Dynamic displacement tracking in viscoelastic solids by actuation stresses: a one-dimensional analytic example involving shock waves

• Autorzy: Irschik Hans , Krommer Michael

Identyfikatory

DOI

10.24425/bpasts.2023.144616

• Języki publikacji: EN

Abstrakty

EN

A one-dimensional (1D) analytic example for dynamic displacement tracking in linear viscoelastic solids is presented. Displacement tracking is achieved by actuation stresses that are produced by eigenstrains. Our 1D example deals with a viscoelastic half-space under the action of a suddenly applied tensile surface traction. The surface traction induces a uni-axial shock wave that travels into the half-space. Our tracking goal is to add to the applied surface traction a transient spatial distribution of actuation stresses such that the total displacement of the viscoelastic half-space coincides with the shock wave produced by the surface traction in a purely elastic half-space. We particularly consider a half-space made of a viscoelastic Maxwell-type material. Analytic solutions to this tracking problem are derived by means of the symbolic computer code MAPLE. The 1D solution presented below exemplifies a formal 3D solution derived earlier by the present authors for linear viscoelastic solids that are described by Boltzmann hereditary laws. In the latter formal solution, no reference was made to shock waves. Our present solution demonstrates its validity also in the presence of singular wave fronts. Moreover, in our example, we show that, as was also indicated in our earlier work, the actuation stress can be split into two parts, one of them producing no stresses, and the other no displacements in two properly enlarged problems.

Słowa kluczowe

EN

displacement tracking; eigenstrains; actuation stresses; linear viscoelasticity; shock waves; symbolic computation

PL

śledzenie przemieszczenia; naprężenia własne; lepkosprężystość liniowa; fale uderzeniowe; obliczenia symboliczne; naprężenie uruchamiające

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2023
• Tom: Vol. 71, nr 3
• Strony: art. no. e144616
• Opis fizyczny: Bibliogr. 18 poz., rys.

Twórcy

autor

Irschik Hans

• Institute of Technical Mechanics, Johannes Kepler University of Linz, Austria

autor

Krommer Michael

• Institute of Technical Mechanics, Johannes Kepler University of Linz, Austria

• https://orcid.org/0000-0003-2753-1090

Bibliografia

• [1] H. Parkus, Instationäre Wärmespannungen. Vienna: Springer, 1959.
• [2] W. Nowacki, Dynamic Problems in Thermoelasticity. Dordrecht: Springer, 1975.
• [3] J. Holnicki-Szulc and J.T. Gierlinski, Structural Analysis, Design and Control by the Virtual Distortion Method. Chichester: Wiley, 1995.
• [4] J. Holnicki-Szulc, Ed., Smart Technologies for Safety Engineering. Chichester: Wiley 2008.
• [5] J. Nowacki, Static And Dynamic Coupled Fields in Bodies With Piezoeffects or Polarization Gradient. Berlin: Springer, 2006.
• [6] Y.C. Fung, P. Tong, and X. Chen, Classical and Computational Solid Mechanics, 2nd ed. New Jersey: World Scientific, 2017.
• [7] H. Irschik, M. Krommer, and Ch. Zehetner, “Displacment tracking of pre-deformed smart structures,” Smart Struct. Syst., vol. 18, no. 1, pp. 139–154, 2016.
• [8] H. Irschik and M. Krommer, “Dynamic Displacement Tracking of Force-Loaded Linear Elastic or Viscoelastic Bodies by Eigenstrain-Induced Actuation Stresses,” in IDETC/CIE 2005, ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference 2005, Long Beach, Cal., USA, 2005, ASME Paper No. DETC2005-84835, pp. 681–688.
• [9] M.J. Leitman and G.M.C. Fisher, “The Linear Theory of Viscoelasticity,“ in: Encyclopedia of Physics, vol. VIa/3, S. Flügge, Ed. Berlin: Springer, 1973, pp. 1–124.
• [10] “Maple Product History,” MapleSoft. [Online]. Available: https://de.maplesoft.com/products/maple/history/ [Acc.: 8. Sept. 2022].
• [11] M.E. Gurtin, “The Linear Theory of Elasticity,” in: Encyclopedia of Physics, vol. VIa/2, S. Flügge, Ed. Berlin: Springer, 1972, pp. 1–296.
• [12] H. Irschik and F. Ziegler, “Dynamic processes in structural thermo-viscoplasticity,” Appl. Mech. Rev., vol. 48, pp. 301–316, 1995.
• [13] E.H. Lee and I. Kanter, “Wave propagation in finite rods of viscoelastic material,” J. Appl. Phys., vol. 24, no 9, pp. 1115–1122, 1953.
• [14] R.V. Churchill, Operational Mathematics, 3rd ed. New York: McGraw -Hill, 1972.
• [15] H. Irschik and A. Brandl, “On Control of Structural Displacements by Eigenstrains in the Presence of Singular Waves,” in: Contributions to Advanced Dynamics and Continuum Mechanics, H. Altenbach, H. Irschik and V.P. Matveenko, Eds., Chalm: Springer Nature 2019, pp. 95–109.
• [16] Y. Vetyukov, E. Staudigl, and M. Krommer, “Hybrid asymptotic-direct approach to finite deformations of electromechanically coupled piezoelectric shells,” Acta Mech., vol. 229, pp. 953–974, 2018.
• [17] A. Humer, A. Pechstein, M. Meindlhumer, and M. Krommer, “Nonlinear electromechanical coupling in ferroelectric materials: large deformation and hysteresis,” Acta Mech., vol. 231, pp. 2521–2544, 2020.
• [18] M. Krommer, S. Klinkel, and T. Wallmersperger, “Editorial to Special Issue “Advanced non-linear modeling and numerical methods for smart materials and structures”,” Acta Mech., 2022, doi: 10.1007/s00707-022-03431-z.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).