⚡ electrical engineering

Skull Perforation by Projectile Impacts to the Head

This study refines and validates the Perforation Criterion (PC) for predicting skull fracture by deriving a highly accurate empirical formula for 50% fracture risk velocity based on 38 ballistic impact conditions, demonstrating its superior fit to experimental data compared to the Blunt Criterion (BC).

Original authors: David C Viano, Larry M Sturdivan

Published 2026-10-09

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CC BY 4.0

Original authors: David C Viano, Larry M Sturdivan

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

When a hard object strikes the human head, the skull acts as a protective shell, but it has limits. For decades, scientists have tried to understand exactly when that shell fails. The question is not just about how hard a blow must be, but how the size of the object, its weight, and the thickness of the bone all work together to cause a fracture. This is a matter of life and death, relevant to everything from military safety to the design of protective gear. Researchers have long known that a small, fast projectile behaves differently than a large, slow one, even if they carry the same amount of energy. The challenge has been to find a single, reliable way to predict when a skull will crack under the force of an impact, turning complex physics into a clear rule that can be applied to real-world situations.

In a study published in late 2026, David C. Viano and Larry M. Sturdivan revisited a massive collection of data from the 1970s and 1980s to solve this puzzle. They focused on a specific type of injury called a depressed skull fracture, where a piece of bone is pushed inward rather than simply cracking or shattering. The researchers began by analyzing 1,273 ballistic tests originally conducted by the U.S. Army, where different projectiles were fired at fresh human skull caps held in a stand. These early tests used a variety of metal shapes, including spheres and cubes, ranging from tiny grains of sand to small metal blocks, traveling at speeds between 50 and 530 meters per second. The goal was to find the exact speed at which there was a fifty-fifty chance of the skull breaking, a threshold known as V50.

The team combined this historical data with more recent experiments involving impacts to the side of the head, creating a comprehensive picture of how skulls fail under stress. They discovered that the risk of fracture is not determined by speed alone, but by a precise balance between the mass of the object hitting the head and the thickness of the bone it strikes. Heavier objects require less speed to break the skull, while thicker bones require more speed to fail. The researchers developed a mathematical formula that describes this relationship with remarkable accuracy, explaining 99.1 percent of the variations seen in their data. This formula works across a vast range of conditions, from tiny, lightweight projectiles to heavy, slow-moving objects, and it holds true for skull thicknesses found in different parts of the human head.

The study also clarified the difference between two ways of measuring impact risk. One method, called the Perforation Criterion, was designed for rigid targets like the skull caps used in the original army tests. The other, the Blunt Criterion, accounts for the fact that a real human head is not a fixed object; it moves and absorbs energy when struck. The researchers found that for the specific tests they analyzed, where the head was heavy and the impact was nearly inelastic, both methods gave very similar results. This confirmed that the underlying physics of how the bone breaks is consistent, whether the target is a fixed piece of bone or a moving human head. The new formula they derived provides a straightforward way to calculate the danger of a specific impact without needing complex computer simulations, offering a clear, practical tool for assessing injury risk.

Ultimately, the work confirms that the human skull fails in a predictable way when struck by a projectile. The fracture happens when the energy of the impact is concentrated on the edge of the contact area, shearing the bone like a cookie cutter. The study shows that the thickness of the bone plays a more significant role than the weight of the projectile in determining the speed required to cause a break. By refining the old data and adding new tests, the authors have provided a reliable guide for understanding when a blow to the head will result in a serious injury. This knowledge helps engineers and safety experts design better protection and assess the risks of various impacts, ensuring that the limits of the human skull are understood with clarity and precision.

Technical Summary: Skull Perforation by Projectile Impacts to the Head

Problem Statement
This study addresses the need to quantify the risk of depressed skull fractures caused by ballistic projectile impacts. While the U.S. Army conducted extensive ballistic testing on skull caps in the 1970s and 1980s, the resulting data and the theoretical "Perforation Criterion" (PC) developed during that era remained largely unpublished in open literature. Specifically, the Army used 1,273 ballistic impacts to determine the velocity for 50% perforation ( $V_{50}$ ), which were then analyzed to establish 37 specific test conditions for developing the skull fracture criterion. There was a need to integrate this historical data with more recent experimental findings (specifically Raymond et al., 2009a,b) and to compare the PC with the modern "Blunt Criterion" (BC) used for blunt head impacts. The primary objective was to document the historical research, refine the theoretical formulation, and derive a robust empirical relationship for predicting skull fracture risk across a wide range of projectile masses, diameters, and velocities.

