H A Israr

N Hongkarnjanakul

S Rivallant

C Bouvet

Post-Impact Investigation of CFRP Laminated Plate

Keywords: CFRP, Impact, permanent indentation, delamination

This paper deals with low-velocity impact and quasi-static indentation tests on carbon fibre reinforced plastic (CFRP) laminates that classically used in aeronautical applications. The aim is to carry out post-impact analysis using Vic 3-D digital image correlation to investigate the behaviour of permanent indentation which is an important condition for the damage tolerance justification in composite. Result shows the indentation depth decreased with the time due to relaxation of the specimen until at certain position. The microscopic observations were carried out to understand the phenomenon. Besides that, the post-mortem analyses on the impacted plate were also carried out through microscopic observation to determine the delamination area at each interface which is normally difficult to extract from the C-Scan result. The observations were made at three different sections cut (0°, 90° and 45°). The results of delamination area obtained from both microscopic and C-scan analysis is compared and there are rather satisfying to each other.

1.0 INTRODUCTION

Composite materials have been used progressively to replace metallic materials, particularly in structural applications of aeronautical industries because of their interesting characteristics, like low specific weight, enhanced mechanical strength and durability. However, their behaviour under impact loading is one of the major concerns as a single impact can even produces several different damage modes simultaneously [START_REF] Zheng | Low Velocity Impact Analysis of Composite Laminated Plates[END_REF][START_REF] Hongkarnjanakul | Validation of low velocity impact modelling on different stacking sequences of CFRP laminates and influence of fibre failure[END_REF]. This is important because impacts do occur during manufactures, normal operations, maintenances, that can decrease the structure's life even though the damages are not visible to the naked eyes. However, internal damage may already have been created including damage characteristics, such as matrix cracks, delamination and fibre ruptures.

On the other hand, impact damage tolerance deals with the effect of existing impact damage on the strength and stability of the structures [START_REF] Coutellier | Presentation of a methodology for delamination detection within laminated structures[END_REF][START_REF] Alderliesten | Damage tolerance of bonded aircraft structures[END_REF][START_REF] Dilger | Eurofighter a safe life aircraft in the age of damage tolerance[END_REF]. It is essential to define a damage tolerance demonstration as soon as a new project begins. Damage tolerance is proposed to ensure the accidental damage occurring within the operational life of the aircraft, the remaining structure can withstand reasonable loads without failure or excessive structural deformation until the damage is detected. The accidental damage is characterised by its visual detect ability and compared to barely visible impact damage (BVID) which is a very important concept in relation to the damage tolerance. According to the JAR25.571 which was introduced in year 1978 for aircraft structures, the effect of damage on the strength of the structure must be analysed and controlled through the application of a proper design philosophy and proper maintenance and repair [START_REF] Elias | Experimental analysis of damage creation and permanent indentation on highly oriented plate[END_REF].

Therefore, a lot of studies have been carried out to help understand and improve the impact response of composite materials and structures [START_REF] Elias | Experimental analysis of damage creation and permanent indentation on highly oriented plate[END_REF][START_REF] Henkhaus | Overview of research on composite material impact behaviour[END_REF][START_REF] Aktas | An experimental investigation of the impact response of composite laminate[END_REF][START_REF] Wiggenraad | Damage propagation in composite structural elements -Analysis and experiments on structures[END_REF][START_REF] Petit | Impact and compression after impact experimental study of a composite laminate with a cork thermal shield[END_REF][START_REF] Broll | Experimental Studies On The Damage Mechanisms of CFRP-Structures[END_REF]. However, there are few results concerning the forming of the permanent indentation [START_REF] Elias | Experimental analysis of damage creation and permanent indentation on highly oriented plate[END_REF][START_REF] Bouvet | Low velocity impact modeling in composite laminates capturing permanent indentation[END_REF][START_REF] He | Prediction of permanent indentation due to impact on laminated composites based on an elasto-plastic model incorporating fiber failure[END_REF], which is one of the main criteria for the definition of damage tolerance in composites. Thus, the aim of this paper is to improve the knowledge of the permanent indentation behaviour after a certain period of time. Generally, the indentation just after the test, which is the temporary indentation, is always bigger than the indentation after a certain period due to the relaxation of the impacted composites. Some of researchers neglected this effect and assume the indentation became permanent after 48 hours [START_REF] Elias | Experimental analysis of damage creation and permanent indentation on highly oriented plate[END_REF][START_REF] Broll | Experimental Studies On The Damage Mechanisms of CFRP-Structures[END_REF].

In this study, the indentation of the impacted composites will be measured longer than 48 hours in order to observe the indentation depth evolution whether it has become permanent after certain period or it keep reducing with time. To support this analysis, microscopic observations are also carried out to study the inside rupture areas of the impacted plate. This project deals with low velocity impacts on the composite structures to study this phenomenon.

Besides that, C-scan technology is often used to determine the global delamination area of plate being impacted as reported in the literature [START_REF] Hongkarnjanakul | Validation of low velocity impact modelling on different stacking sequences of CFRP laminates and influence of fibre failure[END_REF]. However, it is difficult to extract the delamination area between the interfaces. To overcome this limitation, the availability of microscopic observation results has been used to identify the delamination area at each interface. Results of delamination areas from both C-scan and microscopic observations have been compared and discussed in this paper.

2.0 MATERIALS AND METHODS

Two different tests were carried out in this study. The first one is impact test (dynamics test) and the second test is indentation test (quasi-statics test). Both tests are using different test set-up. A charge-coupled device (CCD) camera system was used to obtain the images of specimen surface in three dimensional. Thus, the surface deformation after the test can be observed to measure the relaxation of indentation depth.

Specimen Fabrication

The specimens used were made from high strength carbon fibers with an epoxy resin as matrix which is the unidirectional (UD) oriented fibres of the type T700 and matrix of M21. The specimens consist of 16 layers of plies with the symmetric stacking sequence [0°2/45°2/90°2/45°2]s. The panels were cut into specimens of 150 x 100 mm in dimension with an average thickness of 4.16 mm. This is standard size of specimens prescribed by the industry. Only two specimens were used in this study, T-dyna and T-static for impact and indentation tests respectively.

