Statistics of energy dissipation in the hypervelocity impact shock failure transition

International Journal of Impact Engineering 137 (2020) 103435

Contents lists available at ScienceDirect

International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng

Statistics of energy dissipation in the hypervelocity impact shock failure transition

T

Dennis Grady Applied Research Associates, 4300 San Mateo Blvd NE, Albuquerque, NM, 87110, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Shock wave Compaction Spall Statistics Action

In the hypervelocity impact event, shock waves subject material to failure transitions with the attendant dissipation of the imparted energy. Under shock compression, failure and dissipation entail intense inelastic shear and compaction. Through shock interactions, states of dynamic tension are achieved and further failure dissipation involves fracture and fragmentation. The nature of failure of solids in the shock environment has encouraged considerable experimental effort through the past several decades. Such efforts have yielded results that suggest a level of universality in the shock failure transition over significant spans of shock intensity and solid material types. Examples include the fourth-power relation between pressure and strain rate in solidmaterial compressive shock waves, and power-law relations capturing spall fracture strength and fragmentation size scale in dynamic tensile failure. Comparable power-law relations also describe the shock compaction of distended solids. The present paper explores a statistical perspective of the underlying micro failure dynamics for the purpose of achieving better understanding of the macro failure trends noted above. A statistical correlation function description of the random micro velocity field is introduced. Through the attendant kinetic dissipation, the statistical fluctuation-dissipation principle is applied to the shock failure transition. From this statistical approach, power-law relations for compressive and tensile shock failure emerge that replicate the reported experimental behaviors. Results are compared with selected experimental data.

1. Introduction The hypervelocity impact event entails the transfer of kinetic energy of the impacting body into internal energy and kinetic energy of the impacted material. Energies and energy rates are extreme. Subsequent to the impact shock event, the energy not remaining as transport kinetic energy resides as forms of internal and thermal energy, the latter a consequence of energy dissipation in the shock wave. The nature of energy dissipation in the shock wave is complex, and resides at the heart of much of shock wave physics. Energy dissipation is heterogeneous, consummated through the collective dynamics of localized dissipative structures. Dissipative structures are unique to the material of concern and individually selforganize on the elemental scale in response to the dissipation and dissipation rates dictated by the encompassing shock wave e.g. [1]. Terms in the preceding sentence are adopted from the landmark works of Prigogine e.g. [2]. Dissipative structures can be, for example, thermal adiabatic shear bands in metals, shear and tensile fractures in brittle compounds, or planar compaction zones in porous matter. The formation and dynamics of dissipative structures are integral to the shock failure transition in solid matter.

With some exceptions, little is known as to how dissipative structures self-organize on the local level. Where failure in the shock is accommodated through dislocation plasticity in metals and crystalline solids, a measure of understanding is emerging. Dissipative structures originate as molecular-level slip-bands that nucleate and grow in extent from the sub-nano to the microscale. Dislocation cross-slip plays an integral role in slip-band thickening [3]. On the sub-nano scale, temperature increase from dislocation phonon dissipation is initially modest as phonon mean free path exceeds slip-band dimensions. As shear slip bands thicken thermal transport properties become increasingly germane. Thermal energy leads to softening and even melting in the slip-band region [4]. When microscale slip-band dimensions are achieved, evidence supports continuum thermal conduction, thermal softening, and specific heat properties, joined with a competition of local thermal and momentum diffusion, participating in the self-organization dissipation processes govern optimum shear-band dimensions [5,6]. Tensile or shear fracture in crystalline and metallic solids is also a common dissipative structure. Details are complex as to the physics whereby material defects activate and self-organize into localized fractures. Mature fractures, however, incur measurable dissipation.

E-mail address: [email protected]. https://doi.org/10.1016/j.ijimpeng.2019.103435 Received 25 June 2019; Received in revised form 31 October 2019; Accepted 31 October 2019 Available online 02 November 2019 0734-743X/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Impact Engineering 137 (2020) 103435

D. Grady

compression through a one-dimensional shock wave. Alternatively, planar impacts can be design such that wave interactions lead to increasing states of dynamic tension within an interior planar region of the material. In either, impacts of sufficient intensity can induce stress levels that are in excess of the strength of the material, and transient failure occurs. In the first case, failure proceeds through processes of compressive shear fracture, dislocation plasticity or, perhaps, dynamic compaction if the material is initially distended. In the second case, failure occurs through brittle or ductile tensile spall cavitation and fracture. Through the continuum one-dimensional transport equations of mass, momentum and energy, joined with an appropriate constitutive relation, the continuum time-histories of stress, strain and velocity through the transient shock process can be calculated. Dynamic failure is, however, inherently heterogeneous on the microscale. Failure proceeds through the activation and the growth of discrete dissipative structures (e.g., fractures, slip bands, compaction zones). Dissipative structures self-organize on the elemental scale and interact collectively on the material scale, as needed to accommodate the energy dissipation and dissipation rates required of the shock failure event. A sensible extension of the continuum material response through the shock transition is a statistical Langevin decomposition of the governing transport equations e.g. [9]. Forces and motions are partitioned into the one-dimensional continuum solution and the random microscale variations about the continuum. As such, powerful tools of Boltzmann transport statistics come into play with potential application to the shock transition. A full Langevin analysis of the shock transition in a solid material is not attempted. Rather, generally applicable results that emerge from Langevin statistics are employed in analysis of the shock transition. Continuum particle velocity within the shock transition emerges from the continuum transport equations of motion. For a steady structured compressive shock wave, the particle velocity as the continuum material state transits the Rayleigh line is determined by the Hugoniot conservation relations. In a tensile spall event, the states of increasing continuum particle velocity are commonly captured through the tensile strain rate imparted to the material by the wave interaction. At any point on the microscale within the material, as shock passage proceeds, particle velocity within a planar shock wave can be decomu (x , t ) , with μ(t) the expected value posed into the vector sum μ (t ) + → of the velocity at any time over a sufficient span of a plane with normal in the shock direction. Namely, this statistical decomposition is the continuum particle velocity μ(t), as dictated by the continuum transport → solution, and a variant velocity u (x , t ) characteristic of the microscale variation in velocity about the mean at that point and time. The continuum velocity μ(t) is normal to the plane of the one-dimensional → shock. The randomly oriented variant velocity field u (x , t ) , as a function of space and time, is integral to the correlated failure dynamics within the shock wave event. On any normal plane within the shock, the expected value of the variant velocity over a sufficient span of the plane is necessarily zero. Correspondingly, the expected value of the →2 square of the variant velocity within the plane 〈 u (x , t ) 〉 = σ 2 (t ) is 2 nonzero. The statistical variance σ (t) of the velocity field, with standard deviation σ(t), is identified with kinetic dissipation within the structured shock wave.

