Giulia Gastaldo

Younes Rafik

Marc Budinger

Valérie Pommier-Budinger

Philippe Olivier

Pablo Navarro

PROPOSITION OF A NEW QUALIFICATION IMPACT TEST FOR AEROSPACE SMART STRUCTURES

Keywords: BVID, Smart structures, PZT, Impact Nomenclature BVID Barely Visible Impact Damage CAI Compression After Impact FEM Finite Element Method PZT Lead Zirconate Titanate SHM Structural Health Monitoring

Piezoelectric materials are employed in several different aerospace applications thanks to their actuating and sensing abilities. They can be adopted to control vibration, stress distribution, aeroelastic stability, shape changes, and in structural health monitoring systems. Given the level of reliability required in aeronautics, it is of vital importance to ensure their correct functioning during all aircraft life and for critical events such as impacts. This paper proposes a novel test standard method to measure and qualify the damage resistance of smart PZT structures to a drop-weight impact event. Procedures for the quantification of damage through measurements of electromechanical characteristics are introduced and verified by impact tests.

Introduction

Piezoelectric materials are widely employed in aerospace applications, thanks to their ability to behave both as actuator and as sensors. In the last decade, they have been studied for aircraft [START_REF] Chattopadhyay | Aeroelastic tailoring using piezoelectric actuation and hybrid optimization[END_REF] [2], helicopter [START_REF] Grohmann | Actuation concepts for morphing helicopter rotor blades[END_REF] and UAV [START_REF] Vos | Post-buckled precompressed elements: a new class of flight control actuators enhancing morphing wing uavs[END_REF] actuation applications, for electromechanical ice protection systems [START_REF] Budinger | Electromechanical resonant ice protection systems: Energetic and power considerations[END_REF], as well as for structural health monitoring systems [START_REF] Staszewski | Health monitoring of aerospace structures: smart sensor technologies and signal processing[END_REF], [START_REF] Giurgiutiu | Structural health monitoring: with piezoelectric wafer active sensors[END_REF], [START_REF] Montalvao | A review of vibration-based structural health monitoring with special emphasis on composite materials[END_REF]. Despite the many benefits of adopting smart structures solutions, there are some critical aspects regarding piezoelectric transducers maintenance and functioning that must be taken into account. In fact, the deterioration or complete failure of the aforementioned transducers due to the external loading and environmental conditions negatively affect the performance of the system. Above all others, a threat to these embedded systems is the frequent occurrence of impact loads. When impact occurs, piezoelectric components may be subject to fracture, and this would lead to the degradation of their mechanical and electrical properties [START_REF] Liu | Losses in ferroelectric materials[END_REF], [START_REF] Elahi | Electromechanical degradation of piezoelectric patches[END_REF]. Moreover, the adhesive bonding between the PZT sensors and the structure may be compromised as well, thus deteriorating the overall functioning of the system. Therefore, it is of utmost importance to ensure the integrity of piezoelectric components.

Many studies focus on the real-time failure detection. For the general evaluation of PZT structural conditions, Friswell and Inman introduced in [START_REF] Friswell | Sensor validation for smart structures[END_REF] the concept of sensor validation for smart structures, which is the process of assessing the proper sensor functioning during operation. They were able to determine the functioning state of the sensors by comparing the subspace of the response and the one generated by the lower modes of the structure. Kershen et al. presented a sensor validation method based on the principal component analysis [START_REF] Kerschen | Sensor validation using principal component analysis[END_REF]. Another technique is based on the electromechanical impedance method firstly pioneered by Liang et al. [START_REF] Liang | Coupled electromechanical analysis of adaptive material system-determination of actuator power consumption and system energy transfer[END_REF] and Sun et al. [START_REF] Sun | Experimental modal testing using piezoceramic patches as collocated sensor-actuators[END_REF] and then adopted for the detection of faulty sensors in structural health monitoring systems in the work proposed by Giurgiutiu et al. firstly in [START_REF] Giurgiutiu | Electro-mechanical (e/m) impedance method for structural health monitoring and non-destructive evaluation[END_REF] and then in [START_REF] Giurgiutiu | Piezoelectric wafer embedded active sensors for aging aircraft structural health monitoring[END_REF]. Park et al. proposed a sensor diagnostic process based on electrical admittance measurement [START_REF] Park | Performance assessment and validation of piezoelectric active-sensors in structural health monitoring[END_REF]. They produced an impact in a composite plate, and they then evaluated the shift in the imaginary component of the measured admittance. In [START_REF] Sun | Closed-loop based detection of debonding of piezoelectric actuator patches in controlled beams[END_REF], Sun and Thong addressed the detection of debonding of piezoelectric actuators performing vibration control. The method is based on a control system which is sensitive to frequency changes due to the partial debonding of a PZT patch. Other changes can be measured to detect failures in piezoelectric patches. Wang et al. [START_REF] Wang | Dynamic piezoelectric properties of pzt-5h under shock compression[END_REF], [START_REF] Wang | Experimental research on dynamic response of pzt-5h under impact load[END_REF] evaluated the changes on the dynamic piezoelectric constants to detect impact in a single PZT sensor. In [START_REF] Elahi | Electromechanical degradation of piezoelectric patches[END_REF] [START_REF] Elahi | Response of piezoelectric materials on thermomechanical shocking and electrical shocking for aerospace applications[END_REF], the authors proposed a novel mechanism for characterizing smart structures using mechanical quality factor.

