J P Goulmy

E Rouhaud

P Kanoute

L Toualbi

S Kruch

V Boyer

J Badreddine

D Retraint

A calibration procedure for the assessment of work hardening part I: Effects of the microstructure and load type

Keywords: Inconel 718, work hardening, calibration, XRD, EBSD

This paper presents a methodology to define and quantify the level of work hardening locally in a material. The methodology is proposed after a thorough experimental study based on three complementary experimental techniques for microstructural characterizations: microhardness, X-ray diffraction (XRD) and Electron Backscatter Diffraction (EBSD) applied on Inconel 718 samples. In our analysis, several loading histories including single tension, single compression, high strain rates and low cycle fatigue have been investigated. The effects of the microstructure have been further investigated by modifying the size of the grains and the size of the strengthening precipitates.

Experimental tests have also been simulated to choose a model variable able to represent work hardening. A reciprocal link between work hardening and experimental characterizations has then been established. Correlation curves have been proposed that enable to quantify the level of work hardening from the knowledge of the experimental data. Accuracy and complementarity of the three experimental approaches are discussed as well as the impact of the microstructure of the material on the measured quantities.

Introduction

In order to accurately predict fatigue behavior and crack initiation in metallic components, it is fundamental to possess a realistic picture of the initial state of the material. The processes used to manufacture the parts have indeed introduced plastic deformation, work hardening and also may have changed the local microstructure of the metal. The strain incompatibilities hence generated, have further induced a residual stress field. Residual stresses are not the only quantity that influences fatigue life *1+-*9+, another relevant quantity being the level of work hardening accumulated in the material. Indeed, for the same amount of residual stress observed locally, the level of work hardening may vary. It is further observed that the evolution of the mechanical fields during thermo-mechanical cyclic loadings, the evolution of the stress state, in particular, is strongly influenced by this initial work hardening level as detailed by Prevey *1+. Thus, the initial state to be considered has to be specified, not only with the knowledge of the residual stress field, but also with the knowledge of the consequences of the load history of the material during manufacturing. In the end, both the residual stress field and work hardening, have to be taken into account to define an initial state that is far from being the "neutral configuration" that one would wish to consider to evaluate fatigue.

The term "hardening" has several (related) meanings depending on the communities. It is a term used to refer to the increase in yield stress to refer to model types and variables (like for example isotropic or kinematic hardening…), or to describe the level of disorder in a local microstructure.

Further, work hardening and its effects are not only a function of the amplitude of the applied load, but seem to be influenced by other factors like, for example, the imposed strain rate *10+. This quantity is also strongly impacted by the local microstructure of the material that may further present a gradient in grain size, precipitate size, and that may also evolve in service *1+, *11+, *12+. However, open questions remain concerning the level of work hardening reached in the material as a function of the initial microstructure and during the life cycles of a component. It is particularly interesting to investigate the influence of the local microstructure in regard to the temperature and the strain rates endured by the material.

The purpose of this paper is thus to propose a methodology to define and quantify the level of work hardening locally in a material, taking into account the influence of the type of loading, the temperature, the strain rate and the microstructure. The methodology is proposed after a thorough experimental study based on three different experimental techniques: microhardness, XRD and EBSD applied on Inconel 718 samples. This material is widely used for aeronautical applications such as high-pressure turbine disks. The knowledge of the mechanical state is of primary importance for fatigue life estimation. An original method, allowing defining work hardening with respect to a variable of a behaviour law, is then presented. For this, numerical simulations of experimental tests are performed. A reciprocal link between this variable and the experimental characterizations is then established. The range of applicability of the procedure along with the influence of the experimental parameters taken into account are finally discussed.

Methodology

It is not possible to directly measure work hardening experimentally. Indeed, this term characterizes a set of physical quantities and mechanical phenomena related to the level of disorder reached in the microstructure of the material. In order to build a calibration procedure able to evaluate work hardening, four steps have been followed as described in the next paragraphs:

J o u r n a l P r e -p r o o f First, a large set of mechanical tests has been performed to evaluate the influence of the following parameters:

-The type of loading: Classical uniaxial tensile tests have been performed along with cyclic tests and compressive tests. The objective was to investigate different types of loadings at various levels of plastic deformation. The compressive tests enabled to reach higher strain rates. Fatigue tests have also been performed to evaluate the influence of cyclic loading.

-The temperature:

Two temperatures have been investigated the ambient temperature (20°C) and 550°C. The latter has been chosen to represent the life cycle conditions of the material and to enable comparison with experimental data available in the literature *21+.

-The strain rate:

A quasi-static strain rate of 10 -3 s -1 has been chosen for tensile and compressive tests. The cyclic tests have then been performed with the same 10 -3 s -1 for comparison. Compressive tests with higher rates have been finally completed to consider conditions representative of what could be encountered during manufacturing processes.

-The microstructure: Three microstructures have been investigated to evaluate the influence of grain and precipitate size on work hardening and the sensibility of the calibration procedure.

J o u r n a l P r e -p r o o f

The second step consists in identifying experimental quantities able to represent work hardening in the material. Three experimental methods: microhardness, XRD and EBSD, have been applied to the samples for characterization. Several quantities have been measured that are related to the level of defects in the microstructure. These quantities are the microhardness, the peak width of the X-ray diffracted signal (FWHM, full width at half maximum), the kernel average misorientation (KAM) issued from EBSD measurements. Both the intensity of each quantity and its evolution with the load have been monitored. These three experimental quantities have been chosen because they give a complementary evaluation of the state of the material. The experimental conditions for these three techniques are detailed in section 2.3 along with their definitions and the justification for this choice.

Table 1 presents the type of experimental measurements performed on each sample.

We have next identified a model variable able to quantify work hardening. The third step consists thus in modelling each of the mechanical tests performed on the different samples, as described above in step 1. An elasto-plastic model has been chosen to describe the material and several variables have been identified as potential candidate to represent work hardening in the material.

The numerical results obtained after computation have been correlated with the experimental data obtained in step 2. Then, the model variable that best correlates the experimental results has been selected to represent work hardening in the material. This is presented in section 3.

Finally, a correlation between each experimental quantity and the variable chosen to model work hardening has been established. A set of functions has been built, fitting the data (see section 4). This set of functions can then be used to correlate experimental measurements in a given material to the variable chosen to represent work hardening.

In the end, we dispose of a calibration method to evaluate work hardening locally in the chosen material.

J o u r n a l P r e -p r o o f

Inconel 718

All the samples have been extracted from an Inconel 718 Direct Aged (DA) turbine disk, the composition of which is given in Table 2. Three microstructures of this alloy have been investigated.

The first microstructure (named Direct aged microstructure in the following) is the reference microstructure, obtained directly after forging and corresponding to the "as-received" material. The two other microstructures correspond to the reference microstructure further modified by applying J o u r n a l P r e -p r o o f either one of the following heat treatments:

-The second microstructure is a coarse grain microstructure obtained through an annealing at 1040 °C during 30 minutes (temperature above the δ phase solvus), followed by a δ phase precipitation treatment at 955 °C for one hour, next followed, by the conventional γ'+γ'' precipitation treatment, performed at 720 °C during 8 hours, itself followed by a cooling at 50 °C/h and finally, an aging at 620 °C for 8 hours.

-The third microstructure includes coarse grain and coarse strengthening precipitates obtained with the same heat treatment sequence as the second one, except for the final precipitation treatment of the γ' and γ'' phases: an over aging at 750 °C for 50 hours is applied instead, in order to modify the size of the precipitates.

Note that a color code is used to reference the three microstructures as presented in Table 3; this code will be used throughout this article.

The characterization of the microstructures has been carried out by scanning electron microscopy observations and backscattered electron analyses (see Table 3). The mean grain size and the mean size of the strengthening precipitates of these three microstructures are further detailed in Table 3. A grain size of about 5 µm is obtained for the direct aged microstructure while grains of about 35 µm are obtained for the two other microstructures. The strengthening precipitates have similar sizes in the direct aged microstructure and the coarse grain microstructure (10 to 20 nm), whereas the size of the strengthening precipitates is ranging from 100 to 200 nm for the coarse grain and coarse strengthening precipitate microstructure. In addition to the strengthening precipitates, Inconel 718 is composed of a δ phase, a stable version of the γ'' phase that controls the grain size. Nitrides and carbides are also present and have a size in the order of 10 to 15 µm. Note that heat treatments performed to modify the microstructure do not influence the presence of the nitrides and carbides.

The delta phase, which is dissolved during the annealing at 1040°C, is re-precipitated with a size and a volume fraction which is similar to the initial ones.

J o u r n a l P r e -p r o o f

Thus, comparing the reference microstructure (direct aged microstructure) and the coarse grain microstructure enables to evaluate the influence of the grain size. The comparison of the coarse grain microstructure with the coarse grain and coarse strengthening precipitate microstructure enables to evaluate the influence of the size of the hardening precipitates.