Energy dissipation, as quantified through strain energy release or planestrain fracture toughness, has been measured for many materials. The dissipation of energy by the dynamic dissipative structure is dichotomous. First, dissipation is frictional, occurring interior to the dissipative structure, contributing to the thermal phonon field and local temperature rise, as well as local athermal dissipation. Second, as perhaps first recognized by Mott [7], dissipation is kinetic, proceeding through acoustic phonon radiation outward from the dissipative structure. Radiation dissipation and frictional dissipation are comparable in magnitude, and governed by the same failure and dissipation properties of the material. Acoustic radiation dissipation has been identified with the dissipative action within the shock wave failure transition [1,8]. The elemental dissipative structure both contributes to, and is affected by, the radiation acoustic phonon field. The dynamic history of any individual dissipative structure is dictated, not only by the prevailing stress field imparted by the subsuming shock wave, but also by the incident random acoustic phonon stress field, a consequence of the collective dynamics of the active dissipative structure system. Thus, dynamics of an individual dissipative structure is perhaps best described by a Langevin governing equation e.g. [9] where the impelling stress is decomposed into the continuum stress imposed by the shock wave and the random stress imparted by the fluctuating acoustic phonon field. Consequently, the dynamics of any dissipative structure exhibits a statistically history. Within the framework of Langevin dynamics, the statistics-based fluctuation-dissipation theorem has application to the shock-wave failure transition. Through application of the fluctuation-dissipation theorem, the kinetic dissipation; namely the radiation acoustic phonon field, is identified with the viscosity responsible for structuring the shock wave. Through application of the fluctuation-dissipation theorem, the correlated kinetic dissipation necessary to achieve the shock failure transition is determined. From application of the fluctuationdissipation theorem emerge previously developed and tested relations for the dependence of spall stress and fragment size on the intensity of the imparting shock wave, and the fourth-power relation experimentally displayed by steady structured shock waves. The present effort pursues the necessary background that describes this statistical perspective of the shock physics underlying extreme energy dissipation within the hypervelocity impact shock-wave event. The next section presents a statistics-based theoretical framework appropriate to the shock failure transition. In the subsequent section experimental shock compression data, including steady structured shockwave measurements on both fully dense and distended solid materials, and impact spall strength and fragmentation measurements on metals, are presented for the purpose of supporting the statistical development. The final technical section offers comparisons of experimental shock measurements and theoretical results. 2. Dissipation dynamics and the shock failure transition Through the use of statistical principles that are central to other areas of physics, a statistical framework is constructed that is descriptive of the shock failure transition dynamics within the structured shock wave. Through this framework, expressions are developed for the viscosity appropriate to the spatial structuring of the shock wave. Through this framework emerge critical state relations that determine the shock-induced failure transition. Relations are developed that replicate the experimental power-law behavior observed in structure shock-wave measurements in solids, and the rate dependence of spall strength and fragment size observed in the dynamic tensile failure of condensed materials.

2.2. Dissipation in the shock transition

→2 Kinetic dissipation Ek (t ) = 〈 u (x , t ) 〉/2 = σ 2 (t )/2 , associated with the variant velocity field, is the random acoustic phonon energy that is a consequence of the collective dissipative structure activity underway as dissipation and failure proceeds in the shock transition. This kinetic dissipation will ultimately thermalize. Within the interim, the kinetic dissipation has an integral role to play in the correlation dynamics of the shock failure transition. It is noteworthy that this decomposition of

2.1. The variant velocity field The planar impact onto a solid material leads to dynamic 2

International Journal of Impact Engineering 137 (2020) 103435

D. Grady

a temporal variant velocity correlation function on that plane, referenced to an arbitrary position x = 0 within that plane, is the expected u (0, t ) → u (s, t ) 〉. The temporal coordinate s value expression K (s, t ) = 〈→ is related to an arbitrarily directed scalar position coordinate x within the plane through the acoustic impedance Z = ρc with s = ρx / Z . The variant velocity correlation function K(s, t) is a measure of the temporal correlation of kinetic dissipation as dynamic failure ensues through the shock transition. Note that K (0, t ) = σ 2 (t ) is the particle velocity variance (twice the kinetic dissipation) at time t within the shock wave. Here, a powerful result emerging from the applications of Boltzmann statistics and Langevin analysis to complex systems e.g. [18] is applied to the shock failure transition. Namely, the integral of the variant velocity correlation function equates to a continuum transport property of the material,

