Thermostructural characterization and structural elastic property optimization of novel high luminosity LHC collimation materials at CERN

The CERN Large Hadron Collider is currently being upgraded to operate at a stored beam energy of 680 MJ through the High Luminosity upgrade. The LHC performance is dependent on the functionality of beam collimation systems, essential for safe beam cleaning and machine protection. A dedicated beam experiment at the CERN High Radiation to Materials facility is created under the HRMT-23 experimental campaign. This experiment investigates the behavior of three collimation jaws having novel composite absorbers made of copper diamond, molybdenum carbide graphite, and carbon fiber carbon, experiencing accidental scenarios involving the direct beam impact on the material. Material characterization is imperative for the design, execution, and analysis of such experiments. This paper presents new data and analysis of the thermostructural characteristics of some of the absorber materials commissioned within CERN facilities. In turn, characterized elastic properties are optimized through the development and implementation of a mixed numerical-experimental optimization technique.


I. INTRODUCTION
The LHC is currently being upgraded to operate at twice the stored beam energy of previous runs through the high luminosity LHC (HL-LHC) upgrade [1,2].Operating at unprecedented stored beam energies necessitates improved machine protection and response assessment of beam intercepting devices (BIDs).Collimators, as shown in Fig. 1, form a major part of the BIDs and protective system.These ensure that the LHC running conditions are stable with minimal damaging events, as has been successfully implemented during the LHC first run.As beam bunches circulate within the LHC ring, a slow diffusion of particles out of the beam core is instigated via long range beam-beam effects, intrabeam scattering, and orbit feedback systems, amongst others [3][4][5][6][7].Particles at the edge of the spatial distribution tend to escape from the intended beam trajectory, consequently forming a beam halo.
The LHC collimation system targets over 99% efficient regular and irregular beam halo cleaning during the entire beam cycle at all stages of operation [9].The hierarchical collimation structure consists of a multistage beam cleaning process, ranked according to beam proximity.The setup consists of primary collimator jaws positioned closest to the beam, followed by secondary collimators (TCS) retracted farther away from the beam and finally, tertiary collimators found at the farthest position from the beam.Although carbon fiber composite (CFC) presents an astonishing robustness in case of particle beam impact, its poor electrical conductivity impedes its use in primary and secondary FIG. 1. Cutaway of a secondary collimator as a beam intercepting device with the active component made of a fiber-reinforced carbon jaw [8].
collimators due to the high rf impedance generated on the particle beams.On the other hand, the high-density tungsten alloy adopted in tertiary collimators cannot sustain the accidental impact scenarios of HL-LHC [10].The molybdenum-graphite composite recently developed at CERN and Brevetti Bizz (Italy) [11] is proposed as a replacement of CFC for primary and secondary collimators, balancing a good thermomechanical robustness and stability during normal operation with a good electrical conductivity.On the other hand, copper diamond is proposed as a much higher resistant replacement for the tungsten alloy adopted in tertiary collimators.
The collimator materials, from the thermomechanical point of view, should possess high thermal conductivity, specific heat, melting temperature, and ductility; at the same time, coefficient of thermal expansion and density should be minimized.Additional requirements include high electrical conductivity, machinability, radiation hardness, and low magnetic susceptibility [12].To classify and rank potential materials against this large number of requirements, it is useful to introduce figures of merit which permit several material properties related to a specific requirement to be condensed into a single indicator: the higher the figure of merit, the better the material performance against that specific requirement [13].
An index called the thermomechanical robustness index (TRI), is proposed to assess the material robustness against particle beam impacts.Given that thermal shock problems are, to a large extent, governed by the thermal deformation induced by a sudden temperature increase, it appears reasonable to base this index on the ratio between material admissible strain or strain to failure ε adm and the actual working strain as defined by Eq. (1): where ε ref and ΔT q are the strain and the temperature increase generated by a reference energy deposition, T m is the melting (or degradation) temperature, and m is a coefficient related to the material loss of strength with temperature increase.Another index, called the thermal stability index (TSI) provides an indication of the ability of the material to maintain the geometrical stability of the component under steady-state particle losses.This is particularly important for components such as collimators and long absorbers, which are required to interact with the halo of the particle, maintaining their longitudinal straightness to a fraction of a beam transverse sigma.TSI is defined by Eq. ( 2) as where CTE is the coefficient of thermal expansion [defined later in Eq. ( 9)], ρ the density, λ the thermal conductivity, X g the geometric radiation length, and C s a material constant related to thermal stability.Finally, the rf impedance index RFI is defined by Eq. ( 3) as where σ is the electrical conductivity and μ the magnetic permeability.
