A Method of Determining the Strain-Rate Tensor at the Surface of a Glacier

AbstractThe rate of strain tensor at a point on the surface of a glacier may be determined by setting up a number of stakes in a pattern and measuring the rate of change of the distances between them. A suitable pattern consists of four stakes at the corners of a square with one stake at the center. Five such patterns were used on Austerdalsbreen, Norway, in August 1956. The problem is to deduce the best values of the 3 independent components of the strain-rate tensor from the 8 measured quantities, and, for this purpose, a least-squares method, invented by Bond for the analogous problem in crystal physics, is used. The principal strain-rates are found to within about ±0.005 yr.−1and their directions relative to the stake system to within about ±0.5°. The directions and magnitudes of the principal stresses are then deduced from Glen’s flow law and a suitable general theory. The directions of the principal strain-rates are in good agreement with the directions of the crevasses, but the experiment is inconclusive on the question of the magnitude of the stress needed to form a crevasse.

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Strain Rates On Rutford Ice Stream, Antarctica (Abstract)

Annals of Glaciology ◽

10.1017/s0260305500001592 ◽

1986 ◽

Vol 8 ◽

pp. 207

Author(s):

N. Stephenson ◽

C.S.M. Doake

Keyword(s):

Strain Rate ◽

Velocity Gradient ◽

Surface Strain ◽

Lateral Strain ◽

Strain Rate Tensor ◽

Strain Rates ◽

Average Value ◽

Divergent Flow ◽

Ice Stream ◽

Thickness Variations

In a study of the Rutford Ice Stream, strain rates were measured on a transverse section. Magnitudes ranged up to 40 × 10−3 a−1 but were typically in the order of 3 × 10−3 a−1 with an error of 0.1 χ 10−3 a−1. Variations in the strain rate between adjacent stakes of 0.2 χ 10−3 a−1 to 2 × 10−3 a−1 were matched to the thickness variations on the glacier. For each set of three adjacent stakes, the velocity gradient components of the surface strain rate tensor were calculated by assuming that the gradients were linear over the distance between adjacent stakes. When plotted against distance across the ice stream, each strain rate component revealed different aspects of the flow field. The longitudinal strain rate was compressive, with an almost constant magnitude of 10−3 a−1. The lateral strain rate is extensive, with an average value of 1.1 × 10−3 a−1 which agreed with the angle between the divergent flow lines observed on a Landsat image. Peaks in the lateral strain rate, corresponding to longitudinal bands of thicker ice, showed that these thicker bands were spreading more rapidly at the expense of thinner areas. The two velocity gradient components of the shear rate tensor also reflected differences in ice thickness.

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Rhombus and Rhomboid Parallelogram Patterns on Glaciers: Natural Indicators of Strain

Journal of Glaciology ◽

10.1017/s0022143000014246 ◽

1979 ◽

Vol 22 (87) ◽

pp. 247-261 ◽

Cited By ~ 1

Author(s):

Charles J. Waag ◽

Keith Echelmeyer

Keyword(s):

Strain Rate ◽

Flow Direction ◽

Acute Angle ◽

Principal Strain ◽

Obtuse Angle ◽

Strain Rates ◽

Ice Flow ◽

Glacier System ◽

Glacier Ice

AbstractSubtle rhombus and rhomboid parallelogram patterns occur on Vaughan Lewis Glacier and the Gilkey Glacier System, Juneau Icefield, Alaska. The patterns are within the firn at the firn-ice interface, are formed by differential recrystallization within narrow preferred zones, and are apparently manifestations of stresses transferred upward from the glacier ice. On the glaciers of the Gilkey System the patterns occur where intense lateral shortening is indicated by abrupt convergence of medial moraines and an abundance of extension crevasses. The short axes of the rhombi and the obtuse angle bisectors of the rhomboids are subparallel to the strike of extension crevasses, therefore to the axis of shortening. The long axes of the rhombi and the acute angle bisectors of the rhomboids are parallel to the foliation, and ice-flow direction. The angles of the parallelograms are variable locally, but average 105° and 75°; the variation seems to reflect intensity and duration of stress. Similar parallelograms occur within the troughs of wave bulges below the Vaughan Lewis Icefall. In the wave bulges, the foliation arcs parallel the wave. The long axes of the rhombi and acute angle bisectors of the rhomboids parallel the foliation around the foliation arc. The short axes of the rhombi and the obtuse angle bisectors of the rhomboids parallel the strikes of radial crevasses, are perpendicular to the direction of extension, and form a fan divergent down-stream. The precise mechanisms and conditions of formation of the parallelograms are not yet understood. Preliminary strain-rate measurements suggest, however, that correlations exist between the orientations of the principal strain-rates and the axes of the patterns, and between the magnitude of the strain-rates and the axial lengths of the patterns.