The approach developed in this article is different from real-time failure detection during operations. The proposed method aims to quantify the impact resistance of piezoelectric components before commissioning, in order to qualify and certify them for applications with high reliability requirements, such as aeronautical applications. This method is based on the current barely visible impact damage (BVID) test procedure combined with electromechanical characteristics measurements of smart structures. The novel procedure will be validated through experimental tests.

Extension of the BVID test for the qualification of smart structures

This section first recalls the BVID qualification test for composite laminates and then presents the proposed extension for the qualification of smart structures.

BVID test for the qualification of composite laminates

As previously mentioned, one sensitive aspect for aerospace structures is the occurrence of lowvelocity impact events. This phenomenon has been thoroughly studied for composite laminates, as they suffer from important reduction in their mechanical properties when a localized damage is present in the structure. This type of damage is often caused by low-velocity impacts with objects, producing local indentation, delamination, or fiber/matrix cracking, which are hardly detectable under visual inspection. Barely visible impact damage (BVID) can have serious consequences on structural integrity, especially if the damage size grows under cyclic loading [START_REF] Sudevan | Post-impact fatigue response of cfrp laminates under constant amplitude and programmed falstaff spectrum loading[END_REF]. It mostly occurs in the form of local delamination, and it has a critical effect on the strength of composite materials. To evaluate the effect of barely visible impact damage on composites, one of the most used indicators adopted is the Compression After Impact (CAI) strength [START_REF] Talreja | Assessment of damage tolerance approaches for composite aircraft with focus on barely visible impact damage[END_REF]. In fact, the CAI event can reduce the component's strength up to 60% in comparison to pristine samples [START_REF] Abrate | Modeling of impacts on composite structures[END_REF], [START_REF] Hawyes | Effect of impact damage on the compressive response of composite laminates[END_REF], [START_REF] Sanchez-Saez | Compression after impact of thin composite laminates[END_REF], potentially leading to the complete failure of the structure. Given its great impact on the mechanical performance, the CAI strength is considered a critical design parameter which has to be evaluated in order to assess the tolerance to BVID in composite laminates. For composite materials, the experimental studies are based on existing test methods and standards, such as the Airbus test method AITM 1-0010 [27] and the ASTM standard ASTM D7136 [START_REF]Standard test method for measuring the damage resistance of a fiber-reinforced polymer matrix composite to a[END_REF], which are specific for impact events. Once the impact tests have been carried out, the samples are further subject to compression after impact tests in accordance with ASTM D7137 [START_REF] D ; M-17 | Standard Test Method for Compressive Residual Strength Properties of Damaged Polymer Matrix Composite Plates[END_REF] and CAI strengths are then evaluated.

BVID test for the qualification of piezoelectric smart structures: parameters and procedure

To enable the study of impact resistance of smart structures, it is necessary to adapt the test procedure and propose indicators significant of the operation of smart structures. In this paper, smart structures under consideration are structures integrating electromechanical components and in particular piezoelectric transducers. Consequently, the indicators should be chosen for their ability to measure the electromechanical properties of smart structures.

First, let us analyze the electromechanical behavior of the smart structure. As it has been shown in previous articles by Pommier-Budinger et al. [START_REF] Pommier-Budinger | Analysis of piezoelectric ice protection systems combined with ice-phobic coatings[END_REF], [START_REF] Pommier-Budinger | Electromechanical resonant ice protection systems: initiation of fractures with piezoelectric actuators[END_REF], the performance of smart structures is linked to the value of the force factor characteristic of the electromechanical coupling. In case of piezoelectric materials, this parameter can be obtained from the electrical charges produced by a mechanical solicitation.

