Hamidreza Abdolvand

Karim Louca

Charles Mareau

Marta Majkut

Jonathan Wright

On the nucleation of deformation twins at the early stages of plasticity

Keywords: Hexagonal close-packed crystals, Zirconium, Synchrotron X-ray Diffraction, crystal plasticity finite element, twinning

Understanding the deformation mechanisms of hexagonal close-packed (HCP) polycrystals at the grain scale is crucial for developing both macro and micro scale predictive models. Slip and twinning are the two main deformation mechanisms of HCP polycrystals at room temperature. In this paper, the development of grain-level stress tensors during nucleation and growth of twins is investigated. A pure zirconium specimen with HCP crystals is deformed in-situ while the centre-of-mass, orientation, elastic strain, and stress of individual grains are measured by three-dimensional synchrotron X-ray diffraction (3D-XRD). The observed microstructure is subsequently imported into a crystal plasticity finite element (CPFE) model to simulate the deformation of the polycrystal. The evolution of stress in twin-parent pairs at the early stages of plasticity, further into plasticity zone, and unload is studied. It is shown that twins do not relax very much at the nucleation step, but the difference between the measured stress in the twin and parent increases further into plastic zone where twins relax. While at the early stages of plasticity all six twin variants are active, a slightly better estimation of active variants is obtained using the measured grain-resolved stress tensors.

Introduction

Understanding the deformation mechanisms of polycrystalline materials has been at the centre of many studies for decades. In metals and alloys, plastic deformation is controlled by the movement of dislocations on a particular plane in a particular direction, i.e. slip. In the absence of "easy" slip systems, deformation twinning may become active where a discrete domain of the crystal reorients to accommodate the applied strain through the twin transformation strain. In hexagonal close-packed (HCP) polycrystals deformation by twinning is often active. When all twinned zones orient towards one unique direction, a large macroscopic strain equal to the twin transformation strain will be observed; however, when different variants of the same twin system become active, the effects of the transformation strain from different twins will be homogenised. While it is known that the formation of a twin alters the local state of stress within the parent grain, it has proven to be challenging to quantify the state of deformation within the parent and the twin. This paper focuses on quantifying the evolution of stress in twin and parent pairs, and on how different twin variants become active while deforming a polycrystalline zirconium specimen.

Since deformation twins nucleate and grow with mechanical loading, it is challenging to measure elastic strains within twins, especially at the early stages of plasticity when the size of twins is small. Diffraction based experimental techniques are mainly used to quantify load partitioning between twins and parents [START_REF] Aydiner | Evolution of stress in individual grains and twins in a magnesium alloy aggregate[END_REF][START_REF] Balogh | Spatially resolved in situ strain measurements from an interior twinned grain in bulk polycrystalline AZ31 alloy[END_REF][START_REF] Oddershede | Measuring the stress field around an evolving crack in tensile deformed Mg AZ31 using three-dimensional X-ray diffraction[END_REF][START_REF] Lind | Tensile twin nucleation events coupled to neighboring slip observed in three dimensions[END_REF][START_REF] Bieler | In situ characterization of twin nucleation in pure Ti using 3D-XRD[END_REF][START_REF] Zhang | Multiscale mechanisms of twinning-detwinning in magnesium alloy AZ31B simulated by crystal plasticity modeling and validated via in situ synchrotron XRD and in situ SEM-EBSD[END_REF][START_REF] Agnew | In-situ neutron diffraction of a quasicrystal-containing Mg alloy interpreted using a new polycrystal plasticity model of hardening due to {10.2} tensile twinning[END_REF][START_REF] Schuren | New opportunities for quantitative tracking of polycrystal responses in three dimensions[END_REF][START_REF] Naragani | Investigation of fatigue crack initiation from a non-metallic inclusion via high energy x-ray diffraction microscopy[END_REF]. For example, the development of internal lattice strains during twinning and detwinning of magnesium alloys is studied using neutron diffraction [START_REF] Wu | Twinning-detwinning behavior during fatigue-crack propagation in a wrought magnesium alloy AZ31B[END_REF][START_REF] Wu | Internal stress relaxation and load redistribution during the twinning-detwinning-dominated cyclic deformation of a wrought magnesium alloy, ZK60A[END_REF][START_REF] Wu | Deformation dynamics study of a wrought magnesium alloy by real-time in situ neutron diffraction[END_REF]. It is generally shown that changing the direction of applied load directly affects twinning and detwinning. The lattice strains measured with neutron diffraction have significantly improved our understanding of load sharing between twin and parent grains, yet in such measurements, the local interactions between grains is usually homogenized. Such interactions can be characterized using higher resolution techniques. For example, three dimensional synchrotron micro-Laue X-ray diffraction was used to study twinning and de-twinning in Mg alloys where it was shown that lattice strains measured inside a parent decrease towards the twin interface [START_REF] Wu | Intragranular twinning, detwinning, and twinning-like lattice reorientation in magnesium alloys[END_REF]. Using the same technique, Kumar et al [START_REF] Kumar | Deformation twinning and grain partitioning in a hexagonal close-packed magnesium alloy[END_REF] measured the stress tensor within a parent grain and showed that the stress is localized at the vicinity of the twin boundary. While these studies provide valuable insight into understanding twinning, they are mainly conducted on a few twin-parent pairs.

A better statistical analysis of twins can be provided by the use of electron diffraction [START_REF] Lentz | In-situ, ex-situ EBSD and (HR-)TEM analyses of primary, secondary and tertiary twin development in an Mg-4wt%Li alloy[END_REF][START_REF] Capolungo | Nucleation and growth of twins in Zr: A statistical study[END_REF][START_REF] Jiang | The orientation and strain dependence of dislocation structure evolution in monotonically deformed polycrystalline copper[END_REF][START_REF] Barnett | Influence of orientation on twin nucleation and growth at low strains in a magnesium alloy[END_REF]. For this purpose, the high angular resolution electron back scatter diffraction (HR-EBSD) technique is mainly used [START_REF] Abdolvand | Assessment of residual stress fields at deformation twin tips and the surrounding environments[END_REF][START_REF] Guo | On the nucleation and growth of {112¯2} twin in commercial purity titanium: In situ investigation of the local stress field and dislocation density distribution[END_REF][START_REF] Paramatmuni | Effect of twin crystallographic orientation on deformation and growth in Mg alloy AZ31[END_REF]. For example, Guo et al [START_REF] Guo | Growth of {112¯2} twins in titanium: A combined experimental and modelling investigation of the local state of deformation[END_REF] used HR-EBSD to measure localized stress fields around twins in titanium. They showed that, a better estimation of active twin variant can be achieved if the measured local stresses are used for determining Schmid factor. With the use of the same technique, Khosravani et al [START_REF] Khosravani | Nucleation and propagation of { 1 0 1 ¯ 2 } twins in AZ31 magnesium alloy[END_REF] showed that, while both first and second predominant twin variants are active in magnesium, slip transfer from a soft grain to a hard grain can trigger formation of twins in the hard grain. In this context, grains that have "easy" slip systems and undergo plastic deformation by prism or basal are called plastically "soft" whereas those that lack easy slip systems are plastically "hard" grains. In addition, nano-beam electron diffraction was recently used to understand how strain fields around twin boundaries are affected by twin morphologies. It was observed that elastic strains accumulate at the vicinity of semi-coherent twin boundaries within the parent grain [START_REF] Chen | Quantifying elastic strain near coherent twin interface in magnesium with nanometric resolution[END_REF].

In addition to experimental measurements, numerical modelling techniques have been used to understand the interactions between twins and parent grains [START_REF] Abdolvand | On the effects of reorientation and shear transfer during twin formation: Comparison between high resolution electron backscatter diffraction experiments and a crystal plasticity finite element model[END_REF], twins and twins [START_REF] Kumar | Characterizing the role of adjoining twins at grain boundaries in hexagonal close packed materials[END_REF], and twins and slip bands [START_REF] Kumar | Grain neighbour effects on twin transmission in hexagonal close-packed materials ´ 1[END_REF]. For modelling twins at meso-scales, crystal plasticity is used in the self-consistent [START_REF] Tomé | A model for texture development dominated by deformation twinning: Application to zirconium alloys[END_REF][START_REF] Agnew | Validating a polycrystal model for the elastoplastic response of magnesium alloy AZ31 using in situ neutron diffraction[END_REF][START_REF] Muránsky | Investigation of deformation mechanisms involved in the plasticity of AZ31 Mg alloy: In situ neutron diffraction and EPSC modelling[END_REF][START_REF] Wang | Crystal plasticity modeling and neutron diffraction measurements of a magnesium AZ31B plate: Effects of plastic anisotropy and surrounding grains[END_REF], fast Fourier transform (FFT) [START_REF] Kumar | Numerical study of the stress state of a deformation twin in magnesium[END_REF], and finite element frameworks [START_REF] Kalidindi | Incorporation of deformation twinning in crystal plasticity models[END_REF][START_REF] Staroselsky | A constitutive model for hcp materials deforming by slip and twinning: Application to magnesium alloy AZ31B[END_REF]. For example, Want et al [START_REF] Wang | Crystal plasticity modeling and neutron diffraction measurements of a magnesium AZ31B plate: Effects of plastic anisotropy and surrounding grains[END_REF] have recently developed a self-consistent model for simulating twinning and de-twinning in magnesium. They found that less internal strain develops in hard grains surrounded by hard neighbouring grains. By adapting the self-consistent framework, Barrett et al [START_REF] Barrett | Predicting deformation behavior of α-uranium during tension, compression, load reversal, rolling, and sheet forming using elasto-plastic, multi-level crystal plasticity coupled with finite elements[END_REF] studied the deformation behaviour of α-uranium and found that twinning and de-twinning control the response of uranium during a load reversal. In addition, Grilli et al [START_REF] Grilli | Characterisation of slip and twin activity using digital image correlation and crystal plasticity finite element simulation: Application to orthorhombic α-uranium[END_REF] used crystal plasticity finite element (CPFE) to characterise slip and twin activity in the same material. It was shown that for replicating the experimental data measured by digital image correlation, it was important to consider the interactions between coplanar twins in full-field models. More recently, the CP-FFT full-field modelling scheme was used for studying twins [START_REF] Nagra | A new micromechanics based full field numerical framework to simulate the effects of dynamic recrystallization on the formability of HCP metals[END_REF][START_REF] Mareau | Micromechanical modelling of twinning in polycrystalline materials: Application to magnesium[END_REF]. For example Paramatmuni and Kanjarla [START_REF] Paramatmuni | A crystal plasticity FFT based study of deformation twinning, anisotropy and micromechanics in HCP materials: Application to AZ31 alloy[END_REF] have developed a CP-FFT model to study the interaction of twins and their corresponding parents. This model shows that the evolution of stress in parents and twins depends on their misorientation with respect to the loading direction.