Impact Test

Impact tests were performed using a drop tower system with an impactor with 8 mm radius and 2 kg mass as shown in Figure 1a. The test is according to the guidelines from the standards ASTM D7136/D7136M [START_REF]Standard Test Method for Measuring the Damage Resistance of a Fiber-Reinforced Polymer Matrix Composite to a Drop-Weight Impact Event[END_REF]. The specimen was placed under the drop tower with simply supported on a of 75 mm x 125 mm frame, as shown in Figure 1b. The test was performed at impact energy of 25 J. This energy provides sufficiently different damage mechanisms around the BVID threshold and does not reach the perforation level [START_REF] Hongkarnjanakul | Validation of low velocity impact modelling on different stacking sequences of CFRP laminates and influence of fibre failure[END_REF][START_REF] Bouvet | Low velocity impact modeling in composite laminates capturing permanent indentation[END_REF]. During the test, the optical sensor, accelerometer and force sensor were installed to measure the impact forces, accelerations and optical signal for plate deflection. All the data were recorded using Yokogawa DL708 digital oscilloscope and acquired at the maximum rate of 200 kHz because the impact test is very dynamic that gives a wide range of frequencies.

Indentation Test

The indentation test (static test) was carried out after the analyses of the impact test were done. This is because from the analysis of impact test, the total displacement can be obtained which is the parameter that can be set equal to the indentation test. In other words, during the indentation test the displacement of the impactor can be controlled. The indentation test was done by using the Instron 4206 test machine. The set-up of the test can be seen in the Figure 2. The impactor is the same used in the impact tests. Similarly, the specimen was placed on a simple rectangular frame support as in impact test. The displacement of the impactor was controlled by the feed of the test machine's crosshead and the loading speed was set at 2 mm/min. The displacement was measured using Linear Variable Differential Transformer (LVDT) as this instrument allows more accurate values than the displacement measured by the test machine. The LVDT was installed parallel to the vertical axis and attached to the crosshead. It gave out a voltage signal depending on the vertical displacement of the impactor. The load cell in the test machine provides the force signal that acts on the specimen. As in the impact tests, the force and displacement data were recorded using the Yokogawa DL708 acquisition system. The indentation test took longer times than the impact test, thus no large frequencies will appear. Therefore, the range is set to a minimum of 500 Hz and the data were recorded for every 0.5s. The CCD-cameras system was place behind the test machine to observe the specimen's surface just after the test and the following three days.

Post-Mortem Analysis

In this study, two different types of post-mortem analyses have been carried out namely C-scan and microscopic analyses. The objective is to observe any important marks such as fibre breakages or matrix cracking which is invisible to the images taken during the tests that can support the results obtained from the impact or indentation tests.

C-Scan Analysis

C-scan analyses were done on the impacted specimens to detect the delamination areas which could not be seen through the naked eyes. The ultrasonic waves were sent towards the interior surface to detect delamination. The travelling time was measured and it could be converted to the distance of reflected waves. Then, the area of delamination in each inter-laminar if it occurred could be examined. An interpretation from the results of C-scan was not difficult, the range of colour showed clearly the delamination.

The results from C-scan were the pictures of the impact specimen which had a wide range of colour as presented in the following section. From this colour ban, the delamination in each layer can be distinguished. Moreover, a photo of C-scan on the other side is necessary to make sure that the areas of delamination are correctly detected. The microscopic analysis was done by using an optical microscope on both impacted specimens, T-dyna and T-static. First, the specimens were cut along the 0° and 90° through the impacted area as shown in Figure 3. After the microscopic observations have been done at both sections, only then the 45° section was cut. However, before cutting, the specimens are immersed in resin together with chemical hardener (catalyse polyester peroxide) to make sure the debris will not be removed during the cutting and polish processes as demonstrated in [START_REF] Israr | Experimental and Numerical Investigation of CFRP Composite Laminates under Crushing[END_REF]. These microscopic observations were made to study the debris behaviour inside the crack areas and also to measure the crack length between the interfaces that occurred after the plate being impacted. The measurement of the crack length can be used to validate with the C-scan result. Besides that, the microscopic observation can be useful to study the general behaviour of damage that occurred after plate being impacted.

Microscopic Analysis

3.0 RESULTS AND DISCUSSIONS

Force-Displacement Curves

After all the data have been analysed, a few curves have been plotted to show more precise the phenomenon of this impact test. The most interesting curve to study is the force-displacement curve of the test. It gives information about the specimen's rigidity, which is the slope of the curve, and about the appearance of damages [START_REF] Wiggenraad | Damage propagation in composite structural elements -Analysis and experiments on structures[END_REF]. In Figure 4, the force curve for impact test increased continuously, under the formation of further small damages. When the maximal displacement is reached the force decreases steadily until almost zero. From the curves, it showed the maximum displacement is 5.7 mm. This maximum displacement value is important for the indentation test because this is the maximum displacement value fixed for the indentation test. This value was set at the linear variable differential transformer (LVDT). The maximum energy obtained in this test is almost same as the impact test which is 24.8 J. Similarly, the force for the indentation tests increased linearly at the beginning due to no damage occurs on the specimen. However, after reaching 2.3 mm the damage starts to occur on the specimen due to the specimen started to lose in the stiffness. Another marked appearance of damage can be seen clearly at the displacement of 4 mm. The sudden loss of energy from 8.5 kN to 7.2 kN is caused by the fibre cracks.

Figure 4 The comparison of force-displacement curves between impact and indentation tests

Between the impact and the indentation test, there is one parameter in common, namely the maximal displacement of the impactor. Therefore, a good possibility is given to compare the results of these two different tests. Although the maximal displacement of 5.7 mm was shown by the LVDT machine, however Figure 4 shows that the maximal displacement is only achieved by 5.4 mm. This is probably caused by error in setting LVDT machine at the beginning of the test. This difference is neglected for further analysis. Besides that, the force curve increased linearly longer in the indentation test than in the impact test. Moreover, the force at the same respective displacement is larger in the case of the impact test than the indentation test. From previous studies [START_REF] Broll | Experimental Studies On The Damage Mechanisms of CFRP-Structures[END_REF], this phenomenon is explained due to the inertia effect that occurs during the impact test because of the deformation loading speed in the impact test is higher compared to the indentation test. Thus, the inertia of the dynamically loaded specimen is larger and it reacts stiffer upon the loading. This result is compatible with the other authors that have also obtained the similar trend of the force curves of impact and indentation tests [START_REF] Hongkarnjanakul | Validation of low velocity impact modelling on different stacking sequences of CFRP laminates and influence of fibre failure[END_REF][START_REF] Bouvet | Low velocity impact modeling in composite laminates capturing permanent indentation[END_REF][START_REF] Suh | Compression behavior of stitched stiffened panel with clearly visible stiffener impact damage[END_REF][START_REF] Belingardi | Influence of the laminate thickness in low velocity impact behavior of composite material plate[END_REF][START_REF] Nettles | The impact response of carbon/epoxy laminates[END_REF].

