2. Fundamentals of impact biomechanics

Occupant injury biomechanics rests on three interlocking pillars: how the body moves (kinematics), why it moves that way (kinetics), and how its materials respond when loaded rapidly (dynamic tissue behavior). A restraint engineer who understands only the first pillar can describe a crash but cannot explain why a chest deflection of 40 mm is tolerable at one velocity and injurious at another. This chapter builds the physical vocabulary needed for every subsequent analysis in the course.

Reference Frames and the Kinematic Description of Impact

Before any quantity is computed, the coordinate convention must be fixed and held constant. In vehicle crash analysis the standard is the SAE J211 convention: the x-axis points forward, y to the right, z downward, with rotations about these axes positive according to the right-hand rule. Acceleration is positive when directed forward. Every signal, filter, and injury criterion in this course inherits this convention; mixing sign conventions between a sled pulse and an occupant response is one of the most common sources of erroneous results.

Impact kinematics is conveniently described in three phases. During the pre-impact phase the occupant travels at vehicle velocity with essentially zero relative motion to the cabin. At t₀ the vehicle undergoes a rapid velocity change, but the unrestrained occupant continues at initial velocity — the relative displacement that accumulates is the "ride-down" distance available to the restraint. In the restraint phase the occupant decelerates over that distance, and the peak acceleration the body experiences is inversely related to how much ride-down distance the system provides. This is the kinematic foundation of the energy-management principle: a = v²/(2s) for constant deceleration, so doubling the stopping distance halves the mean acceleration.

Two kinematic quantities dominate injury outcomes. Delta-v (Δv) is the total velocity change of the vehicle; it sets the kinetic energy that must be absorbed. Peak acceleration and its duration determine the inertial loads transmitted to organs. A 56 km/h frontal barrier test produces roughly 15.6 m/s of delta-v, but the occupant compartment pulse typically peaks near 30–40 g for 60–80 ms — the shape of that pulse, not just its peak, governs restraint performance.

Kinetics: Forces, Moments, and Load Paths

Kinetics translates motion into the forces that actually damage tissue. Newton's second law applied to the occupant gives the inertial force F = m·a; for a 75 kg occupant exposed to a 40 g pulse, the inertial load is approximately 29 kN — a figure that immediately explains why belt geometry and load spreading matter. Beyond whole-body inertia, injury depends on local force concentration: the same 5 kN applied over a 100 cm² belt contact patch produces 500 kPa, while concentrated on a 5 cm² buckle edge it produces 10 MPa, well into soft-tissue disruption range.

Moments matter as much as forces. The head-neck system behaves as an inverted pendulum; a horizontal acceleration at the torso generates a bending moment at the occipital condyles proportional to the head mass times the acceleration times the moment arm. This is why whiplash injury correlates with delta-v and with the difference between head and torso acceleration, not with either alone. Load paths — belt-to-clavicle-to-ribcage, airbag-to-face-to-skull — determine which structures receive the load and therefore which injury patterns emerge.

Dynamic Mechanical Behavior of Biological Tissues

Biological tissues are viscoelastic: their stress response depends on strain rate, not strain alone. Three consequences follow directly.

• Rate stiffening. Cortical bone, ligaments, and muscle all exhibit higher elastic modulus and higher failure stress at high strain rates. A rib loaded quasi-statically fails near 3–4 mm deflection; at rates typical of belt loading (order 1 m/s), the same rib tolerates greater deflection before fracture but stores more energy and fails more abruptly.
• Hysteresis and energy dissipation. The loading and unloading curves of soft tissue do not coincide; the enclosed area is energy absorbed as heat and internal damage. This is why a single high-rate loading event can leave tissue preconditioned and weaker for subsequent cycles.
• Relaxation and creep. Under constant strain, stress decays with time (relaxation); under constant stress, strain grows (creep). In a crash lasting 100 ms, relaxation is modest but not negligible for organs such as the liver and brain.

The standard mathematical idealization is the Maxwell–Wiechert or generalized Kelvin–Voigt model: a spring (elastic modulus E) in parallel with a dashpot (viscosity η), giving a relaxation time constant τ = η/E. When the loading duration is much shorter than τ, the tissue responds stiffly (glassy); when much longer, it responds compliantly (rubbery). For brain tissue τ is on the order of tens of milliseconds, placing typical crash pulses squarely in the transition region — which is precisely why brain injury criteria must incorporate rate, not just peak acceleration.

Putting the Three Pillars Together: A Worked Example

Consider a 75 kg occupant in a 56 km/h frontal impact with a compartment pulse idealized as a half-sine of 35 g peak and 70 ms duration. The kinematic requirement is to absorb Δv ≈ 15.6 m/s. If the restraint system allows 0.35 m of occupant ride-down, the mean deceleration is a = v²/(2s) = 15.6²/(2×0.35) ≈ 348 m/s² ≈ 35.5 g — consistent with the assumed pulse, confirming the geometry is plausible. The kinetic consequence: peak inertial force ≈ 75 × 35 × 9.81 ≈ 25.8 kN, distributed across two belt paths and, if present, an airbag. The tissue consequence: chest deflection rate during belt loading is typically 1–3 m/s; at these rates rib and costal cartilage behave stiffly, and the chest compression criterion (maximum deflection normalized to chest depth) becomes the governing injury metric rather than force alone.

Practical Implications for Restraint and Countermeasure Design

The three pillars translate directly into design levers. From kinematics: maximize ride-down distance and shape the pulse to avoid high-frequency spikes that excite structural resonances. From kinetics: spread loads over large areas (wide belts, load limiters that cap force rather than displacement) and route load paths through structures that tolerate them — the clavicle and pelvis rather than the ribs and neck. From tissue behavior: account for rate effects when interpreting test data; a component validated at quasi-static rates may behave very differently at crash rates, and injury criteria calibrated on one rate may not transfer to another.

A practical checklist for any impact analysis follows from this framework. First, confirm the reference frame and sign convention against SAE J211. Second, compute Δv and the available ride-down distance before examining any force signal. Third, convert measured accelerations to inertial forces and check contact pressures against tissue tolerance. Fourth, verify that the loading rate falls within the range for which the chosen tissue model and injury criterion were validated. Each step is a guard against the most common analytical errors: sign confusion, energy-bookkeeping mistakes, and rate-inappropriate criteria.

With these fundamentals established — a fixed coordinate convention, the kinematic accounting of delta-v and ride-down, the kinetic translation to force and pressure, and the viscoelastic, rate-dependent nature of biological tissue — subsequent chapters can treat specific body regions and injury criteria as applications of a single coherent physical framework rather than a collection of empirical rules.