K. L. Johnson^ Department of Mechanical Engineering, Center for Advanced Vehicular Systems (CAVS), Mississippi State University, Mississippi State, MS 39762
M. W. Trim Navai Surface Wartare Center, 9500 MacArthur Bivd, Bethesda,MD 20817
M. F. Horstemeyer Department ot Mechanical Engineering, Center for Advanced Vehicuiar Systems (GAVS), Mississippi State University, Mississippi Stafe, MS 39762
N.Lee Center for Advanced Vehicular Systems (CAVS), 200 Research Bivd, Mississippi State, MS 39762; Agriculfure and Bioiogicai Engineering, Mississippi State University, Mississippi State. MS 39762
L N.Williams Agriculture and Bioiogicai Engineering, Mississippi Stale University, Mississippi State, MS 39762
J. Liao Agriculture and Bioiogicai Engineering, Mississippi State University, Mississippi State, MS 39762
H. Rhee Center for Advanced Vehicular Systems (CAVS), 200 Research Bivd, Mississippi State, MS 39762
R. Prabhu
Geometric Effects on Stress Wave Propagation The present study, through finite element simulations, shows the geometric effects of a bioinspired soiid on pressure and impulse mitigation for an elastic, plastic, and viscoelastic material. Because of the bioinspired geometries, stress wave mitigation became apparent in a nonintuitive manner such that potential real-world applications in human protective gear designs are realizable. In nature, there are several toroidal designs that are employed for mitigating stress waves; examples include the hyoid bone on the back of a woodpecker's jaw that extends around the skull to its nose and a ram's horn. This study evaluates four different geometries with the same length and same initial cross-sectional diameter at the impact location in three-dimensional finite element analyses. The geometries in increasing compiexity were the following: (1) a round cylinder, (2) a round cylinder that was tapered to a point, (3) a round cylinder that was spiraled in a two dimensional piane, and (4) a round cylinder that was tapered and spiraled in a twodimensional plane. The results show that the tapered spiral geometry mitigated the greatest amount of pressure and impulse (approximately 98% mitigation) when compared to the cylinder regardiess of material type (elastic, plastic, and viscoeiastic) and regardless of input pressure signature. The specimen taper effectively mitigated the stress wave as a result of uniaxial deformational processes and an induced shear that arose from its geometry. Due to the decreasing cross-sectionai area arising from the taper, the local uniaxial and shear stresses increased along the specimen length. The spiral induced even greater shear stresses that help mitigate the stress wave and also induced transverse displacements at the tip such that minimal wave reflections occurred. This phenomenon arose although oniy longitudinal waves were introduced as the initial boundary condition (BC). In nature, when shearing occurs within or between materials (friction), dissipation usually results helping the mitigation of the stress wave and is iilustrated in this study with the taper and spiral geometries. The combined taper and spiral optimized stress wave mitigation in terms of the pressure and impulse; thus providing insight into the ram's horn design and woodpecker hyoid designs found in nature. [DOI: 10.1115/1.4026320] Keywords: guided wave propagation, elastic wave interaction and reflection, finite element analysis, dispersion in waveguides, stress wave mitigation, bioinspired design, geometric effects
Center for Advanced Vehicuiar Systems (CAVS), Agriculture and Bioiogicai Engineering, 200 Research Bivd, Mississippi State, MS 39762
Introduction The inspiration for this study stems from the curious geometries often found in biological structures that are subjected to dynamic loads. One such geometry is the natural spiral. Examples of the appearance of the spiral in natural shock absorbing systems include the ram's horn, seashells, and the woodpecker's hyoid bone. Does the reoccurrence of this curious shape throughout nature have some significance in regards to energy dissipation and shock absorption abilities inherent to its geometry? We seek to answer the following question: Can geometries affect stress waves to change the energy states and in the end the forces (or stresses)? 'Corresponding author. Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received September 5. 2013; final manuscript received December 13, 2013; accepted manuscript posted December 24, 2013; published online February 5, 2014. Editor: Beth Winkelstein.