The further development of full-field numerical models requires the measurement of comprehensive experimental data sets that can explain the interaction between twins and their corresponding parents. In this paper, three-dimensional synchrotron X-ray diffraction is used to measure the evolution of stress in individual twin and parent pairs. The centre-of-mass (COM), orientation, relative volume, elastic strain, and stress for individual grains is measured during in-situ deformation of a pure zirconium specimen. Post-processing procedures are developed to find twins and their corresponding parents and track them during early stages of plasticity, further into the plasticity zone, as well as unload. Special focus is given to evolution of stress within the identified pairs and variant selection at the early stages of macroscopic plasticity.

Experiment

The experiment

Specimens were cut from a pure zirconium bar that was heat-treated at 650 °C for two days and subsequently slow cooled to room temperature. The heat treatment was done to recrystallize grains and reduce as much as possible the residual stresses that existed in the bar. The tensile specimen was cut from the radial-transverse face of the bar with the c-axis of HCP crystals oriented towards and perpendicular to the loading direction. The pole figure of the specimen, measured with EBSD prior to the 3D-XRD experiment, is shown in Fig. 1a. The specimen was cut using electrical discharge machining with the geometry shown in Fig 1b . The final cross section of the specimen after mechanical and electro-polishing was 0.78mm x 0.67mm.

The in-situ 3D-XRD experiment was conducted at ID11 at the European Synchrotron Radiation Facility (ESRF), France. The coordinate system of the specimen is shown in Fig. 1b and is defined based on the loading direction (LD), X-ray beam direction (BD), and transverse direction (TD). The experiment was carried out by illuminating the specimen with a planar beam of monochromatic X-rays with the energy set at 78.395 keV. The beam height was initially set at 40 µm where 25 layers of the specimen were measured to cover 1 mm height of the specimen. To avoid overlapping of broadened diffraction peaks, the height of the beam was reduced to 25 µm towards the end of the experiment and the number of scanned layers increased to 40 to cover the same specimen volume. The specimen was mounted on an ADMET tensile rig and was deformed with the average strain rate of 10 -5 s -1 . The axial force was measured by a load cell and the applied strain was monitored by a strain gauge attached to the specimen. For collecting diffraction patterns, the specimen was rotated around the LD in the angular range of [-234.5 o to -125. Diffraction patterns were collected at 9 different loading steps; in the first step, Step-1, the specimen was preloaded to 8 MPa (see Fig 2c). The collected patterns at this step were used to calibrate the single crystal lattice parameters of pure zirconium. In the Step-2, the specimen was deformed to an axial strain of 0.257% , which is in the elastic zone but close to the onset of plasticity. This was followed by three more measurements performed at the early stages of macroscopic plasticity, i.e., at the applied strains of 0.355%, 0.565%, and 1.087% for Step-3, Step-4, and Step-5, respectively. In addition, three more measurement steps were performed in the plastic zone where the specimen was deformed up to 2.6%. The specimen was finally unloaded in Step-9 and the residual stresses were measured. For each measurement step, firstly the positions of the grips of the deformation rig were fixed, the specimen was realigned, and diffraction patterns were subsequently collected. During the alignment and pattern collection, the specimen relaxed, and creep was observed in the gauge length of the specimen. Both macroscopic stress and strain were recorded for CPFE modelling.

Post-processing procedure

The post-processing of the collected patterns was mainly done by developing python codes to use the subroutines embedded in Fable (https://github.com/FABLE-3DXRD) and ImageD11 for peaksearching and indexing grains. Once grains of each layer were identified, they were used to reconstruct the 3D grain maps by adding layers and merging the common grains. Peaks assigned to each grain were used to determine 12 grain properties: three positions, three orientations, and six strain components or lattice parameters (a ,b, c, α, β, γ) [START_REF] Poulsen | Three-Dimensional X-ray Diffraction, Mapping Polycrystals and their Dynamics[END_REF]. On average, more than 97 peaks were assigned to each grain allowing to determine both grain properties and "errors" associated with them. The method used for calculating errors is explained in Oddershede et al [START_REF] Oddershede | Determining grain resolved stresses in polycrystalline materials using three-dimensional X-ray diffraction[END_REF]. At the preload, a set of initial lattice parameters were defined to calculate elastic strains for individual grains. The weighted average of the corresponding stress was then calculated and used for adjusting the "stress-free" lattice parameters. This procedure was conducted for the seven layers of the probed volume. An average of the acquired lattice parameters from the seven layers was subsequently used for calculating strains across the all 25 layers. When the stress from all layers was equal to the applied stress, the initial lattice parameters were deemed to be converged. The fitted lattice parameters for preload were used, without any alteration, for all other loading steps to determine elastic strain within individual grain. Also, at the preload and unload steps, any deviation from the "stress-free" lattice parameters was assumed to arise from residual stresses.

Since the scope of this study is to investigate the evolution of stress within individual grains, the measured 3D maps from each step were compared to other steps to match grains following the procedure described in [START_REF] Abdolvand | Strong grain neighbour effects in polycrystals[END_REF][START_REF] Abdolvand | On the state of deformation in a polycrystalline material in three-dimension: Elastic strains, lattice rotations, and deformation mechanisms[END_REF]. The grains that were not matched between any two consecutive steps were considered as potential twins that nucleated in between the steps. For determining twin-parent BD [START_REF] Wu | Internal stress relaxation and load redistribution during the twinning-detwinning-dominated cyclic deformation of a wrought magnesium alloy, ZK60A[END_REF] LD [START_REF] Kalidindi | Incorporation of deformation twinning in crystal plasticity models[END_REF] TD [START_REF] Guo | Growth of {112¯2} twins in titanium: A combined experimental and modelling investigation of the local state of deformation[END_REF] pairs, the layer at which the twin was found was firstly determined and codes were developed to search for a parent grain in the same layer in the vicinity of the twin. A twin-parent pair is defined when (a) one of the variants of {101 ̅ 2} tensile twinning system could be identified as the common habit plane, (b) the misorientation between the c-axis of the twin and the parent was 85.25°±1.5°, (c) the distance between the twin and parent was less than the summation of their radii and (d) only one unique parent could be identified. With the procedures developed for matching twin-parent pairs in this paper, it was possible to determine grains with multiple twins.

Crystal plasticity model

The development of accurate constitutive models for simulating nucleation and propagation of twins in the finite element framework is an active research area [START_REF] Kasemer | A finite element methodology to incorporate kinematic activation of discrete deformation twins in a crystal plasticity framework[END_REF][START_REF] Liu | On the interaction of precipitates and tensile twins in magnesium alloys[END_REF][START_REF] Paudel | Characterization and modeling of { 10 1[END_REF][START_REF] Yaghoobi | Multiscale modeling of twinning and detwinning behavior of HCP polycrystals[END_REF][START_REF] Bong | Predicting forming limit diagrams for magnesium alloys using crystal plasticity finite elements[END_REF]. In this paper, however, CPFE modelling is used to investigate the evolution of stress only in the parent grains and to simulate how the crystallographic texture evolves with formation of twins. A detailed numerical study discussing constitutive modelling of twins is underway. The details of the model used for simulating plastic deformation by slip and twinning can be found in [START_REF] Abdolvand | Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2[END_REF], only a brief description and the changes made to the model is provided here. The total velocity gradient (𝑳) can be decomposed to its elastic (𝑳 𝑒 ) and plastic (𝑳 𝑝 ) parts, where in the presence of twinning, 𝑳 𝑝 at each integration point (IP) can be written as:

𝑳 𝑝 = (1 -𝑓 𝑡𝑤 ) ∑ 𝑺 𝛼 𝛾̇𝛼 𝑁 𝑠 𝛼=1 + ∑ 𝑺 𝑡 𝛾̇𝑡 𝑁 𝑡𝑤 𝑡=1 (1)
where 𝑺 represents the Schmid tensor that can be calculated using the dyadic product of the normal to the slip/twinning plane and slip/twinning direction. 𝑁 𝑠 and 𝑁 𝑡𝑤 represent the number of slip and twinning systems and 𝑓 𝑡𝑤 is the relative volume of the twin at a given IP. Here, we assume that no plasticity takes place in the twinned zone and that the stress in the twin is equal to that in the parent.

The shear strain rate (𝛾̇𝛼) on each slip or twinning system is calculated using a rate-dependent equation [START_REF] Asaro | Texture development and strain hardening in rate dependent polycrystals[END_REF]:

𝛾̇𝛼 = 𝛾̇0 𝛼 | 𝜏 𝛼 𝑔 𝛼 | 𝑛 𝑠𝑖𝑔𝑛( 𝜏 𝛼 𝑔 𝛼 ) (2)
where 𝛾̇0 𝛼 is the reference shear rate of the slip or twin system, 𝜏 𝛼 is the resolved shear stress acting on the slip or twinning plane and 𝑔 𝛼 is the current critical resolved shear stress (CRSS) for the slip or twining system. It can be assumed that the evolution of twin volume fraction (𝑓 ṫ𝑤 ) correlates with 𝛾̇𝑡, i.e. 𝑓 ṫ𝑤 = 𝛾̇𝑡 𝛾 0 𝑡𝑤 , where 𝛾 0 𝑡𝑤 is the characteristic twin shear, which is 0.169 for zirconium. The resolved shear stress (𝜏 𝛼 ) on each slip system is calculated using the symmetric part of the Schmid tensor (𝑷 𝛼 ) and the Kirchhoff stress (𝜳):

𝜏 𝛼 = 𝑷 𝛼 : 𝜳 (3)
The Jaumann rate of Kirchhoff stress (𝜳 ̌) is calculated as: 𝜳 ̌= ℂ: 𝑫 𝑒 where 𝜳 ̌= 𝜳 ̇-𝜴 𝑒 𝜳 + 𝜳𝜴 𝑒 (4) where ℂ is the stiffness tensor, and 𝑫 𝑒 and 𝜴 𝑒 are the elastic parts of the deformation rate and spin tensor, respectively. They can be calculated using the symmetric and asymmetric parts of 𝑳 𝑒 .

The strength of each slip system (𝑔 𝛼 ) is assumed to follow an extended Voce hardening rule [START_REF] Tome | The relation between macroscopic and microscopic strain hardening in F.C.C. polycrystals[END_REF]:

𝑔 𝛼 = 𝑔 0 𝛼 + (𝑔 1 𝛼 + 𝜃 1 𝛼 Γ) (1 -𝑒𝑥𝑝 (- 𝜃 0 𝛼 Γ 𝑔 1 𝛼 )) (5)
where 𝑔 𝛼 is the current CRSS, 𝑔 0 𝛼 is the initial CRSS, Γ is accumulated shear on all slip/twinning systems, 𝜃 0 𝛼 is initial hardening rate, and 𝑔 1 𝛼 and 𝜃 1 𝛼 determine asymptotic characteristics of hardening.

