Elastic Plate Model Research Articles

Flexural behavior of the continental lithosphere in Italy: Constraints imposed by gravity and deflection data

A procedure is developed that systematically matches observations of plate flexure and gravity anomalies at simple convergent boundaries to the bestfitting two‐dimensional flexural geometry obtained by a thin elastic sheet of specified elastic strength. The procedure allows analytical solution for three independent variables, one of which is the initial (preflexure) elevation of the reference surface and two of which are related to the vertical forces and bending moments that act on the plate. Application to four profiles constructed across the Apennine convergent system in Italy shows that the observed depth of the basal Pliocene surface (which marks the base of the foredeep basin) and the observed Bouguer gravity anomalies can be fit by a two‐dimensional elastic plate model with an effective elastic thickness of about 20 km on three profiles and about 10 km on the fourth profile. The initial elevation, prior to flexure, is determined to have been about 1000 m below sealevel on all profiles. Large bending moments and/or vertical shear forces are required to act at the effective plate ends in order to maintain the observed deflection (moments between 0.5×1017 and 3×1017 N, and shear forces between + 3×1012 and −3×1012 N/m). Gravity modelling indicates that, on the two northernmost profiles, these terminal forces can not be supplied by a static mass at crustal depths and suggests that, on all four profiles, the terminal forces are supplied by anomalously dense material present at depths of 75–150 km beneath and west of the west coast of Italy. This dense material probably corresponds to westward‐subducted lithosphere of the Adriatic plate, extending from the northern Apennines to the southern Apennines and Calabria. This study suggests that in subduction systems, like the Apennines, that are characterized by backarc extension, forces acting on subducted lithosphere at depth can be expressed at the surface in the geometry of the foredeep basin that forms adjacent to the subduction zone.

Deformational models of rifting and folding on Venus

Tectonic features on Venus with characteristic widths and/or spacings can be used to place quantitative constraints on the structure of the mechanical lithosphere through geophysical modeling. We develop theoretical models for both compressional (buckling) and extensional (rifting) deformation based on classical elastic plate models with characteristic spacings determined by the flexural parameter, modified to account for the effects of a realistic rheology. This rheology consists of a relatively weak crust overlying a more creep‐resistant mantle. In both layers the upper portions deform by frictional yielding on preexisting faults with a yield stress that increases linearly with depth, and the lower portions deform by non‐Newtonian temperature‐dependent creep. For extensional deformation a graben model is used in which the primary structure of the rift is formed in the strong mantle layer and secondary faulting occurs in the brittle crustal layer. The introduction of a weak zone in the lower crust that acts to decouple these two layers allows the formation of rifts with the large widths and depths observed on Venus. The compressional folding model utilizes an elastic buckling approach in which the non‐elastic yielding of the upper and lower portions of the layer modifies the elastic folding wave‐length. This approach avoids the unreasonably high stresses that are required for conventional buckling models. The two scales of deformation observed in radar images, 15–25 km and 100–300 km, can be related to the elastic flexural properties of the crust and mantle, respectively, which are separated by an intermediate weak zone in the lower crust. These results imply that for regions in which both scales of deformation are present the crust is 5–15 km thick and the thermal gradient is 10°–15°km−l. In regions such as Ishtar Terra, where only the smaller features are observed, the crust may be considerably thicker. The apparent structure seen in high‐resolution images of the Beta Regio rift, with the major faults located at the edges of an extended linear central depression, lends support to the assumption of a graben mechanism of rift formation.