Indentation Depth Analysis

A further criterion to assess the damage caused by an impact test or static test is the indentation depth of the plate at where the impactor has knocked. The characteristics of the permanent indentation after certain relaxation are one of the main interests in this study. Therefore, the indentation depth of both specimens; Tdyna of impact test and T-static of static test were measured. A series of 3D images of the damaged specimen has been taken for a few days after the tests to measure the plate's indentation by using the CCD-cameras. The indentation was observed directly after the test. From the CCD-camera systems, it gave the contour plot images of the specimen by using Vic Snap and Vic 3-D Digital Image Correlation as shown in Figure 5a. These contour plot images were used for the determination of the indentation. The indentation can be seen in the middle of the plates as shown in Figure 5b. Besides that, a global deformation of the specimens can be marked. The specimen has a twisted shape after being damaged because of the plasticity of the resin (Figure 5b) even though the resin's characteristic is brittle. However, the plasticity can occur in this case. The taken images were marked as Z001, Z002, Z010 and so on. The number 001 or 006 are representing the first image or the sixth image taken by the CCD camera and Z is representing the displacement in Z-direction. One tangent line was drawn approximately in the middle between the curves. The indentation depths were measured from the point on the tangent line to the lowest peaks of each curve as in the Figure 6. All the results of the indentation depths of each curve for the impact and the static test were put in Tables 1 and2. As mentioned before, all the images are representing the time when the images were took by the CCD-Camera system. The times distance between the images were also included in the tables below. Thus, by knowing these data the curve of relaxation of the indentation depth were plotted as shown in the Figure 7a. As a result, the characteristics of indentation depth after certain period of time were as expected, the depth will decrease due to relaxation of the specimen. The difference between temporary and permanent indentation depth for impact test is ≈10 % and ≈18 % for the static test. In reality, the initial indentation depth (D0) for both cases is difficult to obtain, especially for the impact test.

Hence, in this study the D0 value is assumed as the indentation depth of the first picture which was taken just after either the impact or the static test. Therefore, there is always a gap between the original D0 and the assumed D0 depending on how fast the first picture was taken. For both tests, when the plate gets unloaded, the deformation which occurred on the plate tries to back its initial location due to the elasticity of the material that is called the plate relaxation. The debris can prevent that relaxation, the better it is distributed the higher is that effect. It is obvious a better distribution can occurs during a static test because the debris has more time to fill the free space of the cracks [START_REF] Alderliesten | Damage tolerance of bonded aircraft structures[END_REF][START_REF] Wiggenraad | Damage propagation in composite structural elements -Analysis and experiments on structures[END_REF]. Referring the speed of decrease for both tests, during the first 5 hours the indentation decreased dramatically but after that it started to decrease steadily. It is almost constant after 45 hours of the tests. Figure 7b has shown that the indentation depth of the plate performed under static test has greater decrease of speed than the impact test. The reason for this phenomenon was not yet found. The counting of fragment or debris of each plate through the microscopic observation could give some idea of this behaviour.

Debris Observation

The microscopic observations on the 90 and 0 sections cut have shown that there are plentiful of debris exists in the delaminated areas in both sections. Besides the delamination between the interfaces, matrix cracking and fibres rupture are also exists in many areas as shown in Figure 8. Similarly, for the 45 section cut, there are also a lot of debris exists in the delaminated area. Therefore, it seems difficult to count the debris with the naked eyes. Hence, it can be concluded that the existing of these debris will tend to block the fibres and matrices of the specimen being impacted to return to its initial location. It will affect the relaxation of indentation. This could give the general explanation why in previous section the indentation depth became constant after certain period of times. When the crack areas became smaller with time due to relaxation, it will stop when the debris inside the crack areas cannot move anymore. The hypothesis of this mechanism may be due to the creep phenomenon of the little debris which prevents the broken fibers and matrices moving back to its initial location.

Delamination Area Analysis

Delamination area of impacted plate taken from the C-scan analysis was measured for each interface as shown in Figure 9. The delaminated area of each interface presents in Figure 9b was approximately extracted from the Figure 9a. From the color ban (Figure 9a), the delamination area in each layer can be distinguished. Moreover, the C-scan result on the non-impacted side (Figure 9a) is necessary to make sure the areas of delamination were properly extracted. There was no delamination if the plies were in the same orientation such as four plies of -45 in the middle of the laminates.

After that, the measurements of delamination length for each interface of C-scan result were compared with the delamination length calculated from the microscopic observation as presented in Figure 10.

Generally, the delamination length measured from both microscopic and C-scan analyses have some differences. For a better comparison, the measurements from these analyses were plotted in Figure 11. As can be seen in Figure 10 and Figure 11a, the longest crack measured through the microscopic image along the 90° section cut was between the -45°/90° interface which is around 17.1 mm. Meanwhile for the C-scan, three different interfaces have almost the same longest crack around 16.2 mm. This is may be due to the minor error of the approximation process of delaminated area from the Figure 9a. Moreover, the Cscan results have shown that there are five interfaces encountered with delamination but from the microscopic observation, only four interfaces where the delamination can be clearly seen. The 45°/90° interface has only small delamination near to the impacted area.

The microscopic observation on 0 section cut has shown that the longest crack occurred at 45°/0° interface (38.1 mm) which is close to the C-scan result (35.5 mm) as presented in Figure 10b and Figure 11b. Besides that, the number of interfaces having delamination in both analyses is also similar with the delamination length of the other interfaces are compatible to each other.

For both microscopic observations on the 90 and 0 sections cut, the results obtained were rather satisfying compared to the Cscan result. However, for 45 section cut the results obtained by microscopic observation has shown that there are significant differences with the C-scan result. Referring to Figure 10c and Figure 11c, the longest crack occurred at 90°/45° interface with the length of 22.3 mm while for C-scan, the longest crack occurred at 0°/45° interface with length 16.5 mm. Besides that, the second longest crack is also different in both results where for the microscopic observation is 15.7 mm at -45 0 /90 0 interface while for C-scan at the same interface, the approximation made are only 6.5 mm. Another important point to discuss is there was a short crack at 0°/45° interface in the microscopic observation. These major differences may be caused by the difficulties to see exactly the delaminated area on each interface from the general result of C-scan (Figure 9a). The characteristics of the permanent indentation depth after certain relaxation of impacted plates are one of the main interests in this project. As expected the indentation depth decreased due to relaxation of the specimen and it has become almost constant after 48 hours of the tests. From the microscopic observations, this may be due to the existing of debris in the crack areas that tend to prevent the broken fibres and matrixes back their initial location. Moreover, the specimen has been slightly twisted after being damaged and the statically loaded plate suffer more damage in terms of the indentation than impacted one. A reason for this might be due to the longer times taken by the impactor to hit the specimen during the indentation test. Besides that, the indentation depth of plate tested under the static test has also experienced greater decrease of speed than the impact test. The reason for this phenomenon has still not yet found and reported in the open literature.