the total kinetic energy, at any plane within the shock wave, into the continuum transport kinetic energy μ2(t)/2, as dictated by the governing continuum equations, and the random kinetic energy about this plane σ2(t)/2, is a result emerging from fundamentals of classical mechanics e.g. [10]. Kinetic dissipation in the shock is incurred through both elastic and inelastic responses of the material. Elastic dissipation is brought about by wave scattering as the shock progresses through the inherently heterogeneous material. This aspect of kinetic dissipation has previously been addressed in part [11,12]. Here, emphasis is on the kinetic dissipation sustained through inelastic processes in the shock transition. Although momentum diffusion through acoustic phonon transport is central to this presentation, momentum diffusion through mass transport can also play a prominent role in selected shock transitions. Mass transport, for example, dominates wave structuring viscosity in the early shock compaction transition of an initially highly distended solid. Mass transport relating to viscosity in the shock transition is addressed in Chapter 9 of reference [6]. Concurrently, frictional deformation dissipation Ef occurs within the interior of elemental active dissipative structures. Depending on the material and mode of deformation in the shock transition, frictional dissipation is a collective term that can include fracture bond breakage, dislocation plasticity and distended matter compaction among others. Frictional dissipation within the elemental dissipative structure is characterized by a dissipation constant Γ with dimensions of energy per unit area. Familiar is that of fracture, where Γ is a measure of the fracture interface dissipation as commonly specified through measurements of the material fracture toughness. Less familiar is application of the dissipation constant Γ to other forms of failure dissipation including plastic shear slip or distended matter compaction e.g. [5,13]. Further complexities with Γ are noted, with different dissipation mechanisms at play with passage of the structured shock wave through the material. In a metal for example, elastic scattering and perhaps deformation twinning early in the shock wave, transitioning to intense adiabatic shear slip later in the shock, through the crest and to shock completion. Here the dissipation constant Γ is identified with the dominant shock failure dissipation mechanism active through the failure transition, and is assumed sensibly constant through the span of shock intensities examined. Dissipation dichotomy is central to the statistical physics governing the shock failure transition. Frictional dissipation and kinetic dissipation are both a consequence of the same underlying microstructural failure mechanics [1,7]. Both frictional dissipation and kinetic dissipation scale with the dissipation constant Γ. Dissipation dichotomy was perhaps first noted in an analysis of dynamic fracture by Mott [7]. Mott assumed a dimensionless constant relating frictional and kinetic dissipation. More substantive discussions of dissipation dichotomy are explored in the study of Archambeau and Minster [14]. A parity of frictional dissipation and kinetic dissipation is central to an earlier energy-based description of spall fragmentation [15]. Comparable dissipation dichotomy emerges from calculations of elemental adiabatic shear band dynamics [16]. More recently, Bazant and Caner [17] address dissipation dichotomy within the comminution region of brittle solids accommodating impact penetration. Archambeau and Minster [14] calculate that for sufficiently fast failure transition, frictional dissipation and kinetic dissipation are equal. Here, for present purposes, Ek approximately equal to Ef is assumed.

∞

D (t ) =

∫

→ u (0, t ) → u (s, t ) ds=

0

∞

∫ K (s, t ) ds. 0

(1)

Eq. (1) is identified as the fluctuation-dissipation theorem e.g. [9,18] and, in this form, more commonly identified with the GreenKubo equation e.g. [19]. The fluctuation-dissipation relation has wide application in relating continuum transport properties of a system to the underlying microscopic dynamics [18]. In the present treatment this expression relates the correlated microscopic kinetic acoustic dissipation to a continuum phonon momentum diffusion property D(t) at any time t within the advancing shock wave. As such, D(t) is identified with the acoustic phonon viscosity within the shock transition. The property D(t) is integral to the structuring of the time history of the wave through the shock transition. The variant velocity correlation function plays a further role in the shock failure transition. The differential expression dm = Zds is an element of areal mass. Failure transition within the shock is achieved when the correlated kinetic dissipation Ek(t)dm attains the requisite shock transition frictional dissipation as expressed through the dissipation constant Γ within a time span of τ. Dissipation dichotomy through the shock transition is realized by equating the frictional dissipation constant Γ to the correlated kinetic dissipation through the shock transition period τ,

Γ=

1 2

∞

∫ 0

1 → u (0, τ ) → u (s, τ ) dm = 2

∞

∫ K (s, τ ) Zds. 0

(2)

A functional form for the velocity correlation function is not known. Selected statistical processes (e.g., Gaussian, Markovian), common to other random physical applications, dictate an exponential decay for the temporal correlation e.g. [19]. Application of an exponential variant velocity correlation function, with characteristic temporal correlation τ, is accepted here,

K (s, τ ) = → u (0, τ )2 e−s / τ = σ 2 (τ ) e−s / τ .

(3)

Integration of Eq. (2) yields,

Γ=

1 2 σ (τ ) Zτ . 2

(4)

σ 2 (τ )/2

is the specific kinetic dissipation neAs expressed, Ek (τ ) = cessary to the shock failure transition. Eq. (4), relating frictional and kinetic dissipation, takes the elemental energy-time form,

Γ/ Z = Ek (τ ) τ = 2.3. Correlation dynamics in the shock transition

1 2 σ (τ ) τ . 2

(5)

Eq. (5), barring uncertainties of a proportionality constant of order unity, is an energy-time criterion descriptive of the shock failure transition. Eq. (5) follows from considerations of the underlying microstatistics failure and correlation dynamics that bring about the macro shock-failure properties observed in the shock-wave experiment. Such observations, among others, include the rated dependence of the shockinduced spall transition, and the fourth-power dependence of steady

→ Within the shock transition the variant velocity field u (x , t ) is a statistically random function of position and time. On any plane within the shock, with normal collinear to the planar shock direction, the variant velocity is statistically stationary; that is, no point on the plane is statistically preferred. At any time t within the structured shock wave, 3

International Journal of Impact Engineering 137 (2020) 103435

D. Grady

Fig. 1. Steady-wave strain rates and shock widths versus Hugoniot pressure for structured shock-wave measurements on aluminum oxide ceramic, uranium metal and unreacting HMX based energetic material. Open symbols are for single crystal HMX and aluminum oxide (sapphire). Note times three vertical scaling for PBX-9501 and single crystal HMX. Horizontal dashed lines on the right identify nominal grain size.

distributed over Fourier modes larger than ω = 1/ τ and diminishes through the higher frequency modes. As a final note to this section, the terms self-organization and criticality are applied in the present theoretical development. This should not be construed as having any relationship to the well-known theory of self-organized criticality [24,25]. Some considerations on possible applications of this latter overarching theory to the shock transition are, however, perhaps warranted.