A cubic volume with the side length of 400 𝜇𝑚, located in the centre of the 3D-XRD probed volume was meshed for CPFE modelling. First order cubic elements (C3D8) were used for meshing the simulation cube. To assign a grain orientation to each element, the weighted Voronoi tessellation scheme described in Abdolvand et al [START_REF] Abdolvand | On the deformation twinning of Mg AZ31B: A three-dimensional synchrotron X-ray diffraction experiment and crystal plasticity finite element model[END_REF] was used:

𝐶 𝑖 = {𝑋 ∈ 𝑅 𝑑 |‖𝑋 -𝑠 𝑖 ‖ 2 -𝑤 𝑖 2 <‖𝑋 -𝑠 𝑗 ‖ 2 -𝑤 𝑗 2 , 𝑖 ≠ 𝑗} ( 6
)
where X is the position vector of the element 𝐶 𝑖 in the cube, 𝑠 𝑖 is the position vector of the seed point of the grain 𝑖, and 𝑤 𝑖 is the radius of the same grain measured by 3D-XRD. To determine the seed point of each grain, Eq. 6 is solved in two steps. In the first step, 𝑠 𝑖 is replaced by the measured COMi of the grain Gi. This results in a new COMb for Gi which does not necessarily coincide with the actual measured COMi. In the second iteration, 𝑠 𝑖 is set equal to 2COMb-COMi. An example of the simulation cube imported into Abaqus FE solver is shown in Fig. 2a where 10 6 IPs were used to discretise 1335 grains, i.e., more than 749 IPs per grain.

To simulate the experiment, the model was first cooled down to room temperature to model the thermal residual stresses that develop during the heat treatment. The coefficients of thermal expansion for single crystal zirconium are 5.3 x10 -6 and 10.1x10 -6 1/°C along the crystal a-and c-axis, respectively [START_REF] Xu | Modeling lattice strain evolution during uniaxial deformation of textured Zircaloy-2[END_REF].

Although the positions of the grips were fixed during each measurement step and total elongation of the specimen was constant, deformation creep was observed over the gauge of the specimen. Hence, the "as-measured" strain rate of each step was applied to the model as a velocity. This was done by dividing each measurement step into three sub-steps; in the first sub-step, the specimen was deformed to a peak load, in the second sub-step the specimen relaxed while realigning, and in the third sub-step the specimen relaxed during the collection of diffraction patterns. The recorded strain and time at the beginning and end of each sub-step were used to calculate the applied velocity to the model. In The slip systems considered in the modelling are prismatic〈112 ̅ 0〉, basal 〈112 ̅ 0〉, and pyramidal 〈112 ̅ 3〉 with the {101 ̅ 2} 〈101 ̅ 1〉 twinning system included as pseudo slip. The strain rate and hardening parameters used for each slip or twinning system are provided in Table 1. The elastic constants of zirconium single crystal were taken from Fisher and Renken [START_REF] Fisher | Single-crystal elastic moduli and the hcp ??? bcc transformation in Ti, Zr, and Hf[END_REF]: C11=143.5 GPa, C33=164.9 GPa, C12=72.5 GPa, C13=65.4 GPa, and C44=32.1 GPa. No twins existed in the model in the first increment, but nucleation was allowed when the relative volume fraction of a "possible" twin at an IP reached a critical value, i.e. 0.03. At this stage, the orientation of the twin was determined by reflecting the parent orientation across the twin habit plane as explained in Abdolvand et al [START_REF] Abdolvand | Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2[END_REF]. No stress calculation was done for the twinned domain and it was simply assumed that stress in the twin was equal to that of the parent.

Results

In all results presented, 𝜎 11 , 𝜎 22 , 𝜎 33 , represent the normal stresses measured along the beam direction, the transverse direction, and the loading direction, respectively.

General trends

The statistics of the 3D-XRD experiment for the measured volume are shown in Table 2. At the preload (Step-1), 11703 grains were indexed with an average of 119 peaks per grain. It is shown that all volume average stress components are close to zero except for 𝜎 33 , which increases with the macroscopic stress. The grain weighted average for 𝜎 33 falls in between the maximum macroscopic stress measured for each step and the one at the end of the pattern collection when the specimen is relaxed. Due to the conservative criteria imposed in the grain indexing codes and plasticity-induced peak smearing, the number of measured grains drops with increasing strain, except for the unload step (Step-9) where elastic strain and rotations are restored, and it was possible to index more grains. The average number of peaks per grain, however, is more than 97 for all steps. The measured error bar for each stress component is also provided in Table-2. In Fig. 3. the propensity of the misorientation between the HCP crystals c-axis and the beam direction (𝛽) for different steps are shown. The specimen was cut so that HCP crystals have their c-axis both along the LD and TD so that with loading, a combination of both twinned and un-twinned grains can be studied. The twinned grains are mainly the "hard" grains with their c-axis along the LD and since tensile twinning results in 85° reorientation, twins will appear along both TD and BD. In the undeformed specimen, only a few grains had their c-axis along BD (see Table 3), hence, we plot the propensities with respect to BD to avoid mixing parents with twins. For each step, propensities are determined by the number of grains measured for a given 𝛽 range to the total number of grains indexed for the same step. In Fig. 3a, the propensity of 𝛽 for all grains is shown. It can be seen that, while the propensity of grains with 𝛽 > 75° decreases, that of grains with 𝛽<25° increases. This shows that twins form with tensile loading, where a fraction of parent grains transforms into twins. Since 𝛽 for twins spread from 0° to 90°, i.e. from BD to TD, the propensity of twins is overshadowed by those from parents. Hence, the number of twins as a function of 𝛽 is plotted in Fig. 3b. In this figure, only the twins that were successfully matched with parent grains are shown. It can be seen that in Step-2 only a few twins exist, but with further loading, the number of twins increases where a peak is observed at 𝛽 = 25°. At the early stages of plasticity, a rapid growth in the number of twins is observed, e.g in steps 3, 4, and 5, but the number of captured twins has relatively stayed the same for steps further into the plasticity zone, e.g., steps 6, 7, and 8. The number of captured twins from the developed matching method is provided in Table 3. In order to confirm the validity of this method, the number of matched twins is compared with the total number of grains with 𝛽 less than 30° and 40°. This can be considered as a "validation" test for the developed matching method as 𝛽 was not used for defining a twin or a pair. In Table-3, it is shown that it was possible to track 16 pairs back to Step-1. We believe that these pairs formed during polishing of the specimen. This number falls in between 7 and 53 grains that have their 𝛽 less than 30° and 40°. It is worth mentioning that twins form with their c-axis both along BD and TD and the numbers presented in the second and third columns of Table 3 only show a fraction of twins, not all of them. Nevertheless, the observed trends are consistent from the two methods; for example, the number of twins jumps in Steps 3, 4, and 5 and stays unchanged in step 6, 7, and 8. The reduction observed in Step-7 is due to plasticity and significant peak smearing that reduces the number of indexed grains; however, due to reducing the X-ray beam height in the subsequent step (Step-8), a better grain indexing rate was achieved and the number of pairs returned close to what it was in Step-6.

In order to determine the smallest twin size indexed in this experiment, the measured relative volumes for twin-parent pairs are used. Grain size for parent grains is defined as grain diameter assuming that grains are spherical and space filling. Since twins are lenticular, if we assume that they cut through their parent grain, their thickness can be determined as t T = 4V T πD P 2 , where 𝑉 𝑇 is the volume of the twin from peak analysis, and 𝐷 𝑃 is the diameter of its corresponding parent grain. With the use of this analysis, the thickness of the smallest twins at Step-2 and Step-3 are 0.051 μm and 0.059 μm respectively. If these twins were assumed to be spherical, their diameters would respectively be 11.5 𝜇𝑚 and 14.3 𝜇𝑚.

The distribution of 𝜎 33 and evolution of texture are shown in Fig. 4. The results shown in Fig. 4a-4h are the (0002) pole figures from 3D-XRD, where each circle represents a grain, with the colour being proportional to the measured 𝜎 33 and the size being proportional to the measured relative volume for the grain. Similarly, the results for the modelled grains are shown in Fig. 4i-4l. To better show the variations, the limits of the colour bars are rescaled for each of the sub-figures. It is shown that, at the preload, crystals with their c-axis oriented towards the LD generally have tensile 𝜎 33 , whereas those oriented towards TD have compressive 𝜎 33 . These are the residual stresses that developed during the heat treatment. The trends observed in the experiment at the preload are consistent with those calculated with CPFE considering thermal contraction during cooling as shown in Fig. 4i. Further, in section 5.2, it is shown that the tensile residual 𝜎 33 observed in some grains will promote nucleation of twins as early as Step-2, before the onset of macroscopic plasticity. In Fig. 4b the measured poles at Step-2 are shown. The poles located towards TD are elastically and plastically "soft" grains that exhibit low stresses. This is further captured in the CPFE modelling as shown in Fig. 4j. In Fig. 4c, new poles are observed at ~25° from the BD, which are also observed in the results of CPFE modelling (Fig. 4k) confirming the nucleation of twins at the onset of macroscopic plasticity. The stresses calculated with CPFE are slightly higher, consistent with the simulated macroscopic response. This is due to the assumption that stress within the twin is equal to that of the parent and underestimating the effects of the twin transformation strain. Specifically, the constitutive model, which uses a pseudo-slip formulation, does not explicitly consider twinned domains. It therefore does not allow evaluating how stresses are partitioned between parent and twinned domains. A rapid increase in the number of new poles is observed in Fig. 4d and 4e reflecting formation of more new twins. The same trend is observed in the results of CPFE where new poles are clearly formed close to TD with many others at ~25° from the BD. In steps 6 to 8, the distributions of 𝜎 33 and (0002) poles do not change significantly (Fig. 4f-4h). This is also reflected in the macroscopic stress-strain curve where almost no hardening is observed between these steps.

Twin-parent pairs: early stages of plasticity

The development of stress in the matched twin-parent pairs is shown in Fig. 5. Only steps 3 to 5 are discussed in this section, i.e., the nucleation steps at which a jump in the number of twins is observed. In Fig. 5a, the measured axial stress 𝜎 33 of twins at Step-3 is plotted as a function of that of the corresponding parents. The blue circles represent the pairs while the red dot represents the average value calculated for all pairs. The black dashed lines are to bisect the figures and highlights when the quantity of the vertical axis is equal to that of the horizontal axis. It can be seen that, on average, twins are generally less stressed than their parents, although some twins develop higher 𝜎 33 than their parents.

The measured 𝜎 33 in twin and parent pairs for steps 4 and 5 are respectively shown in Fig 5b and5c. While more twin and parent pairs are captured, on average twins are generally less stressed. However, with applying more strain, the red dots are distancing from the dashed black line. That is, 𝜎 33 in twins is closer to their corresponding parents, and twins are more stressed in Step-3 than Step 4 and 5.