The reduced dynamical model of a resonant structure with one degree of freedom, thus having one piezoelectric actuator, can be written as:

M q + D s q + Kq = NV q c = Nq +C 0 V (1)
The system of equations is constituted of a mechanical equation and an electrical equation. q is the modal displacement of the selected point for the reduced model, M the modal mass, K the modal stiffness, D s the damping coefficient, N the force factor, q c the electrical charge, V the voltage, and C o the blocked capacitance. It is possible to compute [START_REF] Pommier-Budinger | Analysis of piezoelectric ice protection systems combined with ice-phobic coatings[END_REF] the force factor by calculating the electrical charge for a short circuit condition, V = 0 and using the second equation:

N = q c q (2)
The evaluation of the force factor evolution is certainly the most interesting parameter for assessing the performance of smart structures. Indeed, in the case of actuation systems integrated in the structure, a decrease in this factor results in a loss of actuation capacities, such less effort or less displacement. In the case of sensors integrated in the structure, a decrease in the force factor induces a lower signal-to-noise ratio.

Second, let us consider the equivalent electric scheme of a piezoelectric transducer, as represented in figure 1. It is constituted by two different parallel branches: a static branch, which represents the purely electric behavior of the piezoelectric material, formed by the capacitance C 0 and the resistance R 0 , and a second motional branch, which is constituted by a resonant RLC circuit. C 0 is proportional to the surface of the transducer, so a decrease of C 0 can represent a part disconnected from the electrode due to a crack. The measurement of the turned-off capacitance C 0 is thus interesting since it is a good indicator of the structural health of the piezoelectric transducer.

Figure 1 -Equivalent circuit [START_REF] Budinger | Contribution à la conception et à la modélisation d'actionneurs piézoélectriques cylindriques à deux degrés de liberté de type rotation et translation[END_REF] Note: the parameter representing the dielectric losses within the piezoelectric transducer R 0 is assumed greater than the resistance of the motional branch R s . Thus, R 0 will not be taken into account in the following sections.

To conclude, two characteristics of a smart piezoelectric structure interesting for quantifying its performance are the force factor and the turned-off capacitance. They will be considered as indicators of the structural integrity of the piezoelectric smart structure.

Measurement of performance indicators

The proposed method for the qualification of smart piezoelectric structures relies on an impact test and on the assessment of indicators which are representative of the performance of the smart structure. As explained in section 2. , the objective is to evaluate two characteristic quantities obtained through non-destructive tests:

• the force factor N

• the turned-off electrical capacitance C 0

The testing procedure will be detailed in the following paragraphs.

Force factor (N)

The force factor is measured at the resonant frequency of the smart structure at a given mode. The sinusoidal supply voltage signal must be maintained until the steady state is reached.

From equation 2 deriving q c and q with respect to time, it is possible to compute N using this formula:

N = I m q (3)
where I m is the current going through the motional branch of the piezoelectric transducer and q is the speed of a given point of the plate. For flexural modes, this point will be located on the plate surface, at an antinode of the selected mode. For an extensional mode, it will be located on one chosen edge of the plate.

The force factor N is different for every mode, and depends on the coupling of the piezoelectric transducer with the structure. This value is influenced by the location of the transducer and by the considered modal shape.The interest is to maximize N. For this purpose, if a flexural mode is considered, the piezoelectric transducer should be located at an antinode and the order of the mode should be selected to fulfill this condition. In our case, the PZT patch is located in the center of the plate, thus flexural symmetric modes are selected. On the contrary, for extensional modes, the PZT transducer should be located at a node of the mode shape.

Figure 2 -Test setup

The setup to realize the measurements is presented in figure 2. The piezoelectric transducer is excited by a sinusoidal signal at the resonant frequency of the selected mode. The frequency can be estimated using FEM models of the sample and determined precisely with a sweep around the estimated frequency. The speed at the maximum displacement of the mode is measured with the laser vibrometer positioned either to measure the displacement on the sample surface in case of flexural modes or the displacement on the sample edge in case of extensional modes. The motional current is measured using a power analyzer.

