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Archive of Applied Mechanics
Published in: Archive of Applied Mechanics 8/2018

06-04-2018 | Original

Longitudinal impact into viscoelastic media

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

Published in: Archive of Applied Mechanics | Issue 8/2018

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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.
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Appendix
Available only for authorised users
Footnotes
1
Since the flyer and backing impedances are equal, \(\theta =1/2\) in Eq. (71), so that \(v(x,\infty )= V_0/2\).
 
2
cf. Gurtin and Sternberg [10], Christensen [11], Wineman and Rajagopal [12], Walton [13] and Willis [14].
 
3
\(\mu \) 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.
 
4
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.
 
5
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.
 
6
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.
 
7
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.
 
8
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.
 
9
This relation is not valid for a finite thickness target. The appropriate impact boundary condition for that case is considered in Sect. 6.
 
10
cf. Widder [21], Fodor [15], Kaplan [22], LePage [23], Wylie [24] and Doetsch [25].
 
11
See Eq. (119) in “Appendix C,” with \(\sigma _1\) replaced by \(u_1\).
 
12
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].
 
13
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.
 
14
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.
 
15
The finite flyer length \(k=5\) in the table is used for the results in Sect. 6.1.
 
16
Unless \(\dot{v_1}(x,t)\) is regarded as a generalized derivative; cf. Fodor [15, Eq. (6.31)].
 
17
cf. Fodor [15, Eq. (6.32)], LePage [23, Eqs. (12–35)], and for single jumps Chadwick and Powdrill [31] and Martin [32].
 
18
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.
 
19
cf. Coleman, Gurtin and Herrera [19] and Nunziato et al. [17, 18].
 
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Metadata
Title
Longitudinal impact into viscoelastic media
Authors
George A. Gazonas
Raymond A. Wildman
David A. Hopkins
Michael J. Scheidler
Publication date
06-04-2018
Publisher
Springer Berlin Heidelberg
Published in
Archive of Applied Mechanics / Issue 8/2018
Print ISSN: 0939-1533
Electronic ISSN: 1432-0681
DOI
https://doi.org/10.1007/s00419-018-1372-z

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