Unlike 𝜎 33 , the stresses in the TD (𝜎 22 ) and BD (𝜎 11 ) within twins are higher than their corresponding parents (see Figs. 5d and5e). These two stress components are only provided for Step-3, but the observed trends hold for the rest of the steps. For example, while the average 𝜎 22 and 𝜎 11 for parent grains are nearly zero, the corresponding values for twins are positive and moderately increasing from

Step-3 to Step-5 (Fig. 5f).

With the method developed for matching twin-parent pairs it was possible to determine which one of the six variants of {101 ̅ 2} 〈101 ̅ 1〉 twinning system was active. Once the active variant was determined, it was possible to rotate the measured stress tensor to the twin coordinate system. In Figs. 5g to 5i, 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 𝑇𝑤𝑖𝑛 and 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 𝑃𝑎𝑟𝑒𝑛𝑡 represent the stresses normal to the twin habit plane calculated using the measured stresses within the twin and parent, respectively. Likewise, in Figs. 5j to 5l, 𝜏 𝑅𝑆 𝑇𝑤𝑖𝑛 and 𝜏 𝑅𝑆 𝑃𝑎𝑟𝑒𝑛𝑡 represent the shear stress calculated on the twin habit plane in the twinning direction using the measured stresses within the twin and parent, respectively. In all figures, the normal to the habit plane and shear direction from the parent grain is used to project stresses into a given direction. A nearly "perfect" linear relationship between 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 𝑇𝑤𝑖𝑛 and 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 𝑃𝑎𝑟𝑒𝑛𝑡 is observed which holds for all measurement steps. For example, the average of 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 𝑃𝑎𝑟𝑒𝑛𝑡 at Step-3 is 121.3 MPs while that of 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 𝑇𝑤𝑖𝑛 is 119.9 MPa. With increasing the applied strain and proceeding to Step-4, this average for parents and twins is respectively, 135.8 MPa and 127.9 MPa. In all measurement steps, 𝜎 𝑛𝑜𝑟𝑚𝑎𝑙 for both twin and parent grains is positive and varies between 50 MPa and 300 MPa.

It is shown in Fig. 5j that while the resolved shear stress within both twin and parent is positive, 𝜏 𝑅𝑆 𝑃𝑎𝑟𝑒𝑛𝑡 is two times higher than 𝜏 𝑅𝑆 𝑇𝑤𝑖𝑛 . Proceeding to Step-4 and Step-5, a wider distribution of 𝜏 𝑅𝑆 within the parent grains is observed, but not so within the twins. This suggests that, at the early stages of plasticity, parent grains undergo more plastic deformation than the twins, and since twins are thin, they deform mainly elastically. In addition, on average, 𝜏 𝑅𝑆 in both twin and parent grains increases with increasing the applied stress. To better visualize the state of 𝜏 𝑅𝑆 within the captured twin-parent pairs at Step-5, a 3D plot of the pairs is shown in Fig. 6a. , where the denominator represents the norm of stress tensor and 𝑃 𝛼 is the symmetric part of the Schmid tensor for the active variant [START_REF] Abdolvand | Multi-scale modeling and experimental study of twin inception and propagation in hexagonal close-packed materials using a crystal plasticity finite element approach; Part II: Local behavior[END_REF]. The same equation is used for calculating the global SF, but the applied stress was used instead of the grain resolved stress tensor. Once the SFs for all six twin variants were calculated, they were sorted from the highest to the lowest and the rank of the twin variant was determined. The rank of the twinned variants for global and local SFs was determined independently.

In Figs 6b and6c, it is shown that all twin variants are active, but the first three variants are more frequent, regardless of using global or local stresses in determining SFs. The twins captured in Step-3 are generally from the parent grains with their c-axis nearly parallel to the LD and hence, all of the six twin variants are equally active. However, with the use of the local stress for calculating SFs, a better estimation of the active variants is obtained since the frequency of the strongest variants is higher. With proceeding to Step-4, similar trends are observed for the distribution of twin variants using local and global SFs.

Twin-parent pairs: plasticity

Between Step-6 and Step-8, the macroscopic strain varies from 1.52% to 2.6 % while the macroscopic axial stress remains quasi-constant, which corresponds to the absence of work hardening. The development of stresses in twin-parent pairs for steps 6 to 8 are shown in Fig. 7. It is shown that, similar to steps 3 to 5, the axial stress 𝜎 33 in twins is generally less than that of parents (Figs. 7a to 7c). Due to the absence of work hardening, the average values (red dots) for the stress components 𝜎 33 , 𝜎 22 and 𝜎 11 do not change significantly between steps 6 and 8.

The normal stresses (𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 ) acting on the twin habit plane for steps 6 to 8 are shown in Figs. 7d to 7f. Results are consistent with those measured in steps 3 to 8 where a linear relationship between 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 within parents and twins is captured. In all cases 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 is positive and varies between 50 MPa to 350 MPa, a slightly broader range than that measured for steps 3 to 5. 𝜏 𝑅𝑆 measured in twinparent pairs for steps 6 to 8 are shown in Figs 7g to 7i. Consistent with previous results, 𝜏 𝑅𝑆 within parents are higher than 2𝜏 𝑅𝑆 𝑇𝑤𝑖𝑛 . Also, with further loading from Step-6 to Step-8, the distribution of 𝜏 𝑅𝑆 for parents further broadens, whereas for twins not much broadening is observed. This is also reflected in the calculated average error bars and indicates that twin and parent grains accommodate the applied strain at different rates.

Twin-parent pairs: Unload

After deforming the specimen to 2.6% strain, the specimen was unloaded to nearly zero applied stress at the unload step (Step-9). The measured (0002) pole figure for all grains measured at Step-9 is shown in Fig 8a . The range of 𝜎 33 at unload is similar to that of preload, i.e. -150 MPa to 150 MPa, but a less distinct variation of 𝜎 33 from TD to LD is observed at the unload step. This means that after deforming the specimen to 2.6% applied strain, the effects of thermal residual stresses disappear. In addition, at unload, the grains located at TD are slightly in compression while those located at LD are in tension. This follows the trends observed at Step-8 where the soft grains with their crystal c-axis pointing towards TD are less stressed than hard grains (Fig. 4h).

The stress state within twin-parent pairs at unload is shown in Fig. 8b-8d. It is shown that upon unloading, the average of 𝜎 33 for both twins and parent grains is nearly zero (Fig. 8b) while 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 in twins is still linearly correlated to that of the corresponding parents (Fig. 8c). At unload, unlike the other 7 steps, 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 is not systematically positive and varies between -100 MPa to 100 MPa. Interestingly, while the average of 𝜏 𝑅𝑆 in the twins is -36.9 MPa, that in parent grains is positive and is 6.8 MPa (Fig. 8d). An important conclusion that can be drawn from these data is that 𝜏 𝑅𝑆 in both twin and parent can be negative while the twin is still not annihilated.

Discussion

General discussion

So far the state of stress within twin and parent pairs of each step is studied independently from other steps and from CPFE results. The stress-strain curve for all twin-parent pairs measured in the 3D-XRD experiment is shown in Fig. 9a. The blue and black lines represent individual parent and twin grains, respectively, while the green and red line respectively represent the average stress measured for parents and twins. In section 4.2, it was shown that 𝜎 33 within twins at Step-3 is close that in parents, but with proceeding to step 5, the difference between the stresses increases and twins relax. In Fig 9a, it is shown that the red line nearly coincides with the green line at Step-3. This could be due to a size effect, i.e. that twins of Step-3 are smaller than those in steps 4 and 5 and hence are harder. To further investigate this aspect, the state of stress for the 64 twins that were captured in Step-2 is studied. These 64 twins are the ones that nucleated in the hard grains with positive thermal residual stresses that helped nucleation of twins in Step-2. The average stress for all twins of each step are calculated and reduced from those from the parents and results are provided in Table 4. It can be seen that the difference between 𝜎 33 of the pairs increases from Step-3 to Step-5. While this difference at Step-3 is 28.5 MPa, at Step-2 is only 16 MPa indicating that twins are not much relaxed at the early stages of plasticity.

Similarly, the difference between the resolved shear stress within twin and parent in Step-2 is minimum yet increases with further loading and increasing to Step-5. This observation supports the previous numerical studies that concluded that directional Hall-Petch effects [START_REF] Proust | Modeling the effect of twinning and detwinning during strain-path changes of magnesium alloy AZ31[END_REF], as well as non-local effects [START_REF] Paudel | Characterization and modeling of { 10 1[END_REF] should be incorporated into modelling of twins. In addition, it suggests that twin and parent grains accommodate the applied strain at different rates. Proceeding further into the plasticity zone, no significant changes in the difference between measured 𝜎 33 of twins and parents is observed in steps 5 to 8 (Table 4). Interestingly, 𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 in all steps, including unload, stays the same in parents and twins. The CPFE results for the stress-strain curve for all parent grains are shown in Fig. 9b. Results are for the average stresses calculated from all IPs assigned to each grain so that a like-to-like comparison can be made and data are extracted at the end of each measurement step. Generally, a good agreement with experimental results is achieved for the average response of the cluster of modelled parent grains. The stress-strain curve for some of the grains that did not twin in the experiment and model are plotted in Fig. 9c. This figure shows that in the absence of twinning, a good agreement can be achieved at the grain level, however, the current constitutive equation should be modified to capture a better response for the twinned grains (see below).

Stress development: individual pairs

In this section, the behaviour of individual twin-parent pairs is studied to understand the origin of the trends reported in the previous sections for the average behaviour of the pairs. In Fig. 10, the measured stress-strain curves for several twin-parent pairs are provided. For these pairs, the parent grains were imported into CPFE model and it was possible to track them back to Step-1. Only the steps that the pairs were indexed, matched, and tracked with high confidence are shown. Modelling results for each step are the average 𝜎 33 calculated from all IPs assigned to the parent. In Figs. 10a to 10c, a twin was captured in Step-3 where in all cases, a distinct drop in the stress of the parent is observed. This means that upon nucleation of twins, a discernible relaxation takes place within the parent grain which is not observed in any of the CPFE results. The stress relaxation is due to twin transformation strain that is underestimated in the current constitutive equations. This stress relaxation in combination with the hard behaviour of twins at the nucleation step, due to their small sizes, are the reasons for the high stress measured for the twin of the grain 5311 (Fig. 10a). A similar trend is observed in grain 4828 and 5623 (Figs. 10c and10d), where 𝜎 33 in the parent is relaxed at the nucleation step, but 𝜎 33 in twins in steps 3 and 4 are close to those of their corresponding parents. With further loading, the difference between the stress of twin and parent increases. In addition, two more pairs are shown in Figs. 10d and10e, where in both cases, 𝜎 33 of the twin coincides with that of the corresponding parent at the nucleation step, but similar to the previous cases, the difference between the two increases with further loading.

