Matthew J Warbritton

Neal R Iverson
email: niverson@iastate.edu

France Lagroix

Anders Schomacker

Strain patterns in glacitectonically thrusted sediments and conditions during thrusting

Keywords: Glacitectonic deformation, Thrust sheets, Anisotropy of magnetic susceptibility, Permafrost, Pore pressure

published or not. The documents may come

Strain patterns in glacitectonically thrusted sediments and conditions during thrusting

Introduction

Glacitectonic thrust complexes with along-flow lengths of kilometers and depths of tens to a few hundreds of meters (e.g., [START_REF] Pedersen | Architecture of Glaciotectonic Complexes[END_REF][START_REF] Phillips | Glacitectonics[END_REF] are among the most spectacular of glacial geologic structures. Interest in them derives from their geomorphic expression, commonly as composite ridges that form end moraines or as the hills of hill-hole pairs (e.g., [START_REF] Aber | Glaciotectonism[END_REF]. They are also of interest as analogs for orogenic belts [START_REF] Croot | Glacio-tectonic structures: a mesoscale model of thin-skinned thrust sheets?[END_REF][START_REF] Hudleston | Glacial movement and thrust emplacement[END_REF][START_REF] Bennett | The morphology, structural evolution and significance of push moraines[END_REF][START_REF] Van Der Wateren | Processes of glaciotectonism[END_REF][START_REF] Phillips | Glacitectonics[END_REF] and because of the short periods over which they can form [START_REF] Benediktsson | The 1890 surge end moraine at Eyjabakkajökull, Iceland: a reassessment of a classic glaciotectonic locality[END_REF]. Environmental effects of thrust complexes include their influence on subsurface water flow (e.g., [START_REF] Klint | A polymorphological landform approach for hydrogeological applications in heterogeneous glacial sediments[END_REF], on rates of mass wasting and erosion of coastal cliffs [START_REF] Pedersen | Prediction and risk evaluation of chalk cliff collapse: the PROTECT project[END_REF][START_REF] Kołodziejczyk | The problem of stabilization of landslides in glaciotectonically disturbed areas. Case study: road engineering structure[END_REF], and on site selection for future nuclear waste repositories (Talbot, 1999;[START_REF] Iverson | Glacier-bed geomorphic processes and hydrologic conditions relevant to nuclear waste disposal[END_REF].

Importantly, glacitectonic thrust complexes may also provide information about whether climates at the margins of former glaciers were sufficiently cold to sustain permafrost. Thrust complexes form beneath glacier margins and can extend in front of them more than 5 km in exceptional cases [START_REF] Aber | Glaciotectonism[END_REF]. Although complexes sometimes contain thrust blocks consisting of sedimentary rocks, some or all of thrusted material is usually weaker, unlithified, glacigenic sediment. As emphasized in reviews by [START_REF] Bennett | The morphology, structural evolution and significance of push moraines[END_REF] and [START_REF] Van Der Wateren | Processes of glaciotectonism[END_REF], such sediments are commonly inferred to have been frozen during thrusting (e.g., [START_REF] Mathews | Deformation of soils by glacier ice and the influence of pore pressures and permafrost[END_REF][START_REF] Moran | Glacier-bed landforms of the prairie region of North American[END_REF][START_REF] Thomas | The origin of the glacio-dynamic structure of the Bride Moraine, Isle of Man[END_REF][START_REF] Menzies | Sand intraclasts within a diamicton mélange, southern Niagara Peninsula, Ontario, Canada[END_REF][START_REF] Mooers | Ice-marginal thrusting of drift and bedrock: thermal regime, subglacial aquifers, and glacial surges[END_REF][START_REF] Boulton | The sedimentary and structural evolution of a recent push moraine complex: Holmströmbreen, Spitsbergen[END_REF][START_REF] Benn | Pleistocene glacitectonic landforms and sediments around central Magellan Strait, southernmost Chile: evidence for fast outlet glaciers with coldbased margins[END_REF][START_REF] Clayton | Effects of late Pleistocene permafrost on the landscape of Wisconsin[END_REF][START_REF] Burke | Imbricate thrust stack model for the formation of glaciotectonic rafts: an example from the Middle Pleistocene of north Norfolk[END_REF].

Although proglacial thrusting can definitively occur in the absence of permafrost (e.g., [START_REF] Ingólfsson | Large-scale glaciotectonic deformation of soft sediments: a case study of a late Weichselian sequence in western Iceland[END_REF][START_REF] Van Der Wateren | Processes of glaciotectonism[END_REF][START_REF] Kuriger | Episodic reactivation of large-scale push moraines in front of the advancing Taku Glacier, Alaska[END_REF][START_REF] Benediktsson | Formation of submarginal and proglacial end moraines: implications of ice-flow mechanism during the 1963-64 surge of Brúarjökull, Iceland[END_REF], we explore whether the distribution of strain in individual thrust blocks of complexes can be used to test whether sediments were frozen during thrusting. Specifically, frozen sediments are expected to be stronger (e.g., [START_REF] Moore | Deformation of debris-ice mixtures[END_REF] and less subject to ductile strain than unfrozen sediments (e.g., [START_REF] Feeser | On the mechanics of glaciotectonic contortion of clays[END_REF], particularly at temperatures well below the freezing point, at which weakening by pre-melted water is minimal [START_REF] Moore | Deformation of debris-ice mixtures[END_REF]Wang and Nishimura, 2017).

Herein we investigate strain patterns within the fine-grained sediments of selected thrust sheets of two glacitectonic complexes of coastal Denmark: Rubjerg Knude and Møns Klint (e.g., Johnstrup, 1874[START_REF] Johnstrup | Om de geologiske forhold i den nordlige del af Vendsyssel[END_REF][START_REF] Jessen | Kortbladene Skagen, Hirtshals, Frederikshavn, Hjørring og Løkken. Danmarks Geologiske Undersøgelse I[END_REF][START_REF] Pedersen | Architecture of Glaciotectonic Complexes[END_REF]. Strain patterns are determined by measuring the anisotropy of magnetic susceptibility (AMS) of the sediments. Determination of the magnetic mineralogy, results of ring-shear experiments in which sediments were sheared, and models of AMS fabric development in orogenic fold and thrust belts are used to interpret the AMS data. We find that thrust blocks behaved too rigidly to reset fabrics during thrusting, except within decimeters of bounding thrust surfaces. Considering shear stresses on thrust faults indicates, however, that a combination of positive pore-water pressure and a proglacial surface sloping down-flow above thrusts can cause thrusting of rigid slabs without added shearing resistance provided by pore ice.

Field settings

The Rubjerg Knude and Møns Klint glacitectonic complexes of Denmark consist of décollement surfaces that dip gently up-glacier, as inferred from reconstructions based on exposed stratigraphy and structures and isostatic depression by ice (Pederson, 2005;[START_REF] Pedersen | Structural development of Maglevandsfald: a key to understanding the glaciotectonic architecture of Møns Klint, SE Denmark[END_REF]. The décollements are overlain by thrust faults consisting of ramps and flats and associated anticlines and synclines. Both complexes formed in front of the advancing margin of the Scandinavian Ice Sheet (e.g., [START_REF] Houmark-Nielsen | Southwest Scandinavia, 40-15 kyr BP: palaeogeography and environmental change[END_REF][START_REF] Pedersen | Architecture of Glaciotectonic Complexes[END_REF] and are structurally similar to thrusted and folded orogenic belts. These complexes lie in the region that Sollid and Sorbel (1988) thought was a sub-freezing marginal zone of the ice sheet, based on relict ice-cored moraines and ice-wedge polygons.

Rubjerg Knude

The Rubjerg Knude complex, first recognized by [START_REF] Johnstrup | Über die Lagerungsverhaltnisse und die Hebungs-phänomene in den Kreidefelden auf Mön und Rügen[END_REF][START_REF] Johnstrup | Om de geologiske forhold i den nordlige del af Vendsyssel[END_REF] and studied in detail by [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF], is located in northwestern Denmark along the northwest coast of Jutland (Fig. 1). The complex is well exposed for 6 km along a coastal cliff that is 60 m above sea level at its highest point. The complex consists entirely of Weichselian or younger deposits, including marine mud, glaciolacustrine mud, glaciofluvial sand and gravel, and till.

Of interest here, with decreasing age, are the Stortorn Formation (SF), Lønstrup Klint Formation (LKF), and Rubjerg Knude Formation (RKF). The SF consists of predominantly clay of the Older Yoldia Sea of the Middle Weichselian [START_REF] Jensen | Late Weichselian deglaciation pattern in the southwestern Baltic: Evidence from glacial deposits off the island of Møn, Denmark[END_REF]. The sea existed until about 32 ka BP [START_REF] Seidenkrantz | Middle Weichselian to Holocene palaecology in the eastern Kattegat, Scandinavia: foraminifera, ostracods and 14C measurements[END_REF] when the outlet to the North Sea was blocked by the Norwegian Channel Ice Stream. This created an ice-dammed lake that persisted from ~32 to 29 ka BP [START_REF] Sadolin | Lacustrine sedimentation and tectonics: an example from the Weichselian at Lønstrup Klint, Denmark[END_REF][START_REF] Houmark-Nielsen | A lithostratigraphy of Weichselian glacial and interstadial deposits in Denmark[END_REF] and deposited the glaciolacustrine mud that dominates the LKF. Close to 29 ka BP, the lake drained catastrophically, resulting in an unconformity between the LKF and overlying glaciofluvial sand and gravel of the RKF [START_REF] Sadolin | Lacustrine sedimentation and tectonics: an example from the Weichselian at Lønstrup Klint, Denmark[END_REF]. At ~29 ka BP, during deposition of the RKF, the Kattegat Ice Stream [START_REF] Houmark-Nielsen | Testing OSL failures against a regional Weichselian glaciation chronology from southern Scandinavia[END_REF], advanced south-southeast across the northern tip of Jutland (Fig. 1), initiating deformation at Rubjerg Knude along a décollement ~40 m below sea level in the marine clay of the SF [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF]. Faulting and folding extended 12 km in front of the glacier margin and resulted in average shortening of 50%; thrust faults generally dip more steeply in the northern and central areas where the décollement is deepest (Fig. 2A) [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF].

In this study we focus on a distal section of the complex, called Ulstrup [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF], and a central section called Grønne Rende, with steeply dipping thrust sheets (Fig 2). The Ulstrup section consists of predominantly horizontal layers of the LKF overlying the RKF. The section was considered to be undeformed until [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF] found evidence for a horizontal décollement at the southern end of the section along which the flat-lying sediment layers have been displaced ~500 m. Otherwise the stratigraphy in this distal part of the complex is mostly intact. In contrast, the Grønne Rende section consists of 13 steeply-dipping, and in some cases near-vertical, thrust sheets. The LKF forms the most conspicuous sheets (Fig. 3), owing to the ability of its mud to support vertical exposures. The sand and gravel of the RKF is interpreted to have been syntectonically deposited in between and on top of thrust sheets during progressive faulting and sheet rotation during longitudinal shortening [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF]. The average displacement along these faults is 70 m [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF].

