Mechanics of Thermal Contraction Cracks and Ice-Wedge Polygons in Permafrost

Abstract

L e f f i n g w e l l 's contraction-crack theory of ice-wedge polygons in permafrost has been examined from the point of view of mechanics. A nonlinear viscoelastic model of thermal stress in permafrost leads to results consistent with the theory within the limits of existing information on polygon dimensions, crack depths, temperature, and mechanical properties of ice and permafrost. Stresses that cause crack ing are evidently generated not only by low temperature but also by rapid cooling. The size of the polygons can be explained in terms of the stress-perturbation due to a single crack and the distribution of mechanical flaws. The polygonal patterns can be classified accord ing to whether or not the intersections are predominantly orthog onal. It is proposed that orthogonal polygons evolve by progres sive subdivision, nonorthogonal ones by successive branching of cracks attaining high propagation velocities.Much of the discussion is general and applies directly to other types of contraction-crack polygons such as columnar basalt joints and mud cracks. 1 List of Symbols E i Principal components of distortional strain tensor i f Principal components of distortional strain-rate tensor Principal components of distortional stress tensor Principal components of distortional stress-rate tensor (x , % > tz Principal components of thermal strain Tx , Ty , Tj Principal components of thermal stress © Temperature referred to mean annual temperature as zero a Linear coefficient of thermal expansion K Elastic bulk modulus Y Young's modulus fj, Elastic shear modulus v Poisson's ratio rj Quasi-viscous parameter rj Viscosity X, ¡jf, rf Viscoelastic parameters t Time $ Time required for stress to reach 90 per cent of equilibrium value under a step increase of strain rate from zero stresst Horizontal thermal stress ( t " , or t z) T* = t ( x ) -pgx Sum of horizontal thermal stress and lithostatic stress ro = r(0 ) = r*(0 ) Horizontal thermal stress at ground surface, x = 0 Sx , Sv , S a , Sxy Tensor components of stress perturbation due to a long straight crack (collectively : Sti) Say = Sy(0, y) ^-component of stress-relief at ground surface (x = 0) Soz = S s(0, y) ^-component of stress-relief at ground surface (x = 0) S Combined horizontal stress-relief of all pre-existing cracks a " Stress depth," thickness of surficial layer with uniform stress b Crack depth p Density g Acceleration of gravity k Crack-edge stress-intensity factor k© Contribution to crack-edge intensity factor due to thermal stress np Contribution to crack-edge intensity factor due to weight of overburden G Crack extension force Gc Value of G necessary for initiation of fast fracture Go Value of G necessary for sustained crack propagation K0 Value of k necessary for sustained crack propagation A Intensity parameter for step approximation to t * ( x ) B Intensity parameter for linear approximation to t * ( x ) y = K /A y /b or k/ B \ / b Normalized crack-edge stress-intensity factor N Nominal tensile strength N -e Tensile strength of a flaw SOME M ECHANICAL ...

Lateglacial and Holocene isotopic and environmental history of northern coastal Alaska – Results from a buried ice-wedge system at Barrow

“…Therefore, we conclude first, that the radiocarbon dates of organic matter in the ice-wedge system are valid and second, that the ice wedges grew continuously and regularly with older ages at the edges and younger ages in the middle of the ice wedge. The symmetry of the stable isotope profiles nicely confirms the hypothesis of icewedge growth introduced by Lachenbruch (1962) and Dostovalov and Popov (1966).…”

“…The remaining land surface is composed of tundra covered drained-lake basins (up to 50%), and polygonal ground. The latter is the result of intensive icewedge growth in the upper 2e4 m. The ice wedges grow over many thousands of years as snow melt water penetrate thermal contraction cracks that develop during the winter as the ground contracts (Lachenbruch, 1962). The soils, lake-basin sediments, and the upper permafrost zone contain relatively large amounts of organic carbon (Hinkel et al, 2003).…”

Starch columns: Analog model for basalt columns

“…In a more extended starch layer, several first-generation cracks are generated. For some time, they continually subdivide the layer into plates of decreasing size, following the orthogonality rule; these properties are well-known [e.g., Lachenbruch, 1962;Walker, 1986]. Rupturing by firstgeneration cracks stops when in the interior of plates contraction stresses remain below the tensile strength, which itself increases with time.…”

“…According to Lachenbruch [1962], who refers to earlier fracture mechanics literature [e.g., Irwin, 1958], the mechanism causing polygons is accelerated horizontal crack lengthen-'ing, until crack tip velocities of about half the shear wave velocity are attained. Then branching into two cracks occurs whose directions differ from the direction of the generating crack by about 60 ø .…”

Polygon pattern geomorphometry on Svalbard (Norway) and western Utopia Planitia (Mars) using high-resolution stereo remote-sensing data

“…These polygonal networks are formed by thermal contraction fissures (Lachenbruch, 1962(Lachenbruch, , 1966 after an initial crack is reactivated recurrently (Mackay, 1974(Mackay, , 1992 and widened through subsequent filling by water, sand, or soil (Péwé, 1959;Black, 1976;Sletten et al, 2003;French, 2007). The frozen ground cracks under volumetric tension due to long cold periods and a severe soil temperature drop in winter (Yershov, 2004;Christiansen, 2005;Fortier and Allard, 2005).…”