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Longitudinal impact into viscoelastic media

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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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Notes

  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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Acknowledgements

We would like to thank our colleague, Dr. Rob Jensen (ARL), for providing the torsional DMA PC data plotted in Fig. 8a.

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Correspondence to George A. Gazonas.

Appendices

Appendix A: Verification of the numerical ILT solutions with the closed-form results of Lee and Kanter [5]

In this “Appendix A,” we determine the relative error between the exact solution derived by Lee and Kanter [5] and the numerical ILT solution using the Mathematica algorithm listed in the appendix in [4]. Lee and Kanter [5] study the problem of a finite length l target composed of a Maxwell material that is subjected to an arbitrary velocity time history v(0, t) on the “impact” surface \(x=0\), and a stress-free back face at \(x=l\) (Fig. 14). If we specialize the velocity to a Heaviside step in velocity \(v(0,t) = V_0\,H(t)\) the solution is

$$\begin{aligned} \begin{gathered} \sigma (x,t)\, = \,z \, V_0 \,e^{-\frac{\mu \, E \, t}{2}} \sum _{n=0}^\infty (-1)^n \left\{ H\left( t-\frac{2\, n\, l+x}{c}\right) I_0\left( \frac{\mu E}{2} \sqrt{t^2-\left( \frac{2 n l+x}{c}\right) ^2}\right) \right. \\\\ \quad \left. -\,H\left( t-\frac{2 (n+1) l-x}{c}\right) I_0\left( \frac{\mu E}{2} \sqrt{t^2-\left( \frac{2 (n+1) l-x}{c}\right) ^2}\right) \right\} . \end{gathered} \end{aligned}$$
(111)
Fig. 14
figure 14

Stress history [Eq. (111) with \(n = 2\) terms] on the face \(x=0\) of Maxwell viscoelastic target of length \(l = 10\) mm subjected to a constant velocity input history \(v(0,t) = V_0 H(t) = 5\) m/s. The back surface \(x=l\) of the target is stress-free. All target properties are from Table 1 with \(\mu = 1/\eta _1\) and \(E = G(0)\)

In Eq. (111), \(I_0\) and H denote the modified Bessel function of the first kind and Heaviside function, respectively. The upper limit of the summation in Eq. (111), i.e., \(n = \infty \), is in fact a finite limit for fixed t since the contributions from the Heaviside functions are nil, i.e., \(H({arg}) = 0\) for \(arg \, < \, 0\). Lee and Kanter [5] derive Eq. (111) through analytical inversion of

$$\begin{aligned} {{\overline{\sigma }}}(x,s)\, = \,\frac{V_0 z \left( e^{\frac{(l-x) \sqrt{s^2+E \mu s}}{c}}-e^{-\frac{(l-x) \sqrt{s^2+E \mu s}}{c}}\right) }{\sqrt{s^2+E \mu s} \left( e^{-\frac{l \sqrt{s^2+E \mu s}}{c}}+e^{\frac{l \sqrt{s^2+E \mu s}}{c}}\right) } . \end{aligned}$$
(112)

If \(\sigma (x,t)\) represents the exact value of the target stress given by Eq. (111), and \({\widetilde{\sigma }}(x,t)\) is the approximate target stress determined by using the numerical ILT algorithm listed in the appendix of [4], then the relative error, relerr, in \({\widetilde{\sigma }}(x,t)\) is given by,

$$\begin{aligned} \text {relerr} = 1 - \frac{{\widetilde{\sigma }}(x,t)}{\sigma (x,t)}. \end{aligned}$$
(113)

It can be shown that \(-\,\text {Log}_{10}(\text {Abs(relerr)})\) is a measure of the approximate number of significant digits in the solution. Note that relerr is defined only if \(\sigma (x,t)\, \ne \, 0\) (Fig. 15).

Fig. 15
figure 15

Numerical ILT accuracy of the modified DAC stress solution of Eq. (112) relative to the exact solution of Eq. (111) depicted in Fig. 14 for n = 2048 terms and tol = \(10^{-5}\) used in the ILT algorithm over the time interval \(0 \le t \le 30\, \upmu \mathrm{s}\); over this interval, the numerical solution accuracy ranges from about 4 to 6 significant digits. Loss in ILT accuracy is seen at \(t = 12 \, \upmu \mathrm{s}\) and \(t = 24 \, \upmu \mathrm{s}\) where jumps occur in \(\sigma (0,t)\)

Appendix B: Sixteen-term Prony series fit to PC data

The relaxation data illustrated in Fig. 8b are fit with \(N=16\) terms in the Prony series (Eq. (32) by assigning the relaxation times \(\tau _n\) at uniform decades that range from \(10^{-13}\) to 100 s and then using MATLAB’s [28] constrained nonlinear multivariable function fmincon to determine the \(g_n\) and the instantaneous elastic modulus G(0); see Table 3.

We also could have found a best fit to the relaxation data by simultaneously determining G(0) and both the \(g_i\) and \(\tau _i\) parameters, but Knauss and Zhao [30] have shown that there is no distinction in the global fit to the data by doubling the number of unknown parameters (Table 3).

Table 3 Sixteen-term Prony series fit to the PC data in Fig. 8b; \(G(0)=1.47306\) GPa

Appendix C: A correct derivation of the Laplace transform of the balance of momentum Eq. (50)

For a function f(t) that is only piecewise continuous and piecewise smooth, rule (46) for the Laplace transform of its derivative \({\dot{f}}\) is not valid.Footnote 16 The correct relation isFootnote 17

$$\begin{aligned} {{\mathscr {L}}} \big \{ \dot{f\,} \big \}(s) = s\,\overline{\! f\,}\!(s) - \sum _{n\ge 1}\, [[ f ]] (t_n)\, e^{-s\,t_n} , \end{aligned}$$
(114)

where \(t_1< t_2 < \cdots \) is the (finite or infinite) sequence of times at which f suffers a jump discontinuity, and \( [[ f ]] (t_n) = f(t_{n}^{+}) - f(t_{n}^{-})\). If the only jump in f occurs at time \(t_1=0\), and assuming (as usual) that \(f(t)=0\) for \(t<0\), then the sum in (114) reduces to \(f(0+)\), and we recover relation (46).