Mechanical tests

The samples have been submitted to different types of loading able to generate plastic deformation.

During the tests, the mechanical fields are controlled to remain homogenous throughout the gauge length of each sample. Precautions have thus been taken to insure that the samples are free of residual stresses at the end of the test. Stress and strain levels were monitored as a function of time throughout the tests. The experimental settings, specific to each applied load, are detailed below.

The uniaxial tensile tests and fatigue tests have been performed on a MTS machine with a force cell of 10 kN, a controlled strain rate of 10 -3 s -1 and an extensometer to determine the strain. The samples were cylindrical with a diameter of 4.37 mm. For the tensile tests, five levels of plastic deformation have been investigated; Table 4 presents the values that has been reached in each sample. Figure 1 presents the stress strain curves for each test. The cyclic tests were strain controlled and performed with the same rate as the tensile tests, for two levels of plastic deformation as specified in Table 1 and for a strain ratio equal to zero.

The quasi-static compressive tests have been performed on an Instron 5584 machine with a force cell of 150 kN. The samples were cylindrical specimens with a diameter of 6 mm and a height of 6 mm.

The strain rate was controlled at 10 -3 s -1 . The results are presented in Table 4.

Material characterization

After deformation, the different samples have been characterized using microhardness technique, J o u r n a l P r e -p r o o f XRD and EBSD. The experimental conditions for these measurements are presented in the following sections.

Microhardness measurement technique

This technique, relatively easy to implement, is sensitive to several parameters and in particular to the microstructure, the work hardening level and the residual stresses in the material. Microhardness has long been used to evaluate the yield strength of polycrystalline materials *23+-*25+. The link between microhardness and yield strength could be explained by strain hardening and the associated increase in dislocation density that raises the resistance to plastic strain.

The samples have been extracted from the test specimens by cutting; it has been verified that they were free of residual stresses. All cross sections have been prepared by mechanical polishing up to 1 μm with diamond paste followed by colloidal silica in order to achieve mirror polished flat surfaces free from damage.

The tests have been performed on a BUEHLER device using a Vickers indenter (pyramidal shape) with a load of 100 grams for 10 seconds. The observation of the indents showed that their diagonal was 20 m on average. This load was chosen to obtain indentation sizes allowing the calibration method to be used to evaluate microhardness gradients at depth. To evaluate the hardness of a given sample, 10 indents have been performed on its surface and averaged. The state of the material is thus evaluated on a volume of approximately 1300 µm 3 . This volume includes more than one grain for the DA microstructure and less than one for the coarse grain microstructure and coarse grain and coarse strengthening precipitate microstructure. The results are presented in section 4.1. The sample preparation was identical to the one performed for X-ray diffraction and microhardness measurements followed by a final ion polishing to eliminate any strain hardening induced by previous mechanical polishing steps.

X-ray Diffraction

Electron BackScatter Diffraction

The data has been collected using a NORDIF CD camera operating at 30 fps coupled to a MERLIN Scanning Electron Microscope operating at an accelerating voltage of 20 kV and a nominal current of 40 nA. The square pattern size corresponds to 1024 x 512 pixels for 512 x 256 µm 2 fields, with a step size of 0.5 µm. Under these conditions:

-10 pixels are dedicated to each grain on average, for the finest microstructure (remember that the finest microstructure has a mean grain size of 5 µm),

-20 grains are investigated in the width of the EBSD window on average for the overall field for the coarse microstructure (remember that the coarse microstructure has a mean grain size of 40 µm).

The chosen resolution is thus sufficiently refined to enable accurate measurements and analyses.

Further, the influence of the parameter N in the KAM computation has been evaluated. The value of N has been varied and the results show that there is no influence of this parameter in the present

Choice of a model variable to evaluate hardening

The expression work hardening describes a complex reality associated to the set of physical phenomena observed in metals when submitted to irreversible deformations. To establish a calibration procedure, it is necessary to select a model quantity that is able to capture enough of these phenomena to be able to characterize a work hardening state. The objective of this section is thus to explore different classical variables defined in constitutive models for plasticity in order to test whether they could be qualified to quantify work hardening. All the experimental tests described in Table 1 have thus been numerically simulated and the results correlated with the experimental measurements.

Model for plasticity

The chosen constitutive model is an elasto-visco-plastic model proposed by Chaboche et al. *21+. In this model, the total strain tensor 𝜺 is additively decomposed into the elastic 𝛆 𝒆 and plastic 𝛆 𝒑 strain tensors. The elastic domain is defined by f≤0 in the stress space  with:

J o u r n a l P r e -p r o o f f=J(-X)-R Eq. 1

where the function J returns the von Mises invariant, R indicates the size of the instantaneous yield condition and X is the back stress tensor. The plastic strain rate tensor 𝜺̇𝒑 is obtained with the flow rule, using the normality assumption:

𝜺̇𝒑 = 𝝀 ̇𝛛𝒇 𝛛𝝈 Eq. 2
where the dot represents the differentiation with respect to time and the plastic multiplier 𝜆 ̇ is given by the Norton power law:

𝝀 ̇ = 〈 𝒇 𝑲 〉 𝒏 Eq. 3
To model visco-plasticity. The equivalent plastic strain rate 𝑝̇ is:

𝒑̇ = √ 𝟐 𝟑 𝜺̇𝒑: 𝜺̇𝒑 Eq. 4
A non-linear isotropic hardening R is defined as the sum of several contributions indexed with "i":

𝑹 = ∑ 𝑹 𝒊 𝒊 𝒘𝒊𝒕𝒉 𝑹 ̇𝒊 = 𝒃 𝒊 (𝑸 𝒊 -𝑹 𝒊 )𝒑̇ Eq. 5
where the b i and Q i are material parameters associated to contribution i. Similarly, the back stress X is decomposed as the sum of several contributions, indexed with "k": and where the 𝜓(𝑿 𝑘 ) is:

𝝍(𝑿 𝒌 ) = 〈 𝑫 𝒌 𝑱(𝑿 𝒌 )-𝝎 𝒌 𝑪 𝒌 𝟏-𝝎 𝒌 〉 𝟏 𝑫 𝒌 𝑱(𝑿 𝒌 )
Eq. 8

where 𝐶 𝑘 , 𝐷 𝑘 and ω k are material parameters associated to each contribution k. Note that the functions 𝜓(𝑿 𝑘 ) contain a threshold effect.

Candidate variables to represent work hardening

Simulation

Each experimental test (see Table 1) has been simulated using the model detailed above with the material parameters for Inconel 718 given in *21+. Note that the fields are homogeneous in the test volume of the samples. Thus, the simulations reduce to solving numerically the chosen constitutive model at a given point of the test volume, given a temperature and loading history identified with the results of the experimental campaign. The software Zset (codeveloped by Onera and Ecole des Mines de Paris) has been used for these simulations *31+.

Results

The values obtained for each selected model variable have been correlated with the FWHM of the diffracted X-ray peaks and the KAM parameter computed from the EBSD measurements, and the harness values. All the tests performed in the experimental campaign are uniaxial. It is thus possible to correlate the experimental data and the model variable for a given plastic deformation. Figure 3 presents respectively the FWHM, KAM and the microhardness obtained for each sample as a function of the four values of the variables identified to represent work hardening and resulting from the simulations. These results are presented for the DA microstructure and for all types of loading; the J o u r n a l P r e -p r o o f loading is identified with the color of each symbol as defined in Table 1, one symbol corresponding to one test sample.

First, it is remarkable to note that, although the quantities that are evaluated by FWHM, by KAM and by the microhardness are different, the comparison between these quantities for each respective variable leads to very similar results. These figures further show that it is not possible to construct a reverse method using isotropic hardening and cumulative plastic strain. Indeed, for the cyclic tests, the cumulative plastic strain is, logically, more important than in the other samples but the FWHM and KAM remain in the lower range of the graph. For isotropic hardening, the values obtained for the cyclic tests and the compressive tests are very close, whereas the FWHM and the KAM clearly lead to distinct evaluations. The values of the von Mises plastic strain and the equivalent kinematic hardening make it possible to distinguish between all the experimental data.

Both thus seem to be able to quantify a work hardening state. In this study, we chose the von Mises plastic strain because it is equal to the plastic deformation in tensile tests.

