# American Institute of Mathematical Sciences

March 2019, 8(1): 231-246. doi: 10.3934/eect.2019013

## Anti-plane shear Lamb's problem on random mass density fields with fractal and Hurst effects

 1 Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 2 Sandia National Laboratories, Albuquerque, NM 87185, USA 3 Department of Mechanical Science & Engineering, also Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

* Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energys National Nuclear Security Administration under contract DE-NA0003525

Received  October 2017 Revised  March 2018 Published  January 2019

This paper reports a study of transient dynamic responses of the anti-plane shear Lamb's problem on random mass density field with fractal and Hurst effects. Cellular automata (CA) is used to simulate the shear wave propagation. Both Cauchy and Dagum random field models are used to capture fractal dimension and Hurst effects in the mass density field. First, the dynamic responses of random mass density are evaluated through a comparison with the homogenerous computational results and the classical theoretical solution. Then, a comprehensive study is carried out for different combinations of fractal and Hurst coefficients. Overall, this investigation determines to what extent fractal and Hurst effects are significant enough to change the dynamic responses by comparing the signal-to-noise ratio of the response versus the signal-to-noise ratio of the random field.

Citation: Xian Zhang, Vinesh Nishawala, Martin Ostoja-Starzewski. Anti-plane shear Lamb's problem on random mass density fields with fractal and Hurst effects. Evolution Equations & Control Theory, 2019, 8 (1) : 231-246. doi: 10.3934/eect.2019013
##### References:
 [1] J. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Co., Amsterdam, 1976. [2] J. Dally and S. Thau, Observations of stress wave propagation in a half-plane with boundary loading, International Journal of Solids and Structures, 3 (1967), 293-300. doi: 10.1016/0020-7683(67)90031-5. [3] T. Gneiting and M. Schlather, Stochastic models that separate fractal dimension and the hurst effect, SIAM review, 46 (2004), 269-282. doi: 10.1137/S0036144501394387. [4] R. K. Hopman and M. J. Leamy, Arbitrary geometry cellular automata for elastodynamics, Sound, Vibration and Design, 15 (2009), 535-547. doi: 10.1115/IMECE2009-11222. [5] E. Kausel, Fundamental Solutions in Elastodynamics: A Compendium, Cambridge University Press, 2006. doi: 10.1017/CBO9780511546112. [6] M. J. Leamy, Application of cellular automata modeling to seismic elastodynamics, International Journal of Solids and Structures, 45 (2008), 4835-4849. doi: 10.1016/j.ijsolstr.2008.04.021. [7] B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco, Calif., 1982. [8] J. Mateu, E. Porcu and O. Nicolis, A note on decoupling of local and global behaviours for the dagum random field, Probabilistic Engineering Mechanics, 22 (2007), 320-329. doi: 10.1016/j.probengmech.2007.05.002. [9] V. V. Nishawala, M. Ostoja-Starzewski, M. Leamy and E. Porcu, Lamb's problem on random mass density fields with fractal and Hurst Effects, Proc. R. Soc. A, 472 (2016), 20160638, 14pp. doi: 10.1098/rspa.2016.0638. [10] V. V. Nishawala and M. Ostoja-Starzewski, Acceleration waves on random fields with fractal and hurst effects, Wave Motion, 74 (2017), 134-150. doi: 10.1016/j.wavemoti.2017.07.004. [11] V. V. Nishawala, M. Ostoja-Starzewski, M. J. Leamy and P. N. Demmie, Simulation of elastic wave propagation using cellular automata and peridynamics, and comparison with experiments, Wave Motion, 60 (2016), 73-83. doi: 10.1016/j.wavemoti.2015.08.005. [12] M. Ostoja-Starzewski, Ignaczak equation of elastodynamics, Mathematics and Mechanics of Solids, 2018. doi: 10.1177/1081286518757284. [13] E. Porcu, J. Mateu, A. Zini and R. Pini, Modelling spatio-temporal data: A new variogram and covariance structure proposal, Statistics and Probability Letters, 77 (2007), 83-89. doi: 10.1016/j.spl.2006.05.013. [14] M. Schlather, Simulation and analysis of random fields, R News, 1 (2001), 18-20. [15] L. Shen, M. Ostoja-Starzewski and E. Porcu, Bernoulli-Euler beams with random field properties under random field Loads: Fractal and Hurst effects, Archive of Applied Mechanics, 84 (2014), 1595-1626. doi: 10.1007/s00419-014-0904-4. [16] L. Shen, M. Ostoja-Starzewski and E. Porcu, Elastic rods and shear beams with random field properties under random field loads: fractal and hurst effects, Journal of Engineering Mechanics, 141 (2015), 04015002. doi: 10.1061/(ASCE)EM.1943-7889.0000906. [17] L. Shen, M. Ostoja-Starzewski and E. Porcu, Harmonic oscillator driven by random processes having fractal and hurst effects, Acta Mechanica, 226 (2015), 3653-3672. doi: 10.1007/s00707-015-1385-4. [18] L. Shen, M. Ostoja-Starzewski and E. Porcu, Responses of first-order dynamical systems to matérn, cauchy, and dagum excitations, Mathematics and Mechanics of Complex Systems, 3 (2015), 27-41. doi: 10.2140/memocs.2015.3.27. [19] J. Von Neumann, Theory of self-reproducing automata, IEEE Transactions on Neural Networks, 5 (1966), 3-14.

