The axisymmetric response of a fluid-filled spherical shell to a local radial impulse—A model for head injury

doi:10.1016/0021-9290(69)90089-X

Journal of Biomechanics

Volume 2, Issue 3, July 1969, Pages 325-341

https://doi.org/10.1016/0021-9290(69)90089-X Get rights and content

Abstract

This investigation is concerned with determination of the dynamic response of a fluid-filled spherical shell subjected to a local radial impulsive load. From the application point of view, such a fluid-shell system is considered to be a simple, but to date, the most improved theoretical model representing the human head when subjected to impulsive external loads. Analysis is based on linear shell theory which includes both membrane and bending effects. The motion of the fluid is assumed to be governed by the wave equation. Laplace transform technique is used in obtaining the transient axisymmetric response of the system to a local radial impulsive load. The solutions thus obtained for the velocity potential of the fluid and the displacement components of the shell mid-surface are the Green's functions of the problem with respect to time. Some numerical results for the theoretical model are obtained for a set of appropriate data. The comparison is made for the stress distributions at various times in the shell for both the empty and the fluid-filled cases. In the fluid-filled case the excess pressure propagation in the fluid is also discussed. The possible locations of brain damage and skull injury are indicated on the basis of the numerical computations.

References (7)

A. Anzelius
The effect of an impact on a spherical liquid mass
Acta. path. microbiol. scand. Suppl.
(1943)
A.E. Engin
The axisymmetric response of a fluid-filled spherical shell
A dissertation submitted to the Graduate School in The University of Michigan
(1968)
F.G. Evans
Stress and Strain in Bones

There are more references available in the full text version of this article.