Small-scale structures may provide insight into the formation of the complex. Ball-andpillow structures within the LKF may indicate where rapid loading caused pore-water pressure to increase, leading to local diapir-like deformation of bedding surfaces [START_REF] Lowe | Water escape structures in coarse-grained sediments[END_REF]. Frost wedges observed by [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF] in the RKF near the paleo-ground surface indicate that some frozen sediments were present during deformation. Also present in the RKF and perhaps indicative of frozen sediments are kilelags [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF], fissures that have opened during initial brittle deformation and are filled with water-transported, syntectonically-deposited sand [START_REF] Sjørring | The glacial stratigraphy of the island of Als, southern Denmark[END_REF].

Møns Klint

Møns Klint, in southeastern Denmark at Høje Møn (Fig. 1), has been a foundational location for study of glacitectonics (e.g., [START_REF] Puggaard | Møens Geologie. Universitetets Guldmedaille Priisskrift[END_REF][START_REF] Aber | Kineto-stratigraphy at Hvideklint, Møn, Denmark and its regional significance[END_REF][START_REF] Aber | Model for glaciotectonism[END_REF][START_REF] Aber | Glaciotectonism[END_REF]. The exposure, in coastal cliffs that reach heights of 130 m and span 4 km, consists of Upper Cretaceous chalk thrusted over Weichselian sediments (Fig. 4). The sedimentary succession includes the Ristinge Klint Till deposited ~60 ka BP by the Baltic ice advance from the southeast, glaciolacustrine mud deposited after the recession of the Baltic ice, and till deposited 25-18 ka BP by ice advances from the northeast and glaciofluvial sand [START_REF] Houmark-Nielsen | Pleistocene stratigraphy and glacial history of the central part of Denmark[END_REF][START_REF] Pedersen | Structural development of Maglevandsfald: a key to understanding the glaciotectonic architecture of Møns Klint, SE Denmark[END_REF].

The initial deformation at Møns Klint commenced during the advance of the Young Baltic Ice Stream from the south-southeast at ~18 ka BP (Houmark-Nielson and Kjaer, 2003).

Deformation occurred along a décollement 80-100 m below sea level [START_REF] Pedersen | Structural development of Maglevandsfald: a key to understanding the glaciotectonic architecture of Møns Klint, SE Denmark[END_REF] and extended approximately 9 km in front of the advancing glacier. A later ice advance from the northeast out of Sweden (16-17 ka) (Houmark-Nielsen, 2007) superimposed deformation onto the eastern edge of the northern part of the complex from Slotsgavlene to Maglevandsfald (Fig. 4) [START_REF] Pedersen | Structural development of Maglevandsfald: a key to understanding the glaciotectonic architecture of Møns Klint, SE Denmark[END_REF]. The southernmost expression of the superimposed deformation is near Maglevandsfald where a till wedge within chalk is folded along an axis trending NW-SE [START_REF] Pedersen | Structural development of Maglevandsfald: a key to understanding the glaciotectonic architecture of Møns Klint, SE Denmark[END_REF]. Farther south in the proximal part of the complex [START_REF] Pedersen | Superimposed deformation in glaciotectonics[END_REF], there is no evidence of the advance from the northeast, with thrust faults striking approximately perpendicular to the Baltic Ice Stream advance from the south-southeast (Fig. 4).

Methods

We use AMS, as a proxy for particle alignment, to interpret strain patterns in thrusted blocks.

This method has been applied to tills deformed subglacially, using ring-shear experiments to calibrate AMS to strain magnitude and direction [START_REF] Thomason | Deformation of the Batestown Till of the Lake Michigan Lobe[END_REF][START_REF] Shumway | Magnetic fabrics of the Douglas Till of the Superior lobe: exploring bed-deformation kinematics[END_REF][START_REF] Vreeland | Magnetic fabrics of drumlins of the Green Bay Lober, southeastern Wisconsin[END_REF][START_REF] Mccracken | Origin of the active drumlin field at Múlajökull, Iceland: New insights from till shear and consolidation patterns[END_REF][START_REF] Ives | Genesis of glacial flutes inferred from observations at Múlajökull, Iceland[END_REF], and applied to glacier ice with debris (Fleming et al., 2013a) and to sediments with glaciotectonic structures (Fleming et al., 2013b). Using AMS in glacial sediments requires collecting multiple, closely-spaced, intact sediment specimens, so a fabric representative of the sampled region can be determined. Tests to determine magnetic mineralogy of the sediments (Appendix) and a ringshear experiment on mud from Rubjerg Knude also contribute to our interpretations.

AMS Sampling

To collect samples a planar face is created by scrapping an exposure sufficiently to expose undisturbed sediment. Aluminum, 18 mm, cubic boxes, open on two ends, are inserted in the face in as tight a grid as was possible without disturbing adjacent boxes (Fig. 5), with 18-25 samples used to determine a fabric. Plastic boxes usually used for such sampling (e.g., [START_REF] Mccracken | Origin of the active drumlin field at Múlajökull, Iceland: New insights from till shear and consolidation patterns[END_REF] are too weak to be pressed into the hard, consolidated muds of glacitectonic complexes. After measuring the orientation and position of each box, it is excavated, with excess mud trimmed from the open ends of the box with a wire saw. With a plunger the mud is extruded into a plastic box of the same size and the box capped, labelled, and sealed in plastic to inhibit moisture loss. In a few cases sandier layers can be sampled by pushing plastic boxes into the sediment directly.

At Rubjerg Knude, sampling was focused on thrust sheets in the Grønne Rende section (Fig. 2B), but additional samples were collected within the mostly intact stratigraphy of the distal Ulstrup section (Fig. 2C). The distal end of the complex was chosen for sampling to determine strain patterns in the absence of pervasive thrusting, so the effects of thrusting on strain in the Grønne Rende section could be isolated. Samples at both sections were mainly from the glaciolacustrine mud of the LKF because its cohesion allowed the collection of minimally disturbed samples, unlike the friable sand of the RKF. At Ulstrup, flat-lying mud (striking 273°, dipping 3° south) of the LKF (Fig. 2B) was sampled along two vertical transects (three fabrics per transect), beneath the unconformity with the RKF.

At Grønne Rende the primary objective was to sample thrust blocks along transects normal to fault surfaces. Three thrust sheets (GR10, GR07, and GR05, Fig. 2a) with conspicuous, nearvertical exposures of the LKF were chosen for study based largely on their accessibility and safety concerns. These sheets have strikes of 266-285° in agreement with thrusting driven by the Kattegat Ice Stream advancing from the north and dips of 56° N to near vertical. Also accessible at GR10 and GR07 were dip-slopes of the LKF eroded into the cliff-face by mass wasting processes along bedding planes, so these surfaces were also sampled, taking care in all cases to measure the orientation of samples.

Our principal concern is how strain varied with distance from the surfaces of thrust faults, so through-going fault surfaces needed to be identified. At Rubjerg Knude we used slickensides that result from P shears and Riedel shears that grow and intersect during slip, forming asymmetric steps on fault surfaces [START_REF] Petit | Criteria for the sense of movement on fault surfaces in brittle rocks[END_REF][START_REF] Doblas | Slickenside kinematic indicators: Tectonophys[END_REF] near the base of the LKF (Fig. 6). In thrust sheets GR10 and GR05, slickensides could at some locations be identified and were used to reference fault-normal transects of AMS samples (Fig. 7A,B). No slickensides were found at thrust sheet GR07. Although some samples for AMS fabrics were gathered in thrust sheets in the absence of adjacent slickensides [START_REF] Warbritton | mineral composition of the analyzed samples. Specifically, assessing whether paramagnetic minerals, ferrimagnetic minerals or a combination of both control the AMS is crucial. The AMS of paramagnetic minerals is dictated by the mineral's crystal symmetry and is said to have a[END_REF], we report data from fault-normal transects only where slickensides provided an indicator of thrust position (13 fabrics). We also consider AMS fabrics collected from two eroded dip-slopes (6 fabrics) distant from thrust faults. At Møns Klint, glaciolacustrine mud of the Middle Weichselian in the proximal part of the complex (Fig. 4), overlain along a sharp, gently dipping contact by Maastrichtian chalk (Fig. 5), was sampled along a 4 m long section (Fig. 8). The fault between the chalk and mud strikes 79° (±7°) and dips 18° (±4°), consistent with thrusting driven by the Young Baltic ice stream advance from south-southeast (Pederson and Gravesen, 2009). Samples were gathered in four adjacent vertical transects, 0.9-2.0 m high, referenced to the contact and sufficient for 17 AMS fabrics.

AMS analysis

AMS was measured with a KLY-3S Kappabridge at the University of Wisconsin-Madison.

This device subjects a sample to a magnetic field of constant strength, H, in 15 different orientations. The intensity of magnetization, M, is defined as kH, where k is the susceptibility [START_REF] Jelínek | Statistical processing of anisotropy of magnetic susceptibility[END_REF][START_REF] Tarling | The Magnetic Anisotropy of Rocks[END_REF]. In three dimensions, the susceptibility is described with a second-rank tensor, and the principal susceptibilities of this tensor, k1, k2, and k3, are represented as the long, intermediate, and short axes, respectively, of an ellipsoid. In materials that are anisotropic, the susceptibility varies with sample orientation. For non-equant magnetic grains that are multi-domain or pseudo-single-domain and have shape anisotropy, such as magnetite, the susceptibility is strongest parallel to the long axes of grains [START_REF] Tarling | The Magnetic Anisotropy of Rocks[END_REF]. Deformation of sediment preferentially aligns grains and creates AMS. As described in the Appendix, mineral magnetic analyses were conducted at the Institut de Physique du Globe de Paris to determine the mineralogy and magnetic grain size of the dominant AMS carrier of the muds from the two complexes.

If the orientations of principal susceptibilities are measured for each of multiple till samples, they together define a fabric that helps indicate the magnitude and orientation of strain [START_REF] Hooyer | Magnetic fabric of sheared till: A strain indicator for evaluating the bed deformation model of glacier flow[END_REF]. The orientations of maximum clustering of k1, k2, and k3 are each characterized with a V1 eigenvector, and the strength of the preferred orientation about V1 is characterized with an S1 eigenvalue [START_REF] Mark | Analysis of axial orientation data, including till fabrics: Bull[END_REF]. We thus analyze orientations of principal susceptibilities separately, rather than using tensor statistics. For poorly clustered data, our technique can lead to V1 eigenvectors for k1, k2, and k3 orientations that are not orthogonal. However, principal susceptibility orientations from our field samples are well-clustered, so we elect to follow the tradition in glacial geology of using eigenvectors and eigenvalues to characterize fabrics. S1 = 1.0 indicates perfect alignment and S1 = 0.33 indicates an isotropic distribution (i.e., no alignment).

Orientation data are plotted in lower hemisphere, equal-area stereonets. Importantly, in stereonets displaying fabrics from dipping thrust sheets, the plane of the primitive circle is taken to be the attitude of the thrust sheet, as indicated by the orientations of thrust faults adjacent to sampling locations or as indicated by bedding attitude in the case of samples from dip-slopes.

This allows easier comparison with experimental and field data in which shear or bedding is horizontal. At sections studied where both fault and bedding attitudes could be measured, they tended to be nearly the same (< 3 ⸰ differences in strike and dip), although elsewhere at Rubjerg Knude that is not necessarily true (Pederson, 2005). In the massive mud sampled at Møns Klint, no bedding could be identified.