In Figs. 10f and 10g, a twin was captured as early as Step-2 where in both cases, 𝜎 33 of twins are nearly equal to those of their corresponding parents. It can be seen that both parent grains had tensile thermal residual stresses at the preload step, which promoted nucleation of twins in Step-2. Further, with the matching method that was developed in this paper, it was possible to capture multiple twins for a single parent as shown in Fig 10g . Two distinct stress relaxation events are observed in the grain 3685 (Fig. 10g); the first in Step-3 after the nucleation of the first twin and the second in Step-5, upon nucleation of the second twin. Interestingly, the stress in the second twin is also very close to that of parent confirming that twins at nucleation are not very much relaxed.

In Figs. 10h and10i, two pairs are shown where the twins were captured in Step-4 and were relaxed. In the parent grain 5380 (Fig. 10h), a stress relaxation is observed in Step-5 which is due to twinning. The difference between the measured 𝜎 33 for twin and parent is minimum in Step-4, but increases with further loading. This is also observed in grain 7459 (Fig. 10i).

In all results presented in this paper, the measured and simulated stresses are the average values for twins or parents. The state of the stress within a twin can vary significantly particularly close to twin tips and boundary [START_REF] Barnett | Plastic relaxation of the internal stress induced by twinning[END_REF][START_REF] Siska | Numerical analysis of twin thickening process in magnesium alloys[END_REF]. Likewise, stress within the parent grain can significantly vary close to twin boundaries. In section 4.4, we showed that upon unloading, 𝜏 𝑅𝑆 within twins and parents can be negative while the twin is not yet annihilated. Higher resolution experimental techniques as well as numerical modelling are required to further study the stress field in the vicinity of twins.

Conclusions

A three-dimensional synchrotron X-ray diffraction experiment was conducted on a pure zirconium specimen to study the evolution of stress in twin-parent pairs. The measurement was conducted during an in-situ uniaxial tensile experiment and in nine steps: preload, one step in the elastic zone but close to onset of plasticity, three steps that covered the onset of plasticity, three more steps in the plastic zone, and finally at unload. The measured microstructure was imported into a crystal plasticity finite element model to simulate deformation of individual grains. A new method was developed to match the newly nucleated twins with their corresponding parents and track them over different loading steps. It is shown that [START_REF] Aydiner | Evolution of stress in individual grains and twins in a magnesium alloy aggregate[END_REF] The difference between the stress along the loading direction (𝜎 33 ) measured for twins and parents is at minimum at the early stages of plasticity but increases with more plastic deformation.

(2) Twins are generally not relaxed, rather stressed in the first step that they are captured.

Relaxation due to plasticity takes place within twins with further loading, but at a different rate relative to their corresponding parent grains. (3) The other two normal stress components, 𝜎 11 and 𝜎 22 , in the twins are slightly higher than their corresponding parents. (4) A distinct relaxation is observed in the curve of the parent grains at the twin nucleation step. [START_REF] Bieler | In situ characterization of twin nucleation in pure Ti using 3D-XRD[END_REF] The normal stress acting on the twin habit plane (𝜎 𝑁𝑜𝑟𝑚𝑎𝑙 ) is similar for both parent and twin grains. (6) When twins are nucleated with their c-axis perpendicular to the tensile loading direction, the resolved shear stress acting on the twin plane in the twinning direction (𝜏 𝑅𝑆 ) within the twin is less than half of the 𝜏 𝑅𝑆 measured in the corresponding parent grain.

(7) For a subset of twin-parent pairs, 𝜏 𝑅𝑆 at the unload was negative in both twin and parent grains while the twin was not annihilated. [START_REF] Schuren | New opportunities for quantitative tracking of polycrystal responses in three dimensions[END_REF] The tensile thermal residual stresses within parent grains promote nucleation of twins as early as the beginning of the onset of plasticity. (9) Thermal residual stresses in zirconium disappear with unloading from 2.6% applied strain. [START_REF] Wu | Twinning-detwinning behavior during fatigue-crack propagation in a wrought magnesium alloy AZ31B[END_REF] The first three twin variants with higher Schmid factors are the most active ones, regardless of using the local measured stress tensor or the applied stress in determining Schmid factors. A better estimation of the active variants is obtained at the early stages of plasticity with the use of the local stresses for calculating Schmid factors. [START_REF] Wu | Internal stress relaxation and load redistribution during the twinning-detwinning-dominated cyclic deformation of a wrought magnesium alloy, ZK60A[END_REF] With the method developed here, it was possible to capture multiple twins per parent. It is shown that the evolution of stress within the second twin follows the trends reported, i.e. stressed at nucleation but relaxed with loading.

Fig. 1 .

 1 Fig. 1. (a) The measured (0002) pole figure of the undeformed specimen. (b) A schematic of the specimen with the coordinate system used in this paper. (c) An example of the twin-parent pairs found in Step-4. Grains are colour coded with respect to their c-axis misorientation with the loading direction and the size of each sphere is proportional to the measured relative volume of the grain. In (a) and (b) LD, BD, and TD refer to loading, X-ray beam, and transverse directions.

Fig. 2 .

 2 Fig. 2. (a) The simulated microstructure imported into the finite element solver. (b) A comparison between the insitu measured strain and the calculated strain from the model. (c) A comparison between the in-situ measured and simulated macroscopic stress strain curve.

Fig. 3 .

 3 Fig. 3. The distribution of 𝜷 for (a) all grains and (b) only twins, at different measurement steps. 𝜷 is the rotation angle between the HCP crystal c-axis and the beam direction.

Fig. 4 . 1 , 4 .

 414 Fig. 4. The evolution of texture and stress for individual grains. 3D-XRD (𝟎𝟎𝟎𝟐) pole figures for (a) Step-1, (b) Step-2, (c) Step-3, (d) Step-4, (e) Step-5, (f) Step-6, (g) Step-7, and (h) Step-8. CPFE results for (i) Step-1, (j) Step-2, (k) Step-3, (l) Step-4. Each circle represents a grain where the colour of the circle is proportional to the measured or calculated stress along the loading direction (𝝈 𝟑𝟑 ), as shown in the colour bar in the right-hand side of the figure. In (a) to (h) the size of the circles is proportional to the measured relative volume of the grain.

Fig. 5 . 5 ;

 55 Fig. 5. Stress in the twin as a function of stress in the corresponding parent: 𝝈 𝟑𝟑 at (a) Step-3, (b) Step-4, and (c) Step-5; 𝝈 𝟐𝟐 at (d) Step-3, and 𝝈 𝟏𝟏 at (e) Step-3, and (f) Step-5; 𝝈 𝑵𝒐𝒓𝒎𝒂𝒍 at (g) Step-3, (h) Step-4 and (i) Step-5; 𝝉 𝑹𝑺 at (j) Step-3, (k) Step-4, and (l) Step-5. 𝝈 𝑵𝒐𝒓𝒎𝒂𝒍 and 𝝉 𝑹𝑺 are the normal and shear stresses acting on the twin habit plane and in the twin direction, respectively. The dashed back lines are to bisect the figures and indicate when the quantity of the vertical axis is equal to that of the horizontal axis. Red crosses are the average values for each population with their corresponding average error bars.

Fig. 6 .

 6 Fig. 6. (a) a 3D view of twin-parent pairs captured in Setp-5. The size of the spheres is proportional to the measured relative volumes and colours are proportional to the measured resolved shear stresses on the twin habit planes. The number of twins for a given twin variant at (b) Step-3 and (c) Step-4. In (b) and (c) Global SF and local SF represent the Schmid factors calculated using the measured global or local stresses and twin variants are ordered from highest to the lowest.

Fig. 7 . 8 ;

 78 Fig. 7. Stress in the twin as a function of stress in the corresponding parent: 𝝈 𝟑𝟑 at (a) Step-6, (b) Step-7, and (c) Step-8; 𝝈 𝑵𝒐𝒓𝒎𝒂𝒍 at (d) Step-6, (e) Step-7 and (f) Step-8; 𝝉 𝑹𝑺 at (g) Step-6, (h) Step-7, and (i) Step-8. 𝝈 𝑵𝒐𝒓𝒎𝒂𝒍 and 𝝉 𝑹𝑺 are the normal and shear stresses acting on the twin habit plane and in the twin direction, respectively. The dashed back lines are to bisect the figures and indicate when the quantity of the vertical axis become equal to that of the horizontal axis. Red crosses are the average values for each population with their corresponding average error bars.

Fig. 8 .

 8 Fig. 8. 3D-XRD results for unload (Step-9): (a) (𝟎𝟎𝟎𝟐) pole figure; stress in the twin as a function of stress in the corresponding parent (b) 𝝈 𝟑𝟑 , (c) 𝝈 𝑵𝒐𝒓𝒎𝒂𝒍 , and (d) 𝝉 𝑹𝑺 . The color of each pole in (a) is proportional to the measured 𝝈 𝟑𝟑 and the size of the pole is proportional to the measured relative volume.

Fig. 9 .

 9 Fig. 9. (a) The measured stress-strain curve for twin-parent pairs from 3D-XRD. The green and red lines represent the average values for each population. (b) CPFE results for paent grains. (c) Comparison between CPFE and 3D-XRD results for the grain that did not twin. In all sub-figures strains on the horizontal axis are the macroscopic applied strains.

Fig. 10 .

 10 Fig. 10. Stress-strain curve for individual twin-parent pairs. Parent IDs are provided in the titles of figures, and IDs are generated during grain indexing. Blue curves represent 3D-XRD results for parents, black and green curves represent 3D-XRD results for twins, and red curves represent CPFE results for parent grains. In all sub-figures strains on the horizontal axis are the macroscopic applied strains.

Table 1 : The single crystal properties of zirconium used in CPFE modelling

 1 [START_REF] Abdolvand | Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2[END_REF]

		𝒏	𝜸̇𝟎(𝒔 -𝟏 )	𝒈 𝟎 𝜶 (GPa)	𝒈 𝟏 𝜶 (GPa)	𝜽 𝟎 𝜶 (GPa)	𝜽 𝟏 𝜶 (GPa)
	Prism	20	3.5×10 -4	0.12	0.33	0.01	0
	Basal	20	3.5×10 -4	0.168	0.22	0.05	0
	Pyramidal	20	1.0×10 -4	0.331	0.27	0.62	0.28
	Twinning	20	1.0×10 -4	0.25	0.2	0.05	0

Table 2 : 3D-XRD measurement steps and statistics.