Table I provides these figures of merit for CFC and tungsten alloy currently adopted in LHC collimators, and the requirements for HL-LHC.
The latest efforts to optimize the operation of collimators target novel absorbent insert composites that can sustain severe beam impacts.The high-energy particle-matter interactions impart a nonuniform temperature increase in the impacted structure, subsequently leading to physical deformations and stress to the possible detriment of the structure functionality.To analyze this behavior, tests are carried out within the High Radiation to Materials (HiRadMat) testing facility at CERN, which allows submitting targets to intense proton or ion beam impacts [14].Distinct experiments are attempting to replicate collimation jaw behavior exposed to direct interaction with high-energy particles, while taking into account the jaw robustness, performance, fabrication, installation, beam operation, and machine maintenance.In conjunction with the design, running, and analysis of any experiment such as HiRadMat, knowledge of the thermophysical temperature-dependent properties of absorbers, experiencing energy deposition like in the HL-LHC, is essential to allow representative material behavior modeling.In light of this, several experiments have been commissioned at the HiRadMat facility under the HRMT-23 experimental campaign.This paper investigates the properties of the materials of the three jaws analyzed in the HRMT-23 campaign.The aim of this campaign is to investigate the thermomechanical properties of three independent jaws in order to guarantee robustness and geometric stability against beam impact, thus qualifying or disqualifying them for use in future fullscale HL-LHC prototypes.The jaws in question consist of two TCSPM and one TCSP jaw, generally distinguished by variations in the absorbent insert considered (Fig. 2).The main features of each jaw consist of the absorbers located at the jaw center, experiencing direct interaction with the particle beam.The inserts consists of novel copper diamond (CuCD), molybdenum carbide graphite (MoGr), and carbon fiber carbon composites, behaving in an isotropic, transversely isotropic, and orthotropic manner, respectively, as determined by Mariani [15].
CuCD consists of a diamond reinforced metal matrix composite developed to optimize the thermal management properties of both the copper and diamond constituent.CuCD is volumetrically composed of 60% diamond powder at a diameter of ∼100 μm for 90% of the powder and a rougher diameter of 325 to 400 μm for the remaining 10% [11].The composite is developed by means of hot processing powder technologies utilizing several stages of hot pressing and subsequent heat treatments, to provide an inherently isotropic and stress relieved composite.The composite depicts a rough structure where the weakest element is the interface between the copper matrix and diamond particles, as shown by fracture surfaces, observed as prominent voids.These voids indicate preferential fracture across the brittle diamond interfaces, without visible signs of plastic deformation of the copper matrix.CuCD is particularly appealing for use in LHC collimators due to its ability to maximize geometric stability and transient shock response, while exhibiting high electrical and thermal conductivities as well as low coefficient of thermal expansion and low densities.The lack of diamond degradation and relatively low densities also allows for a smoother energy deposition profile, while generating high beam impact strength, essential for superior resistance to beam impacts.However, CuCD is limited by the low melting temperature of the copper matrix, especially when compared to other absorbent inserts, reducing overall robustness.
MoGr is a graphitic matrix composite, consisting mostly of spheroidal graphitic flakes.It exhibits the highest chemical purity and degree of graphitization of lamellar graphite flakes with a diameter of 45 μm, having a distinctly regular and round shape that assists in powder compaction and reduces the final anisotropy of the composite.The remainder of the composite is volumetrically composed of 4.5 vol % molybdenum.Spark plasma sintering is carried out with processing temperatures ranging from 1700 to 2600 °C.Because of the raw purely metallic molybdenum and graphitic powders being chemically affine to one another, sintering creates a highly integrated composite.During sintering recrystallization of graphitic grains also occurs and, because of the pressing process, graphite orients itself parallel to the sintering plate.This in turn creates an inherently transversely isotropic composite with preferential properties along the directions parallel to the sintering plane [11].
