# Approximation result for non-autonomous and non-local rock fracture models

• E. F. Doungmo Goufo
• A. Kubeka
Original Paper Area 1

## Abstract

The increasing number of earthquakes in the world and the unusual behavior observed in rock fracture processes across the globe have recently raised number of concerns. The way rocks undergo fractures in nature (sometime caused by human activities) is not always the idealized one as wished by geologists. This makes their mathematical modeling and analysis quite complex. In this paper a non-local and non-autonomous rock fracture model where sizes of clusters are discrete and rock fracture rate is time, position and size dependent is investigated by means of forward propagators. Such a model, where in addition, new particles are spatially randomly distributed according to some probabilistic law, has not yet been analyzed in the same work. Our analysis consists of approximating the solution of a discrete, non-local and non-autonomous rock fracture model by a sequence of solutions of cut-off problems of a similar form. Then, we make use of the classical argument of Dini (Lemma 4, Kato in Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften. Band, vol 132. Springer, New York, 1984) to show existence of strong solutions of the model in the class of Banach spaces (of functions with finite higher moments) $$\mathcal {X}_{r}:=L_{1}(\mathfrak {I},X_{r})$$, with $$X_{r}=\{g:\mathbb {R}^{3}\times \mathbb {N}\ni (x,n)\rightarrow g(x,n), \Vert g\Vert _{r}:=\int _{\mathbb {R}^{3}}\sum _{n=1}^{\infty }n^{r}|g(x,n)|dx<\infty \}.$$ Numerical implementations of the approximation scheme show the fracture evolution of the shear stress resulting from rock fracture as a function of slip. A comparison with the standard well-know results reveals, as expected, that the approximation scheme maintains a sharp weakening of the shear stress in the real fracture case compared to the idealized (homogenous) case. This great observation together with this approximation scheme may contribute to the proof of uniqueness of a strong solution to the discrete, non-local and non-autonomous rock fracture model which remains open and unsolved.

## Keywords

Evolution system Forward propagator Non-autonomous rock fracture Non-local model Discrete model Integro-differential equations

46Txx 47Txx

## References

1. 1.
Ohnaka, M.: A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intact rock, and earthquake rupture. J. Geophys. Res. Solid Earth 108(B2) (2003).
2. 2.
Ordoñez, A., Peñuela, G., Idrobo, E.A., Medina, C.E.: Recent advances in naturally fractured reservoir modeling. Ciencia, Tecnologia y Futuro 2(2), 51–64 (2001)Google Scholar
3. 3.
Rice, J.R.: Constitutive relations for fault slip and earthquake instabilities. Pure Appl. Geophys. 121(3), 443–475 (1983)
4. 4.
Daub, E.G., Carlson, J.M.: Friction, fracture, and earthquakes. Annu. Rev. Condens. Matter Phys. 1(1), 397–418 (2010)
5. 5.
Moore, J.R., Gischig, V., Button, E., Loew, S.: Rockslide deformation monitoring with fiber optic strain sensors. Nat. Hazards Earth Syst. Sci. 10, 191–201 (2010)
6. 6.
Andrews, D.J.: Rupture velocity of plane strain shear cracks. J. Geophys. Res. 81(32), 5679–5687 (1976)
7. 7.
Beeler, N.M., Tullis, T.E., Blanpied, M.L., Weeks, J.D.: Frictional behavior of large displacement experimental faults. J. Geophys. Res. Solid Earth 101(B4), 8697–8715 (1996)
8. 8.
Falk, M.L., Shi, Y.: Strain localization in a molecular-dynamics model of a metallic glass. MRS Online Proc. Libr. Arch. 754 (2002)Google Scholar
9. 9.
Falk, K.L., Langer, J.S.: Shear transformation zone theory elasto-plastic transition in amorphous solids. Phys. Rev 57, 7192–7204 (1998)
10. 10.
Marone, C.: Laboratory-derived friction laws and their application to seismic faulting. Annu. Rev. Earth Planet. Sci. 26(1), 643–696 (1998)
11. 11.
Zheng, G., Rice, J.R.: Conditions under which velocity-weakening friction allows a self-healing versus a cracklike mode of rupture. Bull. Seismol. Soc. Am. 88(6), 1466–1483 (1998)Google Scholar
12. 12.
Falk, M.L., Langer, J.S.: From simulation to theory in the physics of deformation and fracture. MRS Bull. 25(5), 40–45 (2005)
13. 13.
Lois, G., Lemaître, A., Carlson, J.M.: Numerical tests of constitutive laws for dense granular flows. Phys. Rev. E 72(5), 051303 (2005)
14. 14.
Daub, E.G., Carlson, J.M.: A constitutive model for fault gouge deformation in dynamic rupture simulations. J. Geophys. Res. Solid Earth 113(B12) (2008)Google Scholar
15. 15.
Doungmo Goufo, E.F., Oukouomi Noutchie, S.C.: Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates. C. R. Math. C.R Acad. Sci. Paris Ser. I (2013).
16. 16.
McLaughlin, D.J., Lamb, W., McBride, A.C.: Existence results for non-autonomous multiple-fragmentation models. Math. Methods Appl. Sci. 20, 1313–1323 (1997)
17. 17.
Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (Book 194). Springer, Berlin (2000)Google Scholar
18. 18.
Doungmo Goufo, E.F.: A mathematical analysis of fractional fragmentation dynamics with growth. J. Funct. Spaces 2014, 7. Article ID 201520 (2014).
19. 19.
Doungmo Goufo, E.F.: Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: basic theory and applications. Chaos Interdiscip. J. Nonlinear Sci. 26(8) (2016).
20. 20.
Doungmo Goufo, E.F.: Stability and convergence analysis of a variable order replicator-mutator process in a moving medium. J. Theor. Biol. 403, 178–187 (2016).
21. 21.
Oukouomi Noutchie, S.C., Doungmo Goufo, E.F.: Global solvability of a continuous model for nonlocal fragmentation dynamics in a moving medium. Math.Probl. Eng. Article ID320750 2013, 8 (2013).
22. 22.
Kato, T.: Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften. Band, vol. 132. Springer, New York (1984)Google Scholar
23. 23.
Da Prato, G., Grisvard, P.: Sommes d’opérateurs linéaires et équations différentielles opérationnelles. J. Math. Pures Appl. (9) 54(3), 305–387 (1975)
24. 24.
Neidhardt, H., Zagrebnov, V.A.: Linear non-autonomous Cauchy problems and evolution semigroups. AMS, Mathematical Physics (2007)
25. 25.
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. In: Applied Mathematical Sciences, vol. 44. Springer (1983)Google Scholar
26. 26.
Yosida, K.: Functional analysis, 6th edn. Springer, Berlin (1980)
27. 27.
Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)