Computational ballistic analysis of the cranial shot to John F. Kennedy

Almost 60 years after the assassination of John F. Kennedy in 1963 the majority of Americans are still reluctant to believe the official reports of commissions from 1964 and again in 1976 that determined the direction of the shot resulting in the fatal head injury. Long-withheld, confidential government files released in 2017 reignited the controversy. The present investigation computationally simulated projectile-skull impacts from the direction specified in official reports and from three other directions. Detailed geometric models of the human head and ammunition, as well as known parameters from the assassination site served as the supportive base for analysis. Constitutive mathematical models for the impact of projectile material with skull tissues at supersonic speed were employed to analyze bone and bullet fragmentation mechanics. Simulated fracture characteristics of the bone and the bullet were compared with photographic and X-ray evidence. The most likely origin of the fatal shot was determined based on the degree of corresponding deformation and fragmentation between simulation and documented evidence. Computational corroboration could be established as physically consistent with high-speed impact from the rear, as established by the official commissions. Simulations of three other speculative shot origins did not correspond to the documented evidence.

The President's Commission on the Assassination of President Kennedy, unofficially the Warren Commission, released its final report in 1964 and attributed the assassination of President John F. Kennedy (JFK) to three shots fired from a single marksman behind the presidential limousine with two bullets striking the president from the rear. While the first shot missed, the second shot caused a neck wound to Kennedy. The third and fatal shot hit Kennedy in the head [1].
Controversy concerning the commission report arose due to a physician in the overall disarray describing the neck wound from the second shot as a bullet entrance from the front throat [2][3][4][5]. Considerable doubt regarding the shot direction was further raised by the testimony of over 40 eye witnesses, including treating doctors, who unanimously reported a large wound in the right back of Kennedy's head [5][6][7][8][9], contradicting the official version of a large wound in the temple region of the right frontal bone [6]. Further contradiction was added by the film from a civilian bystander [10], known as the Zapruder Film, showing the president's head snapping violently backward upon impact with the third shot. Intuitive interpretation of this backward movement led to a lay verdict that the projectile must have come from the front.
Due to diminishing public trust in the 1964 report and to resolve controversy, the United States House of Representatives Select Committee on Assassinations (HSCA) was convened as a second official panel in 1976. The HSCA confirmed the main findings of the Warren Commission and concluded that JFK was hit by two shots from behind him. However, it was also concluded that with "high probability" at least two gunmen were involved [11]. Intense controversy thus remained and conspiracy theories still linger, many of them involving other or multiple possible shot directions as considered likely by the HSCA report [12]. In opposition to both official commissions that located the 6th floor of the Texas School Book Depository building to the rear of President Kennedy as the origin of the fatal shot, various conspiracy theories claim different sites as shooter locations [6,7,[13][14][15][16][17]. The disagreement about the exact circumstances of the assassination, including possible shot directions, is reflected to date in over 40,000 published books and articles [18].
The present investigation compares the official findings related to the fatal shot origin to conspiracy related hypotheses by using state-of-the-art numerical software simulating the complex scenario of high-dynamic three-dimensional nonlinear bullet-skull penetration.
Various shot origins were simulated using detailed models of the head and the bullet and these were compared with photographic and forensic evidence, particularly the documented sharp skull bone defect in Kennedy's right temple and the deformed and fragmented bullet jacket. Also, the mechanical cause of the non-intuitive violent post-impact retrograde head snap is elucidated.

Potential shot origins
At time of the fatal shot John F. Kennedy was situated in the presidential limousine on Elm Street in Dallas, Texas, cf.  head are depicted to scale in Fig.1 and, also to scale but enlarged in Fig.2. Geometric details of Fig.1 and Fig.2 including traces from the potential shot origins were generated using schematic scale drawings together with a true to scale top view picture of the assassination site [12,19,20]. Data of the presidential limousine was based on a commercially available digital model as well as scale drawings [5,12].  Shot traces were also used to spatially orient the bullet model relative to the head model in the simulation.

