The Gurney Formula: Variations on a Theme by Lagrange

Jacobs, S. J.

doi:10.21236/ad0783941

Cited by 14 publications

(11 citation statements)

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“…From each of the Equations (19) and (20) in Paper 1, the resulting second-order Burgers' equation is obtained as expressed in Equation (16). When γ ∼ = 0, Equation (16) reduces to the firstorder Burgers' equation.…”

Section: Modeling Of Variation Of Pressure In Terms Of U (T X)mentioning

confidence: 99%

“…We now consider an additional cylindrical explosive blast forcing term [18,20], of E P = , which we may assume, after this energy has entered the brain, decays exponentially in the form of q (t,…”

Section: Calculation Of Induced Temperature Variation and Energies Wimentioning

confidence: 99%

“…Since we have already defined: E P = 1 2 v 2 m C + 1 2 and using data provided by Jacobs [20], and others such as Gerber [21], if we take v = 1,732, 2,500, 3,162 and 4,000 m/s (as based on the referenced literature) and Integrating Equation (260) with respect to ω on the interval 0 ≤ ω ≤ π L , and substituting the above (and additional) energies into the solution, with an initial scaled peak-value oft = L 2 γ π 2 , yields several linearly increasing maximum, "peak, " temperatures (at x = 0.088 m). Some of these are displayed in Table 1.…”

Section: Calculation Of Induced Temperature Variation and Energies Wimentioning

confidence: 99%

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The Modeling of Shock-Wave Pressures, Energies, and Temperatures Within the Human Brain Due to Improvised Explosive Devices (IEDs) Using the Transport and Burgers' Equations

Mason¹

2018

Front. Appl. Math. Stat.

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This second paper adopts a more rigorous, in-depth approach to modeling the resulting dynamic-pressures in the human brain, following a transitory improvised explosive device (IED) shock-wave entering the head. Determining more complicated boundary conditions, a set of particular-solutions for both Burgers' and the Transport equations has been obtained to describe the highly damped neurological pressures, complete with respective graphical plots. Many of these two-dimensional solution-curves closely resemble the Friedlander curve [1][2][3][4], not only illustrating enormous over-pressures that result almost immediately after the initial impact, but under-pressures experimentally depicted in all cases, due to oscillatory motion. It appears, given experimental evidence, that most-if not all-of these models can be aptly described by damped sinusoidal functions, these facts being further corroborated by existing literature, referencing models expounded by Friedlander's seminal work [1][2][3][4]. Using other advanced mathematical techniques, such as the Hopf-Cole Transformation, application of the Dirac-delta function and the Heat-Diffusion equation, expressions have been determined to model and predict the associated energies and temperatures within this paper.Keywords: traumatic brain injury, PDEs, IED-blast, pressure, under-pressure, energy, temperature, Heat-Diffusion equation INTRODUCTIONIn Paper 1, a one-dimensional model of the human head, through which a shock-wave due to an improvised explosive device (IED) blast traveled, was developed. Within this scenario, the initial shock-wave was modeled with a partial differential equation (PDE) known as Burgers' equation and, from the assumed initial boundary-condition, several shock-wave solution-curves were graphically plotted using experimental data derived from existing literature. It was hoped that, given this primary data, the theoretical calculations would match the experimental observations. Thus, future predictions about as-yet unknown IED neurological shock-waves should be possible. In setting up the preparatory conditions to tackle the challenges of this paper, it was indicated that the sheer violence of the shock-wave may have significantly harmful effects upon the human brain during and after battle, and a physics-based discussion about some of these issues followed. Mason IED Neurological Shock-Waves and EnergiesIn addition to high-velocity longitudinal pressure-waves, observed to cause significant brain trauma, it is mentioned that shear-or transverse-waves can be explained by similar physics-based mathematics, in terms of the way in which waves behave during their propagation through the brain.In continuing this mathematical discourse, we turn to further worked solutions of Burgers' equation by first considering damped sinusoidal motion of the resulting, dynamic, variation in pressure following initial transitory shock-wave propagation through the human head. MODELING OF VARIATION OF PRESSURE IN TERMS OF U (T, X)From each of the Equations (19) and (20...

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Section: Modeling Of Variation Of Pressure In Terms Of U (T X)mentioning

confidence: 99%

Section: Calculation Of Induced Temperature Variation and Energies Wimentioning

confidence: 99%

Section: Calculation Of Induced Temperature Variation and Energies Wimentioning

confidence: 99%

See 1 more Smart Citation

The Modeling of Shock-Wave Pressures, Energies, and Temperatures Within the Human Brain Due to Improvised Explosive Devices (IEDs) Using the Transport and Burgers' Equations

Mason¹

2018

Front. Appl. Math. Stat.

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“…In his detailed analysis of cylinder test data, Jacobs [13] found that M ¼ 0.5, 0.3, and 0.2 for the copper, steel, and aluminum circular cylinders, respectively. Moreover, it should be pointed out that the above-described Gurney formula is derived under the rigid-body assumption for the metal shell.…”

Section: Gurney Formula and Gurney Energymentioning

confidence: 99%

Theoretical Prediction of Expansion History of Metal Cylinder for Multicomponent Explosives

Liu

Duan

et al. 2014

Journal of Energetic Materials

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As a traditional experimental approach, the cylinder test has been widely applied to evaluate the ability to accelerate metal, acquire the specific dynamic energy, and calibrate the equation of state of the detonation product for an explosive. In this article, based on the constant metal-density Gurney formula, a theoretical approach to predicting the expansion history of a metal cylinder shell driven by the detonation product of a multicomponent explosive under any explosive mixture ratio is proposed, providing that the corresponding theoretical density and the initial internal energy and the Jones-Wilkins-Lee parameters of each explosive component in the multicomponent explosive are known. Based on this predicted expansion history, the ability to accelerate metal, the specific dynamic energy, and the equation of state of a multicomponent explosive under any explosive mixture can all be acquired without an extra cylinder test for the multicomponent explosive itself. For verification, numerical results for the expansion histories of a copper cylinder driven by the detonation products of a PBX-C03 and a Comp-B multicomponent explosive were calculated and found to be in reasonably good agreement with previous cylinder test data.

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“…Figure 2, taken from Kennedy 5 , is a plot of this-expression and of the formula for an asynmmetric plane case as well. Table 2, taken from Jacobs 6 , is a recent compilation of values of V = (2E) 1 /2 from analysis of measurements in experiments conducted at the Naval Ordnance Laboratory (NOL) and at the Lawrence Livermore Laboratory (LLL).…”

Section: Initial Ve~locitymentioning

confidence: 99%

Fragment and Debris Hazards

Zaker¹

1975

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The Gurney Formula: Variations on a Theme by Lagrange

Cited by 14 publications

References 1 publication

The Modeling of Shock-Wave Pressures, Energies, and Temperatures Within the Human Brain Due to Improvised Explosive Devices (IEDs) Using the Transport and Burgers' Equations

The Modeling of Shock-Wave Pressures, Energies, and Temperatures Within the Human Brain Due to Improvised Explosive Devices (IEDs) Using the Transport and Burgers' Equations

Theoretical Prediction of Expansion History of Metal Cylinder for Multicomponent Explosives

Fragment and Debris Hazards

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