Spall Fracture: The Far Face Blows Off Before the Near Face Breaks

A Flat Plate, a Gas Gun and a Square Pressure Pulse

Spall studies begin with the cleanest experiment in shock physics: the symmetric plate impact. A single-stage light-gas gun accelerates a flat flyer disc down a 30-100 mm bore at anywhere from ~100 to ~1,200 m/s; two-stage light-gas guns reach 8 km/s, and magnetically launched flyers on Sandia's Z machine have reached ~45 km/s. The flyer strikes a parallel target of the same material, aligned to within a few milliradians. Because both sides are identical, the impact plane is a plane of symmetry, so each side must move at half the closing speed and the particle velocity behind the shock is simply u_p = V_imp/2.

Everything else follows from the Rankine-Hugoniot jump conditions plus one stubborn empirical fact: over a wide range, the shock speed in a metal is linear in particle velocity, U_s = c₀ + s·u_p. For 6061-T6 aluminium (ρ₀ = 2,703 kg/m³, c₀ = 5,350 m/s, s = 1.34) a 500 m/s symmetric impact gives u_p = 250 m/s, U_s = 5,350 + 1.34 × 250 = 5,685 m/s, and a shock stress

P = ρ₀ U_s u_p = 2,703 × 5,685 × 250 ≈ 3.8 GPa — roughly 38,000 atmospheres, held for about a microsecond.

The front is not one wave but two. An elastic precursor runs ahead at the longitudinal sound speed c_L ≈ 6.4 km/s in aluminium (~5.9 km/s in steel), carrying only as much stress as uniaxial-strain elasticity allows — the Hugoniot elastic limit, σ_HEL = Y(1−ν)/(1−2ν) ≈ 0.5-0.6 GPa for 6061-T6. The plastic shock follows just behind at 5.7 km/s. A 6 mm target is therefore crossed in about 1.05 µs, and the duration of the compressive plateau is set by the round trip inside the flyer, 2h_f/U_s — about 1.05 µs for a 3 mm flyer.

Why a Free Surface Turns a Push Into a Pull

A stress wave meeting an interface divides according to the mismatch in acoustic impedance, Z = ρc. The reflected stress amplitude obeys σ_r/σ_i = (Z₂ − Z₁)/(Z₂ + Z₁). Aluminium has Z = ρ₀c₀ ≈ 1.45 × 10⁷ kg m⁻² s⁻¹; the air or vacuum behind the target has Z ≈ 4 × 10², which is zero to five significant figures. So Z₂ = 0, the reflection coefficient is exactly −1, and the compressive pulse comes back as a tensile pulse of equal magnitude. Stated in velocities instead, the coefficient is +1: incident and reflected particle velocities add, and the rear face leaps forward at

u_fs ≈ 2u_p ≈ 500 m/s — for a symmetric impact, the free surface ends up moving at essentially the original impact velocity. (At finite amplitude u_fs exceeds 2u_p by a few percent, because the release path off the Hugoniot is not quite its mirror image; the small excess is a standard correction.)

Tension at the surface itself does nothing — the surface is already free, and free surfaces cannot break. The damage happens where two release fans cross. The flyer is finite, so a rarefaction launched from the flyer's own rear free surface chases the shock forward through the target. Meanwhile the target's rear free surface sends a rarefaction backward. In the region where they overlap, both fans have already unloaded the material to zero stress and each keeps pulling: the superposition is net triaxial tension in the interior, with no nearby free surface to relieve it. For a symmetric impact with a target more than about twice the flyer thickness, the crossing region sits roughly one flyer-thickness in from the target's back face.

Two details make this lethal rather than merely interesting. First, the loading is uniaxial strain: lateral inertia forbids the material from contracting sideways on a microsecond timescale, so a metal that would neck down and stretch to 20% elongation in a tensile machine has no such escape route. Second, the tension is applied in nanoseconds, which is where dynamic strength comes in.

Dynamic Tensile Strength: Why Metal Is Stronger for a Microsecond

The spall strength σ_sp is the tensile stress the material sustains before it separates. In ductile metals, failure is nucleation-growth-coalescence of voids: microscopic cavities open at second-phase particles, inclusions and grain boundaries, expand by viscoplastic flow of the surrounding shell, and merge. Carroll and Holt's 1972 hollow-sphere pore model, carried over to tensile void growth by J. N. Johnson in 1981, gives a growth rate of the form dR/dt ∝ R(σ − σ_g)/η, with η an effective viscosity; the Curran-Seaman-Shockey NAG models developed at SRI International through the 1970s and 80s turned that into an engineering damage law. In brittle materials — glass, ceramics, very hard steels — the same tension instead opens penny-shaped cleavage cracks. Either way the process runs to completion in tens of nanoseconds and leaves a flat, disc-shaped internal fracture: the scab, which departs intact.