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Vibration-based elastic parameter identification of the diploë and cortical tables in dry cranial bones
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Experimental vibration studies on in vivo human heads are relatively rare, likely due to the difficulty and cost of performing such experiments, while the scarcity for dry cranial bone studies can be explained in part by lack of direct applicability of such results to topics like hearing by bone conduction. On the modeling side, many early papers employed elastic fluid-coupled spherical shell models (Engin, 1969; Hickling and Wenner, 1973), and similar idealized models have also been used more recently (El Baroudi et al., 2012; Charalambopoulos et al., 1997). However, due to the geometric and mechanical complexities involved, numerical models have mainly been developed using the finite element method (FEM), with the paper by Nickell and Marcal (1974) being one of the pioneering studies specifically devoted to the study of skull vibrations.
Various human skull models feature a layered cranial structure composed of homogeneous cortical tables and the inner diploë. However, there is a lack of fundamental validation work of such three-layer cranial bone models by combining high-fidelity computational modeling and rigorous experiments. Here, non-contact vibration experiments are conducted on an assortment of dry bone segments from the largest cranial bone regions (parietal, frontal, occipital, and temporal) to estimate the first handful of modal frequencies and damping ratios, as well as mode shapes, in the audio frequency regime. Numerical models that consider the cortical tables and the diploë as domains with separate isotropic material properties are constructed for each bone segment using a routine that identifies the cortical table–diploë boundaries from micro-computed tomography scan images, and reconstructs a three-dimensional geometry layer by layer. The material properties for cortical tables and diploë are obtained using a Hounsfield Unit-based mass density calculation combined with a parameter identification scheme for Young’s modulus estimation. With the identified parameters, the average error between experimental and numerical modal frequencies is 1.3% and the modal assurance criterion values for most modes are above 0.90, indicating that the layered model is suitable for predicting the vibrational behavior of cranial bone. The proposed layered modeling and identified elastic parameters are also useful to support computational modeling of cranial guided waves and mode conversion in medical ultrasound. Additionally, the diploë elastic properties are rarely reported in the literature, making this work a fundamental characterization effort that can guide in the selection of material properties for human head models that consider layered cranial bone.
Development of a child head analytical dynamic model considering cranial nonuniform thickness and curvature – Applying to children aged 0–1 years old
2018, Computer Methods and Programs in Biomedicine
Citation Excerpt :
In previous studies, Anzelius [14] first proposed a head analytical model that considered the human head as a rigid spherical shell filled with non-stick liquid. This model was further improved in Engin's study by assuming the head to be a fluid-filled elastic spherical shell [15]. Young [18] developed an analytic model of a spherical shell impacting with a solid sphere through combining the Hertzian contact stiffness and the shell contact stiffness.
Although analytical models have been used to quickly predict head response under impact condition, the existing models generally took the head as regular shell with uniform thickness which cannot account for the actual head geometry with varied cranial thickness and curvature at different locations. The objective of this study is to develop and validate an analytical model incorporating actual cranial thickness and curvature for child aged 0–1YO and investigate their effects on child head dynamic responses at different head locations.
To develop the new analytical model, the child head was simplified into an irregular fluid-filled shell with non-uniform thickness and the cranial thickness and curvature at different locations were automatically obtained from CT scans using a procedure developed in this study. The implicit equation of maximum impact force was derived as a function of elastic modulus, thickness and radius of curvature of cranium.
The proposed analytical model are compared with cadaver test data of children aged 0–1 years old and it is shown to be accurate in predicting head injury metrics. According to this model, obvious difference in injury metrics were observed among subjects with the same age, but different cranial thickness and curvature; and the injury metrics at forehead location are significant higher than those at other locations due to large thickness it owns.
The proposed model shows good biofidelity and can be used in quickly predicting the dynamics response at any location of head for child younger than 1 YO.
Analytical models for the impact of a solid sphere on a fluid-filled spherical shell incorporating the stress wave propagation effect and their applications to blunt head impacts
2017, International Journal of Mechanical Sciences
Citation Excerpt :
These models were improved in Engin [10] by regarding a head as a fluid-filled elastic spherical shell subjected to a radial impulsive load described by a Dirac delta function. The model of Engin [10] was modified in Kenner and Goldsmith [28] by extending it to loadings of finite duration. Young [56] developed an analytical model for blunt head impacts through studying the impact of a solid sphere on a fluid-filled spherical shell based on the Hertz contact theory [21] and the Reissner spherical shell theory [44].
A new analytical (non-linear) model for the impact of a solid sphere on a fluid-filled spherical shell is developed by including the stress wave propagation effect in addition to the Hertzian contact deformations and the shell membrane and bending actions. The expressions for determining the maximum mutual approach and impact duration are analytically derived using the principle of energy conservation. A simplified (linearized) model incorporating the elastic energy loss due to the stress wave propagation is then formulated by using a linear force-deflection relation, which leads to a closed-form expression for the impact duration. It is shown that the new non-linear model reduces to that of Young (2003) and the linearized model recovers that of Mansoor–Baghaei and Sadegh (2011) when the stress wave propagation effect is not considered. By directly applying the newly obtained non-linear and linearized models, three representative problems simulating blunt head impacts are analyzed. The first problem characterizes the blunt impact of a human head on the ground or on an automobile, the second one simulates the blunt impact of a non-lethal projectile on a human head, and the third one represents two football players’ head collision. Numerical results for the maximum deflection, maximum impact force and impact duration are provided to quantitatively show their variations with the impact velocity and shell thickness. The values predicted by the current new models are also compared with those given by the two existing models and with available finite element simulation results, and a good agreement is observed.
Numerical study on the mechanical response of brain under the impact loading based on elastic-viscoelastic model
2016, Applied Mathematics and Computation
Citation Excerpt :
It can help us understand the damage mechanism of brain injury, and reduce brain injury. There are the three ways to study the bio-mechanics of brain impact injury, such as experimental methods [7–9], theoretical models [10–13], and numerical simulations [14–16]. Biomechanists always construct a particular system, and get the physical and geometric properties.
The mechanical response of the brain on the impact process, and the relationship of brain injury and load are investigated rarely due to the complexity of brain. An elastic–viscoelastic model for analyzing the brain impact injury is presented. The elastic–viscoelastic correspondence principle is used to obtain the stress in the brain under impact. This mechanical model is in a good agreement with experimental results, and it can be applied to simulate and analyze brain injury. The results show that the stress of the brain is estimated under the different impact loads. The maximum stress of the brain may reduce by one order of magnitude, whether someone wears the safety helmet or not. Based on the interesting and rational mechanical model, it can be applied in designing the reasonable thickness of safety helmet for protecting brain and provide theoretical basis to determine injury index.
A closed form solution for the impact analysis of elastic ellipsoidal thin shells
2015, Thin-Walled Structures
In this paper the impact of a hollow elastic ellipsoidal thin shell with an elastic flat half space is analyzed. Due to the nonlinearity and complexity of the impact equation, classical numerical solutions cannot be employed. A closed-form solution for this problem is accomplished, using Hertzian theory [6] and the Vella’s analysis [17] in conjunction with the Newtonian method and a linearization scheme. The closed-form solution facilitates a parametric study of this problem. In next step, the closed-form analytical solution is validated by other research studies and explicit finite element method. Good agreement between the closed-form solution and other numerical results is observed. Based on the closed-form solution the maximum total deformation of the shell, the maximum transmitted force and the time duration of the contact were determined. The variations of the total deflection and the transmitted force as a function of the shell thickness and the different geometry of the shell were also determined. Finally it is concluded that the closed form equations are trustworthy and appropriate to investigate the impact of elastic ellipsoidal shells.
Elastic spherical shell impacted with an elastic barrier: A closed form solution
2011, International Journal of Solids and Structures
In this paper, the impact of a thin-walled elastic spherical shell with an elastic barrier is investigated. Hertzian and Reissner (membrane–bending deformation) theories were employed for the deformation equations. Due to the complexity of the equations a linearization of the equations was proposed and a closed form solution of the problem was obtained. Newtonian method is applied in order to obtain the impact force and the time duration. The closed-form solution enables one to parametrically study the impact and the related quantities. Finally the results from the analytical solution are validated by the finite element method and also are compared with the results presented in Young (2003). The comparison of the results reveals a good agreement. It is concluded that the proposed closed-form solution can be used to parametrically assess the impact of elastic spherical shells to elastic half space.

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^☆: Presented at the ASME Third Biomechanical and Human Factors Division Conference at the University of Michigan, Ann Arbor, June 12–13, 1969.

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The axisymmetric response of a fluid-filled spherical shell to a local radial impulse—A model for head injury☆

Abstract

The effect of an impact on a spherical liquid mass

Acta. path. microbiol. scand. Suppl.

The axisymmetric response of a fluid-filled spherical shell

A dissertation submitted to the Graduate School in The University of Michigan

Stress and Strain in Bones