Scaling contact force parameters across body size, limb count, and number of contact spheres

Dynamic simulations of contact interactions involve a trade-off between realism and numerical performance. Contact interactions in musculoskeletal simulations are often modelled using Hertz theory applied to contact spheres, with Hunt-Crossley based dissipation. Suitable contact parameters are highly dependent on the morphology, scale, materials, and movement in question. Inappropriate parameter choices can manifest in unpredictable ways, potentially resulting in misinterpretations or failed simulations. Here, I demonstrate that the plane strain modulus and dissipation parameters are not scale invariant. I derive equations to scale the contact parameters in dimensionless form, which allows accounting for differences in body size, contact sphere radius, number of contacting elements (e.g., limb count), and number of spheres. As a demonstration of this approach, I scale the contact parameters of a 62 kg human to a 500 kg human, a mouse (0.02 kg), an emu (37.8 kg), a horse (545 kg), and a giraffe (1190 kg). Geometrically and dynamically similar contact behaviour is achieved in all cases. The equations presented here can be used to derive contact parameters in a variety of ways, including scaling parameters between different models (useful when simulating the effects of allometric scaling), and estimating suitable contact parameters directly from force-displacement curves. The limitations of Hertz-Hunt-Crossley contact models are discussed, and I provide some strategies to navigate the trade-offs inherent to their use in biomechanical simulations. Lastly, I derive dimensionless expressions and scaling guidelines for the smoothed contact force implementation "SmoothSphereHalfSpaceForce" in the popular simulator OpenSim.