 2

	Step ID	Description	Applied Strain %	Applied stress (MPa) Start* End**	No. of grains	No. of Peaks per grain	Grain weighted average of stresses with estimated errors (MPa) 𝝈 𝟏𝟏 𝝈 𝟐𝟐 𝝈 𝟑𝟑 𝝈 𝟐𝟑 𝝈 𝟏𝟑 𝝈 𝟏𝟐	Beam height (𝝁𝒎)
	1	Step-1	0.001	8	8	11703	119	-4.0 ±17.5	0.3 ±13.3	1.6 ±11.4	-0.1 ±3.2	0.3 ±3.4	-1.0 ±4.3
	2	Step-2	0.257	180	172	11308	118	-6.3 ±16.8	-6.2 ±12.6	169.4 ±10.8	0.2 ±3.1	1.4 ±3.3	-1.4 ±4.1
	3	Step-3	0.355	233	226	11087	117	-4.4 ±15.9	2.6 ±11.2	227.1 ±10.3	0.7 ±2.9	1.5 ±3.11	-1.4 ±4.0
	4	Step-4	0.565	297	274	10995	115	-2.8 ±15.5	1.6 ±11.7	287.1 ±10.0	0.9 ±2.8	1.3 ±3.03	-1.6 ±3.9	40
	5	Step-5	1.087	321	295	10578	109	-2.1 ±21.0	-1.8 ±15.7	301.3 ±13.5	1.1 ±3.9	0.6 ±4.1	-1.5 ±5.2
	6	Step-6	1.524	329	304	9071	105	5.9 ±26.6	1.1 ±19.6	315.3 ±17.0	1.7 ±4.8	0.3 ±5.2	-1.6 ±6.6
	7	Step-7	1.890	335	310	8042	102	12.7 ±29.5	5.5 ±21.7	325.6 ±18.9	0.8 ±5.4	1.3 ±5.8	-1.3 ±7.5
	8 9	Step-8 Step-9	2.600 2.271 x	344 2	306 3	7312 8455	97 102	20.3 ±44.0 17.8 ±43.3	3.1 ±32.2 32.2 ±31.4	327 ±28.0 -3.9 ±27.3	2.4 ±8.0 0.8 ±7.6	0.3 ±8.6 -0.2 ±8.4	-1.4 ±11.0 -2.7 ±10.7	25
	*

Measured at the end of reloading; ** Measured at the end of pattern collection. X Measured strain at unload

Table . 3. Results from twin-parent matching process

 .

	Step	Number of grains with 𝜷 < 𝟑𝟎°	Number of grains with 𝜷 < 𝟒𝟎°	Number of matched pairs
	Step-1	7	53	16
	Step-2	18	82	80
	Step-3	34	100	113
	Step-4	87	189	282
	Step-5	187	341	499
	Step-6	196	354	506
	Step-7	181	303	410
	Step-8	212	348	492
	Step-9	255	431	591

Table 4 . The average difference between measured stresses in twins and parents

 4

	Step	𝝈 𝟑𝟑 𝑷𝒂 -𝝈 𝟑𝟑 𝑻𝒘 (MPa)	𝝈 𝑵𝒐𝒓𝒎𝒂𝒍 𝑷𝒂 (MPa) -𝝈 𝑵𝒐𝒓𝒎𝒂𝒍 𝑻𝒘	𝝉 𝑹𝒔 𝑷𝒂 -𝝉 𝑹𝒔 𝑻𝒘 (MPa)
	Step-2	16	1.4	62.7
	Step-3	28.5	1.4	70.2
	Step-4	46	7.9	74.6
	Step-5	51.3	7.8	78.0
	Step-6	49.4	8.2	80.0
	Step-7	45.7	11.5	79.1
	Step-8	35.8	3.6	81.8
	Step-9	3.5	5.5	43.6

Acknowledgment

We acknowledge the ESRF for allocating beamtime for the experiment MA-3613. This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).


Francesco Di

Giuliano Salviati

Ingrid Houssaye Michienzi

Dominique Cardon
email: cardon.dominique@wanadoo.fr

Sourcing and consumption of alum in the dyeing company "

Keywords:

de la Légion d'Honneur (2015) et de Chevalier de l'Ordre des Arts, des Lettres et de la Culture de Madagascar (2017).

Résumés

Cette étude, basée sur les informations fournies par trois livres de comptes conservés au sein de l'Archivio Salviati (Pise), offre un aperçu de l'importance de l'alun utilisé dans les processus de mordançage, dans une entreprise de teinture en grand teint, spécialisée dans les colorants à mordants, à Florence à la fin du XV e siècle. Elle fournit des preuves supplémentaires confirmant le nouvel essor de la production florentine de draps dans la seconde moitié du XV e siècle, en grande partie lié à la conquête des immenses marchés de l'Empire ottoman par les négociants florentins. mots-clés : alun -compagnie de teinture -Florence -livres de comptes -Salviati Based on evidence provided by three manuscript account books preserved in the Archivio Salviati (Pisa), this study offers a direct insight into the importance of alum used in mordanting processes, in a dyeing company specialized in fast mordant dyes in Florence in the end of the 15 th century. It provides further evidence confirming the new rise in the Florentine broadcloth production in the second half of the 15 th century, largely linked to the conquest of the huge markets of the Ottoman Empire by Florentine merchants.

Introduction

This study offers a direct insight into the importance of alum used in mordanting processes, in a dyeing company specialized in fast mordant dyes (Arte maggiore) in Florence in the end of the 15 th century. It is based on evidence provided by the study of three manuscript account books preserved in the Archivio Salviati, at the Scuola Normale Superiore, in Pisa (Italy). The Archivio Salviati is one of the most important archive in Europe for economic and social history. It contains thousands of account books and documents written from the 14 th century until the 17 th century, concerning textile production, bank and international trade with many places in Europe and beyond. The present study was but a small part of a vast interdisciplinary research program based on the study and publication of the Archive, under the acronym ENPrESa 1 . Our contribution to this joint effort includes a focus on the documents of this Archive that concern the history of dyeing. The three account books mentioned above are the only documents that have been preserved, concerning the dye company of "Francesco di Giuliano Salviati e Comp., tintori d'Arte Maggiore in Firenze". Francesco's father, Giuliano Salviati, was involved, with two of his cousins (Iacopo and Alamanno Salviati) in the production of wool broadcloth of high and medium qualities. Iacopo Salviati was also involved in the production of silk textiles. Francesco Salviati, through his dye company, was thus completing the activities of an important family business that operated in textile production. The two first books -Reg. 394, Libro azzuro, seg. A (Fig. 1), corresponding to the years 1483 to 1486 and Reg. 395, Libro giallo, seg. B (Fig. 2), corresponding to the years 1486 to 1491record all the company's commercial dealings organized by products and providers (buying dyes and mordants, paying furnishers, dyers, local custom duties, etc.). Those two books corresponded to the main ledgers (often called libro mastro, libro grande or libro debitori e creditori). In the Tuscan accounting systems of the 14 th -16 th centuries, that kind of books was the result of a particularly complex accounting chain. In fact, the Florentine agents multiplied the number of preparatory entries, most often recorded on a daily basis. The third register we studied, Reg. 396, is a Giornale (Fig. 3), recording all daily commercial activities from 1490 to 1498. It was indeed a giornale, which registered the day-to-day operations, that, later, were debited and credited in the main ledger. The books refered to each other through constant writing games. The only way to really understand the operations is to cumulate the information contained in the different registers. Generally speaking, the books that are most focused on a specific activity, commercial or technical, are those that offer the most detail. As the information goes to the ledger, it is more and more synthesised. Consequently, the way to have the maximum amount of information is to juggle between the registers, thanks to internal cross-references. These complex, seemingly dry sources are extremely rich and shed new light on the economic and social history of the time. Unfortunately, no book on the technical aspects of the dyeing activity of the firm having been preserved, all that can be known about Francesco Salviati's company therefore is: what dyestuffs, mordants and connected products were bought and used, in what quantities each year, how and from where they were obtained, at what prices. All this information allows to better understand the kind of dyeing he specialized in.

The lack of any mention of blue dyestuff, either woad or indigo, shows that the company did not include a vat-dyeing branch. It obviously specialised in dyeing in fast dyes in boiling baths. The commodities most frequently and massively bought are, on one hand, the typical ingredients of mordanting baths as described in contemporary Italian treatises and regulations on dyeing: alum (alume), tartar (gromma) and bran (crusca); and on the other hand, red and yellow dyestuffs. Sources of reds include madder (robbia), kermes (grana), a local Porphyrophora cochineal (chermisi di Marche), sappanwood (verzino), processed orchil and orchil-producing lichens (oricello and raspa). Sources of yellows include saw-wort (ceretta), weld (gualda), young fustic (scotano) and a still unidentified yellow dye-plant, guilicie 2 .

"... li tentori di panni & lane, alli quali non è manco necessario chel pane al'huomo... "

As vital to wool dyers of broadcloths, or of wool in the fleece, as bread is to humans: this is how Vannoccio Biringuccio presents alum in the sixth chapter of the second book of his treatise Pirotechnia, first published in Venice in 1540: "Dell'alume di rocca, et sua minera" [START_REF][END_REF] . No wonder then, that alum is the third product to be bought, on november 20 th 1483, six days after Francesco di Giuliano Salviati has established his dye company. It appears at folio 5 of Reg. 394, the Libro azzuro. The first had been madder, recorded at folio 3, the earliest purchase of it being dated november 19 th . Next to be purchased is saw-wort, recorded at folio 4, with the first consignment dated november 20 th , the same day as alum. A source of red and a source of yellow: the two basic colours of the art of dyers in fast dyes, the Arte Maggiore. From then on, recordings of purchases of the complete range of red and yellow dyestuffs used in Italy at the time will keep appearing regularly in the three books of the company until the end, in parallel with massive purchases of alum. This in itsef is telling, since alum was not necessary in all branches of dyeing. It was useless, particularly, for blue dyeing in the woad vat used for wool or in the indigo vat used for silk, because the indigo from woad or from the exotic product, reduced to its colourless soluble form in these vats, precipitates into and onto the wool or silk fibres and takes back its blue form, simply through the action of oxygen from the air. The only additions needed in these processes are bran, which helps maintain the bacterial fermentation responsible for the reduction of the indigo, and ashes or lime, to maintain the alkaline pH necessary to the survival and proliferation of the reducing, alkaliphile, bacteria 4 . Since the dawn of the art of dyeing, however, dyers had noticed that some dyestuffs would not bind strongly enough with textile fibres to give intense, fast colours. Their empirical intuition helped them understand somehow that this had to do with the nature of these colorants -their chemical compositionand inspired them to use "mordants" to "bite" into the fibres and help such colorants get fixed more tightly to them. The mordants mostly used in such cases were metallic salts that facilitated the formation of a complexion between colorant, metal and the fibre to be dyed [START_REF] Cardon | [END_REF] . Alum was one of the most anciently and most massively used mordants, both because the diverse aluminium salts that were commonly referred to as "alum" were quite easily available -aluminium being the most common metal found in the earth's crust -and because mordanting with aluminium did not alter the colour of the dyes 6 . Since all the colorants he is going to use belong to the group of dyestuffs which need the textile to undergo a mordanting process, Francesco di Giuliano Salviati has to make sure that, from the start, his dyeworks are supplied with the indispensable alume.