Journal of Biomechanical Engineering
The fundamentals of the physics of stress waves have been well studied [1-3]. As a premise, the context of shock wave physics is warranted. If one were to neglect surface waves, then two main types of waves can propagate through isotropic solids: longitudinal waves and shear waves. Longitudinal (also called dilatational, pressure, primary, or P-) waves propagate with a characteristic wave speed and represent a volumetric change. Their motion is parallel to the direction of the wave propagation. Shear (also called secondary, S-, or distortional) waves represent no volume change and propagate at a slower wave speed with respect to longitudinal waves. Their motion is normal to the direction of propagation [4,5]. When either a longitudinal or shear wave impinges on a boundary, new waves are generated due to the reflective nature of waves. In a solid body with finite dimensions, these waves bounce back and forth between the bounding surfaces and interact with one another. These interactions can lead to wave amplification, cancellation, and other wave distortions.
Copyright © 2014 by ASIUIE
FEBRUARY2014, Vol. 136 / 021023-1
The pressure wave can be integrated over time leading to an impulse. The impulse is equal to the change in momentum of the body. It is possible for a very brief force, due to a shock, to produce a larger impulse when compared to a smaller force acting over a larger time period. Therefore, it is important to consider the transient forces, particularly those associated with wave phenomena. The question remains then: How do the longitudinal and shear waves that arise from impacts induce associated pressures and impulses in different solid geometries? If the geometries found in natural impact scenarios serve a purpose in pressure wave mitigation, then we would expect to see different geometries admitting different pressures and impulses in a solid. The four geometries included in this study comprise a cylindrical bar, a tapered cylindrical bar, a spiral with a circular cross section, and a tapered spiral (also with a circular cross section). The cylindrical bar serves as a "base-line" case. By comparing the response of the tapered cylinder to that of the uniform cylinder, we gain insight into how reducing the cross-sectional area influences the transient response of the structure. Similarly, comparing the spiral geometry to the uniform cylinder leads to an understanding of the effects of increasing curvature on the wave propagation. Finally, analysis of the tapered spiral allows us to understand the coupled influence of increasing curvature and decreasing cross-sectional area on wave propagation and reflection.
Cylinder 1
1
¿
Spiral .1
1
'»•••II^^HHI d,
Tapered Cylinder 1 1
d,
^
Tapered Spiral
1
1 d¡
Fig. 1 Schematic representation of the four finite element meshes illustrating the four different geometric configurations with the same length (and the same bar diameter where the pressure was applied) used in the analysis
A purely computational approach employing finite element analysis has been employed to study the wave propagation and Table 1 Geometric dimensions used in finite element analysis reflection characteristics of these different geometric bodies. Finite element analysis is the most efficient technique to perform Total Fixed-end Free-end Fixed-end Free-end these types of studies and has become a widely accepted analysis Length, L Diameter, Diameter, Area, Area, A2 tool [6-10]. Geometry (mm) di (mm) d2 (mm) Aj (mm^) (mm^) Although there are few studies on geometric stress waves in 704 30 30 707 ion solids, some studies (mostly experimental and/or numerical since Cylinder 704 30 5.3 707 22 closed-form analytical solutions do not exist) have been per- Tapered cylinder 704 30 30 707 107 formed on geometric effects on shock waves in gases. Setchell Spiral 704 30 5.3 707 22 et al. [U] conducted experiments with a conical converging ge- Tapered spiral ometry that demonstrated a shock strengthening from the walls focusing the shock wave in air. Lind [12] numerically studied shock waves with air in a cowl geometry illustrating that shock wave weakening could occur. Bond et al. [13] conducted simulations Table 2 Material properties used in finite element analysis employing an Eulerian framework and validated the simulations with experiments with a wedge design in carbon dioxide and Density Young's Poisson's Modulus (GPa) (kg/m') Ratio nitrogen. The wedge was essentially a two-dimensional linearly Material convergent geometry that focused the incoming shock repeatedly 1780 44 0.35 as multiple reflections increased the incoming pressure wave simi- AM30 1200 2.4 0.37 lar to the Setchell et al. [11] results. Inoue [14] numerically stud- Polycarbonate 3100 0.14 410 ied the geometry of a logarithmic spiral (log-spiral) duct to clarify Silicon carbide the vortex formation