Electrical capacitance (C 0 )

A method to estimate C 0 is to measure the electrical capacitance of the piezoelectric component at low frequency, far from the first resonant mode. Indeed, as shown in figure 3, the capacitance is almost constant at low frequencies. For this measurement, an impedance meter is used. At low frequencies, it is possible to measure the turned-off capacitance as:

C 0 = lim ω→0 Im(Y (ω)) ω s (4)

Test results

In this section, the procedure proposed for the qualification of the resistance to impact of piezoelectric structures is assessed. One set of samples is tested and the measurement of the indicators is carried out.

The objective is to realize measurements in order to verify the evolution of the indicators N and C 0 according to impact energy. Furthermore, the results obtained from the proposed procedure will be compared with the ones from conventional tests to assess the better suitability of this method to qualify smart structures.

Impact test

The impact tests are carried out according to AITM 1-0010 [27] and ASTM D7136 [START_REF]Standard test method for measuring the damage resistance of a fiber-reinforced polymer matrix composite to a[END_REF] standards. These correspond to the BVID standard, of which figure 5 gives a schematic representation. In the BVID standard, the specimen to test is a 150x100 mm plate supported on a 125x75 mm window. The aluminum plate thickness is 2 mm, while the thickness for the composite plates is 4.2 mm. These dimensions are kept to test smart structures, but samples are equipped with piezoelectric components. The idea is to replace the compression after impact (CAI) test by the measurements of electromechanical properties to assess the integrity of the smart structure and quantify the tolerance to damage. For these measurements, the samples are removed from the impact machine and tested in free conditions to avoid dispersion due to the boundary conditions.

Samples characterization and testing

The experimental set is constituted by four different samples, which are shown in figure 6: • Sample 1: Aluminum plate + piezoceramic (PIC155)

• Sample 2: Composite plate + piezoceramic (PIC155)

• Sample 3: Composite plate + piezocomposite (DuraAct patch)

• Sample 4: Composite plate + piezofiber (Macro Fiber Composite MFC)

Figure 6 -Samples tested

The idea is to test the different technologies of piezoelectric actuators, illustrated in figure 7 : The piezoceramic corresponds to a bulk piezoelectric component.The piezocomposite is the DuraAct patch made by PI, which is a bulk ceramic encapsulated between two kapton layers, allowing a prestress in compression in the ceramic. The piezofiber is similar in structure to the piezocomposite, but the bulk ceramic is replaced by fibers of PZT. These Macro Fiber Composite patches are made by Smart Material company.

PROPOSITION OF A NEW QUALIFICATION IMPACT TEST FOR AEROSPACE SMART STRUCTURES

The measurement of the force factor N is carried out for 3 modes, 2 symmetric flexural modes and 1 extensional mode, shown in figure 8. The goal is to measure the evolution of the indicators N and C 0 according to the impact energy. An iterative approach is used (figure 9). Initially, low values of impact energy are tested (0.2 J or 0.5 J) and these values are increased until a noticeable drop in the actuator performance is detected. For this sample, a significant drop for N and C 0 is noticed after an impact of 2 J, which indicates damage in the smart structure. The measurement of C 0 is easier to repeat than the one for N. A first conclusion from these results is that there is no interest in measuring N for several modes as N decreases for all the modes. Thus, it is preferable to select only one mode which will allow an easy computation of the force factor. The measurement for extensional modes is difficult, as the laser has to be positioned in order to measure the displacement at a point located on the edge of the plate (2 or 4.2 mm thick). As a consequence of these remarks confirmed with the tests of the other samples, it is possible to conclude that the measurement of N for the first flexural mode is sufficient and recommended since it is the easiest and the most accurate way to measure the performance of the actuator.

-43% -4% -2%
Sample 2 is equipped with a bulk piezoceramic transducer. The ceramic material and the electrodes on top and bottom are brittle. This transducer is the most sensitive to impact as N and C 0 dropped for a value of impact energy of 0.5 J. For this transducer, if a cohesive crack is present, only the connected part will be operating. The loss of C 0 is proportional to the lost surface. Also, for the tests performed in this study, the ceramic is fixed to the composite substrate with an electrically conductive glue having lower mechanical properties compared to a typical epoxy glue. Therefore, it is easier to detach the transducer, which results in a lower N.