On folio 5 of the Libro azzuro (Fig. 4) are recorded all the purchases of alum made by the Company between november 1483 and june 1485. The transcript below presents those made between the date of the beginning of the company and the end of that year, that is from November 20 th 1483 to March 8 th 1484 (the Florentine year then began on March 25 th , the day of the Annunciation). It will serve as a representative example for all entries, since these are all in the same standardized and much abbreviated format. These accounting data were copied from another type of record, the journal or giornale, where transactions were recorded daily.

In each reference the following information is given: -the quantity of alum purchased by the company, in Arabic numerals, written in the left margin in order to facilitate final additions.

-the date of the purchase -the amount of the purchase, in Roman numerals as well as in Arabic numerals, and the currency in which it was expressed. Different types of florins were then in use in Florence 7 . Prices here are expressed in florin a oro larghi, which was a gold currency.

-the quantity expressed in pounds, in Arabic numerals -the name of the supplier -the last part mentions the accounting counterpart of the creditor, with the corresponding folio number. The notion of double-entry implies that each entry in one account must find a symmetrical counterpart in another account. In fact, any amount charged to the accounts will be recorded twice: once as a debit to one account and a second time as a credit to another account. This principle ensures the balance of accounting transactions since any amount credited to one account will have to be offset by an identical transaction debiting another account.

-finally, the amount was again written on the right margin, to facilitate final additions.

Table 1. Register 394, f°5, left page (considering the standardized, repetitive structure of the entries, an example of translation needs only be given for the first two occurrences). "200 Alume per nostro conto de' dare a dì XX di novenbre 1483 fiorini tre larghi per libbre 200 chonpramo da l'arte della lane ; a lei avere in questo carta 2 f. III s." Translation: « 200 Alum for our account must give on November 20 th 1483, florins three larghi for pounds 200 we bought from the arte della lana ; hers to have in this book folio 2. f. III s."

"216 E a dì detto fiorini tre soldi IIII° denari 9 a oro larghi per libbre 216 d'alume da l'arte della lana ; a lei avere in questo carta 2 f. III s. IIII° d. VIIII°" Translation: "216 same day, florins three sols IIII° deniers 9 a oro larghi for pounds 216 of alum from the arte della lana ; hers to have in this book folio 2. f. III s. IIII° d. The three books thus allow to follow the consumption of alum by the company during twelve years. Most entries, and the most important quantities, concern purchases of alum. However, in a few occasions Francesco's company resold some alum to other dyers and, only once, to a clothier. In table 2, therefore, the quantities resold have been subtracted from the totals for each year, in order to offer a more precise picture of the amount of alum that was actually used in the company. All quantities are recorded in Florentine pounds. Our conversions are based on the equivalence: 1 Florentine pound = 0.339 kg 8 .

356.65

The sourcing of alum The picture emerging from the three books of Francesco Salviati's dye company is that of a rather strict administrative control of the alum trade in Florence during that period. This is shown by two characteristics of the records. Firstly, the commodity appears to be nearly exclusively in the hands of provveditori dell'Arte della lana, providers from the guild of clothiers : 220,461 pounds of alum (= 74,736 kg, more than 90% of the total quantity purchased by Francesco's company) are obtained from them, directly or through intermediaries who turn out to be clothiers, members of the Arte, such as Guido and Niccolò Mannelli e comp., lanaioli. In a few cases (amounting to a total quantity of 416 pounds or 141 kg), alum is simply mentioned as being "bought from the Arte della lana". In a few other instances a more precise mention seems to imply a special organization concerning the sourcing and distribution of alum within the Arte della lana: the providers are identified as provveditori dell'allume e dell'Arte della Lana (the corresponding transactions amounting to 1,386 pounds or 470 kg) or as provveditori dell'Arte della Lana e dell'allumeria (this concerns purchases for a total of 2,575 pounds or 873 kg). Six individual providers are mentioned for 15 purchases: they can be identified as clothiers and dyers from Florence and the price at which alum is obtained from them is the same as in the other transactions.

The remarkable stability of the prices recorded during the twelve years documented by the three books is another sign -and result -of the administrative control on alum. The cost of the mordant only varies from its constant -obviously regulated -value of 359/360 denari per 100 pounds in the last year, 1495, when its price suddenly drops. It had remained the same, around 359/360 denari, until April 1495 and then, the last 22 entries concerning alum in the Giornale, dated from May to September 1495, mention prices in the range of 288 denari per 100 pounds. As these are the last accounting records of alum in the Salviati archive, it is impossible to know from this source whether the market price fell and then remained stable, or whether it recovered afterwards. Fortunately, the records of a Medici dyeing company, preserved at Harvard University, complement these data and shed further light on the prices of alum until the early 16 th century. These are three account books, all giornali, where transactions were recorded on a daily basis 9 . They come from the company of Rafaello di Francesco de' Medici (1498-1505), then of Giovenco e Giovanni di Giuliano de' Medici (1508-1511). All these businessmen involved in craft activities, like Francesco Salviati, were tintori d'arte maggiore. The city of Florence had numerous arti. The term 'arts' was used to describe the unions of craftsmen, merchants and workers, who exercised the same profession and were subject to certain rules. There were major arts (Arti maggiori) grouping the professions involved in the most important economic, commercial and financial activities, and minor arts (Arti minori) concerning less important sectors. The art of the Tintori, born during the Ciompi revolt in 1378, in a context of radical questioning of the power relations between all the different professions of the textile industry, starting with wool merchants, was shortlived. It was suppressed in 1382, as soon as merchants and bankers returned to power. The dyers were thus dependent on other arts linked to the textile activities and divided into several categories.

In the oldest account book, concerning purchases made in 1498-1499, the price of alum constantly stays around 288 denari for 100 pounds of product. These references indicate that the price of alum has remained stable since 1495. The same policy was now applied to alum in order to secure the supply: Franco Franceschi has identified a period around 1441-1442 when the Consuls of the Arte della Lana fear a shortage of alum, caristiam aluminum, and do their best to replenish the stocks 11 . The stability of its price during most of the period documented by the Salviati archive seems to place this product out of competition and out of the interplay of supply and demand, and results from a desire for control by the authorities through the corporations. One of the objectives was certainly to avoid both the shortage of raw materials and sudden rises or falls in prices which could have major repercussions and cause major and sometimes dangerous tensions in a society whose economy and wealth were based on the textile industry.

The three books of Francesco Salviati's dye company unfortunately do not provide all the elements that determined and explained the price of alum on the Florentine market and its fluctuations at the end of the 15 th century. There obviously was a close link between the origin of alum, the control of the supply chains and the resulting price, but what particularly is missing in the records is information on the type(s) or provenance(s) of alum bought: native alum from Egypt, from the Solfatare of Pozzuoli, or from the Lipari and Aegean islands ? Potash alum from the alunite mines of Phocaea or La Tolfa ? By contrast, some elements about the origin of alums appear in the account books of the Salviati commercial company of Florence, kept during the same period (Averardo di Alamanno e Iacopo di Giovanni Salviati e compagnia di Banco di Firenze). These records refer to purchases of alum, in very large quantities, from Tuscan deposits, for export purposes. Different places of alum extraction are mentioned 12 . In 1473, for example, they bought 900 cantares of alum of Montaone (nowadays Montioni where the alum mine of the Lord of Piombino was located) from Zaccaria di Francesco Strozzi of Siena for 1890 gold florins, immediately loaded on Rinaldo Altoviti's boat bound for the coasts of Provence 13 . Other sources of alum are mentioned: alum from Piombino (in 1488) 14 , from Massa (in 1491) 15 , or alum from Campiglia shipped to Flanders during the years 1494-1497 16 . These alums all come from Tuscany 17 . Throughout the 15 th century, merchants never ceased to try and bring the raw material closer to the place of consumption, with the obvious aim of reducing costs. The exploitation of new alum deposits made it possible to free the local textile industry from supply constraints, in the same way as what happened with the development of sericulture in Tuscany. Alum and silk were then fundamental raw materials for the textile industries. The silk used in Florentine workshops, at first imported mainly from Persia and Spain, then came mostly from Calabria, and then from places closer and closer to Florence through the development of sericulture in Marche, Abruzzo and Tuscany.

In a parallel way, and at the same time, the exploitation of alum mines closer to Florence made it possible to compete with the discovery of the mines of Tolfa, even if they did not provide enough alum of the required quality for the needs of the textile industries. While some accounts suggest that Tuscan deposits were exploited, these alums are mostly purchased from Sienese intermediaries for long-distance export. No alum purchased for the Florentine market is recorded in the registers of the commercial company Salviati of Florence, nor in the registers of the Salviati company of wool workers (Francesco d'Alamanno Salviati e fratelli, lanaioli in San Martini di Firenze), although they belong to the Arte della lana. They are not mentioned in the registers of the dyeing company of Francesco Salviati either. These account books -just like the Medici ones -thus definitely appear not to be the kind of sources in which to look for a better understanding of the relative importance and chronology of the different regions of production of alum at the end of the Middle Ages. They nevertheless raise interesting questions for further research. Concerning the drop in the price of alum recorded in 1495, for instance, it might be connected with the fact that, that same year, the mine of Campiglia was incorporated into the Arte della lana 18 .

Alum as production indicator

Do the three account books prove more useful sources to understand the business volume of Francesco Salviati's dye company and its evolution over the years? In a broad prospect, they do. The quantities of alum bought are plainly massive: not taking into account the data for the uncomplete first year 1483, one can see that during the following year, already, the weight of alum bought and not resold adds up to nearly three and a half metric tons, even after deduction of a small amount of 27 pounds (9 kg) resold to another dyer. After 1484, the figures will constantly range between five and more than ten metric tons in the peak year 1492. The quantities of alum resold to other dyers do not alter much the overall picture, because they are usually small (two times 27 pounds, once 40 pounds or 13.56 kg), only exceptionally bigger (once 228 pounds or 77 kg, two times 200 pounds or 68 kg). The purpose of these resales obviously was to help colleagues out in occasions when they had not planned a sufficient supply of alum, in the same way as, in fifteen occasions, Francesco's company had to purchase additional quantities of alum from individual providers, as mentioned above.

