. F., HU, Y., HUANG, M. J., JI, Q. X., LU, P., LIU, J. Q., CHEN, W., and WU, Y. C. An autonomous soft actuator with light-driven self-sustained wavelike oscillation for phototactic self-locomotion and power generation. Advanced Functional Materials, 30, 1908842 (2020) |
| [70] | OLHAN, S., ANTIL, B., MAIMÍ, P., and BEHERA, B. K. Advances in 4D printing of polymeric composite smart materials. Composite Structures, 377, 119859 (2026) |
| [71] | ZHANG, M., LU, X. X., YE, T. T., HU, H. D., SCARBROUGH, J. P., WANG, R., YAO, Y. X., LI, S. C., and WANG, X. G. Liquid crystal elastomers for solar, mechanical, thermal, and electrochemical energy applications. Journal of Materials Chemistry A, 14, 3185–3209 (2026) |
| [72] | FENG, C. R., RAJAPAKSHA, C. P. H., and JAKLI, A. Ionic elastomers for electric actuators and sensors. Engineering, 7, 581–602 (2021) |
| [73] | SU, J. T., HE, K., LI, Y. Z., TU, J. Q., and CHEN, X. D. Soft materials and devices enabling sensorimotor functions in soft robots. Chemical Reviews, 125, 5848–5977 (2025) |
| [74] | SHI, P. J., CHEN, C., WANG, Z., HONG, W., DUAN, S. D., LIU, Z. X., HOU, W. H., SI, M. Q., ZHANG, C. W., and HE, X. M. Air plastron-enabled heat management for enhanced photothermal actuation in underwater soft robots. Science Advances, 11, eadx7189 (2025) |
| [75] | BOOTHBY, J. M., GAGNON, J. C., MCDOWELL, E., VAN VOLKENBURG, T., CURRANO, L., and XIA, Z. Y. An untethered soft robot based on liquid crystal elastomers. Soft Robotics, 9, 154–162 (2022) |
| [76] | WU, H. Y., HOU, H. Y., QIU, Y. L., GE, D. L., and LI, K. Light-powered self-rolling liquid crystal elastomer rovers between two hollows. Communications in Nonlinear Science and Numerical Simulation, 156, 109723 (2026) |
| [77] | WEI, L., JIANG, X., HU, X. A., GE, D. L., and LI, K. Gravity-independent self-rolling of a photothermal liquid crystal elastomer rod for microgravity actuation. Communications in Nonlinear Science and Numerical Simulation, 157, 109764 (2026) |
| [78] | YANG, H. X., ZHANG, C., CHEN, B. H., WANG, Z. J., XU, Y., and XIAO, R. Bioinspired design of stimuli-responsive artificial muscles with multiple actuation modes. Smart Materials and Structures, 32, 085023 (2023) |
| [24] | PANDA, S., HAJRA, S., MISTEWICZ, K., IN-NA, P., SAHU, M., RAJAITHA, P. M., and KIM, H. J. Piezoelectric energy harvesting systems for biomedical applications. Nano Energy, 100, 107514 (2022) |
| [25] | ZHOU, H. L., ZHANG, Y., QIU, Y., WU, H. P., QIN, W. Y., LIAO, Y. B., YU, Q. M., and CHENG, H. Y. Stretchable piezoelectric energy harvesters and self-powered sensors for wearable and implantable devices. Biosensors and Bioelectronics, 168, 112569 (2020) |
| [26] | WANG, Y. C., DANG, A. L., ZHANG, Z. F., YIN, R., GAO, Y. C., FENG, L., and YANG, S. Repeatable and reprogrammable shape morphing from photoresponsive gold nanorod/liquid crystal elastomers. Advanced Materials, 32, 2004270 (2020) |
| [27] | SUN, J. H., WANG, Y. P., LIAO, W., and YANG, Z. Q. Ultrafast, high-contractile electrothermal-driven liquid crystal elastomer fibers towards artificial muscles. Small, 17, 2103700 (2021) |
| [28] | XU, Z. T., CHEN, Y. Z., ZHU, L. R., GE, Q., WU, Z. L., QU, S. X., and XIAO, R. Tailored helix morphing of 3D-printed liquid crystal elastomer bilayers. Cell Reports Physical Science, 6, 102835 (2025) |
| [29] | YANG, H. X., YIN, X. F., ZHANG, C., CHEN, B. H., SUN, P., and XU, Y. Weaving liquid crystal elastomer fiber actuators for multifunctional soft robotics. Science Advances, 11, eads3058 (2025) |
| [30] | BAI, C. P., KANG, J. T., and WANG, Y. Q. Light-induced wrinkling in annulus anisotropic liquid crystal elastomer films. Physical Review E, 111, 015421 (2025) |