For sample 3, the piezocomposite is a piezoceramic encapsulated between two kapton layers. The fabrication process allows a prestress in compression that increase the maximum admissible stress in the ceramic. Also, the piezocomposite is fixed to the substrate with an epoxy resin with enhanced mechanical properties. For these reasons, sample 3 has a higher impact resistance and a noticeable drop of N is observed only after an 8 J impact. The electrode of the piezocomposite transducer is meshed, more resilient to impact compared to the electrodes of the piezoceramic patch. This explains why C 0 decreased less than N.

In sample 4, the piezofiber patch has a mesh for both the electrode and the ceramic parts, increasing once more the impact resistance. The impact at 10.5 J results in the damage shown in figure 12 (a). However, despite the visible damage from the impact, the actuation function is only very slightly deteriorated as shown in figures 13 and 14. The last impact for this sample is performed at 15.7 J. As shown in figure 12 (b), the crack propagates further. The plate performances are evaluated, but the test resulted in a shortcut that prevents from retrieving any results. This is probably due to the loss of the whole piezoelectric transducer and the consequent deterioration of the contact between the plate and the PZT actuator. Considering the complete loss of electromechanical properties at 15.7 J, it is assumed that the impact resistance of the structure falls in the interval between 10.5 J and 15.7 J.

Comparison of the results with other measurement methods

Tomography A tomography was realized on sample 4 after the impact at 8 J in order to scan through the thickness of the sample. Figure 15 illustrates the image given by the tomography inside the piezofiber transducer. The tomography shows the cracks in the piezofiber patch. It is possible to notice that the main crack is vertical with respect to the impact location, and more cracks can be found at ±45 • . After the 8 J impact, the piezofiber actuator was still healthy as N and C 0 respectively dropped by only -2% and -1%. This indicates that the tomography is too pessimistic for assessing the electromechanical performances of piezoelectric actuators after an impact.

CAI test Compression after impact (CAI) tests were performed on samples 2 and 3 using a pulling machine. Figure 16 illustrates the loading curves applied to samples 2 and 3. The maximum load applied on the sample corresponds to its resistance. It is possible to notice that, despite the different impact energy applied to the plates (0.5 J for sample 2, 8 J for sample 3), the resistance is close in value. Therefore, CAI test is not suitable to evaluate the functional damage on the piezoelectric transducers, as it is mostly influenced by the substrate structural health. The evaluation of the electromechanical quantities allows the detection of the changes in performances for the two specimens, which are mostly influenced by the type of piezoelectric actuator mounted on the composite substrate.

Conclusions

This paper aims at proposing a novel test method to measure and qualify the damage resistance of smart PZT structures to a drop-weight impact event. The method is based on the BVID method for composite materials subject to impact and adapted to smart structures. It relies on the measurements of the electromechanical characteristics of the piezoelectric transducer, the force factor N and the capacitance C 0 both in undamaged and damaged conditions. The damage is detected when a decrease of these properties occurs. Four samples considering different substrates and different actuator technologies are tested, allowing a comparison of impact resistance. A first conclusion is that the bulk piezoceramic patch is the most brittle (samples 1 and 2). The samples 3 and 4 show an improvement in the performances when adopting piezocomposites and piezofiber technologies, which are more resistant than the piezoceramic components. The piezofiber transducer has shown the highest impact resistance, but it is also more expensive if compared to the other actuators. For industrial applications, a tradeoff between the performance and the cost of application is necessary.

The approach is compared with other measurement methods, the compression after impact (CAI) test and the tomography. Tomography results appear to be too pessimistic, as they indicate damage before the degradation of the actuator electromechanical performances. CAI tests are not suitable either, as the compression resistance depends mostly on the substrate structural condition and does not measure the health of the piezoelectric transducer.

In future studies, the proposed qualification test could be further validated by using a single-impact test instead of increasing the energy step by step to assess the resistance of the smart structures. The value of the impact energy would be chosen based on the results obtained in the present study.

Figure 3 -

 3 Figure 3 -Evolution of the capacitance according to the frequencyThe setup for the measurement of C 0 is shown in figure4. The piezoelectric transducer is plugged in the impedance meter and the plate is in free boundary condition.

