Shuai Qi
email: qishuailw@163.com

Yu-Jun Cui

Jean-Claude Dupla

Ren-Peng Chen

Han-Lin Wang

Yu Su

Francisco Lamas-Lopez

Jean Canou

Mr Shuai Qi

Investigation of the parallel gradation method based on the response of track-bed materials under cyclic loadings

Keywords: interlayer soil, cyclic triaxial tests, permanent deformation, resilient modulus, parallel gradation method

Ballast/fines mixture is often found in railway substructure and the overall behaviour of tracks is strongly dependent on the mechanical behaviour of this mixture. Since directly testing such mixture with large grains is limited with common laboratory equipment, the consideration of a model material at smaller size is advisable. Parallel gradation method is widely used for this purpose. This study assesses the validity of this method in the case of ballast/fines mixture. Large-scale cyclic triaxial tests were carried out on the ballast/fines mixture at six volumetric coarse grain contents. The results obtained were analysed, together with those obtained previously from small-scale triaxial tests on a microballast/fines mixture whose microballast grain size distribution was determined by applying the parallel gradation method. The cyclic parameters (permanent strain and resilient modulus) were obtained for the two types of mixture. Results show that for all mixtures two distinct soil fabrics can be identified according to the variations of permanent strain and resilient modulus with the volumetric content of coarse grains fv: a fine-fine contact structure for fv ≤ 20% and a grain-grain contact structure for fv ≥ 35%. In the case of fine-fine contact structure, the permanent strains and resilient modulus values of the ballast specimens are consistent with those of the microballast specimens, evidencing the validity of the parallel gradation method. In the case of grain-grain contact structure, the permanent strains and resilient modulus values are found to coincide globally at the two scales, also justifying the validity of the parallel gradation method, the slight differences between the two scales being attributed to the irregular grain sliding and the 3 distribution of the fines soils.

Introduction

Ballast/fines mixture is often found in railway substructure [START_REF] Duong | Effects of fines and water contents on the mechanical behavior of interlayer soil in ancient railway sub-structure[END_REF][START_REF] Indraratna | A new parameter for classification and evaluation of railway ballast fouling[END_REF]. For instance, in the French conventional railway lines (representing 94% of the total railway network), there is an interlayer which was formed by interpenetration of ballast and subgrade soils under the effect of train circulation [START_REF] Duong | Effects of fines and water contents on the mechanical behavior of interlayer soil in ancient railway sub-structure[END_REF]. During the renewable program for conventional lines, this layer was maintained since the dry density of the interlayer was as high as 2.4 Mg/m 3 and thus a favourable mechanical behaviour was expected [START_REF] Cui | Investigation of the hydro-mechanical behavior of fouled ballast[END_REF]. According to the in-situ investigation, the soil fabric of the interlayer is found to be greatly dependent on the content of ballast grains [START_REF] Trinh | Comportement hydromécanique des matériaux constitutifs de plateformes ferroviaires anciennes[END_REF][START_REF] Trinh | Mechanical characterisation of the fouled ballast in ancient railway track sub-structure by large-scale triaxial tests[END_REF]: the ballast grains are in contact with each other for the upper part with high ballast content, but floated in the fines matrix for the lower part with lower ballast content. Since the ballast/fines mixture plays an important role in the overall behaviour of rail tracks, it is important to investigate the mechanical response of such mixture under cyclic loading in terms of permanent deformation and resilient modulus. Considering the large size of ballast grains, large-scale triaxial apparatus is normally needed for this purpose. However, since the large testing system is complex and costly, scaling the ballast grains down to a model ballast material with smaller size and performing tests using standard apparatus appear quite promoting.

Three methods are commonly used for the scaling purpose: the scalping method [START_REF] Zeller | The shear strength of the shell materials for the Göschenenalp Dam, Switzerland[END_REF] which consists in discarding the grains larger than a specified size from the prototype materials, the parallel gradation method [START_REF] Lowe | Shear strength of coarse embankment dam materials[END_REF] which consists in creating a model material at smaller size with its grain size distribution parallel to that of the prototype material, and the replacement method [START_REF] Frost | Some testing experiences and characteristics of boulder-gravel fills in earth dams[END_REF] which consists in replacing the grains from a specified size to the maximum size with the grains from this specified size to a smaller size. Among them, the parallel gradation method is regarded to be the most appropriate because it can consider a large range of prototype gradation shape; thus, it has been adopted mostly [START_REF] Aingaran | Experimental investigation of static and cyclic behavior of scaled railway ballast and the effect of stress reversal[END_REF][START_REF] Indraratna | Large-scale triaxial testing of greywacke rockfill[END_REF][START_REF] Qi | Influence of grain size distribution of inclusions on the mechanical behaviors of track-bed materials[END_REF][START_REF] Varadarajan | Testing and modeling two rockfill materials[END_REF][START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF][START_REF] Wang | Investigation on the mechanical behavior of track-bed materials at various contents of coarse grains[END_REF][START_REF] Xiao | Bounding surface model for rockfill materials dependent on density and pressure under triaxial stress conditions[END_REF]. The validity of this method under cyclic loadings was assessed by Sevi and Ge [START_REF] Sevi | Cyclic behaviors of railroad ballast within the parallel gradation scaling framework[END_REF] based on the response of the prototype and model grains of ballast grains. Their results in terms of resilient modulus, permanent axial strain and permanent volumetric strain were unfortunately not conclusive for validating the parallel gradation method. In addition, only pure ballast grains were tested without fines fraction.

In this study, the parameters characterizing the cyclic response (permanent strain and resilient modulus) of ballast/fines mixtures representing the interlayer soils in the conventional French railway track-beds were determined by conducting large-scale cyclic triaxial tests. Six volumetric ballast contents (0%, 5%, 10%, 20%, 35%, 45%) were considered. The results obtained were analysed together with those obtained previously by Wang et al. [START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF] from small-scale cyclic triaxial tests on microballast/fines mixture with microballast prepared by applying the parallel gradation method. Comparison of the mechanical parameters obtained at two scales allowed the validity of the parallel gradation method to be assessed on coarse grain/fines mixtures.

Materials

The ballast grains having the same grain size distribution as the interlayer soil in "Sénissiat site" were used [START_REF] Trinh | Comportement hydromécanique des matériaux constitutifs de plateformes ferroviaires anciennes[END_REF], as shown in Fig. 1. The minimum grain size D min b and the maximum grain size D max b are 20 mm and 63 mm, respectively. The microballast was tested previously by Wang et al. [START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF]; its maximum grain size D max m was chosen as 20 mm to adapt to a small-scale triaxial cell of 100 mm with a value of 5 as the ratio of specimen to maximum grain size. This is consistent with the recommendation of Fagnoul and Bonnechere [START_REF] Fagnoul | Shear strength of porphiry materials[END_REF], Nitchiporovitch [START_REF] Nitchiporovitch | Shearing strength of coarse shell materials[END_REF] and Pedro [START_REF] Pedro | De l'étude du comportement mécanique de sols hétérogènes modèles à son application au cas des sols naturels[END_REF]: a ratio larger than 5 must be adopted to minimize the specimen size effect. The grain size distribution of the microballast is transited from that of the ballast by following the parallel gradation method. In this method, the correlative grain sizes of ballast and microballast share the same percentage of grain passing. The grain sizes of these two materials are correlated according to the following equation:

(1)

D b -D min b D m -D min m = D max b -D min b D max m -D min m = A
where D max , D min and D are the maximum grain size, the minimum grain size, and a given grain size, respectively. The superscripts b and m stand for ballast and microballast, respectively. A is a constant.

To prepare the microballast, three angular-shaped commercial coarse grains were used. Visually, the grain shape of the chosen commercial grains was similar to that of the field ballast from "Sénissiat site". The minimum grain size of the commercial grains D min m was 1.6 mm, defining a value of 2.337 for parameter A. For a given grain size of microballast D m , 10.0 mm for instance, equation (1) gives a correlated ballast grain size D b of 39.6 mm. As the ballast grain passing percentage is 59.1 % for D b = 39.6 mm, according to the parallel gradation method, the microballast grain passing percentage is also 59.1 % for D m = 10 mm. Following this procedure, the complete grain size distribution of the microballast was obtained, as shown in Fig. 1. This target (parallel gradation) curve was verified by measurement and it is observed that the measured grain size distribution and the target one show a good agreement (Fig. 1).

For the fines soils, since obtaining enough quantity from the field was difficult, they were fabricated in the laboratory. Nine commercial soils were mixed to simulate the fines soils from "Sénissiat site", which included five kinds of medium sands, two kinds of fine sands and two kinds of clay soils (kaolinite and bentonite). Table 1 listed the properties of these soils. The mass proportion of each constituting soil was calculated according to the grain-size distributions of the fines from "Sénissiat site" (Fig. 2). The calculated results were also shown in Table 1. With the proportion and the grain size distribution of each soil, the grain size distribution of the fabricated fines can be calculated, which was plotted in Fig. 2 as well. During the mixing procedure, the materials to be mixed were carefully added into the mixer following the sequence of medium sands, water, fine sands, kaolinite and bentonite. The mass of the added water corresponded to a small water content of 4% of the total mixed soils.

The aim of water addition was to prevent the small particles loss during the mixing, occurring when fine sands, kaolinite and bentonite were dry mixed with others.

Bentonite was added last since it was the most reactive soil with water. With this sequence, most kaolinite and bentonite particles were expected to be adhered to the surface of sands. Using this procedure, a homogeneous mixture can be obtained [START_REF] Lamas-Lopez | Field and laboratory investigation on the dynamic behaviour of conventional railway track-bed materials in the context of traffic upgrade[END_REF].

Wang et al. [START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF] also adopted this method when preparing specimens of microballast and fines mixture. After preparation, the fabricated fines soil was compared with the field one in terms of grain size distribution and plasticity. As can be observed from Fig. 2, the grain size distribution of the fabricated fines showed a good agreement with that of the in situ one. In addition, the liquid limit and the plasticity index of the fabricated fines, which were 32% and 20%, respectively, matched the target liquid limit (35.5%) and plasticity index (17.3%) well. This soil is classified as lean clay (CL) according to ASTM D2487-11 [START_REF]D2487-11: Standard practice for classification of soils for engineering purposes (unified soil classification system)[END_REF]. Standard Proctor compaction tests were carried out on the fines material following ASTM D698-12 [START_REF]Standard test methods for laboratory compaction characteristics of soil using standard effort[END_REF].