J o u r n a l P r e -p r o o f

Calibration curves for work hardening

The analyses of the results presented in section 3 lead to the choice of the von Mises equivalent plastic strain to represent the state of work hardening in the samples. It is referred as "the equivalent plastic strain" for simplicity in the rest of this article. This section presents the correlation between the three experimental quantities (microhardness, FWHM and KAM) with this equivalent plastic J o u r n a l P r e -p r o o f strain for all the samples of the experimental campaign: see Figures 4, 5 and7. Note that each point on these figures corresponds to one mechanical test of Table 1. The results are analyzed and correlation curves are proposed up to 50% of equivalent plastic strains, fitted to the experimental data with a least-square method. Based on the work of *16+, a power law has been chosen to fit the experimental data presented in Figure 4: 𝐻𝑣 0,1 (𝜀 𝑝 ) = 𝐻𝑣 0,1 (0) (1 + 𝐴𝜀 𝑝 𝐵 ) Eq. 9

Microhardness

where 𝐻𝑣 0,1 (0) is the microhardness in the sample without plastic deformation. Because of the dispersion of the microhardness values, the fit has been performed considering that the three microstructures followed the same power law. With this hypothesis only the Hv 0.1 value is function of the microstructure. The values of the parameters obtained after the fitting are given in Table 5. In Figure 4, the curves representing the microhardness have been extrapolated up to 50% of equivalent plastic strain (dotted lines) with the function established by the fitting to allow a correlation at high equivalent plastic deformations.

J o u r n a l P r e -p r o o f It is thus possible to build a correlation curve able to relate the microhardness to the equivalent plastic strain as a function of the microstructure of the material. Due to the dispersion of the measurements obtained during the tests, the use of the calibration curve to deduce an equivalent plastic strain from a measured microhardness is not the most reliable. One of the issues is the size of the indent, and thus the size of the volume analyzed by the method. In the present study, the indent has a rather small size, the analyzed volume being approximately of the same size as the grains for the coarse microstructure and the coarse microstructure with strengthening precipitates. This fact might explain the irregularity of the measured hardness that, in this case, is more sensitive to the position of the indents (over a grain boundary or in the center of a grain). Also, the microhardness gives a global evaluation on the material state below the indent, sensitive to the hardening state but also to the residual stresses *32+. All the samples tested in this calibration study were free of residual stresses. If residual stresses are present in the part, the use of the microhardness to evaluate the work hardening level might be more problematic. For all these reasons, the microhardness measurements were performed only on the samples submitted to tensile tests at 20°C. Nevertheless, this technique is relatively easy to perform and further investigations on its capacity to evaluate the hardening state of the material might be of interest.

J o u r n a l P r e -p r o o f

FWHM from X-ray Diffraction

The correlation between the FWHM obtained by X-ray diffraction and the equivalent plastic strain is presented in Figure 5. The results show that the FWHM increase with the plastic strain and that this is function of the strain rate and the microstructure. The increase of the FWHM is important for low equivalent plastic strains (below 2%) and becomes moderate and linear for higher strains. For the DA microstructure and for a strain rate of 10 -3 s -1 , the temperature and loading conditions (tensile, compressive or fatigue tests) have no influence on the measured FWHM value. Indeed, experimental points resulting from these tests induce a common evolution (thin blue line). In contrast, for a given equivalent plastic strain, the FWHM is higher when the strain rate increases (thick blue line). The measured FWHM is sensitive to the strain rates at which the tests have been performed in this study.

Further, the two modified microstructures present a similar evolution of the FWHM that is smaller than the FWHM obtained for the DA microstructure for a given equivalent plastic strain. Figure 5.b

shows that the FWHM also seems to be slightly sensitive to the intragranular microstructure, i.e. to the size of the strengthening precipitates.

J o u r n a l P r e -p r o o f A function including an exponential increase followed by a straight line has been chosen to fit the data for calibration with the FWHM values *1+:

𝑭𝑾𝑯𝑴(𝜺 𝒑 ) = 𝑨 𝑭𝑾𝑯𝑴 (𝟏 -𝒆 -𝑩 𝑭𝑾𝑯𝑴 𝜺 𝒑 ) + 𝑪 𝑭𝑾𝑯𝑴 𝜺 𝒑 + 𝑫 𝑭𝑾𝑯𝑴 Eq. 10

where A FWHM , B FWHM , C FWHM and D FWHM are parameters that have been evaluated for each microstructure; the results are given in Table 6. Two fits have been established for the DA microstructure to take into account the sensitivity to the strain rate. For the microstructure with coarse grains and coarse strengthening precipitates, the straight line for high equivalent plastic strains has been extrapolated parallel to the one corresponding to the coarse grain microstructure. The FWHM value is driven by several factors related to microstructural aspects: grain size, strengthening precipitates size and volume fraction. At a microscopic scale, the size of the coherent domains is preponderant: the smaller they are, the wider the diffraction peak. At a mesoscopic scale, the dislocation arrangements and the several phases impact the FWHM value. Finally at a macroscopic scale, the crystallographic structure and the grain size are the main driving factors *26+.

The microstructure has then a major influence on the FWHM values. Therefore a microstructure dependent work hardening calibration is needed for the XRD method as demonstrated in Figure 5.

It should also be noted that X-ray diffraction methods enable to evaluate residual stresses with the shift of the peak, independently of FWHM measurements. This is a real advantage compared to the microhardness measurements.

KAM from Electron BackScatter Diffraction

KAM maps obtained for single tension, single compression and cyclic tests performed at 20°C on the three microstructures are presented in Figure 6 for several levels of equivalent plastic strain. An evolution of the cartography is observed as a function of the plastic deformation. On this scale, there are no significant differences in the distribution of KAM values in the map according to the type of solicitation applied (cyclic test, tensile test and compressive test). On the other hand, the equivalent plastic strain reached in the sample has a major impact on the KAM maps: the higher the strain, the higher the KAM values are. Moreover, the local misorientation tends to increase near grain J o u r n a l P r e -p r o o f boundaries for the three microstructures; it is quite remarkable on coarse grain microstructures as it has also been observed in previous studies *14+, *30+.

Figure 6 demonstrates that, even if the KAM cartography has a size that includes several grains, it is nevertheless possible to extract a global information dependent on the equivalent plastic strain. This is related to the fact that each cartography is different from the other and is rather homogeneous for a given plastic deformation. The average value of the KAM parameter has thus been computed for each cartography and appears to be a meaningful quantity. The relationship between the mean KAM parameter and the equivalent plastic strain is presented in Figure 7 for single tension, single compression and cyclic tests performed at 20°C, for the three studied microstructures. As observed previously on the detailed maps of Figure 6, the mean value of the KAM parameter increases as the plastic deformation rises. For the three microstructures, the most significant increase in the mean value of the KAM parameter occurs for a value of the equivalent plastic strain between 0 and 15%. After 15%, the evolution of the KAM parameter is less J o u r n a l P r e -p r o o f pronounced for the Direct Aged microstructure.

The two microstructures featuring coarse grains present the same evolution for the KAM parameter as a function of the equivalent plastic strain, smaller than the KAM values obtained for the DA microstructure. Thus, the grain size seems to be a predominant factor in the EBSD calibration. These results are to be compared with the density of grain boundaries, that is higher in the case of the DA microstructure with a smaller grain size. Indeed, since the values of the KAM parameter are higher close to grain boundaries, the mean KAM values are necessarily more important for a microstructure with a higher grain boundary density, I.e. a smaller grain size.

The difference in the values of the KAM at 0% of plastic strain could be related to the thermomechanical history of the samples that is different between the DA microstructure and coarse grain microstructures. Indeed, in the case of coarse grain microstructures, an annealing at 1040 °C during 30 minutes has been performed, generating a decrease in the plastic deformation induced during the manufacturing stages performed to produce the reference microstructure (notably during forging).

J o u r n a l P r e -p r o o f

Eq. 11

Where A EBSD , B EBSD , C EBSD and D EBSD are the parameters to be determined with the fitting. A EBSD is the value of the KAM at the origin, for a value of the equivalent plastic strain equal to zero, B EBSD is the initial slope, C EBSD corresponds to the position of the transition region and D EBSD corresponds to the positions of the asymptote for large equivalent plastic deformation *33+. The values of these constants obtained for each microstructure are listed in Table 7. No difference has been made for the fitting of the coarse grain microstructure.

J o u r n a l P r e -p r o o f The EBSD technique allows the detection of geometrically necessary dislocations that represent around 15 to 30 % of the overall dislocations *29+, but are of primary importance in order to understand the plastic deformation and then describe the evolution of the work hardening. As the geometrically necessary dislocations density is mainly sensitive to activities close to the grain boundaries, the KAM value measured by EBSD will be greater for fine microstructures which are characterized with higher grain boundary densities (see the higher KAM levels for the DA microstructure in Figure 7). A work hardening calibration depending on the microstructure is therefore of interest to characterize work hardening.

Conclusion

The objective of this article is to propose a methodology to define and quantify the level of work hardening locally in a material, taking into account the influence of the type of loading, the temperature, the strain rate and the microstructure. We thus proposed an experimental campaign including a wide range of conditions: in particular several loading histories that enabled to establish the calibration up to an equivalent plastic strain of 50%. Cyclic loadings and high strain rates have also been J o u r n a l P r e -p r o o f considered to determine the sensitivity of the correlation method to these factors. Three microstructures have been tested to evaluate the influence of the size of the grains and strengthening precipitates. The results demonstrate that the measurements performed with the three techniques (microhardness, FWHM of XRD and KAM of EBSD) are sensitive to the microstructure and that this sensitivity depends on the characterization technique.