structured shock waves in solids. A comparison of Eq. (2) with Eq. (1) reveals the ratio Γ/Z equates to a critical state amplitude of the self-diffusion constant and kinematic viscosity D(τ) within the temporal span of the shock transition. This critical state constant is identified with the dissipative action A associated with the shock failure transition [8]. Namely, D (τ ) = A = Γ/ Z . As such, this equality is recognized as a generalization to the shock wave event of the classic relation emerging from the statistical analysis of Brownian diffusion that is currently independently attributed to Einstein [20], Sutherland [21], and Smoluchowski [22]. In that application, a constant of diffusion is equated to a product of the intensity of the kinetic thermal field, as captured through the temperature T, and a measure of mobility appropriate to the system. Here, a critical diffusion constant, a kinematic viscosity identified with the dissipative action, is equated to the product of a measure of the acoustic kinetic dissipation, captured through the dissipation constant Γ, and the phonon mobility 1/Z. As noted in the work of Kubo [18], the fluctuation-dissipation relation is useful in two ways: The nature of fluctuations is understood and the relation is used to calculate continuum diffusivity, viscosity, or other transport properties of the material. Alternatively, the latter continuum properties are known and, through the fluctuation-dissipation relation, used to assess fluctuation characteristics of the system. Classic application of the latter is the Nyquist theory of electrical conductivity [23] where material resistivity is known and the fluctuationdissipation relation is used to assess the nature of the Johnson noise current and voltage fluctuations. Alternatively, in dilute fluids knowledge of the fluctuation and scattering physics is accessed to determine continuum diffusion and viscosity properties of the material. Both perspectives have application to Eq. (5) and the shock transition. In one application properties on the left are known and used to assess failure energies, failure stresses and characteristic failure times. In another application, constancy of the left-hand side, joined with knowledge of energies and times on the right, allow determination of wave structure trends in the shock failure transition. Additional perspective to the present development is provided through a spectral representation of the variant velocity correlation function spanning the shock transition,

3. Experimental measurements of shock failure transitions Selected shock-wave experiments on solid materials are discussed here that have all been previously reported. These shock wave data are further examined for the purpose of exploring consequences of the analysis of the previous section. This re-examination of earlier experimental data is nonetheless independently noteworthy. Systematic trends in these data are noted that are informative as to the nature of the shock failure event. 3.1. Structured shock waves in solid matter Measurements of steady structured shock waves in solids e.g. [26,27] have demonstrated regularities that have encouraged diverse explanations. Results of such measurements are shown in Figs. 1 and 2 for selected solids; namely, aluminum oxide ceramic, uranium metal, and an unreacting HMX explosive and binder mixture (PBX-9501). Also included with these data are structured wave measurements for single crystal sapphire and single crystal HMX. On the left in Figs. 1 and 2 is the more common display of the Hugoniot pressure step versus the structured-wave strain rate. (Note that the pressure step initiates from the Hugoniot elastic limit (HEL) pressure state in this presentation of the data.) These data replicate the fourth-power dependence previously reported for these and other full density solid materials [28]. Such fourth-power dependence has been shown to follow from an assumption of a constancy of the Hugoniot energy-time product [26], a constant that has subsequently been identified with the shock dissipative action of the material [8]. On the left in Fig. 2 is shown a display contrasting unalloyed αuranium metal [29] with U6Nb uranium alloy [30]. Although data for the latter are sparse, both tend towards fourth power at higher pressures. The intercept difference between the two metals attests to the reduced wave structuring shock viscosity of the U6Nb as compared to α-uranium. U6Nb uranium is thermodynamically metastable, with a crystal structure that is monoclinic distortion of the orthorhombic αuranium structure characteristic of unalloyed uranium [31]. This crystal structure difference, however, gives rise to markedly different elastic properties and strength characteristics. The lower shear modulus of U6Nb (30 GPa as compared with 86 GPa) is suspect in affecting dynamic strength and wave structure viscosity in this uranium metal. Note

∞

K (s, τ ) =

∫ J (ω, τ )cos ωsdω. 0

(6)

Accepting the simplified correlation function as expressed in Eq. (3), a spectral density of the form,

J (ω, τ ) =

2 σ2 , π (1 + (ωτ )2)

(7)

yields the spectral distribution of the correlated kinetic dissipation. The Eq. (7) spectral density would imply, when sensibly averaged over the span of the shock transition, that kinetic dissipation is equitably 4

International Journal of Impact Engineering 137 (2020) 103435

D. Grady

Fig. 2. Steady wave shock pressure step versus strain rate for uranium metal [29,30] on the left and for brittle granular compounds [8] on the right. Note times two pressure for tungsten carbide to separate from silicon dioxide data.

impact tests [29,38]. Fragment size data for U6Nb uranium are from expanding ring tests [39]. The power-law relations that overlie the spall stress and fragment size data result from an energy-based theory of dynamic fracture and fragmentation [40]. A plane-strain fracture toughness of 60 MPa m1/2 is applicable to both uranium metals. On the right in Fig. 3 are shown results of exploding cylinder experiments for two steels [41]. Through variable heat treatment a span of plane-strain fracture toughness values was achieved. In each test fragment size distributions were determined and the Mott distribution size scale assessed. Strain rate at fragmentation is assessed from the reported expansion velocity of the cylinders. The calculated curve is described in the next section.