Flexural isostasy and uplift of the Sierra Nevada of California

The Sierra Nevada of California has experienced several kilometers of uplift in the last 10 m.y., but its present elevation is isostatically compensated by a crustal root that has probably existed since Late Cretaceous. Flexural isostasy can reconcile these observations: Basin and Range faulting and extension starting 10 m.y. ago allowed rapid uplift of an eroded and regionally compensated magmatic arc underlain by an excess crustal root. A quantitative model of an unbroken elastic plate buoyed up by the excess root provides an explanation for the estimated 10‐m.y. paleotopography of the range. The same model fits the present topography when broken (shear and fiber stresses not transmitted) at the location of the Owens Valley fault zone. The broken‐plate model predicts the appropriate amount of westward tilt of the Sierra Nevada and can explain the pronounced eastward tilt of the White and Inyo mountains. The position where paleo and present drainages in the Sierra cross over in elevation is the most sensitive indicator of the elastic thickness required of the lithosphere. An effective elastic thickness of 50 km best fits the observed crossover distances and is consistent with the amount of post‐10 m.y. uplift. Calculated fiber and shear stresses suggest that actual extension of the crust is more important in permitting the uplift than the mere presence of normal faults. Part of the residual isostatic gravity anomaly pattern (high gravity over the eastern Great Valley and western Sierra Nevada, low residual gravity over the high Sierra) may be explained by the continuing flexural support of departure from local isostatic equilibrium. The model implies that before 10 m.y. ago the Sierra should have been marked by a negative isostatic gravity anomaly of more than −100 m Gal, flanked by isostatic gravity highs. Such large isostatic anomalies have not been observed to date over eroded arcs, but the Sierra has uniquely low heat flow and, presumably therefore, high lithospheric strength, and the Sierran root is at the upper limit of size that could be isostatically suppressed.

Isostasy of the Northern Bay of Biscay continental margin

Summary. Spectral analysis of eight marine gravity profiles and seven SEASAT profiles, combined with corresponding bathymetric data over the Northern Bay of Biscay origin, yield identical admittance functions for wavelengths greater than 120 km. The resulting admittance function has been interpreted in terms of an Airy model of compensation for wavelengths greater than 250 km and in terms of an elastic plate model of compensation for shorter wavelengths. The Airy model corresponds to a crustal thickness variation across the margin. The plate model with an elastic thickness of 8 km is associated with the regional compensation of a sedimentary load which was probably emplaced during and just after rifting.

Effects of lithospheric in‐plane stress on sedimentary basin stratigraphy

In‐plane lithospheric stress, though known to exist, has generally been ignored in the quantitative modeling of basin stratigraphy. However, low levels of intraplate lateral stress can induce observable plate deformations (10–100 m of vertical motion) if the lithosphere contains a preexisting deformation (such as a sedimentary basin). The importance of in‐plane stress in modifying the stratigraphy of a sedimentary basin has been assessed using an elastic plate model for the lithosphere. A compressive in‐plane stress generally induces basin subsidence with peripheral uplift (shoreline regression), while a tensile in‐plane stress induces basin uplift and peripheral subsidence (shoreline transgression). These simple results are complicated by variations in crustal thickness, as now two interfaces are involved, that is, the sediment/basement interface of the sedimentary basin and the Moho topography; the isostatic state of a sedimentary basin therefore ultimately controls the resultant deformation induced by in‐plane stress. The implications of in‐plane stress modification of basin stratigraphy are profound: regionally correlatable transgressions and regressions of basin interiors (e.g. cyclothems) and passive continental margins (the third‐order variations of the Vail et al. [1977a] coastal onlap curve), may be tectonically produced by the interaction of stress‐induced base level changes and a basin tectonic driving subsidence. In‐plane stress, its generation, magnitude, and variation, is considered a consequence of plate boundary reconfigurations during continental collisions, rifting, and subduction/obduction.

Stress patterns in an interplate shear zone: an effective anisotropic model and implications for the transverse ranges, California