It may be concluded that the three techniques offer a certain complementarity:

-

The microhardness gives a global information on the mechanical state of the analyzed volume. The correlation of the measurements with an equivalent plastic deformation is possible, even if the dispersion of the data is important. The fact that the microhardness also depends on the residual stress can become problematic when evaluating work hardening in a structure.

-The equivalent plastic strain is well correlated with FWHM of XRD and a correlation taking the strain rate into account has been established. Its resolution is compatible with the evaluation of gradients in a complex mechanical part. In addition, since the same measurements can be used to characterize plastic deformation and residual stresses this technic presents a great interest.

-The average KAM parameter of EBSD is well correlated with the equivalent plastic deformation. The advantage of this method resides in the fact that it gives very local information that stays meaningful when averaged.

Simulations have been performed for each experimental test with a pertinent elasto-visco-plastic model. A correlation between the model variables and the experimental parameters (FWHM, KAM and microhardness) enables to choose the equivalent plastic strain as a variable representative of the work hardening state in the material. This methodology has been applied on Inconel 718 but we expect it to remain relevant for any metals. The chosen experimental techniques are classically used on metals with different crystal structures. The constitutive model can be similarly declined on a large J o u r n a l P r e -p r o o f range of metallic materials. By implementing this approach, it is therefore possible, with certain precautions, to use the calibration curves and determine the work hardening state in a mechanical part.

The evaluation of the work hardening induced by different manufacturing processes (forging, rolling, shot-peening, ...) could be an interesting application of the proposed methodology.

  The recent improvements of experimental techniques such as X-ray diffraction (XRD) and electron backscatter diffraction (EBSD) have rendered possible numerous studies correlating work hardening to other experimental data. Several authors have proposed procedures to quantify work hardening J o u r n a l P r e -p r o o f *13+, *14+, *10+, *15+-*18+. These studies present a significant complementarity between experimental techniques like XRD, EBSD and microhardness *13+-*16+, *19+, *20+. In particular, Soady et al. compare these three techniques and propose a conversion between plastic strain and work hardening *16+.

  Figure 1. Experimental stress strain curves of the tensile tests for the three microstructures and for the various end-levels of plastic deformation.

X

  -ray diffraction techniques allow an indirect evaluation of work hardening with the determination of the peak width of the diffracted signal. Several parameters can lead to a variation in this width, coming from instrumental and experimental contributions. The experimental contribution depends J o u r n a l P r e -p r o o f in particular on the size of the grains and substructures, the work hardening of the material and the different phases involved *26+. Because the analyzed volume is usually large compared to the size of the grains, the values of the full width at half maximum of the peaks (FWHM) display information on both intergranular and intragranular work hardening.A Seifert PTS diffractometer with a Co-K tube radiation ( Co-K = 1.79 Å) has been used to perform the measurements on the 311 diffraction peak at a 2θ angle of about 111° (accelerating voltage: 20 kV, nominal current: 4 mA). The sample preparation is identical to the one performed for microhardness measurements. With these conditions, the X-ray penetration depth is estimated to be between 2 and 4 µm and the size of the analyzed volume is a few millimeters wide.The X-ray data have been processed with classical methods *26+. The diffraction peaks have been modelled by a pseudo-Voigt function, and the FWHM of this function has been evaluated. The classical sin²ψ method combined with material removal by electrochemical etching has been used to determine the residual stress profile in each studied specimen (results not presented in this paper) as well as the in-depth line broadening profile defined by the FWHM. For each depth, an average taking into account eleven ψ angles have been used to estimate the FWHM parameter. The results are presented in section 4.2.

  This technology is used to identify crystallographic phases, grain size and crystallographic local orientations. Several authors have highlighted a relationship between local misorientations determined by EBSD and the accumulation of geometrically necessary dislocations *27+. We have chosen to evaluate a kernel average misorientation (KAM) parameter from the EBSD measurements.The KAM parameter represents the average value of the misorientation between the considered pixel and its N first neighbors, i.e. the Nth first adjacent pixels (Figure2). A correlation between KAM and plastic strain has been found in several papers for nickel base alloys and steel materials *14+, *18+, J o u r n a l P r e -p r o o f *28+.The volume analyzed by EBSD is of a few micrometers in depth for surfaces up to hundreds of square micrometers. This is much smaller than the volume analyzed with X-ray diffraction. This technique gives thus a very local information on the material, and, depending on the device, the resolution can be as fine as hundreds of nanometers. It gives access to a partial quantification of the dislocation density, since only the geometrically necessary dislocations are detected *27+, *29+. These dislocations are characteristic of a non-uniform plastic deformation of the material, preferentially localized on the grain boundaries *14+, *30+. As the geometrically necessary dislocations density is mainly sensitive to activities close to the grain boundaries, the KAM value measured by EBSD are sensitive to the microstructure.

J

  o u r n a l P r e -p r o o f study. Its value has thus been set to one. The KAM values displayed in sections 3 and 4 correspond to the mean values of the observed fields. The results are presented in section 4.3.

Figure 2 .

 2 Figure 2. Evaluation of the KAM parameter: misorientation between the considered pixel, at the center, with respect to its N first neighbors (N = 3 for the neighbors in red).

  𝜺̇𝒑 -𝑫 𝒌 𝝍(𝑿 𝒌 )𝑿 𝒌 𝒑̇ Eq. 7

Four

  scalar quantities have been identified in the model presented above able to represent work J o u r n a l P r e -p r o o f hardening and for comparison with the experimental data: -The von Mises equivalent plastic strain 𝜀 𝑝 = √ 2 3 𝜺 𝒑 : 𝜺 𝒑 , this variable represents an instantaneous level of plasticity, that can increase or decrease; -The cumulative plastic strain p: 𝑝(𝑡) = ∫ 𝑝̇(𝑠)

Figure 3 .

 3 Figure 3. Correlation of the model variables chosen to represent work hardening with the values of the experimental parameters FWHM, KAM and microhardness obtained for each homogeneously loaded specimens for DA microstructure.

Figure 4

 4 Figure4presents the evolution of the microhardness as a function of the equivalent plastic strain for

Figure 4 .

 4 Figure 4. Microhardness as a function of the equivalent plastic strain obtained for the tensile tests at 20°C with the respective calibration curves.

Figure 5 .

 5 Figure 5. FWHM as a function of the equivalent plastic strain for several loading history, temperature, strain rates and microstructures with the respective calibration curves. b) Close-up of Figure a) for small plastic strains.

Figure 6 .

 6 Figure 6. Local misorientation EBSD maps of Direct Aged, coarse grain and coarse grain and coarse strengthening precipitate microstructures for several equivalent plastic strain levels.

Figure 7 .

 7 Figure 7. KAM parameter as a function of the equivalent plastic strain for different loading history and microstructures with the respective calibration curves.

Table 1

 1 presents an exhaustive description of the experimental campaign completed in this study. To facilitate the interpretation of the data, colored symbols have been associated to each test condition; this graphic code is applied throughout this article. Section 2.2 details the experimental conditions of the tests.

Table 1 . Test conditions for calibration specifying the microstructure, the applied load, the range of plastic deformation that has been reached, the temperature, the strain rate, the experimental techniques used on each sample and the graphic code used in this article. A total of 30 tests has been performed. Microstructure Test condition Quantity Range of plastic strain Temperature (°C) Strain rate (s -1 ) Technical analysis Symbol

 1

		Tensile	5	*0.05-0.15+	20	10	-3	XRD, EBSD, Microhardness
		Tensile	2	*0.01-0.015+	550	10	-3	XRD
	Direct Aged	Compressive	3	*0.3-0.4+	20	10	-3	XRD, EBSD
		Compressive	5	*0.1-0.3+	20	> 10	3	XRD
		Cyclic	1	*0.0008, 0.009+	20	10	-3	XRD, EBSD
		Cyclic	1	*0.0006+	550	10	-3	XRD
	Coarse grain	Tensile	5	*0.05-0.15+	20	10	-3	XRD, EBSD, Microhardness
		Compressive	3	*0.3-0.4+	20	10	-3	XRD, EBSD
	Coarse grain and
	coarse strengthening	Tensile	5	*0.05-0.15+	20	10	-3	XRD, EBSD, Microhardness
	precipitates
	2 Experimental campaign

Table 2 . Composition of the Inconel 718 DA Alloy provided by the manufacturer (Wt%).

 2

	Ni	Fe	Cr	Mo	Al	Ti	Nb	Si	C
	54.18 17.31 17.97 2.97	0.56	1	5.39	0.1	0.023

Table 3 . Scanning electron microscopy observations and mean grain size and size of the strengthening precipitates for the microstructures investigated. Throughout this article, the reference microstructure is always referenced in blue, the coarse grain microstructure in red and coarse grain and coarse strengthening precipitate microstructure in orange.