Tests carried out within the HRMT facility in previous years utilized CFC as a control material having previously survived under full LHC beam impacts.CFC is a commercial carbon-matrix composite consisting of two dimensional planes of short carbon fiber reinforcements embedded within a graphitic matrix.Short carbon fibers are randomly dispersed, creating a homogeneous layer along the plane of deposition.Several carbon fiber planes are stacked on top of one another and bundled into homogeneous layers in the perpendicular direction to the fiber orientation, creating a composite with outstanding mechanical and thermal properties with respect to isotropic graphite.After successive layering and hot pressing cycles, several stages of heat treatment follow.Subsequently, bar extrusion is carried out to increase the final density and provide a higher flexural strength between the two plane directions, resulting in an inherently orthotropic composite.CFC has been chosen for application in the LHC collimation system due to its good resistance to high thermal shocks and low CTE.This ensures high geometric stability and robustness, as required by the design specifications.CFC maintains high strength and low coefficient of thermal expansion at temperatures above 2000 °C and displays excellent thermal shock resistance.
The orthogonal coordinate system shown in Fig. 2 for the HRMT-23 jaws is maintained for the remainder of this work, where X, Y and Z represent transversal (T), perpendicular (P), and longitudinal (L) directions respectively.TCSPM jaws are fairly similar and mainly distinguished by the differing inserts considered, where CuCD and MoGr are utilized for TCSPM-C and TCSPM-M jaws, respectively.The final TCSP jaw exhibits distinct structural features in comparison to TCSPM jaws, while containing CFC inserts.

II. THERMAL CHARACTERIZATION
The thermal characterization campaign considers three techniques: laser flash analysis (LFA), differential scanning calorimetry (DSC), and dilatometry (DIL) with specimens having the dimensions denoted in Fig. 3.The experimental setup has been calibrated as per the required standards, utilizing thermal gauges with an accuracy of AE0.05%.

A. Laser flash analysis
The transient LFA method introduced by Parker et al. [17], based on the theory of Carslaw and Jaeger [18], is aimed at measuring diffusivity of a material specimen under noncontacting analysis.The experimental setup consists of a NETZSCH LFA 427 system, operating within a thermal range of 22 to 2000 °C according to the ASTM E1461-13 standard [19].The plane parallel disc specimens used in the test are spray coated with graphitic deposition, which improves the emissivity behavior prior to insertion within an inert argon atmosphere at pressures up to 10 −5 mbar with a gas flow rate of 50 ml/min.The pure gas operating environment is considered, to prevent unwanted corrosive reactions between specimen and setup components, while operating under isothermal (δT/δt < 0.1 K/min) conditions.
A nitrogen-filled protective cooling tube is used while the surrounding furnace is activated.After the specimen reaches a predetermined temperature, a high-intensity light laser energy pulse is fired towards the bottom side of the sample, inducing a temperature rise.The thermal rise induced as a function of time is recorded by a noncontact infrared detector on the opposing sample side, recording the time-dependent energy transmission rate.
