Analysis and Design of Visco-Elastic Cushions for Delicate Objects

Abstract

Efficient protective structures are needed to ensure safe transport of fragile devices, cushioning of delicate objects in mechanically hazardous environments, like spacecraft launch, and protection of humans during collisions in athletics and automotive accidents. Compliant delicate objects, that are enclosed in rigid shells, are protected from indentation, but are vulnerable to impacts that involve large accelerations and high coefficients of restitution. The human brain is an example of such a delicate object. Early theories of traumatic brain injury identified two kinematic injury regimes, which are separated by a characteristic time. Depending on the duration of the impact, either a maximum tolerable acceleration or velocity governs risk, and common cushioning strategies that rely solely on reducing force may be inadequate for impacts that are shorter than the object's natural period.

First, we introduce a dissipative cushioning strategy that uses the fractional derivative visco-elastic model known as the fractional Standard Linear Solid (FSLS). The FSLS is capable of describing the complex modulus of a variety of materials accurately with a smaller number of parameters than a multi-term Prony series. Material selection criteria and a framework for optimizing cushions for minimum size are also presented. The cushions that dissipate most efficiently have relaxation times which are slightly shorter than the impact, and have high loss factors. The acceleration of the impact can be reduced more efficiently with a visco-elastic cushion than a purely elastic collision. The successful implementation of this cushioning strategy requires knowledge of the characteristic time of the object and threshold values for its maximum tolerable velocity and acceleration.

Therefore, the calculation of the characteristic time and approximate kinematic tolerances for the brain, obtained by analyzing an idealized two-dimensional cylindrical model, are sought in the second part of this thesis. Both rotation and linear translation of the cylinder's shell are evaluated, and the resulting pressure and shear strain are calculated. This approach is based on the theory that dangerous strain arises from rotational motion, and that dangerous levels of pressure come from linear motion. The rotation of the skull primarily produces shear stress and strain. Linear motion would produce both shear stress and strain, but the high bulk modulus of the brain suppresses volumetric strain and the low shear modulus suppresses high shear stress. The strain from linear motion is found to be relatively small.

The final chapters, which consider the strain due to rotation, provide scaling laws for estimating shear strain from rotational loads. In particular, shear strain is either dependent on angular velocity or the angular velocity times a fractional power of the impact duration, when the impact is short. The points of transition between different types of response are identified as the characteristic times of the brain, and their dependence on material properties and dimensions of the brain are discussed. Knowing which types of motion bring about more or less deformation is important for knowing which aspects of the impact should be mitigated. This can be paired with the analysis of the cushion response, addressed initially, to design cushions that meet design criteria based on the kinematic damage criteria. Therefore, the correlations between strain and kinematics can be coupled with the cushion design framework to optimize the design of helmets and evaluate their performance with simple experiments that measure easily-observable kinematic quantities rather than internal strain.