The results of delaminated area obtained by both microscopic observation and C-scan are rather satisfying to each other despite of some erroneous due the difficulties to see exactly the delaminated area on the 45° cut on the general result of C-scan (Figure 9a).

Figure 1

 1 Figure 1 (a) The arrangement of the impact test (b) Boundary condition of the impact test

Figure 2

 2 Figure 2 The indentation test arrangement on the test machine

Figure 3

 3 Figure 3 Specimen preparations for microscopic observation (a) the plate after being cut (b) The areas that were polished and observed (section no.2)

Figure 5

 5 Figure 5 The Image from Vic 3-D Digital Image Correlation (a) Coordinate system and size (b) The twisted shape of specimen after being damage

Figure 6

 6 Figure 6 The indentation depth measurements (a) impact test) (b) indentation test

Figure 7 Figure 8 Figure 9 Figure 10

 78910 Figure 8 Some example of debris in the delamination areas and the matrices cracking for (a) Near to the impact of 90 cut and (b) Near to the impact of 0 cut

  (a) 90° section cut (b) 0° section cut (c) 45° section cut

Figure 11

 11 Figure 11The results comparison of microscopic observation and Cscan

Table 1

 1 The evolution of indentation depth of impact test on plate T-dyna

	No. of	Time	Indentation	Speed of
	Picture	(Hours)	Depth (mm)	Decrease
	Z001 (ref)	0	0.67264	-
	Z006	0.23	0.65872	-0.090330
	Z010	0.43	0.64631	-0.031523
	Z020	1.15	0.63928	-0.012081
	Z030	1.98	0.63372	-0.005776
	Z050	4.07	0.62795	-0.002257
	Z070	7.97	0.62463	-0.001134
	Z090	16.63	0.61836	-0.000648
	Z120	40.58	0.61041	-0.000455
	Z127	47.58	0.60890	-0.000009
	Z130	286.667	0.60886	-

Table 2

 2 The evolution of indentation depth of indentation test on plate Tstatic

	No. of	Time	Indentation	Speed of
	Picture	(hours)	Depth (mm)	Decrease
	Z003 (ref)	0.017	0.94222	-
	Z005	0.033	0.92765	-0.8500745
	Z010	0.067	0.90183	-0.7178192
	Z015	0.083	0.89354	-0.4076438
	Z020	0.100	0.88892	-0.1713831
	Z030	0.300	0.85826	-0.0770489
	Z040	1.000	0.82303	-0.0367393
	Z050	2.000	0.79891	-0.0056559
	Z070	9.717	0.77618	-0.0012278
	Z091	21.717	0.77591	-0.0000046
	Z125	93.300	0.77581	-

Acknowledgement

The authors would like to thank the Department of Material and Structure Mechanic of Institut Supérieur de l'Aéronautique et de l'Espace for providing the research facilities and materials to undertake this work.


Alison B Duncan
email: alison.duncan@umontpellier.fr.

Cassandra Marinosci

Céline Devaux

Sophie Lefèvre

Sara Magalhães

Joanne Griffin

Adeline Valente

Ophélie Ronce

Isabelle Olivieri

Transgenerational cues about local mate competition affect offspring sex ratios in the spider mite Tetranychus urticae

Keywords: local mate competition, sex-ratio, sib-mating, haplodiploid

In structured populations, competition for mates between closely related males, termed Local Mate Competition (LMC), is expected to select for female-biased offspring sex ratios. However, the cues underlying sex allocation decisions remain poorly studied. Here, we test for several cues in the spider mite Tetranychus urticae, a species that was previously found to adjust the sex ratio of its offspring in response to the number of females within the local population, i.e. a patch. We here investigate whether the offspring sex ratio of T. urticae females changes in response to 1) the current number of females in the same patch, 2) the number of females in the patches of their mothers and 3) their relatedness to their mate. Single females on patches produced similar sex ratios to those of groups of 15 females; their mothers had been in identical conditions of panmixia. The offspring sex ratios of females mated with their brother did not differ from those of females mated with an unrelated male. Females however produced a more female-biased offspring sex ratio if their mothers were alone on a patch compared to 15 other females. Thus, maternal environment is used as a cue for the sex allocation of daughters. We discuss the conditions under which the maternal environment may be a reliable predictor of LMC experienced by grand-sons.

INTRODUCTION

In structured populations, dispersal of one sex can cause competition to be more intense between kin of the nonispersing sex than between kin of the dispersing one. This asymmetry in competition can cause a bias in the offspring sex ratio to reduce competitive interactions between brothers or sisters.

If females are the dispersing sex, related males can compete for mates, a phenomenon termed Local Mate Competition (LMC). Female-biased offspring sex ratios are expected to evolve to reduce this competition between males, both through a decrease in the number of male competitors and an increase in the number of mates per son [START_REF] Hamilton | Extraordinary Sex Ratios[END_REF][START_REF] Smith | The evolution of sex[END_REF][START_REF] Taylor | Intra-sex and inter-sex sibling interactions[END_REF]Charnov, 1982;Frank, 1985;[START_REF] Herre | Sex ratio adjustment in fig wasps[END_REF]. In addition to LMC, for haplodiploid species, a more femalebiased offspring sex ratio can arise because mothers that mate with a brother are more related to their daughters than they are to their sons [START_REF] Greeff | Alternative mating strategies, partial sex ratios in haplodiploid species sibmating and split[END_REF][START_REF] Reece | Kin discrimination and sex ratios in a parasitoid wasp[END_REF]. Inbreeding does not change the relative relatedness of mothers to sons as they only receive genes from their mothers [START_REF] Greeff | Alternative mating strategies, partial sex ratios in haplodiploid species sibmating and split[END_REF].