Ring-shear experiment

To help establish the relationship between shear strain orientation and the AMS signature, an experiment with a ring-shear device was conducted with mud of the LKF. The device [START_REF] Iverson | A ring-shear device for the study of till deformation: tests on tills with contrasting clay contents[END_REF] has been used extensively to shear subglacial till with the same objective [START_REF] Hooyer | Magnetic fabric of sheared till: A strain indicator for evaluating the bed deformation model of glacier flow[END_REF][START_REF] Shumway | Magnetic fabrics of the Douglas Till of the Superior lobe: exploring bed-deformation kinematics[END_REF][START_REF] Vreeland | Magnetic fabrics of drumlins of the Green Bay Lober, southeastern Wisconsin[END_REF][START_REF] Mccracken | Origin of the active drumlin field at Múlajökull, Iceland: New insights from till shear and consolidation patterns[END_REF][START_REF] Ives | Genesis of glacial flutes inferred from observations at Múlajökull, Iceland[END_REF]. Sediment with its pore water drained to atmospheric pressure is sheared within a ring-shaped chamber (O.D.= 0.6 m, width = 0.115 m) between horizontal platens. The upper platen, which can move up or down as sediment porosity changes, applies a steady stress normal to the shear plane. During shear, the lower platen and lateral walls of the lower half of the sample chamber rotate at a constant speed. Shearing occurs in a zone, < 35 mm thick, centered on the break between the rotating lower walls and static upper walls.

The LKF mud was first vigorously mixed with distilled water so aggregates would break down and grain orientations would be randomized. The mixture was then added to the sample chamber as a slurry to a thickness of 0.06 m. A normal stress (65 kPa) was applied for a period of 11 days to consolidate the mud. The amount of consolidation (14.5%) was determined by using two dial gauges to measure the downward movement of the upper platen, which resulted from the decrease in specimen thickness. The test was then run at a constant shear rate of 1.27 × 10 -5 m/s to a displacement of 0.6 m along the sample centerline. After the test was completed, 53 samples were collected for AMS analysis. A smooth, flat surface was exposed just below the uppermost extent of the shear zone, as inferred from past experiments on tills [START_REF] Hooyer | Magnetic fabric of sheared till: A strain indicator for evaluating the bed deformation model of glacier flow[END_REF], and samples were collected by pressing into the mud plastic boxes like those used for the field measurements.

Results

Experiments

Tests aimed at assessing magnetic mineralogy indicate that the dominant carrier of magnetic susceptibility of the glaciolacustrine muds from the two thrust complexes is multi-domain or pseudo-single domain magnetite (Appendix). Such magnetite has normal shape anisotropy [START_REF] Tarling | The Magnetic Anisotropy of Rocks[END_REF], so the orientations of principal susceptibilities should reflect grain orientation.

This expectation is confirmed at Rubjerg Knude by the ring-shear experiment conducted on the mud sampled there (Fig. 9A). The V1 orientation based on 53 k1 orientations indeed parallels the shearing direction and plunges "up-glacier" 11°, consistent with past ring-shear experiments on AMS of tills with magnetite as the dominant carrier mineral, including tills derived from remobilized lake sediments (Fig. 9B) [START_REF] Hooyer | Magnetic fabric of sheared till: A strain indicator for evaluating the bed deformation model of glacier flow[END_REF][START_REF] Thomason | Deformation of the Batestown Till of the Lake Michigan Lobe[END_REF][START_REF] Vreeland | Magnetic fabrics of drumlins of the Green Bay Lober, southeastern Wisconsin[END_REF][START_REF] Ives | Genesis of glacial flutes inferred from observations at Múlajökull, Iceland[END_REF]. Also consistent with past experiments are the orientations of V1 for k2 and k3: k2 orientations are horizontal and perpendicular to shear, and k3 orientations plunge steeply (~80°) "down-glacier." However, unlike past results of experiments on tills, including on those consisting of remobilized lake sediments (Fig. 9B), clustering of k1 and k2 orientations is weak, with S1= 0.59 in each case as opposed to S1 > 0.9 for fine-grained till [START_REF] Hooyer | Magnetic fabric of sheared till: A strain indicator for evaluating the bed deformation model of glacier flow[END_REF]. Rather than clustering strongly, these principal susceptibilities form asymmetric girdles that dip gently "up-glacier" (Fig. 9A).

Unlike many studies of the magnetism of deformed rocks, we do not use various indices of the magnitude of anisotropy of individual samples to infer strain; ring-shear experiments conducted to wide a range of strains on tills that contain magnetite as the source of AMS show no systematic dependence of these indices on strain magnitude [START_REF] Hooyer | Magnetic fabric of sheared till: A strain indicator for evaluating the bed deformation model of glacier flow[END_REF][START_REF] Vreeland | Magnetic fabrics of drumlins of the Green Bay Lober, southeastern Wisconsin[END_REF].

Rubjerg Knude

Principal-susceptibility orientations for the six fabrics from the distal end of the Rubjerg Knude thrust complex (Ulstrup section) display a relatively uniform pattern (Fig. 10). Here, the mud and overlying sand units are horizontal and with little macroscopic evidence of deformation.

A stereonet combining principal-susceptibility orientations from these fabrics (Fig. 11A) demonstrates strong clustering of the three principal susceptibilities (S1 ≥ 0.81). Orientations of k1 cluster near perpendicular to the glacier flow direction (south-southwest, [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF], k2 orientations are close to horizontal and parallel to the flow direction, and k3 orientations plunge near vertically.

In thrust sheets of the Grønne Rende section, four fabric transects, one in GR10 and three in GR05, were measured normal to bounding thrust faults with well-exposed fault slickensides (Fig. 12,13). In all cases, orientations of principal susceptibilities are clustered (S1 ≥ 0.77). However, two distinct fabric patterns are evident depending upon distance from the fault. Within 0.1 m and ~0.7 m of the GR10 and GR05 fault surfaces, respectively, orientations of principal susceptibilities are similar to those of ring-shear experiments (Fig. 9), if data are plotted relative to the attitude of the fault surfaces. If all data from close to faults are combined in a single such stereoplot, the similarity is clearest (Fig. 11B, compare with Fig. 9A,B): the V1 of k1 orientations is roughly parallel to the glacier flow direction (SSW) and plunges gently up-glacier relative to the plane of the fault. Orientations of k2 cluster perpendicular to the shear direction. Orientations of k3 are clustered the strongest (S1 = 0.92) and steeply plunge down-glacier relative to the plane of the fault. A difference between the field and ring-shear AMS data is that k2 and k3 clusters are rotated ~10º around the axis of the k1 cluster In contrast, farther away from faults, orientations of principal susceptibilities cluster similarly to fabrics from the distal part of the thrust complex at Ulstrup (Fig. 11C, compare with Fig. 11A): k1 orientations cluster in the fault plane and sub-perpendicular to the flow direction; k2 orientations also cluster in the fault plane but sub-parallel to the flow direction. Orientations of k3 cluster normal to the fault surface. Unlike fabrics from Ulstrup, k1 and k2 clusters are rotated counterclockwise 27° from being perpendicular and parallel to the regional glacier flow direction, respectively. Possibly significant, however, is that five of the seven fabrics used to compile Figure 11C are from thrust sheet GR05, which strikes in a direction consistent with a local thrusting direction that is similarly rotated (Fig. 13).

All fabrics from samples collected from eroded dip-slope surfaces away from faults and plotted on stereonets relative to the dip of the bedding also have AMS fabrics similar to those at the distal end of the thrust complex. Combining the susceptibility orientations from these fabrics on one stereonet shows this clearly (Fig. 11D, compare with Fig. 11A). Orientations of k1 cluster near perpendicular to the glacier flow direction; k2 orientations cluster in the plane of the bedding and parallel to the flow direction, and k3 orientations cluster parallel to the glacier flow direction and plunge near-normal to bedding. A slight difference is that k2 and k3 clusters from dip-slope fabrics indicate they dip up-glacier relative to bedding, approximately 15°.

Møns Klint

Fabrics from the exposure at Møns Klint, if plotted on stereonets relative to the plane of the fault and grouped according to their distance from it (Fig. 14), are similar to some fabrics at Rubjerg Knude. Orientations of k1 cluster either roughly perpendicular to the glacier-flow direction (NNW, as inferred by [START_REF] Pedersen | Structural development of Maglevandsfald: a key to understanding the glaciotectonic architecture of Møns Klint, SE Denmark[END_REF] from the strikes of thrust faults in the complex) or at an angle less than ~45° to the flow direction. Orientations of k3 tend to cluster with steep plunges up-glacier; k2 orientations are the most weakly clustered but tend either to plunge slightly down-glacier, where k1 clusters are perpendicular to flow (Fig. 14, < 0.36 m, 1.19-1.8 m), or to trend at an angle greater than ~50° to the flow direction where k1 clusters are more closely aligned with the flow direction (Fig. 14,0.51-0.98 m). The former fabrics are similar to those of the intact distal stratigraphy of the Rubjerg Knude complex (Fig. 11A), as well as parts of thrust sheets in that complex away from faults (Fig. 11C,D).

Discussion

AMS fabric interpretation

In addition to our results indicating that the AMS of the thrust sheets is controlled by multidomain or pseudo-single domain magnetite (Appendix), we have two means of interpreting AMS fabrics. Results from ring-shear experiments on the LKF mud and on tills provide a model for AMS development in till deformed predominately in simple shear. In addition, models of AMS fabric development in larger-scale compressional tectonic environments, tested experimentally, provide guidance for interpreting fabric development during layer-parallel shortening.

Experimental shearing of the LKF mud yields V1 eigenvectors for the three principal susceptibilities that are similar to those obtained in identical experiments conducted on finegrained tills in which magnetite was the dominant magnetic mineral (Fig. 9). Eigenvectors based on k1 orientations are close to the shearing direction (within 11º) and plunge gently up-glacier; those based on k2 orientations are nearly perpendicular to the shearing direction (within 7º) in the plane of the fault, and those based on k3 plunge steeply down-glacier.

Dissimilar, however, is that in the experiment on mud, k1 and k2 orientations did not cluster but formed asymmetric girdles (Fig. 9A), despite shearing displacements that were just as large as in the till experiments. The lack of clustering of k1 and k2 orientations likely reflects strain localization in the mud in thin shear bands, a characteristic behavior of clay-rich soils in geotechnical experiments in which strain commonly localizes in zones 0.3-10 mm thick [START_REF] Morgenstern | Microscopic structures in kaolin subjected to direct Shear[END_REF][START_REF] Hicher | Microstructural analysis of strain localization in clay[END_REF]. AMS samples, in contrast, were collected after the experiment over a mud thickness of 16 mm (the sampling box dimension), which would have resulted in the collection of significant volumes of unsheared mud. This unsheared mud, owing to its consolidation prior to the experiment and associated rotation of particles toward the horizontal, would have had a k1-k2 girdle in the plane of shear and a vertical cluster of k3 orientations (e.g., [START_REF] Tarling | The Magnetic Anisotropy of Rocks[END_REF][START_REF] Parés | Sixty years of anisotropy of magnetic susceptibility in deformed sedimentary rocks[END_REF]. The asymmetric k1-k2 girdle that is observed, therefore, likely reflects a significant volume of unsheared mud in the AMS samples. Thus, although eigenvectors from the mud experiment are fully consistent with those developed from simple shear of tills, clustering of k1 and k2 orientations from experiments with fine-grained till probably better reflects the AMS response of the mud to simple shear, as long as thicknesses of shear zones sampled in the field exceeded the thickness of sample boxes.