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##### References:
 [1] J. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Co., Amsterdam, 1976. [2] J. Dally and S. Thau, Observations of stress wave propagation in a half-plane with boundary loading, International Journal of Solids and Structures, 3 (1967), 293-300. doi: 10.1016/0020-7683(67)90031-5. [3] T. Gneiting and M. Schlather, Stochastic models that separate fractal dimension and the hurst effect, SIAM review, 46 (2004), 269-282. doi: 10.1137/S0036144501394387. [4] R. K. Hopman and M. J. Leamy, Arbitrary geometry cellular automata for elastodynamics, Sound, Vibration and Design, 15 (2009), 535-547. doi: 10.1115/IMECE2009-11222. [5] E. Kausel, Fundamental Solutions in Elastodynamics: A Compendium, Cambridge University Press, 2006. doi: 10.1017/CBO9780511546112. [6] M. J. Leamy, Application of cellular automata modeling to seismic elastodynamics, International Journal of Solids and Structures, 45 (2008), 4835-4849. doi: 10.1016/j.ijsolstr.2008.04.021. [7] B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco, Calif., 1982. [8] J. Mateu, E. Porcu and O. Nicolis, A note on decoupling of local and global behaviours for the dagum random field, Probabilistic Engineering Mechanics, 22 (2007), 320-329. doi: 10.1016/j.probengmech.2007.05.002. [9] V. V. Nishawala, M. Ostoja-Starzewski, M. Leamy and E. Porcu, Lamb's problem on random mass density fields with fractal and Hurst Effects, Proc. R. Soc. A, 472 (2016), 20160638, 14pp. doi: 10.1098/rspa.2016.0638. [10] V. V. Nishawala and M. Ostoja-Starzewski, Acceleration waves on random fields with fractal and hurst effects, Wave Motion, 74 (2017), 134-150. doi: 10.1016/j.wavemoti.2017.07.004. [11] V. V. Nishawala, M. Ostoja-Starzewski, M. J. Leamy and P. N. Demmie, Simulation of elastic wave propagation using cellular automata and peridynamics, and comparison with experiments, Wave Motion, 60 (2016), 73-83. doi: 10.1016/j.wavemoti.2015.08.005. [12] M. Ostoja-Starzewski, Ignaczak equation of elastodynamics, Mathematics and Mechanics of Solids, 2018. doi: 10.1177/1081286518757284. [13] E. Porcu, J. Mateu, A. Zini and R. Pini, Modelling spatio-temporal data: A new variogram and covariance structure proposal, Statistics and Probability Letters, 77 (2007), 83-89. doi: 10.1016/j.spl.2006.05.013. [14] M. Schlather, Simulation and analysis of random fields, R News, 1 (2001), 18-20. [15] L. Shen, M. Ostoja-Starzewski and E. Porcu, Bernoulli-Euler beams with random field properties under random field Loads: Fractal and Hurst effects, Archive of Applied Mechanics, 84 (2014), 1595-1626. doi: 10.1007/s00419-014-0904-4. [16] L. Shen, M. Ostoja-Starzewski