Population structure can vary in time, and consequently so might the intensity of LMC and sib-mating. In such a situation, selection should favour females that facultatively (plastically) adjust their sex allocation in response to cues in the environment that reflect the intensity of LMC their sons will experience as well as whether they have mated with kin (Frank, 1985;[START_REF] Herre | Sex ratio adjustment in fig wasps[END_REF][START_REF] Nagelkerke | Hierarchical levels of spatial structure and their consequences for the evolution of sex allocation in mites and other arthropods[END_REF]Macke et al., 2011a). For example, females could adjust their offspring sex ratios in response to the number of females laying eggs (foundresses) in the same patch or to the level of relatedness of their mate. Consistent with expectations that females respond to current cues in the environment, more female biased offspring sex ratios are observed in patches formed by a single foundress as compared to patches founded by multiple foundresses, across a wide range of species [START_REF] Shuker | Sex ratios under asymmetrical local mate competition: Theory and a test with parasitoid wasps[END_REF][START_REF] West | Sex-ratio adjustment when relatives interact: a test of constraints on adaptation[END_REF][START_REF] Sato | Can the extremely female-biased sex ratio of the social spider mites be explained by Hamilton's local mate competition model?[END_REF][START_REF] Wang | Non-quantitative adjustment of offspring sex ratios in pollinating fig wasps[END_REF]. Several studies, however, found that females do not adjust their offspring sex ratio in response to different numbers of foundresses on a patch [START_REF] Innocent | Lethal combat and sex ratio evolution in a parasitoid wasp[END_REF]Abe et al., 2014;[START_REF] Lievens | Maladaptive sex ratio adjustment in the invasive brine shrimp Artemia franciscana[END_REF]. Moreover, there is currently no empirical support for the prediction that sib-mating results in more female biased offspring sex ratios than does random mating [START_REF] Reece | Kin discrimination and sex ratios in a parasitoid wasp[END_REF]Burton-Chellew et al., 2008).

Alternatively, LMC and inbreeding may change in a predictable fashion over generations. This scenario is predicted by haystack models of LMC in which sub-divided populations may persist over multiple generations, and relatedness between females sharing a patch may increase across generations [START_REF] Nagelkerke | Hierarchical levels of spatial structure and their consequences for the evolution of sex allocation in mites and other arthropods[END_REF]. This increased relatedness can select for even more female biased offspring sex ratios due to more intense LMC (Frank, 1985;Gardner et al., 2009). If change in the sub-divided population structure is predictable through time, transfer of information from mothers to daughters could allow for a better adjustment of sex allocation.

Discrepancies among studies investigating sex allocation may be caused by the underlying cues being unknown and thus overlooked when designing experiments. In principle, these cues could stem from both the current or past environment, i.e. factors experienced while a female is laying her eggs, or those encountered as a juvenile, or even in previous generations. For example, females of the parasitoid wasp Nasonia vitripennis change their sex allocation in response to immediate cues in the environment, including the current number of foundresses, female odours and the presence of eggs in the environment [START_REF] Shuker | Sex ratios under asymmetrical local mate competition: Theory and a test with parasitoid wasps[END_REF][START_REF] Shuker | Information use in space and time: sex allocation behaviour in the parasitoid wasp Nasonia vitripennis[END_REF]. In contrast, it is the environment that house wrens experience as juveniles, not the current environment, that affects the sex ratio of their own offspring: they produce a more female-biased offspring sex ratio when raised in a smaller brood (Bowers et al., 2017). Also, female mice exposed to stress in utero produce more daughters (Edwards et al., 2016).

Here we investigate the impact of immediate and past cues on sex allocation in the spider mite Tetranychus urticae. An earlier study showed that spider mites evolving under constant, high LMC lost the ability to facultatively adjust their offspring sex ratio, whilst those evolving under weaker LMC retained the ability for facultative sex allocation in response to the number of foundresses (Macke et al., 2011a). In that study, sex ratio was measured after mites had spent two generations on patches containing the same number of foundresses. Thus, both mothers and their daughters experienced cues indicating the same intensity of LMC for their sons. Therefore, the set-up did not allow disentangling the current vs past cue(s) used by females that retained the ability for facultative sex allocation. Other studies have shown that maternal cues impact a number of traits in spider mites including sex allocation, dispersal distance, fecundity and diapause [START_REF] Oku | Different maternal effects on diapause induction of tetranychid mites, Tertranychus urticae and T. kanzawai (Acari: Tetranychidae)[END_REF][START_REF] Magalhães | Environmental effects on the detection of adaptation[END_REF]Bitume et al., 2014;[START_REF] Marinosci | Effects of host plant on life-history traits in the polyphagous spider mite Tetranychus urticae[END_REF]. T. urticae females also respond to current cues and produce more female-biased offspring sex ratios when there are fewer females sharing a patch [START_REF] Wrensch | Effects of density and host quality on rate of development, survivorship and sex ratio in the Carmine spider mite[END_REF][START_REF] Roeder | Sex ratio response of the two-spotted spider mite[END_REF] and they are able to recognise kin [START_REF] Tien | Mate choice promotes inbreeding avoidance in the two-spotted spider mite[END_REF]Bitume et al., 2014).

We aim at uncovering the interplay between immediate and past cues in the environment in determining sex ratios in T. urticae. We investigate whether females alter their sex allocation in response to the number of females in the same patch (i.e. current indicators of LMC that their sons will experience) or the number of females in the patch of their mothers (i.e. past LMC conditions). Because sib-mating is negatively correlated with the number of foundresses on a patch, and a certainty in our high LMC treatments, we evaluate the effect of sib-mating on sex allocation in a separate experiment.

MATERIAL AND METHODS

Biological model

The two-spotted spider mite, T. urticae Koch (Acari: Tetranychidae), is a generalist hebivore (Migeon & Dorkeld, 2006-2017;[START_REF] Helle | Spider mites: Their biology, natural enemies and control[END_REF] with a short generation time (less than 15 days at 20°C on a suitable host; [START_REF] Helle | Spider mites: Their biology, natural enemies and control[END_REF][START_REF] Riahi | Temperature effects on development and life table parameters of Tetranychus urticae (Acari: Tetranychidae)[END_REF]. It has an arrhenotokous haplodiploid life cycle in which females produce haploid sons parthenogenetically and diploid daughters from fertilised eggs. After the egg stage, all individuals pass through a larval and two nymphal (protonymph and deutonymph) stages, with short quiescent stages between each, before reaching maturity. Mature females are approximately twice as big as, and rounder than, mature males. The sex ratios measured in this study are tertiary sex ratios defined as the proportion of adult males on a patch; offspring sex ratio is a trait of the mother.

Study population

This study uses the same base population as Macke et al. (2011a). This population was collected from a cucumber greenhouse in Pijnacker, The Netherlands, in May 1994. Mites were transferred to a climate chamber at the University of Amsterdam where they were maintained on the same host plant. In September 2007, after ~240 months (~480 generations), approximately 5000 mites were sampled from the Amsterdam population to create two populations on cucumber plants at the University of Montpellier. Mite populations were maintained under the same conditions in a single climate chamber at 25°C ± 2°C, with a photoperiod of 14L:10D. A new, more genetically variable, base population was created in February 2011 by mixing and transferring approximately 200 adult females from the two previous populations. This new base population was maintained on 4 -6 cucumber plants contained in a single plastic box (505 mm length x 310 mm width x 380 mm height) with a hole in the lid covered in a fine mesh to avoid condensation. Plants were watered every week and rotten plants removed and replaced with young, mite-free plants; plants were placed beside infested ones to facilitate mite dispersal.

