{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:08:56Z","timestamp":1760058536786,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,5]],"date-time":"2025-04-05T00:00:00Z","timestamp":1743811200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>When two axisymmetric stress waves of the same strength propagate radially at the same velocity, the stress wave wavefronts collide and interact along a specific locus, which is the perpendicular bisector between the two sources. The maximum principal stress occurs along this perpendicular bisector, and the tensile stresses result in crack bridging between the two source points. This symmetric wave propagation behavior allows us to use the Voronoi tessellation diagram and its symmetric dual graph, the Delaunay triangulation, to gain first-order insight into complex wave propagation phenomena for an arbitrary distribution of wave propagation sources. The inherent symmetry of these simultaneous wave propagation mechanics allows us to rapidly visualize and predict the stress wave propagation and interactions, and the resultant crack bridging patterns that arise from random blast sources. The current work is focused on rock blast fracture mechanics, but the visualization scheme can be implemented for any application where waves propagate axisymmetrically and interact.<\/jats:p>","DOI":"10.3390\/sym17040555","type":"journal-article","created":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T12:05:56Z","timestamp":1744200356000},"page":"555","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Using Voronoi Tessellation Diagrams to Visualize the Mechanical Response of Interacting Axisymmetric Simultaneous Propagating Waves"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8832-8435","authenticated-orcid":false,"given":"Zherui","family":"Martinez-Guo","sequence":"first","affiliation":[{"name":"School of Aeronautics & Astronautics, Purdue University, West Lafayette, IN 47907, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s00193-015-0567-2","article-title":"Interaction and coalescence of multiple simultaneous and non-simultaneous blast waves","volume":"26","author":"Qiu","year":"2016","journal-title":"Shock Waves"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"031302","DOI":"10.1063\/5.0138156","article-title":"Fundamental review on collision of blast waves","volume":"35","author":"Almustafa","year":"2023","journal-title":"Phys. Fluids"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1080\/13855149709408402","article-title":"Blast wave propagation in rock mass\u2014Part I: Monolithic medium","volume":"1","author":"Rossmanith","year":"1997","journal-title":"Fragblast"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"185187","DOI":"10.1109\/ACCESS.2020.3030253","article-title":"Investigation of Stress Wave Interaction and Fragmentation in Granite During Multihole Blastings","volume":"8","author":"He","year":"2020","journal-title":"IEEE Access"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1803","DOI":"10.1007\/s00603-015-0876-x","article-title":"Stress Wave Interaction Between Two Adjacent Blast Holes","volume":"49","author":"Yi","year":"2016","journal-title":"Rock Mech. Rock Eng."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Chew, L.P., and Dyrsdale, R.L. (1985, January 5\u20137). Voronoi diagrams based on convex distance functions. Proceedings of the First Annual Symposium on Computational Geometry\u2014SCG\u201985, Baltimore, MD, USA.","DOI":"10.1145\/323233.323264"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Zhang, Z.-X. (2016). Rock Fracture and Blasting: Theory and Applications, Elsevier.","DOI":"10.1016\/B978-0-12-802688-5.00003-8"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1007\/s11803-019-0533-6","article-title":"Numerical study on the fracturing mechanism of shock wave interactions between two adjacent blast holes in deep rock blasting","volume":"18","author":"Yuan","year":"2019","journal-title":"Earthq. Eng. Eng. Vib."},{"key":"ref_9","first-page":"2124694","article-title":"In Situ Stress Effects on Smooth Blasting: Model Test and Analysis","volume":"2020","author":"Yang","year":"2020","journal-title":"Shock Vib."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"104622","DOI":"10.1016\/j.ijrmms.2021.104622","article-title":"Stress wave superposition effect and crack initiation mechanism between two adjacent boreholes","volume":"138","author":"Ding","year":"2021","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1076\/frag.6.1.104.8854","article-title":"The Use of Lagrange Diagrams in Precise Initiation Blasting. Part I: Two Interacting Blastholes","volume":"6","author":"Rossmanith","year":"2002","journal-title":"Fragblast"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Tang, H.-L., Yang, J., and Yu, Q. (2023). Experimental Investigation of the Effect of Delay Time on Rock Fragmentation in Multi-Hole Bench Blasting. Appl. Sci., 13.","DOI":"10.3390\/app13127329"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/BF02538964","article-title":"Wave propagation, damage evolution, and dynamic fracture extension. Part II. Blasting","volume":"32","author":"Rossmanith","year":"1996","journal-title":"Mater. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Needham, C.E. (2018). Blast Waves, Springer International Publishing.","DOI":"10.1007\/978-3-319-65382-2"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1115\/1.3629724","article-title":"Transient Compressional Waves in an Infinite Elastic Plate with a Circular Cylindrical Cavity","volume":"31","author":"Scott","year":"1964","journal-title":"J. Appl. Mech."},{"key":"ref_16","unstructured":"Miklowitz, J. (1984). The Theory of Elastic Waves and Waveguides, North Holland Publishing Company."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"E33","DOI":"10.1190\/1.2424543","article-title":"A comparison of Heelan and exact solutions for seismic radiation from a short cylindrical charge","volume":"72","author":"Blair","year":"2007","journal-title":"Geophysics"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"349","DOI":"10.2118\/8215-PA","article-title":"Explosive Fracture Studies on Oil Shale","volume":"20","author":"Grady","year":"1980","journal-title":"Soc. Pet. Eng. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/0148-9062(82)90886-5","article-title":"Explosive energy coupling in geologic materials","volume":"19","author":"Larson","year":"1982","journal-title":"Int. J. Rock Mech. Min. Sci. Geomech. Abstr."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1137\/141000671","article-title":"Julia: A Fresh Approach to Numerical Computing","volume":"59","author":"Bezanson","year":"2017","journal-title":"SIAM Rev."},{"key":"ref_21","unstructured":"(2025, March 26). MethodOfLines.jl: Automated Finite Difference for Physics-Informed Learning. Available online: https:\/\/docs.sciml.ai\/MethodOfLines\/stable\/."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/0148-9062(71)90018-0","article-title":"On the fracture process in blasting","volume":"8","author":"Kutter","year":"1971","journal-title":"Int. J. Rock Mech. Min. Sci. Geomech. Abstr."},{"key":"ref_23","unstructured":"Whittaker, B.N., Singh, R.N., and Sun, G. (1992). Rock Fracture Mechanics: Principles, Design, and Applications, Elsevier."},{"key":"ref_24","unstructured":"(MATLAB Software, 2023). MATLAB Software, Version R2023b."},{"key":"ref_25","unstructured":"(2024, November 02). Voronoi Diagrams\u2014MATLAB & Simulink. Available online: https:\/\/www.mathworks.com\/help\/matlab\/math\/voronoi-diagrams.html."},{"key":"ref_26","unstructured":"(ABAQUS Software\/Standard User\u2019s Manual, 2014). ABAQUS Software\/Standard User\u2019s Manual, Version 2014."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/555\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:10:54Z","timestamp":1760029854000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/555"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,5]]},"references-count":26,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040555"],"URL":"https:\/\/doi.org\/10.3390\/sym17040555","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,4,5]]}}}