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A Corotational Elastoplastic Formulation With Subloading Surface Model for Hardening Materials

Volume 1 ◽

10.1115/esda2004-58401 ◽

2004 ◽

Author(s):

Ali Reza Saidi ◽

Koichi Hashiguchi

Keyword(s):

Constitutive Model ◽

Strain Rate ◽

Simple Shear ◽

Deformation Theory ◽

Elastoplastic Deformation ◽

Strain Tensor ◽

Surface Model ◽

Strain Rate Tensor ◽

Hencky Strain ◽

Stress Components

In this paper a corotational constitutive model for the large elastoplastic deformation of hardening materials using subloading surface model is formulated. This formulation is obtained by refining the large deformation theory of Naghdabadi and Saidi (2002) adopting the corotational logarithmic (Hencky) strain rate tensor and incorporating it into the subloading surface model of Hashiguchi (1980, 2003) falling within the framework of the unconventional plasticity. As an application of the proposed constitutive model, the large Elastoplastic deformation of simple shear example has been solved and the results have been compared with classical elasto-plastic model using the Hencky strain tensor. Also the effect of the choice of corotational rates on stress components has been studied.

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Recording Wire Strainmeters on the Barnes Ice Cap, Baffin Island, Canada

Journal of Glaciology ◽

10.3189/s0022143000013940 ◽

1978 ◽

Vol 20 (83) ◽

pp. 409-423 ◽

Cited By ~ 1

Author(s):

K. Evans ◽

D.J. Goodman ◽

G. Holdsworth

Keyword(s):

Strain Rate ◽

Periodic Variation ◽

Strain Rates ◽

Baffin Island ◽

Rate Data ◽

Rapid Acquisition ◽

Ice Cap ◽

Good Agreement

AbstractThe report describes an experiment to evaluate the use of geophysical (Cambridge-type) wire strainmeters for the rapid acquisition of strain-rate data and to compare strains on a large ice mass over gauge distances of 5 m, 50 m, and 1 km.Three continuously recording wire strainmeters were installed at the centre of two separate arrays of strain poles 10.6 km and 19.5 km from the ice divide on the Barnes Ice Cap. Data was collected between 24 April and 15 May 1976. The 1 km strain arrays had previously been measured in 1974 and 1975. The results show good agreement between the strainmeter data and the larger strain arrays at the 10.6 km site but differ at the 19.5 km site. When the daily means are calculated for the strainmeters at the 19.5 km site, the strain-rates show a possible periodic variation with an apparent period ofabout 11 d. Since there appears to be no direct correlation between the strainmeter signal and either temperature or pressure, the result is assumed to represent real varying strain within the ice.

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Rhombus and Rhomboid Parallelogram Patterns on Glaciers: Natural Indicators of Strain

Journal of Glaciology ◽

10.3189/s0022143000014246 ◽

1979 ◽

Vol 22 (87) ◽

pp. 247-261

Author(s):

Charles J. Waag ◽

Keith Echelmeyer

Keyword(s):

Strain Rate ◽

Flow Direction ◽

Acute Angle ◽

Principal Strain ◽

Obtuse Angle ◽

Strain Rates ◽

Ice Flow ◽

Glacier System ◽

Glacier Ice

AbstractSubtle rhombus and rhomboid parallelogram patterns occur on Vaughan Lewis Glacier and the Gilkey Glacier System, Juneau Icefield, Alaska. The patterns are within the firn at the firn-ice interface, are formed by differential recrystallization within narrow preferred zones, and are apparently manifestations of stresses transferred upward from the glacier ice. On the glaciers of the Gilkey System the patterns occur where intense lateral shortening is indicated by abrupt convergence of medial moraines and an abundance of extension crevasses. The short axes of the rhombi and the obtuse angle bisectors of the rhomboids are subparallel to the strike of extension crevasses, therefore to the axis of shortening. The long axes of the rhombi and the acute angle bisectors of the rhomboids are parallel to the foliation, and ice-flow direction. The angles of the parallelograms are variable locally, but average 105° and 75°; the variation seems to reflect intensity and duration of stress. Similar parallelograms occur within the troughs of wave bulges below the Vaughan Lewis Icefall. In the wave bulges, the foliation arcs parallel the wave. The long axes of the rhombi and acute angle bisectors of the rhomboids parallel the foliation around the foliation arc. The short axes of the rhombi and the obtuse angle bisectors of the rhomboids parallel the strikes of radial crevasses, are perpendicular to the direction of extension, and form a fan divergent down-stream. The precise mechanisms and conditions of formation of the parallelograms are not yet understood. Preliminary strain-rate measurements suggest, however, that correlations exist between the orientations of the principal strain-rates and the axes of the patterns, and between the magnitude of the strain-rates and the axial lengths of the patterns.

