Principal Strain Rate Research Articles

Strain accumulation in western Washington

The Juan de Fuca plate is subducted beneath the North American plate off the coast of Washington at a rate of about 40 mm/yr N68°E. The average principal strain rates (extension reckoned positive) measured in northwestern Washington are as follows: Olympic peninsula 25 km south of Port Angeles from 1982 through 1990, and and near Seattle from 1972 through 1985, and . Both strain measurements are consistent with uniaxial contraction in the direction of plate convergence. Uplift rates inferred from tide gage recordings are about 4 mm/yr on the Pacific coast and near 0 mm/yr farther inland near Seattle. These deformation rates are consistent with a model of the Cascadia subduction zone in which the plate interface beneath the continental slope and outer continental shelf is locked but free to slip farther landward. The limited downdip extent of the locked segment of the plate interface is consistent with a shallow depth (∼20 km) of the isotherm (∼450°C) that defines the brittle‐ductile transition. Small thrust events diagnostic of seismic subduction should then occur only offshore and at shallow depths. The principal strain rates measured from 1972 through 1983 in the back arc region near Richland, Washington, are and .

Ice‐ocean interaction on Ronne Ice Shelf, Antarctica

Detailed glaciological studies have been completed at 28 sites lying on an approximate flow line, extending 760 km across Ronne Ice Shelf. Parameters measured at each location include ice velocity, thickness, principal strain rates, surface elevation, temperature, and accumulation rate. The data have been used in a steady state model to derive the basal mass flux and the temperature profile with depth at each site, from the principles of mass and energy conservation. These calculations indicate basal melting in excess of 1 m yr−1 over the first 100 km of the flow line downstream of the grounding line, where the ice shelf is between 1200 m and 1600 m thick. The maximum melt rates in this region occur near the inland margin and exceed 4 m yr−1. Melting continues at a lesser rate over the next 200 km before freezing commences. Freezing then dominates up to the final 100 km before the ice front, resulting in the accumulation of a layer of basal sea ice up to 50 m thick. This is rapidly removed as melt rates increase to over 6 m yr−1 at the ice front. The cumulative effect of this pattern of basal accumulation and ablation is the wastage by melting of 85% of the mass discharged across the grounding line before it reaches the ice front. Supporting evidence for a layer of saline ice underlying the ice shelf is provided by the strength of basal radar reflections, observed during radio echo sounding of ice thickness. In the region where mass balance calculations suggest an accretion of basal sea ice, reflection coefficients are consistently low, ranging from −6 dB to −38 dB. Most of these weak reflections are believed to originate from the true base of the ice shelf, the additional energy loss being the result of increased attenuation of the radar signal within the saline layer. The derived pattern of basal melting and freezing is consistent with a simple model of sub‐ice‐shelf oceanic circulation, involving a deep thermohaline convection cell. Dense, saline water, which is formed over the continental shelf during winter when the sea surface freezes, drains into the deepest parts of the sub‐ice‐shelf cavity. At the inland margin, where this water mass comes into contact with the ice shelf, its temperature is 1°C above the local pressure freezing point. Melting of ice results, producing a buoyant outflow of cold, relatively fresh water, along the ice shelf base. Basal freezing occurs towards the ice front, where the ascending water becomes supercooled. This circulation has important implications for the production of Antarctic Bottom Water and for the response of the ice shelf to driving stresses, through the temperature dependent viscosity of ice.

Three-dimensional folding of an embedded viscous layer in pure shear

A thick-plate analysis is carried out to first-order in interface slope for three-dimensional folding of a single linear viscous layer of thickness H and viscosity η embedded in a uniform viscous medium with viscosity η 1, Layer and medium are subject to a basic state of homogeneous pure shear with principal directions of strain-rate x and y lying in the plane of the layer. The solution is explicitly carried out for a fold perturbation in interface shape that is symmetric with respect to the principal strain-rate axes, f cos (lx) cos (my), but it is shown to apply to an arbitrary perturbation in the x,y plane. The growth rate of the perturbation is found to be ▪ are the principal strain-rates of the basic state. The wavenumber of the most rapidly growing perturbation, λ d , is the same as that obtained for a cylindrical perturbation ( m = 0) in a basic state of plane strain ϵ ̄ yy = 0) . For maximum rate of shortening parallel to x, the cylindrical fold form with axis normal to x, m/ l = 0, grows most rapidly for any ratio of ϵ ̄ yy/ ϵ ̄ xx < 1. If ϵ ̄ yy = ϵ ̄ xx all fold forms grow at the same rate and in particular, an ‘egg-carton’ fold form is not preterentially amplified. The velocity field for the perturbing flow consists of a poloidal field, which solely determines the growth of the fold form, and a toroidal field, with non-zero component of vorticity about the axis normal to the layer. The latter is required to satisfy both of the two independent shear traction continuity conditions at interfaces. There is no coupling between the two fields. Three-dimensional fold forms in the Appalachian Plateau province in western Pennsylvania are described.