Thermal diffusivity (α) is obtained from the empirical relation developed by Parker et al. [17] using the mathematical solution of Carslaw and Jaeger [18].For Eq. ( 4), t 1/2 represents the time in which the temperature rise on the specimen side opposite to where heat is applied reaches half of the maximum temperature, and c represents specimen thickness.Thermal conductivity (k) is determined via the empirical relation in Eq. ( 5), as a function of temperature [20],

B. Differential scanning calorimetry
Differential scanning calotimetry (DSC) is an experimental technique used to determine the specific heat capacity of a material.The technique consists of measuring the energy required to obtain a near-to-zero temperature difference between the specimen being tested and a reference inert specimen, which has known properties.The reference specimen has properties that are well known for the tested temperature range.Since the two samples experience the same thermal history, the specific heat for the unknown specimen under investigation can be calculated.Testing is set up in accordance with the ASTM E1269-11 standard [21] functioning within a thermal range of 40 °C to 500 °C.Above 500 °C, the specific heat is assumed to remain constant in the scope of the thermo-mechanical model built.The setup operates dynamically under an argon atmosphere with a gas flow rate of 50 ml/ min and a heating rate of 20 K/ min.The heat flux principle considers a sample (S) and a reference (R) specimen, typically sapphire, subjected to a pre-determined temperature programme including heating and cooling cycles.The measurement process consists of three main stages, where the behavior of the empty calorimeter is initially determined as a base line (B).This is followed by measurement of the reference material and finally measurement of the sample under analysis.
Specific heat capacity is defined in Eq. ( 6) together with the mathematical description of heat content of a material as a function of temperature, denoted in Eq. (7).Combining both equations leads to the derivation of the dynamic calorimeter calculation in Eq. ( 8), where ρ represents specimen density, V represents specimen volume, and DSC refers to the sensor signal, allowing specific heat C p calculation.The results of the LFA and DSC techniques are observed in Figs.4-9, displaying mean plots as a result of repetitive testing to ensure statistical repeatability and control data scattering, with thermal parameters shown in Table II, C. Dilatometry Dilatometry (DIL) measures longitudinal expansion of a reference specimen, with a known coefficient of thermal expansion (CTE), and longitudinal expansion of an unknown specimen.The experimental setup consists of a high-temperature horizontal pushrod NETZSCH DIL 402C dilatometer operating within a thermal range of 22 to 2000 °C, performing measurements in accordance with the ASTM E228-11 standard [22].A cylindrical specimen is placed in a furnace containing a controlled helium environment with a gas flow rate of 50 ml/ min and a heating rate of 2 K/ min.The thermal testing ranges and incremental divisions are set at specific time intervals, with temperature recorded via a thermocouple adjacent to the specimen.During the heating cycle, the specimen expands and imparts a pressure proportional to the imparted displacement on the graphitic pushrod.The signal is transferred to a linear variable differential transformer system and converted into numerical strain.Both the connecting rod and the sample holder are exposed to the same atmospheric temperature as the specimen, with the final results being a summation of length variations of all the setup components.To cater for this, the equipment expansion behavior is carried out utilizing the reference material, followed by the specimen measurement.The CTE for an unknown specimen material is defined by Eq. ( 9), where ΔL L 0 represents the change in specimen length corresponding to a specified change in temperature ΔT.The results are depicted in Figs.10-14, showing both the dimensional specimen length-temperature change and the calculated CTE against temperature during both heating (H) and cooling (C) cycles.Residual deformation is observed by the discrepancies at the heating start and cooldown end cycles.The CTE is obtained by using the plots of dimensional change against temperature for the cooling cycles from start to end points, with results shown in Table II 10) is Hooke's law for an orthotropic material, where ε is strain, E is Young's modulus, v is Poisson's ratio, G is the shear modulus of elasticity, and σ is stress: for all quantities the direction is denoted by the subscript.The strain and stress tensor are linked by the compliance matrix [S], which contains the following elastic constants: elastic modulus (E), shear modulus (G), and Poisson's ratio (v) [23].Parameters occurring along the main matrix diagonal are obtained via structural characterization testing, whereas the off-diagonal parameters may be obtained through relations with the main diagonal parameters.For an orthotropic case, Eq. ( 10) indicates nine independent elastic constants with five independent constants for a transversely isotropic case and three constants for an isotropic case, due to the introduction of symmetry planes.This is reflected in the parameters for a transversely isotropic case being identical along the (L) and (P) orientations, whereas for an isotropic case the parameters are identical along all the three orthogonal planes.