behind the reflected shock wave in air. The contribution of this study is to show the geometric effects on stress waves transmitted through three different solid materials (elastic, plastic, and viscoelastic) and different input pressure pro- polymer (polycarbonate denoting a common viscoelastic matefiles as the loading condition with an objective of garnering infor- rial), and ceramic (silicon carbide (SiC) denoting a common elasmation for design of impulse mitigating structures like those tic material) were investigated. These materials were also of observed in nature. In particular two different geometric effects, a interest due to their applications in impact scenarios. SiC has been round tapered cone and a spiral, are presented with an analysis on previously used in ceramic plates in bulletproof vests. Polycarbonthe pressures and impulses. ate is the most common material used in football helmets, and AM30 has been investigated as a possible material for automobile crash rails. Table 2 lists material properties for these materials. Methodology Material data for the plastic behavior of AM30 and viscoelastic Figure 1 depicts the four geometries that were studied along behavior of polycarbonate [16] were used in ABAQUS. SiC was with the load and prescribed boundary conditions. Table 1 pro- modeled as purely elastic due to its brittle nature. A compressive vides the dimensions used in the finite element analysis. The total pressure pulse was applied in four different loading conditions to length and cross-sectional diameters at the starting end were the end of each bar as illustrated in Eigs. 2{a)-2{d). The peak maintained among the four geometries. The ratio of the large- and amplitude was set as 130 MPa and the duration of pressure small-end diameters was also consistent for the tapered application was calculated to ensure the area under the pressure versus time curve (impulse) remained consistent between all four geometries. The finite element program ABAQUS/Explicit v6.11 [15], a loading conditions. These four different loading applications were stress wave dynamics code, was used as the numerical model in studied to ensure that impulse mitigation was not dependent on a this study. To demonstrate material independence, material prop- single loading condition. Eor brevity, all results other than normalerties for three different materials were used. The materials are ized impulse and maximum transverse displacement were found metal (AM30 magnesium denoting a common plastic material). using the consecutive increasing and decreasing pressure pulse. 021023-2 / Vol. 136, FEBRUARY 2014
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M. W. Trim Navai Surface Wartare Center, 9500 MacArthur Bivd, Bethesda,MD 20817
M. F. Horstemeyer Department ot Mechanical Engineering, Center for Advanced Vehicuiar Systems (GAVS), Mississippi State University, Mississippi Stafe, MS 39762
N.Lee Center for Advanced Vehicular Systems (CAVS), 200 Research Bivd, Mississippi State, MS 39762; Agriculfure and Bioiogicai Engineering, Mississippi State University, Mississippi State. MS 39762
L N.Williams Agriculture and Bioiogicai Engineering, Mississippi Stale University, Mississippi State, MS 39762
J. Liao Agriculture and Bioiogicai Engineering, Mississippi State University, Mississippi State, MS 39762
H. Rhee Center for Advanced Vehicular Systems (CAVS), 200 Research Bivd, Mississippi State, MS 39762
R. Prabhu
Geometric Effects on Stress Wave Propagation The present study, through finite element simulations, shows the geometric effects of a bioinspired soiid on pressure and impulse mitigation for an elastic, plastic, and viscoelastic material. Because of the bioinspired geometries, stress wave mitigation became apparent in a nonintuitive manner such that potential real-world applications in human protective gear designs are realizable. In nature, there are several toroidal designs that are employed for mitigating stress waves; examples include the hyoid bone on the back of a woodpecker's jaw that extends around the skull to its nose and a ram's horn. This study evaluates four different geometries with the same length and same initial cross-sectional diameter at the impact location in three-dimensional finite element analyses. The geometries in increasing compiexity were the following: (1) a round cylinder, (2) a round cylinder that was tapered to a point, (3) a round cylinder that was spiraled in a two dimensional piane, and (4) a round cylinder that was tapered and spiraled in a twodimensional plane. The results show that the tapered spiral geometry mitigated the greatest amount of pressure and impulse (approximately 98% mitigation) when compared to the cylinder regardiess of material type (elastic, plastic, and viscoeiastic) and regardless of input pressure signature. The specimen taper effectively mitigated the stress wave as a