 3

Table 4 . Values of the plastic deformation reached in the samples submitted to tensile and compressive tests as illustrated in Figure 1 and for the three microstructures.

 4

	Plastic deformation

Table 5 . Values of the constants obtained for the calibration of the microhardness as a function of the equivalent plastic strain fitted with equation 9 for the three microstructures.

 5

				Coarse grain and coarse
		DA microstructure	Coarse grain microstructure	strengthening
				precipitate microstructure
	A Hv		0.3
	B Hv		0.53
	Hv 0.1 (0)	514.7	509.2	483.1

Table 6 . Values of the constants obtained for the calibration of the FWHM for the three microstructures as defined by equation 10.

 6

		DA microstructure Low strain rates High strain rates	Coarse grain micro-structure	Coarse grain and strengthening precip-itate microstructure
	A FWHM	0.66	0.85	0.56	0.35
	B FWHM	31.12		23.92	20.00
	C FWHM	0.99			0.90
	D FWHM	1.04		1.00	1.18

J o u r n a l P r e -p r o o f

Table 7 . Values of the constants obtained for the calibration curves as defined by equation 11 with the fitting of the KAM parameter for the three microstructures.

 7

			Coarse grain microstructure
			and
		DA microstructure	Coarse grain and strength-
			ening precipitate micro-
			structure
	A EBSD	0.30	0.18
	B EBSD	1.39	1.03
	C EBSD	0.12	0.31
	D EBSD	2.30	2.30

Acknowledgement

This work was conducted with the help of the French Technological Research Institute for Materials, Metallurgy and Processes (IRT M2P) under the CONDOR project. The authors would like to acknowledge IRT M2P and all the partners of the project led by IRT M2P. Safran is warmly thanked for its precious collaboration in this work.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to legal or ethical reasons.


Manuel Brazidec
email: manuel.brazidec@gmail.com

Vincent Perrichot
email: vincent.perrichot@univ-rennes1.fr

The first fossil Parascleroderma (Hymenoptera: Bethylidae): a new species in mid-miocene zhangpu amber

Keywords: Chrysidoidea, Bethylidae, Pristocerinae, Cenozoic, Miocene, China

The recently discovered amber biota from Zhangpu, China, offers a unique window into the tropical ecosystem of an Asian rainforest during the Mid-Miocene Climatic Optimum (MMCO). Here, the first flat wasp (Bethylidae) from this deposit is documented: Parascleroderma palaeosinica n. sp. is described and figured from a single male, and is the first fossil Parascleroderma known to date. The new extinct species is characterized by its minute size, the pedicel longer than the flagellomeres, the posterior ocelli inserted much posteriad on head, the notauli not impressed on anterior third of anteromesoscutum, and the hypopygium simple, with posterior margin slightly incurved medially. Parascleroderma is a cosmopolitan genus within the Pristocerinae and it is therefore not surprising to find its record in the Zhangpu biota, simultaneously with the northern expansion of the southeastern Asian tropical forest during the MMCO.

Introduction

With up to 3000 current species arranged in 96 valid genera [START_REF] Azevedo | Global guide of the flat wasps (Hymenoptera, Bethylidae)[END_REF], the Bethylidae form the largest family among the Chrysidoidea. This family of wasps displays its greatest diversity in tropical regions but is cosmopolitan in its global distribution. Bethylid wasps are generally considered as external parasitoids of lepidopteran and coleopteran larvae that they paralyze and drag into cracks [START_REF] Evans | A synopsis of the American Bethylidae (Hymenoptera, Aculeata)[END_REF][START_REF] Rubink | Notes on the nesting behavior of the bethylid wasps, Epyris eriogoni Kieffer, in southern Texas[END_REF].

Within Chrysidoidea, Bethylidae are generally retrieved as the sister-group of Chrysididae in phenotypic studies [START_REF] Brothers | Phylogeny of Aculeata: Chrysidoidea and Vespoidea (Hymenoptera)[END_REF][START_REF] Carpenter | What do we know about chrysidoid (Hymenoptera) relationships?[END_REF][START_REF] Brothers | A new Late Cretaceous family of Hymenoptera, and phylogeny of the Plumariidae and Chrysidoidea (Aculeata)[END_REF][START_REF] Brothers | Re-evaluation of the Cretaceous family Plumalexiidae and its relationships (Hymenoptera: Chrysidoidea)[END_REF], but sister to Plumariidae in phylogenomic studies [START_REF] Branstetter | Phylogenomic insights into the evolution of stinging wasps and the origins of ants and bees[END_REF][START_REF] Peters | Evolutionary history of the Hymenoptera[END_REF]. [START_REF] Grimaldi | Evolution of the Insects[END_REF] indicate that the narrowly transverse and nearly completely hidden metanotum would be the synapomorphy of the clade (Bethylidae + Chrysididae). However, in the bethylid subfamily Pristocerinae, the metanotum is developed medially and tends to overlap the mesoscutellum [START_REF] Azevedo | Global guide of the flat wasps (Hymenoptera, Bethylidae)[END_REF]. Bethylids are recognizable by the prognathous head, the antennae with the same number of antennomeres in males and females (11 or 10, rarely 8 or 7 flagellomeres), the pronotum with anterior flange, the prosternum small, transverse, and the six or seven exposed metasomal terga [START_REF] Finnamore | Superfamily Chrysidoidea[END_REF]. The flattened body, evolved to sneak into their host habitats, is also retained as a characteristic feature of the family, hence their nickname 'Flat wasps'. Nine subfamilies were recently recognized in the Bethylidae [START_REF] Colombo | Phylogenetic overview of flat wasps (Hymenoptera, Bethylidae) reveals Elektroepyrinae, a new fossil subfamily[END_REF]: the extant Bethylinae, Epyrinae, Mesitiinae, Pristocerinae and Scleroderminae, and the extinct Elektroepyrinae, Holopsenellinae, Lancepyrinae and Protopristocerinae. Since then, however, Holopsenella [START_REF] Engel | A new bethylid wasp in Lebanese Early Cretaceous amber (Hymenoptera: Chrysidoidea), with comments on other Mesozoic taxa[END_REF], type genus of the Holopsenellinae, has been removed from Bethylidae and elevated to a family left as incertae sedis in the Aculeata [START_REF] Lepeco | The wasp genus †Holopsenella in mid-Cretaceous Burmese amber (Hymenoptera: †Holopsenellidae stat. nov.)[END_REF].

With more than 90 species in seven of the eight subfamilies (no fossil Mesitiinae has been described yet), the fossil record of Bethylidae spans nearly 130 million years from the Lower Cretaceous to the Holocene (see [START_REF] Martynova | New records of Upper Eocene chrysidoid wasps from basins of Styr and Stokhod rivers (Rovno amber)[END_REF]; and the posterior discoveries of: [START_REF] Engel | A holopsenelline wasp in mid-Cretaceous amber from Myanmar (Hymenoptera: Bethylidae)[END_REF]Falières and Nel, 2019a, 2019b, 2019c, 2020;[START_REF] Colombo | Phylogenetic overview of flat wasps (Hymenoptera, Bethylidae) reveals Elektroepyrinae, a new fossil subfamily[END_REF]Colombo et al., , 2021aColombo et al., , 2021b;;[START_REF] Jouault | New Burmese amber fossils clarify the evolution of bethylid wasps (Hymenoptera: Chrysidoidea)[END_REF]Jouault et al., , 2021;;Colombo and Azevedo, 2021;[START_REF] Jouault | A new protopristocerine wasp (Hymenoptera: Bethylidae) from the mid-Cretaceous Burmese amber[END_REF][START_REF] Tribull | A new genus and species of Pristocerinae (Hymenoptera, Bethylidae) from Upper Eocene Baltic amber with a review of conspecific association from insect fossils[END_REF]. Despite this relatively high richness compared to other chrysidoid lineages (e.g., Chrysididae), lots of new species are likely to be described from Cretaceous or Cenozoic deposits, particularly from the recently discovered Miocene amber of Ethiopia and China [START_REF] Bouju | A review of amber and copal occurrences in Africa and their paleontological significance[END_REF][START_REF] Wang | The mid-Miocene Zhangpu biota reveals an outstandingly rich rainforest biome in East Asia[END_REF]. Even from the long-surveyed deposits such as Baltic amber, new taxa may be recovered, as exemplified by the recent discovery of a new pristocerine genus [START_REF] Tribull | A new genus and species of Pristocerinae (Hymenoptera, Bethylidae) from Upper Eocene Baltic amber with a review of conspecific association from insect fossils[END_REF].

Here, we report the first Bethylidae from the rich mid-Miocene amber biota of Zhangpu, in Fujian Province, China, and we discuss its taxonomic affinities.