The striking number is how strong the material is. Annealed OFHC copper with a quasi-static UTS near 220 MPa spalls at 1.2-1.5 GPa. 6061-T6 aluminium at 310 MPa static spalls at 1.0-1.5 GPa. Armour steels reach 2-3 GPa. The factor of three to five comes from strain rate: a laboratory tensile test runs at ~10⁻³ s⁻¹, while the release fan in a plate impact imposes 10⁴-10⁷ s⁻¹. Void nucleation and growth are kinetic processes; if you pull faster than the voids can grow, the stress overshoots.

Grady's 1988 energy-balance model makes this quantitative. Equating the kinetic energy released by fragmentation with the fracture surface energy gives

σ_sp = (3 ρ₀ c₀ K_Ic² ε̇)^1/3

For aluminium with K_Ic ≈ 30 MPa·m^1/2 and ε̇ = 10⁵ s⁻¹ this evaluates to about 1.6 GPa — squarely in the measured band. The cube-root scaling captures the trend, but it over-predicts at extreme rates: measured exponents across 10⁴ to 10⁹ s⁻¹ are closer to 0.1-0.25 than to 0.33, and reconciling continuum, mesoscale and atomistic descriptions across that span is unfinished business.

Reading the Pullback: VISAR, PDV and What the Trace Says

You cannot watch the inside of the plate, but you can watch the rear surface, and it tells you everything. Louis Barker and Ray Hollenbach at Sandia published VISAR — the Velocity Interferometer System for Any Reflector — in the Journal of Applied Physics in 1972. Light reflected from the moving free surface is split; one leg is delayed by an etalon of delay τ and the two are recombined, so fringes count velocity rather than displacement. The velocity per fringe is λ/(2τ(1+δ)), with δ the etalon dispersion correction, giving nanosecond-scale time resolution on velocities up to ~10 km/s at percent-level precision.

The trace has a characteristic shape. It jumps to the plateau u_fs, falls as the release arrives — and then, instead of continuing to zero, it reverses and climbs again. That reversal is the spall pullback. When the material separates internally, it creates a brand-new free surface inside the plate, which immediately radiates a compression wave into the scab and re-accelerates it. The depth of the dip, Δu_fs, measured from the first maximum to the minimum, gives the strength directly in the acoustic approximation:

σ_sp ≈ ½ ρ₀ c_b Δu_fs

For aluminium with Δu_fs = 160 m/s: ½ × 2,703 × 5,350 × 160 ≈ 1.16 GPa. Corrections for elastic-plastic release (c_L ≠ c_b) and for finite wave rise time shift this by 10-20%, and different published corrections disagree — a real source of scatter in the literature. The subsequent ringing gives a bonus: the reverberation period 2h_s/c_L is 313 ns for a 1 mm aluminium scab, so the trace reports the scab thickness without anyone cutting the sample.

Modern practice adds photonic Doppler velocimetry (Strand and colleagues, Lawrence Livermore, 2006), a 1550 nm telecom-fibre heterodyne technique that is cheaper, multi-point and free of fringe-count ambiguity. Soft-recovery fixtures with momentum-trapping rings let metallurgists retrieve incipient spall — arrays of voids frozen mid-growth. And in-situ X-ray phase-contrast imaging at the Dynamic Compression Sector of the Advanced Photon Source (Argonne, operating since 2016), at LCLS-MEC and at the European XFEL now resolves micron-scale voids while they grow, alongside 800 MeV proton radiography at LANSCE.

From Hopkinson's Bar to Martian Meteorites

Bertram Hopkinson demonstrated the effect in 1914 (Philosophical Transactions of the Royal Society A 213, 437), measuring pressure pulses from guncotton detonations and rifle bullets. A short cylindrical pellet was held on the end of a long steel bar by nothing but a film of grease; when the pulse reflected off the free end, the pellet flew away carrying the momentum of the leading part of the wave. That is spall made mechanical and reversible — the same reflection, but at a joint weak enough to open without fracturing metal. Hopkinson was killed flying with the Royal Flying Corps in 1918. John S. Rinehart at the US Naval Ordnance Test Station quantified scabbing in 1951 (J. Appl. Phys. 22, 555), proposing a critical-normal-stress criterion and, the following year, explaining multiple scabbing when the pulse is long enough to spall a second and third layer.

