nach oben

Archive of Applied Mechanics

Erschienen in:

06.04.2018 | Original

Longitudinal impact into viscoelastic media

verfasst von: George A. Gazonas, Raymond A. Wildman, David A. Hopkins, Michael J. Scheidler

Erschienen in: Archive of Applied Mechanics | Ausgabe 8/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We consider several one-dimensional impact problems involving finite or semi-infinite, linear elastic flyers that collide with and adhere to a finite stationary linear viscoelastic target backed by a semi-infinite linear elastic half-space. The impact generates a shock wave in the target which undergoes multiple reflections from the target boundaries. Laplace transforms with respect to time, together with impact boundary conditions derived in our previous work, are used to derive explicit closed-form solutions for the stress and particle velocity in the Laplace transform domain at any point in the target. For several stress relaxation functions of the Wiechert (Prony series) type, a modified Dubner–Abate–Crump algorithm is used to numerically invert those solutions to the time domain. These solutions are compared with numerical solutions obtained using both a finite-difference method and the commercial finite element code, COMSOL Multiphysics. The final value theorem for Laplace transforms is used to derive new explicit analytical expressions for the long-time asymptotes of the stress and velocity in viscoelastic targets; these results are useful for the verification of viscoelastic impact simulations taken to long observation times.

Vorheriger Artikel Complete vibrational bandgap in thin elastic metamaterial plates with periodically slot-embedded local resonators

Nächster Artikel Nonlinear dynamical behaviors of a complicated dual-rotor aero-engine with rub-impact

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

Since the flyer and backing impedances are equal, \(\theta =1/2\) in Eq. (71), so that \(v(x,\infty )= V_0/2\).

cf. Gurtin and Sternberg [10], Christensen [11], Wineman and Rajagopal [12], Walton [13] and Willis [14].

\(\mu \) is the stress relaxation function in shear, i.e., \(\mu (t)\) is the shear stress at time \(t\ge 0\) due to a “unit” Heaviside step in shear strain at time 0; cf also Eq. (12). A physical interpretation of \(\lambda \) is discussed in Sect. 2.2.

Simpler but analogous results hold for an isotropic linear elastic material. The momentum balance equation (19) is unchanged, and the only nonzero stress components are \(\pm \,\sigma _{11} = G\, \varepsilon _{11}\) and \(\pm \,\sigma _{22} = \pm \,\sigma _{33} = \lambda \,\varepsilon _{11}\), where G and \(\lambda \) are constants.

Actually, Eq. (21) is valid if \({\dot{\varepsilon }}_{11}\) is interpreted as a generalized derivative (cf. page 74 of Fodor [15]), although we will not use this interpretation here.

In section 6.2 of Gazonas et al. [1], it was shown that tensile stresses can arise temporarily in an elastic target in certain cases; it is necessary but not sufficient that the impedance of the target exceed the impedance of the elastic backing. A similar situation is expected for viscoelastic targets as well. That this tensile state would necessarily be temporary follows from a result established later in this paper, namely that the long-time asymptote of the longitudinal stress in a viscoelastic target is compressive.

This is a common convention in the shock physics literature; cf. Nunziato et al. [17, 18]. The strain is often taken positive in compression as well, although we do not do so here.

cf. Coleman, Gurtin and Herrera [19] and Theorem 5 of Fisher and Gurtin [20]. This speed governs shock waves and continuous waves (e.g., acceleration waves) in linear viscoelastic materials. Observe that it is independent of the deformation ahead of the wave front.

This relation is not valid for a finite thickness target. The appropriate impact boundary condition for that case is considered in Sect. 6.

cf. Widder [21], Fodor [15], Kaplan [22], LePage [23], Wylie [24] and Doetsch [25].

See Eq. (119) in “Appendix C,” with \(\sigma _1\) replaced by \(u_1\).

cf. Fodor [15, §10(b)], Kaplan [22, §6.11], LePage [23, §10.16, 12.7], Wylie [24, §7.4] and Doetsch [25, §34, 35, 37].

Proofs can be found in Kaplan [22, Thm. 13 in §6.11] for the case where f is piecewise continuous and in Doetsch [25, Thm. 34.3] for the general case.

