在前人工作的基础上，考虑月表向内吸收的太阳光照热量和向外释放的月球热辐射热量，以及与温度、深度相关的月壤热力学参数的非线性，开发了适合研究月壤温度、变形和热应力时空演化的热弹性耦合有限元并行程序，利用4组有限元模型研究不同的月壤特征厚度对月表温度、变形和热应力时空演化的影响。计算结果表明，月表月壤温度呈现1个月球日(29.5个地球日)的周期性变化，赤道月表的温度在100~385 K之间变化，变化幅度随深度增加指数衰减，影响深度约达50 cm。温度的周期性变化引起月表垂直位移也呈现上升和下降的周期性变化，以及月表水平正应力的挤压和拉张的周期性变化。总体而言，水平应力白昼是挤压，夜晚是拉张，其中拉应力增长速率最快的时刻是18:00，拉应力最大的时刻是06:00。月壤特征厚度对温度、水平正应力的时空分布的影响较大。热应力的量级有可能达到月表岩土抗张强度，张应力增长最快和幅值最大时段与观测到的月球浅表热事件发生的早晚频度较高相吻合。
On the basis of the previous work, considering the solar heat absorbed by the lunar surface inward and the lunar thermal radiation heat released outward, as well as the nonlinearities of the thermodynamic parameters of the lunar soil related to the temperature and depth, we have developed a thermo-elastic coupled finite element parallel program suitable for the study of the temporal and spatial evolutions of the temperature, deformation, and thermal stress of the lunar soil, and have utilized the four sets of finite element models to investigate the effects of the characteristic thicknesses of the lunar soil on the temporal and spatial evolutions of the temperature, deformation and thermal stresses of the lunar surface. The computational results show that the temperature of the lunar surface varies periodically over one lunar day (29.5 Earth days), and the temperature of the equatorial lunar surface varies from 100 to 385 K, with the variation decaying exponentially with the increase of the depth, and the depth of influence reaches to about 50 cm. The temperature cyclic changes also cause the vertical displacement of the lunar surface to rise and fall, and the horizontal normal stress of the lunar surface in the form of compression and tension. In general, the horizontal stresses are compressed during the day and tensile during the night, with the fastest increase in tensile stress at 18:00 and the highest tensile stress at 06:00. The characteristic thickness of the lunar soil has a strong influence on the temporal and spatial distributions of the temperature and the horizontal positive stresses. The magnitude of thermal stresses may reach the tensile strength of the lunar surface. The fastest growth of tensile stress and the period of maximum amplitude coincide with the observed high frequency of thermal events on the lunar surface in the morning and evening.
[1] Nakamura Y. Farside deep moonquakes and deep interior of the Moon[J]. Journal of Geophysical Research: Planets, 2005, 110(E1): E01001. DOI:10.1029/2004je002332.
[2] Nakamura Y, Latham G, Dorman H J, et al. Passive seismic experiment, long period event catalog, final version (1969 day 202-1977 day 273, ALSEP stations 11, 12, 13, 14, 15, and 16)[R]. Institute for Geophysics, 1981. DOI:10.15781/t2ff3mh78.
[3] Duennebier F, Sutton G H. Thermal moonquakes[J]. Journal of Geophysical Research, 1974, 79(29): 4351-4363. DOI:10.1029/jb079i029p04351.
[4] Dimech J L, Knapmeyer-Endrun B, Phillips D, et al. Preliminary analysis of newly recovered Apollo 17 seismic data[J]. Results in Physics, 2017, 7: 4457-4458. DOI:10.1016/j.rinp.2017.11.029.
[5] 赵娜. 月震特征及与地震的对比[J]. 空间科学学报，2020，40(2)：264-272. DOI:10.11728/cjss2020.02.264.
[6] Keihm S J, Peters K, Langseth M G, et al. Apollo 15 measurement of lunar surface brightness temperatures thermal conductivity of the upper 1 1/2 meters of regolith[J]. Earth and Planetary Science Letters, 1973, 19(3): 337-351. DOI:10.1016/0012-821X(73)90084-8.
[7] Heiken G, Vaniman D, French B. Bookreview：lunar sourcebook：a user’s guide to the moon [J]. Sky & Telescope, 1991, 82:498. DOI:10.5860/choice.29-3868.
[8] Criswell D R, Lindsay J F. Thermal moonquakes and booming dunes[C]//Abstracts of the Lunar and Planetary Science Conference: Houston, 1974.
[9] 欧阳自远. 月球科学概论[M]. 北京：中国宇航出版社，2005.