How much textile could be mordanted with such quantities of alum? This is a more difficult question, which the Salviati account books do not allow to answer directly nor indeed very precisely. They do not allow to answer it directly because one first has to relate these weights of alum used with the quantitative data recorded for those red and yellow dyestuffs which are always used after previously mordanting the cloth or wool in the fleece with alum. But a further problem is that the proportions of alum employed varied according to the colorants used and to the colours desired, as shown by other sources, such as regulations and technical treatises on dyeing. For instance, the tariff of the Florentine wool dyers, dated 1463, mentions the use of 25 pounds (8.5 kg) of alum per piece of cloth dyed scarlet in the style of Lucca ("panni luchesini con 25 lbr. d'allume pro panno per alluminare…") or dyed peony or rose colours with kermes ("Paghonazzi e rosati di grana") 19 . And we know from occasional records that the company of Francesco Salviati dyed cloths in these colours with kermes, for instance in April 1486 and in March 1487 20 . But in the same tariff of Florentine wool dyers, 10 pounds (3.39 kg) of alum only are required to dye a cloth in imitation scarlet with madder ("Panni scarlatini per fini con 10 lbr. d'allume e 20 lbr. di ciocchi") and 8 pounds (2.712 kg) to dye cloths in shades of green, tan, yellow, flax flower, peach flower and tan ("Panni Verdi, tanè, gialli, fiordilino et fiordipescho o ranci con 8 lbr. pro panno"). While the three account books reveal that Francesco Salviati's dye company bought and used enormous quantities of madder and of yellow dye-plants (weld, saw-wort and young fustic), they do not mention how many cloth pieces were dyed in these different colours. It nevertheless is interesting to relate some data offered by the three books on the quantities of dyestuffs bought, with the consumption of alum the use of these colorants would have involved for their mordanting. Taking madder as an example, one has to subtract a certain amount of resales (44,237 pounds or 14,996 kg) from the enormous quantities of madder bought by the Company between 1483 and 1495, i.e. 210,308 lb or 71,294 kg. As a result, the Company kept for its own use as much as 166,000 lb of madder (= 56,298 kg). Three mediaeval Italian sources mention an identical amount of madder -22 lb -as the quantity to be used per piece of cloth to be dyed in imitation scarlet [START_REF]Dyebook of the Compagnia della Tinta of Francesco di Marco Datini and Nicolo di Piero del Giunta in Prato, Archivio di Stato di Prato, Datini, register 312[END_REF] . With the total amount of madder mentioned above -166,000 lb -. 7,545 cloth pieces could therefore be dyed into mock scarlet. Since the same sources mention that 10 to 13 pounds of alum per piece should be used in the mordanting process, mordanting these 7,545 scarlattini cloths would have required 75,450 to 100,348 pounds of alum (= 25,578 to 34,018 kg). Examining the quantities involved for mordanting the yellow and green cloths dyed with saw wort and weld is equally revealing. On a total of 37,123 lb (= 12,585 kg) of saw wort bought by the Company, 664 lb (= 225 kg) are resold. Since 80 pounds of dry plant are used per piece of cloth, it implies that 456 cloths could have been dyed with the 36,459 lb (= 12,360 kg) of saw wort kept and used by the Company. The quantities of weld bought are much bigger: 436,111 lb (= 147,842 kg) and even after important resales, for a total amount of 10,977 lb (= 3,721 kg), the quantity of weld kept by the Company for its own use, i.e. 425,134 lb (= 144,120 kg) would have allowed to dye 4,251 cloths yellow or green, using 100 lb of dry plant per piece as ordered in the Florentine tariffs. The same amount of alum -8 lb per piece -being used in the mordanting processes of the 456 pieces dyed with saw wort and the 4,251 pieces dyed with weld, mordanting these 4,707 pieces would have required 37,659 pounds or 12,766 kg of alum. Nearly 13 metric tons! A reliable and reasonably cheap supply of alum was indeed vital to a dye company.

Final thoughts

The significance of the quantitative data obtained through the patient deciphering of the three books is easiest to understand once they are arranged, as above, in a table allowing to visualize at a glance the massive quantities of alum used yearly in the dye company "Francesco di Giuliano Salviati e Comp., tintori d'Arte Maggiore". They are particularly impressive when one takes into account that Francesco Salviati's dye company was neither the only one nor the most important in Florence at the time: in his three books, at least thirty other dyers then in business in Florence are mentioned. The precise data provided by these three books open vertiginous prospects on the huge need of mordant and dyestuffs that accompanied the massive development of wool broadcloth production in many parts of Europe during the Middle Ages. The three Salviati books thus provide further evidence confirming the new rise in the Florentine broadcloth production which Hidetoshi Hoshino argued happened in the second half of the 15 th century, in contrast with Alfred Doren's vision of the "decadenza" of this industry at that period

  XI d. IIII 267 E a dì 8 detto fiorini III soldi II denari I a oro larghi per libbre 207 d'alume da proveditori ; a loro avere in questo carta 7 f. III s. II d. I 200 E a dì X detto fiorini tre larghi per libbre 200 d'alume da proveditori ; a loro avere in questo carta 7 f. III s. ------403 E a dì 26 di gennaio [1484] fiorini sei soldi -denari 10 a oro larghi per libbre 403 da proveditori ; a loro avere in questo carta 7 f. VI s. -d. X 224 E a dì 16 di febraio [1484] fiorini tre soldi VII denari 2 a oro larghi per libbre 224 da proveditori ; a loro avere in questo carta 7 f. III s. VII d. II 209 E a dì 19 di detto fiorini tre soldi II denari VIII a oro larghi per libbre 209 da proveditori ; a loro avere in questo carta 7 f. III s. II d. VIII 202 E a dì 27 di febraio [1484] fiorini tre soldi -denari VII a oro larghi per libbre 202 da proveditori ; a loro avere in questo carta 7 f. III s. -d. VII 222 E a dì 3 di marzo [1484] fiorini tre soldi VI denari VII a oro larghi per libbre 222 da proveditori ; a loro avere in questo carta 7 f. III s. VI d. VII 201 E a dì detto fiorini tre soldi -denari 3 a oro larghi per libbre 201 da proveditori ; a loro avere in questo carta 7 f. III s. -d. III 1 177 E a dì 8 di marzo [1484] fiorini diciasette soldi XIII denari I a oro larghi per libbre 1 177 d'alume auto da proveditori per Ghuido e Nicholò Mannegli e compagnia ; posto debino avere in questo carta 22 f.

VIIII°" Continuation of the transcript: "431 E a dì 2 di dicembre fiorini sei soldi VIIII° denari III a oro larghi per libbre 431 da proveditori de l'arte ; a loro avere in questo carta 7 f. VI s. VIIII° d. III 34 E a dì 4 di dicenbre soldi XI denari IIII° a oro larghi per libbre 34 d'alume da Giovanbatista di Nerozo ; da proveditori de l'arte ; a loro in questo avere carta 9 f. -s. XVII s. XIII d. I."

Table 2 . Weights of alum bought and not resold by the Company of Francesco Salviati from 1483 to 1495, in Florentine pounds (upper line) converted into kilograms (lower line) ; and corresponding prices Source Year Weights Pounds Kilograms Prices in denari Average price for 100 pounds in denari

 2

	Reg. 394	1483	1 288	4 649	360.95
	(1483-1486)		437
		1484	10 151	36 505	359.62
			3 441
		1485	14 742	52 998	359.50
			4 998
		1486	17 686	63 648	359.88

TOTAL 1483-1495 244 937 83 033 873 573 (F. 3 639 s.17 d.9)

Table 3 Purchases of alum in the first register of Rafaello de' Medici (1498-1499).

 3 Prices then increased to an average of 371 denari per 100 pounds in the second register (purchases made between 1502 and 1505), and even went up to 571 denari per 100 pounds in the last book preserved (purchases made between 1508 and 1511), in spite of the fact that the supplier was always the same, i. e. the arte della lana. Such administrative control over a raw material of strategic importance as is revealed by the Salviati archive obviously brings to mind the parallel with couched woad, the only source of fast blues used by the dyers of wool broadcloths and wool in the fleece at that time. The Statutum Universitatis Artis Lanae of Florence had already ordered in 1317 that a warehouse dedicated to the storing of woad be created in order to better control the trade in this dyestuff of prime importance: « statutum et firmatum est quod pro arte lane fiat et fieri debeat fundacus, in quo vendatur et ematur et vendi et emi debeat totum guadum quod conducetur et emetur in civitate, burgis et subburgis 10 ».

	Period	Weights	Prices	in	Average price for
		Pounds	denari		100 pounds
		Kilograms			in denari
	July-August 1498	3 939	11 349		288.12
		1 335
	September 1498	2 516	7 251		288.2
		853
	October-November	2 698	7 776		288.21
	1498	914
	November 1498	3 131	9 024		288.21
		1 061
	January 1499	3 359	9 696		288.66

  22 . Hoshino attributed this new development largely to the conquest of the huge markets of the Ottoman Empire by Florentine merchants. The increased need of alum to mordant cloths then dyed in varied shades of reds, and of yellows to produce greens on various blue grounds, again confirms this view. An illustration of this has been provided by Ingrid Houssaye Michienzi's recent study of three account books written between 1491 and 1493 from Ottoman Constantinople by Giovanni di Marco Salviati, a distant cousin of Francesco's[START_REF][END_REF] .Figure1The blue book, signed A (Libro azzuro, segnato A) Reg. 394 in the Salviati Archive, registrations from November 1483 to the end of December 1486. Figure 2 The yellow book, signed B (Libro giallo, segnato B) Reg. 395 in the Salviati Archive, registrations from December 1486 to November 1491.Figure 3 The Giornale, diary, Reg. 396 in the Salviati Archive, recording all daily activities from October 1490 to June 1498.Figure 4 Folio 5 of Reg. 394: first records of purchases of alum. De la pirotechnia. Libri X dove ampiamente si tratta non solo di ogni sorte & diversità di miniere, ma anchora quanto si ricerca intorno a la prattica di quelle cose di quel che si appartiene a l'arte de la fusione over gitto de metalli come d'ogni altra cosa simile a questa, Curtio Navo, Venise, 1540. marchés ottomans à la fin du XV e et au début du XVI e siècle », Mélanges de l'École française de Rome -Moyen Âge 127-2, 2015, [https://journals.openedition.org/mefrm/2753]. REBORA, Giovanni, Un Manuale di Tintoria del Quattrocento, Giuffré, Milan, 1970. SINGER Charles, The earliest chimical Industry, an essay in the historical relations of economics and technology illustrated from the alun trade, The Folio Society, London, 1948.

	Bibliography
	SOURCE
	BIRINGUCCIO Vannoccio,

Acknowledgments

We are deeply grateful to Maddalena Taglioli, Curator of the Salviati Archive, and to all the team at the Scuola Normale Superiore in Pisa for their unfailing kindness and helpfulness and to all our colleagues and friends of the ANR ENPrESA program, for very fruitful exchanges.

Notes and references