حل تحلیلی انتشار امواج انفجار در سنگ با استفاده از تئوری الاستودینامیک

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشکده مهندسی معدن و متالورژی، دانشگاه یزد، یزد، ایران

2 دانشکده مهندسی معدن، دانشگاه تهران، تهران، ایران

چکیده

عملیات انفجار در سنگ کاربردهای مختلفی در موارد گوناگون دارد. بیشترین کاربرد این عملیات در خردایش سنگ در معادن بوده که برای استخراج ماده معدنی انجام می‌شود. حفاری کانال، ترانشه، تونل و شیروانی‌های سنگی نیز از دیگر کاربردهای عملیات انفجار در سنگ است. پدیده‌های دینامیکی مانند زلزله و انفجار در سنگ عوامل انتشار امواج دینامیکی تنشی در سنگ هستند. ازاین‌رو انتشار امواج و مطالعه چگونگی انتشار آنها از اهمیت بالایی در شناخت اثرات بعدی پدیده‌های دینامیکی در سنگ برخوردار هستند. به‌منظور حل چنین مسئله‌ای می‌توان از روش‌های مختلفی استفاده کرد، ولی در میان این روش‌ها حل تحلیلی مسئله ازنظر دقت نسبت به سایر روش‌ها برتری دارد. در مطالعه حاضر یک تابع گرین الاستودینامیک دوبعدی برای حل مسئله انتشار امواج انفجار در سنگ توسعه داده شده است. ابتدا از معادلات حرکت و انفجار به‌عنوان معادلات حاکم استفاده شده و درنهایت تابع گرین مربوطه برحسب جابجایی به دست آمده است. همچنین میدان سرعت ذره‌ای بر اساس تابع حاصله توسعه داده شده‌اند. بر همین اساس یک چال انفجاری در یک محیط الاستیک همگن همسانگرد ایزوتروپ بی‌نهایت سنگی مورد تحلیل قرارگرفته و نتایج حاصل از آن ارائه شده است. سپس به‌منظور بررسی صحت نتایج حاصل از روند تحلیلی ارائه‌شده، یک مسئله واقعی با استفاده از راه‌حل پیشنهادی مدل‌سازی و حل شده است. مسئله موردنظر پیش‌ازاین توسط محققی دیگر به‌صورت تجربی و با استفاده از یک روش عددی حل شده است. نتایج این تحقیق در قالب حداکثر سرعت ذره‌ای در هر دو روش عددی و داده‌های برداشت‌شده از یک انفجار واقعی با یکدیگر مقایسه شده‌اند. درنهایت نتایج حاصل از پژوهش حاضر با نتایج مطالعه مذکور و در قالب همان حداکثر سرعت ذره‌ای مقایسه شده است. این مقایسه نشان داده است که حداکثر سرعت ذره‌ای حاصل‌شده از روش پژوهش حاضر تطابق خوبی با واقعیت دارد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Analytical solution of the explosion-induced wave propagation in rock using elastodynamic theory

نویسندگان [English]

  • Meysam Lak 1
  • Mohammad Fatehi Marji 1
  • Alireza Yarahmadi Bafghi 1
  • Abolfazl Abdollahipour 2
  • mehdi pourghasemi sagand 1
1 Dept. of Mining and Metallurgy Engineering, Yazd University, Yazd, Iran
2 Dept. of Mining Engineering, University of Tehran, Tehran, Iran
چکیده [English]

Summary
In this study, an analytical solution has been introduced to solve the explosion-induced wave propagation phenomenon in rock using elastodynamic theory.
Introduction
The explosion of explosives in the blast-hole can apply very high pressure to the surrounding rock media. The pressure can cause to propagate cracks and fractures and eventually fragment and destroy them. Rock blasting, like other dynamic phenomena, needs to propagate waves and dynamic stresses through the body. Therefore, the study of the propagation of waves is of great importance in understanding the subsequent effects of rock blasting. For this purpose, an analytical algorithm is used to obtain the appropriate Green functions to achieve the best description of the problem. To solve the blast-induced wave propagation problem.
Methodology and Approaches
A two-dimensional elastodynamic Green's function has been analytically derived. Firstly, Navier's equations of motion and explosion equations have been used as governing equations. Then, using the Helmholtz theory, the Navier equations have been separated into two scleral and vector fields in terms of displacement. Explosion pressure as a body force has been added to the equations using the Poisson function. After performing the relevant calculations and solving the above equations, the corresponding Green function in terms of displacement is obtained. Moreover, a time-dependent green function is developed for the particle velocity using the basic functions and the resulting green function. After finding the appropriate Green functions, a typical problem consisting of a blast-hole in an elastic homogenous isotropic infinite rock media has been investigated and modeled. Finally, to validate, a real problem has been analytically modeled and solved using the proposed solution.
Results and Conclusions
Its results have been compared with results from a finite element-discrete element numerical method and the explosion reality results. Based on the results of the comparison, there is a good agreement between the analytical, experimental, and numerical results at different distances from the blast-hole. A good match between the analytical results of this study and the experimental and numerical results of previous studies shows the validity of the proposed method and confirms the Green function obtained in this study. In general, it can be said that this method can be used to study the propagation of elastic waves in rock and rock-like materials. Also, it can be utilized from the resulting Green functions in other methods for different purposes as an input dynamic function.

