The constitutive model relates stress to strain, strain rate, internal energy, and damage (degree of fracturing) in a number of equations approximat- ing a given rocks behaviour to differential stress. The strength of rock materials is complex, as revealed by laboratory experiments which have
shown confining pressure, temperature, strain, strain rate and porosity are all factors affecting the critical stress of rock at the onset of failure (yield strength). To accurately model target material behaviour in impact simu- lations, experimentally-derived material strength data is used and fitted to the equations described below.
Rocks can be intact or fractured when external forces act upon them. Intuitively, fractured rock is weaker than intact rock, as blocks within the fractured mass are able to move. The two states are therefore considered separately. For intact rock, the yield strength Yiis formulated by (Lundborg, 1968) Yi= Yo+ P µi 1 + P µi Ym− Yo (3.11) where Yo and Ym are material parameters, P is the pressure and µ is
the coefficient of friction.
A damage parameter, D, is used to define the amount of rock fracturing (Collins et al.,2004). This parameter is a function of plastic strain and varies between 0 (for completely intact, undamaged rock) and 1 (completely frac- tured, damaged rock). The higher the total accumulated strain, the greater the damage to the material. The yield strength of completely damaged rock Yd is
Yd= P µd+ Ydo (3.12)
where µd is the coefficient of friction of the damaged rock and Ydo is a
constant. For partially damaged rock ( 0 < D ≤ 1), yield strength is defined as
Y = (1 − D)Yi+ DYd (3.13)
In the survivability study described in Chapter 4, only the initial contact and compression stage is modelled; damage was therefore not considered as it is irrelevant for impact velocities greater than a few km/s (Collins et al., 2004). As the complete crater formation process was modelled for
the basin-forming impacts described in Chapters 5 and 6 damage was taken into account, as material strength plays an important role in collapse and ultimate structure of impact craters.
Material weakening - temperature dependence
Geological materials lose strength as their temperature increases, with all shear strength lost upon melting (e.g.Jaeger & Cook,1969). In iSALE this relation is approximated by the equation (after Ohnaka,1995)
Yt/Y = tanh[ξ(Tm/T ) − 1] (3.14)
where Yt is the material’s yield strength, Y is material strength at low
temperature (its ‘cold’ strength), T is ambient temperature, Tm is the melt
temperature and ξ is a material constant. As the ambient temperature approaches the melt temperature the bracket on the right hand side tends to zero, thus the material’s strength drops to zero. Melt temperature is calculated using the Simon approximation (Poirier,1991)
Tm = T0(P/a + 1)1/c (3.15)
where Tm is the melt temperature at pressure P, T0 is the melt temper-
ature at normal pressure, and a and c are material constants, derived from fits to experimental data.
Additional material weakening
In large-scale crater formation, it is generally assumed that some additional weakening mechanism(s) act(s). The presence of central peaks and rings in large-scale craters suggest some kind of fluid-like motion within the collapse phase. However, as the morphology of these features is retained, the weak- ening must act over a timescale similar to that of crater collapse (Melosh & Ivanov,1999).
Acoustic fluidisation
Acoustic fluidisation, conceived by Melosh (1979), outlines a method by which impact rock debris can behave in a fluid-like manner on a timescale suitable for crater collapse. The fundamental idea of acoustic fluidisation is that, for a coulomb material, yield stress is a linear function of overburden pressure (to a first order approximation) with a proportional coefficient (the coefficient of internal friction). Under normal conditions, material at depth under too high an overburden pressure will not be able to fail and flow.
Impact events create seismic waves and induce ground shaking; this shak- ing may well affect debris around an impact crater (Melosh & Gaffney,1983). Acoustic fluidisation therefore suggests that the material, normally under too high a pressure to fail, may flow if vibrations of a random seismic wave field, generated by the impact shock, propagating through impact-fractured rocks temporarily reduce overburden pressure. Explosion tests have shown stress fluctuations near craters can exceed the overburden pressure (Gaffney & Melosh, 1982); if reduced below a certain threshold, material is able to flow with a viscosity. As vibrations dissipate, the fluctuations in pressure decrease, and once overburden pressure exceeds the local pressure, material is prevented from moving, becoming frozen in place. The strain rate in the acoustically fluidised material can be calculated from
˙ = τ ρλβ 2 erf c[χ]− 1 −1 (3.16) where τ is the shear stress, ρ is the density, λ is the dominant wavelength of the acoustic field, and β is the s-wave velocity in the debris; erfc is the complementary error function; χ = (1 − Ω)/Σ, where Ω is the dimensionless driving force (τ/τstatic) which varies from 0 to 1; τstatic= µp and represents
the force needed to initiate failure in the absence of vibration, Σ is the amplitude of vibration (also dimensionless), where Σ = σ/p. χ varies from 0-1.
