text: the physics of stress waves , and all other wave types , are governed by three fundamental , conservation laws : conservation of mass , momentum , and energy . neglecting surface waves , there are two main types of waves that propagate through elastic , 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 propagation of the wave . 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 . see , for example , davis j l . wave propagation in solids and fluids . new york , n . y . : spring - verlag inc ., 1988 ; zukas j a , nicholas t , swift h f , greszczuk l b , curran d r , impact dynamics . malabar , f l : krieger publishing co ., 1992 ; and achenbach j d . wave propagation in elastic solids : north - holland , 1993 , all of the foregoing publications of which are incorporated herein by reference in their entirety . when either a longitudinal or shear wave impinges on a boundary , new waves are generated due to the reflective nature of waves . in a 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 . in the invention described here , both the spiral geometry and wavy embedded structure introduce deleterious shear waves that disperse , attenuate , and dissipate the input pressure . when the cross - sectional area of a cylindrical bar is reduced , a geometric impedance difference arises despite the intrinsic impedance of the material remaining unaltered . when a compressive elastic wave produced by a dynamic load or impact reaches the free end of the bar , it reflects back from that surface as a tensile wave . this reflected tensile wave can have detrimental effects on the medium through which it travels . impulse is defined as the integral of a force with respect to time . the impulse is equal to the change in momentum of the body . it is possible for a very brief force to produce a larger impulse than a force acting over a much larger time period if that force is sufficiently large . therefore , it is important to consider these transient forces . a fast - acting force can often be more detrimental to a structure than one that is more dispersed with respect to time . to reiterate , geometry plays a critical role in the response of a structure to a dynamic load . the four spiral geometries included in this disclosure comprise a cylindrical bar , a tapered cylindrical bar , a spiral with a cylindrical cross - section , and a tapered spiral with a cylindrical 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 , comparison of 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 . the embedded wavy material is a also a geometric effect that plays a critical role in structures under dynamic loads . the embedded wavy material or gap is compared to a baseline embedded straight line showing the much greater dissipation by way of lower pressures and lower impulses . with the exception of the simple cylinder , obtaining exact solutions for these geometries is unpractical , if not impossible . furthermore , the main goal of this analysis was to provide more of a qualitative understanding of how the transients are affected by only geometric differences . for these reasons , a purely computational approach employing the finite element ( fe ) method has been chosen to study the wave propagation and reflection characteristics of these bodies . the fe method is the most efficient technique to perform these types of studies and has become a widely accepted analysis tool . see , for example , demma a , cawley p , lowe m , pavlakovic b ., the effect of bends on the propagation of guided waves in pipes . journal of pressure vessel technology , transactions of the asme 2005 ; 127 : 328 ; gavric l ., computation of propagative waves in free rail using a finite element technique . journal of sound and vibration 1995 ; 185 : 531 ; treyssède f . elastic waves in helical waveguides . wave motion 2008 ; 45 : 457 ; mace b r , duhamel d , brennan m j , hinke l . finite element prediction of wave motion in structural waveguides , journal of the acoustical society of america 2005 ; 117 : 2835 ; and abaqus v 6 . 10 user documentation . providence , r . i . : dassault systemes simulia corp ., 2010 , all of the foregoing of which are incorporated herein by reference . fig1 a depicts the four geometries that were studied along with the load and boundary conditions that were prescribed . the length and cross - sectional dimensions of each model were kept consistent . the actual dimensions used in the finite element analysis are provided in table 1 , set forth hereafter . the ratio of total length to cross - sectional diameter was also maintained among the four geometries , i . e . l / d 1 = 10 . the ratio of the large and small - end diameters was also consistent ; d 1 / d 2 = 2 for the tapered geometries . the finite element program abaqus / explicit v6 . 10 [ 10 ] was used as the numerical model in this study for all simulations . it is anticipated that any finite element code would give similar results to all of the solutions generated here . linear elastic material properties typical of steel were used ; i . e . mass density , poisson &# 39 ; s ratio , v = 0 . 3 , and young &# 39 ; s modulus , e = 207 gpa . all geometries were meshed with 3 - dimensional , 8 - noded , continuum , linear , brick elements with reduced integration and hourglass control ( c3d8r ). a ramped , compressive , pressure pulse was applied to the end of each bar . the peak amplitude and duration were set as 1 × 10 5 pa and 38 . 