Initial conditions prior to bullet impact
The Carcano projectile, officially identified as the assassination ammunition [5,12], was determined to have an average muzzle velocity of 670 m/s with a clockwise twist rate of 1 x 21.5 cm [21]. The velocity is 638 m/s after travelling 24 m, which was the distance between JFK's head and the location of the SD, 633 m/s at a distance of 30 m from the fence on the GK, and approximately 590 m/s at a distance of 82 m from the 6th floor window of the TSBD, and an average 574 m/s for a distance of 80 -90 m for the SK-area [5,19,21,22]. The trajectory declination angles were approximately 0.2°-5°, 5°, 8°, and 15° from the SK, GK, SD, and TSBD, respectively, to the position of Kennedys' head. The ground distance to the 6 th floor of the TSBD was assumed 18.5 m, the approximate vertical distance from the GK fence to street level 2.5 m, 0 m from the SD and 4 -5 m from the SK area. These distances and angles determine spatial bullet position and orientation to the head.
Approximate spatial orientation of the head relative to the limousine was based on Frame  Positioning of the bullet and head model with respect to a shot from the TSBD was based on the digitalized trajectory as well as on information from the Clark and HSCA forensic panel report stating that the wound on the back of JFK's head was located about 100 mm above the external occipital protuberance, and between 18 -25 mm to the right of the head midline [12,23]. Bullet positioning for the other potential shot origins -SK, SD and GK -was based on trace/trajectory geometries as previously delineated.

Projectile geometry and mass distribution
To obtain computer-aided design data of the Carcano projectile (6.5×52 mm "Parravicini-Carcano"), scale drawings [24] were digitalized. In addition, bullets samples were milled to obtain section cut samples, Fig.4a. Using a 3D-digital microscope bullet dimensions were verified: jacket outer length and diameter at the bearing surface were 30.4±0.2 mm and = 6.6±0.2 mm, respectively.  Mechanically, this design is described as a "single cell, one-end closed tube structure with closed thin wall section". This is important to comprehend bullet stability and deformation behavior.

Skull geometry
Precise data specifically pertaining to Kennedy's head anatomy do not exist. Thus, a human skull exhibit, as well as a plastinated brain, were scanned using computed tomography (CT) to acquire digitalized anatomical data. Scan images were reconstructed using MIMICS software (Materialise, Leuven, Belgium) providing digital data used for finite element (FE) head modeling including cortical and cancellous bone, cerebrospinal fluid (CSF) volume, brain and skin tissue. Thickness of the parietal/ occipital bone was estimated from Kennedys' autopsy X-ray photographs together with his head circumference, i.e. 61 cm [12,25]. The FE model was scaled accordingly to fit these values.
The peak strain rate corresponding to a typical defense-related ballistic impacts is in the order of 10 5 -10 6 s −1 . Material data in that range is sparse. Bullet, scalp skin, bone, and brain tissue material properties thus were deduced based on the literature.