For the structural characterization campaign, four techniques target the elastic response for all material orientations over a low strain-rate range, at room temperature.These consist of flexural testing, compression testing, impact excitation testing (IET), and ultrasonic (US) testing,

A. Flexural testing
The elastoplastic response is given by the stress-strain response of a material tested under bending with a four point flexural setup, which consists of a universal Zwick Roell Z-100 machine with a maximum load and travel speed of 100 kN and 5 mm/s, respectively, operating in accordance with the ASTM C1161-02c standard [24].A strain gauge is attached to the lower part of a rectangular specimen, experiencing tensile effects, to record the stress variation at predefined loading steps until specimen failure.A load is applied at the midpoint of the loading member, imparting the load equally amongst both the upper loading pins until failure, ensuring that the center span is stressed in bending to avoid the presence of shear stresses.The results are depicted in Figs.[15][16][17] showing the incremental stress variation against strain, with elastic moduli results shown in Table III obtained in the initial linear regions of each respective plot.

B. Compression testing
Compression testing is a destructive test utilized to establish behavior of materials under the influence of a compressive force until failure.The experiment operates in accordance with the ASTM C695-91 standard [25].A short cylindrical specimen with attached rosette strain gauges is vertically set within a universal Zwick Roell Z-100 testing machine, set up with two parallel supporting contacting steel blocks.A load is gradually imparted at preset steps to the loading member with a rate of 2 mm/s, while the corresponding load against strain data is recorded via strain gauges until specimen rupture.The results are depicted in Figs.18-20, with elastic moduli results tabulated in Table III, obtained in the initial linear regions of each respective plot, reflecting the isotropic, transversely isotropic, and orthotropic nature of the CuCD, MoGr, and CFC composite, respectively.first vibrational modes, denoted as f f and f t , respectively.Rectangular specimens with a high length-to-thickness ratio are subjected to minute strains via a light strike with an impulse tool.The specimen supports, impulse locations, and signal pick-up points are selected to induce and measure selective vibrational modes.The resulting signal is picked up through a noncontacting microphone transducer and converted into an electrical signal.The signal is then amplified, converted into a quantifiable numerical frequency, and read out to a computer-aided postprocessing software package.This leads to the determination of dynamic elastic properties at or near the origin of the stress-strain curve, with minimum possibility of fracture.For dynamic elastic moduli, the empirical relation represented in Eq. ( 11) is applied in accordance with the analytical resonance theories of Timoshenko, Goens, and Pickett [26].This is created such that numerical comparisons of these combined theories, up to four significant figures, are justified, assuming an arbitrary Poisson's ratio of 0.292 as specified in the relevant standard [28,29].Equation ( 11) is considered reliable through the inclusion of a shape correction factor (T l ), able to account for the finite bar thickness and Poisson's ratio variations [28].Pickett identified a general empirical relation for shear moduli obtained from the first torsional frequency by using two correction factors A o and B o , as shown in Eq. ( 12) [30].It is assumed that the relationship between resonance frequencies and dynamic moduli is valid for a material that is homogeneous, elastic, and isotropic, as per the relevant standards.Considering this, it is implied that an improved Poisson's ratio value is obtained by Eq. ( 13), The frequency and corresponding elastic parametric results are shown in Table IV together with the accompanying standard deviation (SD).
For CuCD, a small SD results for both E and G, with a contrasting significantly high SD for v, reflecting a lack of accuracy amongst sample measurements.A similar pattern is observed for both MoGr and CFC along all orientations, where Poisson's ratios have the highest SD.Typically Poisson's ratios vary within a range of 0-0.5, where excessively large or negative values beyond this range are not considered physically possible.Table IV indicates that both negative and excessively large Poisson's ratios result from experimentally obtained elastic and shear moduli.Lauwagie et al. [31] noted similar results when carrying out experimental resonance beam tests on commercially available orthotropic 6082 aluminum alloy and 304 stainless steel.The results from resonance beam testing failed to accurately model the directional variation of both E and G, leading to an inaccurate Poisson's ratio identification when compared to other testing techniques.This is due to the fact that the relation depicted in Eq. ( 13) is based on a fully isotropic material model, which is not always the case.