An optimum water content wopt-f = 13.70% and a maximum dry density ρdmax-f = 1.82

Mg/m 3 were determined.

Experimental methods

Specimen preparation

In order to quantify the coarse grains in a specimen, a parameter namely volumetric content of coarse grains fv was adopted, which was defined as the ratio of the total volume of coarse grains to the total specimen volume [START_REF] Pedro | De l'étude du comportement mécanique de sols hétérogènes modèles à son application au cas des sols naturels[END_REF][START_REF] Qi | Influence of grain size distribution of inclusions on the mechanical behaviors of track-bed materials[END_REF][START_REF] El Dine | Mechanical characterization of matrix coarse-grained soils with a large-sized triaxial device[END_REF][START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF][START_REF] Wang | Investigation on the mechanical behavior of track-bed materials at various contents of coarse grains[END_REF]. The total specimen volume consisted of two parts: the coarse grain volume and the fines volume. All pores were assumed to be included in the fines soils. For the upper part of the natural interlayer soils at "Sénissiat site", the fv value was 52.9% [START_REF] Trinh | Comportement hydromécanique des matériaux constitutifs de plateformes ferroviaires anciennes[END_REF]. In this study, six fv values (0%, 5%, 10%, 20%, 35%, 45%) were considered for the specimens prepared with both ballast and microballast. For convenience, in further analysis, the specimens prepared with ballast and microballast are referred to as ballast specimens and microballast specimens, respectively. The dimensions of the ballast specimens are 300 mm diameter and 600 mm height, while the dimensions of microballast specimens are 100 mm diameter and 200 mm height.

For the preparation of the ballast specimen, the total volume was first calculated with the specimen dimensions (300 mm diameter and 600 mm height). For a given fv value, the volumes of both ballast grains and fines can be obtained based on this total specimen volume. After that, the mass of ballast was obtained using the dry unit mass of ballast (2.68 Mg/m 3 ). The fines in all specimens were kept at the same state defined by the optimum water content wopt-f = 13.70% and the maximum dry density ρdmax-f = 1.82 Mg/m 3 . The optimum state was selected aiming to simulate the heavily compact state in the field condition [START_REF] Duong | Investigation of the hydro-mechanical behavior of ancient railway platforms in scope to reinforcement by soil-mixing[END_REF][START_REF] Lamas-Lopez | Field and laboratory investigation on the dynamic behaviour of conventional railway track-bed materials in the context of traffic upgrade[END_REF][START_REF] Trinh | Mechanical characterisation of the fouled ballast in ancient railway track sub-structure by large-scale triaxial tests[END_REF]. Accordingly, the masses of dry fines and water can be determined. Water was firstly added into the fines by spray to reach the optimum water content wopt-f = 13.7%. Then, the wetted fines soil was stored in hermetic containers for 24 h for moisture homogenization. Afterwards, the homogenized wetted fines soil was equally divided into eight parts (by weight) and so were the oven-dried ballast grains, after which they were mixed separately. Finally, the ballast specimen was compacted using a layered compaction method, which was widely adopted by many researchers to ensure reasonable uniformity of the specimen [START_REF] Cai | Influence of grain gradation on permanent strain of unbound granular materials under low confining pressure and high-cycle loading[END_REF][START_REF] Lenart | Large-scale triaxial tests of dense gravel material at low confining pressures[END_REF][START_REF] Suiker | Static and cyclic triaxial testing of ballast and subballast[END_REF][START_REF] Wichtmann | Prediction of permanent deformations in pavements using a high-cycle accumulation model[END_REF]. The mixed materials were compacted in eight layers using a vibration hammer, each with a thickness of 75 mm.

Similar protocol was applied for the preparation of the microballast specimen (100 mm diameter and 200 mm height). The masses of the dry microballast, dry fines soil and water were firstly determined. Afterwards, the fines soil was wetted with the sprayed water to reach the optimum water content of 13.7%, stored in hermetic container for 24 h and then mixed with the microballast. Finally, this mixture was compacted dynamically to reach the target height, also following the layered compaction method to ensure reasonable uniformity of the specimen. More details can be found in Wang et al. [START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF]. Note that the test results at five fv values (5%, 10%, 20%, 35%, 45%) obtained by Wang et al. [START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF] were taken for comparison with the corresponding ones of the ballast samples in this study.

Cyclic triaxial tests

For the microballast specimen with 100 mm diameter and 200 mm height, a small-scale cyclic triaxial apparatus was used. Details about this device can be found in Wang et al. [START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF]. For testing a ballast specimen (300 mm diameter and 600 mm height), a large-scale cyclic triaxial apparatus developed by Dupla et al. [START_REF] Dupla | Mechanical behavior of coarse grained soils reference[END_REF] was adopted. The loading procedures adopted are the same for the ballast and the microballast specimens. No saturation process was applied and the specimen water content was kept constant during the test. In addition, the drainage valve was kept open during the loading process. All the tests were performed at a confining pressure of 30 kPa, which corresponded to the average horizontal stress estimated for the field condition, considering the effects of the train wheel load, the interlayer depth and the Poisson ratio [START_REF] Duong | Effects of water and fines contents on the resilient modulus of the interlayer soil of railway substructure[END_REF][START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF]. The applied cyclic loads followed a sine-shaped pattern (Fig. 3). The deviator stress amplitude Δq is defined as the difference value between the maximum deviator stress qmax and the minimum deviator stress qmin [START_REF] Wang | Estimating static and dynamic stresses in geosynthetic-reinforced pile-supported track-bed under train moving loads[END_REF][START_REF] Wang | Full-scale model study on variations of soil stress in geosynthetic-reinforced pile-supported track-bed with water level change and cyclic loading[END_REF][START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF]. A frequency of 1.78 Hz corresponding to a low train speed of 50 km/h was chosen, aiming to guarantee the quality of controlling the pre-defined loading shape. Note that this low frequency did not reflect the high train speed normally encountered in the field. This selection was based on the hypothesis that the train speed does not influence the assessment of the validity of the parallel gradation. During the loading procedure, axial stress and strain were monitored.

The application of cyclic loads included two consecutive stages. The loading stage 1 was focused on the aspect of the permanent deformation under a large number of loading cycles. The loading stage 2 was focused on the aspect of the resilient modulus or stiffness property under a wide range of stress amplitudes, including some large amplitudes possibly applied on the interlayer soils. In stage 1, a multi-step loading procedure proposed by Gidel et al. [START_REF] Gidel | A new approach for investigating the permanent deformation behavior of unbound granular material using the repeated load triaxial apparatus[END_REF] was adopted (Fig. 4a): the deviator stress amplitude Δq = qmaxqmin was increased stepwise from 10 kPa to 30 kPa with an increment of 5 kPa. At each amplitude, 90,000 cycles were applied, which were thought to be enough for the stabilization of the permanent strain [START_REF] Duong | Effects of fines and water contents on the mechanical behavior of interlayer soil in ancient railway sub-structure[END_REF][START_REF] Gidel | A new approach for investigating the permanent deformation behavior of unbound granular material using the repeated load triaxial apparatus[END_REF][START_REF] Lamas-Lopez | Field and laboratory investigation on the dynamic behaviour of conventional railway track-bed materials in the context of traffic upgrade[END_REF][START_REF] Trinh | Mechanical characterisation of the fouled ballast in ancient railway track sub-structure by large-scale triaxial tests[END_REF]. This multi-stage loading procedure allowed the application of several stress amplitudes on the same specimen, with which not only the number of tests could be reduced, but also the effect of the variability of soil specimens on the testing results can be minimised. Thus, this procedure has been widely used in the study of the permanent deformation behaviour [START_REF] Cao | Long-term behavior of clay-fouled unbound granular materials subjected to cyclic loadings with different frequencies[END_REF][START_REF] Duong | Effects of fines and water contents on the mechanical behavior of interlayer soil in ancient railway sub-structure[END_REF][START_REF] Jing | Permanent deformation behaviour of a granular material used in low-traffic pavements[END_REF][START_REF] Lamas-Lopez | Field and laboratory investigation on the dynamic behaviour of conventional railway track-bed materials in the context of traffic upgrade[END_REF][START_REF] Tang | Dynamic behaviours of railway's base course materials subjected to long-term low-level cyclic loading: experimental study and empirical model[END_REF][START_REF] Trinh | Mechanical characterisation of the fouled ballast in ancient railway track sub-structure by large-scale triaxial tests[END_REF][START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF]. These loading amplitudes were selected based on the vertical stress measured at the equivalent depth of interlayer in the field [START_REF] Lamas-Lopez | In-situ investigation of the behaviour of a French conventional railway platform[END_REF][START_REF] Lamas-Lopez | Field and laboratory investigation on the dynamic behaviour of conventional railway track-bed materials in the context of traffic upgrade[END_REF]. In stage 2 (Fig. 4b), a procedure proposed by Lamas-Lopez [START_REF] Li | Method for railroad track foundation design. I: Development[END_REF] was adopted.

The deviator stress amplitude Δq was firstly increased following a Δq sequence (10 kPa, 30 kPa and 50 kPa) and then decreased following the reverse sequence.

Afterwards, Δq was increased following a sequence from 10 kPa to 100 kPa and then decreased to 10 kPa. Finally, Δq was increased in steps to 200 kPa and then decreased to 10 kPa. At each stress level, 100 cycles were applied. The aim of firstly increasing

Δq and then decreasing it was to obtain the resilient modulus at both elastoplastic stage and pure elastic stage. Note that the large stress amplitudes (50 kPa, 100 kPa and 200 kPa) were defined according to the vertical stress of 40-90 kPa at similar depth of the interlayer observed by Selig and Waters [START_REF] Selig | Track geotechnology and sub-structure management[END_REF], Jain and Keshav [START_REF] Jain | Stress distribution in railway formation -a simulated study[END_REF] and Yang et al. [START_REF] Yang | Dynamic stress analysis of a ballasted railway track bed during train passage[END_REF], the vertical stress of 120-138 kPa applied by heavier wagons in some countries [START_REF] Grabe | Effects of principal stress rotation on permanent deformation in rail track foundations[END_REF][START_REF] Jain | Stress distribution in railway formation -a simulated study[END_REF][START_REF] Li | Method for railroad track foundation design. I: Development[END_REF] and the contingent maximum vertical stress at the top of railway structure as high as 200 kPa [START_REF] Lamas-Lopez | Field and laboratory investigation on the dynamic behaviour of conventional railway track-bed materials in the context of traffic upgrade[END_REF].