result of uniaxial deformational processes and an induced shear that arose from its geometry. Due to the decreasing cross-sectionai area arising from the taper, the local uniaxial and shear stresses increased along the specimen length. The spiral induced even greater shear stresses that help mitigate the stress wave and also induced transverse displacements at the tip such that minimal wave reflections occurred. This phenomenon arose although oniy longitudinal waves were introduced as the initial boundary condition (BC). In nature, when shearing occurs within or between materials (friction), dissipation usually results helping the mitigation of the stress wave and is iilustrated in this study with the taper and spiral geometries. The combined taper and spiral optimized stress wave mitigation in terms of the pressure and impulse; thus providing insight into the ram's horn design and woodpecker hyoid designs found in nature. [DOI: 10.1115/1.4026320] Keywords: guided wave propagation, elastic wave interaction and reflection, finite element analysis, dispersion in waveguides, stress wave mitigation, bioinspired design, geometric effects
Center for Advanced Vehicuiar Systems (CAVS), Agriculture and Bioiogicai Engineering, 200 Research Bivd, Mississippi State, MS 39762
Introduction The inspiration for this study stems from the curious geometries often found in biological structures that are subjected to dynamic loads. One such geometry is the natural spiral. Examples of the appearance of the spiral in natural shock absorbing systems include the ram's horn, seashells, and the woodpecker's hyoid bone. Does the reoccurrence of this curious shape throughout nature have some significance in regards to energy dissipation and shock absorption abilities inherent to its geometry? We seek to answer the following question: Can geometries affect stress waves to change the energy states and in the end the forces (or stresses)? 'Corresponding author. Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received September 5. 2013; final manuscript received December 13, 2013; accepted manuscript posted December 24, 2013; published online February 5, 2014. Editor: Beth Winkelstein.
Journal of Biomechanical Engineering
The fundamentals of the physics of stress waves have been well studied [1-3]. As a premise, the context of shock wave physics is warranted. If one were to neglect surface waves, then two main types of waves can propagate through isotropic solids: longitudinal waves and shear waves. Longitudinal (also called dilatational, pressure, primary, or P-) waves propagate with a characteristic wave speed and represent a volumetric change. Their motion is parallel to the direction of the wave propagation. Shear (also called secondary, S-, or distortional) waves represent no volume change and propagate at a slower wave speed with respect to longitudinal waves. Their motion is normal to the direction of propagation [4,5]. When either a longitudinal or shear wave impinges on a boundary, new waves are generated due to the reflective nature of waves. In a solid body with finite dimensions, these waves bounce back and forth between the bounding surfaces and interact with one another. These interactions can lead to wave amplification, cancellation, and other wave distortions.
Copyright © 2014 by ASIUIE
FEBRUARY2014, Vol. 136 / 021023-1
The pressure wave can be integrated over time leading to an impulse. The impulse is equal to the change in momentum of the body. It is possible for a very brief force, due to a shock, to produce a larger impulse when compared to a smaller force acting over a larger time period. Therefore, it is important to consider the transient forces, particularly those associated with wave phenomena. The question remains then: How do the longitudinal and shear waves that arise from impacts induce associated pressures and impulses in different solid geometries? If the geometries found in natural impact scenarios serve a purpose in pressure wave mitigation, then we would expect to see different geometries admitting different pressures and impulses in a solid. The four geometries included in this study comprise a cylindrical bar, a tapered cylindrical bar, a spiral with a circular cross section, and a tapered spiral (also with a circular cross section). The cylindrical bar serves as a "base-line" case. By comparing the response of the tapered cylinder to that of the uniform cylinder, we gain insight into how reducing the cross-sectional area influences the transient response of the structure. Similarly, comparing the spiral geometry to the uniform cylinder leads to an understanding of the effects of increasing curvature on the wave propagation. Finally, analysis of the tapered spiral allows us to understand the coupled influence of increasing curvature and decreasing cross-sectional area on wave propagation and reflection.