کلیدواژه‌ها [English]

  • Blasting
  • Green&rsquo
  • s function
  • Analytical solution
  • Wave propagation
  • Rock Dynamics
[1]                 Gong, W., Wang, Y., Ning, Y., Luo, Y., Wang, Y. "Validating the ability of the discontinuous deformation analysis method to model normal P-wave propagation across rock fractures" International journal for numerical and analytical methods in geomechanics, 2017.
[2]                  Lak, M., Fatehi, M., Yarahmadi, A., "A finite difference modelling of crack initiation in rock blasting" The 2018 World Congress on Advances in Civil, Environmental, & Materials Research (ACEM18), Songdo Convensia, Incheon, Korea, 2018.
[3]                 Lak, M., Fatehi, M., Yarahmadi, A., Abdollahipour, A. "Numerical Modelling of Crack Initiation around a Wellbore Due to Explosion" International Journal of Geological and Environmental Engineering, Vol. 12, No. 6, pp. 420-423, 2018.
[4]                 Gu, W., Nihei, K. T., Myer, L. R. "Numerical simulation of elastic wave propagation in fractured rock with the boundary integral equation method" Journal of geophysical research, Vol. 101, No. B7, pp. 15,933-15,943, 1996.
[5]                 Lak, M., Baghbanan, A., Hashemolhoseini, H. "Effect of seismic waves on the hydro-mechanical properties of fractured rock masses" Earthquake engineering and engineering vibration, Vol. 16, No. 3, pp. 525-536, 2017.
[6]                 Babanouri, N., Kariminasab, S., Mansouri, H. "Modeling of blast waves propagation through jointed rock masses: a case study at the Gol-e-Gohar iron ore mine (Iran)" 46th US Rock Mechanics / Geomechanics Symposium, Chicago, USA, pp. 24-27, 2012.
[7]                 Babanouri, N., Fattahi, H. "Evaluating orthotropic continuum analysis of stress wave propagation through a jointed rock mass" Bull. Eng. Geol. Environ., 2016.
[8]                  Lak, M., Fatehi, M., Yarahmadi, A., Abdollahipour, A. "Discrete element modeling of explosion-induced fracture extension in jointed rock masses" Journal of Mining and Environment, DOI: 10.22044/jme.2018.7291.1579, 2019.
[9]                  Basabe, J. D., Sen, M. K., Wheeler, M. F. "Elastic wave propagation in fractured media using the discontinuous Galerkin method" GEOPHYSICS, Vol. 81, No. 4, 2016.
[10]              Li, D., Han, ZH., Zhu, Q., Zhang, Y., Ranjit, P. G. "Stress wave propagation and dynamic behavior of red sandstone with single bonded planar joint at various angles" International Journal of Rock Mechanics and Mining Sciences, Vol. 117, pp. 162–170, 2019.
[11]               Li, J. C., Li, N. N., Chai, S. B., Li, H. B. "Analytical study of ground motion caused by seismic wave propagation across faulted rock masses" International journal for numerical and analytical methods in geomechanics, pp. 1–5, 2017.
[12]               Rossmanith, H. P., Uenishi, K., Kouzniak, N. "Blast wave propagation in rock mass-Part I: monolithic medium" International Journal of Blasting and Fragmentation, Vol. 1, pp. 317-359, 1997.
[13]               Kouzniak, N., Rossmanith, H. P. "Supersonic detonation in rock mass - Analytical solutions and validation of numerical models - Part 1: Stress analysis" International Journal of Blasting and Fragmentation, Vol. 2, pp. 44-86, 1998.
[14]               Lak, M., Fatehi, M., Yarahmadi, A., Abdollahipour, A. "Analytical and numerical modeling of rock blasting operations using a two dimensional elasto-dynamic Green's function" International Journal of Rock Mechanics and Mining Sciences, Vol. 114, pp. 208–217, 2019.
[15]               Niu, L., Zhu, W., Li, Sh., Guan, K. "Determining the viscosity coefficient for viscoelastic wave propagation in rock bars" Rock Mechanics and Rock Engineering, 2018.
[16]               Hoseininasab, H., Fatehi, M. "A semi-infinite higher-order displacement discontinuity method and its application to the quasistatic analysis of radial cracks produced by blasting" Journal of Mechanics of Materials and Structures, Vol. 2, No. 3, 2007.
[17]               Fan, L. F., Zhou, X. F., Wu, Z. J., Wang, L. J. "Investigation of stress wave induced cracking behavior of underground rock mass by the numerical manifold method" Tunnelling and Underground Space Technology, Vol. 92, 2019.