Figure 4 -

 4 Figure 4 -Setup for the measure of C 0

Figure 5 -

 5 Figure 5 -Schematic representation of the BVID standard ([27])

Figure 7 -

 7 Figure 7 -Technologies of piezoelectric transducers tested

Figure 8 -

 8 Figure 8 -Tested modes

Figure 9 -

 9 Figure 9 -Testing approach

Figure 10 - 1 Figure 11 -

 10111 Figure 10 -Evolution of N for sample 1

( a )Figure 12 -Figure 13 - 4 Figure 14 -

 a1213414 Figure 12 -Sample 4: damage to the piezoelectric transducer after impact

Figure 15 -

 15 Figure 15 -Tomography of sample 4 after 8J impact

Figure 16 -

 16 Figure 16 -Loading curves for samples 2 and sample 3

Table 2 -

 2 Impact resistance -Samples 2,3,4

		Sample 2 Sample 3 Sample 4
	Impact resistance (J) 0.5	8	10.5
	∆N	-53%	-23%	-2%
	∆C 0

Contact Author Email Address

mailto: giulia.gastaldo@isae-supaero.fr mailto: rafik@insa-toulouse.fr

Acknowledgments

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 956703 (SURFICE Smart surface design for efficient ice protection and control).

Copyright Statement

The authors confirm that they, and/or their company or organization, hold copyright on all of the original material included in this paper. The authors also confirm that they have obtained permission, from the copyright holder of any third party material included in this paper, to publish it as part of their paper. The authors confirm that they give permission, or have obtained permission from the copyright holder of this paper, for the publication and distribution of this paper as part of the ICAS proceedings or as individual off-prints from the proceedings.


29,004 pQTLs were dected by GWAS for 95% of the 3,900 (=1,950 proteins x 2 conditions) molecular phenotypes. Only 8.4% of the pQTLs detected in the WW condition had a co-localizing pQTL in the WD condition, suggesting that water deficit has deeply impacted the genetic regulation of protein expression (Fig. 3).

Water deficit has extensively remodeled the proteome Identification of genomic regions involved in multi-scale genetic control

About 75% of the 1,950 quantified proteins showed significant abundance changes in response to water deficit (Fig. 2). These included several stress and drought-response proteins. The evolution of maize yields under drought is of particular concern in the context of climate change and human population growth. To better understand the mechanisms associated with the genetic polymorphisms underlying the variations of traits related to drought tolerance, we used a systems genetics approach integrating high-throughput phenotypic, proteomics and genomics data acquired on 254 half-sibs maize hybrids grown under well watered (WW) and mild water deficit (WD) conditions (Fig. 1). By using a systems genetics approach including MS-based proteomics data, we highlighted several original results. We showed that water deficit, even mild, induced a strong proteome remodeling and a reprogramming of the genetic control of the abundance of many proteins, notably those involved in stress and drought responses. We identified close co-localizations between QTLs and pQTLs in the WD condition, thus highlighting environment-specific pleiotropic loci associated to the co-expression of droughtresponsive proteins and to the variations of phenotypic traits. These findings bring several lines of evidence supporting candidate genes at many loci and provide novel insight into the molecular mechanisms of drought tolerance.

Co-expression networks were build from protein abundances in each environment, showing that stress proteins were specifically co-expressed under WD (Fig. 4).

Proteins Genotype x condition combinations

369 co-expression QTLs (coQTLs obtained by submitting the abundance values representing each co-expression module to GWAS) were detected. Co-localizations between QTLs, pQTLs, and coQTLs indicated the presence of genomic regions involved in the genetic control of traits at different levels of biological complexity and highlighted promising underlying candidate genes (Fig. 5).

Fig 2 :

 2 Fig 2: Heatmap representation of protein abundances showing that water deficit has extensively remodeled the proteome.

Fig 3 :

 3 Fig 3: Example of the Manhattan plots obtained in the WW (A) and WD (B) conditions for a same protein. Black arrows indicate pQTLs detected in only one of the two conditions. Stars indicate the position of the protein encoding gene.

Fig. 4 :

 4 Fig. 4: Representation of the co-expression networks in WW (A) and WD (B). Functional enrichments and condition specificities are indicated.

Fig. 1 :

 1 Fig.1: Schematic representation of the systems genetics approach. Protein abundances were first analyzed by using genome wide association study (GWAS) and co-expression networks. Then, they were integrated with phenotypic data measured for drought-related trait in the same conditions through correlation analysis and search for QTL/pQTL co-localizations.

  Fig. 5. Detailed view of the QTLs, pQTLs, coQTLs and promising candidate genes detected in a region of chromosome 5 representing a hotspot of protein expression regulation. Dots represent the SNPs determining the position of the QTLs and horizontal bars represent the linkage disequilibrium-based window around each SNP.