Strong lateral variations in geological structure within a transcurrent interplate deformation boundary have a substantial influence upon the way in which ambient stress is related to the relief of regional stress within the boundary zone. Much of the crustal deformational structure in southern California and environs consists of a conjugate wrench fault system. The Quaternary fault system consists of a series of parallel and sub-parallel strike-slip faults that are causally related to the horizontal interplate shearing. A prominent crustal structural inhomogeneity is the Transverse Ranges, where fault orientation is east-west, transverse to the dominant northwesterly trend. We investigate some of the consequences of this transverse inhomogeneity on the overall stress and strain field in the southern California region. The activity of the strike-slip (or wrench) system to the south and north of the Transverse Ranges suggests a mechanical model consisting of weak zones with a relatively strong degree of orientation. An effective anisotropy model is constructed based on: (1) a two-component laminate model consisting of competent unfaulted rock adjacent to incompetent faulted rock; (2) theoretical results for the weakening of a plate due to a doubly periodic array of cracks; and (3) finite element treatment of a checkerboard array of cracks. The fundamental parameter for weakening is = 1 - where is a non-dimensional form of Biot’s slide modulus. In the limit of 1 the crust becomes extremely weak and anisotropic, and as A -> 0 the condition of a strong, isotropic crust is recovered. The components of the stiffness (or compliance) matrix are directly related to the mechanical properties of a finite width fault zone, or to the average fault spacing and asperity density within a particular geological province, or both. An elastic plate model that incorporates the stress-strain channelling caused by multiple, oriented fault systems is constructed. The plate is assumed to be stressed by pure shearing forces maintained at infinity. The ambient field then corresponds to the north-south compressions, east-west extensions tectonic regime that dominates North-American-Pacific interplate shear along the San Andreas fault, California. Embedded within the plate is an elliptical inclusion in which multiple fault stress channelling also occurs. The inclusion thus mimics the misaligned structure of the Transverse Ranges in southern California. The boundary value problem associated with the model is treated both analytically and with finite element computations. The simple model predicts (i) the enhanced seismic energy release associated with the Transverse Ranges; and (ii) the clockwise rigid rotation indicated by a palaeomagnetic studies. The relatively simple nature of the model helps to isolate those features of the southern California tectonic stress regime that might be attributed to the transverse orientation of the Transverse Ranges. Stress channelled into the crosscutting tectonic structure from the ambient interplate field is significant. Contradirectionality alone cannot provide an explanation for the enhanced north-south compressive stress relative to east-west extension.

Inverse problems for elastic plates with variable flexural rigidity

Abstract The pulse-spectrum technique (PST), an iterative numerical algorithm, is extended and further developed to solve inverse problems in the dynamic structural identification and the dynamic structural design based on the two-dimensional elastic plate model with variable flexural rigidity. Numerical simulations on several simple examples are carried out to test the feasibility and to study the general characteristics of PST without the real measurement data for the case of dynamic structural identification or with the prescribed design data for the case of dynamic structural design. It is found that PST does give satisfactory results and also fares very well in regard to the given four practical criteria for evaluating numerical methods.

Elastic plates model of domain walls in intercalated graphite

Abstract The recently proposed model of thin elastic plates coupled via harmonic forces for the calculation of domain wall energies in intercalated graphite is extended to higher stages. It is found that whether coherent domains bind to each other depends on the degree of stagger. The effect of the loss of coherence and the interface between different stages are discussed.

Thin elastic and periodic plates

AbstractThis paper is devoted to the study of a three dimensional model of elastic periodic plate when the thickness e of the plate and the size ω of the periods are small. In the three studied limits (e → 0 then ω → 0), (ω → 0 then e → 0) and lately (e and ω → 0 together) the three dimensional equation of elasticity are approached by the two dimensional general equations of a linear anisotropic plate, the stretching and bending being coupled. This study points out the importance of the ratio of the two small parameters, indeed the moduli occuring in the two dimensional equations are different in the three limits. In each case a convergence proof is carried out.

Geometrically nonlinear theory of viscoelastic multilaminate piezoelectric plates and shells

Multilaminate metal--ceramic plates and shells are being widely used in different areas of technology [;3]. The article [15] presented a survey of studies attempting to construct linear models of elastic single-layer piezoceramic plates and shells. We should also note the work done in this direction in [4-6, ;;, 14-16]. The level achieved to date in attempts to construct linear models of elastic multilaminate plates is reported in the article [!3]. It follows from these articles that there is no attempt in the literature to construct a geometrically nonlinear theory of layered metal--ceramic plates and shells, particularly with allowance for the effects of viscosity. Nevertheless, such a theory needs to be developed in connection with the need to solve a whole range of nonlinear static and dynamic problems arising in the case of large deflections, when the strains and displacements are non!inearly related to each other. Also, as experiments show [;2, 2;], the piezoceramic materials presently in use have viscoelastic properties which may significantly affect the electromechanical behavior of elements made of them. In particular, there may be an undesirable increase in the temperature associated with dissipative heating. The dissipative heating of plates and shells under long-term harmonic loads was discussed in [9, I0] within the framework of a geometrically linear theory.