that HEL strength of U6Nb is not known and not accounted for in the Fig. 2 plot. See a brief discussion in the next section. On the right in Fig. 1, the same structured wave data for the aluminum oxide ceramic, uranium metal, PBX-9501 and single crystal solids, are plotted in an alternative Hugoniot pressure step versus shock-width display. Fourth-power dependence translates to thirdpower dependence in this representation of the data. Several observations are noteworthy: For the polycrystalline materials, nominal grain size is shown in the plot. For alumina ceramic and uranium metal, marked disparity in the experimental widths of the shock wave, as compared to that of the grain size, suggests that granularity-induced wave dispersion plays a minimal role in structuring the steady-wave shock. The PBX-9501 material entails a more complex bi-model granularity identified by fine and coarse in the figure. Again, however, structured wave widths appear to be unrelated to microstructure granularity. Also plotted on the right in Fig. 1, are structured shock-wave widths for single crystal c-cut [32,33] and r-cut [33] sapphire, and for two orientations, {110} and {011}, of single crystal HMX [34]. Although tentative, the reasonable agreement between polycrystalline and single crystal materials has profound implications. Namely, crystal plasticity, and not material microstructure, is principally responsible for the viscous structuring of the steady shock wave. In this regard, experiments of Arnold [35] on Armco iron are notable. In the reported shock experiments, a range of initial grain sizes of the test iron metal failed to affect the observed fourth-power slope and intercept for the metal studied. (See also [36].) Observed shock structure was independent of polycrystal metal grain size. A comparable plot of experimental steady structured wave data for shock compaction of brittle granular compounds [8] is shown on the right in Fig. 2. Initial test densities are typical of that of pour densities of the granular media. The range of Hugoniot states achieve densities that are in the partial compaction regime of the initially distended solids. In contrast to the fourth-power character observed for full density materials, power-law dependence ranging from about one to two for Hugoniot pressure versus strain rate is observed for the shock compaction of granular media. Similar results are observed in shock compaction experiments on granular HMX and sugar [37]. Curves overlaying the respective sets of data, as discussed in the next section, result from a calculation based on application of a dissipative action energytime principle comparable to that posed for full density material [8].

4. Universal features of the shock failure transition Experimental observations have shown that measurements of shock failure transitions demonstrate power-law behavior over extended ranges of dynamic intensity. Structured compressive steady shock wave experiments on near full density solids have shown power-law scale invariance with an exponent of four. Similarly, time-resolve structured wave measurements within the Hugoniot compaction range of distended solids also demonstrate power-law behavior. Power-law exponents observed for the latter range over about one to two. In the spall failure and fragmentation of solid materials, where spall stress and characteristic fragment size are measured over a range of imposed strain rates, power-law dependence is also observed. This power-law nature is suggestive of a measure of universality of the mechanisms underlying the shock failure transition. Here, statistical issues developed in the earlier section are related to the experimental shock wave failure transition measurements described in the previous section. 4.1. Structured shock waves and the fourth-power Based on the premise that the dissipation constant Γ characteristic of the elemental dissipative structure does not vary within the span of the experimental data, and similarly that the impedance Z is sensibly constant, invariance of the ratio Γ/Z constrains the kinetic dissipation and characteristic transition time on the right in Eq. (5). A steady shock wave traversing the Rayleigh line path dictates the requisite energy dissipation to a specific Hugoniot state. Consequently, the wave duration τ through the shock transition must adjust to maintain constancy of the relation. Over the span of shock pressures of the data displayed in Fig. 1, the Hugoniot transition for the several materials is adequately described with a linear shock velocity versus particle velocity relation as expressed through the slope S and intercept Co. (Again, note that the Hugoniot elastic limit pressure state is accounted for in this analysis of the data with the Hugoniot pressure step from the HEL state displayed on the ordinate.) To lowest order in the Hugoniot strain, energy

3.2. Dynamic fracture and fragmentation in solid matter In Fig. 3 are shown shock-induced spall strength and dynamic fragmentation data representative of shock-induced tensile failure in solid materials. Test data on the left are for uranium metal. Spall strength measurements for α-uranium are from one-dimensional planar 5

International Journal of Impact Engineering 137 (2020) 103435

D. Grady

Fig. 3. Spall strength and fragment size measurements for uranium metal on the left [8]. Mott fragment size scale versus fracture toughness for steel cylinder fragmentation experiments on the right [41].

HEL strength of U6Nb has not been determined, to this author's knowledge, because of wave structure complications as a consequence of the anomalously low shear modulus [30]. The plot shows the HEL strength on the ordinate versus the dissipation constant assessed from the analysis on the abscissa. Although marked differences in HEL strength for single crystal and polycrystalline materials of the same compound are noted, a clear first power dependence between HEL strength and the dissipation constant is dictated by the data. (Only the dissipation constant value for U6Nb is identified in the plot.) This representation of the data suggests that crystal strength properties of the material responsible for onset of failure at the Hugoniot elastic limit continue to exercise a dominant role in subsequent dissipation and wave structuring in transition to the Hugoniot state. Magnitude of the calculated dissipation constant is also of import; however, a more studied development of the theory assessing the constant of proportionality between the right and left hand relations in Eq. (9) should be a prerequisite. An alternative statistical construction is instructive. Returning to Eqs. (2) and (3), introduce the change of variables m = Zs and dm = Zds . Eq. (2), joined with Eq. (3), develops into,

dissipation through the shock transition is,

Eh =

SCo2 ε 3/3.

(8)

Hugoniot strain is ε = uh / U , with uh the particle velocity step through the Hugoniot shock transition and U the shock velocity. Accepting dissipation dichotomy with Ek = Ef, and hence a kinetic dissipation nominally one-half that of Eq. (8), application of Eq. (5) yields,

Γ/ Z = SCo2 ε 3τ /6.

(9)

With Hugoniot pressure and shock width to lowest order of p = ρo Co2 ε and λ = Co τ , Eq. (9) replicates the third-power dependence of shock pressure step on wave width exhibited by the data on the right in Fig. 1. ˙ , Eq. (9) also reIdentifying a characteristic strain rate through ε = ετ plicates the pressure step versus strain rate fourth-power trend of the data on the left in Figs. 1 and 2. Therefore, energy dissipation as dictated by steady-wave Hugoniot compression, joined with the statistical fluctuation-dissipation principle as presently applied, constrains the shock rise time and the characteristic strain rate to the shock state, yielding the steady-wave fourth-power relation. An informative presentation of Eq. (9) for the data of Figs. 1 and 2 is provided in Fig. 4. The acoustic impedance Z for the materials is adequately known. Using the fourth-power fit to the experimental data and Eq. (9), the dissipation constant Γ can be calculated. Further, for most of the materials, the Hugoniot elastic limit (HEL) has been measured. The

Γ=

1 2

∞

∞

0

0

∫ K (m, μ) dm = 12 ∫ σ 2 (μ) e−m/μdm,

(10)

Where again the simplifying exponential approximation of the correlation function is applied. The correlation parameter transitions to μ = Zτ with dimensions of mass per unit area. Integration of Eq. (10) yields,