The topology of ice-sheet centres

AbstractAn ice sheet will, in general, possess points where the horizontal component of velocity is zero, and some of these will be ice centres, occurring close to summits. The paper examines the possible flow patterns near such points. The corresponding horizontal strain-rate pattern is studied by considering an ice sheet which initially has perfect circular symmetry about a vertical axis. Before perturbation there is an isotropic point for the horizontal surface strain rate at the centre. It may be shown, on purely topological grounds and without any reference to the mechanism of flow, that, when the symmetry is broken, this point, being degenerate and structurally unstable, breaks up into two structurally stable components. The breakup always occurs in essentially the same way. Around the two component points the trajectories of principal strain-rate directions always have the lemon pattern. The contours of equal principal strain rate around them are usually hyperbolic; however, if the unperturbed flow pattern had a very pronounced spiral character, they would be elliptic. This behaviour is in contrast to that of the ice centre itself, which remains unsplit.

Numerical simulations of a compact convergent system of ice floes

The granular nature of the Arctic pack ice, and the plastic nature of the deformation of the pack due to ridging, has long been recognised. However, because of a lack of experimental data, assumptions must be made to define the shape of the yield curve and the associated flow rule which characterize the aggregate rheology of the ice pack. In this work, the results of numerical experiments with a simulated granular viscous-plastic material are presented. The experiments model the granular texture of the ice pack as a dense assembly of non-uniform diameter disks in a rectangular control area. Deformation of the control area is driven by constant strain rates. In the experiments, the ratio of the principal strain rates is varied from isotropic convergence through uniaxial divergence. The stresses computed in the experiments, at the various strain-rate ratios, define a yield curve in principal-stress space. The effects of different coefficients of friction and viscous damping on the shape of the yield curve are explored.

Numerical simulations of a compact convergent system of ice floes

The granular nature of the Arctic pack ice, and the plastic nature of the deformation of the pack due to ridging, has long been recognised. However, because of a lack of experimental data, assumptions must be made to define the shape of the yield curve and the associated flow rule which characterize the aggregate rheology of the ice pack. In this work, the results of numerical experiments with a simulated granular viscous-plastic material are presented. The experiments model the granular texture of the ice pack as a dense assembly of non-uniform diameter disks in a rectangular control area. Deformation of the control area is driven by constant strain rates. In the experiments, the ratio of the principal strain rates is varied from isotropic convergence through uniaxial divergence. The stresses computed in the experiments, at the various strain-rate ratios, define a yield curve in principal-stress space. The effects of different coefficients of friction and viscous damping on the shape of the yield curve are explored.

The topology of ice-sheet centres

AbstractAn ice sheet will, in general, possess points where the horizontal component of velocity is zero, and some of these will be ice centres, occurring close to summits. The paper examines the possible flow patterns near such points. The corresponding horizontal strain-rate pattern is studied by considering an ice sheet which initially has perfect circular symmetry about a vertical axis. Before perturbation there is an isotropic point for the horizontal surface strain rate at the centre. It may be shown, on purely topological grounds and without any reference to the mechanism of flow, that, when the symmetry is broken, this point, being degenerate and structurally unstable, breaks up into two structurally stable components. The breakup always occurs in essentially the same way. Around the two component points the trajectories of principal strain-rate directions always have the lemon pattern. The contours of equal principal strain rate around them are usually hyperbolic; however, if the unperturbed flow pattern had a very pronounced spiral character, they would be elliptic. This behaviour is in contrast to that of the ice centre itself, which remains unsplit.

Movement Patterns of the Ogawamura-Sodechi Landslide, in Nagano, Central Japan

The strain-rate tensor at a point on the surface of a landslide can be determined by setting up 5 stakes and measuring the rate of change of the distances between them . A suitable arrangement of stakes consists of 4 stakes at the corners of a square with 1 stake at the centre . Eight such patterns of stake arrange ment were used in the Ogawamura-Sodechi Landslide in 1987-1988 . Three independent components of the strain-rate tensor and their errors can be estimated by a least-squares method with 4 governing conditions derived from 8 distance measurements . The principal strain-rates are found to within about±6×10-4/day. The surface deformation pattern of the landslide, directions of the cracks and seasonnal variations of the landslide movement are well indicated by the strain-rate tensor and its invariants; namely, magnitude of the dilatation rate and maximum shear strain-rate. The previously reported model of the landslide movement is improved by the strain-rate tensor determianation.