Projectile model
Substantial controversy surrounds the bullet's alloy composition [12,26,27]. In this report it was assumed that the bullet jacket consisted of 90% copper and 10% zinc, and the soft core consisted of common lead [12].
To realistically simulate bullet fragmentation behavior, adequate material modeling is essential. Jacket material was modeled using a modified version of the Johnson-Cook constitutive relation [28]. This model expresses the equivalent flow stress , Eq. (1), considering the effects of rate-independent strain hardening represented by a power law together with strain-rate dependence and thermal softening: (1) with A, B, C, m, and n being experimentally determined constants, is the effective plastic strain, is a dimensionless effective strain rate with where is the reference strain rate used in underlying material tests. with being the initial temperature of the material during testing. is the material's melting temperature, the current temperature. In the model, two material failure criteria combined with the FE erosion method were used: the Cockcroft-Latham criteria [29] as well as a temperature limit, assuming a strength limit at 90% of the material's melting temperature.
Lead core material behavior was represented by the Steinberg-Lund constitutive model [30] where the shear modulus G and the yield strength are given by , . ( The volumetric pressure response was represented by the Mie-Gruneisen EOS, . In Eqs.(2,3) and represent pressure and temperature dependence on the shear modulus and are, together with the initial shear modulus and f, experimentally determined constants. is the initial yield stress, where is the partial differential of with respect to pressure. is the relative volume with and being the initial and the specific volume, respectively; p is the pressure and and are work-hardening parameters. , and are the internal, melting, and cold compression energy per unit volume, respectively. is defined as with R being the gas constant and A the atomic weight, and are the initial and equivalent plastic strain, respectively. In Eq.(4), is the internal energy, is the Gruneisen coefficient with b the volume correction to . C is the bulk sound speed. S 1 , S 2 , S 3 are the Hugoniot slope coefficients and is the compression coefficient.
Since bullet jacket and core materials are only loosely connected, a penalty-based contact was defined between both constituents.

Head model
Based on digitalized CT-scan data the head model includes: scalp skin, cortical bone, cancellous bone, CSF, and brain, Figs.6a-c.

Scalp skin
Scalp skin was included since, in general, skin stiffens significantly at increasing loading rates [36,37]. Experimental human skin data collected under tension at a strain rate of 167 s -1 was used for tissue representation [37]. The isotropic Ogden form was employed to define tissue behavior [38,39], skin anisotropy was neglected. An algorithm based on the simplex strategy was used to optimize model parameters constraining Poisson's ratio to ν = 0.497.
Employed values are shown in Table 6. Effective failure strain was estimated at 0.25, based on strain at peak stress [37]. Scalp skin tissue was modeled using thick shell elements with two elements over the thickness. Skin thickness was assumed to average 6.5 mm [40,41].
In contrast to dry bone, wet bone can pass through a plastic phase [57]. Thus, a rate-dependent elasto-plastic material model with plastic strain to failure criteria was used, based on both tension and compression yield stress versus effective plastic strain. Failure strain was estimated based on findings for cranial cortical bone and for femoral cortical bone [45,56]. A factor of 1.4 between the response in mechanical strength to tension and compression of cancellous bone was deduced [58,59] as well as for cortical bone [60,61]. Employed bone material data is shown in Table 7, including Young's modulus for tension/ compression, E t /E c , Poisson's ratio ν, yield stress during tension/compression / , and plastic failure strain during tension/compression, / (assumed equal). Volumetric densities were adopted from Sharma and Fry et al. [62,63]. Cranial suture structures were not included in the model since findings are mixed regarding suture strength in adults compared to adjacent bone [64][65][66]. Stacked thick shell elements with two elements over the thickness were used to model cortical and cancellous bone layers.

Cerebrospinal fluid
CSF was considered a Newtonian fluid with properties close to water. It was modeled with tetrahedral continuum elements employing zero-stiffness material definition in addition to a linear polynomial EOS to accommodate for high-pressure shock loading. Coefficients were C 1 = 2190 MPa, C 2 = 9224 MPa and C 3 = 8767 MPa fitting Hugoniot data for water [67][68][69][70].
The subarachnoid space was set to a constant thickness of 2 mm. A penalty contact between outer and inner CSF surface to inner skull bone and outer brain surface was assigned. A dynamic shear viscosity coefficient of = 3.5E-09 [MPas] was used.