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INTEGRATION OF GNSS-GPS NETWORKS (cGPS) FOR OBTAINING STRESS AND STRAIN MODELS FOR THE SPINA REGION (SOUTH OF THE IBERIAN PENINSULA AND NORTH AFRICA)

10.5194/egusphere-egu2020-19570 ◽

2020 ◽

Author(s):

Alejandro Pérez-Peña ◽

Alberto Fernández-Ros ◽

Belen Rosado ◽

Amós De Gil ◽

Gonzalo Prates ◽

...

Keyword(s):

Strain Rate ◽

Iberian Peninsula ◽

Least Squares Method ◽

Weighted Least Squares ◽

Ductile Deformation ◽

Series Data ◽

Strain Rate Tensor ◽

Shear Strain Rate ◽

Geodynamic Processes ◽

Gnss Data

Nowadays, both, the number of observations and the accuracy of satellite-based geodesic measurements, like GNSS, have increased. Therefore, GNSS provides more data as displacement values and velocities. This paper demonstrates that GNSS data analysis is a powerful tool to study geodynamic processes.In this study, the analyzed GNSS data correspond to continuously recorded GPS (CGPS) stations, what we call the SPINA network. These stations are located in a region called Ibero-Maghrebian which includes the southern areas of the Iberian Peninsula and northern Africa.The CGPS stations are included in the following organizations: RENEP (National Network of Permanent Stations), RAP (Andalusian Positioning Network), the Murcia Region CGPS Networks, ERVA (Valencian Reference Stations Network), IGN (National Geographic Institute) and the network TOPOIBERIA. The velocity was obtained in two steps: (1) preprocessing position time-series data of daily GPS measurements and (2) applying a combined model using the weighted least-squares method.The prior knowledge of the crustal strain rate tensor provides a description of geodynamic processes such as the fault strain accumulation.Based on the distribution of the GNSS stations, several grid sizes were tested to identify the best resolution. A Python script was used to compute the full two-dimensional velocity gradient tensor by means of inverting the GNSS velocities. The tensorial analysis provides different aspects of deformation, such as the maximum shear strain rate, including its direction, and the dilatation strain rate. These parameters can be used to characterize the mechanism of the current deformation.Based on the computations from the GNSS-data model of components of horizontal deformations, the rates of both principal, values and axes, of the Earth&#8217;s crust deformation were found. Deformations measured in the Ibero-Maghrebian region with GPS could be interpreted in terms of either elastic loading or ductile deformation.

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Relating the occurrence of crevasses to surface strain rates

Journal of Glaciology ◽

10.1017/s0022143000015926 ◽

1993 ◽

Vol 39 (132) ◽

pp. 255-266 ◽

Cited By ~ 99

Author(s):

David G. Vaughan

Keyword(s):

Tensile Strength ◽

Strain Rate ◽

Impurity Content ◽

Surface Strain ◽

Strain Rates ◽

Principal Stresses ◽

Ice Dynamics ◽

Ice Shelves ◽

Creep Flow ◽

Failure Envelopes

AbstractThe presence of crevasses on the surface of ice masses indicates that a fracture criterion has been met. Understanding how crevasses form will provide information about the stress and strain-rate fields in the ice. This study derives a relationship between measurements of strain rate and observations of crevassing on the surface of ice masses. A literature search yielded 17 polar and alpine locations where strain rates had been measured and crevassing recorded. By plotting strain rates (converted to stresses using a creep law) using axes representing the surface-parallel principal stresses, failure envelopes were derived by enclosing measurements where surface crevassing was absent. The derived failure envelopes were found to conform well to theoretical ones predicted by the Coulomb and the maximum octahedral shear stress (von Mises) theories of failure. The derived failure envelopes were scaled by the tensile strength, which was found to vary from 90 to 320 kPa. There was no systematic variation of tensile strength with either temperature at 10 m depth or the method used to locate the crevasses. The observed variation in tensile strength could result from variations in ice properties (e.g. crystal size, impurity content or density) or could be related to uncertainty in the constitutive relation. Creep flow and fracture share a very similar temperature dependence, suggesting similar crystal-scale processes are responsible for both. The observed relationship will provide a supplementary tool with which to verify and test models of ice dynamics against remotely sensed imagery. The study also indicates that a temperature rise of a few degrees throughout the ice column will not result directly in any increase in calving rates from the large Antarctic ice shelves such as the Filchner–Ronne or Ross Ice Shelves.