Results and discussions

Variations of permanent deformation with fv

Fig. 5 plots the deviator stress q versus the axial strain ε 1 during the first 90,000 cycles of loading stage 1 for the ballast specimen at fv = 35%. In a loading-unloading cycle, the permanent axial strain ε 1p represented the irreversible part of the total axial strain. It can be observed that the ε 1p value of the first cycle was particularly large. This phenomenon was also observed for other ballast specimens and the microballast specimens. This large permanent strain can be attributed to the adaptation of the specimen to the loading system, in particular in terms of contact between the specimen and the loading piston. As the initial specimen-piston contact was not identical for all the specimens, this ε 1p value would make impossible the relevant comparison of permanent strain between different specimens. Thus, in further analysis, the first cycle was not accounted for the cumulative permanent strain.

Fig. 6 presents the permanent strain ε 1p against the number of cycles N for six ballast specimens, with the corresponding stress levels. The evolutions of permanent strain with number of cycles N were the same as the ones of microballast specimens [START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF]. It can be observed from Fig. 6 that for each fv value, ε1p increased quickly at the start of each loading level and then tended to stabilize with the increasing cycles.

According to Werkmeister et al. [START_REF] Werkmeister | Pavement design model for unbound granular materials[END_REF], the large deformation in the beginning resulted from a significant rearrangement of the particles. As the cycle number increased, this particle rearrangement became more and more limited, and correspondingly, the permanent strain tended to become stable. As can also be seen from Fig. 6, a smaller fv value led to a larger ε1p, which can be attributed to the involvement of less coarse grains.

Due to the multi-step loading procedure adopted, the ε 1p at each stress level was greatly influenced by the previous loadings (Fig. 6). If this loading history is eliminated (as if the specimen has been subjected to only one loading level), the effect of the ballast contents can be better evaluated. For this purpose, a permanent strain estimation method proposed by Gidel et al. [START_REF] Gidel | A new approach for investigating the permanent deformation behavior of unbound granular material using the repeated load triaxial apparatus[END_REF] and then used by Lamas-Lopez [START_REF] Lamas-Lopez | Field and laboratory investigation on the dynamic behaviour of conventional railway track-bed materials in the context of traffic upgrade[END_REF] and Wang et al. [START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF] was adopted. The application of this method is explained in Fig. 7 by considering two successive loading levels (Loading level M and Loading level M + 1). For the estimated permanent strain ε 1p M+1 of the Loading level M + 1 at the first 90,000 cycles, it is determined as:

(2)

ε 1p M+1 = ε 1p M + δε 1p M+1
where ε 1p M is the measured permanent strain of the Loading Level M; δε 1p M+1 is the translated permanent strain determined by resetting both the initial strain value and the first cycle of the measured permanent strain of the Loading level M + 1 to 0.

For estimating the permanent strain of the Loading level M + 1 after 90,000 cycles, a linear increasing trend between permanent strain and number of cycles is considered, with its slope kept at the same value as the one determined at the last cycles of Loading level M + 1. More details can be found in Wang et al. [START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF].

The estimated permanent strain curves for fv = 0% of the ballast specimen are presented in Fig. 8, together with the measured results. It can be observed that at the last cycle of each stress level, the estimated strain was larger than the measured one.

For example, for Δq = 20 kPa, the measured permanent strain at the last cycle N = 270,000 was 0.182%, while the corresponding estimated one was 0.226%. This phenomenon can be attributed to the different loading histories: for the estimated strain, the specimen was assumed to undergo one higher stress level of 20 kPa. When it came to the measured strain, successive lower stress levels of 10kPa, 15 kPa and 20 kPa were applied, leading to a smaller strain value.

Gidel et al. [START_REF] Gidel | A new approach for investigating the permanent deformation behavior of unbound granular material using the repeated load triaxial apparatus[END_REF] examined this permanent strain estimation method through cyclic triaxial tests, concluding that the permanent strain estimated for a single stress level agreed well with the measured one for the loading cycles applied at each step (90,000 cycles in this study). In other words, in this study, the estimated permanent strain at the first 90,000 cycles was relatively accurate. Fig. 9 depicts the estimated permanent strain evolutions at the first 90,000 cycles under different stress levels for ballast specimen at fv = 0%. As can be observed, a larger stress level led to a larger strain value. In addition, the permanent strain gradually stabilized with the increase of the cycle number. To further illustrate the effect of ballast content fv, the estimated permanent strains for cycle N = 90,000 (end-stage cycle in Fig. 9) at all stress levels were taken and compared (Fig. 10). It can be observed that at each stress level, the permanent strain decreased with the increase of fv. Moreover, a bi-linear decreasing trend could be identified. When fv ≤ 20%, the permanent strain decreased rapidly. By contrast, when fv ≥ 35%, this decreasing rate became smaller. The same observations were made by Wang et al. [START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF] for the microballast specimens.

In the study of Wang et al. [START_REF] Wang | Permanent deformation of track-bed materials at various inclusion contents under large number of loading cycles[END_REF], X-ray microcomputed tomography (μCT) scans were carried out on the as-compacted microballast specimens. Two soil fabrics were identified at different fv values: a fine-fine contact structure at fv = 0%-20% and a distinct grain-grain contact structure at fv = 35%-45%. It is worth noting that at fv = 20%, the coarse grains started to be partially connected without forming a whole skeleton. Unfortunately, this μCT scan could not be conducted on ballast specimens due to their large size. However, basically, it seems plausible to consider the same feature of fabrics for the ballast specimens: also a fine-fine contact structure at low fv and a grain-grain contact structure at high fv. For fine-fine contact structure, the permanent strain results from the compression of the fines, which decreases rapidly with increasing fv. By contrast, for grain-grain contact structure, the rearrangement of the grains plays a dominant role and the permanent strain decreases slightly when fv increases.

Variations of resilient modulus with fv

Typical hysteresis loops at the first cycles of different stress levels are shown in Fig. The evolution of hysteresis loop with number of cycles is illustrated in Fig. 12 by plotting the loops for 100 cycles (from N = 701 to N = 800) at Δq = 100 kPa for the ballast specimen with fv = 20%. A large and unclosed hysteresis loop was observed for the first cycle, due to the stress level of 100 kPa which was larger than all the previous applied stress levels. In addition, with the increase of the cycle number, the hysteresis loop became smaller and smaller and at the end of loading, the loops became closed and relatively unchanged, suggesting a pure elastic behaviour of soil.

As shown in Fig. 13, the resilient modulus Mr is defined as [START_REF] Rollins | Shear modulus and damping relationships for gravels[END_REF][START_REF] Seed | Moduli and damping factors for dynamic analyses of cohesionless soils[END_REF]:

(3) M r = Δq/ε 1r
where Δq is the deviator stress amplitude, and ε 1r is the resilient strain.

The evolution of Mr with the number of cycles N at varying stress levels is presented in Fig. 14 for the ballast specimen at fv = 20%. The results of other ballast specimens and microballast specimens obey the same rule and are not shown for clarity. It can be observed that when the stress amplitude increased, the Mr value immediately decreased and then tended to stabilize. By contrast, with the stress amplitude decreasing, the Mr value increased instantly and then tended to reach a stabilization state. The decrease of Mr with the increase of stress amplitude can be explained by the strain increasing due to the increase of stress amplitude [START_REF] Atkinson | Non-linear soil stiffness in routine design[END_REF].

In order to illustrate the effect of fv on resilient modulus, the Mr values of the last (end-stage) cycles of all stress levels were taken and presented as a function of fv for different stress levels (Fig. 15). The data were named according to the number of cycles N and the corresponding Δq values. It appears that an increasing trend of Mr with fv can be identified for all stress levels. Moreover, this increasing trend followed a bi-linear pattern with the slope at fv ≥ 35% larger than the one at fv ≤ 20%, evidencing that the soil fabrics of the ballast specimens at these two fv ranges were different: fine-fine contact structure at low fv and grain-grain contact structure at higher fv. Same bi-linear variation trend of Mr was identified by Wang et al. [START_REF] Wang | Effects of inclusion contents on resilient modulus and damping ratio of unsaturated track-bed materials[END_REF] for the microballast specimens, similarly, evidencing that the soil fabrics of microballast specimens at fv ≤ 20% and fv ≥ 35% were separately dominated by fine-fine contact structure and grain-grain contact structure.

Comparison of permanent strain between ballast and microballast specimens

The estimated permanent strains for cycle N = 90,000 (end-stage cycle) at six fv values are presented in Fig. 16 for the ballast and the microballast specimens, for five stress levels (10, 15, 20, 25 and 30 kPa). It can be observed that in the case of fv = 0%, at each stress level, the ε1p value of the ballast specimen coincides well with that of the microballast specimen. For the fine-fine contact structure at fv = 5% and 10%, on the whole, the ε1p values of the corresponding different scale specimens are found to be similar. This phenomenon can be explained by the fact that in this case the fines matrix constitutes the soil skeleton and the permanent strain was dominated by the compression of the fines, giving rise to the permanent strains which were independent of coarse grain size. For the grain-grain contact structure at fv = 35% and 45%, it can be observed that the permanent strains of the ballast specimens and the microballast specimens were almost the same (Fig. 16), in particular at Δq = 10 kPa (Fig. 16a).

Note that although the case at fv = 20% was categorized into the group of fine-fine contact structure, the influence of coarse grain size on the permanent strain followed the same pattern. This can be explained by the fact that at fv = 20%, part of the coarse grains started to be in contact with each other. Further examination shows that at Δq higher than 10 kPa, the strains of microballast specimens were slightly larger than those of the ballast specimens. This phenomenon can be explained as follows: upon loadings, the grains would slide into the pores nearby, giving rise to permanent strain.