D. Ultrasonic testing
Measuring ultrasonic velocities in materials with conventional ultrasonic pulse echo flaw detection equipment is utilized to aid in optimization of elastic parameters.The setup consists of an ultrasonic instrument, Krautkramer USN60, and straight beam longitudinal wave transducers, GE MSW-QC2.25 (∅0.5", 2.25 MHz) and GE Alpha 2 (∅5 mm, 10 MHz), used in a contact mode with a plexiglas delay line and water as couplant.Ultrasonic velocity measurements are carried out by comparing transmission TABLE IV.Mean resonance testing flexural frequency f f [Hz], torsional frequency f t [Hz], and elastic parameters, including elastic modulus E [GPa], shear modulus G [GPa], and Poisson's ratio v, for CuCD, MoGr, and CFC along (L), (T), and (P).times of a longitudinal wave pulse through a specimen to its travel path along the entire specimen span [32].
Results show that homogeneity is present along and between each specimen, with the final sound speeds tabulated in Table V.No results are attainable for CFC due to the difficulties encountered during experimentation, most likely occurring due to the high degree of absorption of the specimens.For MoGr, results show a difference between the sound speeds along (L) and (T), possibly due to variations between stacking of the internal graphitic layers.A higher sound speed is noted along (T), possibly due to the orientation reflecting the material pressing direction during manufacture, leading to a densely packed graphitic structure along the specific direction, allowing for easier sound wave transfer.

IV. STRUCTURAL CHARACTERIZATION OPTIMIZATION A. Optimization technique
The method reported in Sec.III C fails to determine the elastic constants of the orthotropic materials because the relationships used for the elastic constants are relevant to isotropic cases.Therefore, in the most general sense, the problem must be solved numerically, with a modal analysis targeting the first flexural and torsional frequencies, which are also measured experimentally as a function of the parameters of the compliance matrix.This section deals with determination of the elastic constants, including elastic moduli, shear moduli, and Poisson's ratios, for the orthotropic materials.The calculations are done by an educated guess of the compliance matrix parameters and the calculation of the resulting numerical frequencies together with an optimization procedure carried out in ANSYS, whose scope is to minimize the error between numerical and experimental frequencies by manipulation of the matrix constants.
The initial stage of optimization is similar to the approach adopted by Lauwagie et al. [31], where IET and tensile tests are used together with a mixed numerical experimental technique (MNET).The principle of the MNET is to compare experimentally measured frequencies with numerically computed frequencies by implementing a mock numerical model of the experimental setup.The properties obtained are iteratively refined until an acceptable precision limit is reached.This technique is also successfully adopted by Hosten [33] for polyether etherketone-carbon fiber composites and Grediac and Paris [34] for anisotropic metal plates.Pronounced variations are present for Poisson's ratios for all cases, because they were computed from fully isotropic behavioral assumptions, considered during IET.To cater for this, an alteration is applied using the US technique as proposed by Kohlhauser et al. [35], mostly applicable for anisotropic materials.Combining both methodologies, including results from the characterization campaign, an optimization chain is developed.This optimization chain is based on the use of the experimental data and the first flexural and torsional frequencies as the target of the numerical analysis.
Educated values of the elastic constants are the starting point of the numerical analysis; the set of constants is used by ANSYS to determine the compliance matrix, solving then the eigen-frequency problem and comparing the results with the target experimental frequency.The optimization module within ANSYS then automatically updates the elastic constants values and relaunches the analysis, with the objective of minimizing the difference between simulated and target values, until the difference is reduced to 1% or less.The geometric model replicates both the geometry and the free-free boundary conditions experienced by specimens during structural IET and US characterization.Because of inaccurate and nonphysical Poisson's ratios reported in Sec.III, an educated guess of 0.1 was used as the initial Poisson's ratio value, to be later confirmed by US measurements.