Bathymetric prediction from SEASAT altimeter data

The linear response function technique is used to analyze two 1300‐km tracks of SEASAT altimeter data and corresponding bathymetry in the Musician Seamounts region north of Hawaii. Bathymetry and geoid height are highly correlated in the 50‐ to 300‐km wavelength range. A predictive filter is developed which can operate on SEASAT altimetry in poorly surveyed oceanic regions to indicate the presence of major bathymétrie anomalies. Modeling of the bathymetry‐geoid correlation in the Musician region is attempted using the elastic plate model. The flexural rigidity D of the plate is not well constrained by our data but appears to lie in the range 5×1021 N m ‐ 5×1022 N m at the time of loading. Since the Musician Seamounts and the crust on which they lie are both Late Cretaceous in age, this value represents the effective flexural rigidity of very young lithosphere that was ‘frozen in’ at the time of voicanism, The modeling indicates that the general form of a predictive filter will strongly depend on various geologic parameters, especially the effective flexural rigidity. Hence, some a priori geologic constraints are necessary to estimate successfully the bathymetry from the altimeter data. Alternately, if high‐quality bathymetry is available, a crude estimate of the age of loading (i.e., voicanism) can be made from the altimeter data.

Approximation of Non‐Unique Solutions of a Homogeneous Elastic Plate Model

AbstractA dual variational description for a von Kármán plate version, capable of a dual finite‐dimensional approximation of non‐unique solutions is constructed, investigated, and tested on a numerical example.

Flexure of Anadarko Basin: ABSTRACT

The Anadarko basin in Oklahoma has long been a major oil and gas producing region and contains the deepest wells drilled in North America. The region has had a long sedimentary-tectonic history reaching back to the Proterozoic and was the site of an early Paleozoic basin. The present shape of the Anadarko basin, however, was developed in late Paleozoic times as a result of the uplift of the Wichita Mountains. COCORP seismic reflection profiles show at least 8 to 9 km (5 to 5.6 mi) of overthrusting northward, and the Anadarko basin was developed as a result of flexural bending of the lithosphere due to this shortening. Downwarping of the basin can be observed to extend for over 300 km (185 mi) northward, indicating a high flexural rigidity (Te > 40 km [25 mi]). However, nearer the Wichita front, the basin steepens rapidly as the post-Mississippian sediments thicken to over 20,000 ft (6,100 m). The shape of the bending is such that it cannot be explained by the use of a constant rigidity elastic plate model. We have modeled the post-Mississippian development of the Anadarko basin as the result of flexure of an elastic-plastic plate due to vertical and horizontal loading caused by the Wichita Mountains. Implications of these results for the development of the Anadarko basin and the mechanical properties of continental lithosphere will be discussed. End_of_Article - Last_Page 552------------

Lithospheric flexure and the evolution of sedimentary basins

The sediments that have accumulated in sedimentary basins during geological time represent a load on the lithosphere that should respond by flexure. Simple elastic and viscoelastic (Maxwell) plate models have been used to examine quantitatively the contribution of flexure to basin formation. The models have been used to predict the stratigraphy and gravity anomalies associated with basins for different thermal and loading histories. The predictions of the models have been compared with observed stratigraphy and free-air gravity anomalies from interior and cratonic basins. The best overall fit to the observations is for an elastic plate model in which the flexural strength of the lithosphere increases with age. A similar model has recently been used successfully to explain observations from continental margin basins, oceanic islands and seamounts, and deep-sea trench - outer rise systems. This model explains the increase in the overall width of basins during their evolution as well as the stratigraphy of the basin edges. The apparent decrease in the widths of some basins through time can be explained by the model if sediment deposition is followed by erosion of the basin and its edges. The models results suggest that the flexural properties of continental and oceanic lithosphere are generally similar and that flexure is an important factor to consider in backstripping studies in which the tectonic subsidence of a basin is isolated and stratigraphic studies in which relative changes of sea level are estimated.