and E. Porcu, Elastic rods and shear beams with random field properties under random field loads: fractal and hurst effects, Journal of Engineering Mechanics, 141 (2015), 04015002. doi: 10.1061/(ASCE)EM.1943-7889.0000906. [17] L. Shen, M. Ostoja-Starzewski and E. Porcu, Harmonic oscillator driven by random processes having fractal and hurst effects, Acta Mechanica, 226 (2015), 3653-3672. doi: 10.1007/s00707-015-1385-4. [18] L. Shen, M. Ostoja-Starzewski and E. Porcu, Responses of first-order dynamical systems to matérn, cauchy, and dagum excitations, Mathematics and Mechanics of Complex Systems, 3 (2015), 27-41. doi: 10.2140/memocs.2015.3.27. [19] J. Von Neumann, Theory of self-reproducing automata, IEEE Transactions on Neural Networks, 5 (1966), 3-14.
(a) cell (i, j) and its neighbors for Cellular Automata discretization (b) the neighbor stresses acting on cell (i, j)
(a) Computational domain (b) Anti-plane triangular load
At 92$\mu$s: CA displacement and stress responses
White noise RFs with carying coarseness, mean = 1300 $\mathrm{ kg/m^{3}}$ and $\mathrm{CV_{RF}}$ = 0.124. Legends are density in $\mathrm{ kg/m^{3}}$
WN RFs responses with varying coarseness (a): mean and SD of random field responses versus theoretical solution and homogenerous results; (b) CV of response; (c) SNR of response
Cauchy RFs with $\left\langle \rho \right\rangle = 1, 300$ $\mathrm{ kg/m^{3}}$, $\mathrm{CV_{RF}}$ = 0.124. Legends are density in $\mathrm{ kg/m^{3}}$
Cauchy RFs with mean = 1300 $\mathrm{\ kg/m^{3}}$ and $CV_{RF}$ = 0.124, (a, d, g) $\beta$ = 0.2; (b, f, i) $\beta$ = 1.0; (c, f, i) $\beta$ = 1.8. (a, b, c) $\alpha$ = 1.8; (d, f, g) $\alpha$ = 1.0; (h, i, j) $\alpha$ = 0.2
Cauchy RFs with $\mathrm{CV_{RF}} = 0.124$: Comparison of $\mathrm{SNR_{R}}$ and $\mathrm{SNR_{RF}}$ for varying $\alpha$ and $\beta$. The boundary between $\mathrm{ SNR_{R}}$ less than or greater than $\mathrm{ SNR_{RF}}$ is represently by the dotted line
Dagum RFs with $\left\langle \rho \right\rangle = 1, 300$ $\mathrm{ kg/m^{3}}$, $\mathrm{CV_{RF}}$ = 0.124. Legends are density in $\mathrm{ kg/m^{3}}$
Dagum RFs responses (at $92$ $\mu s$) with $\alpha$ = 0.8 (D = 2.9) and varying $\beta$. (a): mean and SD of random fields responses versus theoretical solution and homogenerous results; (b) CV of response versus CV of RF; (c) SNR of response versus SNR of RF
Dagum RFs with $\mathrm{CV_{RF}}$ = 0.124: Comparison of $\mathrm{ SNR_{R}}$ and $\mathrm{SNR_{RF}}$ for varying $\alpha$ and $\beta$. The boundary between $\mathrm{\ SNR_{R}}$ less than or greater than $\mathrm{\ SNR_{RF}}$ is represently by the dotted line
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