Cylinder 1
1
¿
Spiral .1
1
'»•••II^^HHI d,
Tapered Cylinder 1 1
d,
^
Tapered Spiral
1
1 d¡
Fig. 1 Schematic representation of the four finite element meshes illustrating the four different geometric configurations with the same length (and the same bar diameter where the pressure was applied) used in the analysis
A purely computational approach employing finite element analysis has been employed to study the wave propagation and Table 1 Geometric dimensions used in finite element analysis reflection characteristics of these different geometric bodies. Finite element analysis is the most efficient technique to perform Total Fixed-end Free-end Fixed-end Free-end these types of studies and has become a widely accepted analysis Length, L Diameter, Diameter, Area, Area, A2 tool [6-10]. Geometry (mm) di (mm) d2 (mm) Aj (mm^) (mm^) Although there are few studies on geometric stress waves in 704 30 30 707 ion solids, some studies (mostly experimental and/or numerical since Cylinder 704 30 5.3 707 22 closed-form analytical solutions do not exist) have been per- Tapered cylinder 704 30 30 707 107 formed on geometric effects on shock waves in gases. Setchell Spiral 704 30 5.3 707 22 et al. [U] conducted experiments with a conical converging ge- Tapered spiral ometry that demonstrated a shock strengthening from the walls focusing the shock wave in air. Lind [12] numerically studied shock waves with air in a cowl geometry illustrating that shock wave weakening could occur. Bond et al. [13] conducted simulations Table 2 Material properties used in finite element analysis employing an Eulerian framework and validated the simulations with experiments with a wedge design in carbon dioxide and Density Young's Poisson's Modulus (GPa) (kg/m') Ratio nitrogen. The wedge was essentially a two-dimensional linearly Material convergent geometry that focused the incoming shock repeatedly 1780 44 0.35 as multiple reflections increased the incoming pressure wave simi- AM30 1200 2.4 0.37 lar to the Setchell et al. [11] results. Inoue [14] numerically stud- Polycarbonate 3100 0.14 410 ied the geometry of a logarithmic spiral (log-spiral) duct to clarify Silicon carbide the vortex formation behind the reflected shock wave in air. The contribution of this study is to show the geometric effects on stress waves transmitted through three different solid materials (elastic, plastic, and viscoelastic) and different input pressure pro- polymer (polycarbonate denoting a common viscoelastic matefiles as the loading condition with an objective of garnering infor- rial), and ceramic (silicon carbide (SiC) denoting a common elasmation for design of impulse mitigating structures like those tic material) were investigated. These materials were also of observed in nature. In particular two different geometric effects, a interest due to their applications in impact scenarios. SiC has been round tapered cone and a spiral, are presented with an analysis on previously used in ceramic plates in bulletproof vests. Polycarbonthe pressures and impulses. ate is the most common material used in football helmets, and AM30 has been investigated as a possible material for automobile crash rails. Table 2 lists material properties for these materials. Methodology Material data for the plastic behavior of AM30 and viscoelastic Figure 1 depicts the four geometries that were studied along behavior of polycarbonate [16] were used in ABAQUS. SiC was with the load and prescribed boundary conditions. Table 1 pro- modeled as purely elastic due to its brittle nature. A compressive vides the dimensions used in the finite element analysis. The total pressure pulse was applied in four different loading conditions to length and cross-sectional diameters at the starting end were the end of each bar as illustrated in Eigs. 2{a)-2{d). The peak maintained among the four geometries. The ratio of the large- and amplitude was set as 130 MPa and the duration of pressure small-end diameters was also consistent for the tapered application was calculated to ensure the area under the pressure versus time curve (impulse) remained consistent between all four geometries. The finite element program ABAQUS/Explicit v6.11 [15], a loading conditions. These four different loading applications were stress wave dynamics code, was used as the numerical model in studied to ensure that impulse mitigation was not dependent on a this study. To demonstrate material independence, material prop- single loading condition. Eor brevity, all results other than normalerties for three different materials were used. The materials are ized impulse and maximum transverse displacement were found metal (AM30 magnesium denoting a common plastic material). using the consecutive increasing and decreasing pressure pulse. 021023-2 / Vol. 136, FEBRUARY 2014
Transactions of the ASME
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