8 μs , respectively . the prescribed load history is shown in fig2 . the nodes along the outer perimeter of the load - end were pinned ( u 1 = u 2 = u 3 = 0 ) for each case . no additional constraints were prescribed . the resulting stress wave was allowed to propagate through the structure for 800 μs prior to terminating the calculation . post - processing of data was performed using abaqus / cae v6 . 10 [ 10 ]. wave propagation plots were generated by defining a path through each model that extended from the cross - sectional center of the fixed end to the cross - sectional center of the free end . pressure and displacement response histories at the free - ends were generated by averaging the respective output of each node lying on the cross - section of the free end . fig1 b shows the wave dispersion of the pressure once a wave was initiated at the left end of the block . there is a gap between the upper and lower material the gap being in a wave form . the speed at which a longitudinal , elastic wave travels through a cylindrical , isotropic bar is given by c l =√{ square root over ( e / ρ )}, where e and ρ are the young &# 39 ; s modulus and mass density , respectively . similarly , an elastic , shear wave travels through the same media at a speed given by c s =√{ square root over ( g / ρ )} where the shear modulus , substitution of the typical steel values given above yields c l = 5 . 152 × 10 3 m / s and c s = 3 . 196 × 10 3 m / s . displacement contour and wave propagation plots for the cylinder , tapered cylinder , spiral , and tapered spiral are shown in fig3 . the plots for t = 40 μs show the initial wave immediately after the pressure load is released . at t = 104 μs , the wave is traveling in the + z direction . the wave reaches the free end of the tapered cylinder at t = 184 μs . at t = 256 μs , the reflected wave , is traveling in the − z direction on its way back to the fixed end . and at t = 328 μs , the wave peak reaches the fixed end of the cylinder where it had originated . similar plots for pressure and the von mises stress invariants are provided in fig4 and fig5 , respectively . fig6 ( a ) shows the pressure response at the free end of the cylinder , tapered cylinder , spiral , and tapered spiral . the free - end displacement response for the four geometries is shown in fig6 ( b ). on the lower abscissa , τ l = t ·( c l / l )= 1 is the time at which the longitudinal wave first reaches the free end . the first and second reflected longitudinal wave arrive back at the free end at τ l = 3 and τ l = 5 , respectively . similarly , on the upper abscissa , τ s = t ·( c s / l )= 1 corresponds to the time at which the shear wave reaches the free end and τ s = 3 represents the arrival of the reflected wave back to the free end . fig7 ( a ) compares the normalized impulse at the free end . the impulse is calculated by multiplication of the free - end pressure history by the respective free - end area followed by integration of the resulting force history ( where negative values are neglected ). fig7 ( b ) is a comparison of the normalized free - end displacement . free - end displacement is taken as the area under the free - end displacement history curve . the free - end impulse and displacement values of the cylinder are used to normalize the results and provide simple comparison . fig9 shows the different scenarios of the embedded material or gaps within the block of material representing a simple structure . it is anticipated that any structural geometry with the embedded wavy material or gap would generate similar results . the different colors illustrate the effect of the reflections of the various boundaries along with the embedded wavy material or gap . fig1 shows the dramatic drop in the impulse from when the embedded wave was introduced with a single wave , a single wave with an out - of - phase wavy boundary , and a single wave with an in - phase wavy boundary . clearly , the interactions of the embedded wavy geometries reduces dramatically the impulses ( integrated pressure - time histories ) much more than the straight line baseline case . from fig3 , we see that at t = 40 μs , the wave front is at z / l = 0 . 3 for the cylinder and tapered cylinder . comparing that to the position of the wave at t = 104 μs , we see that prior to any reflection from the free end , the wave travels through the cylinder and tapered cylinder at approximately the same velocity . however , the displacement amplitude is magnified by the reduction in area of the tapered cylinder . the displacement wave reaches the free end of the tapered cylinder at t = 184 μs . at this same time , the wave has already reflected from the free end of the uniform cylinder and is traveling in the − z direction . in the two spiral geometries , there is a slight bump in the displacement at t = 104 μs and z / l = 0 . 