We conclude, therefore, that the clustered orientations of principal susceptibilities obtained in ring-shear experiments on tills, formed subglacially through remobilization of glaciolacustrine mud and with their AMS controlled by magnetite (Fig. 9b), is an appropriate model for the muds of this study, if they have been subjected to a sufficient strain.

On that basis, a solid conclusion is that the fabric pattern close to thrust faults (< ~0.7 m) at Rubjerg Knude (Fig. 11b) reflects shear of the mud during thrusting. Shear must have been to a sufficiently high strain close to faults to fully overprint the AMS fabric that existed prior to thrusting. At Møns Klint, on the other hand, despite collection of AMS samples as close as 3 cm to the fault between the chalk and underlying mud, this fabric pattern indicative of simple shear is absent.

Additional insights into the fabric patterns at Rubjerg Knude and Møns Klint come from studies of AMS fabric development in deformed rocks (see Borrradaile and Henry, 1997; Parés, 2015, for reviews) and sediments (e.g., [START_REF] Collombat | Magnetic fabric as a strain indicator in unconsolidated sediments of the Chile triple junction area[END_REF][START_REF] Housen | Magnetic fabrics from the Costa Rica margin: sediment deformation during the initial dewatering and underplating process: Earth Planet[END_REF][START_REF] Caricchi | Distinct magnetic fabric in weakly deformed sediments from extensional basins and fold-and-thrust structures in the Northern Apennine orogenic belt (Italy)[END_REF] in crustal tectonic environments. A leading model developed from such field studies (notably by [START_REF] Averbuch | Magnetic fabric as a structural indicator of the Deformation path within a fold-thrust structure: a test case from the Corbières (NE Pyrenees, France)[END_REF][START_REF] Borradaile | Tectonic applications of magnetic susceptibility and its Anisotropy[END_REF][START_REF] Bakhtari | Magnetic fabrics of Tertiary sandstones from the Arc of Fars (Eastern Zagros, Iran)[END_REF][START_REF] Saint-Bezar | Magnetic fabric and petrographic investigation of hematite-bearing sandstones within ramp-related folds: examples from the South Atlas Front (Morocco)[END_REF] and more recently tested in analogue sandbox experiments [START_REF] Almqvist | Bulk strain in orogenic wedges based on insights from magnetic fabrics in sandbox models[END_REF] allows interpretation of fabrics resulting from layer-parallel shortening (LPS) in fold and thrust belts and accretionary prisms. The model (Fig. 15) describes the change in AMS fabric pattern from a depositional fabric, found in forefield regions, to fabrics indicative of progressively higher-strain LPS towards the hinterland. The depositional fabric, with k3 orientations clustering vertically and k1 and k2 orientations in a horizontal girdle (fabric 1 of Fig. 15), reflects the horizontal settling and subsequent consolidation during burial of oblate, clay-sized particles [START_REF] Tarling | The Magnetic Anisotropy of Rocks[END_REF]. Common in fold-thrust environments are intermediate fabrics in which this depositional fabric has been overprinted by a small amount of LPS. This fabric (fabric 2 of Fig. 15) reflects consolidation greater than LPS, such that k3 orientations remain vertical. Orientations of k1, however, rotate into the direction of maximum extension, which is horizontal and perpendicular to the shortening direction. With still more LPS, a vertical k2-k3 girdle forms parallel to shortening (fabric 3), which transitions at still greater strain to k2 and k3 clusters that are vertical and parallel to the shortening direction, respectively (fabric 4). This highest-strain AMS fabric has been measured in till interpreted to have been extruded upward into basal crevasses [START_REF] Ankerstjerne | Origin of a washboard moraine of the Des Moines Lobe inferred from sediment properties[END_REF].

AMS fabrics measured at Rubjerg Knude in the distal Ulstrup section and away from thrusts in the Grønne Rende section, as well as most fabrics measured at Møns Klint, are similar to fabric 2 of this model. Orientations of k1 cluster in the plane of the bedding or fault and perpendicular or sub-perpendicular to the thrusting direction; k2 orientations also cluster in the plane of the fault or bedding but are parallel or sub-parallel to thrusting, and k3 orientations either cluster vertically or are steeply dipping. Therefore, prior to thrusting of mud sheets at Rubjerg Knude and Møns Klint the mud was consolidated (fabric 1, Fig. 15) and underwent minor shortening parallel to bedding. More importantly, thrusting did not change this fabric at Grønne Rende, except within decimeters of fault surfaces where simple shear reset the LPS fabrics. At Møns Klint, there is no evidence of thrusting resetting this fabric, likely indicating that shear was focused in the ~3 cm of mud closest to the fault where the mud was not sampled (Figs. 5 and14).

AMS of sediments across basal décollements of accretionary prisms sometimes indicate a similar lack of detectable overprinting by shear deformation [START_REF] Housen | Strain decoupling across the decollement of the Barbados accretionary prism[END_REF]. In contrast, the superimposed deformation onto the eastern edge of the northern part of the complex from Slotsgavlene to Maglevandsfald (Fig. 4), which occurred later, recumbently folded the glacial sediments [START_REF] Pedersen | Structural development of Maglevandsfald: a key to understanding the glaciotectonic architecture of Møns Klint, SE Denmark[END_REF], likely resetting LPS fabrics.

The clear implication is that during thrusting sheets were rotated, with little internal deformation except very near faults. As noted, a leading hypothesis for glacitectonic thrusting of rigid sheets of unlithified sediment is that permafrost made thrusting of coherent sheets possible through some combination of increasing their strengths and serving as a barrier for groundwater flow. In that case, the narrow zones of shear near thrust surfaces might be interpreted as the result of heat dissipation during slip and the associated warming or melting of pore ice there. [START_REF] Warbritton | mineral composition of the analyzed samples. Specifically, assessing whether paramagnetic minerals, ferrimagnetic minerals or a combination of both control the AMS is crucial. The AMS of paramagnetic minerals is dictated by the mineral's crystal symmetry and is said to have a[END_REF] showed that heat dissipation from the average displacement on faults at Rubjerg Knude estimated by Pedersen (2005, ~70 m) was sufficient to warm permafrost substantially near slip surfaces. Potentially this softened the permafrost through production of pre-melted water, enabled strain localized in narrow zones, and reduced shear stresses sufficiently to allow the slip of mostly intact sheets.

Thrusting of intact sheets without permafrost

We now step back and consider more generally whether thrusting of largely intact sediment sheets can occur under fully thawed conditions, by appealing to the effects of pore-water pressure and thrust geometry rather than to permafrost. Consider a thrust sheet (Fig. 16) with the wedge geometry commonly considered in studies of crustal faulting (e.g., Turcotte and Schubert, 2014). Wedge-shaped thrust sheets were also considered by [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF] in his reconstructions of thrusting at Rubjerg Knude. The upper surface of the thrusted wedge slopes at an angle, α, toward the distal end of the wedge, as expected if layer-parallel shortening is largest near the glacier margin and is accompanied perhaps by proximal thrusting in the wedge. The thrust ramp dips in the opposite direction at an angle, β. The thickness of the wedge, H, is defined normal to the thrust surface. The wedge is pushed from upstream along the x coordinate, defined parallel to the thrust ramp, by a force that reflects the depth integral of the stress, σx (Fig. 16). In the opposite direction is a shear stress, τb, acting along the base of the thrust ramp.

In their study of the mechanics of fold and thrust belts, [START_REF] Davis | Mechanics of fold and thrust belts and accretionary wedges[END_REF] balanced forces on the wedge and considered the stresses required for its Coulomb failure to estimate τb for the special case of a wedge deforming internally at its Coulomb strength. In that case, shear failure occurs under a traction, S = S0 + µ (σz -pw), where S0 is the sediment cohesion, σz is the total stress, pw is the pore-water pressure, and µ is an internal friction coefficient commonly expressed in terms of a friction angle 𝜑𝜑 such that µ = tan 𝜑𝜑. For the small values of α and β relevant to this problem, for which sin α ≈ α and sin β ≈ β, and assuming the wedge is saturated with water everywhere, [START_REF] Davis | Mechanics of fold and thrust belts and accretionary wedges[END_REF] derived the shear stress on the thrust fault required to deform the wedge:

𝜏𝜏 𝑏𝑏 = (𝜌𝜌 -𝜌𝜌 𝑤𝑤 )𝑔𝑔𝑔𝑔𝑔𝑔 + (1 -𝜆𝜆)𝐾𝐾𝜌𝜌𝑔𝑔𝑔𝑔(𝑔𝑔 + 𝛽𝛽), ( 1
)
where ρ is the density of the sediment grains of the wedge, and ρw is the density of pore water.

Two dimensionless quantities are particularly important in Eq. ( 1): the water pressure ratio, λ=pw/σz and the parameter, K, which scales with the frictional strength of the wedge. The shear stress at the base must balance the down-slope component of gravity acting on the sloping surface of the wedge, as given by the first right-hand term, and must also balance the force on the upstream end of the wedge as limited by the wedge's Coulomb strength and given by the second right-hand term. In deriving Eq. ( 1), [START_REF] Davis | Mechanics of fold and thrust belts and accretionary wedges[END_REF] assumed zero cohesion in the wedge, so for the fine-grained and likely cohesive muds of this study, Eq. ( 1) provides a minimum value of the stress required to deform the wedge.

For the wedge to slip along its basal fault without deforming the wedge internally, the frictional stress resisting slip, Sb, at the base of the wedge must be less than τb. Defining µb and λb as the friction coefficient and water pressure ratio, respectively, at the base of the wedge,

𝑆𝑆 b = 𝜇𝜇 b (1 -𝜆𝜆 b )𝜌𝜌𝑔𝑔𝑔𝑔.
(2)

Thus, the condition for slip without internal deformation of the wedge is

𝜇𝜇 b (1 -𝜆𝜆 b )𝜌𝜌𝑔𝑔𝑔𝑔 < (𝜌𝜌 -𝜌𝜌 w )𝑔𝑔𝑔𝑔𝑔𝑔 + (1 -𝜆𝜆)𝐾𝐾𝜌𝜌𝑔𝑔𝑔𝑔(𝑔𝑔 + 𝛽𝛽). (3)
To maximize slip resistance on the fault and to thereby optimize the potential for deforming a wedge of a given geometry and strength, we assume that λb =λ, despite the possibility that porewater pressure might be elevated along the fault relative to the wedge overall. Assuming λb =λ, and rearranging Eq. ( 3) yields the pore-pressure condition required for slip without internal deformation of the wedge:

𝜆𝜆 > 1 - 𝛼𝛼�1- 𝜌𝜌 𝑤𝑤 𝜌𝜌 � 𝜇𝜇 b -𝐾𝐾(𝛼𝛼+𝛽𝛽) . ( 4
)
Despite the possibility that sliding friction was less than internal friction, we set the friction coefficient of the fault equal to the internal friction coefficient of the wedge, µb = µ = tan 𝜑𝜑, to further optimize the potential for deforming it. In that case, as shown by [START_REF] Davis | Mechanics of fold and thrust belts and accretionary wedges[END_REF], a good approximation for the Coulomb strength factor is

𝐾𝐾 = sin 𝜑𝜑 1-sin 𝜑𝜑 + sin 2 𝜑𝜑 cos 2 𝜑𝜑 . ( 5
)
Eqs. ( 4) and ( 5) illustrate that if the glacier forefield sloped down-glacier at even a small angle α, a wide a range of pore-pressure ratios could have resulted in thrusting without internal deformation of the thrust sheet. Fields of λ and α above and to the right of the plots of Figure 17 indicate thrusting without internal deformation. For the thrust-ramp inclination that characterized initial thrusting in the Grønne Rende section at Rubjerg Knude (18º, Pedersen, 2005, p. 118), even hydrostatic pore-water pressure (i.e., no upward groundwater flow) could have resulted in fault slip without internal deformation, as long as α > 4.3º. Still smaller surface slopes would have been required for later stages of faulting at Grønne Rende when thrust ramps became steeper than 30º [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF]. Also, given that upward groundwater flow is common in glacier forefields (e.g., [START_REF] Boulton | Groundwater flow beneath ice sheets: Part II-its impact on glacier tectonic structures and moraine formation[END_REF], pore pressures could have easily exceeded hydrostatic. Assumptions made in the derivation of Eqs. ( 4) and (5) (i.e., no internal cohesion, λb =λ, and µb = µ = tan 𝜑𝜑 also make the estimates of λ and α in Figure 17 required for thrusting without internal deformation conservatively large.