Indeed, by Theorem 34.1 in Doetsch [25], this assumption on f implies that \(s{\overline{f}}(s)\thicksim \,a\,\Gamma (b+1)/s^{\,b}\) as \(s\rightarrow 0\), where \(\Gamma \) denotes the gamma function. Then the requirement that \(\lim _{\,s\rightarrow 0} \, s{\overline{f}}(s)\) exists implies that either \(b=0\) or \(b<0\), in which case the final value relation (65) holds with both limits equal to a or 0, respectively.

The finite flyer length \(k=5\) in the table is used for the results in Sect. 6.1.

Unless \(\dot{v_1}(x,t)\) is regarded as a generalized derivative; cf. Fodor [15, Eq. (6.31)].

cf. Fodor [15, Eq. (6.32)], LePage [23, Eqs. (12–35)], and for single jumps Chadwick and Powdrill [31] and Martin [32].

Observe that up to this point we have not made use of the fact that \(\sigma _1\) is a stress component; hence, Eq. (119) holds for other piecewise continuous and piecewise smooth functions with jumps across the shock only. See also the interesting papers by Chadwick and Powdrill [31] and Martin [32] for the multi-dimensional analogue of Eq. (119) for a single jump; i.e., they do not treat the case of multiple reflections. Also, by (119) it follows that we may interchange the Laplace transform and the partial derivative with respect to x for continuous and piecewise smooth functions.

cf. Coleman, Gurtin and Herrera [19] and Nunziato et al. [17, 18].

Gazonas, G.A., Scheidler, M.J., Velo, A.P.: Exact analytical solutions for elastodynamic impact. Int. J. Solids Struct. 75–76, 172–187 (2015)CrossRef

Gazonas, G.A., Velo, A.P., Wildman, R.A.: Asymptotic impact behavior of Goupillaud-type layered elastic media. Int. J. Solids Struct. 96, 38–47 (2016)CrossRef

Gazonas, G.A., Wildman, R.A., Hopkins, D.A.: Elastodynamic impact into piezoelectric media. U.S. Army Research Laboratory, ARL-TR-7056, Sept (2014)

Gazonas, G.A., Wildman, R.A., Hopkins, D.A., Scheidler, M.J.: Longitudinal impact of piezoelectric media. Arch. Appl. Mech. 86, 497–515 (2016)CrossRefMATH

Lee, E.H., Kanter, I.: Wave propagation in finite rods of viscoelastic material. J. Appl. Phys. 24(9), 1115–1122 (1953)CrossRefMATH

Berry, D.S., Hunter, S.C.: The propagation of dynamic stresses in visco-elastic rods. J. Mech. Phys. Solids 4, 72–95 (1956)MathSciNetCrossRefMATH

Behring, A.G., Oetting, R.B.: On the longitudinal impact of two, thin, viscoelastic rods. J. Appl. Mech. 37(2), 310–314 (1970)CrossRef

Musa, A.B.: Numerical solution of wave propagation in viscoelastic rods (standard linear solid model). In: 4th International Conference on Energy and Environment 2013 (ICEE 2013), IOP Conference Series: Earth and Environmental Science, vol. 16; Putrajaya, Malaysia, pp. 1–6 (2013)

COMSOL Multiphysics Reference Manual, Version 4.4. COMSOL, Inc., Burlington (2013)

10.

Gurtin, M.E., Sternberg, E.: On the linear theory of viscoelasticity. Arch. Ration. Mech. Anal 11, 20–356 (1962)MathSciNetCrossRefMATH

11.

Christensen, R.M.: Theory of Viscoelasticity: An Introduction. Academic Press, New York (1982)

12.

Wineman, A.S., Rajagopal, K.R.: Mechanical Response of Polymers: An Introduction. Cambridge University Press, New York (2000)

13.

Walton, J.R.: On the steady-state propagation of an anti-plane shear crack in an infinite general linearly viscoelastic solid. Q. Appl. Math. 40(1), 37–52 (1982)CrossRefMATH

14.

Willis, J.R.: Crack propagation in viscoelastic media. J. Mech. Phys. Solids 15, 229–240 (1967)CrossRefMATH

15.

Fodor, G.: Laplace Transforms in Engineering. Akadémiai Kiadó, Budapest (1965)MATH

16.