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Vorticity, strain-rate and dissipation characteristics in the near-wall region of turbulent boundary layers

Journal of Fluid Mechanics ◽

10.1017/s0022112097006770 ◽

1997 ◽

Vol 350 ◽

pp. 29-96 ◽

Cited By ~ 62

Author(s):

ANANT HONKAN ◽

YIANNIS ANDREOPOULOS

Keyword(s):

Strain Rate ◽

Joint Probability ◽

Rate Of Change ◽

Small Scale ◽

Wall Region ◽

Strain Rate Tensor ◽

Viscous Effects ◽

Vorticity Vector ◽

Vorticity Flux ◽

Near Wall

Experimental results are presented that reveal the structure of a two-dimensional turbulent boundary layer which has been investigated by measuring the time-dependent vorticity flux at the wall, vorticity vector, strain-rate tensor and dissipation-rate tensor in the near-wall region with spatial resolution of the order of 7 Kolmogorov viscous length scales. Considerations of the structure function of velocity and pressure, which constitute vorticity flux and vorticity, indicated that, in the limit of vanishing distance, the maximum attainable content of these quantities which corresponds to unrestricted resolution, is determined by Taylor's microscale. They also indicated that most of the contributions to vorticity or vorticity flux come from the uncorrelated part of the two signals involved. The measurements allowed the computation of all components of the vorticity stretching vector, which indicates the rate of change of vorticity on a Lagrangian reference frame if viscous effects are negligible, and several matrix invariants of the velocity gradient or strain-rate tensor and terms appearing in the transport equations of vorticity, strain rate and their squared fluctuations. The orientation of vorticity revealed several preferential directions. During bursts or sweeps vorticity is inclined at 35° to the longitudinal direction. It was also found that there is high probability of the vorticity vector aligning with the direction of the intermediate extensive strain corresponding to the middle eigenvector of the strain-rate matrix. The results of the joint probability distributions of the vorticity vector orientation angles showed that these angles may be related to those of hairpin vortex structures. All invariants considered exhibit a very strong intermittent behaviour which is characterized by large-amplitude bursts which may be of the order of 10 r.m.s. values. Small-scale motions dominated by high rates of turbulent kinetic energy dissipation and high enstrophy density are of particular interest. It appears that the fluctuating strain field dominates the fluctuations of pressure more than enstrophy. Local high values of the invariants are also often associated with peaks in the shear stress.

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Crustal Deformation Prior to the 2017 Jiuzhaigou, Northeastern Tibetan Plateau (China), Ms 7.0 Earthquake Derived from GPS Observations

Remote Sensing ◽

10.3390/rs10122028 ◽

2018 ◽

Vol 10 (12) ◽

pp. 2028 ◽

Cited By ~ 2

Author(s):

Weiwei Wu ◽

Xiaoning Su ◽

Guojie Meng ◽

Chentao Li

Keyword(s):

Tibetan Plateau ◽

Strain Rate ◽

Crustal Deformation ◽

Multiscale Approach ◽

Deformation Pattern ◽

Strain Rate Tensor ◽

Strain Rates ◽

Shear Strain Rate ◽

Jiuzhaigou Earthquake ◽

Epicentral Area

The 2017 Jiuzhaigou Ms 7.0 earthquake occurred on the northeastern margin of the Tibetan Plateau, with no noticeable rupture surface recognized. We characterized the pre-seismic deformation of the earthquake from GPS (Global Positioning System) data at eight continuous and 73 campaign sites acquired over the 2009–2017 period. With respect to the Eurasian plate, the velocity field showed a noticeable decrease, from west of the epicenter of the Jiuzhaigou earthquake to the western edge of the Longmenshan fault, in the southeast direction. The total northwest west–southeast east shortening rate in the vicinity of the epicentral area was in the range of 1.5 mm/y to 3.1 mm/y. With a GPS velocity transect across the Huya fault (HYF), where the epicenter was located, we estimated the activity of the HYF, showing a dominant left-lateral slip rate of 3.3 ± 0.2 mm/y. We calculated strain rates using a spherical wavelet-based multiscale approach that solved for the surface GPS velocity according to multiscale wavelet basis functions while accounting for spatially variable spacing of observations. Multiscale components of the two-dimensional strain rate tensor showed a complex crustal deformation pattern. Our estimates of strain rate components at the scale of seven and eight revealed extensional strain rate on the northern extension of the HYF. The Jiuzhaigou earthquake occurred at the buffer zone between extensional and compressional deformation, and with significant maximum shear rates being 100–140 nanostrain/y. In addition, a maximum shear strain rate of 60–120 nanostrain/y appeared around the epicenter of the 2013 Ms 6.6 Minxian–Zhangxian earthquake. These findings imply that inherent multiscale strain rates could be separated to identify strain accumulation related to medium- and large-sized earthquakes.

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