[10] Lozano A. Development of a lunar regolith thermal energy storage model for a lunar outpost [D]. Luleå: Luleå University of Technology, 2016.
[11] Mitchell D L, de Pater I. Microwave imaging of mercury's thermal emission at wavelengths from 0.3 to 20.5 cm[J]. Icarus, 1994, 110(1): 2-32. DOI:10.1006/icar.1994.1105.
[12] Vasavada A R, Paige D A, Wood S E. Near-surface temperatures on mercury and the moon and the stability of polar ice deposits[J]. Icarus, 1999, 141(2): 179-193. DOI:10.1006/icar.1999.6175.
[13] Shkuratov Y G, Bondarenko N V. Regolith layer thickness mapping of the moon by radar and optical data[J]. Icarus, 2001, 149(2): 329-338. DOI:10.1006/icar.2000.6545.
[14] Fa W Z, Wieczorek M A. Regolith thickness over the lunar nearside: results from earth-based 70 cm Arecibo radar observations[J]. Icarus, 2012, 218(2): 771-787. DOI:10.1016/j.icarus.2012.01.010.
[15] Hayne P O, Bandfield J L, Siegler M A, et al. Global regolith thermophysical properties of the moon from the diviner lunar radiometer experiment[J]. Journal of Geophysical Research: Planets, 2017, 122(12): 2371-2400. DOI:10.1002/2017je005387.
[16] 徐向华，梁新刚，任建勋. 月球表面热环境数值分析[J]. 宇航学报，2006，27(2)：153-156，200. DOI:10.3321/j.issn: 1000-1328.2006.02.001.
[17] 任德鹏，贾阳，彭松. 月面热环境的反演研究[J]. 宇航学报，2018，39(4)：435-441. DOI:10.3873/j.issn.1000-1328.2018.04.010.
[18] Schreiner S S, Dominguez J A, Sibille L, et al. Thermophysical property models for lunar regolith[J]. Advances in Space Research, 2016, 57(5): 1209-1222. DOI:10.1016/j.asr.2015.12.035.
[19] Kovach R L, Watkins J S. The velocity structure of the lunar crust[J]. The Moon, 1973, 7(1/2): 63-75. DOI:10.1007/BF00578808.
[20] 张翔，张金海. 月震研究进展与展望[J]. 地球与行星物理论评，2021，52(4)：391-401. DOI:10.19975/j.dqyxx.2021-032.
[21] Liou K N, Liou K N. An Introduction to atmospheric radiation [M]. Amsterdam， Boston: Academic Press, 2002.
[22] Keihm S J. Interpretation of the lunar microwave brightness temperature spectrum: feasibility of orbital heat flow mapping[J]. Icarus, 1984, 60(3): 568-589. DOI:10.1016/0019-1035(84)90165-9.
[23] Vasavada A R, Bandfield J L, Greenhagen B T, et al. Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment[J]. Journal of Geophysical Research: Planets, 2012, 117(E12): E00H18. DOI:10.1029/2011je003987.
[24] 廖翼. 月球表面物理温度分布模型及数值计算[D]. 武汉：华中科技大学，2011.
[25] Colwell J E, Batiste S, Horányi M, et al. Lunar surface: dust dynamics and regolith mechanics[J]. Reviews of Geophysics, 2007, 45(2): RG2006. DOI:10.1029/2005rg000184.
[26] Alshibli K A, Hasan A. Strength properties of JSC-1A lunar regolith simulant[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2009, 135(5): 673-679. DOI:10.1061/(asce)gt.1943-5606.0000068.
[27] Ledlow M J, Zeilik M, Burns J O, et al. Subsurface emissions from Mercury：VLA radio observations at 2 and 6 centimeters[J]. The Astrophysical Journal Letters, 1992, 384: 640. DOI:10.1086/170906.
[28] Malla R B, Brown K M. Determination of temperature variation on lunar surface and subsurface for habitat analysis and design[J]. Acta Astronautica, 2015, 107: 196-207. DOI:10.1016/j.actaastro.2014.10.038.
[29] Turcotte D, Schubert G. Geodynamics [M].3th ed. Cambridge: Cambridge University Press, 2014.
[30] 占伟，李斐. 月球内部构造研究综述[J]. 地球物理学进展，2007，22(3)：737-742. DOI:10.3969/j.issn.1004-2903.2007.03.012.
[31] Langseth M G, Keihm S J, Peters K. Revised lunar heat-flow values[C]//Lunar Science Conference. Houston, 1976: 3143-3171.