Brain tissue
Based on the similarities of human and porcine brain tissue, porcine brain is often used for testing. Based on tests with varying strain rate in porcine brain tissue samples during compression (≤ 90 s -1 ), shear (≤ 120 s -1 ) and tension (≤ 90 s -1 ) instantaneous shear moduli in the range of 12-27 kPa were deduced [71][72][73][74][75]. Similarly, based on tests including porcine brain at shear rates up to 800 s -1 , a mean instantaneous shear modulus of 14 kPa was derived [76]. Shear tests on bovine brain tissue at a shear strain rate of 700 s -1 led to an instantaneous shear modulus of 54 kPa [77]. In this report, a viscoelastic Maxwell model was used to describe brain tissue mechanical properties: .
The deviatoric response was modeled based on experimental data of white matter tested in shear up to 6300 Hz [72]. From this data, instantaneous and long-term shear moduli of 62 kPa and 0.27 kPa, respectively, can be deduced. Employed Prony series shear relaxation moduli and shear decay constants are provided in Table 8. Brain tissue was assumed isotropic and modeled with under-integrated constant stress hexahedral continuum elements with hourglass stabilization. The element erosion method was defined to account for material failure, assuming principal failure strain to be in the range of 40% [79]. To conserve mass and momentum, Lagrangian solid elements used for brain modeling and exceeding the failure strain were eroded from the mesh and defined to transform to freely-moving SPH particles. These in turn were set in contact definition to adjacent Lagrangian elements of brain and skull to continue their contribution to the fragmentation process.

Bullet fragments
Following the assassination, three small bullet lead fragments and two larger jacket fragments, Figs.7,8, were found in the limousine (exhibits CE567 and CE569, [5]). components of the same bullet could not be determined [12].
Fragments CE569 (Figs.7a,b) and CE567 (Figs.8a,b) exhibit unique features that were used for comparison with simulation results. From fragment CE569 it is evident that -the lead core separated from the jacket base, Fig.7a -the jacket is folded (flapped) back on one side, several millimeters above the cannelure,  The skull X-ray photographs of JFK [5,12] have been variably interpreted, including possible alteration [81,82]. The HSCA photographic panel, however, unanimously agreed that all X-rays (and autopsy) photographs had not been altered in any manner [12]. Computer enhanced versions of the original X-ray photographs are shown in Figs.9a,b, as has been shown in the HSCA report [12]. They show that the top right side of the skull is extensively damaged with extreme loss of bone (Fig.9a) and multiple fracture lines (Fig.9b), [77].

Autopsy photographic evidence
In this report the following images were partly masked, due to their graphic content.
Autopsy photographs Figs.10a,b show Kennedy's head and face from both sides. In Fig.10a, a large defect in the temple region of the right frontal bone resembling a sharply demarcated 'V' is apparent. No damage is visible to the left side of the head, Fig.10b. The autopsy photograph Fig.10c shows a circular wound indicated with {E} (cf. also [6] and exhibit F-48 in [12]), identified as the bullet entrance site by the HSCA.  (Note: reported simulation times always refer to time ∆t after initial bullet-head contact)  Figs.11d,e. In this investigation, only bullet jacket deformation was compared since lead core material deformation was unspecific in geometry.    Fig.12a.

2:
Contact with brain tissue. Here partial asymmetric mushrooming of the jacket occurrs due to asymmetrical initial bone contact (impact angle not orthogonal to bone surface) and partial disconnection of the lead core from the jacket. Disconnection was driven by faster forward movement of the lead core caused by higher contact forces (due to locking and friction) on the jacket by adjacent tissue. Due to asymmetric bullet deformation and resulting asymmetric resistance bullet rotation begins around the vertical z-axis (s. Figs.13 for coordinate system). This caused indentation of the jacket base since the lead core had already traversed forward leaving the jacket base hollow.  Fig.12a in the right temple region, and shown in detail in Fig.12b. At exit the bullet was rotated approximately 35° about the vertical z-axis compared to its pre-impact line of flight. Here progressive damage to the jacket and core occurs (again asymmetrically, due to unilateral contact to the parietal/frontal bone structure).     The image shown in Fig.15b is an enhanced copy of the fractures shown by X-ray (unenhanced X-ray images are presented in the HSCA report [12]) and can be compared to  the simulation model Fig.15a, although exact perspectives differ. Descriptions of single fracture lines of the HSCA radiologists can be found in exhibits F-32,33 [12]. The distinct Sshape along line [G] in Fig.15b correlates with the path in Fig.15a Fig.14b. The front head X-ray, Fig.9a, shows that skull damage extended to the sagittal suture. This is also seen in the simulation results in Fig.13f  bone damage coincides with the testimony of one of the autopsy pathologists stating that during the autopsy the skull was all "falling apart" [85]. The wound characteristics visible from the outside, cf. Fig.14a, could well correlate with the wound location reported by eyewitnesses [6,8] as it largely extends to the right back of the head.  Fig.10b) and not as hypothesized by some [6,8] at the right rear of the head.