Unfortunately values of α are poorly known during thrusting at Grønne Rende. Pedersen's (2005, p. 159) reconstruction indicates α > 5º late during thrusting, but he argued that at the beginning of thrusting the hinterland of the complex sloped toward the glacier 2º, owing to isostatic depression by the glacier. Uncertainty in the value of α and its evolution during thrusting clearly limit mechanical interpretations.

Nevertheless, at Rubjerg Knude and Møns Klint, during phases of thrusting when the proglacial surface sloped away from the glacier, positive pore-water pressure could have caused thrusting of intact sheets of mud in the absence of proglacial permafrost. This interpretation is consistent with the field observations of [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF], who interpreted sand-filled cracks, local brecciation, and small-scale mud diapirism at Rubjerg Knude as reflecting high pore-water pressures. It is also consistent with real-time observations of proglacial thrusting of smaller sheets of unfrozen sediments by modern glaciers with little internal deformation of thrust sheets (e.g., [START_REF] Kuriger | Episodic reactivation of large-scale push moraines in front of the advancing Taku Glacier, Alaska[END_REF].

Our calculations do not rule out that parts of thrust sheets may have been frozen [START_REF] Pedersen | Structural analysis of the Rubjerg Knude glaciotectonic complex, Vendsyssel, northern Denmark[END_REF] but support the viewpoint that permafrost is commonly not necessary for thrusting of coherent sheets (e.g., [START_REF] Van Der Wateren | Processes of glaciotectonism[END_REF]. The observation that shear at Rubjerg Knude occurred only near thrust faults may indicate that in mud immediately adjacent to them the condition λb > λ was met-a reasonable expectation if consolidated mud of the LKF served as a confining unit for groundwater flow.

Conclusions

Although AMS had been applied to sediments where macroscopic structures indicate glaciotectonic deformation (Fleming et al., 2013b), this study is the first attempt to apply AMS to the kinematics of glacitectonic thrusting. crystallographic preferred orientation. Ferrimagnetic mineral particles with high saturation magnetizations, such as magnetite and maghemite, have AMS controlled by the shape of the particle and are said to have a preferred shape orientation [START_REF] Tarling | The Magnetic Anisotropy of Rocks[END_REF]. Rock magnetic analyses at room temperature and low-temperature were conducted at the Institut de Physique du Globe de Paris (France) in order to determine whether magnetite (or maghemite)

were present and at what concentration.

Representative bulk samples from Rubjerg Knude (RK3) and Møns Klint (MUN01) were analyzed. Each bulk sample was homogenized using an agate mortar and pestle and dried at 40°C overnight. The powdered bulk samples were then firmly packed in 0.21 ml gelatin capsules (14.3 mm in length and 5 mm in diameter). The RK3 and MUN01 subsamples contained 434.4 mg and 476.6 mg, respectively, of bulk material.

Two room-temperature experiments were conducted on a Princeton Measurements Model 3900 Vibrating Sample Magnetometer (VSM). First, hysteresis loops and initial magnetizations curves were measured in maximum magnetic fields of ± 1.5 T. Mass-specific low-field magnetic susceptibility was calculated from the initial slope of the first magnetization curves. Saturation magnetization, saturation remanent magnetization, coercivity and high field magnetic susceptibility were derived from the hysteresis loops after correction for the high field slopes.

Second, an isothermal remanent magnetization (IRM) acquisition curve was measured on demagnetized subsamples by acquiring IRMs in progressively increasing magnetic fields from zero to 1.5 T with 200 logarithmically spaced steps.

Two low-temperature experiments were conducted on a Magnetic Properties Measurement System (MPMS) XL-5 Evercool. First a room-temperature (300 K) IRM was acquired in a 2.5 T magnetic field and measured as the subsample was cooled to 10 K and warmed back to 300 K at a rate of 5 K/min. The second experiment measures a low-temperature (10 K) IRM acquired in a 2.5 T magnetic field as the subsample is warmed from 10 K to 300 K at a rate of 5 K/min. The 10 K IRM is acquired following one of two pre-treatments. The zero-field cooled (ZFC) pretreatment cools the subsample from 300 K to 10 K in zero (< 500 nT) magnetic field before acquiring the 10 K IRM and the field cooled (FC) pre-treatment cools the subsample from 300 K to 10 K in a 2.5 T magnetic field before acquiring the 10 K IRM.

The hysteresis loop and first magnetization derived parameters are listed in Table A1 and hysteresis loops before and after correction shown in Figure A1. Low-field magnetic susceptibility, inclusive of all minerals, of RK3 and MUN01 are quite similar 20 x 10 -8 and 17 x 10 -8 m 3 /kg. However, the high-field magnetic susceptibility of RK3, which excludes ferromagnetic (sensus lato) that have saturated (primarily ferrimagnetic minerals), is more than 3 times the value for MUN01. Saturation magnetization values confirm the higher concentration of ferrimagnetic minerals in MUN01 with values twice those of RK3. Squareness and coercivity ratios of RK3 and MUN01 are in agreement with a population of ferrimagnetic magnetic domain sizes larger than single domain. Therefore, inverse AMS fabrics are not expected.

The low temperature experiments shown in Figure A2 confirm that magnetite is the dominant ferrimagnetic mineral present in both RK3 and MUN01 subsamples. The Verwey transition, which is characteristic of stoichiometric magnetite, is observed by a sharp drop in remanence at 120 K in the ZFC and FC 10 K IRM warming curves. The 300 K IRM cooling and warming cycles display more broadly but just as convincingly the Verwey transition. The remanence loss and gain in Figure A2A is identical in both samples, which attests to the very similar magnetic domain states of the magnetite population contributing to the AMS at Rubjerg Knude and Møns Klint.

Since magnetite is shown to be the dominant ferrimagentic mineral, saturation magnetization values, which are independent of magnetic domain size (i.e. particle size) unlike remanent magnetization, can be used to calculate weight percent concentration of magnetite. The intrinsic saturation magnetization of stoichiometric magnetite is 92 Am 2 /kg at room temperature (Dunlop and Özdemir, 1997). Assuming that stoichiometric magnetite is the sole contributor to the saturation magnetization RK3 contains 0.01 wt. % magnetite and MUN01, 0.03 wt. % magnetite.

In conclusion, magnetite is the dominant ferrimagnetic mineral in both RK3 and MUN01. The magnetite population grain-size distributions are similar at both sites and fall well into the pseudo single-domain to small multi-domain particle size range. Inverse AMS fabrics are therefore not expected. Magnetite concentrations are low, 0.01 wt% at Rubjerg Knude and 0.03 wt% at Møns Klint but similar, for example, to the 0.03 wt.% magnetite calculated for the Batestown Till [START_REF] Hooyer | Magnetic fabric of sheared till: A strain indicator for evaluating the bed deformation model of glacier flow[END_REF]) in which magnetite controlled AMS fabrics.

  Fabrics of intact stratigraphy near the distal end of the Rubjerg Knude thrust complex indicate that, before thrusting the LKF mud underwent consolidation with longitudinal horizontal shortening. This shortening was moderate, as indicated by clustering of k1 and k2 orientations perpendicular and parallel to the shortening direction and by the lack of rotation of vertical k3 orientations developed during mud deposition and consolidation (Fig. 15). AMS fabrics of thrust sheets of LKF mud within the Grønne Rende section of Rubjerg Knude, indicate rotation of sheets mostly without internal deformation during thrusting. Within decimeters of fault surfaces, however, mud was sheared to strains sufficient to overprint the LPS fabric. The resultant fabric is observed consistently in ring-shear experiments conducted to large displacements (> 0.5 m), with k1 orientations parallel to shear and plunging mildly relative to the shear plane. In thrusted mud at Møns Klint, only rotation of LPS fabrics is observed, despite samples collected as close as 3 cm to the chalk-mud contact. Slip, therefore, was seemingly focused in a still narrower zone, with most of the thrust sheet behaving too rigidly to reset the fabric. Permafrost was not required for thrusting of intact sheets. If topographic surfaces of thrust sheets sloped gently away from the glacier margin, a wide range of pore-water pressures could have enabled slip on thrusts without internal deformation of individual thrust sheets. Shear deformation observed close to faults at Rubjerg Knude may reflect elevated pore pressures in mud adjacent to faults during thrusting.

Figure captions Figure 1 .

 1 Figure captionsFigure 1. Locations of the two glacitectonic complexes. Black line indicates the "Main

Figure 2 .

 2 Figure 2. (A) Schematic cross-sections of the Rubjerg Knude complex, The three thrust sheets

Figure 3 .

 3 Figure 3. Thrusted mud sheets of the Lønstrup Klint Formation (grey) with intervening sand and

Figure 4 .

 4 Figure 4. (A) Oblique photograph of the east coast of the island of Møn, Denmark (from

Figure 5 .

 5 Figure 5. Aluminum sample boxes pushed into the glaciolacustrine mud at Møns Klint,

Figure 6 .

 6 Figure 6. View normal to the surface of a thrust fault at the base of sheet GR10 in the Grønne

Figure 7 .

 7 Figure 7. (A) Mud and sand of the Lønstrup Klint Formation thrusted along a slickensided fault

Figure 8 .

 8 Figure 8. Upper Cretaceous chalk overlying glaciolacustrine mud of the Middle Weichselian at

Figure 9 .

 9 Figure 9. (A) Orientations of the three principal magnetic susceptibilities of the mud from

Figure 10 .

 10 Figure 10. Principal-susceptibility orientations for the six fabrics from the distal end of the

Figure 11 .

 11 Figure 11. Aggregated principal susceptibility orientations at Rubjerg Knude. (A) Principal-

Figure 12 .

 12 Figure 12. Four fabrics from thrust sheet GR10 from the Grønne Rende section of Rubjerg

Figure 13 .

 13 Figure 13. Nine fabrics from transects A, B, and C in the thrust sheet GR05 from the Grønne

Figure 14 .

 14 Figure 14. Principal-susceptibility orientations for 17 AMS fabrics collected in the four

Figure 15 .

 15 Figure 15. Model for AMS fabric development during deposition/consolidation and layer-

Figure 16 .

 16 Figure 16. Schematic diagram of a thrust wedge subject to a push, σx (modified from Davis et

Figure 17 .

 17 Figure 17. Threshold values of pore-pressure ratio, λ, and topographic slope of the thrust wedge,

Figure A1 .

