Research Papers

Multifunctional Optimization of Viscoelastic Materials Subjected to Spherical Impact

[+] Author and Article Information
Martin Herrenbrück

Fachgebiet Computational Mechanics,
Technische Universität München,
München 80333, Germany
e-mail: herrenbruck@gmail.com

Fabian Duddeck

Fachgebiet Computational Mechanics,
Technische Universität München,
München 80333, Germany
e-mail: duddeck@tum.de

Roman Lackner

Arbeitsbereich für Materialtechnologie,
Leopold-Franzens-Universität Innsbruck,
Innsbruck 6020, Austria
e-mail: Roman.Lackner@uibk.ac.at

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 13, 2015; final manuscript received September 3, 2015; published online September 28, 2015. Assoc. Editor: Weinong Chen.

J. Appl. Mech 82(12), 121009 (Sep 28, 2015) (8 pages) Paper No: JAM-15-1314; doi: 10.1115/1.4031554 History: Received June 13, 2015; Revised September 03, 2015

The event of rigid solids impacting a viscoelastic body is encountered in many engineering disciplines. However, for this problem no analytical solution is available, making the proper design of, e.g., protective bodies a difficult task. As a remedy, generally valid solutions in form of nondimensional response curves are presented in this paper, serving as reference for validation purposes or as design charts for the performance-oriented development of impact absorbers. Hereby, the problem of a frictionless rigid sphere impinging a viscoelastic half-space is investigated by finite-element analyses. The general applicability of the results is assured by a transformation to dimensionless problem parameters. For this purpose, the analytical solution by Hertz (1881, “Über die Berührung fester elastischer Körper,” J. die Reine Angew. Math., 1882(92), pp. 156–171) for the purely elastic impact is taken into account. The chosen nondimensional format leads to a reduced number of system parameters, allowing for a compact representation by so-called master curves. From these, optimal material characteristics are found for three different design objectives.

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Fig. 1

Rheological models for linear viscoelasticity: (a) the Maxwell element and (b) the standard linear solid

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Fig. 2

Finite-element model where use of rotational symmetry was made

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Fig. 3

Example results (normalized to unit range) for the penetration, velocity, kinetic energy, and acceleration over time

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Fig. 4

Maxwell model—each symbol represents the result from one numerical simulation: (a), (c), and (e) dimensional results and (b), (d), and (f) master curves obtained by the use of dimensionless abscissa and ordinate. (a) Dependence of uve on τ, (b) Πu over Πτθ, (c) dependence of vfinal on τ, (d) Πrest over Πτθ, (e) dependence of ave on τ, and (f) Πa over Πτθ.

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Fig. 5

Standard linear solid, where five different values for α were used: (a), (c), and (e) dimensional results and (b), (d), and (f) master curves obtained by the same scaling as in Fig. 4. The Hertzian solution refers to the initial stiffness E0. The horizontal lines given in (b) and (f) for small Πτθ indicate the limit values (1−α)−0.4 and (1−α)+0.4, respectively. (a) Dependence of uve on τ, (b) Πu over Πτθ, (c) dependence of vfinal on τ, (d) Πrest over Πτθ, (e) dependence of ave on τ, and (f) Πa over Πτθ.

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Fig. 6

Standard linear solid (α=0.9): dimensional results from a numerical parameter study for varying characteristic times τ in the range from 0.001 s to 4.0 s. For each of the seven simulations, the acceleration is plotted versus the absolute value of the penetration, where the physical time is a curve parameter. From this, the maximum values for acceleration and penetration are marked by diamonds. For comparison, the dotted line shows the relationship between uel and ael according to the Hertz solution.

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Fig. 7

Standard linear solid: plot of Πa over Πu (i.e., nondimensional representation), where the solid line indicates the Hertzian solution using E=(1−α)E0 for a varying parameter α

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Fig. 8

Maxwell model: multi-objective function D(Πτθ) for different weights wu=1−wa, where the symbols represent result points from numerical simulations corresponding to D=du (i.e., wu=1) and D=da (i.e., wu=0), respectively

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Fig. 9

Optima according to the L2 and the L∞ norm: (a) plot of the normalized distances du and da over the design variable Πτθ. The sum of the squared distances and its square root (L2 norm) are minimum at Πτθ=0.301. The intersection of du and da is located at Πτθ=0.261. (b) L2 and L∞ optimum in the objective function space. All designs are Pareto optimal.

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Fig. 10

Maxwell model: Identification of Πτθ=τ/θ for three different constraints on the maximum penetration, i.e., three different values for du, and the respective result for da

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Fig. 11

Maxwell model: relation between du, uve, da, and ave for the problem parameters listed in Table 3

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Fig. 12

Maxwell model: Sensitivity of uve and ave with respect to τ for the cases ① (uve100%=5 mm, full lines), ② (uve100%=7 mm, dashed lines), and ③ (uve100%=12 mm, dotted lines), see Table 5




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