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Research Papers: Design Automation

# Setting Material Function Design Targets for Linear Viscoelastic Materials and Structures

[+] Author and Article Information
R. E. Corman

Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: corman1@illinois.edu

Lakshmi Rao

Department of Industrial and Enterprise
Systems Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801

N. Ashwin Bharadwaj

Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801

James T. Allison

Assistant Professor
Department of Industrial and Enterprise
Systems Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801

Randy H. Ewoldt

Assistant Professor
Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 9, 2015; final manuscript received January 25, 2016; published online March 21, 2016. Assoc. Editor: Nam H. Kim.

J. Mech. Des 138(5), 051402 (Mar 21, 2016) (12 pages) Paper No: MD-15-1206; doi: 10.1115/1.4032698 History: Received March 09, 2015; Revised January 25, 2016

## Abstract

Rheologically complex materials are described by function-valued properties with dependence on a timescale (linear viscoelasticity), input amplitude (nonlinear material behavior), or more generally both (nonlinear viscoelasticity). This complexity presents a difficulty when trying to utilize these material systems in engineering designs. Here, we focus on linear viscoelasticity and a methodology to identify the desired viscoelastic behavior. This is an early-stage design step to optimize target (function-valued) properties before choosing or synthesizing a real material. In linear viscoelasticity, it is not obvious which properties can be treated as independent design variables. Thus, it is nontrivial to select the most design-appropriate constitutive model, to be as general as possible, but not violate fundamental restrictions. We use the Kramers–Kronig constraint to show that frequency-dependent moduli (e.g., shear moduli $G′(ω)$ and $G″(ω)$) cannot be treated as two independent design variables. Rather, a single function such as the relaxation modulus (e.g., K(t) for force-relaxation or G(t) for stress relaxation) is an appropriate function-valued design variable. A simple case study is used to demonstrate the framework in which we identify target properties for a vibration isolation system. Viscoelasticity improves performance. Different parameterizations of the kernel function are optimized and compared for performance. While parameterization may limit the generality of the kernel function, we do include a nonobvious representation (power law) that is found in real viscoelastic material systems and in the spring-dashpot paradigm would require an infinite number of components. Our methodology provides a means to answer the question, “What viscoelastic properties are desirable?” This ability to identify targeted behavior will be useful for subsequent stages of the design process including the selection or synthesis of real materials.

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## Figures

Fig. 2

The introduction of a viscoelastic connection can change the design space from a discrete arrangement of linear springs and dashpots (left) to a single viscoelastic element with a relaxation kernel, K(t)

Fig. 1

Function-valued viscoelastic properties may enable novel or improved performance (performance example schematics depict vibration damping of structures, vibration isolation of a suspended mass, and pressure sensitive adhesives). An early-stage design question is “what properties are optimal?” (represented here by the left, curved, downward arrow). This paper focuses on a methodology for answering that question with linear viscoelastic materials/systems including the nontrivial problem of selecting design-appropriate modeling using material properties (represented by the right, curved, upward arrow). Successful identification of targeted properties then leads to microstructure and formulation design (material selection or material synthesis) [2528], which is beyond the scope of the work here.

Fig. 3

Parameterizations of the relaxation kernel, K(t), for a general linear viscoelastic fluid model (Fig. 4(c)) into numerous linear viscoelastic models to reduce the complexity of the design space. The viscoelastic models used to parameterize the relaxation modulus of the added viscoelastic component are: (i) a linear dashpot (solid-line), (ii) a Maxwell element (linearspring and dashpot connected in series) single mode (dashed–dotted line), (iii) multimode Maxwell element (short dashed–dotted line), and (iv) a critical gel model, mechanical spring-dashpot analog is not applicable (short-dashed line). Inset plot is double-log plot of the same curves with the Maxwell model characteristic timescale λ (as in Eq. (13)) and the critical gel exponent n (as in Eq. (16)) labeled for reference.

Fig. 4

Design of optimal viscoelastic vibration isolation for a one-dimensional spring-mass system (a); (b) the typical approach of an arrangement of springs and dashpots, and (c) a generalized viscoelastic element with relaxation kernel, K(t). The latter approach increases design freedom and identifies more optimal targets.

Fig. 9

The simple vibration isolator problem is extended to optimize the performance for a range of natural frequencies of the system but a fixed K(t). Increasing the natural frequency of the system (ω1=k/m) can be conceptualized by increasing the spring constant k and/or decreasing the system mass, m, shown in (a)–(c).

Fig. 8

(a) Optimized performance (acceleration amplitude) and (b) the corresponding optimal viscoelastic (VE) design (kernel function). The viscoelastic models used to parameterize the relaxation kernel of the added viscoelastic component (shown in Fig. 4(c)) are: (i) a linear dashpot, solid, (ii) a single mode Maxwell element, dashed–dotted lines, (iii) three-mode Maxwell element, short dashed–dotted lines, and (iv) a critical gel model, short-dashed lines. A multimode Maxwell model uses its additional degrees of freedom to achieve approximately the same relaxation kernel and performance as a single mode, which therefore overlap as the most optimal of these parameterizations.

Fig. 10

The performance (left) and viscoelastic (VE) design (right) for a Maxwell element and Critical gel element in the extended vibration isolation problem. Multimode and single mode system response was identical, thus only a single mode is shown. The system performance is measured by the acceleration response while the viscoelastic design corresponds to the kernel function, K(t), of the viscoelastic element. Dashed lines represent optimization for a reference nondimensional natural frequency ω̃*=1, where ω̃* is as described in Eq. (34); solid lines represent the results optimized for the range of ω̃*. In the left panels (performance), the values of the objective function (fourth order norm of the maximum accelerations of each curve), optimized for a reference ω̃*=1 (dashed lines) and a range of ω̃* values are shown as horizontal lines. The optimizer works to minimize this objective function.

Fig. 5

Design tradeoff from added dashpot in Fig. 4(b) vibration isolator, displacement amplitude (|x|), and acceleration amplitude (|x¨|) responses. Decreasing peak amplitude at a resonant frequency lends to worse signal attenuation at higherfrequencies. Damping coefficients (ζ=c/2mωn) of ζ=0.1,0.5,0.9 are shown. Response of the mass is plotted with respect to ω̃, a frequency normalized by the resonant frequency.

Fig. 6

Design involving a generalized viscoelastic element with relaxation kernel K(t) can eliminate high-frequency acceleration. Here shown with Maxwell element (dashed–dotted line), and critical gel (short-dashed line) compared to dashpot (solid line), or no additional component (dashed line).

Fig. 7

The design space for (a) the single mode Maxwell element and (b) the critical gel to isolate vibrations in a single mass-spring system (given by Eqs. (13) and (16)). Each model increases the design space to two dimensions, compared to the single dimension of a simple linear dashpot. Higher saturation correlates with lower peak normalized accelerations, as given in the scale bar (right), under simple sinusoidal displacement forcing. The white circle identifies a global minimum of peak acceleration.

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