Γ=

1 2 σ μ. 2

(11)

Alternatively, introducing the mass density ρ, and identifying the correlation length scale through λ = μ/ ρ yields from Eq. (11),

1 2 ρσ λ = Γ. 2

(12)

Namely, equality of the kinetic dissipation within the thickness span λ of the shock and the dissipation constant Γ is demonstrated. 4.2. Shock wave spall and fragmentation The tensile spall failure of solids is brought about through the interaction of shock waves that carry the material into tension until material strength is exceeded, and failure through fracture and fragmentation ensues. Commonly the intensity of spall failure is specified through a measure of the rate of strain at which the material is carried into tension. Of course, dynamic tensile failure is also achieved through other means. Fragmentation of uranium metal, for which fragment size data are plotted on the left in Fig. 3, is achieved through an outward

Fig. 4. Hugoniot elastic limit versus the dissipation constant Γ, as assessed from Eq. (9) and the fourth-power presentation of the structured wave experimental data in Figs. 1 and 2. 6

International Journal of Impact Engineering 137 (2020) 103435

D. Grady

media. Again, the integrated kinetic dissipation expressed as the fluctuation-dissipation theorem posed in Eq. (1) is identified with the kinematic viscosity responsible for structuring the compaction shock wave. This viscosity then underlies the strain rate versus Hugoniot pressure display of the compaction data in Fig. 2. Here, however, both mass diffusion, as well as momentum diffusion, contribute to viscous structuring of the shock wave. When the simplifying approximation for the velocity correlation function introduced in Eq. (3) is applied, integration of Eq. (1) again leads to the form of Eq. (5). Here, however, decomposition of the kinematic viscosity (the dissipative action) into the ratio A = Γ/ Z may not be appropriate, and is certainly not known. Nonetheless, dissipation energy and temporal shock width on the right in Eq. (5) are readily extracted from the experimental data. Kinematic viscosity A can be calculated and trends with shock amplitude and material type explored. This examination of the compaction data shown on the right in Fig. 2 was undertaken in reference [8] and is only briefly outlined here. The compaction Hugoniot data are adequately described by the Hugoniot pressure versus compaction strain relation p (ε ) = pc (ε / εc )α . Hugoniot energy is readily calculated and, if rigid-compaction material response is assumed, directly equated to the kinetic dissipation. Internal energy and fracture energy can be demonstrated as negligible. With kinetic dissipation and temporal shock width, the product on the right in Eq. (5), and hence the viscosity A, can be calculated for the Hugoniot data. To relate to the data as displayed in Fig. 2, temporal shock width, ˙ . Hugoniot strain and steady-wave strain rate are related through ε = ετ A relation between strain rate and Hugoniot pressure ε˙ ∼ pn , with n = (α + 2)/ α , is calculated, where the constant of proportionality is dependent on the kinematic viscosity A. Application of the analysis for any of the sets of available compaction Hugoniot data suggests that the viscosity does not vary significantly over the span of the data. An assumption of A equal to a constant leads to the power-law curves shown in Fig. 2, where the power n is provided directly from the compaction Hugoniot parameter α as noted above. That constancy of the product of the shock dissipation energy and the dissipation time, and hence the kinematic viscosity A, merges markedly different data for the shock compaction of granular materials is a remarkable observation. Constancy of the energy-time product forecasts a fourth power dependence of the steady structured wave that is observed in solid materials including metals, compounds, glasses and composites. Here it is shown that constancy of the same energy-time product predicts a starkly different, near-first-power dependence, for steady structured shock waves in the shock compaction of granular solids. Although the underlying responsible physics is not understood, this guiding rule emerging from constancy of the dissipative action provides practical use in modelling dynamic compaction of porous solids.

directed impulse (shock) causing circumferential dynamic straining of the ring-like body until failure and fragmentation occurs. Similarly, fragment size data displayed on the right results for explosive loading (shock) of the hollow steel cylinder samples. Application of Eq. (5) to the spall failure transition relies on knowledge of the left-hand side of the equation, and a dynamic parameter, namely the strain rate, constraining dissipation on the right-hand side. The appropriate acoustic impedance Z is known, and the dissipation constant Γ, through the plane-strain fracture toughness Kc, is available for many materials. With the spall failure transition achieved through elastic tension to failure strain εs, elastic strain energy release partitions into kinetic dissipation and friction (fracture) dissipation. Hence Ek = (1/2)(Z 2/2ρ2 ) εs2 accepting, as before, through dissipation dichotomy comparable levels of kinetic and frictional dissipation. Eq. (5) then yields,

Γ=

1 Z3 2 1 2 εs τ = p τ, 4 ρ2 4Z s

(13)

with the tensile failure stress and strain related through ps = (Z 2/ ρ) εs . Noted, is the p-squared-tau expression familiar to studies of spall failure e.g. [42], as well as other shock transition processes. Relating the time ˙ , Eq. (13) provides an span to spall failure and strain rate through εs = ετ expression for spall strength dependence on strain rate,

ps = (2ZK c2 ε˙)1/3,

(14)

with the dissipation constant and fracture toughness related through 2Γ = ρK c2/ Z 2 . Eq. (14) differs insignificantly from expressions for spall strength that have resulted from earlier theoretical developments e.g. [6,40]. Compare, for example, with the equation for spall strength that is inset in the plot on the left in Fig. 3. Alternatively, in Eq. (5), relate the correlation time span τ to a nominal fracture spacing, or characteristic fragment size, through λs = 2Co τ = 2(Z / ρ) τ . Again, with time and strain rate related through ˙ , relations for the size scale λs resulting from Eq. (14) are, εs = ετ

λs = 2(4Γ/ ρε˙ 2)1/3 = 2( 2 K c / Zε˙)2/3.