 A1 Figure A1. (A) Hysteresis loops for samples from Rubjerg Knude (RK3) and Møns Klint

Figure A2 .FiguresFigure 1 .

 A21 Figure A2. Results of low temperature experiments for samples from Rubjerg Knude (RK3) and

  Figure 12. Four fabrics from thrust sheet GR10 from the Grønne Rende section of Rubjerg Knude. Distances listed are relative to the adjacent fault surface. The fault orientation is shown in a lower-hemisphere stereonet at upper-right, and the cyan arrow shows the iceadvance direction. Large orange symbols show V1 eigenvectors calculated from the principalsusceptibility distributions.

  Figure 14. Principal-susceptibility orientations for 17 AMS fabrics collected in the four transects at Møns Klint (see Figure 8), aggregated for five distance ranges from the fault at the chalk-mud boundary. The fault orientation is shown in a lower-hemisphere stereonet at upperright, and the cyan arrow shows the ice-advance direction. Large orange symbols show V1 eigenvectors calculated from the principal-susceptibility distributions.

Figure 16 .

 16 Figure16. Schematic diagram of a thrust wedge subject to a push, σx (modified from[START_REF] Davis | Mechanics of fold and thrust belts and accretionary wedges[END_REF].

Figure A2 .

 A2 Figure A2. Results of low temperature experiments for samples from Rubjerg Knude (RK3) and Møns Klint (MUN01). (A) 300 K IRM cooling and warming cycles. (B) IRM warming curves.

Acknowledgements

We thank P.J. Hudleston and an anonymous reviewer for their helpful comments on the manuscript, and Z. Kornse for his assistance with the field measurements. We also thank B.

Tikoff for the use of his Kappabridge at the University of Wisconsin-Madison and N. Garibaldi for his help there.

Figure 17. Threshold values of pore-pressure ratio, λ, and topographic slope of the thrust wedge, α, above which slip is expected on thrusts without internal deformation of thrust sheets, for different values of thrust dip, β. Zero internal cohesion, λb =λ, and µb = µ = tan 𝜑𝜑 are assumed, making estimates of λ and α required for slip without internal deformation conservatively large. Also assumed are sin α ≈ α and sin β ≈ β, good to within 5% for the largest angles considered (β = 30 ⸰ ). The hydrostatic value of λ= pw/σz is based on a groundwater table at the ground surface, a sediment porosity of 50%, and rock particle density of 2700 kg m -3 .


M Blondel

Y Abidine

P Assemat

S Palieme

P Swider

Identification of effective elastic modulus using modal analysis; application to canine cancellous bone

Keywords:

Mechanical properties of cancellous bone is of increasing interest due to its involvement in aging pathologies and oncology. Characterization of fragile bone tissue is challenging and available methodolo gies include quasi static compressive tests of small size specimens, ultrasound and indentation tech niques. We hypothesized that modal analysis of flexure beams could be a complementary methodology to obtain Young modulus.

Theoretical and experimental approach

The sampling methodology was adapted such that the uniqueness of the linear dynamic response was available to determine th e elastic modulus from natural frequencies and mode shapes. ln a first step, the methodology was validated using a synthetic bone model as control. Then, water jet cutting allowed col lecting fourteen small beam like specimens in canine distal femurs. X ray microtomography confirrned the microarchitecture preservation, the homogeneity and the isotropy at the specimen scale to derive effective properties. The first natural frequency in clamped free boundary conditions was used to obtain mean values of Young modulus, which ranged from 210 MPa to 280 MPa depending on the specimen col lection site. Experimental tests were rapid and reproducible and our preliminary results were in good agreement with literature data. ln conclusion, beam modal analysis could be considered for exploring mechanical properties of fragile and scarce biological tissues.

Introduction

The exploration of bone mechanical properties has focused on cortical bone, which is considered as the most mechanically corn petent tissue. However, characterization of cancellous bone is of increasing interest due to its involvement in aging pathologies and oncology [START_REF] Morgan | Bone mechanical properties in healthy and diseased states[END_REF][START_REF] Ordway | A Biomechanical Comparison of Four Hip Arthroplasty Designs in a Canine Model[END_REF] i Cancellous bone mechanical properties are subjected to significant changes secondary to trauma, osteoporosis, neoplasia or therapeutic strate gies such as minimally invasive osteosynthesis and chemotherapy [START_REF] Greenwood | The micro-architecture of human cancellous bone from fracture neck of femur patients in relation to the structural integrity and fracture toughness of the tissue[END_REF][START_REF] Goodheart | Simulating activities of daily living with finite element analysis improves fracture prediction for patients with metastatic femoral lesions[END_REF].

Characterization of elastic properties of fragile bone tissue like cancellous bone is challenging. Main methodologies include quasi static compressive tests of small size specimens [START_REF] Keaveny | Systematic and random errors in compression testing of trabecular bone[END_REF][START_REF] Linde | The effect of specimen size and geometry on the mechanical behaviour of trabecular bone specimens[END_REF][START_REF] Odgaard | The underestimation of Young's modulus in compressive testing of cancellous bone specimens[END_REF], ultra sound and indentation techniques [START_REF] Linde | The effect of specimen size and geometry on the mechanical behaviour of trabecular bone specimens[END_REF][START_REF] Rho | An ultrasonic method for measuring the elastic properties of human tibial cortical and cancellous bone[END_REF][START_REF] Dong | Effects of end boundary conditions and specimen geometry on the viscoelastic properties of cancellous bone measured by dynamic mechanical analysis[END_REF]Prat et al., 2016;[START_REF] Vijayakumar | Quantifying the regional variations in the mechanical properties of cancellous bone of the tibia using indentation testing and quantitative computed tomographic imaging[END_REF][START_REF] Daoui | Assessment of trabecular bone tissue elasticity with resonant ultrasound spectroscopy[END_REF][START_REF] Wu | Young's modulus of trabecular bone at the tissue level: A review[END_REF][START_REF] Wear | Mechanisms of Interaction of Ultrasound With Cancellous Bone: A Review[END_REF]. These tech niques are complementary since they do not explore the tissue response at the same scale nor under identical physical stimulus. In those articles, the role of porosity, microarchitecture, anisotropy and boundary conditions on effective Young modulus have been examined and results showed a great sensitivity to samples in term of collection site, size, shapes, heterogeneity and anisotropy.

To address this characterisation challenge, modal analysis can be used to explore the dynamic behavior of structures and materi ais [START_REF] Nashif | Vibration damping[END_REF][START_REF] Ewins | Modal Testing; theory, practice and application[END_REF]. This methodology has been applied in the literature to identify effective properties at the scale of the organ [START_REF] Couteau | Finite element modelling of the vibrational behaviour of the human femur using CTbased individualized geometrical and material properties[END_REF][START_REF] Neugebauer | Experimental modal analysis on fresh-frozen human hemipelvic bones employing a 3D laser vibrometer for the purpose of modal parameter identification[END_REF][START_REF] Scholz | Validation of density-elasticity relationships for finite element modeling of human pelvic bone by modal analysis[END_REF]Henys & Capek,2017) or at tissue scale [START_REF] Swider | Characterization of bone-implant fixation using modal analysis: application to a press-fit implant model[END_REF][START_REF] Miyashita | Evaluation of implant screw loosening by resonance frequency analysis with triaxial piezoelectric pick-up: in vitro model and in vivo animal study[END_REF].

In case of complex dynamic responses and heterogeneous and anisotropie tissues, the problem is often under determined due to numerous unknown elastic coefficients in material laws. In this context, the challenge is to obtain the uniqueness of eigen values and eigen vectors pairs, i.e. natural frequencies and mode shapes. To untangle this problem, numerical methods such FEM associated with inverse numerical methods are often used [START_REF] Swider | Dynamic modelling of a composite plate, a mixed numerical and experimental approach[END_REF][START_REF] Taylor | Determination of orthotropic bone elastic constants using FEA and modal analysis[END_REF]Henys & Capek 2019) as theoretical solutions are rare. Experimentally, modal analysis benefi ts from non contact measurement techniques and excitation force adaptation [START_REF] Swider | Theoretical and experimental dynamic behavior of a golf club made of composite material[END_REF][START_REF] Ewins | Modal Testing; theory, practice and application[END_REF][START_REF] Neugebauer | Experimental modal analysis on fresh-frozen human hemipelvic bones employing a 3D laser vibrometer for the purpose of modal parameter identification[END_REF].

In this article, we propose to adapt the sampling methodology such that the uniqueness of the theoretical dynamic response is available. Therefore, we hypothesize that beam modal analysis could be used to determine the effective elastic properties of bone tissue. This leads to sampling reduced size specimens accurate in geometry with homogeneous microstructure and isotropic behav ior. We used cancellous bone explanted from canine distal femurs as test tissue and the methodology was previously evaluated with polyurethane foam bone model.

Material and methods

In this section, the theoretical background is detailed before introducing the samples preparation method, the type of samples and the experimental method.

Governing equations of beam transverse motion

Static response: Three points bending

The static three points bending test was used as a control test of the proposed methodology. Equation (1) gives the static elastic modulus E s as a function of the applied load F, the specimen length l, the section properties (width b, height h) and the mid span deflection v(l/2) using the Timoshenko beam model [START_REF] Timoshenko | Compressive and shear properties of commercially available polyurethane foams[END_REF]. Shear effect is taken into account in the second term in brackets of equation ( 1). For a rectangular cross section and a span (l/h) greater than 10, shear coefficient a of 0.85 with m lower than 0.5 gives a second order shear contribution.

E s

Fl

3 4bh 3 1 v l=2 ð Þ 1 þ h l 2 2 1 þ t ð Þ a " # ð1Þ
2.1.2

. Theoretical modal analysis

The dynamic bending motion v(x,t) of a uniform straight beam in its transversal plane is given by the governing equation (2) [START_REF] Meirovitch | Elements of vibration analysis[END_REF][START_REF] Lalanne | Mechanical Vibrations for[END_REF] when secondary effects (shear, rotation inertia) and dissipative effects are negligible [START_REF] Huang | The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions[END_REF][START_REF] Meirovitch | Elements of vibration analysis[END_REF]). Geometrical properties are the cross section S, the area moment of inertia I and the beam span l.

Effective material properties are the elastic modulus E and the material density q.

EI @ 4 v x; t ð Þ @x 4 þ qS @ 2 v x; t ð Þ @t 2 0 ð2Þ
The transverse motion v(x,t) is written as the product of a space function or mode shape /(x) by a time function f(t) and equation ( 2) is transformed into equation (3a).

EI @ 4 / x ð Þ @x 4 f t ð Þ þ qS Á / x ð Þ Á @ 2 f t ð Þ @t 2 0 ð3aÞ
with v x; t ð Þ / x ð Þ Á f t ð Þ ð3bÞ
and / x ð Þ Csinbx þ Dcosbx þ Esinhbx þ Fcoshbx with C; D; E and F constant: ð3cÞ
Solutions of f(t) are harmonic functions of angular frequency x n and the roots of the characteristic equation in space are expressed by equation (4a). Then, application of boundary conditions gives the transcendental equation, which is expressed by equation (4b) in case of clamped free conditions.

b n

qSx n 2 EI 4 r a ð Þ 1 þ cosh b n l ð Þcos b n l ð Þ 0 b ð Þ ð4Þ
The clamped free boundary conditions give the lowest frequen cies compared with other types of boundary conditions which is convenient with small, light and stiff specimens and the lowest root of equation (4b) was b 1 = 1.875. Using equation (4a), the effec tive modulus E d was expressed by equation (5a). The mode shape / 1 expressed by equation (5b) was determined using equation (3c) and (4a) together with boundary conditions that were v(0) =o x v(0) = 0 at the clamped section and o 2

x v(l) = o 3 x v(l) = 0 at the free end.

