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- W4313334824 abstract "Abstract We examine geological crack patterns using the mean field theory of convex mosaics. We assign the pair $$left({overline{n } }^{*},{overline{v } }^{*}right)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mfenced> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>v</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> </mml:mfenced> </mml:math> of average corner degrees (Domokos et al. in A two-vertex theorem for normal tilings. Aequat Math https://doi.org/10.1007/s00010-022-00888-0 , 2022) to each crack pattern and we define two local, random evolutionary steps R 0 and R 1 , corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the $$left({overline{n } }^{*},{overline{v } }^{*}right)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mfenced> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>v</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> </mml:mfenced> </mml:math> plane. We prove the existence of limit points for several types of trajectories. Also, we prove that cell density $$overline{rho }= frac{{overline{v } }^{*}}{{overline{n } }^{*}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mfrac> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>v</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> </mml:mfrac> </mml:mrow> </mml:math> increases monotonically under any admissible trajectory." @default.
- W4313334824 created "2023-01-06" @default.
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- W4313334824 date "2022-12-29" @default.
- W4313334824 modified "2023-09-26" @default.
- W4313334824 title "A discrete time evolution model for fracture networks" @default.
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- W4313334824 doi "https://doi.org/10.1007/s10100-022-00838-w" @default.
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