If the difficulty of sliding into large pores for the large grains was the same as that for the smaller grains to slide into smaller pores, identical permanent strains can be expected. It is the case for the ballast specimen and the microballast specimen at Δq = 10 kPa. Apparently, the grains sliding depended on the stress level and the grain sizes;

in the microballast specimen, the grains with smaller sizes would slide into the adjacent pores more easily, leading to a slightly larger strain.

Another possible explanation for the slightly larger strains in the microballast specimen is related to the effect of the fines soils. As shown in Fig. 17, the fines soils can be roughly considered to be distributed at two locations: the pores near the coarse grain and the coarse grain intervals along the stress chain (supposed to be along the specimen height). The applied load was transmitted along the stress chain by compressing the fines between grains. When the load reached a certain level, grain rearrangement occurred [START_REF] Oda | Experimental micromechanical evaluation of strength of granular materials: effects of particle rolling[END_REF][START_REF] Oda | Microstructure of shear band and its relation to the mechanisms of dilatancy and failure of dense granular soils[END_REF]. It can be inferred that the fines in the pores inhibited the coarse grain sliding, while those between grains promoted it. At a given fv value, for the microballast specimen, more grain-fines-grain contacts can be expected along the stress chain, leading to more distribution of fines soils. As a result, relatively larger permanent strain can be expected.

As the difference of permanent strain between the two scale specimens appears quite small (Figure . 16), it can be reasonably concluded that the parallel gradation method is valid for the ballast/fines mixture in terms of permanent strain.

Comparison of resilient modulus between ballast and microballast specimens

The comparison of Mr values between the microballast and ballast specimens with fine-fine contact structure is shown in Fig. 18. At fv = 0%, the Mr values of the ballast specimen were comparable with those of the microballast specimen. This was normal because the scaling question wasnot involved in that case. For higher fv values, except fv = 10%, the Mr values of the ballast and the microballast specimens coincided with each other. The singular case at fv = 10% was probably due to a technical problem.

The Mr values of the ballast specimens at grain-grain contact structure were compared with those of the mciroballast specimens in Fig. 19 (Fig. 19a for fv = 35% and Fig. 19b for fv = 45%). It can be observed that almost the same Mr values were obtained for the two specimens, indicating that the resilient modulus was independent of grain size. Further investigation shows that on the whole the Mr values of the microballast specimens were slightly smaller than those of the ballast specimens.

These phenomena can be explained based on the analysis of Yang and Gu [START_REF] Yang | Shear stiffness of granular material at small strains: does it depend on grain size?[END_REF]. They derived the resilient modulus of a simple cubic array of identical spheres under a certain confining pressure by applying a small shear stress onto it. In the analysis, the induced resilient shear strain of the cubic array was determined by adding up the tangential strains between every two spheres along the vertical stress chain. This tangential strain was calculated using the Hertz-Mindlin contact law, which was a function of the applied shear stress. The expression for the resilient modulus (defined as the ratio of the shear stress to the shear strain) was obtained, showing no dependence on sphere radius. Note that this analysis did not consider any contribution from fine soils, thus not fully reflecting the case of this study. Indeed, as mentioned previously, in the case of mixture, the resilient modulus is affected by the fine soils along the stress chain. For the microballast specimens, as stated before, there are more fines soils along it. Thus, relatively smaller resilient modulus values are expected.

As for the permanent strain, globally, the resilient modulus also showed a good agreement between the two scale specimens albeit the slight difference discussed above, which validates the parallel gradation method in terms of resilient modulus.

Conclusions

The parallel gradation method was assessed on coarse grain/fines mixture by comparing the mechanical parameters (permanent strain and resilient modulus) of the two different scale specimens. Six volumetric coarse grain contents were considered for each kind of specimens. Based on the results obtained, the following conclusions can be drawn.

Two soil fabrics could be defined as fine-fine contact structure at fv = 0-20% and grain-grain contact structure at fv = 35-45%. It was observed that for the ballast and microballast specimens, with the increase of fv value, the permanent strain decreased rapidly when fv ≤ 20% and this decreasing rate slowed down when fv ≥ 35%, while the resilient modulus increased slightly when fv ≤ 20% and increased significantly when fv ≥ 35%.

The consistency between the large-scale and small-scale triaxial tests were found to be satisfactory: at fv = 0%, the permanent strain as well as the resilient modulus values obtained from the two scales showed a good correspondence. In the case of fine-fine contact structure, the parallel gradation method was found to be valid.

The permanent strain and resilient modulus values of the ballast specimens were consistent with the respective ones of the microballast specimens. This could be explained by the fact that in this case the fines matrix governed the soil structure.

In the case of grain-grain contact structure, on the whole, the permanent strains of the ballast specimens coincided with those of the microballast specimens (especially at Δq = 10 kPa). This coincidence was also observed in terms of resilient modulus. For the permanent strain, it could be explained by the fact that the difficulties of the sliding of large ballast grains into the adjacent large pores and the sliding of the small microballast grains into the smaller pores were expected to be the same. For the resilient modulus, it could be explained by the analytical results of Yang and Gu (2013), showing that the resilient modulus is independent of grain size.

Further examination showed that there were slight differences of permanent strain and resilient modulus between two different scale specimens, which could be attributed to the irregular grain sliding and the fines distribution.

List of tables

Table 1. Mass proportions and grain size ranges of nine commercial soils

List of Figures

f v = 0% f v = 5% f v = 10% f v = 20% f v = 35% f v = 45%

  11 for the ballast specimen at fv = 20%. The loops for stress levels of Δq = 10 kPa, Δq = 30 kPa, Δq = 50 kPa, Δq = 100 kPa and Δq = 200 kPa corresponded to cycle numbers of N = 1, N = 101, N = 201, N = 701 and N = 1401, respectively. It appears that for the low stress levels of 10 kPa and 30 kPa, the hysteresis loops were pretty small and closed. This was due to the application of 450,000 loading cycles under Δq ≤ 30 kPa in loading stage 1. By contrast, for stress levels larger than 30 kPa, permanent strain developed, leading to larger and unclosed hysteresis loops, in particular under stress levels as high as Δq = 100 kPa and 200 kPa.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .Fig. 16 .Fig. 17 .Fig. 18 .Fig. 19 .Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 12345678910111213141516171819123456 Fig. 1. Grain size distributions of the ballast and the microballast (modified from

25 kPa 40 Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .-Fig. 16 .Δq = 15 Fig. 17 .FinesFig. 18 .Fig. 19 .Table 1 .

 254078910111213141516151718191 Fig. 7. Illustration of the method to eliminate the loading history effect

Acknowledgements

The support from the Chinese Scholar Council (CSC) is greatly acknowledged.


Clemens Grünsteidl
email: clemens.gruensteidl@recendt.at

Martin Ryzy

Mike Hettich

Thomas Berer

Istvan Veres

Evaluation of elastic wave attenuation in the GHz range using zero-group-velocity resonances

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

We recently showed, that the material caused attenuation of elastic waves in seemingly homogeneous samples can significantly deviate from thermo-elastic theory at GHz frequencies [START_REF] Grünsteidl | Measurement of the attenuation of elastic waves at GHz frequencies using resonant thickness modes[END_REF]. For the development of micro-acoustic devices, for example surface-or bulk acoustic wave filters, precise knowledge of the mechanical properties of the materials used is of advantage. We therefore attempt to experimentally determine the attenuation. Material caused attenuation of elastic waves in the frequency range of interest around 2 GHz is poorly documented, and its measurement challenging. Published methods have either been demonstrated in a too low [START_REF] Ryzy | Measurement of coherent surface acoustic wave attenuation in polycrystalline aluminum[END_REF], [START_REF] Aussel | Measurement of ultrasound attenuation by laser ultrasonics[END_REF] or to high [START_REF] Maznev | Lifetime of high-order thickness resonances of thin silicon membranes[END_REF], [START_REF] Cuffe | Lifetimes of Confined Acoustic Phonons in Ultrathin Silicon Membranes[END_REF] frequency range.

Our approach is based on ZGV Lamb waves, which are locally resonant modes of vibration in plates. It has been shown that these modes can be accessed extraordinary well using LUS and many application for non-destructive testing have been developed [START_REF] Clorennec | Local and noncontact measurements of bulk acoustic wave velocities in thin isotropic plates and shells using zero group velocity Lamb modes[END_REF]- [START_REF] Mezil | Investigation of interfacial stiffnesses of a tri-layer using Zero-Group Velocity Lamb modes[END_REF]. Balogun, Murray and Prada demonstrated the laser-based measurement of ZGV resonances around 700 MHz in 4 μm thick tungsten samples. This inspired us to test the feasibility of a LUS response measurement with thinner samples. We found that ZGV resonances can be excited and detected with high SNR and a resolution down to 1 Hz up to ~4 GHz.

LASER-ULTRASOUND SETUP

For the plate response measurements we adapted a LUS setup consisting of a stabilized Michelson interferometer and an intensity modulated excitation laser to work in the GHz regime. The combination of intensity modulated sources and low-noise narrow-band detection [START_REF] Ryzy | Measurement of coherent surface acoustic wave attenuation in polycrystalline aluminum[END_REF], [START_REF] Murray | High-sensitivity laser-based acoustic microscopy using a modulated excitation source[END_REF], [START_REF] Grünsteidl | Spatial and temporal frequency domain laser-ultrasound applied in the direct measurement of dispersion relations of surface acoustic waves[END_REF] showed to achieve high SNR at low excitation intensities, which is beneficial when investigating delicate micro-scale samples. The setup is shown in Fig. 1. For the detection laser (beam path shown in green) a 532nm cw-laser is used. The excitation (1550 nm, shown in red) is achieved with an electroabsorptive modulated laser diode (EML) amplified by an erbium doped fiber amplifier (EDFE). The principle of the setup is described in detail elsewhere [13], [START_REF] Grünsteidl | Effects and applications associated with zero group velocity Lamb waves[END_REF]. To achieve the high bandwidth we selected an according photo-detector (Alphalas UDP-50-SP), amplifier (Miteq AFS4-00100600-13-10P-4) and used an 8 GHz vectornetwork-analyzer (Rhode&Schwarz ZNB8) in transmission mode, replacing the lock-in-amplifier and the signal generator from previous setups. A moving knifeedge technique was used to ensure that the excitation and detection spot were co-aligned and focused in the sample plane. Both spots had a full-width half maximum diameter below 3μm in the focal plane. The samples we investigated are micro-scale tungsten (W) plates. They were produced by photo-lithographic methods, similar as applied in the production processes of micro-acoustic devices. The dimensions are ~1.095μm x 500μm x100μm, the micro-plate is sustained on two opposite edges as shown in Fig. 1. To accurately position the confocal detection and excitation spots on the sample, we implemented a microscope in the setup, which allowed us to view sample and laser spots from the top (see Fig. 1).