To start the experiment (Generation 0), 12 independent samples from the base population consisting of 40 adult females each were created; this ensured that females seeding the experiment were of the same age. Each sample was placed on a whole cucumber leaf resting on water-saturated cotton wool in a plastic box (255 mm x 185 mm x 77 mm) to prevent both leaf dehydration and mite dispersal. Females were allowed to lay eggs and, 14 days later, their mature adult daughters were sampled to start the experiment 'sex allocation in response to LMC across generations'. In the 'sib mating' experiment, slightly older daughters were used because the experiment was started 20 and 21 days after females started laying eggs in generation 0. In all experiments, mites were maintained in the same plastic boxes on leaves or leaf fragments placed on saturated cotton wool in climate chambers at 25 ± 2ºC with a 16L:8D cycle. Prior to starting and during the experiments, we confirmed that our T. urticae populations were Wolbachiafree (see Supplementary Materials, Appendix 1 for details).

Sex allocation in response to LMC across generations

This experiment investigated the sex allocation of females over two generations. In Generation 1 (the 'maternal generation'), we measured the offspring sex ratios in response to high (1 foundress) versus low (15 foundresses) LMC experienced by laying females (these were the daughters of females from Generation 0). Each patch from Generation 0 contributed equal numbers of females, arbitrarily sampled, to the high and low LMC treatments. In Generation 2 (the 'daughter generation'), we investigated the sex allocation under high LMC of daughters from females of Generation 1, which had experienced either high or low LMC. Thus, in Generation 2 we tested whether LMC experienced by mothers influenced the offspring sex ratios of their daughters (see Figure S1 for an outline of the experimental design).

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Generation 1 (maternal generation):

In the low LMC treatment, a total of 360 females, in groups of 15 females, were placed on 60 cm 2 cucumber leaf patches. We had 6 boxes, each containing 4 low LMC patches. In the high LMC treatment, 240 females were individually placed on a 4 cm 2 leaf patch, and spread across 5 boxes, each containing 48 patches. Female density was the same for both low and high LMC conditions (0.25 females × cm 2 ). Each box was divided into four quarters, each quarter containing 12 patches of single females in high LMC boxes and one patch of 15 females in low LMC boxes. This was done to get the same numbers of replicates (quarters) for each treatment with similar numbers of females laying eggs (12 for low LMC and 15 for high LMC). Note that we did not control for female mortality, i.e. there may have been fewer females contributing to the next generation in each treatment. In both LMC treatments, females laid eggs, and 14 days later we counted the number of emerging adults on each patch, and determined their sex using a stereo light microscope. Only measurements from a subset of 3 high LMC boxes and 2 low LMC boxes were included in the statistical analyses due to experimenter error.

Generation 2 (daughter generation):

Independently of LMC levels, 12 females were randomly collected from each quarter of each box from Generation 1 and individually placed under high LMC on a 4 cm 2 leaf patch (same density as in Generation 1). Due to differential fecundity, some females from Generation 1 may have contributed more offspring to Generation 2 than did others. All patches were randomly spread across 6 boxes (48 patches per box). Note that some females in Generation 2 were sampled from patches that were not used in the analysis of sex ratio in Generation 1 due to experimenter error. Females were allowed to lay eggs and their offspring sex ratio was measured 14 days after transfer. In total, we measured the offspring sex ratio and number of emerging adults on 114 patches from 20 quarters in boxes testing the effect of low maternal LMC (5 -6 females per quarter), and 116 patches, from 14 quarters, in boxes testing the effect of high maternal LMC (3 -12 females per quarter). Some patches were lost due to female mortality before they laid eggs or to rotting of leaf patches.

Sex allocation adjustment in response to sib mating

This experiment investigated whether females that mated with their brothers produced a more female biased offspring sex ratio than did females that mated with unrelated males. In total, 96 females sampled from Generation 0 were individually placed on 4 cm 2 patches spread across 2 boxes (48 patches per box). This experiment was done in temporal blocks, over two days. Females were allowed to lay eggs and 9 days later quiescent daughters (2 from each patch, later allocated to each of the two treatments) were individually placed on 4 cm 2 patches to ensure all mites assigned to our mating treatments were virgin. The same was done for emerged or quiescent sons. Mating pairs were formed three days later, when all quiescent females were adult. Females were assigned to one of two mating treatments: (1) the sib-mating treatment, in which one arbitrarily sampled brother was added to their patch or (2) the unrelated mating treatment, in which a male from another patch was added to the female's patch. Males were removed after three days, females were allowed to lay eggs and 14 days later the sex ratio and number of emerging adults was measured. In total, we formed 24 patches with unrelated mate pairs and 28 patches with sibling mate pairs.

Statistical analyses

For all analyses, except for the sex ratio of females from Generation 1 (see below), we analysed variation in the number of sons (offspring sex ratio) and the total number of offspring that emerged as adults using generalised linear mixed models, assuming a binomial distribution and a Poisson distribution, respectively. These models were performed in SAS using the GLIMMIX model procedure with the lsmeans statement to account for unbalanced designs (SAS script provided in Appendix 2, Supplementary Materials).

Sex allocation and offspring production in response to LMC across two generations

At Generation 1, the sex ratio was measured for single females for the high LMC treatment and for groups of 15 females for the low LMC treatment. To analyse variation in the number of sons across groups of females of similar sizes, we used a two-tailed permutation test, randomly sampling groups of 12 and 15 females (corresponding to the experimental design) under the null hypothesis of no effect of LMC. We generated 10,000 empirical values of the z statistics and could thus obtain the p-value for the effect of LMC on sex ratio. The observed statistic was generated using the number of sons produced in quarters of boxes by groups of 12 females for the high LMC treatment and by groups of 15 females for the low LMC treatment.

At Generation 1, we analysed variation in mean individual offspring production in each quarter as the number of offspring reaching adulthood in 14 days (this corresponds to ~4 days of laying eggs) accounting for LMC treatment as a fixed effect, and the box containing each quarter as a random effect nested within LMC treatment. At Generation 2, we tested the effect of the LMC conditions experienced by the mother at Generation 1 on male production and offspring production. We accounted for the LMC treatment of the mothers as a fixed effect, and quarter from which each female originated as a random factor nested within LMC treatment.

Sex allocation adjustment in response to sib mating

In the sib-mating experiment, we accounted for the mating treatments of the mothers as a fixed effect, and the patch of each mother and temporal block as random factors. Note that block could not be included in the analysis for offspring production because the model did not converge.