Schapery, R.A.: Viscoelastic behavior and analysis of composite materials. In: Sendeckyj, G.P. (ed.) Mechanics of Composite Materials, vol. 2, pp. 85–168. Academic Press, New York (1974)

17.

Nunziato, J.W., Schuler, K.W.: Shock pulse attenuation in a nonlinear viscoelastic solid. J. Mech. Phys. Solids 21, 447–457 (1973)CrossRef

18.

Nunziato, J.W., Walsh, E.K., Schuler, K.W., Barker, L.M.: Wave propagation in nonlinear viscoelastic solids. In: Truesdell, C. (ed.) Handbuch Der Physik; Volume VIa/4 Mechanics of Solids IV, pp. 1–108. Springer, New York (1974)

19.

Coleman, B.D., Gurtin, M.E., Herrera, R.I.: Waves in materials with memory, I. The velocity of one-dimensional shock and acceleration waves. Arch. Ration. Mech. Anal. 19, 1–19 (1965)MathSciNetCrossRefMATH

20.

Fisher, G.M.C., Gurtin, M.E.: Wave propagation in the linear theory of viscoelasticity. Q. Appl. Math. 23(3), 257–263 (1965)MathSciNetCrossRefMATH

21.

Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)

22.

Kaplan, W.: Operational Methods for Linear Systems. Addison-Wesley, Reading (1962)MATH

23.

LePage, W.R.: Complex Variables and the Laplace Transform for Engineers. Dover Publications, New York (1980)MATH

24.

Wylie, C.R.: Advanced Engineering Mathematics, 3rd edn. McGraw-Hill, New York (1966)MATH

25.

Doetsch, G.: Introduction to the Theory and Application of the Laplace Transform. Springer, New York (1974)MATH

26.

Mathematica Edition: Version 10.3. Wolfram Research, Champaign (2015)

27.

Laverty, R., Gazonas, G.A.: An improvement to the Fourier series method for inversion of Laplace transforms applied to elastic and viscoelastic waves. Int. J. Comput. Meth. 3(1), 57–69 (2006)MathSciNetCrossRefMATH

28.

MATLAB 9.2.0. The MathWorks, Inc., Natick, Massachusetts, United States (2017)

29.

Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)MathSciNetCrossRefMATH

30.

Knauss, W.G., Zhao, J.: Improved relaxation time coverage in ramp-strain histories. Mech. Time Depend. Mater. 11, 199–216 (2007)CrossRef

31.

Chadwick, P., Powdrill, B.: Application of the Laplace transform method to wave motions involving strong discontinuities. Proc. Camb. Philos. Soc. 60, 313–324 (1964)MathSciNetCrossRefMATH

32.

Martin, P.A.: The pulsating orb: solving the wave equation outside a ball. Proc. R. Soc. A 472, 20160037 (2016)MathSciNetCrossRefMATH

33.

Faciŭ, C., Molinari, A.: On the longitudinal impact of two phase transforming bars. Elastic versus a rate-type approach. Part II: the rate-type case. Int. J. Solids Struct. 43, 523–550 (2006)CrossRefMATH

34.

Critescu, N., Suliciu, I.: Viscoplasticity. Martinus Nijhoff Publishers, The Hague-Boston-London (1982)

Titel: Longitudinal impact into viscoelastic media
verfasst von: George A. Gazonas
Raymond A. Wildman
David A. Hopkins
Michael J. Scheidler
Publikationsdatum: 06.04.2018
Verlag: Springer Berlin Heidelberg
Erschienen in: Archive of Applied Mechanics / Ausgabe 8/2018
Print ISSN: 0939-1533
Elektronische ISSN: 1432-0681
DOI: https://doi.org/10.1007/s00419-018-1372-z

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 8/2018

Nonlinear dynamical behaviors of a complicated dual-rotor aero-engine with rub-impact

Dynamic study of viscoelastic rotor: a comparative study using analytical and finite element model considering higher-order system

Complete vibrational bandgap in thin elastic metamaterial plates with periodically slot-embedded local resonators

Numerical and experimental analysis of the vibroacoustic behavior of an electric window-lift gear motor

On the parametric and external resonances of rectangular plates on an elastic foundation traversed by sequential masses

Pochhammer–Chree waves: polarization of the axially symmetric modes

Premium Partner

Marktübersichten