RESPONSE OF THIN PLATES

The left side of Figure 2 shows two typical response spectra recorded with the setup of Fig. 1 and processed as explained below. The red curve corresponds to a 1.095 μm W sample. It features a dominant peak at ~2.1 GHz. This peak is due to the ZGV resonance and corresponds to a local minimum in the dispersion curves shown on the right side, which we calculated for a thickness h=1.095 μm, longitudinal sound velocity cL=4972.2 m/s, Poisson's ratio ν=0.31 and mass density ρ=19300 kg/m³. Another (much smaller) peak shows at ~3.55 GHz. It also corresponds a ZGV resonance which is hardly visible as a minimum in the dispersion curves. As a comparison, we also show the response and dispersion curves for an aluminum sample (blue curves, h=1.389 μm, cL=6420 m/s, ν=0.363, ρ=2700 kg/m³). It features a ZGV peak around 2 GHz and additionally a significant peak corresponding to the thickness-stretch mode, which we evaluated recently for a range of aluminum samples [START_REF] Grünsteidl | Measurement of the attenuation of elastic waves at GHz frequencies using resonant thickness modes[END_REF]. Evaluating the L1 peak has the advantage of directly resulting in a Q value for longitudinal waves (and not a Q value of the ZGV mode, composed of normal and shear strains). However, the L1 peak in W shows only as a tiny hump in the response and cannot be evaluated practically. We hypothesize that this is a result of the thermal properties of the materials and we plan to clarify this in future work. To evaluate W, for now we focus on the dominant ZGV mode. Here the transient signals cause a peak immediately after excitation at t=0 before they propagate away from the detection spot. If we discard this section and backtransform only the signal for t>t0 (show as red), we obtain a spectrum which is not offset from zero and comes close to a pure ZGV resonance (see red curve in top row of Fig. 3). We applied this "time-gating" to obtain the responses shown in Fig. 2, which provides a clearer picture of the resonance peaks for tungsten and aluminum. The temporal decay of ZGV modes was described by Prada et al. [START_REF] Prada | Power law decay of zero group velocity Lamb modes[END_REF] and Laurent et al. [START_REF] Laurent | Temporal behavior of laser induced elastic plate resonances[END_REF]. They showed that ,

Where uZ is the out-of-plane displacement of the surface at the point of excitation, fZGV is the frequency of the ZGV resonance and α is accounting for material caused attenuation. In order to obtain α we divide the time-gated signal by . The resulting time-and frequency domain signals are shown as orange curves in Fig. 3. The resulting signal uZ' is an exponentially decaying harmonic oscillation: ,

We fitted Eq. 3 to the complex values spectral response. The magnitude of the resulting fit is shown as a dashed purple line in Fig. 3 (top). For the fit we omitted the right flank of the peak, as we see ripples in the measurement in the range, which we think are due to standing waves emerging from reflections off the edge of the sample.

The fit shows excellent agreement to the experimental data for the left flank of the peak, yielding αfit = 1.6442 10 6 /s, which is equivalent to Q = 638.6.

CHALLENGES AND OUTLOOK

We currently work on 3D finite element simulations reproducing the response, including the ripples on the right flank of the peak, which trouble the evaluation process. The simulations will be used to estimate the maximum error introduced by the ripples, depending on the used fit range and to confirm measures to avoid them. We think the ripples are interferences from other modes, standing across the sample width. Thus, using larger samples, such that the waves decay beyond detectability on their way to the edge and back, would be a simple solution for future experiments.

To obtain frequency trend of the Q values, we investigate a range of sample thicknesses, tailored to provide resonances in a range of frequencies. To obtain Q values for pure longitudinal and transversal waves, as opposed to a Q value for the ZGV mode, we attempt to extend our investigation to additional resonances (see peak at 3.55 GHz in Fig. 2). However, disentangling the different attenuations of different resonances into primary attenuation coefficients, is still a challenge.

ACKNOWLEDGEMENT

This work was funded by the Austrian research funding association (FFG) under the scope of the COMET program within the research project "Photonic Sensing for Smarter Processes (PSSP)" (contract #871974).

Figure 1 .

 1 Figure 1. LUS setup and sample geometry. Components are labeled as λ/2 -half wave plate, λ/4 -quarter wave

Figure 2 .Figure 3 ,

 23 Figure 2. Left side: Response-spectra obtained for tungsten (red) and aluminum samples; Right side: Corresponding dispersion curves. While aluminum shows a significant L1 peak, the expected peak in tungsten is buried within the ZGV peak

Figure 3 .

 3 Figure 3. Recorded and processed response of 1.095 μm thick tungsten sample in the frequency (top) and time (bottom) domain.

10.48465/fa.2020.0582

e-Forum Acusticum, December 7-11, 2020


J Spytek

Patrycja Pyzik

Aleksandra Ziaja-Sujdak

Lukasz Ambrozinski
email: ambrozin@agh.edu.pl

Ł Ambroziński

Comparison of ultrasonically generated shear and zero-group velocity Lamb waves for damage detection in adhesively bonded aluminum plate

Comparison of ultrasonically generated shear and zero-group velocity Lamb waves for damage detection in adhesively bonded aluminum plates

INTRODUCTION

Adhesively bonded metals are often used for manufacturing aerospace and automotive structures. Critical, load-bearing elements reinforced using stiffeners bonded to the main structure are increasingly used in these applications. Inspecting the bonded parts remains challenging for modern non-destructive testing techniques [START_REF] Adams | Nondestructive testing of adhesively-bonded joints[END_REF]. Since, the limited bandwidth of classical ultrasonic (US) testing systems cannot resolve multiple reflections and mode-conversions occurring at subsequent layers. Laser ultrasound (LU), due to its broadband nature, has shown great potential in high-resolution imaging of carbon-reinforced composites [START_REF] Ambrozinski | Detection and imaging of local ply angle in carbon fiber reinforced plastics using laser ultrasound and tilt filter processing[END_REF]- [START_REF] Pelivanov | Heat damage evaluation in carbon fiberreinforced composites with a kHz A-scan rate fiber-optic pump-probe laser-ultrasound system[END_REF]. For a metal surface, however, excitation of longitudinal waves in the normal direction is inefficient. On the other hand, other elastic modes can be efficiently generated using laser absorption and used to image defects in metallic structures.

Shear waves can be efficiently generated in the thermoelastic regime without surface ablation [START_REF] Davies | Laser-generated ultrasound: Its properties, mechanisms and multifarious applications[END_REF]. Induced waves propagate at an oblique angle in the structure and reflect from each discontinuity laying along their path. Therefore, source-receiver distance has to be optimized in each case to maximize the amplitude of shear wave reflected from a given depth and detected at the sample surface.

In recent years, the zero-group velocity Lamb modes were proposed as an alternative to classical bulk waves for nondestructive testing of plate structures [START_REF] Clorennec | Laser ultrasonic inspection of plates using zero-group velocity lamb modes[END_REF]. The ZGV modes are special case of Lamb modes, that do not propagate away from the excitation point and create local resonances. The ZGV modes were proven highly sensitive to even slight changes in plate thickness, material properties, or bonding quality in multi-layered structures. The detection of defects can be realized by mapping the central frequency of particular ZGV mode to the local coordinates of scanned surface. The resulting frequency shift indicates local structural changes. However, ZGV modes of more complex, multi-layered structures are more numerous and difficult to interpret. Therefore, for such structures the most optimal modes for damage detection should be determined by prior theoretical analysis [START_REF] Spytek | Evaluation of disbonds at various interfaces of adhesively bonded aluminum plates using alloptical excitation and detection of zero-group velocity Lamb waves[END_REF].

METHODS

Zero-Group Velocity mapping

The US waves propagating in plate-like structures can be described in terms of dispersion curves. The curves represented using frequency -group velocity plots were shown in fig. 1. The zero-group velocity (ZGV) modes can be identified as curves that cross the group velocity value at 𝑣 𝑔 = 0 from positive to negative values. Two such points were marked using red dots in fig. 1a for homogeneous aluminum plate. For plate structure consisting of three aluminum layers -assuming dimensions 1, 1.5 and 3.5 mm -bonded using epoxy layers, more ZGV modes were identified in the same frequency range, as shown in the fig 1b.

The primary damage topologies occurring in multilayered plates involve failure at particular bonding layers. Complete delamination can be represented in terms of dispersion characteristics by assuming two plates separated by air gap. Therefore, the negative group velocity curves corresponding to individual separated plates were demonstrated in fig. 1b as colored lines. It is evident, that different failure modes corresponded to shifts in frequencies of particular ZGV modes. Therefore, the frequency shift of selected modes was used as damage indicator.

Shear waves detection

The inspection using shear waves propagating at oblique angle requires assuming a proper stand-off distance between excitation and measurement point. For this purpose, the wave speed and directivity pattern of US waves has to be determined. However, for multi-layered thin structures, the directivity pattern of generated US cannot be evaluated analytically due to near-field effects. Thus, a number of multiphysics finite difference simulations had to be performed to optimize the sourcedetector configuration.

Two 2D models of 20 mm-long structure composed of three aluminum plates with thicknesses of 1, 1.25 and 3.5 mm bonded with epoxy layer were created. The defects simulated as 4 mm-long air gaps were placed in the first and the second interface for the two respective models. An additional simulation of model with no defects was performed to obtain reference signals. For each case, the time histories were collected from the number of points on the surface. To isolate the waves reflected from the defect, the reference signals were subtracted from damaged ones. This led to obtain space-time representations of damage-reflected wave modes (fig. 2). Using these maps, the optimal source-receiver distance was set to 2 mm and 4 mm for the first and the second model interface imaging, respectively, which corresponded to maximum shear wave reflection at those depths.