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RESULTS

In all experiments, mean (± 1 SE) offspring per female was not affected by the intensity of LMC in current (19.01 ± 1.630 for high LMC; 20.16 ± 0.920 for low LMC; F 1, 3 = 0.13, p = 0.7397) or past generations (13.05 ± 0.860 for high LMC; 14.79 ± 0.744 for low LMC; F 1, 32 = 2.37, p = 0.1335), or by the relatedness between mates (14.54 ± 1.088 for sib-matings; 16.42 ± 0.934 for unrelated matings; F 1, 21 = 2.95, p = 0.1005). The absence of an effect on offspring production ensures that the estimates of sex-ratio have the same precision between LMC and sib-mating treatments.

Sex allocation in response to LMC across two generations

At Generation 1, females did not modify their offspring sex ratio according to the number of other females in the same patch (p = 0.938; Figure 1A). In contrast, at Generation 2, the maternal environment affected the sex allocation of their daughters, as females produced a less female biased offspring sex ratio if their mothers had experienced low, compared to high, LMC (F 1, 32 = 10.98, p = 0.002, Figure 1B).

Sex allocation adjustment in response to sib-mating

Mating with a brother or an unrelated male did not change sex allocation (F 1, 21 = 0.06, p = 0.82; Fig . 1C). This means the effect of high LMC over two generations on sex allocation are not due to sib-mating.

DISCUSSION

We found that sex allocation in the spider mite T. urticae does not depend on the number of females on the same patch or on the relatedness between mates. Instead, we found that females alter their offspring sex ratio according to the number of females on their mother's patch, showing that the maternal environment may be important for sex allocation. Additionally, none of the treatments -current or maternal levels of LMC, or mating with a sibling-impacted the number of offspring that were produced over a ~ 4 day period and reached adulthood.

Female T. urticae did not produce a more female-biased sex ratio when they mated with their brother compared to an unrelated male. As spider mites do recognise kin [START_REF] Tien | Mate choice promotes inbreeding avoidance in the two-spotted spider mite[END_REF]Bitume et al., 2013), this calls for an explanation. One possibility may be that there is sexual conflict over the optimal offspring sex ratio [START_REF] Shuker | Sexual conflict, sex allocation and the genetic system[END_REF][START_REF] Macke | Local mate competition mediates sexual conflict over sex ratio in a haplodiploid spider mite[END_REF]. In haplodiploid species, males only pass their genes to the next generation via daughters, thus it is always beneficial for males to produce a more female-biased offspring sex ratio, whereas for females the optimal offspring sex ratio depends on levels of LMC [START_REF] Shuker | Sexual conflict, sex allocation and the genetic system[END_REF][START_REF] Macke | Local mate competition mediates sexual conflict over sex ratio in a haplodiploid spider mite[END_REF][START_REF] Shuker | Evolution: conflict by the sexes, for the sexes[END_REF]). Since T. urticae males can manipulate females to produce more female-biased offspring sex ratios [START_REF] Macke | Local mate competition mediates sexual conflict over sex ratio in a haplodiploid spider mite[END_REF] females may not alter their sex allocation in response to any cue provided by their mate as it may not be selectively advantageous.

As found in a number of other species [START_REF] Reece | Sex allocation and interactions between relatives in the bean beetle, Callosobruchus maculatus[END_REF][START_REF] Innocent | Lethal combat and sex ratio evolution in a parasitoid wasp[END_REF]Abe et al., 2014) we did not find that female T. urticae alter their sex allocation in response to the number of other females in the same patch (but see [START_REF] Wrensch | Effects of density and host quality on rate of development, survivorship and sex ratio in the Carmine spider mite[END_REF][START_REF] Roeder | Sex ratio response of the two-spotted spider mite[END_REF]. For example, in Nasonia vitripennis other cues, specifically egg presence and the relative number of eggs a female contributes to a patch, had a stronger influence on sex allocation than did the number of foundresses [START_REF] Shuker | Information constraints and the precision of adaptation: sex ratio manipulation in wasps[END_REF]Burton-Chellew et al., 2008). It could be, for our population under our experimental conditions, that the number of females in the current patch does not provide reliable information about the LMC level that the offspring will experience. This may be because (a) females often move away from the patch or (b) females have high mortality during the egg laying period. In spider mites, the second hypothesis is possible if females are on low-quality plants [START_REF] Magalhaes | Adaptation in a spider mite population after longterm evolution on a single host plant[END_REF][START_REF] Marinosci | Effects of host plant on life-history traits in the polyphagous spider mite Tetranychus urticae[END_REF].

It may be that females respond to density and not the absolute number of females. In studies that found an effect of foundress number on LMC, density was not controlled [START_REF] Wrensch | Effects of density and host quality on rate of development, survivorship and sex ratio in the Carmine spider mite[END_REF][START_REF] Roeder | Sex ratio response of the two-spotted spider mite[END_REF], but we did control for density in our experiment. Alternatively, the absolute number of females on a patch could have been too low to elicit a response. Spider mites use mechanical and chemosensory processes via web and faeces to detect conspecifics [START_REF] Royalty | Arrestment of male twospotted spider mite[END_REF]Clotuche et al., 2014). T. urticae can also use volatiles released by infested plants to detect conspecifics, but at densities much higher than those in our experiment, and on whole plants, and not leaves as here [START_REF] Pallini | Odour-mediated responses of phytophagous mites to conspecific and heterospecific competitors[END_REF]. These cues may not have been available in our experiment with low densities of females.

In Generation 2 densities were also identical to those of Generation 1 and between treatments, and here we did detect a difference in sex allocation. Adult females whose mothers experienced low LMC produced less female biased offspring sex ratios. In both Macke et al. (2011a) and our experiment sex allocation is more female biased when females experienced at least two generations of LMC. In both experiments sex allocation could be in response to the number of foundresses in the mother's patch, or the number, or relatedness, of individuals in the juvenile environment.

