Setting material function design targets for linear viscoelastic materials and structures

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 and G00) 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.