EXPERIMENTAL SETUP

For both US modalities presented in this work, the excitation was realized by a diode-pumped nanosecond laser to generate probing signals. For detection, a fiberoptic Sagnac interferometer LuxSonics was used [START_REF] Pelivanov | A new fiber-optic non-contact compact laser-ultrasound scanner for fast nondestructive testing and evaluation of aircraft composites[END_REF]. In both setups, the laser source was focused to a narrow line in order to obtain wave directivity and to achieve maximum signal to noise ratio while keeping the fluence below the ablation threshold.

The sample used in this work is a multi-layered plate, consisting of three aluminum layers -1, 1.25 and 3.5 mm-thick. Plates were bonded using epoxy layers at two interfaces. The damage was introduced to the structure by placing a 15×15 mm Teflon patch at each interface to prevent plate bonding.

During the inspection using ZGV modes, the excitation and detection beams were superimposed on the surface of the specimen. For shear waves detection, source-receiver distance were set to 2 mm and 4 mm to receive waves reflected from the first and the second interface, respectively.

For both cases, the surface of the sample was scanned using LUS system in the area 25×25 mm around the predefined defect locations. The sample was fixed on a translation platform, allowing for positioning in the XY plane. The scanning resolution of 0.02 mm was achieved, resulting in 1250×1250 mesh of points. For each point, the time histories sampled at 200 MHz frequency were collected.

ZGV maps

Based on the prior theoretical analysis the most sensitive ZGV modes were identified for detection of defects at each of the two interfaces of the experimental specimen. For damage at 1-1.25 mm interface, the ZGV frequencies at the proximity of 2.85 MHz bandwidth were traced, and for 1.25-3.5 mm interface at the bandwidth near 0.78 MHz. For each temporal signal at coordinate (𝑥, 𝑦), the mean frequency was calculated from the assumed bandwidth and mapped to the particular coordinate. Using this approach, 2D maps of mean frequency values were created, as shown in fig. 4. For the scan involving 1-1.25 mm defect, a clear increase of mean ZGV frequency was detected in the rectangular area, corresponding the size of the Teflon patch. For the scan involving 1.25-3.5 mm defect, a similar effect was observed, but the mean frequency value decreased. These effects corresponded qualitatively to the prior theoretical analysis of damage models in fig. 1.

Shear wave maps

An example of A-scans acquired over pristine and damaged areas for the sample with the defect at the first interface was presented in fig. 5. The waveforms were normalized to the peak amplitude, which corresponded to a surface wave. Although the signals seemed to be very similar, for TOF equal to approx. 0.9 µs the signal acquired over the defect had a significantly larger amplitude than the reference. This was in good accordance with simulations. The amplitude of all recorded signals corresponding to TOF equal to 0.95 µs was imaged in the form of a C-scan, as presented in fig. 6a. For the depth of second interface, the maximum shear wave reflection appeared at TOF equal to approx. 1.9 µs. The corresponding C-scan was shown in fig. 6b. Clearly, the Teflon inserts were successfully detected and imaged for both damage cases. More detailed description of the method and results were provided in the article "Detection of disbonds in adhesively bonded aluminum plates using laser-generated shear acoustic waves" accepted for publication in Photoacoustics journal.

CONCLUSIONS

Although longitudinal wave cannot be efficiently excited in metal plates using non-destructive LU techniques, other wave modes can be used for inspection. We demonstrated that adhesively bonded aluminum plates can be inspected using reflected shear waves and ZGV spectroscopy. The waves are generated in the thermoelastic regime using a thin laser strip excitation.

For shear waves detection, the source-receiver distance has to be adjusted to maximize inspection sensitivity. Optimizing the experimental setup can be facilitated using numerical simulations based on the known geometry of the part to be inspected.

The ZGV modes are very sensitive to even small damage in the structure. However, proper utilization of ZGV spectroscopy requires theoretical analysis to select the most sensitive modes for particular damage topologies. Using the presented approach, the proper ZGV modes were identified using theoretical dispersion curves of undamaged plate and predicted damage modes. The obtained results show good contrast of damaged regions with respect to background.

Figure 1 :

 1 Figure 1: Dispersion curves corresponding to homogeneous aluminum plate (a) and multi-layered aluminum plate (b); corresponding ZGV modes were marked using red dots

Figure 2 :

 2 Figure 2: Left panels: space-time representation of simulated signals obtained as the difference between responses for pristine and damage cases with encircled area of maximum shear wave reflection. Right panels: envelope of signals presented in left panel. Combination of labels L and S denotes reflected or mode-converted longitudinal or shear waves respectively. Panels (a) and (b) correspond to models with defect at the first and the second sample interface, respectively.

Figure 3 :

 3 Figure 3: Diagram of the experimental setup 4. RESULTS

Figure 4 :

 4 Figure 4: The images of mean frequencies mapped in the scans near the 1-1.25 mm defect (a) and 1.25-3.5 mm defect (b)

Figure 5 :

 5 Figure 5: Time histories acquired over pristine and damaged areas for defect at the first interface

Figure 6 :

 6 Figure 6: The images of defects in the form of C-scans for the first (a) and the second (b) sample interfaces.

10.48465/fa.2020.0843

e-Forum Acusticum, December 7-11, 2020


Lin-Lin Wang
email: linlin.wang@cup.edu.cn

Professor Chao-Sheng Tang
email: tangchaosheng@nju.edu.cn

Yu-Jun Cui
email: yujun.cui@enpc.fr

Guang-Qing Zhang
email: zhang0@263.net

Inyang Hilary
email: h.inyang26@gmail.com

Nucleation and propagation mechanisms of soil desiccation cracks

Keywords: Soil desiccation cracking, Drying shrinkage, Digital image correlation, Full-field strain measurement, Crack pattern

Contraction of soil during desiccation gives rise to polygon-patterned cracking. This phenomenon was investigated by continuous full-strain measurements on soil sample surfaces using digital image correlation techniques. The results show that desiccation cracking was of the mode I variety and can be reliably predicted through strain field analyses: the observed cracking was restricted mostly to high stretching domains. The observed high stretching stems from either external restraint or internal flaws of the shrinking soil. The cracks redistribute the stress in their vicinity and lead neighboring cracks to intersect them mostly orthogonally. As desiccation proceeds, the surface strain field ceases to provide useful information for predicting crack locations. Further analysis indicates that although drying occurs from exposed material surfaces, late-generation cracks actually initiate in the subsurface and express themselves subsequently at the surface. These observations offer insights into the intrinsic mechanisms of drying-induced crack growth in soils.

Introduction

Desiccation-induced cracking is a ubiquitous phenomenon in both natural and artificial materials, including dried sediment layers, paints and glazes on ceramics [START_REF] Neal | Giant desiccation polygons of Great Basin playas[END_REF][START_REF] Pollard | Progress in understanding jointing over the past century[END_REF][START_REF] Jenkins | Optimal spacing and penetration of cracks in a shrinking slab[END_REF][START_REF] Zhao | Experimental study of mechani cs and seepage characteristics of sandstones after liquid-nitrogen stimulation[END_REF]. In a dry environment, evaporation of water from soils (particularly, certain soils with high clay content) results in shrinkage of the materials and subsequently generates polygonal crack networks (Figure 1). Polygon-patterned cracks are encountered in a wide variety of scales ranging from a fraction of a millimeter in some drying colloids to several hundreds of meters in playas. Broadly, the morphology of crack networks reflects the drying process and can provide information about ancient climatic conditions. Glennie [1970] observed that desiccation crack patterns can be preserved after their formation if sediment fills the spaces. Conversely, by understanding the conditions necessary for the formation of desiccation patterns, climatic conditions that prevailed during the history of liquid water activity on earth surface can be estimated. For instance, based on the cracks observed on Mars, [START_REF] El-Maarry | Desiccation mechanism for formation of giant polygons on Earth and intermediate-sized polygons on Mars: Results from a pre-fracture model[END_REF], 2013, 2014] made inferences on the paleoclimate of Mars and suggested that liquid water existed on Mars. In addition, surface desiccation cracking patterns are important for quantifying local windblown (aeolian) soil levels [START_REF] Style | Mud peeling and horizontal crack formation in drying clays[END_REF]. The appearance of cracks can dramatically increase soil surface roughness. Surface roughness has been shown to reduce the wind speed threshold at which soil particles are picked up from the earth surface.

In engineering works, cracking breaks up the integrity of soils and may lead to various geotechnical and geo-environmental failures. The reduction in the strength of soils due to desiccation cracking may cause failure of exposed earth structures [START_REF] Morris | Cracking in drying soils[END_REF], landslide [START_REF] Tang | Experimental characterization of shrinkage and desiccation cracking in thin clay layer[END_REF], and mudflow [START_REF] Wang | Effects of wetting drying cycles on soil strength profile of a silty clay in micro-penetrometer tests[END_REF]. Cracks provide preferential channels for mass transfer which adversely affect the functionality of dams, levees, embankments and landfill barriers [START_REF] Albrecht | Eff ect of desiccation on compacted natural clays[END_REF][START_REF] Ye | Experimental investigation on gas migration in saturated Shanghai soft clay[END_REF]. The presence of cracks increases the weathering depth of soil and aggravates soil erosion on slopes [START_REF] Tang | Characterizing drying-induced clayey soil desiccation cracking process using electrical resistivity method[END_REF]. The cracking process also damages plant roots and reduces crop yield [START_REF] Kelly | Preferential flow and transport of nitrate and bromide in a clay pan soil[END_REF]]. In recent years, due to global climate change, droughts occur with an increasing intensity and frequency [START_REF] Dai | Drought under global warming: a review[END_REF][START_REF] Pachauri | Contribution of Working Groups I, II and III to the fifth assessment report of the Intergovernmental Panel on Climate Change[END_REF]. A systematic study of soil desiccation cracking behavior is therefore fundamental to evaluate and predict the influence of droughts on earth structures (i.e. dam, embankment and landfill barriers) and ecological systems.

