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Cratering as a Geological Process Part 1 ： (a) Simple and complex craters; (b) Fundamental concepts of stress waves, plastic waves, and shock waves; (c) Impact and crater modifying processes

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Importance of Crater Studies: 1.Principal process in shaping planetary surfaces. 2.Principal means of determining relative and for some planetary bodies the absolute ages of planetary surfaces. 3.Major impact events may have affected the tectonic, chemical and biological evolution of planetary bodies (e.g., initiation of plate tectonics; alternate mantle convection; extensive melting in the mantle; removal mantle and crustal materials of target planets during major impacts, extinction and possible delivery of organism of planets). 4.Crater morphology may tell us about the rheological structures of the crust and mantle of planetary bodies. 5.Large impacts may have affected planetary rotations and orbits.

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Basic Classification: (1) Simple craters: strength controlled formation process with smooth bowl shapes; relatively higher depth-to-diameter ratios. (2) Complex craters: gravity-dominated modification process, clear sign of wall and floor modifications expressed as central peaks, peak-ring structures; relatively low depth-to-diameter ratios (i.e., relatively shallower basins than simple craters). (3) Multi-ring basins: having peaks in concentric rings on flat floors.

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Moltke Crater (D = 7km): Simple crater with a bowl- shaped interior and smooth walls. Such craters typically have depths that are about 20 percent of their diameters (Apollo 10 photograph AS10-29-4324.)

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Tycho crater (D = 83 km): Complex crater with terraced rim and a central peak.

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Bessel Crater (D = 16 km h = 2km) A transitional-type crater between simple and complex shapes. Slumping of material from the inner part of the crater rim destroyed the bowl-shaped structure seen in smaller craters and produced a flatter, shallower floor. However, wall terraces and a central peak have not developed. (Part of Apollo 15 Panoramic photograph AS15-9328.)

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Mare Orientale (D = 930 km) A lunar multiring basin An outer ring has a D = 930 km Three inner rings with D= 620, 480, and 320 km. Radial striations in lower right may be related to low-angle ejection of large blocks of excavated material. Young lava flow

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Transition size from simple to complex craters on various planetary bodies: (1) Europa: 5 km (2) Mars: 8-10 km (3) Moon: 15-20 km (4) Venus: No craters with diameters < 10 km are scarce, possibly due to thick atmosphere; dominantly complex craters and multi-ringed basins. (5) Earth: 2-4 km (6) Mercury: 10 km

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Simple-to-complex crater transition occurs when the yield strength is related to the gravity (g), density (rho), transient crater depth (h), and a constant c that is less than 1: or Where D tr is the transient crater diameter that can be related to the final crater diameter by That is, D is proportional to yield strength (Y) and inversely proportionally to density (rho) and gravity (g).

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Transverse and longitudinal waves are related to bulk modulus K 0, shear modulus , and density . Wave-induced longitudinal and transverse or perpendicular stress components are Poisson’s ratio = U L is particle velocity C is wave velocity

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Relative importance of longitudinal and transverse waves: Transverse waves are not important in the cratering process, because the shear strength of materials limits the strength of the wave. The strength of the longitudinal waves have no limit, the strength of compression has no upper bound. In most cratering modeling, transverse waves are neglected.

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Compressional and tensional waves are converted at a free surface Free surfaces require both normal and shear stresses are zero, but the particle velocity can be non zero. U L at the free surfaces for compressive wave is: U L = L /C L For tensional waves, L and C L have the opposite sign, and thus U L = L /C L Thus, at the interface, U L is doubled. Deformation generated by this process is called “spalling”.

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Three stages of cratering: (1) Contact and compression stage (initiation of shock waves) (2) Excavation stage (shockwave expansion and attenuation; crater growth; ejection of impactor and target materials) (3) Post-impact modification

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Reflection at an interface from high-velocity material to low- velocity material: Reflected wave is tensile waves and the rest continues into the low-velocity material as compressive wave.

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Reflection at an interface from low- velocity material to high- velocity material: Reflected wave is compressive wave and the rest continues into the high-velocity material as compressive wave.

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Plastic Yielding at “HEL”— the Hugoniot Elastic Limit When stress in the stress wave reached the plastic limit, irreversible deformation will occur. This plastic yield strength affects both the speed and shape of the stress wave. The onset of this behavior is indicated by a characteristic kind in the Hugoniot P-V plot. The corresponding pressure is known as the Hugoniot Elastic Limit (HEL). In the continuum and fracture mechanics sense, when the differential stress L – p = - Y, Y is the yield strength, the material begins to experience “plastic flow”. Shear stress = - ( L – p )/2 Pressure P = - ( L + 2 p )/3

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HEL

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At the failure point, when = - Y/2, the longitudinal stress equals to HEL, that is

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Idealized cross section of a simple crater

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1.Once the wave-induced stress reaches HEL, the shear stress = - ( L – p )/2 remains constant, and thus the increase in L and p must also maintain in such a way that its difference is the same as -2 . 2.When P is much greater than , we neglect the differential stress term and corresponding stress wave becomes strong pressure wave.

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Cautions for the HEL: 1.For porous medium, there might be two HEL points, one for the collapse of the pores, and the other for the onset of ductile flow. 2.Yield strength may not be constant, but a pressure- dependent envelope. 3.Yield strength may be rate-dependent.

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Elastic Wave: L < HEL Longitudinal wave depends on both bulk and shear muduli Plastic Wave: L = HEL Longitudinal wave depends almost completely on the bulk modulus. The wave propagates much slower than the elastic wave and at a speed of the “bulk wave speed” defined by C B = [K(P)/ 0 ] 1/2 Where bulk modulus increases with pressure. Thus, the bulk speed of plastic wave is much higher under high-pressure condition.

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This segment is called strong or shock wave, which is a plastic wave travels faster than elastic wave

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Strong compressive waves: The Hugoniot Equations Conservation of mass Conservation of momentum Conservation energy

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P = P(V, E) is equation of state for shock waves. There are two ways to represent the equation of state for shock waves (P-V and U-u p plots) Shock pressure as a function of specific volume Shock wave velocity as a function of particle velocity

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Release wave or rarefaction wave 1.The high-pressure state induced by an impact is transient, ranging from 10- 3 to 10 -1 sec for projectile of 10 m and 1 km size. 2.The high-pressure in a shock wave is relieved by the propagation of rarefaction, or release waves from free surfaces into the shocked materials. This type of wave from strong compression generally moves faster than the shock wave and is proportional to the slope of the adiabatic release curve on the P-V diagram

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Stage 1: Contact and compression (shockwave generation and projectile deformation): this stage only lasts a few seconds. Rarefaction waves cause projectile to transform into vapor and melts instantaneously. From O’Keefe and Ahrens (1975) Impact of 46-km-diameter projectile at a speed of 15 km/s 1 s after the impact.

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Formation of simple craters

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Examples of Complex craters

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Peak-ring crater on the Moon (Schrodinger crater, D = 320 km)

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Mare Orientale (D = 930 km) A lunar multiring basin An outer ring has a D = 930 km Three inner rings with D= 620, 480, and 320 km. Radial striations in lower right may be related to low-angle ejection of large blocks of excavated material. Young lava flow

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c: Cohesive strength

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Isostatic adjustment after a large impact removing crust and uplift mantle

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