E d l 1:875

4 qS I x 1 2 with x 1 rd=s ð Þ 2pf 1 and f 1 Hz ð Þ : 1 st resonance frequency ð5aÞ
/ 1 cosh 1:875 l x cos 1:875 l x 0:734 sinh 1:875 l x sin 1:875 l x ! ð5bÞ
The response to forced harmonic excitation reaches maximal magnitude when the angular forcing frequency matches the angu lar frequency. Experimentally, natural frequency is found by using a swept sine or random noise in case of linear behaviour, and for low damping, amplitude resonance and phase resonance are iden tical [START_REF] Nashif | Vibration damping[END_REF].

Parameter sensitivity analysis

In the following, variables p and q successively correspond to parameters l, b, h, q, m F and x. Complete first order Taylor expan sion of relation ( 1) and (5a) gives uncertainty DE/E to specimen properties and measurement uncertainties. Details are given in appendix.

2.2. Specimen preparation 2.2.1. Bone model specimen: Polyurethane foam Sawbone Ò models #1522 02 representing a validated alterna tive for testing human cancellous bone were used as preliminary controls for static and dynamic tests (Thompson et al. 2003;[START_REF] Elfar | Composite bone models in orthopaedic surgery research and education[END_REF]. Four straight beams of 4 mm in width (b) and 40 mm in length were cut from a polyurethane sheet of 3 mm thickness (h) with an oscillating saw (IsoMet 1000, Buehler Ò ) because sheet lightness was not adapted to the water jet machine used for canine femurs. For the three points bending test, the sam ple span was 30 mm. Manufacturer data were E s = 123 MPa in com pression and E s = 173 MPa in traction. The homogenous porosity was / = 80% and density was q = 240 kg/m 3 .

Canine cancellous bone specimen

Femurs were harvested from seven adult beagle dogs, which had died for reasons unrelated to orthopedic conditions. According to a validated protocol [START_REF] Linde | The effect of different storage methods on the mechanical properties of trabecular bone[END_REF], bones were cleaned of soft tissues, wrapped in saline soaked gauze and stored frozen at 20 °C for less than 4 months.

The proximal extremity of each femur was embedded in polyur ethane resin (RenCast Ò FC 52/53 Isocyanate/FC 52 Polyol) as shown in Fig. 1a after having been thawed at room temperature and kept moist with saline soaked sponges [START_REF] Brear | Density and temperature effects on some mechanical properties of cancellous bone[END_REF]. For embed ding, each femur was positioned so that a) the cranial and lateral anatomic axis were perpendicular to the mold bottom, b) the cau dal part of both femoral condyles was tangent to the back face of the mold and c) the lateral condyle slightly exceeded the lateral face of the mold. Orientations were controlled using two orthogo nal laser beam devices (PCL20; Bosch, Gerlingen, Germany). Embedded bones were stored at 20 °C and thawed at room tem perature one day before speàmen sampling.

Fourteen specimens were collected from the distal femoral epi physis and metaphysis with a water jet cutting machine (WJA Il; LOSA, Bar Le Duc, France) (Schwieger et al. 2004i Straight beams 4 to 5 mm in width (b) and 2 to 3 mm in height (h) were eut in the medio lateral direction. Seven specimens were classified in group A, and seven in group B, the latter being doser to the artic ular surface as shown in Fig. 1 a. Any remaining cortical bone was removed with a scalpel blade. Geometrical properties were mea sured at three points on each specimen using a digital micrometer (LS 7000 series; Keyenoe SAS, Courbevoie, France) with an accu racy of 2 µm. The effective density was obtained by weighing (Met ter Toledo�) each specimen with an accuracy of 10-6 kg, and by determining the specimen volume.

Fourteen samples were harvested from canine distal femurs; seven were classified in group A, and seven in group B.

Experimental methodology 2.3.1. X ray micro tomography

Specimens were examined by X ray microtomography (Nan otom 180 Phoenix; GE Sensing, Labège, France) to explore the tis sue microarchitecture. The scanner was operated at 80 kV and 90 µA with corrections for noise reduction and beam hardening to obtain images with voxel sizes of 10 x 10 x 10 µm. Images from six zones of iso length per specimen were explored using Avi zo�(Thermo Fisher Scientific, Waltham, MA USA) and BoneJ (lma geJ �) to determine porosity <p and anisotropie degree DA [START_REF] Harrigan | Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor[END_REF][START_REF] Doube | BoneJ: free and extensible bone image analysis in ImageJ[END_REF][START_REF] Greenwood | The micro-architecture of human cancellous bone from fracture neck of femur patients in relation to the structural integrity and fracture toughness of the tissue[END_REF]. No drying protocol was envisaged to preserve the limited number of available specimens, thus mass per unit length pS used in equation (Sa) was not accessible from micro CT.

Static three point bending test

Three point bending tests were performed on bone model spec imens (Sawbone�) using an electric testing machine (10 MH MTS; MTS Corporation�. Eden Prairie, MN USA) [START_REF] Autefage | Effective mechanical properties of diaphyseal cortical bone in the canine femur[END_REF]. The beam span was 10 mm, the non destructive force applied at 0.1 mm/min was 10 N and data sampling was set to 10 Hz (Test work'® 4.088, MTS). Using equation (1 ), three successive measure ments of force and transverse displacements gave the mean effective Young modulus of each specimen

Experimental modal analys is

The specimen was clamped over 2 mm length in a 3D printed clamping fixture with cyanoacrylate polymer as shown in Fig. 1 b, c andd. The free beam span was then measured with a depth gauge accurate to 20 µm. The specimen was clamped over 2 mm length in a 3D printed PlA fixture with cyanoacrylate polymer (CP) as shown in Fig. 1 b, c andd. The fixture mass was 50 g whereas the specimen mass was 0.2 g. The Young modules of PLA and CP were 2.4 MPa and 1 GPa respectively. The significant changing of stiff ness at the interface between the speàmen and the fixture, associ ated with mass ratio reproduced the clamped conditions in term of strain and kinetic energies and matched the theoretical considera tions developed in paragraph (2.1.2).

Specimens frequencies were obtained from the frequency response function (FRF), corresponding with the ratio of the beam transverse displacement t(x,t) over the forced excitation F(.Q) in the frequency space [START_REF] Ewins | Modal Testing; theory, practice and application[END_REF]. To generate the base exàtation, the fixture was mounted on an electrodynamic shaker (K2007E01; PCB ® Piezotronics) using a piezoelectric force gauge (208C02; PCB ® Piezotronics), (Fig. 2a and Fig. 2b). The transverse motion of the speàmen was detected by using an optical displacement probe (CHRocodile unit 3 mm; Preàtec ® , spot diameter 12 µm, vertical resolution 40 nm), combined to a signal conditioner (CHRocodile 2S; Precitec ® ). A custom made 3 axis motorized system using translation actuator (Newport ® ) was used to map the specimen. Excitation, signal acquisition and post treatment were simultane ously controlled with a dynamic analyzer (OR34; Oros ® ). The base displacement was monitored imposing swept sine in the range 10 Hz 2 kHz and the linear behavior was verified by comparing responses with ascending and desoending sweeps.

During vibration testing, cancellous bone specimens were sprayed with saline at room temperature (22 °C) to avoid dehydra tion. At least three FRF with the displacement probe pointed at three different locations along the beam span were performed on each specimen to obtain mean values and standard deviation of magnitude resonances and phase resonances, and to identify the mode shape <Pl-Measurement time was Jess than one minute per speàmen and no effect was detected on FRF. Geometry properties of fourteen biological specimens were b = 4.82 mm SD 0.23 and h = 2.6 mm SD 0.16. Straightness discrep ancies corresponding with the mean value of relative standard deviation was 1.17% for band 1.77% for h which approximately cor responded with 0.06 mm and 0.05 mm. Mas s per unit length pS used in equation (Sa) was 1132 kg/m SD 90 and 1138 kg/m SD 110 for group A and group B, respectively.

-Results

Geometrical properties of spedmens

For the experimental modal analysis, the mean free length of the specimens was 1 = 18.94 mm SD = 3.3 mm. Specific geometrical properties (1, b, h) were used to determine the effective modulus Ed.

X ray microtomography of cancellous bone specimens

The porosity was determined for the six zones on each speci men using X ray tomography images post processing, each zone ranging from 2.4 mm to 3.3 mm in length. Porosity was homoge nous with <p = 0.775 SD 0.058 and degree of anisotropy(DA) tended to isotropie value (Daube et al. 2010) with DA= 0.25 SD 0.05. Since DA and fabric tensor are correlated [START_REF] Zysset | A review of morphology-elasticity relationships in human trabecular bone: theories and experiments[END_REF][START_REF] Zbislaw | Equivalence of mean intercept length and gradient fabric tensors -3d study[END_REF]), isotropy at the beam scale was a validated assumption.

Detailed analysis of the acquisitions revealed two other types of materials. The first material corresponded to the cyanoacrylate polymer used to clamp the specimen, which saturated a few pores locally. The mean penetration length was 1.95 mm SD 0.34 and the reduction of free length was 10.6% SD 1.6. The second material rep resented the abrasive particles used during the cutting process (see

Comparison with control bone models (Sawbone Ò )

Results of the three point bending tests and of modal analysis are summarized in Table 1. The effective modulus obtained from equation (1) relative to bending tests was E s = 100 MPa SD 6 was correlated with manufacturer data ranging between 123 MPa and 173 MPa obtained with tensile tests.

Concerning the modal analysis, variations in frequency mea surements were less than 0.7% (repeatability test) and the effective modulus obtained from equation ( 7) was E d = 128 MPa SD20. This result was in good agreement with bending tests and manufacturer data.

Experimental modal analysis of canine cancellous bone specimens

Geometrical properties and frequencies of group A and group B are listed in Table 2 andTable 3, respectively. The natural frequen cies were obtained from at least three successive measurements of each specimen with ascending and descending swept sine and the relative standard deviation ranged from 0.1% to 2%. The FRF Bode diagram of specimen #5 in Fig. 2c shows a magnitude peak and a phases angle changing at f 1 = 789 Hz describing the first natural frequency. Phase resonance and magnitude resonance showed dis crepancies lower than 1%. The mode shape / 1 (x) described by equation (5b) was checked (Fig. 2d) by translating the confocal sensor along the specimen excited at the first resonance frequency. No other nodal points than the clamped zone and no signal cancel lation or dephasing was found and results were confirmed using portable strobe.

Effective modulus E d obtained from equation (7) were 211.5 MPa SD 18 and 279 MPa SD 52 for group A and group B, respectively. Group B showed significantly higher modulus, i.e. + 29%, with a p value of 0.011 (Student test).

Parametric sensitivity analysis

Variation dp corresponded with relative standard deviation of specimen properties measurement uncertainties calculated as variation ranges divided by mean values. In dynamics, the length sensitivity preceded those of height, frequency and density. Cross section width did not play any role. In statics, length and height showed similar influence before width, force and Poisson ratio. With dp maximized to 2%, uncertainties were almost similar in static and dynamic for the effective Young modulus with DE/E equal to 14%. Direct terms of equation (A2) in appendix were prominent since cross terms were two orders lower.