Methodology
The study employed a combination of theoretical modeling, data re-analysis, and statistical regression:

Theoretical Formulation: The authors revisited the Perforation Criterion (PC) model originally developed by Sturdivan and Bexon (1981). This model conceptualizes skull perforation as the driving of a truncated right circular cone of bone into the brain. The criterion is based on the ratio of impact energy to the stress on the side-wall of the bone plug, expressed as $PC = \ln(X)$ , where $X$ is a function of kinetic energy ( $E$ ), skull thickness ( $t$ ), projectile diameter ( $D$ ), and a conical angle ( $\Psi$ ).
Data Integration: The study digitized data from 37 specific test conditions derived from the 1,273 historical impacts (Miller et al., 1970; Michiewicz, undated) involving various projectile masses (0.055–4.116 gm) and velocities. Additionally, data from 14 recent parietal impacts by Raymond et al. (2009a,b) using a 103 gm impactor were incorporated.
Statistical Analysis: Logistic regression was used to determine the probability of fracture based on the PC and the Blunt Criterion (BC). The BC was adapted for head impacts, incorporating absorbed energy ( $E_3$ ) and effective diameter ( $\hat{D}$ ) to account for momentum transfer and scalp deformation.
Empirical Modeling: A multi-variate power-law regression was performed on the 38 combined impact conditions to derive an empirical formula for $V_{50}$ as a function of impactor mass ( $m$ ) and skull thickness ( $t$ ). The analysis utilized natural logarithmic transformations to linearize power-law relationships.

Key Contributions

Documentation of Historical Data: The paper makes available the analysis of 37 specific test conditions derived from 1,273 ballistic impacts from the 1970s, specifically focusing on the determination of $V_{50}$ for depressed skull fractures.
Unification of Criteria: The study demonstrates that the Perforation Criterion (PC) and the Blunt Criterion (BC) are fundamentally similar, differing primarily in how they account for momentum transfer (two-body dynamics) and contact geometry. It shows that for fixed skull cap tests, PC is sufficient, while BC is necessary for free-moving head impacts where energy absorption and restitution play a role.
Derivation of a New Empirical Formula: The authors derived a highly predictive empirical equation for the 50% risk velocity ( $V_{50}$ ) of depressed skull fractures:
$V_{50} = 14.83 t^{0.716} / m^{0.377}$
(where $t$ is in cm and $m$ is in kg).
Mechanical Interpretation: The paper provides an engineering mechanics explanation for the empirical exponents. It posits that the fracture mechanism is a mixed mode between brittle shear (scaling with $t^{0.5}$ ) and plate bending (scaling with $t^3$ ). The derived exponent of 0.716 reflects the composite structural response of the human skull (cortical tables and diploe) transitioning from elastic bending to brittle fracture.

Results

Model Fit: The empirical relationship fits the experimental data with an $R^2$ of 0.991 and a p-value < 0.0001. The average error between the model and actual $V_{50}$ values for the historical data was 0.72%, with an absolute average error of 4.44%.
Logistic Regression: Both the PC and BC provided good fits to the Raymond et al. (2009a,b) data, with Brier Skill Scores (BSS) of 0.4137 and 0.4114, respectively, indicating "good" predictive accuracy for depressed skull fractures.
Parameter Sensitivity: The analysis confirmed that skull thickness and projectile mass are the dominant factors in fracture risk. The study validated that the conical angle ( $\Psi$ ) could be adjusted (23° for spheres, 30° for cubes, 35° for Raymond's data) to align different projectile shapes on a single predictive line.
Scope: The derived formula is valid for a wide range of ballistic projectiles: masses from 0.055 to 103.0 gm, diameters from 2.4 to 38.0 mm, and velocities from 20 to 530 m/s.

Significance
The paper claims that the derived empirical relationship provides a "convenient and accurate formula" for determining the velocity associated with a 50% risk of depressed skull fracture. By linking the historical Perforation Criterion with modern Blunt Criterion concepts, the study offers a validated framework for assessing skull fracture risks in both ballistic and blunt impact scenarios. The authors emphasize that the terms and powers in the empirical relationship possess physical meaning rooted in engineering mechanics, specifically relating to energy density at the perimeter of the displaced bone plug. The work serves to finalize and validate decades of U.S. Army research on non-lethal and lethal ballistic head injuries, providing a tool for injury assessment without requiring new experimental data for standard conditions.

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