Material and methods

Zhangpu amber is found with plant impressions in two layers of sandy mudstone interbedded with coal seams that belong to the Fotan Group, a geological unit that occurs widely in Zhangpu County, Fujian Province, southeastern China (Wang et al., 2021, fig. 1). Under-and overlying basalt layers allow constrained dating of the amber between 14.8 ± 0.6 Ma and 14.7 ± 0.4 Ma, which corresponds to the middle Miocene (Langhian; [START_REF] Zheng | Age constraint on a Neogene tropical rainforest in China and its relation to the Middle Miocene Climatic Optimum[END_REF].

This study is based on a complete male specimen kindly lent by Prof. Bo Wang (Nanjing, China) to the authors for study, and housed in the Nanjing Institute of Geology and Palaeontology (NIGP), Chinese Academy of Sciences, Nanjing, China.

The fragile amber piece was embedded in a block of epoxy resin (Araldite® 2020) for consolidation and posteriorly polished to facilitate the observation of the specimen, using thin silicon carbide sanding papers on a grinder polisher (Buehler EcoMet 30).

The examination and photographs were conducted with a Leica DMC4500 camera attached to a Leica M205C stereomicroscope. All images are digitally stacked photomicrographic composites of several focal planes, which were obtained using Helicon Focus 6.7 software. Adobe Illustrator CC2019 and Photoshop CC2019 software were used to compose the figures and ImageJ 1.53 for measurements. The description of the characters follows the nomenclature of [START_REF] Lanes | Revisited morphology applied for Systematics of flat wasps (Hymenoptera, Bethylidae)[END_REF].

This published work and its new nomenclatural act are registered in ZooBank with the following LSID (reference): urn:lsid:zoobank.org:pub:570AA2F6-4E8D-4DD0-B01A-9D824F8CD4F6.

Systematic palaeontology

Superfamily

Etymology:

The species name combines 'palaiós', meaning 'ancient' in Greek and 'sinica', in reference to the geographic origin of Zhangpu amber.

Material: Holotype male, NIGP178180.

Type locality: Zhangpu County, Zhangzhou Prefecture, Fujian Province, China.

Horizon: Sedimentary layer II, Fotan Group, middle Miocene: Langhian, 14.7 Ma.

Diagnosis (male):

Body minute-sized, 1.67 mm long, pale brown; pedicel longer than each flagellomere (Fig. 1C); posterior ocelli inserted much posteriad on head (Fig. 1D); notauli not impressed on anterior third of anteromesoscutum, well-marked on posterior two-thirds (Fig. 1D); hypopygium simple with posterior margin slightly incurved medially (Figs. 1A,2). Description: Male. Body flattened and narrow, not visibly sculptured, pale brown.

Head prognathous, longer than high, longer than wide, oval in dorsal view; frons smooth, concave, with faint median sulcus originating from anterior ocellus, frons with sparse long setae; compound eye oval, longer than high, without visible setae, located in anterior part of head; mandibles only slightly overlapping at tip; maxillary palpus with at least 4 palpomeres and long apical setae; 13 antennomeres, all antennomeres bearing numerous long setae at most as long as one third of flagellomere length; scape elongate, 2.5 times longer than wide, slightly curved, pedicel as long as apical flagellomere; all flagellomeres cylindrical, longer than wide, flagellomeres I-X subequal in length, only slightly shorter than flagellomere XI;

anterior ocellus located at back of frons, posterior ocelli pushed at extreme posterior of frons and widely separated from each other; occipital carina present, forming ridge extending from occipital foramen to mandibles.

Discussion

Following the key to the subfamilies of Bethylidae of [START_REF] Azevedo | Global guide of the flat wasps (Hymenoptera, Bethylidae)[END_REF], the new fossil keys out in Pristocerinae because of the following characters: macropterous species, forewing with Rs+M vein absent, metanotum developed medially and overlapping mesoscutellum posteriorly. This last character is usually considered as a synapomorphy of the Pristocerinae.

Following the keys to pristocerine genera of [START_REF] Azevedo | Global guide of the flat wasps (Hymenoptera, Bethylidae)[END_REF], the new fossil keys near Parascleroderma because of the following characters: body smaller than 15 mm, hypopygium not deeply divided into two lobes, outer surface of hypopygium flat, dorsal pronotal area short, forewing with intersection of Rs&M far from pterostigma, profemur narrow, metafemur without spines, hypopygium slightly incurved. Among the diagnostic characters provided by [START_REF] Azevedo | Global guide of the flat wasps (Hymenoptera, Bethylidae)[END_REF] for the genus Parascleroderma, the fossil clearly possesses the body flattened and elongated, the propleuron elongated, the pronotum elongated and anteriorly narrowed, the intersection of Rs&M and Sc+R far from pterostigma and the metafemur enlarged.

Because of the preservation in amber, characters on the clypeus, genitalia, and internal features of hypopygium cannot be observed. Parascleroderma is closely related to Afgoiogfa [START_REF] Argaman | A new subfamily of Bethylidae allied to Pristocerinae (Hymenoptera)[END_REF], Dissomphalus Ashmead, 1893, Foenobethylus Kieffer, 1913, Protisobrachium Benoit, 1957, and Pseudisobrachium Kieffer, 1904[START_REF] Alencar | Opening Pandora's box of Pristocerinae: molecular and morphological phylogenies of Apenesia (Hymenoptera, Bethylidae) reveal several hidden genera[END_REF]. These genera differ from our fossil as follows [START_REF] Azevedo | Global guide of the flat wasps (Hymenoptera, Bethylidae)[END_REF]: pterostigma comma-like, maxillary palpus with 3 palpomeres and a long apical seta or 6 palpomeres with only short setae, notauli large and poorly impressed (Afgoiogfa); body robust, metasomal tergum II with process (Dissomphalus); posterior hypopygial margin abruptly v-incurved, metatrochanter and metafemur usually with spines (Foenobethylus); eyes with long erect setae, R1 present and long after the pterostigma, posterior hypopygial margin strongly incurved (Protisobrachium); forewing with intersection of Rs+M and Sc+R close to pterostigma (Pseudisobrachium). Therefore, our new species is excluded from these genera and, instead, displays features that are all diagnostic of Parascleroderma.

Parascleroderma was erected by [START_REF] Kieffer | Description de nouveaux Dryininae et Bethylinae du Musée Civique de Gênes[END_REF] based on a species from Italy: P. nigriceps [START_REF] Kieffer | Description de nouveaux Dryininae et Bethylinae du Musée Civique de Gênes[END_REF] In the following years, the same author subsequently described species from Hungary, Italy, France and the Seychelles (e.g., [START_REF] Kieffer | Bethylidae[END_REF][START_REF] Kieffer | Hymenoptera, Proctotrupoidea[END_REF]: P. fulviceps [START_REF] Kieffer | Description de nouveaux Hyménoptères[END_REF], P. fuscipennis (Kieffer, 1905) (synonymized from Ceratepyris [START_REF] Kieffer | Description de nouveaux Hyménoptères exotiques[END_REF][START_REF] Argaman | A new subfamily of Bethylidae allied to Pristocerinae (Hymenoptera)[END_REF], P. remota [START_REF] Kieffer | Hymenoptera, Proctotrupoidea[END_REF] (transferred from Apenesia by [START_REF] Azevedo | Global guide of the flat wasps (Hymenoptera, Bethylidae)[END_REF], P. seychellensis [START_REF] Kieffer | Hymenoptera, Proctotrupoidea[END_REF] (transferred from Sclerodermus Latreille, 1809 by [START_REF] Evans | A synopsis of the American Bethylidae (Hymenoptera, Aculeata)[END_REF] and P. sulcatifrons [START_REF] Kieffer | Bethylidae[END_REF] (synonymized from Ceratepyris by [START_REF] Argaman | A new subfamily of Bethylidae allied to Pristocerinae (Hymenoptera)[END_REF]. The genus currently includes 32 extant species and has a cosmopolitan distribution (Fig. 3), but was hitherto undocumented from the fossil record. Among major contributors to the knowledge of Parascleroderma, [START_REF] Evans | A revision of the Apenesia in the Americas (Hymenoptera, Bethylidae)[END_REF][START_REF] Evans | A synopsis of the American Bethylidae (Hymenoptera, Aculeata)[END_REF][START_REF] Evans | New generic records of Bethylidae from South America (Hymenoptera)[END_REF][START_REF] Evans | The Bethylidae of America North of Mexico[END_REF] described four new species from America North of Mexico: P. arivaca [START_REF] Evans | The Bethylidae of America North of Mexico[END_REF], P. carinata Evans, 1964, P. insolita (Evans, 1963) (transferred from Apenesia by [START_REF] Evans | A synopsis of the American Bethylidae (Hymenoptera, Aculeata)[END_REF], P. minima [START_REF] Evans | The Bethylidae of America North of Mexico[END_REF]; and one from Peru: P. pucallpa [START_REF] Evans | New generic records of Bethylidae from South America (Hymenoptera)[END_REF][START_REF] Argaman | A new subfamily of Bethylidae allied to Pristocerinae (Hymenoptera)[END_REF] summarized the known species and erected six new ones from Europe, Turkey, Israel, and Iran: P. cisnora [START_REF] Argaman | A new subfamily of Bethylidae allied to Pristocerinae (Hymenoptera)[END_REF], P. fiturcata Argaman, 1988, P. hindola Argaman, 1988, P. norcasta Argaman, 1988, P. oriana Argaman, 1988and P. varlinda Argaman, 1988[START_REF] Terayama | Discovery of the genus Parascleroderma Kieffer from the Oriental region, with descriptions of four new species (Hymenoptera: Bethylidae)[END_REF][START_REF] Terayama | Insects of Japan, Bethylidae (Hymenoptera)[END_REF], [START_REF] Xu | A new species of the genus Parascleroderma Kieffer, 1904 from China (Hymenoptera: Bethylidae)[END_REF][START_REF] Lim | A taxonomic review of the subfamily Pristocerinae (Hymenoptera: Chrysidoidea: Bethylidae) from Korea with descriptions of two new species[END_REF] described new species from the Oriental region (i.e., Korea, China, Taiwan, and Japan): P. atayal [START_REF] Terayama | Discovery of the genus Parascleroderma Kieffer from the Oriental region, with descriptions of four new species (Hymenoptera: Bethylidae)[END_REF], P. ishama Terayama, 2006, P. jinmo Terayama, 2006, P. maae Xu, He and Terayama, 2002, P. okajimai Terayama, 1998, P. renaiensis Terayama, 1998and P. tetradentica Lim and Lee in Lim et al., 2011. The vast majority of the known Parascleroderma species have been collected in the northern hemisphere but six species occur in the southern hemisphere: one from Peru and two from the Seychelles, mentioned above, P. neounicolor Alencar and Azevedo in Azevedo et al., 2018 (Marquises Islands), P. nigra Brues, 1910 (South Africa) and P. azevedonis (New Zealand). [START_REF] Alencar | Opening Pandora's box of Pristocerinae: molecular and morphological phylogenies of Apenesia (Hymenoptera, Bethylidae) reveal several hidden genera[END_REF] also indicate potential new species in Thailand, Australia and South Africa.