The crustal structure of the Madagascar Ridge

Summary. The Madagascar Ridge is an elongated aseismic plateau in the southwest Indian Ocean, which separates two ocean basins of mid-to late Cretaceous age. From interpretations of a reversed refraction line on the ridge and a continuous gravity profile across it we conclude that it formed during the Cretaceous as thickened oceanic crust, at a mid-ocean ridge which possibly overlay a mantle hot-spot. The seismic data consist of record sections from five recording sonobuoys. Forty-one shots were fired along a 210 km long NNE—SSW line across the central part of the ridge. Simple horizontal-layer analysis of first arrivals reveals a lower crustal refractor with Vp= 7 kms−1 at a depth of 8–8.5 km and the Moho at an apparent depth of 18 km, but fails to characterize the crustal structure as either oceanic or continental. We obtain a definitively oceanic structure by using synthetic seismogram techniques together with Tau(p) analysis and second arrival correlations. The model which best fits the data consists of 2.2 km of water and 1 km of sediments overlying a velocity gradient from 4 to 7 kms−1. Beneath this, at a depth of 7.5 km, is a 9.5 km thick layer with Vp= 6.9–7.1 kms−1. A 7.5 kms−1 layer, not observed in the first arrival analysis, is found at the base of the crust. The Moho is a transition zone 1 km thick and occurs at ∼ 25 km depth. The gravity data consist of a continuous NE—SW profile across the ridge and on to the flanking basins. Two-dimensional gravity modelling allows a structure consistent with the synthetic seismogram results, and supports the existence of a high-density basal layer in the crust. We use in addition a time-series analysis approach to estimate the depth of compensation assuming an Airy model, and lithospheric thickness assuming an elastic plate model. Convolution of bathymetry along the profile with a range of model filters gives minimum rms gravity residuals for a crustal compensation depth of 23 ± 5 km, and a plate thickness of less than 2–5 km.

Earthquakes and bending of plates at trenches

The mechanisms, distribution, and total moment of earthquakes within the bending oceanic plate seaward of trenches constrain possible mechanical models of the lithosphere. The average annual horizontal slip in normal fault earthquakes, as estimated from the cumulative seismic moment, exceeds the total extension predicted by an elastic plate model. Thus bending is not predominantly an elastic process but must include a large amount of permanent deformation. Normal‐faulting earthquakes are associated with every major trench system in the Pacific and Indian oceans, but events indicating horizontal compression are rare. The predominance of normal faults over thrust faults, normal events as deep as 25 km within the lithosphere (well constrained by pP), and the existence of large (Ms > 7.5) normal fault earthquakes suggest that the average oceanic lithosphere is not under regional horizontal compressive stress which is a large fraction of the bending stresses. The location of thrust fault events as deep as 40–50 km within the bending plate requires the lithosphere to have significant strength at least as deep as 50 km. A two‐layer plate model with elastic‐perfectly plastic rheology can satisfy these seismological constraints and fit the observed shape of the trench and outer rise. Large regional stresses are not required, and variations in thickness and strength are not needed to fit individual profiles. In our preferred average model the mechanical plate is 50 km thick, with a weak (1‐kbar yield strength) upper layer and a strong (6 kbar) lower layer. The mechanism for plastic strain at yield is assumed to be slip accompanying earthquakes. Normal fault earthquakes are consistent with Coulomb criterion failure, but some other failure mechanism must be invoked for the thrust fault earthquakes. The bottom of the plate is thought to be defined by a transition to steady state creep, induced either by increasing temperature with depth or by a change from dry to wet rheology. Variations in regional stresses should produce corresponding changes in the shapes of the topographic profiles. An examination of the tectonic settings of a number of profiles suggests that increasing horizontal compression should decrease the curvature of the profile in the vicinity of the zero‐crossing point on the seaward wall of the trench.

