Several investigators have proposed materials for brain surrogates, the most common being a polydimethylsiloxane (PDMS) known commercially as Sylgard. More recently, a hydrogel has been shown to capture brain response over a larger dynamic range. We have used PDMS as a brain surrogate and plaster-of-Paris as a skull surrogate in a preliminary investigation to visualize brain kinematics in 2-D helmeted head impact experiments. The PDMS brain is surrounded by the skull surrogate. The surface of the brain is coated with a speckle pattern of paint. The helmet consists of concentric layers of elastic and visco-elastic polymers around the skull. The structure is impacted by a massive pendulum and filmed during the impact event with high-speed digital cameras (Photron Fastcams). Digital image correlation (DIC) analysis is used to determine the normal and shear displacements of the speckle pattern. Further analysis provides brain strains, velocities, and accelerations (linear and rotational).
The largest positive (red) and negative (purple) shear strains are readily determined from these contours. Moreover, maximum brain velocity and acceleration contours can be generated, and their locations can be correlated with those of strain maxima. These experiments also allow us to correlate the velocity and acceleration of the pendulum with brain kinematics to determine impact parameters in which the skull acceleration causes critical strain measures vs. those parameters for which head velocity is the damage metric. Three visco-elastic materials were tested under impact conditions on a force plate using a drop tower with an anvil set an at initial height such that its kinetic energy upon impact was 7.5 J. The specimens were all the same size (100 mm in diameter and 19 mm in height). The force was recorded as a function of time, from which the force as a function of frequency was determined, using Fourier transforms. The frequency content of the stress waves traveling through these materials differs in this example because of the differences in the mechanical properties of the materials. The storage moduli of the three materials at 50 Hz was found from dynamic mechanical analysis (DMA) to be as follows: A, 1 MPa; B, 5 MPa; and C, 16 MPa.
These experiments demonstrate the coupling between the mechanical properties of an impacted material and its dissipative requirements under given impact conditions. The frequencies traveling through these materials, and hence the frequencies they would transmit through the skull to the brain if they were used in helmet designs, are shown in Figure 2 to truncate by 1000 Hz, eliminating the need to obtain brain properties above 1000 Hz. This important simplifying assumption has also been validated by numerical models of the wave speed though systems with geometries and material properties approximating the skull and brain. Linear visco-elastic solids with a single characteristic relaxation time are known as standard linear solids (SLS) (aka "Kelvin-Maxwell" materials) and may be described in terms of three material properties: the unrelaxed modulus Eu; the relaxed modulus Er; and the relaxation time R. The impact response of an SLS may be computed by solving this equation with the proper boundary conditions for impact loading. As Er/Eu decreases, the damping increases, the peak damping frequency shifts to a lower frequency, and the range of frequencies over which the viscoelastic material dissipates energy efficiently increases.
Both the force and impulse have been normalized by the values that would be transmitted by a perfectly elastic body. It will be noted that the peak force is efficiently reduced by very low values of the relaxed modulus for materials with relatively short relaxation times. While reduction of the impulse is enhanced by a relatively short relaxation time, an efficient reduction requires a careful match of all the parameters in a relatively narrow range. Stated another way, for an application using a material of a given shape, size, and mass, there is an optimal choice for the visco-elastic parameters of the material to minimize the force and impulse. However, with careful matching and design it is possible to dissipate almost all the energy of impact - as indicated by the fact that the curves reach a dimensionless impulse equal to 1, corresponding to a perfectlyplastic collision.
A 2-D analysis of the impact response of helmets was done using the commercial finite element analysis package, ABAQUS Explicit. In these simulations the head was modeled as a skull/brain system and the properties of the helmet were determined through DMA testing. An existing helmet was compared to our preliminary helmet prototype. Linear and rotational accelerations were examined throughout the region of the brain and peak values recorded for comparison. The peak pressure, translational acceleration, and rotational acceleration histories inside the brain are reduced in our helmet prototype compared to
those of the existing design. The peak values occur at different nodes for the various quantities recorded, and for different nodes in each helmet, but in every case, the highest magnitude was searched within the entire brain region and that is what is recorded for comparison. It is evident that in the existing helmet, a single-impact loading-event results in multiple peak-acceleration events and these events are attenuated in our conceptual helmet that is designed to provide nearly optimal damping. The SLS model predicts the modulus to change over a narrower frequency range than that demonstrated in brain tissue. The loss modulus response of an SLS material doesn't remotely match that of brain tissue. These results demonstrate that the use of an SLS model for brain tissue is highly inaccurate, yet SLS models are the most popular of those in the current Literature. More complicated Prony series models require multiple relaxation times and dozens of material parameters to replicate the broad frequency dependent range of the storage and loss moduli of brain tissue. Our preliminary data show that fractional-derivative (non-linear) visco-elastic models are able to capture the viscoelastic response of a variety of real polymeric materials having similarly broad frequency-dependent responses. These data were fitted with a fractional-derivative model, requiring 4 material parameters, and with a 13-term Prony series model that had a total of 28 material parameters. The advantages of the fractional-derivative model are, arguably, a better fit and a tractable number of material design parameters