In geological and geoengineering literature, one can find many descriptions of desiccation crack patterns in nature [START_REF] Weinberger | Initiation and growth of cracks during desiccation of stratified muddy soils[END_REF][START_REF] Li | Study of desiccation crack initiation and development at ground surface[END_REF][START_REF] Zhang | Impact of NaCl on drying shrinkage behavior of low plastic soil in earthen heritages[END_REF], and in laboratory samples [START_REF] Groisman | An experimental study of cracking induced by desiccation[END_REF][START_REF] Yesiller | Desiccation and cracking behavior of three compacted landfill liner soils[END_REF][START_REF] Rodrí Guez | Experimental and numerical analysis of desiccation of a mining waste[END_REF][START_REF] Lakshmikantha | Image analysis for the quantification of a developing crack network on a drying Soil[END_REF][START_REF] Peron | Fundamentals of desiccation cracking of fine grained soils: experimental characterisation and mechanisms identification[END_REF][START_REF] Tang | Experimental characterization of shrinkage and desiccation cracking in thin clay layer[END_REF]Costa et al., 2011;[START_REF] Sanchez | Exploring desiccation cracks in soils using a 2D profile laser device[END_REF][START_REF] Gui | Numerical modelling of laboratory soil desiccation cracking using UDEC with a mix-mode cohesive fracture model[END_REF][START_REF] Li | Experimental study on soil-water characteristic curve for silty clay with desiccation cracks[END_REF][START_REF] Sun | Investigating the microstructure changes for silty soil during drying[END_REF][START_REF] Tollenaar | Observations on the desiccation and cracking of clay layers[END_REF]. However, the dynamics of crack pattern formation and the underlying mechanisms are still unclear. It is commonly postulated that cracking is triggered by tensile stresses that develop due to moisture gradient during desiccation [START_REF] Corte | Experimental research on desiccation cracks in soil-Research Report[END_REF][START_REF] Neal | Giant desiccation polygons of Great Basin playas[END_REF][START_REF] Morris | Cracking in drying soils[END_REF][START_REF] Jenkins | Optimal spacing and penetration of cracks in a shrinking slab[END_REF].

However, up to the present, the spatio-temporal evolution of the tensile stress field in drying soils has not been quantitatively characterized. In past decades, a large number of experiments have been undertaken to determine soil tensile strength through direct and indirect methods [START_REF] Lu | Tensile strength of unsaturated sand[END_REF][START_REF] Venkataramana | A critical review of the methodologies employed for determination of tensile strength of fine-grained soils[END_REF][START_REF] Ebrahim-Zadeh | Investigating the correlation between soil tensile strength curve and soil water retention curve via modeling[END_REF]. Most of the tensile tests were performed on compacted soil samples at specific water contents, where the corresponding test conditions are completely different from soil desiccation conditions. More specifically, the measured tensile strength is actually a macro mechanical parameter.

It cannot adequately reflect the cracking process that initiate from micro damage, such as the breakage of inter-particle bonding [START_REF] Shin | Desiccation cracks in saturated fine-grained soils: particle level phenomena and effective stress analysis[END_REF]. It is well known that soil is a highly heterogeneous but porous material. The distribution of inter-particle bonding strength or the resistance to tensile failure is not uniform. Currently, there is still not an effective method that can accurately measure the micro-mechanical parameters related to soil tensile failure and its distribution characteristics at particle scale.

The study reported herein, focuses on analyses of the dynamic process of desiccation cracking through a continuous full-field strain measurement using digital image correlation. More specifically, the surface of a drying soil was monitored continuously using a digital camera. The evolution of the in-plane strain field during desiccation was obtained and correlated with the observed desiccation cracking patterns. These observations can provide new insights into the initiation and propagation of desiccation cracks in soil. Moreover, the method developed herein, could also be used to investigate the desiccation cracking process in other materials such as paints, plasters and some water-based mixtures.

Material and Experiment method

Material

The tested material is a clayey soil obtained from Nanjing area, China, at the depth of 0.5-1 m. It is widely deposited in the middle and lower reaches of the Yangtze River and is a very important foundation soil involved in numerous construction projects. Its physical and mechanical properties are summarized in Table 1. Grain size analyses indicate that its sand, silt, and clay contents are 24%, 34%, and 42%, respectively. The mineralogical composition of the clay component was found to be 17.9% montmorillonite, 73.6% illite and 8.5% kaolinite.

Experiment method

The tested material was crushed and sieved to grain sizes smaller than 2 mm. Then, it was mixed with distilled water with a solid/water ratio o f 1:1.7. The obtained slurry was poured into a glass dish of 117 mm diameter. The dish was vibrated for 5 min to remove air bubbles and then, covered for at least 72h for sedimentation. After that, the pure water above the slurry surface was removed. The sample was finally pruned to 8 mm thickness, and the measured water content was 115%. Black paint was sprayed on the sample surface to obtain a random speckle pattern for digital image correlation analyses. The speckles, typically sized in of hundreds of micrometers, are sparsely distributed on the sample's surface. Hence, their influence on the characteristics of the original soil can be neglected.

The desiccation test was performed under normal laboratory conditions (25 ± 0.5 • C, 50 ± 3 % relative humidity). The experimental setup is illustrated in Figure 2. A digital camera was fixed above the sample to record images every 30 minutes. The sample was placed on a scale, and the mass loss during drying was monitored. The change in the water content of the sample could then be back-calculated. Several LED lights were placed around the sample to improve the illumination conditions.

The recorded images were analyzed using digital image correlation techniques (DIC). Broadly, DIC entails correlating certain subsets of pixels in the reference image (in position X) with their homologous ones in the deformation image (in position x) [START_REF] Bornert | Corrélation d'images[END_REF], Wang et al., 2017].

Once correlation is obtained, the transformation gradient (F ij = ∂x i /∂X j ) can be assessed approximately by averaging the infinitesimal gradient over a surface, D. In practice, the transformation gradient over a surface, D is evaluated from the displacements at its boundary according to Green's theorem [START_REF] Wang | Microscale insight into the influence of humidity on the mechanical behavior of mudstones[END_REF]:

11 () i ij i j DD j x F X dw x n ds D X D        (1)
where ∂D is the boundary of D with outer unit normal n. The Green-Lagrange strain E is then calculated:

1 () 2 T E F F I    (2)
where T denotes the transpose of a tensor, and I is the second order identity tensor. D can be an assembly of matching subsets with arbitrary shape, and its average deformation can be assessed.

When D is delimited by the neighbors of a given matching subset, the local strain is determined (Figure 3).

In this work, the DIC analysis and subsequent strain measurements were performed using the CMV software developed at Ecole des Ponts ParisTech, France. The correlation subsets are sized at 40×40 pixel. The in-plane components of the local strain tensor (ε xx , ε yy , ε xy ) can be evaluated at some local gauge length, depending on image characteristics and the chosen DIC parameters.

The two in-plane eigenvalues of the 2D Green-Lagrange strain tensor (namely ε 1 and ε 2 with ε 2 > ε 1 ) are then calculated. The convention of continuum mechanics is applied here: tensile strain is positive, while compressive strain is negative. Therefore, ε 2 corresponds to the maximum tensile strain or minimum compressive strain (called major strain), while ε 1 is the maximum compressive strain (called minor strain). Also, the equivalent strain, ε eq = 2(ε 2 -ε 1 )/3, is assessed as an indicator of shear strain.

Results and discussion

Desiccation cracking and deformation modes

The experiments lasted for 5 days. The variation of water content with elapsed time is presented in Figure 4. The water content decreased linearly first, and then reached stabilization gradually.

Images were recorded by the camera every 30 minutes, from which 12 images are chosen to show the initiation and propagation of the first 26 cracks (Figure 5). The 26 cracks are numbered according to their order of appearance. The first crack appears at 26.5h in the middle of the specimen surface, and the corresponding water content is 66.6%. It subsequently propagates across the specimen, dividing the latter into two parts. Most of the cracks initiate at intact spots instead of from previously existing cracks, but their propagations trend to join adjacent cracks (see for example the onset and propagation of crack 5 in steps 6-8 of Figure 5). One exception is crack 4 which seems to initiate from crack 1. The cracks intersect at right angles forming T junctions. The explanation of this T-junction pattern is well recognized: the first crack releases the normal stress in its vicinity; the second crack approaches the first orthogonally, that is along the direction perpendicular to the local maximal tensile stress [START_REF] Weinberger | Initiation and growth of cracks during desiccation of stratified muddy soils[END_REF].

The early-onset cracks such as cracks 1, 2, 3 and 5 normally show large openings and high lengths, and t h e y are named primary cracks herein. Crack 1 is oriented radially, whereas cracks 2, 3 and 5 are circumferential (sub-parallel to the vessel wall). After the appearance of primary cracks, shorter cracks which are considered herein, as being secondary, appear. These secondary cracks typically intersect the primary cracks or earlier secondary cracks at right angles. For instance, crack 6 joins crack 1 and crack 2. Among these secondary cracks, a group of cracks (crack 7, 9-10, 16, 18-21, 24-25) are located between the circumferential primary cracks and the vessel wall.

The ε 1 , ε 2 and ε eq strain maps of the specimen surface for the chosen 12 steps are illustrated in Figures 6, 7 and 8, respectively. In these maps, the local strain of a correlation subset is illustrated by a cross: the color indicates the strain magnitude while the major axis indicates the ε 2 direction.

For better illustration, a zoom of the Figure 7 strain map at step 11 is shown in Figure 9. It is observed that the crack lips involve high tensile strain (in red color) with ε 2 direction perpendicular to them, indicating opening of the cracks. However, in the adjacent zone of the cracks, the ε 2 strains are negative, indicating compression in the corresponding zone due to the opening of cracks. The compression is even higher (in violet color) at the corner of two intersecting cracks. In addition, a shear-deforming band in which ε 2 is oblique to the orientation of the band (see the yellow band in Figure 9), is observed.

From the above discussion, the ε 2 direction can serve as an indicator of the crack mode. For instance, ε 2 is perpendicular to the opening of a I-mode crack (i.e., tensile crack) whereas ε 2 is oblique to the opening of a II-mode crack (i.e., shear crack). Examining the 26 cracks in our experiment, all of them are in I-mode. It should be noted t h a t the cracks propagate not only laterally in the free surface but also vertically inside the sample. Nevertheless, the present 2D measurements focus on the crack patterns of the free surface. Information on deeper-seated geometries is not available.