Velocity field equations and strain localization in rigid-plastic materials

The velocity field equations are set up for a standard rigid-plastic material. They allow a direct determination of the velocity field, and therefore correspond to the Navier-Stokes equations of a Newtonian fluid. Under conditions of plane strain or axisymmetric deformation, the velocity field equations lead, using the Lévy-Huber-von Mises flow law, to the principal lines—i.e. to the trajectories of the principal strain-rates as characteristics. However, an incipient strain localization, combined with a velocity discontinuity, cannot be expected to occur along a principal line. Rather, a degeneration of the velocity field equations, shown to exist for certain rate-sensitive materials. might be responsible for such a localization effect.

Analysis of double shearing in frictional materials

AbstractThe concept that the flow of granular materials is governed by shear on certain critical planes is used to formulate an elastic–plastic model. When the elastic strains are neglected, the Double Shearing model becomes identical to the rigid‐plastic model of de Josselin de Jong which he named the Double Sliding model. After a discussion of the model, the small strain formulation is used to describe general boundary‐value problems. A comparison is then made between the coaxial Mohr–Coulomb yielding and the Double Shearing model on the basis of several numerical simulations. The non‐coaxiality of the axes of principal plastic strain rate and principal stress in the Double Shearing model leads to essentially different behaviour for stress rotations in comparison to the coaxial model. The Double Shearing model predicts in general lower limit loads because, for a given state of stress, it allows for several possible directions of plastic flow rather than a unique direction which derives from a plastic potential.

Biaxial path dependence of deformation substructure of type 304 stainless steel

Although martensitic transformations in austenitic stainless steels have been studied rather thoroughly for uniaxial monotonie and cyclic loading, data are scant for biaxially loaded specimens. In particular, recent nonproportional straining experiments have indicated a significant increase in cyclic hardening beyond that observed in uniaxial tests at equivalent strain levels. In this paper, a link is made between the additional hardening and microstructural uniformity of transformation product. This link is expressed through a micromechanical viewpointvia increased latent hardening associated with rotation of the principal stress and plastic strain rate directions.

Theory of chocolate tablet boudinage

Two-dimensional boudinage in a flattening type of bulk deformation, with equal layer-parallel extensions in all directions, leads to the development of roundish or polygonal outlines of boudins in plan-view. As combined experimental and theoretical studies show, chocolate tablet boudinage with two sets of mutually perpendicular boudin axes may form in different ways. (1) Unequal layer-parallel extension in the matrix results in one set of extension fractures forming perpendicular to the greatest principal stress in the matrix. Once these long narrow boudins are formed, the greatest principal stress in the brittle layer becomes approximately parallel to the long axis of the boudin. As a result a second set of fractures forms normal to the first set. (2) In lineated rocks the anisotropy of tensile strength leads to the sequential formation of two sets of extension fractures, parallel and perpendicular to the lineation. Depending on the orientation of the lineation the boudin axes may or may not be parallel to the principal stresses in the matrix. (3) Boudins with rectangular plan-view may also form when two successive events of unidirectional boudinage are superposed on one another. Irrespective of the direction of principal extensional strain rate in the matrix, the second generation extension fractures are likely to form approximately perpendicular to the first generation boudin axes.

Glacio-geophysical implication of an NNSS positioning in and around Syowa Station, East Antarctica

Utilization of the two-wave NNSS receiver drastically improved the positioning accuracy on the ice sheet of Antarctica. The NNSS positioning gives us 3 m three-dimensional convergence with 25 accepted satellite passes, and is most useful for the measurement of ice flow velocity. The flow velocity vectors along Route S-H-Z on Mizuho plateau, East Antarctica were obtained by estimating positional change of glaciological traverse stations after 7 years' interval. The transformation of the coordinate system was necessary before the comparison of the positioning in 1973 by the geodetic traverse method on the Bessel Reference Ellipsoid with that in 1980 by the satellite Doppler method on the NWL-8E Ellipsoid. NNSS receiving experiments at astronomical datum point in Syowa Station enabled us to estimate the translation of the coordinate origin of the Bessel Reference Ellipsoid against the geocentric NWL-8E Ellipsoid and correction terms for coordinate transformation. The obtained flow velocity is 15 m/a at H17 of around 1000 m a.s.l. and 70 m/a at Z2 of around 2000 m a.s.l. with ±6% uncertainty, and is too large to be explained only by the laminar flow of the ice sheet. The obtained velocity vectors are found to be mostly parallel to the maximum slope of the free-air gravity anomaly contours and can be interpreted as the ice sheet sliding down the slope of the subglacial mound of 2400 m relative height from the average subglacial bedrock topography. If such bedrock sliding occurrs over the whole region of Mizuho Plateau, the related thinning of the ice sheet may be detected by the precise measurement of the height change of the same marker station. By dynamically modelling the ice sheet and substituting the observed parameters such as precipitation, principal strain rate, etc., into the equation of ice thickness change the submergence velocity of around −1 m/a is expected and will be detected by carefully designed repetitive NNSS receiving experiments after several year's interval.