The Zhangpu amber originated from a tropical rainforest with abundant Dipterocarpaceae [START_REF] Wang | The mid-Miocene Zhangpu biota reveals an outstandingly rich rainforest biome in East Asia[END_REF] in the greenhouse context of the Mid-Miocene Climatic Optimum [START_REF] Kasbohm | Rapid eruption of the Columbia River flood basalt and correlation with the mid-Miocene climate optimum[END_REF]. Thus, the presence of Parascleroderma in Thailand nowadays is interesting because southeast Asian dipterocarp forests are the closest extant relatives of the mid-Miocene Zhangpu amber forest in terms of floral assemblage and climate (Wang et al., 2021, fig. 2). This further supports our attribution of the new species to Parascleroderma, whose extant species are present in tropical biomes. Therefore, it is not surprising to discover an extinct species of Parascleroderma northward in the Zhangpu biome (24.2°N), simultaneously with the megathermal climate of the MMCO that allowed the tropical forest to expand toward higher latitudes.

Figure captions

  Parascleroderma palaeosinica n. sp. (Figs. 1, 2) LSID: urn:lsid:zoobank.org:act:1E0E3362-9053-47AA-A6A2-457880270E5E.

  Mesosoma shorter than metasoma; propleuron elongate, slightly exposed dorsally, 'neck-shaped'; pronotal flange developed, dorsal pronotal area longer than anteromesoscutum, narrowed anteriorly; prosternum reduced between procoxae; mesoscutum with notauli impressed on posterior two thirds, slightly converging posteriorly, absent anteriorly; parapsidal signum not marked; mesoscutellum with blunted apex; metanotum large, overlapping mesoscutellum posteriorly, transversally carinate; metapectal-propodeal complex longer than wide, without posterolateral spines, smooth, without sharp longitudinal carinae. Forewing with only costal [C], radial [R] and cubital [1Cu] cells closed by tubular veins; C, Sc+R, M+Cu, A, Rs, cu-a, basal segment of M and Cu tubular; basal segments of Rs and M forming contiguous line; cu-a angled at junction with Cu; Cu nebulous posteriorly to junction with 1m-cu then completely fading; 1m-cu nebulous; pterostigma reduced in length, rapidly vanishing after 2r-rs; R1 not apparent after pterostigma; 2r-rs and Rs continuously arched, without marked angles between sections, distal segment of Rs tubular for much of its length, then reaching wing margin as nebulous vein, thus not closing radial cell [2R1]; distal segment of M completely spectral. Hind wing slightly shorter than forewing; C only pigmented vein; Rs originating from C in anterior half of wing, nebulous to spectral; no closed cells. Legs sparsely pubescent; profemur swollen; first tarsomere longer than combined length of following tarsomeres; proand mesotibiae with two spurs, metatibial spurs not visible; metafemur enlarged, without spines. Metasoma sparsely setose; metasomal tergites without modifications; metasomal tergite I as long as wide; metasomal tergite II shorter than I; hypopygium not divided, with outer surface flat and posterior margin weakly incurved medially; genital structures not visible. Measurements (in mm): Body length 1.67; head length 0.34, width 0.26; scape length 0.10, width 0.04; pedicel length 0.06; flagellomeres I-X length ca. 0.04-0.05; flagellomere XI length 0.06; mesosoma length 0.63; forewing length ca. 1.10; hind wing length 0.98; mesosoma length 0.70.

Fig. 1 .

 1 Fig. 1. Parascleroderma palaeosinica n. sp., holotype, NIGP178180; (A) habitus in lateroventral view; (B) habitus in dorsal view; (C) head in dorsal view; (D) line drawing of head and mesosoma in dorsal view; scale bar = 0.5 mm.

Fig. 3 .

 3 Fig. 3. Distribution map of extant and extinct species of Parascleroderma showing type localities.

Acknowledgments

We warmly thank the guest editor Prof. Bo Wang for its invitation to contribute this special issue of Palaeoworld and for its precious comments on the manuscript.

We are grateful to the reviewers, Prof. Celso O. Azevedo, Prof. Massimo Olmi and an anonymous reviewer for valuable comments and remarks on the earlier manuscript.

Research on the Zhangpu amber was supported by the Strategic Priority Research

Program of the Chinese Academy of Sciences (XDB26000000) and National Natural Science Foundation of China (42125201, 41688103). This work is part of Manuel Brazidec's Ph.D. project CHRYSIS: "The role of greenhouses on the diversification and evolution of chrysidoid wasps".


P Bambade

G Le Meur

C Rimbault

F Touze

K Mönig

D Schulte

Results of the EUROTeV Beam-Beam Simulation (BBSIM) Task

This paper is the deliverable of the EUROTeV Beam-Beam Simulation (BBSIM) task and gives an overview of the published results.

Introduction

In a linear collider, because of very strong beam-beam effects during collisions, the luminosity performance and background rates from secondary particles emitted at the interaction point depend critically on the specified beam parameters. Accurate predictions, important for several aspects of the design of a linear collider, can usually not be obtained analytically and numerical simulations must be used to describe the combination of electromagnetic and quantum processes involved. Two programs are widely used in the community: GUINEA-PIG and CAIN. The main goals of the BBSIM task were to:

! benchmark physics processes in the GUINEA-PIG simulation against known and trusted physics generators, in order to estimate uncertainties relevant for the design of both the accelerator and detector, and ultimately for the physics program, ! implement spin transport into GUINEA-PIG, also comparing with an already existing treatment in CAIN, and quantifying depolarising effects for different beam parameter sets, ! provide a documented version of GUINEA-PIG containing all implemented improvements and suitable for future maintenance and developments.

The results obtained are described below, emphasising the most important achievements.

Study of incoherent pairs and comparison between generators

Secondary e " e # pairs are produced copiously from the interaction of both real and virtual photons radiated during collisions. The bulk of these pairs follow the spent beam and are most likely stopped after being over-focusing in the first magnets of the beam transport, on their way out of the interaction region. A small fraction of them have a large enough transverse momentum to reach the first layers of the micro vertex detector directly, where they constitute a quasi-irreducible background.

The work started by comparing the treatments implemented in the GUINEA-PIG and CAIN simulations for the Landau-Lifchitz process, one of the processes involved, with that in BDK, a standalone four-fermion generator which was extensively tested in the context of the LEP experiments at CERN. While in the former, the equivalent photon approximation is used for the (virtual) photon radiation and requires proper adjustment of some internal parameters, the latter relies on fully massive expressions to provide an exact first-order treatment. The study [START_REF] Rimbault | Incoherent pair generation in a beam-beam interaction simulation[END_REF] enabled validating the GUINEA-PIG implementation of the Landau-Lifshitz process and to explain the origin of a long-standing factor-of-two discrepancy with the CAIN simulation observed in predictions for direct hits in the micro vertex from incoherent pairs (see Figure 1).