For acoustic fluidisation to operate λ d, where d is the grain size (Melosh & Ivanov,1999). However, a quantitative prediction of the rheology associated with the process of acoustic fluidisation cannot be made without further assumptions as λ cannot be predicted. The size at which acoustic fluidisation acts has been shown to have a 1/g dependence, which helps to explain the simple-to-complex crater transition and the existence of some temporary weakening mechanism (Melosh & Ivanov,1999).
The block model
Acoustic fluidisation is implemented into iSALE by the block model (Melosh & Ivanov, 1999; Wünnemann & Ivanov, 2003). This is a one dimensional approximation of acoustic fluidisation in which rocks deform as a system of discrete blocks, rather than as a continuum (which the original acoustic
Figure 3.3: The block model: (a) the forces acting on the theoretical block (see text for description), (b) the acoustic waves alter the pressure acting on the block, once the pressure drops below the static pressure µp, the block will slide (after
fluidisation model assumes). The block model can be described visually (Figure 3.3) as a block sliding along a flat surface. The block is subjected to a normal force, the overburden pressure, p. The dry frictional force acting on the block is µp. Acoustic energy sets the block oscillating in a vertical manner with a period, T , and amplitude, Sv, resulting in a sinusoidal variation in
the normal stress. The block will remain stationary if p + Svsin(2πt/T ) >
traction (µp). Strain rate ˙ can be shown to equal δ/T h, where δ is the block displacement per cycle, and h is the block characteristic length. The block model therefore gives the strain rate as:
˙ = (τ − YB)T 2π2ρh2 r 1 + χ 1 − χ− χ 1 − χcos −1 χ cos−1χ (3.17)
where χ = (1 − Ω/Σ) and Σ = Sv/p. For the block to slide, traction must be
greater than YB which is equal to µ(p − Sv), at which point it will flow with
a viscosity, η = 2πh2/ρT. The block’s velocity will drop to zero as soon as
the frictional force exceeds the driving force.
Use of the block model and appropriate parameterisations found it could be used to produce two main features of impact crater collapse: a critical diameter at which complex craters begin to form, and a gradual change in crater morphology with increasing size (e.g. central peaks to peak rings) (Ivanov & Kostuchenko, 1998; Melosh & Ivanov, 1999). Ivanov & Kos- tuchenko (1998) assumed the oscillation of the vibration intensity decayed as 1/r2 and followed an exponential decay. Therefore, in simple craters, the
rapid decay of vibrations prevented uplift, hence why a critical threshold determines whether block oscillations facilitate uplift.
An advantage of the block model is the characteristic block length, h, can be determined from observations; drilling at 40 km-scale terrestrial craters showed block sizes of, on average, 100 m (Ivanov et al., 1996), whereas in the original acoustic fluidisation model, the dominant wavelength was unde- termined). However the time period, T , has be be determined from other assumptions or observations. Figure 3.4 shows plots normalised strain rate against normalised shear stress for the flow laws of the original acoustic flu- idisation modelMelosh(1979) and the block model (Melosh & Ivanov,1999); the different methods produce similar results as the ultimate mechanisms for fluidisation in both models is similar.
Figure 3.4: Comparison of rheological behaviour predicted by acoustic fluidisation and the block model (after Melosh & Ivanov, 1999). On the y-axis strain rate has been normalised by maximum strain rate. See text for definition of Ω. Σ is the dimensionless ratio between the amplitude of vibrations and the overburden pressure. The two differing methods produce similar rheological behaviour.
Effective viscosity
Within this thesis, the melt temperature refers to the solidus, the point at which a material begins to transition from completely solid to liquid. The material will not become a complete liquid until reaching the liquidus. Once a material exceeds its solidus, solid clasts within the material melt. This con- sequently lowers the material’s viscosity - its resistance to flow (shear stress). Experiments have shown viscosity to alter by many orders of magnitude (in a sigmoidal fashion) between the solidus and liquidus (e.g. Caricchi et al.,
2007;Costa et al.,2009). Using Equation 3.14, a super-solidus material will not possess any shear strength and therefore behave as an inviscid fluid (have no resistance to flow). However a super-solidus, sub-liquidus material will be a mixture of solid clasts and melt, with the solid clasts providing some resistance to shear. To account for this shear resistance, an effective viscosity was assigned to super-solidus material; see section 5.3.5 for a discussion of this.
0 50 100 150 200 250 300 350 400 -1000 -950 -900 -850 -800 Pressure (GPa) Depth (m) a1=0.0, a2=0.0 a1=1.0, a2=0.0 a1=1.0, a2=1.0 a1=1.0, a2=0.2 a1=0.5, a2=0.5
Figure 3.5: The effect of artificial viscosity (values a1 and a2) on a shock front.