(15)

Eq. (15) does not differ significantly from expressions for fragment size developed previously e.g. [6,15,40]. Compare with the relation for fragment size that is inset on the left in Fig. 3. Eq. (15) is applied directly to the remarkable data of Weimer and Rogers [41] on the right in Fig. 3, in which plane-strain fracture toughness was varied through selected heat treatment of several steels, and the Mott fragment size scale parameter determined from exploding cylinder fragmentation testing. This comparison closely matches an earlier calculation based on the same data [15]. It is noteworthy that the responsible energy, identified in that early analytic study as the local kinetic energy [15], is here identified with the correlated kinetic dissipation as expressed on the right in Eq. (2). The present brief discussion begs consideration of the shock-induced failure and fragmentation of hard brittle solids such as glasses and ceramics e.g. [6]. Power-law distributions in fragment particle size that can span several decades are the rule. The correlation statistic pursued here would appear to have application. Parallel relationships of brittle fracture and fragmentation to hydrodynamic turbulence have been noted [43]. Potentially more appropriate to this statistical pursuit, would be application of the alternative correlation structure functions as developed for turbulence by Kolmogorov [44].

5. Closure Constancy of the dissipative action (energy-time product) is an experimentally observed consequence of the several shock failure transitions explored. The steady shock wave structure in the Hugoniot transition in both solid and distended solids, and the dynamic tensile spall fracture of solid materials, are consistent with constancy of the dissipative action over an extended span of dynamic intensity. An explanation for this universal nature to the shock transition is not apparent. There are circumstantial arguments as to why invariance of this action property might be expected. Invariance of, or the constancy of, an action as the state of a physical system is altered has history in other areas of physics. Action invariance appears in the dynamics of periodic systems. (Oscillating pendulums, or a gravitationally bound orbiting body, are classic examples.) In the Hamiltonian description of a periodic system, transformation to an alternative canonical set of

4.3. Structured shock waves and the compaction of distended solids Measurements of Hugoniot pressure versus structured wave strain rate for selected granular materials [8] within the compaction range are shown on the right in Fig. 2. Unlike the fourth power behavior observed in full-density compounds, metals, and mixtures, markedly different power-law character is observed in the shock compression of granular 7

International Journal of Impact Engineering 137 (2020) 103435

D. Grady

interests or personal relationships that could have appeared to influence the work reported in this paper.

momentum-position coordinates yields benefits in describing dynamics of the system. Alternative canonical coordinates appropriate to periodic systems are the action and angle variables. In particular, the alternative momentum coordinate; namely the action variable, is found invariant under adiabatic changes in the periodic system. Noteworthy in this regard is the Planck electromagnetic spectrum, which demonstrates action invariance as is succinctly expressed through Wien's displacement law e.g. [9]. Correspondingly, the equilibrium spectral energy distribution of low-frequency thermal phonons has the same Planck distribution, and similarly demonstrates action invariance through Wien's displacement law [45]. The acoustic phonon field that constitutes the kinetic dissipation is, however, neither adiabatic nor in equilibrium. Nonetheless, that the acoustic phonon field also demonstrates action invariance is in keeping with action invariance of these related periodic systems. Although far from equilibrium, one can suppose that the acoustic phonon field strives towards transient metastable statistical states that reflect the Wien's law statistical distribution. Invariance of the dissipative action remains a topic for further investigation. The verification that the steady structured shock wave is an adiabatic invariant would be a remarkable achievement. The broader purpose of the present paper, however, is the undertaking of an effort to impose a statistical framework on the shock failure transition. Within this statistical framework, the intent is to unify commonalities observed in diverse applications of shock-induced failure in solid matter. The shock failure transition in a solid is heterogeneous on the microscale. The microdynamics of shock failure dissipation are a consequence of the collective activity and interaction of elemental selforganizing dissipative structures. Such shock failure demonstrates universality with power-law descriptions of macroscale strength properties that are applicable over spans of shock intensity. Universality is a consequence of invariance in the failure dynamics of elemental dissipative structures over comparable spans of shock intensity. Failure dissipation within elemental structures is here captured through a constant of dissipation Γ. Dissipation of an elemental structure is dichotomous, expressed through internal friction dissipation and external kinetic dissipation, the latter manifest here principally through acoustic phonon radiation. (It is noted that both mass and momentum diffusion participate in structuring the shock compaction wave.) The intensity of the two dissipation modes are comparable and both determined by the dissipation constant Γ. Kinetic dissipation is quantified through a variant velocity field; namely, the spatial variation of the particle velocity over the time span of the shock failure transition. In particular, the second moment of the variant velocity field specifies the specific kinetic dissipation. Of particular import is the variant velocity correlation function that determines the correlated temporal and spatial span of the kinetic dissipation. The integral of the velocity correlation function provides the fluctuation-dissipation relation with relevance to the shock failure transition. First, the fluctuation-dissipation relation determines the acoustic phonon diffusion, or kinematic viscosity, that accounts for wave structuring in the shock failure event. Second, the fluctuation-dissipation relation determines the critical-state correlated kinetic dissipation necessary to the shock failure transition. Through application of the fluctuation-dissipation relation, as constructed for the shock failure transition, power-law functional relations emerge that are descriptive of experimental observation of shock wave failure, including the fourth power nature of steady structured shock waves in solids, the structure of compaction shock waves in distended materials, and the spall fracture and fragmentation of solid matter.

Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijimpeng.2019.103435. References [1] Grady DE. Principles underlying the fourth power nature of structured shock waves. Proc. APS Topical Group on Shock Compression of Condensed Matter, St. Louis, MO, July 10-14, 2017; AIP Conf. Proc., 1979 2018:070014https://doi.org/10.1063/1. 5044823. 2018. [2] Prigogine I. Introduction to thermodynamics of irreversible processes. 2nd ed. New York: Interscience; 1961. OCLC 219682909. [3] Wang ZQ, Beyerlein IJ, LeSar R. Slip band formation and mobile dislocation density generation in crystals. Phil Mag 2008;88(9):1321–43. [4] Coffey CS. Localization of energy and plastic deformation in crystalline solids during shock or impact. J Appl Phys 1991;70:4248–54. [5] Grady DE, Kipp ME. The growth of unstable thermoplastic shear. J Mech Phys Solids 1987;35(1):95–118. [6] Grady DE. Physics of shock and impact. Bristol, UK: IOP Publishing; 2017. Volume I and II. [7] Mott NF. Brittle fracture in mild steel plates. Engineering 1948;165:16. [8] Grady DE. Unifying role of dissipative action in the dynamic failure of solids. J Appl Phys 2015;117:165905. [9] Reif F., (1965) Fundamentals of statistical and thermal physics, McGraw Hill. [10] Goldstein H. Classical mechanics. Addison-Wesley; 1965. [11] Grady DE. Scattering as a mechanism for structured shock waves in metals. J. Mech Phys Solids 1998;46:2017–32. [12] Mescheryakov YI. Particle velocity non-uniformity and steady-wave propagation. Shock Waves 2017;27(2):291–7. [13] Grady DE. Dissipation in adiabatic shear bands. Mech Materials 1994;17:289–93. [14] Archambeau C.B. and Minster J.B., (1978) Theory of failure as a transition process in a stressed medium, in workshop elastic waves in solids, edited by Achenbach JD, Pao YH, Tiersten HF, Natl. Sci. Found., Washington, DC, p. 196–266. [15] Grady DE. Local inertial effects in dynamic fragmentation. J Appl Phys 1982;53:322–5. [16] Grady DE. Properties of an adiabatic shear-band process zone. J Mech Phys Solids 1992;40:1197–215. [17] Bazant ZP, Caner FC. Impact comminution of solids due to local kinetic energy. J Mech Phys Solids 2014;64(223–235):236–48. [18] Kubo R. The fluctuation-dissipation theorem. Rep Prog Phys 1966;29:255–84. [19] Resibois P, De Leener M. Classical kinetic theory of fluids. Wiley InterScience Publ.; 1977. [20] Einstein A. Investigations on the theory of the brownian movement. Ann Phys [Leipzig] 1905;322:549. [21] William Sutherland. LXXV. A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. Philosophical Mag, Series 1905;6(9):54. [22] Smoluchowski M. Zur kinetischen theorie der brownschen molekularbewegung und der suspensionen. Ann der Phys 1906;326(14):756–80. [23] Nyquist H. Thermal agitation of electric charge in conductors. Phy Rev 1928;32:110–3. [24] Bak P, Tang C, Wiesenfeld K. Self-Organized criticality: explanation of the 1/f noise. Phys Rev Lett 1987;59:381. [25] Watkins NW, Pruessner G, Chapman SC, Crosby NB, Jensen HJ. Twenty five years of self-organized criticality: concepts and controversies. Space Sci Rev 2016;198:3–44. [26] Grady DE. Strain-Rate dependence of the effective viscosity under steady-wave shock compression. Appl Phys Lett 1981;38:825–6. [27] Swegle JW, Grady DE. Shock viscosity and the prediction of shock wave rise times. J Appl Phys 1985;58:692–701. [28] Grady DE. Structured shock waves and the fourth-power law. J Appl Phys 2010;107:013506. [29] Grady DE. Steady-Wave rise-time and spall measurements on uranium (3–15 GPa). In: Murr LE, Staudhammer KP, Meyers MA, editors. Metallurgical applications of shock-wave and high-strain-rate phenomena. Marcel Dekker, Inc.; 1986. p. 763. [30] Hixson RS, Vorthman JE, Gustavsen RL, Zurek AK, Anderson DL, Tonks DL. Spall response of U-NB (6%) alloy. In: Furnish MD, Chhabildas LC, Hixson RS, editors. Shock compression of condensed matter - 1999. New York: Am. Inst. Physics; 2000. p. 489–92. [31] Brown DW, Hackenberg RE, Teter DF, Bourke MA, Thoma D. Aging and deformation of uranium-niobium alloys 30. Los Alamos Science; 2006. Number. [32] Reinhart WD, Chhabildas LC, Vogler TJ. Investigating phase transitions and strength in single-crystal sapphire using shock–reshock loading techniques. Int J Impact Engng 2006;33:655–69. [33] Kanel GI, Nellis WJ, Savinykh AS, Razorenov SV Rajendran AM. Effect of crystalline anisotropy on shock propagation in sapphire. Shock compression of condensed matter. American Institute of Physics; 2009. p. 851–4. [34] Dick J.J., Martinez A.R., Hixson R.S., (1998) Plate impact response of pbx 9501 and its components below two GPa, Los Alamos National Laboratory Rept. LA-13426MS, April. [35] Arnold W. Dynamisches werkstoffverhalten von armco-eisen bei

Declaration of Competing Interest The authors declare that they have no known competing financial 8

International Journal of Impact Engineering 137 (2020) 103435

D. Grady

1988;36(3):353–84. [41] Weimer RJ, Rogers HC. Dynamic fracture phenomena in high‐strength steels. J Appl Phys 1979;50:8025. [42] Tuler FR, Butcher BM. A criterion for the time dependence of dynamic fracture. Int J Frac Mech 1968;4:431–7. [43] Grady DE. Length scales and size distributions in dynamic fragmentation. Int J Fracture 2010;163:85–99. https://doi.org/10.1007/s10704-009-9418-4. [44] Kolmogorov AN. The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers. Doklady Akad Nauk, SSSR 1941;30:301–5. [45] Maasilta I, Minnich AJ. Heat under the microscope. Phys Today 2014;67(8):27.

stosswellenbelastung. Duesseldorf: VDI-Verlag; 1992. [36] Armstrong RW, Arnold W, Zerilli FJ. Dislocation mechanics of copper and iron in high rate deformation tests. J Appl Phys 2009;105:1–7. [37] Sheffield SA, Gustavsen RL, Anderson MU. Shock loading of porous high explosives, in high-pressure shock compression of solids IV. In: Davison L, Horie Y, Shahinpoor M, editors. Springer; 1997. p. 23–61. [38] Cochran S, Banner D. Spall studies in uranium. J Appl Phys 1977;48:2729–39. [39] Grady DE, Olsen ML. A statistical and energy based theory of dynamic fragmentation. Int J Impact Engng 2003;29:293–306. [40] Grady DE. The spall strength of condensed matter. J Mech Phys Solids

9