| [31] | KANG, J. T., BAI, C. P., LIU, S. X., and JIA, Y. G. Light-induced nontethered rolling of liquid crystal elastomer and carbon nanotube composite ring. ACS Applied Polymer Materials, 6, 2709–2718 (2024) |
| [32] | SUN, X., ZHOU, K., CHEN, Y. Q., GAO, J. F., and XU, P. B. Self-oscillation chaotic motion of a liquid crystal elastomer pendulum under gradient-stabilized illumination. Chaos, Solitons & Fractals, 193, 116128 (2025) |
| [33] | KANG, W. D., CHENG, Q., LIU, C. Y., WANG, Z. J., LI, D. F., and LIANG, X. D. A constitutive model of monodomain liquid crystal elastomers with the thermal-mechanical-nematic order coupling. Journal of the Mechanics and Physics of Solids, 196, 105995 (2025) |
| [79] | BAUMANN, A., SÁNCHEZ-FERRER, A., JACOMINE, L., MARTINOTY, P., LE HOUEROU, V., ZIEBERT, F., and KULIĆ, I. M. Motorizing fibres with geometric zero-energy modes. Nature Materials, 17, 523–527 (2018) |
| [80] | LIANG, H., LIU, Y. W., XU, H. T., YANG, Y., HE, E. J., YANG, Z. J., WEI, Y., and JI, Y. Thiol-acrylate catalyst enabled post-synthesis fabrication of liquid crystal actuators. Chinese Journal of Polymer Science, 41, 1656–1662 (2023) |
| [1] | . [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(2): 283-302. |
| [2] | Bohua SUN. Incompatible deformation field and Riemann curvature tensor[J]. Applied Mathematics and Mechanics (English Edition), 2017, 38(3): 311-332. |
| [3] | Heng XIAO. Quantum enigma hidden in continuum mechanics[J]. Applied Mathematics and Mechanics (English Edition), 2017, 38(1): 39-56. |
| [4] | Xiaodong LI, Baohong BAI, Yingbo XU, Min JIANG. Research status and trend of wind turbine aerodynamic noise[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(S1): 43-50. |
| [5] | Haoyuan GUO, Liyuan ZHANG, YajunYIN, Yongxin GAO. Relations between cubic equation, stress tensor decomposition, and von Mises yield criterion[J]. Applied Mathematics and Mechanics (English Edition), 2015, 36(10): 1359-1370. |
| [6] | Lidan YU, Tianfu JIN, Zhengnan YIN, Heng XIAO. Multi-axial strain-stiffening elastic potentials with energy bounds: explicit approach based on uniaxial data[J]. Applied Mathematics and Mechanics (English Edition), 2015, 36(7): 883-894. |
| [7] | B. AMIRIAN;R. HOSSEINI-ARA;H. MOOSAVI. Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(7): 875-886. |
| [8] | 朱永安, 王璠, 刘人怀. Thermal buckling of axisymmetrically laminated cylindrically orthotropic shallow spherical shells including transverse shear[J]. Applied Mathematics and Mechanics (English Edition), 2008, 29(3): 291-300 . |
| [9] | 王志乔;兑关锁. Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions[J]. Applied Mathematics and Mechanics (English Edition), 2007, 28(9): 1249-1257 . |
| [10] | 戴天民. Renewal of basic laws and principles for polar continuum theories (XI)---consistency problems[J]. Applied Mathematics and Mechanics (English Edition), 2007, 28(2): 163-172 . |
| [11] | 戴天民. RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅹ) —MASTER BALANCE LAW[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(2): 167-174 . |
| [12] | 戴天民. RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅸ)— THERMOMECHANICS[J]. Applied Mathematics and Mechanics (English Edition), 2005, 26(6): 709-715 . |
| [13] | . MOVING INCLINED LOAD AT BOUNDARY SURFACE[J]. Applied Mathematics and Mechanics (English Edition), 2005, 26(4): 476-485 . |
| [14] | 戴天民. RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅷ)—PRINCIPLE OF TOTAL WORK AND ENERGY[J]. Applied Mathematics and Mechanics (English Edition), 2005, 26(3): 313-318 . |
| [15] | 梁建术;陈予恕;梁以德. ROBUST CONTROL OF PERIODIC BIFURCATION SOLUTIONS[J]. Applied Mathematics and Mechanics (English Edition), 2004, 25(3): 263-271. |
| 阅读次数 | ||||||
|
全文 |
|
|||||
|
摘要 |
|
|||||

Email Alert
RSS