Precursor of desiccation cracking

Shrinking is heterogeneous within the sample, even before the appearance of cracks (see step 1 of Figure 6). A red (compaction) ring is observed at the specimen edge. Considering its form, this shrinkage ring should be caused by the external restraint provided by the vessel wall. Interestingly, tensile strains appear during the shrinking process (see the yellow and red colors in step 1 of Figure 7). In particular, the ε 2 map at step 1 is characterized by a red ring, as well as four red points which involve high tensile strains. The tensile-straining ring appears at the immediate inner side of the compaction ring. Thus, the two rings are compatible and should derive from the same mechanism. In the subsequent steps, cracks 2 and 3 initiate from this high-tensile-straining ring.

Crack 3 is a good example of the key role played by the high-tensile-straining ring in the observed cracking phenomena. The corresponding domain in crack 3 has already suffered from high tensile straining in step 1 and is cracked at step 5. Then, crack 3 propagates, following exactly the path of the ring before deviating towards crack 2 (see steps 5-7 in Figure 7). Crack 5 does not initiate from the high-tensile-straining ring. Nevertheless, when it approaches the ring, it subsequently propagates along the ring's path.

In addition to the high-tensile-straining ring, four isolated points involving pronounced tensile deformation, are also found before the onset of cracking. They are designated as A-D in step 1 of Figure 7. These points arise from the flaws of the material (e.g., inclusions, pores, grain boundaries). It is found that point A gives rise to the initiation of crack 1 at step 2, while point D leads to the onset of crack 6 at step 7. For the other two points, point B disappears at step 2, because the opening of crack 1 releases the normal stress at its vicinity; similarly, cracks 4 and 6 relieve the high-straining of point C (see steps 6-7 in Figure 7).

To examine whether desiccation cracking might arise in mode II, the ε eq maps are shown in Figure 8. Compared to those found in ε 1 and ε 2 maps, no more high-straining zones are found in ε eq maps.

Two high ε eq rings are found at the boundary of the specimen: the outer one mainly arises from high compaction, and the inner one results from high tensile straining. The outer high ε eq ring might explain the initiation of the outer radial cracks near the vessel wall, exemplified by cracks 7, 18 and 25. To examine this, the evolution of the strains in the domain of crack 7 (see the definition of the domain at reference stage of Figure 6) is shown in Figure 10. It is found that before the initiation of the crack at step 7, the ε eq of the domain increases gently until reaching 1.5% at step 6 (29.5h).

Considering that the ε eq in the domain of crack 4 attains 7.4% without crack initiation, the shear in the domain of crack 7 is too small to activate the cracking. Hence, the strain at the surface cannot provide precursory information on the initiation of crack 7; the latter probably initiates inside the specimen and propagates to the free surface afterwards.

In summary, the strain maps before the onset of cracking already contain precursory information on the cracking phenomenon that would subsequently occur (the early generation of cracks particularly) in mode I. The high tensile straining stems from either the external restraint or from the flaws of the shrinking material.

Propagation of desiccation cracks

Desiccation cracks that are observed in this study do not always propagate along a straight path but are curved such as cracks 2, 3, 5, 6, 11 and 17. As discussed previously, the curved cracks 2, 3, 5, 11 and 17 are related to the high-tensile-straining ring. In addition, the curved propagation is also affected by the existence of adjacent cracks: when a new crack propagates toward an early-generated crack, the former seems to be "attracted" by the latter, and the two finally join orthogonally. Crack 6 is a good illustration of this phenomenon. The initial propagation path of this crack is perpendicular to crack 1 (at step 7). However, when approaching crack 2, the crack deviates from the initial path and finally intersects crack 2 at right angles (at step 8). To gain some insights into the mechanism of this deviating path, the ε 2 strain field around crack 6 at steps 6-8, are zoomed (see the definition of the zoomed zone in step 8 of Figure 7) and shown in Figure 11.

Before the arrival of crack 2, the ε 2 directions in the zone of crack 6 are rather randomly distributed. However, these ε 2 directions are redistributed when crack 2 passes through: the ε 2 directions in the crack lips are perpendicular to the length of crack 2, whereas they are parallel to the length in the vicinity. These redistributed ε 2 directions define the propagation path of crack 6 deviating towards crack 2 at right angles, because the crack should propagate along the direction perpendicular to the major tensile stress (consistent with the ε 2 direction).

Initiation positions of desiccation cracks

Only a small fraction of cracks (i.e., cracks 1, 2, 3, 6) seems to be linked with the strain field measured at the sample surface. To confirm this quantitatively, the evolution of the strains in the initiation domains of cracks 4 and 17 are given in Figure 10. The definitions of the two domains are shown in Figure 6, and their strains are calculated from the displacements of their boundaries using Equation 1).

From Figure 10, the strains in the two domains do not always change monotonously with time.

This is mainly because the onset of neighboring cracks would lead to stress release. Regarding crack 4, the tensile strain at its initiation domain already reaches 10.3% at step 1 without cracking (see Figure 10a). Because of the opening of crack 1, the strain even falls to negative (-0.2%) at step 4, prior to crack initiation at step 5. The tensile strain is 13.6% when crack 4 appears. Thus, it is deduced that the tensile strain threshold for that cracking is between 10.3% and 13.6%. Crack 17 appears inside the high-tensile-straining ring. Nevertheless, the evolution of strain at its initiation domain shows that, before the initiation at step 10, the maximum tensile strain appears at step 5

(5.9%) instead of step 9 (5.7%). The two declining portions at steps 6 and 8 stem from the stress releases induced by the initiation of crack 5 and 9, respectively.

From the analysis above, it can be inferred that a significant portion of desiccation cracks might initiate at depth and propagate upwards to the surface. This is contradictory to the prevailing claim that such cracks initiate from the surface and propagate downwards because the tensile stress due to self-restraint is maximal at the surface [START_REF] Jenkins | Optimal spacing and penetration of cracks in a shrinking slab[END_REF]. Actually, in addition to the self-restraint effect, the heterogeneous strain fields suggest that material heterogeneity also plays a key role in cracking process.

When desiccation proceeds, the drying surface penetrates inside the specimen and may encounter heterogeneous points such as flaws. The resulting concentrated stress due to strain incompatibility could give rise to cracking at depth. Moreover, when the drying front reaches the vessel bottom, the restraint of the vessel bottom on the shrinking of the material also results in tensile stresses that eventually cause cracking. The cracks nucleate at the bottom then propagate upwards. This kind of cracking mechanism is confirmed by the picture shown in Figure 12. Base on the discussions above, we suggest that, although evaporation of water begins at the exposed surface, desiccation cracks can nucleate inside the sample and propagate upwards.

Conclusions

The initiation and propagation of polygonal desiccation cracks in soil were investigated through full-field strain measurements on the desiccation surface. It is found that desiccation cracking can be reliably predicted by the full-field strain analysis of surfaces because they already exhibit Tables Table 1. Physical and mechanical properties of the tested soil.

Soil properties Value

Specific gravity (g/cm 3 )

Consistency limit

Liquid limit (%) The convention is: tensile strain is positive, while compressive strain negative. Step 6

Step 7

Step 8

Initial unordered distribution of

  high-stretching before the appearance of cracks. The high stretching stems from either external restraint or internal flaws of the shrinking soil. Desiccation cracks do not always propagate in straight lines. Some of the curved cracks are controlled by the prevailing boundary conditions (e.g., the circle-shaped restraint from the vessel in this work). Other controlling factors are related to the stress redistribution of the neighboring cracks, which requires the cracks to intersect each other orthogonally. The final geometrical characteristics of crack pattern are actually determined by the interactions between cracking development and stress redistribution. Although drying begins from the exposed surface, cracks (particularly the ones generated later) could nucleate in the subsurface, and propagate upwards to the surface.

Figure 1 .

 1 Figure 1. Desiccation crack polygon in natural soil.

Figure 2 .

 2 Figure 2. Schema of experimental setup.

Figure 3 .

 3 Figure 3. Implementation of DIC analysis. The correlation subset is in size of 40×40 pixel, and its

Figure 4 .

 4 Figure 4. Change of water content with elapsed time during drying.

Figure 5 .

 5 Figure 5. Evolution of crack patterns on the specimen surface.

Figure 6 .

 6 Figure 6. ε 1 (most compressive strain) maps of the specimen surface at different desiccation

Figure 7 .

 7 Figure 7. ε 2 (most tensile strain) maps of the specimen surface at different desiccation moment. The

Figure 8 .

 8 Figure 8. ε eq (shear strain) maps of the specimen surface at different desiccation moment. The

Figure 9 .

 9 Figure 9. A zoomed domain of  2 map at step 11. The local strain of a correlation subset is

Figure 10 .

 10 Figure 10. Strain history of the domains of interest: a)  2 , b)  eq . The three domains are defined in

Figure 11 .

 11 Figure 11. Deviated propagation of crack towards the adjacent crack.

Figure 12 .

 12 Figure 12. A crack nucleated at the bottom of the sample and propagated upwards.

FiguresFigure 1 .

 1 Figures

Figure 2 .Figure 3 . 2 Figure 4 .

 2324 Figure 2. Schema of experimental setup.

Figure 5 .

 5 Figure 5. Evolution of crack patterns on the specimen surface.

Figure 6 .

 6 Figure 6. ε 1 (most compressive strain) maps of the specimen surface at different desiccation

Figure 7 .

 7 Figure 7. ε 2 (most tensile strain) maps of the specimen surface at different desiccation moment.

Figure 8 .

 8 Figure 8. ε eq (shear strain) maps of the specimen surface at different desiccation moment. The

Figure 9 .

 9 Figure 9. A zoomed domain of  2 map at step 11. The local strain of a correlation subset is

Figure 10 .

 10 Figure 10. Strain history of the domains of interest: a)  2 , b)  eq . The three domains are defined in

Figure 11 .

 11 Figure 11. Deviated propagation of crack towards the adjacent crack.

Figure 12 .

 12 Figure 12. A crack nucleated at the bottom of the sample and propagated upwards.

Acknowledgments

The authors would like to acknowledge Dr. Michel Bornert at Ecole des Ponts ParisTech for using CMV software for digital image correlation analyses. This work was supported by the Science Foundations of China University of Petroleum, Beijing (Grant No.2462016YJRC021, No.2462018BJC002), National Natural Science Foundation of China (Grant No. 41572246, 41772280, 41322019), Natural Science Foundation of Jiangsu Province (BK20171228, BK20170394), Key Project of National Natural Science Foundation of China (Grant No.