Discussion and conclusion

We hypothesized that modal analysis of uniform straight beams in bending was a methodology adapted to the identification of effective elastic modulus (or Young modulus) of cancellous bone. Exact solutions of frequencies and mode shapes being available, difficulties due to complex structural geometry were circumvented to focus on material properties themselves. The relationship between eigen values eigen vectors and elastic properties had a unique solution. Water jet cutting allowed collecting beam like specimens in canine distal femurs with accuracy and agility while preserving tissue microarchitecture. Tissue integrity, homogeneity and isotropy were confirmed by X ray micro tomography to vali date the theoretical framework.

Base excitation on clamped free straight beam was well adapted to small, light and stiff structure. We used the first natural frequency of samples for which secondary effects of shear and rotation inertia had no relevant influence. The second frequency about six times higher was detected with some samples but the frequency dependence of elastic properties was not targeted.

The theoretical and experimental methodology has been vali dated with synthetic bone model used as control in statics and dynamics. Samples mimicking biological specimens were tested in static bending whereas data manufacturer were obtained from tensile tests. The local effects of boundary conditions and strain distribution patterns in cross sections could explain discrepancies. In return, modal analysis gave similar effective elastic modules since local effects were smoothed by the balance of modal strain energy and modal kinetic energy.

For tissue, modal analysis was prioritized to prevent alteration by static tests since specimens were available. The shaping of biological samples allowed using the framework of transverse vibrations of flexure beams to get directly effective elastic modules from natural frequency measurement since mode shapes were pre dictable, unique and reproducible. Mean values of elastic modulus varying between 210 MPa and 280 MPa, were in agreement with uniaxial compressive tests ranging from 210 MPa and 315 MPa [START_REF] Kang | Mechanical properties and bone densities of canine trabecular bone[END_REF][START_REF] Kuhn | The limitations of canine trabecular bone as a model for human: a biomechanical study[END_REF][START_REF] Sumner | Distribution of Young's modulus in the cancellous bone of the proximal canine tibia[END_REF][START_REF] Vahey | Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur[END_REF].

We proposed a preliminary application with a relative statisti cal power given the limited number of samples. Anyway, it seemed relevant enough to highlight property differences depending upon location and a significant increase of effective modulus when approaching the articular surface was found. This result, in agree ment with the literature [START_REF] Vahey | Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur[END_REF][START_REF] Sumner | Distribution of Young's modulus in the cancellous bone of the proximal canine tibia[END_REF], could be explained by a local mechanobiological remodelling due to increased strain energy which induced changes in tissue microarchitecture, porosity and stiffness.

The methodology for elastic modulus determination that we suggested was based on the exploration of one intrinsic property of a structure, which is its modal basis, i.e. mode shapes associated to natural frequencies [START_REF] Lalanne | Mechanical Vibrations for[END_REF][START_REF] Meirovitch | Elements of vibration analysis[END_REF]. The bending strain and stress fields into the beam like specimens were predominant, which allowed concluding that the effective elastic modulus (or Young modulus) was effectively the main actor of the structural response. For weakly dissipative structures, Lagran gian conservation smoothes the local influence of boundary condi tions in modal strain energy by the presence of the modal kinetic energy. Moreover, a resonance can be revealed with minimal input energy while being measured with great accuracy. The first order sensitivity analysis using maximum value of parameter uncertain ties concluded to a global uncertainty of tissue Young modulus lower than 14% which was a satisfactory result.

A limitation could concern the concept of effective elastic mod ulus underlying the proposed methodology. Indeed, cancellous bone was considered as a biphasic material, result of the minimal combination of a strongly porous medium drained by a fluid bone marrow. In static, it is possible to consider a negligible role of the fluid phase, which can be reasonably supposed, especially in com pressive tests, if fluid flow is drained at boundary conditions. In dynamic oscillations of low magnitudes, we can consider that the fluid phase acts mainly in mass but weakly contributes to the structure stiffness since the beam cross section is quasi constant during cycles. The mass effect is taken into account in the effective mass per unith length of samples.

In return, fluid structure interaction at the microscale may play a significant role in dissipative energy. These effects might modify the natural frequencies while shifting amplitude resonance from phase resonance. The derivation of elastic modulus can be updated with specific theoretical and experimental modal analysis of damped systems, if however this represents a significant impact compared to the uncertainties of measurement. This question can be addressed by implementing modal analysis of fresh tissue and dehydrated monophasic tissue. It could also help to interpret the differences between static and dynamic properties of biological tis sue. In addition, the second and third frequencies of beams are about six times and 17 times higher. Even if the second frequency was detected with some samples, it was generally more difficult to obtain them and exploring the frequency dependence of elastic properties was not the initial aim of the study.

In conclusion, beam modal analysis showed compelling aspects for the identification of mechanical cancellous bone properties. The methodology focused on Young modulus will be completed by Poisson ratio determination directly from torsion resonances. Inverse algorithms could also be implemented to simultaneously identify several elastic coefficients [START_REF] Henyš | Material model of pelvic bone based on modal analysis: a studyon the composite bone[END_REF][START_REF] Swider | Dynamic modelling of a composite plate, a mixed numerical and experimental approach[END_REF][START_REF] Swider | Interactions between numerical and experimental approaches in composite structure dynamics[END_REF]. In all cases the increased data base of biological specimen tested by modal analysis and static tests should confirm the methodology relevance.

The proposed methodology was rapid and reproducible, non destructive and adaptable to variable structure size. This could be reasonably exploited to characterize other fragile tissues, mate rials which elastic properties evolve rapidly in time, and small vol umes materials or tissues.

Appendix:

The first order Taylor expansion of function Young modulus E at point (p 0 ,q 0 ) is expressed by equation (A1), dp and dq being the small change in variables p and q. In static, variables p and q repre sent specimen parameters and mechanical responses, i.e. force F and displacement v for static three point bending tests. In dynam ics, p and q represent specimen parameters and the natural fre quency x 1 . E p 0 þ dp; q 0 þ dq ð Þ E p 0 ; q 0 ð Þþdp@ p E p 0 ; q 0 ð Þ þ dq@ q E p 0 ; q 0 ð Þþ dpdq@ 2 p;q E p 0 ;

q 0 ð Þ þ O d 2 p; d 2 q À Á ðA-1Þ
Finally, the generic form of modulus uncertainty is expressed by equation (A 2).

DE

E dp Á @ p E=E þ dq Á @ q E=E þ dpdq@ 2 p;q E=E þ O d 2 p ; d 2 q ðA-2Þ with : @ b E d =E d 0; @ h E d =E d 2 h ; @ l E d =E d 4 l ; @ q E d =E d 1 q ; @ x E d =E d 2 x ;
@ 2 E d @h@l =E d 8 hl ; @ 2 E d @h@q =E d 2 qh ; @ 2 E d @h@x =E d 4 hx ;

@ 2 E d @l@q =E d 4 lq ; @ 2 E d @l@x =E d 8 lx

@ 2 E d @q@x =E d 2 qx and :

@ b E s =E s 1 b ; @ h E s =E s 3 h ; @ l E s =E s 3 l ; @ F E s =E s 1 F ; @ v E s =E s 1 v ; @ 2 E s @b@h =E s 3 bh ; @ 2 E s @b@l =E s 3 bl ; @ 2 E s @b@l =E s 3 bl ; @ 2 E s @b@F =E s 1 bF ; @ 2 E s @b@v =E s 1 bv ; @ 2 E s @h@l =E s 9 hl ;
@ 2 E s @h@F =E s 3 hF ; @ 2 E s @h@v =E s 3 hv ; @ 2 E s @l@F =E s 3 lF ;

@ 2 E s @l@v =E s 3 lv ; @ 2 E s @F@v =E s 1 Fv

Fig. 1 .

 1 Fig. 1. (a) Medio-lateral X-ray image of the femur embedded into a polyurethane resin block after water cutting of specimen A (distal zone of the articular cartilage) and specimen B (proximal zone of the articular cartilage); (b) Cancellous bone specimen in clamped-free boundary conditions into the 30-printed clamping fixture; (c) Microarchitecture of the specimen and ( d) Partial view of a X-ray tomographie slice showing: pores -a, cancellous bone -b, clamping cyanoacrylate polymer -1 and abrasive particles -2 used in water-cutting process.

  Properties of polymeric foam beams (Sawbone ® ) used as con trol were: b = 4.93 mm SD 0.1, h = 3.42 mm SD 0.1 and 1 = 40 mm SD 0.2. Measured density was p = 248 kg/m 3 SD 16. transverse displacement v(x,t) confocal transmitted force, F (t ) •• •. � :---•• • • •• ,;. •; •• • specimen force gage

  Fig. 2. (a) Diagram of the experimental dynamic devic e. (b) Experimental set-up, (c) FRF (magnitude and phase angle) of specimen #5 withf, • 789 Hz and (d) associated mode shape ,J,,(x).

Table 1

 1 Effective Young modulus E of control specimens (#1522-13 Sawbone Ò ) in 3 points bending test (static) and experimental modal analysis. Control values* provided by the manufacturer were obtained with static compressive or traction tests.

		Control value *		Present study
		compression	traction	3 pts bending test	modal analysis
	E (MPa)	123	173	100 SD 6	128 SD 20

Table 2

 2 Geometrical properties, natural frequencies and effective Young modulus E d of specimens A collected from the zone more distal to the articular surface of the femoral epiphysis.

	specimen	l (10 3 m)	b (10 3 m)	h (10 3 m)	qS (10 2 kg/m)	f 1 (Hz)	E d (MPa)
	1	13.8	4.9	2.83	1.435	1076.9	208.25
	2	12.24	5.43	2.65	1.593	1284.4	222.67
	3	15.33	4.97	2.51	1.388	805.0	241.25
	4	15.66	4.81	2.74	1.562	759.9	210.95
	5	14.53	4.67	2.54	1.438	789.0	199.82
	6	11.23	4.65	2.64	1.547	1391.3	212.48
	7	16.54	4.79	2.63	1.276	661.6	184.70
					mean E d = 210 MPa SD 19

Table 3

 3 Geometrical properties, natural frequencies and effective Young modulus E d of specimens B collected from the zone more proximal to the articular surface of the femoral epiphysis.

	specimen	l (10 3 m)	b (10 3 m)	h (10 3 m)	qS (10 2 kg/m)	f 1 (Hz)	E d (MPa)
	9	17.65	4.47	2.4	1.418	590.1	298.71
	10	16.78	4.86	2.55	1.505	755	324.84
	11	21.93	5.1	2.16	1.310	337.5	257.47
	12	15.42	4.93	2.81	1.554	771.25	182.61
	13	22.15	4.88	2.7	1.439	475.62	314.17
	14	19.85	4.71	2.63	1.252	534.42	248.18
	15	20.98	4.60	2.61	1.225	541.63	326.59
					mean E d = 271.0 MPa SD 53

Acknowledgements

The French Minister of Education and Research, the French National Center for research (CNRS) and the National Veterinary School of Toulouse (France) are acknowledged for their assistance. The study was partially funded by the Occitanie Region (France), grant RBIO 2015 INPT IMFT # 14054303.

Declaration of Competing Interest

The authors declare that they have no known competing finan cial interests or personal relationships that could have appeared to influence the work reported in this paper.