5 . but , the main displacement wave in the spiral geometries lags behind the main wave in the cylinders . also , in the spirals , there are more wave interactions as the waves reflect off of the surfaces , which cause the waves to be more dispersed . the displacement wave reaches the free end of the tapered cylinder first , at t = 184 μs . at t = 256 μs , the cylinder leads the tapered cylinder . the reflected wave in the tapered cylinder travels slower . the shear wave travels slower than the longitudinal wave . therefore , when the waves arrive at the boundary at different times , this leads to dispersion and / or cancellation and lower impulse near the free end of the rods . for the spirals t = 184 μs is an interesting time because the longitudinal wave has reached the free end but the shear wave has not . pressure ( or hydrostatic stress ), as plotted in fig4 , is the stress that tends to change the volume of the body . compressive stress is taken as positive and tensile stress is negative . the von mises stress that is used to construct fig5 is the second deviatoric stress invariant , i . e ., the von mises stress is the part of stress tensor that tends to distort the body and is independent of the hydrostatic stress component . the spiral shaped element and the embedded wavy material or gap are two useful ways in dissipating energy imposed upon it by an object . the energy is dissipated as a shear wave is induced by vibration of the spiral shaped element and also by the embedded wavy material or gap . furthermore , the tapered spiral shaped element is better at dissipating impact energy than the spiral shaped element having uniform circular cross section throughout its length . also , when multiple embedded wavy materials or gaps are introduced within a material , more dissipation occur as well . the impact can occur from any direction ( and any angle ), and the spiral shaped element and embedded wavy material or gap will dissipate the impact energy . the spiral shaped elements and the embedded wavy material or gap can be made out of numerous possible materials . any material that will enable vibration can be used , including but not limited to , elastic , viscoelastic , and / or plastic . shock mitigating materials can be manufactured to include one or more of the spiral shaped elements or structures with embedded wavy materials or gaps . for example , in the case of a football helmet , a helmet layer or football helmet pad insert can be produced with numerous spiral shaped elements in order to dissipate energy when a football player wearing the helmet is impacted and the outer shell can have embedded wavy materials or gaps included in the design . in the shock mitigating materials , the spiral shaped elements can be situated in or surrounded by air , liquids , gel , elastic , viscoelastic , plastic , or any other material that permits the spiral shaped element to vibrate for the purpose of dissipating impact energy . furthermore the embedded wavy material can include , air , liquids , gels , viscoelastic , plastic , or any other material that admits the wave to dissipate . it should be emphasized that the above - described embodiments of the present invention , particularly , any “ preferred ” embodiments , are merely possible examples of implementations , merely set forth for a clear understanding of the principles of the invention . many variations and modifications may be made to the above - described embodiment ( s ) of the invention without departing substantially from the spirit and principles of the invention . all such modifications and variations are intended to be included herein within the scope of this disclosure and the present invention . the spiral shaped elements in the shock mitigating material can take many different shapes and sizes , depending upon design and / or manufacturing preferences . also , the embedded wavy material can also take different wave forms ( sinusoid , blocks , triangles , etc .) with different amplitudes and periods . in some embodiments of shock mitigating materials , each spiral shaped element has a consistently shaped cross section ( e . g ., circular , polygonal , triangular , square , rectangular , trapezoidal , etc .) throughout its length and is tapered either from a large outside end to a small inside end or from a small outside end to a large inside end . the amplitude and the period of the embedded wavy material may also change within the structure . in other embodiments of shock mitigating materials , each of the spiral shaped elements is configured in the shape of a helix ( or corkscrew ). moreover , the helix in this configuration may be tapered or nontapered . finally , each element can be in the shape of a conical helix , conical toroid , cylinder helix , or other helix . the embedded wavy material may also have three dimensional helical attributes as well . in other embodiments of shock mitigating materials , each of the spiral shaped elements reside ( are coiled ) in a single plane . the elements can be placed side by side in the materials . in other embodiments of shock mitigating materials , each of the spiral shaped elements is a sheet that is disposed in a rolled configuration so that its cross section along the span of the elongate structure is spiral . the sheet can be tapered or nontapered from an outside end to an inside end . furthermore , each of the elements can be nonuniform along the elongated span of the rolled configuration , for example , it could be conical . in other embodiments of shock mitigating materials , there is a mix of different types of spiral shaped elements , as previously mentioned .