Simulated shots from the SK-direction
Three possible shooter origins, cf. Fig.1, were analyzed. Common to all sites is an assumed entrance site over the right supra-orbital ridge {1} resulting in an exit site at the right rear of the head, {2} in Fig.18a.  While for locations (1) and (2) Fig.14a,b, as well as brain matter drives the head into retrograde motion and not forward. This was first proposed in 1976 [87], recently confirmed by Nalli [88]. Simulation shows that 50 g of brain

Discussion
Due to controversy surrounding official reports in regard to the assassination of John F.
Kennedy, the fatal head shot was analyzed in the present report by means of numerical x y z simulation and outcome of different simulated shot directions were compared to forensic evidence. Shot directions included the officially designated shot origin site from the Texas School Book Depository to the back of the president's head, as well as three additional sites promoted by conspiracy theories. It was postulated that the specific characteristics and complexity of anatomic and physical evidence existent in the form of X-ray and autopsy photographs as well as recovered bullet fragments and post-impact head motion can be reproduced by simulation from only one shot direction. Specifically, unique features were the sharply demarcated defect in the temple region of the right frontal bone, evident in autopsy photographs, as well as the specific skull bone crack formation evident in X-rays.
Deformation of the bullet jacket material was also particularly characteristic. Comparing all simulation results with the evidence, it could be shown that only a bullet fired from a direction in Kennedy's rear, specifically originating from the direction of the Texas School Book Depository at a ground height approximately matching the 6 th floor correlates well with the evidence. The sharply demarcated V-shaped defect in the temple region of the right frontal bone, which at first glance appears artificial and had been proposed as the result of manipulation to cover a wound resulting from a shot from the front [81], was shown to be a direct consequence of the particular loading situation. For all simulated cases it is shown that though the bullet deforms, fractures, and rotates, it generally follows a straight path in the soft brain tissue, such that the exit site is in line with the pre-impact bullet flight path. This rigorously excludes hypotheses including locations other than the TSBD as shot origin, cf.
shot traces in Fig.2 as they contradict autopsy evidence, Figs.10b,c.

Conclusions
The presented simulations support the official commission reports regarding origin and direction of the fatal head shot to John F. Kennedy. From a mechanical standpoint, based on computational simulation, JFK's head was hit by a single bullet from the rear originating from the Texas School Book Depository. Deformation patterns regarding skull fracture and bullet jacket deformation as shown in photographs could be matched by simulation with unambiguous consistency, leaving little room for further speculation about the origin of the projectile. Simulated post-impact head motion corresponds to evidence provided by the Zapruder Film. Simulation results together with forensic evidence exclude the sites "Grassy Knoll, South Knoll, and storm drain", as potential origins of the fatal shot.
The modeling approach employed here, based on computer-aided simulation, could be applied to similar cases in the forensic field. Simulation shows that most fragments -including those depicted in Figs.11a,c -travelled towards the driver's seat. Smaller fragments of the bullet jacket material were deflected at head contact and moved towards the windshield frame and the windshield.
Even though simulated traces are approximations (they depend on initial projectile velocity, all material properties, initial head position, contact interactions and the impact wound location used in the model) they coincide with the fact that fragments were recovered from the left front of the presidential limousine, whereas others hit the windshield [1].