Our finding could be a true maternal effect in sex allocation, meaning that the mother's experience of her environment influences her offspring's phenotype [START_REF] Mousseau | The adaptive significance of maternal effects[END_REF]. Maternal effects have a role in determining many offspring traits including offspring size and juvenile survival [START_REF] Pick | Disentangling genetic and prenatal maternal effects on offspring size and survival[END_REF], resistance to parasites [START_REF] Little | Maternal transfer of strain-specific immunity in an invertebrate[END_REF] and competitive ability (Bentz et al., 2016). In T. urticae maternal effects partly determine offspring phenotypes for dispersal distance (Bitume et al., 2014), diapause incidence [START_REF] Oku | Different maternal effects on diapause induction of tetranychid mites, Tertranychus urticae and T. kanzawai (Acari: Tetranychidae)[END_REF] and offspring life-history traits [START_REF] Magalhães | Environmental effects on the detection of adaptation[END_REF][START_REF] Marinosci | Effects of host plant on life-history traits in the polyphagous spider mite Tetranychus urticae[END_REF]. To the best of our knowledge, parental effects have not been previously shown to play a role in determining offspring sex allocation. An exception may be studies in which stress during the embryonic period influences offspring sex ratio. For example, female mice produce a more female biased offspring sex ratio if they experience stress in utero (Edwards et al., 2016), and paternal incubation temperature influences offspring sex ratio in a lizard [START_REF] Warner | Transgenerational sex determination: The embryonic environment experienced by a male affects offspring sex ratio[END_REF]. Whether these can be considered as maternal (or paternal) effects is debatable, as the developing individual is itself experiencing the stress. If our results are due to a maternal effect the same one might be responsible for the results observed in the selection experiment by Macke et al. (2011a). Indeed, in that experiment, mothers and daughters experienced the same predictable LMC level, so sex ratio adjustment may be explained by the maternal environment.

If female sex allocation in T. urticae is in part influenced by maternal cues for LMC this could come about via the manipulation of a daughter's egg production. Egg size is thought to be one way by which female T. urticae could plastically adjust their sex allocation, as fertilisation probability increases with egg size, bigger eggs being more likely to develop into females (Macke et al., 2011b). Furthermore, T. urticae evolving under high LMC produce larger eggs and a more female biased sex ratio [START_REF] Macke | Sex-ratio adjustment in response to local mate competition is achieved through an alteration of egg size in a haplodiploid spider mite[END_REF]. Female T. urticae not peer-reviewed) is the author/funder. All rights reserved. No reuse allowed without permission. possess a single ovary within which eggs develop and are fertilised (Feiertag-Koppen & Pijnacker, 1982). Female larvae that have recently emerged from their eggs possess cells in their ovaries undergoing meiosis (Feiertag-Koppen & Pijnacker, 1982;[START_REF] Mothes-Wagner | Postembryonic development of the twospotted spider mite Tetranychus urticae with special reference to the Int[END_REF]; the conditions or resources of the mothers could influence the development and size of their daughters' eggs, which could translate into a given sex ratio.

Another possibility is that the juvenile environment acts as a possible proximal cause of sex allocation, which itself depends on maternal LMC. When mothers experience high LMC, and are alone on a patch, females will develop with their siblings whereas under low LMC, juveniles will share their patch with both related and unrelated individuals. Furthermore, although we controlled for density across treatments, under low LMC there is a higher absolute number of juveniles than there is under high LMC. Offspring sex ratio has been linked to the environment a mother experiences during the juvenile period in wrens and voles; when mothers experienced juvenile resource limitation they produced less male biased offspring sex ratios [START_REF] Helle | Life in varying environments: Experimental evidence for delayed effects of juvenile environment on adult life history[END_REF]Bowers et al., 2017). Nevertheless, if females respond to cues experienced whilst juvenile and before they disperse to find their own patch, the conclusion remains that factors due to maternal LMC influence a daughter's sex allocation.

Why, then should individuals use maternal cues, or cues available from the juvenile environment, rather than those present in the immediate environment to adjust the sex ratio of their offspring? It might be that our observed response is not adaptive. Indeed, maternal effects do not necessarily increase offspring fitness [START_REF] Sheriff | Determining the adaptive potential of maternal stress[END_REF]Barks & Laird, 2015). However, it might be adaptive if levels of LMC change in a predictable fashion across generations (for example, increasing or decreasing LMC) allowing a female to integrate information provided by mothers [START_REF] Mousseau | The adaptive significance of maternal effects[END_REF]. It has been hypothesised that T. urticae life-history follows predictable population groupings across multiple generations on the same host plant [START_REF] Nagelkerke | Hierarchical levels of spatial structure and their consequences for the evolution of sex allocation in mites and other arthropods[END_REF]. If this is the case, maternal cues may provide reliable information about the current state of a population, especially if relevant cues are not available in the patch. Mitchell (1972) showed that patches in experimental populations can be relatively small (1 cm 2 ) compared to the size of a leaf, and the patch sizes we used. Patch size and population structure remain to be measured in natural populations.

We can conclude from our experiment and others that multiple cues act together to influence sex allocation decisions, with the contribution of each depending on context. Maternal cues (maternal LMC or juvenile environment) may provide a more integrative measure of population structure, which females use for their sex allocation decisions at low densities. Clearly, more studies are needed to assess the relative value of cues informing on population structure and on the precise scale at which competition for mates actually occurs. differentiation of the female reproductive system.

Figure 1 .

 1 Figure 1. Proportion of male offspring for (A) females experiencing high (1 female) or low (15 females) LMC, (B) females whose mothers experienced high or low LMC and (C) females mated with their brother or an unrelated male. Figures show empirical means ± 2 standard errors (confidence interval).

  reviewed) is the author/funder. All rights reserved. No reuse allowed without permission.

ACKNOWLEDGEMENTS

We would like to thank David Carbonell for growing and maintaining plants for this experiment. This project was funded by a joint grant from the Agence Nationale de la Recherche and the Fundação para a Ciência e a Tecnologia to IO and SM (FCT-ANR/BIA-EVF/0013/2012). This is ISEM contribution number ISEM 2018-096.


Joseph Morlier

Hanno Niemann

Amir Shahdin

Topology optimization for robust damage localization using aggregated FRFs statistical criteria.

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(f ) ü + u + u = (t) k l H kl (Ω) = n i=1 ψ ik ψ il m i (ω 2 i -Ω 2 + 2jζ i ω i Ω) mm [(0/90/45/ -45) 3 ] s f P r p : Φ(χ) = f (χ) + P (r p , (χ), (χ)) χ e χ e = ρ e ρ 0 0 ≤ χ min ≤ χ ≤ 1 p q ω ω ǫ H Ω j g H j = |H kl (Ω j )| -|H kl (Ω j )| 2 < ǫ H j = 1, 2, ..., n H 8J n χ avg = n i=1 a i • χ i a i a i 1 a i = nω j=1 |ω j -ω j |

To link to this article:

  This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 12276 DOI: 10.4028/www.scientific.net/AMM.629.513 URL: http://dx.doi.org/10.4028/www.scientific.net/AMM.629.

513 To cite this version: Morlier, Joseph and Niemann, Hanno and Shahdin

  , Amir Topology optimization for robust damage localization using aggregated FRFs statistical criteria. (2014) AppliedMechanics and Materials, vol. 629. pp. 513-518. ISSN 1662-7482