Geodetic measurement of horizontal crustal deformation in eastern Taiwan

Four medium-aperture trilateration networks in eastern Taiwan have been surveyed three to four times since 1981. One is the Ilan network crossing an elongated active seismic zone, the others are the Hualien, Yuli and Taitung networks located at the northernmost, middle, and southernmost portions of the Longitudinal Valley, respectively. Based on changes of the observed line length, the three components of the surface strain rate tensor for each of the networks are obtained by a least squares adjustment technique. Then the principal strain rates are calculated. The Ilan network gives a principal strain rate of uniaxial extension at 2.3 μstrain/yr in the direction of N45°W. The Hualien network has a principal strain rate of 1.2 μstrain/yr extension in N62°E and 1.2 μstrain/yr contraction in N152°E. The Yuli network yields essentially a principal strain rate of uniaxial contraction at 8.4 μstrain/yr in N117°E, whereas the Taitung network has a principal strain rate of 1.5 μstrain/yr extension in N24°E and 3.9 μstrain/yr contraction in N114°E. The directions of contractionof both the Yuli and Taitung networks are consistent with the direction of the maximum compressive stress of this area. Furthermore, average velocities of the relative motion between geodetic stations in the Central Range and the Coastal Range are estimated from the average rates of changes in line length. Stations in the Hualien network show a left-lateral relative motion in a direction more or less parallel to the strike of the Longitudinal Valley, while stations in the Yuli and Taitung networks move toward each other in the direction approximately perpendicular to the trend of the Longitudinal Valley.

Finite strain calculations of continental deformation: 2. Comparison with the India‐Asia Collision Zone

Numerical experiments on a thin viscous sheet model for deformation of continental lithosphere subjected to an indenting boundary condition yield distributions of crustal thickness, of stress and strain rate, and of latitudinal displacements that may be compared with observations in the India‐Asia collision zone. A simple indenting boundary condition applied to initially laterally homogeneous sheets obeying a power law rheology produces results that are in broad agreement with the observations, provided that the power law exponent is three or greater and the sheet can support vertically integrated stress differences of 2×1013 (±5 × 1012) N m−1 in regions in front of the indenter. Under these conditions, the calculated deformation shows accommodation of convergence primarily by crustal thickening, to produce a plateau in front of the indenter. Palaeomagnetic data from India and Tibet, and the observed distribution of topography, suggest that much of the post‐Eocene convergence of India with Asia has been taken up by deformation within Asia that involved crustal thickening. The principal difference between calculation and observation is the absence from the calculated strain rate fields of east‐west extension of the plateau in front of the indenting boundary. The calculations show that once such a plateau is formed, the buoyancy force associated with the crustal thickness contrast inhibits further thickening and the plateau strains at less than half the rate of its immediate surroundings. Seismically determined regional strain rates exhibit a similar distribution, with the Tibetan plateau straining at about one quarter the rate of the Tien Shan and Ningxia‐Gansu regions. Calculated principal compressive stress orientations and regional strain rates agree with the seismically determined quantities in the Mongolia‐Baikal, Tien Shan, Tibet, and Ningxia‐Gansu regions of Asia, to within the uncertainty of the latter. The vertically integrated stresses that are calculated for the viscous sheet are comparable with those that can be supported by a Theologically stratified continental lithosphere obeying laboratory‐determined flow laws. We suggest that the thin viscous sheet model, described in this paper and its companion, gives a simple and physically plausible description of the observed deformation in central Asia; in this description the predominant mechanism of accommodation of continental convergence is diffuse crustal thickening, with shear on vertical planes playing a subsidiary role once large crustal thickness contrasts have been established.