[18]               Buchen, P. W. "The elastodynamic Green's tensor for the 2D half-space" Journal of Austral. Math. Soc., Vol. 20, pp. 385-400, 1978.
[19]               Tadeu, A. J. B., Kausel, E. "Green’s functions for two-and-a-half-dimensional elastodynamic problems" Journal of Engineering Mechanics, Vol. 126, No. 10, 2000.
[20]               Sun, L. "Time–harmonic elastodynamic Green functions of plates for line loads" Journal of Sound and Vibration, Vol. 264, No. 2, pp. 337-348, 2001.
[21]               Sanchez-Sesma, F. J., Perez-Ruiz, J. A., Campillo, M. "Elastodynamic 2D Green function retrieval from cross-correlation: Canonical inclusion problem" Geophysical research letters, Vol. 33, 2006.
[22]               Wapenaar, K. "A single-sided representation for the homogeneous Green’s function of a unified scalar wave equation" Journal of Acoustical Society of America, Vol. 141, No. 6, pp. 4466–4479, 2017.
[23]              Aspel, R. J., Luco, J. E. "On the Green's functions for a layered half-space. Part II" Bulletin of the Seismological Society of America, Vol. 73, No. 4, pp. 931-951, 1983.
[24]              Tewary, V. K. "Computationally efficient representation for elastostatic and elastodynamic Green's functions for anisotropic solids" Physical review B, Vol. 51, No. 22, 1995.
[25]              Liu, G. R., Lam, K. Y. "Two-dimensional time-harmonic elastodynamic Green's functions for anisotropic media" International Journal of Engineering Science, Vol. 34, No. II, pp. 1327-1338, 1996.
[26]              Varycuk, V. "Exact elastodynamic Green functions for simple types of anisotropy derived from higher-order ray theory" Studia geoph. et geod., Vol. 45, pp. 67-84, 2001.
[27]              Filho, A. C., Ravasi, M., Curtis, A., Meles, G. A. "Elastodynamic Green’s function retrieval through single-sided Marchenko inverse scattering" Physical review E, Vol. 90, 2014.
[28]              Varycuk, V. "Seismic moment tensors in anisotropic media: A review" S. D’Amico (ed.), Moment Tensor Solutions, Springer Natural Hazards, Springer International Publishing, 2018.
[29]              Zhan, Q., Zhuang, M., Fang, Y., Liu, J. G., Liu, Q. H. "Green’s function for anisotropic dispersive poroelastic media based on the Radon transform and eigenvector diagonalization" Royal Society Publishina, Proceeding A, 2019.
[30]              Hutchings, L., Wu, F. "Empirical Green's functions from small earthquakes: A wave form study of locally recorded aftershocks of the 1971 San Fernando earthquake" Journal of geophysical research, Vol. 95, NO. B2, pp. 1187-1214, 1990.
[31]              Wang, C. Y., Achenbach, J. D. "Three-dimensional time-harmonic elastodynamic Green’s functions for anisotropic solids" Proc. R. Soc. Loud., Vol. 449, pp. 441-458, 1995.
[32]              Dineva, P. S., Manolis, G. D., Wuttke, F. "Fundamental solutions in 3D elastodynamics for the BEM: A review" Engineering Analysis with Boundary Elements, Vol. 105, pp. 47–69, 2019.
[33]              Trivino, L. F., Mohanty, B. "Assessment of crack initiation and propagation in rock from explosion-induced stress waves and gas expansion by cross-hole seismometry and FEM–DEM method" International Journal of Rock Mechanics & Mining Sciences, Vol. 77, pp. 287–299, 2015.
[34]              Jimeno, E. L., Jimeno, C. L., Carcedo, A. "Drilling and Blasting of Rocks" London: Taylor & Francis, 1995.
[35]              Duvall, W. "Strainwave shapes in rock near explosions" Geophysics, Vol. 18, No. 2, pp. 310–323, 1953.
[36]              Sharpe, J. A. "The production of elastic waves by explosion pressures. I. Theory and empirical field observations" Geophysics, Vol. 7, No. 2, pp. 144–154, 1942.
[37]              Banerjee, P. K., Ahmad, S., Manolis, G. D. "Transient elastodynamic analysis of threedimensional problems by boundary element method" Earthq. Eng. Struct. Dyn., Vol. 14, No. 6, pp. 933–949, 1986.
[38]              Stokes, G. G. "On the variation of gravity on the surface of the earth" Trans. Camb. Philos. Soc., Vol. 8, pp. 672–695, 1848.
[39]              Rice, J. R. "The mechanics of earthquake rupture" North Holland: Italian Phys. Soc., pp. 555–649, 1980.
[40]              Aki, K., Richards, P. G. "Quantitative Seismology" 2nd ed. California: University Science Books, 2002.