As in Sect. 2.2, let \( t_1(x)< t_2(x) < \cdots \,\) denote the infinite sequence of times at which the shock arrives at the point x in the target. That is, \(t_n(x)\) is the instant at which the shock reaches x on its nth one-way trip across the target. Then on applying (114) to \(f(t)=v_1(x,t)\), we obtain

$$\begin{aligned} {{\mathscr {L}}} \left\{ {\dot{v}}_1 \right\} (x,s) = s\,{\overline{v}}_1(x,s) - \sum _{n=1}^{\infty }\, [[ v_1 ]] \big (x,t_n(x)\big )\, e^{-s\,\,t_n(x)} . \end{aligned}$$
(115)

This is the correct expression for the term \({{\mathscr {L}}} \left\{ {\dot{v}}_1 \right\} \) that appears on the right-hand side of Eq. (49) for the Laplace transform of the momentum balance.

Now consider the term on the left-hand side of (49). We have

$$\begin{aligned} {{\mathscr {L}}} \{ \sigma _1 \} (x,s) = \int _{0}^{\infty } \sigma _1 (x,t)\, e^{-s\,t}\mathrm{d}t = \sum _{n=1}^{\infty } \int _{t_{n-1}(x)}^{t_n(x)} \sigma _1 (x,t)\, e^{-s\,t}\mathrm{d}t, \end{aligned}$$
(116)

where \(t_0(x)\equiv 0\). It is not hard to see that for \(n\ge 1\),

$$\begin{aligned} t_n(x) = {\left\{ \begin{array}{ll} \dfrac{ (n-1)\,l+x }{c_1} &{}\quad \text {if}\;{n=1,3,5,\ldots ;} \\ \dfrac{ n\,l-x }{c_1} &{}\quad \hbox { if} \;n=2,4,6,\ldots , \end{array}\right. } \end{aligned}$$
(117)

where \(c_1\) is the shock wave speed [cf. Eq. (37)]. The minus sign on x in the bottom expression reflects the fact that for n even the wave is traveling in the negative x-direction after reflection from the back face of the target. By (117),

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\,t_n(x) = \frac{ (-1)^{n-1} }{ c_1 }, \quad n=1,2,3,\ldots , \end{aligned}$$
(118)

independent of x. Next, we take the partial derivative of (116) with respect to x and use Leibniz’s rule for the derivative of an integral with variable limits of integration. Then on using (118) in the result and rearranging terms, we obtainFootnote 18

$$\begin{aligned} \frac{\partial }{\partial x}\, {{\mathscr {L}}} \{ \sigma _1 \} (x,s) = {{\mathscr {L}}} \left\{ \frac{\partial \sigma _1}{\partial x} \right\} (x,s) + \frac{1}{c_1} \,\sum _{n=1}^{\infty } \,(-1)^n\, [[ \sigma _1 ]] \big (x,t_n(x)\big )\, e^{-s\,\,t_n(x)} . \end{aligned}$$
(119)

On using Eqs. (115) and (119) in Eq. (49) for the Laplace transform of balance of momentum, we obtain

$$\begin{aligned} \frac{\partial {\overline{\sigma }}_1}{\partial x} (x,s) + \rho _1 s\,{\overline{v}}_1(x,s) = \sum _{n=1}^{\infty } \left( \frac{ (-1)^n}{c_1}\, [[ \sigma _1 ]] + \rho _1 [[ v_1 ]] \right) \!\! \big (x,t_n(x)\big )\, e^{-s\,\,t_n(x)} . \end{aligned}$$
(120)

We claim that the right-hand side of Eq. (120) is zero. Indeed, when stress is taken positive in compression the jumps in stress and particle velocity across a shock are related byFootnote 19

$$\begin{aligned}{}[[ \sigma _1 ]] = \rho _1 U [[ v_1 ]] , \end{aligned}$$
(121)

where U is the intrinsic or referential velocity of the wave front,

$$\begin{aligned} U(t) = \frac{\mathrm{d}}{\mathrm{d}t}\, Y(t), \end{aligned}$$
(122)

and Y(t) is the position of the shock front at time t. For a linear viscoelastic material and finite thickness target, \(|U(t)|=c_1\), independent of the time t, but during the nth one-way trip of the shock,

$$\begin{aligned} U(t) = (-1)^{n+1} c_1. \end{aligned}$$
(123)

U(t) is positive for odd n, when the wave is traveling in the positive x-direction (i.e., toward the back face); and U(t) is negative for even n, when the wave is traveling in the negative x-direction (i.e., toward the front face). From Eqs. (121) and (123), we see that

$$\begin{aligned}{}[[ \sigma _1 ]] \big (x,t_n(x)\big ) = \rho _1 (-1)^{n+1} c_1 [[ v_1 ]] \big (x,t_n(x)\big ). \end{aligned}$$
(124)

When this is substituted into Eq. (120), the terms inside the large parentheses cancel so that the right-hand side is zero, which yields Eq. (50).

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Gazonas, G.A., Wildman, R.A., Hopkins, D.A. et al. Longitudinal impact into viscoelastic media. Arch Appl Mech 88, 1275–1304 (2018). https://doi.org/10.1007/s00419-018-1372-z

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