Strain accumulation in the Rocky Mountain states

Strain accumulation in trilateration networks in Montana, Utah, and New Mexico has been measured for over a decade. The network in New Mexico spans the Rio Grande rift near Socorro. The absence of any observed strain accumulation in that network places an upper limit (2 standard deviations) of 3 mm/a on east‐west spreading across the 100‐km‐wide rift. The network in Utah spans the Wasatch fault near Ogden. Although the 1972–1984 average rate of strain accumulation in that network is not significant at a 2‐standard‐deviation detection threshold of 0.03 μstrain/a, short‐term strain fluctuations of marginal significance have been detected in the network. The network in Montana spans the epicentral area of the 1959 Hebgen Lake earthquake (magnitude 7.1). The principal strain rates observed in that network for the 1973–1984 interval were and , extension reckoned positive. The principal extension is perpendicular to the 1959 rupture trace. The deformation of this network suggests rather rapid reloading of the 1959 rupture zone, but whether this is steady strain accumulation or still postseismic relaxation is not known. The strain rate does appear to be constant over the 1973–1984 interval.

Multiaxial elongation of polyisobutylene with various and changing strain rate ratios

The multiaxial elongational rheometer equipped with rotary clamps is modified such that in addition to simple, equibiaxial and multiaxial elongations also tests with new modes of elongation can be performed. As an example, polyisobutylene is elongated with a ratio of the principal strain rates of\(m = \dot \varepsilon _{22} /\dot \varepsilon _{11} = 0.5\) and magnitudes of the maximum strain rate\(\dot \varepsilon _{11} = 0.008\), 0.04 and 0.08 s−1. As a test result, the first elongational viscosityµ1(t) is obtained which follows closely the linear viscoelastic shear viscosity\(\mathop \eta \limits^ \circ (t)\). In contrast, the second elongational viscosityµ2(t) remains below\(\mathop \eta \limits^ \circ (t)\). By means of a further modification of the rheometer, the test modes can be varied during the deformation period. This allows one to investigate the influence of a well-defined rheological pre-history on the following rheological behaviour. As an example a variation ofm = 0.5 → 2 was performed. The measured normal-stress differences superpose from the single steps of deformation similar to the linear viscoelastic prediction.

Determination of the Principal Stresses of a Snow Cover on a Mountain Slope Using Snow Pressure Gauges

Principal stresses in a snow cover on a uniform slope were determined by two methods, each using thin pressure gauges to measure snow pressure in the snow. These snow pressures were principal stress σ2 on a vertical plane perpendicular to the contour lines and normal compressive stress σ θ on a plane perpendicular to the vertical plane. In addition, plastic Poisson’s ratio v was estimated in a snow cover on level ground. Estimates of principal strain rates were used to calculate principal stresses and viscosity by two different methods, using estimates of v and the constitutive equations of Yosida (1980) and the derived values of σ2and σθ. For dry and compact snow, σ1 and σ3 calculated by both methods agreed well with each other, and also with values obtained by the hole-mark method reported by Shimizu and Huzioka (1975).

Determination of the Principal Stresses of a Snow Cover on a Mountain Slope Using Snow Pressure Gauges

Principal stresses in a snow cover on a uniform slope were determined by two methods, each using thin pressure gauges to measure snow pressure in the snow. These snow pressures were principal stress σ2on a vertical plane perpendicular to the contour lines and normal compressive stress σθon a plane perpendicular to the vertical plane. In addition, plastic Poisson’s ratio v was estimated in a snow cover on level ground. Estimates of principal strain rates were used to calculate principal stresses and viscosity by two different methods, using estimates of v and the constitutive equations of Yosida (1980) and the derived values of σ2and σθ.For dry and compact snow, σ1and σ3calculated by both methods agreed well with each other, and also with values obtained by the hole-mark method reported by Shimizu and Huzioka (1975).

Regional deformation near Palmdale, California, 1973–1983

The Tehachapi trilateration network spans the intersection of the San Andreas and Garlock faults in southern California in the “Big Bend” region of the San Andreas fault. Analysis of data from 1973–1983 shows strain differences between the northwest and southeast regions of the network and slip at depth on both faults. The Palmdale network, spanning the San Andreas fault entirely within the Tehachapi network, showed increases of about 1 μstrain in both east‐west and north‐south extension in late 1979. The Tehachapi strains also jumped at this time, but the magnitude of the increase was only about one third that of Palmdale. The principal strain rates for Tehachapi over the time interval 1973–1983 are μstrain/yr and μstrain/yr, with the 1 axis directed N76°E. Strains were also computed for two subregions. The principal strain rates for the southeast Tehachapi region are μstrain/yr and μStrain/yr, with the 1 axis directed N73°E. This result differs significantly from the principal strain rates at Palmdale ( μstrain/yr, μstrain/yr, with the 1 axis directed N71°E). For the northwest Tehachapi region, μstrain/yr and μstrain/yr, with the 1 axis directed N93°E.