The evolution of sedimentary basins on a viscoelastic lithosphere: theory and examples

Summary. A systematic approach is suggested for modelling the development of sedimentary basins. The theory, which partitions basin formation into initiating and isostatic adjustment processes, is applicable to all modes of basin formation if these processes are linear, or can he represented with sufficient accuracy in an incrementally linear form. The dynamics of regional isostatic adjustment are characterized by the Heaviside space-time Green functions for the response of elastic and viscoelastic (Maxwell) thin plate models of the lithosphere. It is shown, by convolving the Heaviside—Green functions with cylindrical surface loads, that the rate of isostatic adjustment on a viscoelastic lithosphere is a function of the wavelength of the surface load, long wavelengths being compensated most rapidly. Six archetypal initiating processes for sedimentary basin development are presented. These processes are those responsible for the subsidence of the Earth's surface which creates a depression in which water and sediments collect. Isostatic amplification of subsidence by sediment and water loads is cast in the form of an integral equation with isostatic Heaviside—Green functions as kernel. Specific examples, the basins that result from a graben initiating process, are compared with the largest scale structure of the North Sea Basin, a basin that is known to be underlain by a graben system. A model, in which a 50-km wide graben subsides exponentially with a time constant of 5 × 107yr during the interval 180–100 Myr bp, is shown to be consistent with the largest scale structure of the North Sea Basin if the underlying lithosphere is viscoelastic with a flexural rigidity of ∼5 × 1025 Nm and relaxation time constant ∼ 106 yr.

A detailed study of two earthquakes seaward of the Tonga Trench: Implications for mechanical behavior of the oceanic lithosphere

Two nearby earthquakes seaward of the Tonga trench, one normal faulting and one thrust faulting, were studied in detail by using comparative event, surface wave techniques. The focal depth for the thrust‐faulting event was found to be 49 km below sea level, consistent with the focal depth estimated from the pP‐P delay time. The normal‐faulting event had a depth of 14 km. These are the first precise depth determinations for earthquakes which are associated with the bending of the oceanic lithosphere before subduction. The pattern of horizontal deviatoric tension near the surface and compression within the interior of the lithosphere is consistent with the stresses predicted by an elastic plate model, although the depth of the thrust event suggests that either the elastic layer begins near the base of the crust or the rheological model should be modified to include a plastic layer of finite strength.

Isostatic Compensation on a Continental Scale: Local Versus Regional Mechanisms

Summary. Using the techniques of linear and quadratic programming, it can be shown that the isostatic response function for the continental United States, computed by Lewis & Dorman (1970), is incompatible with any local compensation model that involves only negative density contrasts beneath topographic loads. We interpret the need for positive densities as indicating that compensation is regional rather than local. The regional compensation model that we investigate treats the outer shell of the Earth as a thin elastic plate, floating on the surface of a liquid. The response of such a model can be inverted to yield the absolute density gradient in the plate, provided the flexural rigidity of the plate and the density contrast between mantle and topography are specified. If only positive density gradients are allowed, such a regional model fits the United States response data provided the flexural rigidity of the plate lies between 1021 and 1022 N m. The fit of the model is insensitive to the mantle/ load density contrast, but certain bounds on the density structure can be established if the model is assumed correct. In particular, the maximum density increase within the plate at depths greater than 34 kin must not exceed 470 kg m−3; this can be regarded as an upper bound on the density contrast at the Mohorovicic discontinuity. The permitted values of the flexural rigidity correspond to plate thicknesses in the range 5–10 km, yet deformations at depths greater than 20 km are indicated by other geophysical data. We conclude that the plate cannot be perfectly elastic; its effective elastic moduli must be much smaller than the seismically determined values. Estimates of the stress-differences produced in the earth by topographic loads, that use the elastic plate model, together with seismically determined elastic parameters, will be too large by a factor of four or more.

The Deflection of an Ice Sheet by a Submerged Gas Source

The deflection of an infinite ice sheet by a submerged gas source, as would result from an undersea gas or oil well blowout is analysed utilizing an elastic thin plate model. The results show that fracture may occur either at the bubble center or just beyond the bubble edge, depending upon the bubble depth, the ice thickness, and the material properties assumed for the ice sheet. For ice one meter in thickness and a trapped gas depth greater than 100 mm, fracture at the bubble edge is probable. The critical bubble radius for failure varies rapidly with ice thickness, bubble depth, and the ice properties, which in view of the variability of the latter, makes the prediction of actual bubble radii to cause failure subject to a large degree of uncertainty.