The weaponised version is HESH — High Explosive Squash Head, a British design of the 1940s. A thin-walled shell of plastic explosive flattens against armour and a base fuze detonates it a moment later, once the charge has spread. Nothing penetrates; the shock enters the plate, reflects off the inner face, and throws a scab tens of centimetres across and anywhere from hundreds of grams to several kilograms in mass into the crew compartment at 100+ m/s. The 165 mm L9A1 demolition gun of the Centurion AVRE and the 120 mm L31 round for Chieftain and Challenger are the canonical examples. The countermeasures follow straight from the physics: spaced and composite armour breaks the pulse at every impedance mismatch, and spall liners of aramid (Kevlar), UHMWPE (Dyneema) or glass-reinforced plastic bonded to the hull interior of vehicles such as the Bradley and Stryker do not stop the wave at all — they catch the fragments it makes.

The most consequential natural case is planetary. H. Jay Melosh showed in 1984-85 (Icarus 59, 234) that near-surface spallation during a large impact ejects thin plates of near-surface rock at above escape velocity while subjecting them to only modest shock — tens of GPa — because the free-surface release cancels most of the compression before the material leaves. That is how the shergottite-nakhlite-chassignite meteorites and ALH 84001 got off Mars with their igneous textures intact. The same physics runs constructively in laser shock peening, where a confined plasma drives a GPa shock into a turbine blade to leave beneficial compressive residual stress — push the energy too high and the far face spalls.

Look-alikes, Limits and Open Questions

Several unrelated processes share the word. Nuclear spallation is GeV-class protons — 800 MeV at ISIS and LANSCE, ~1 GeV at the SNS, 2 GeV at the European Spallation Source — knocking of order twenty to thirty neutrons per proton out of tungsten or mercury nuclei; the shared name traces to an old Germanic root meaning to split, but the physics is entirely different. Concrete spalling in a fire is free pore water flashing to steam: pore pressure of a few MPa plus a steep thermal gradient pops the cover off over minutes, a thermo-hydraulic process with no wave in it. Corrosion spalling is rust occupying two to six times the volume of the steel it replaces and jacking off the cover over years. Bearing and rail spalling is rolling-contact fatigue: subsurface Hertzian shear, millions of cycles. Only the first — shock-driven — case is spall fracture in the sense used here.

What remains genuinely unsettled:

• Extracting a number from a trace. The acoustic pullback formula assumes elastic release and a symmetric wave; real elastic-plastic release and finite rise times mean published σ_sp values for the same alloy can differ by tens of percent depending on the correction applied.
• Rate scaling. No single model spans 10³ to 10¹⁰ s⁻¹. Molecular dynamics at 10⁹-10¹⁰ s⁻¹ reaches the ideal strength (of order E/10, ~7 GPa in aluminium) but cannot be run at experimental rates; continuum damage models fitted at 10⁵ s⁻¹ do not extrapolate.
• Temperature and melting. Spall strength falls steeply with temperature and collapses toward zero as the release path approaches the melt line, where spall becomes cavitation in a liquid. Exactly where that crossover sits, and how it behaves in a partially molten mixture, is contested.
• Microstructure. Grain size, crystallographic texture and inclusion statistics dominate nucleation, and additively manufactured alloys, high-entropy alloys and laser-welded joints spall differently from wrought stock, with no predictive model yet.
• Phase transitions. Iron's α-to-ε transition at 13 GPa splits the front into three waves, so the release structure and the tension history are not what a single-Hugoniot analysis assumes — a standing complication for every steel experiment above that pressure.

Dynamic (spall) tensile strength versus quasi-static strength, with the bulk sound speed used to convert a measured pullback into a stress.

Material	Bulk sound speed c₀ (km/s)	Spall strength σ_sp (GPa)	Quasi-static UTS (MPa)
Annealed OFHC copper	3.94	~1.2-1.5	~220
6061-T6 aluminium	5.35	~1.0-1.5	~310
4340 / rolled homogeneous armour steel	~4.6	~2-3	~1,100
Tantalum	3.41	~4-6	~250-400
PMMA (Perspex)	2.60	~0.15-0.2	~70
Single-crystal aluminium, laser-driven (~10⁹ s⁻¹)	5.35	~4-9	~310 (polycrystal)