The variation of such backgrounds with different beam parameters, as the bunch width and length, and for detector configurations with different vertex detector radii and magnetic fields, was also checked. The High Luminosity beam parameter set suggested in the ILC optimisation process was for instance found incompatible with the lowest magnetic field options discussed for the detector, as illustrated in Figure 2.

Study of biases on the luminosity measurement from beambeam effects

Much of the ILC physics program involves precision measurements for which the luminosity must be known accurately, typically at levels between 10 -3 and 10 -4 , depending on the particular physics study. As in other e " e # colliders, Bhabha scattering in the forward region can be used to measure luminosity. In addition to theoretical uncertainties in the calculation of the corresponding cross section, and experimental errors from the measurement procedure, the very strong beam-beam space charge effects present during collisions at ILC lead to large biases in the counting rate, which must be corrected to achieve the required precision. See Figure 3 (left) where the change in scattering angle of the final state Bhabha particles induced by the space charge is illustrated. These effects and their consequences on the luminosity measurement were studied in detail [START_REF] Rimbault | Impact of beam-beam effects on precision luminosity measurements at the ILC[END_REF][START_REF] Rimbault | Bias on Absolute Luminosity Measurements at the ILC from Beam-Beam Space Charge Effects[END_REF].

The C++ version of the GUINEA-PIG program, guineapig++, was modified to properly track Bhabha events from dedicated external generators in the field of the colliding bunch, taking into account the modified kinematics from beamstrahlung radiation and from initial beambeam deflections, in order to quantify changes in counting rates in the defined experimental acceptance. It was found that for nominal ILC beam parameters, after applying specially adapted asymmetrical energy and polar angle selection cuts, the residual relative bias on the luminosity measurement was about 1.5%, two thirds of which were from beamstrahlung emissions and the rest from electromagnetic deflections.

Since both the beamstrahlung emissions and electromagnetic deflections vary with the collision energy, bunch length and horizontal size, so do the biases induced on the integrated luminosity. Reconstructing the luminosity spectrum from the scattered Bhabha angles is a good way to measure the amount of beamstrahlung and allows inferring and correcting the corresponding bias. The precision expected from such a procedure was studied. A reduction to below 10 -3 of the beamstrahlung contribution should be possible. The residual bias after applying such a correction, from electromagnetic deflections, appears however to be harder to control experimentally, in particular when considering variations in bunch length. Keeping luminosity biases from deflections below 10 -3 will be possible if the bunch length and horizontal size can be monitored with a 20% precision.

The evolution of these effects with the collision energy was also studied. For physics running at the Z boson resonance (the GigaZ option), the bias from electromagnetic deflections is typically several %, i.e. about hundred times the accuracy on the luminosity needed for precision measurements at that energy. An example of this is shown in Figure 3 (right).

Implementation of beam-beam depolarising effects

A precise knowledge of the polarisation state of the beams at the interaction point is required in future linear collider experiments. As for many other beam characteristics, the polarisation is modified by beam-beam effects. The spin motion in an electromagnetic field is influenced by two effects: the classical spin precession, described by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation and the spin-flip effect arising from beamstrahlung, called the Sokolov-Ternov (ST) effect.

Both of these effects have been implemented as a new development of guineapig++, the C++ version of GUINEA-PIG, and tested through comparisons with the CAIN simulations [START_REF] Rimbault | Implementation of beam-beam depolarising effects in GP++[END_REF]. The comparisons were done both for the polarisation after full beam-beam collisions (see Figure 4) and for the luminosity-weighted polarisation of interacting particles. Results obtained for the various beam parameter sets considered at the ILC showed reasonable agreement between guineapig++ and CAIN predictions, at 90% and 95% levels for the luminosity-weighted and final depolarisations, respectively.

For the nominal ILC conditions at 500 GeV, the simulations predict a luminosity-weighted depolarisation of ~0.23%, dominated by the spin precession contribution. The effect from spin-flip is about five times smaller in this case. However, this ratio depends on the strength of beam-beam effects at the IP. For some beam parameter sets, it can reach up to ~40% of the total depolarisation.

Studies of the energy dependence were also performed, as well as of the sensitivity to beam parameter variations.

The beam-beam simulation guineapig++ is now available to study depolarising effects at future linear colliders, for either fully or partially polarised input beams, as a complementary tool to CAIN.

guineapig++, the GUINEA-PIG simulation in C++

An object-oriented C++ version of the beam-beam interaction program GUINEA-PIG has been developed: guineapig++ [START_REF] Meur | Description of guineapig++, the C++ upgraded version of the GUINEA-PIG beam-beam simulation program[END_REF]. The aim was to facilitate the maintenance and further development of this code already designed in C, making use of up-to-date standards and tools. The development has been achieved in two directions: restructuring of the existing code and implementation of new functionalities.

Concerning the restructuring [START_REF] Rimbault | GUINEA-PIG++: An Upgraded Version of the Linear Collider Beam-Beam Interaction Simulation Code GUINEA-PIG[END_REF], the algorithms have been kept, but have been inserted in a system of inter-related "classes" such as: BEAM, PARTICLEBEAM, PARTICLE, GRID etc. Conformity with the C-version of the code has been carefully checked. Some improvements have also been introduced. The random generation of beam particles was extended to 64-bit machines and the seed mechanism is now under the control of the user. An interface with the latest version (3.1) of the high-performance fast Fourier transform FFTW was implemented in order to accelerate the field calculations. An abstract input/output interface has been designed. This interface concerns all the input and output files which are, at the moment, in ASCII format. The data are now decoupled from the input/output formats (the latter defined as classes) so that formats other than ASCII can be easily plugged.

As a new functionality the beam-beam effects can be applied to Bhabha events that are provided in a dedicated input file. An automatic grid sizing has been designed in order to avoid errors due to unmatched grids. Indeed, when the size of the grid is less than the size of the beam, the field is not calculated for the more ex-centred particles and the computation can be fully meaningless. With the new function an approximate matching is insured. This function takes into account the size and divergence of the initial beam together with the disruption angle. The number of grid cells is also adjusted. This procedure is still being tested and improved based on feedback from users.

The most significant new functionality is the treatment of the polarisation which did not exist at all in the C-version of GUINEA-PIG. Two possibilities are proposed at the moment (for electrons/positrons only): the semi-classical effects as described by the T-BMT equation and quantum effects known as "spin-flip". An initial polarization can be specified by the user or the whole polarized beam can be entered as an input file.

The maintenance, development, documentation and distribution of the code are achieved using dedicated computer tools. The versioning is ensured by Subversion (SVN; http://subversion.tigris.org). The code configuration is managed by CMT (Configuration Management Tool; https://trac.lal.in2p3.fr/CMT/). Through the TRAC tool (http://trac.edgewall.org/), a dedicated wiki web site is available in order to provide the documentation, downloads and support.

List of publications

Figure 1 :

 1 Figure 1: (left) Comparison of GUINEA-PIG (coloured continuous line) and BDK (dashed thick line) photon virtuality spectra for the Laudau-Lifshitz process. (right) Two-dimensional photon virtuality spectrum in BDK. Highlighted points are events with a final state particles reaching the vertex detector. The dashed lines in both plots indicate the value of the electron mass used as a cut in the implementation of the equivalent photon approximation in CAIN.

Figure 2 :

 2 Figure 2: P t versus $ plane for the background electrons and positrons in the vertex detector, for High Luminosity beam parameter set. The black points are all the incoherent e " e # generated with GUINEA-PIG. Blue square-shaped points correspond to the subset reaching the innermost layer of the vertex detector, assuming a radius r = 15mm, for a magnetic field in the detector of 5 Tesla. Adding the green circle-shaped (red triangle-shaped) ones corresponds to the case of a 4 (3) Tesla. For comparison purposes, the thick line indicates the position of the hard edge of generated incoherent particles for the Nominal beam parameter set.

Figure 3 :

 3 Figure 3: (left) Change in Bhabha scattering final state polar angle due to the deflection induced by the space charge of the opposite bunch as a function of the polar angle at production. (right) Reduction in Bhabha counting rate in the acceptance of the luminosity counter (see [2-3] for details) as function of energy, keeping all other beam parameters constant. The contributions due to beamstrahlung (thin line) and electromagnetic deflections (dashed line) are shown as well as the combined effect (top thick line).

Figure 4 :

 4 Figure 4: Comparison of projections of particle spin vectors after on the longitudinal axis after collisions, taking into account both precession and spin-flip effects, for Nominal (upper left), Now N (upper right), Large Y (lower left) and Low P (lower right) beam parameter sets. Distributions in red and blue show the predictions from guineapig++ and CAIN, respectively.

EUROTeV-Report-2008-065

Acknowledgement

This work is supported by the Commission of the European Communities under the 6 th Framework Programme "Structuring the European Research Area", contract number RIDS-011899.