The optimization algorithm used to minimize the difference between numerical and experimental frequencies starts with a design of experiment.The initial step involves identification of the design parameters under experimentation, including identification of both the input and output parameters of interest.The numerical ranges within which the input parameters may lie are selected based on physical considerations, and, in particular, by imposing that no constant can be negative or higher than 1000 MPa, while Poisson's ratios are constrained between 0 and 0.5.From the available combinations, an adequate amount of design points are automatically selected from the maximum and minimum boundaries of the output parameters, creating a design space within which optimization occurs.The design points are mapped to create a response surface, graphically depicting the relation between any desired inputs and outputs, as shown in Fig. 21.To carry out the optimization, specific objectives and constraints are set for output resonant frequencies.The targeted objectives set are the experimental resonant frequencies.For the target constraints, resonant frequencies are allowed to vary within a maximal frequency range set lower than the standard deviation results identified in Table IV.These constraints allow the optimization technique to provide results that exhibit lower deviation than those obtained experimentally.Additional constraints are introduced through parametric relationships, reflecting the inherent characteristic material behavior.For example, for MoGr both Young's and shear moduli are considered equal along the plane of isotropy.
The multiobjective genetic algorithm was used.This goal-driven optimization technique represents a class of algorithms that simulates the process of natural evolution via crossover and mutation techniques, specifically beneficial for multiple conflicting objectives, allowing for flexible problem formulation and availability of continuous input parameters, as shown in Fig. 22 [36].
Once the genetic algorithm is successful, a combination of final candidate sets are given, out of which the best results are selected.Using a multiobjective form of optimization does not provide a single solution for any input parameters but rather gives a set of possible solutions.
However, this technique is still desirable for problems involving multiple conflicting objectives and highly complex design spaces due to its ability to handle multiple goals, the flexible problem formulation, and the availability of continuous input parameters.
The general material stress-strain relationship is portrayed in terms of compliance [S] and stiffness [C] matrices, which is its inverse, Eqs. ( 14) and ( 15), respectively.This general case is applicable for orthotropic CFC, with symmetry along the 11-22 plane considered for MoGr and identical parameters along all orientations for CuCD.The optimized elastic and shear moduli obtained from numerical modal analysis are inserted into the compliance matrix to define the constitutive material law.For a general orthotropic case, equations are available that relate ultrasonic sound velocities with compliance tensors along the relative directions [37].This approach is adopted by Hosten et al. [33] where the product of stiffness and compliance tensors is used to numerically solve for off-diagonal stiffness tensor components.Knowing that the compliance tensors are the inverse of the stiffness matrix, Poisson's ratios are obtained utilizing both [S] and [C] matrices while ensuring that the variation does not exceed an acceptable limit of 1%,

B. Results and evaluation
The optimized elastic parameter results displayed in Figs.23 and 24 indicate that CuCD has the highest elastic and shear moduli, reflecting significant stiffness when compared to both MoGr and CFC, which exhibit rather similar elastic behavior, possibly due to the similar microstructural graphitic nature.Figure 24 shows that Poisson's ratios are rather similar for each material along each direction and lie within a physically acceptable range of 0-0.5, except for MoGr displaying a slight increase for v yz .

V. CONCLUSIONS
In modern high-energy particle beam accelerators, accidental beam deviations are frequently experienced, imparting rapid internal heating and, in turn, inducing a dynamic response.The LHC accelerator allows for highenergy circulating beams, necessitating adequate machine protection devices during operation, such as collimators.Collimation systems may experience significant particlematter interactions with the highest energy deposition imparted onto absorbent materials.Understanding the thermophysical behavior of absorbers under such circumstances is critical to assist with the development of representative material models.
The thermostructural characterization campaign carried out within CERN research facilities for collimator jaw absorbers provides an overview of the three composite insert types along all orientations.Initial assumptions, focused on isotropic material response, were considered during the initial phases of the characterization.The need to obtain the full elastic parameters for transversely isotropic and orthotropic materials from the initial structural campaign data leads to the computation of the parameters by means of a combined Finite Element Method model and optimization procedure.Combining both experimental and numerical approaches a MNET is created.This approach indicates similarity of frequency mode results with impact excitation characterization results, reflecting a stable and robust approach and providing optimization of experimentally determined elastic properties.The optimized elastic parametric results indicate that CuCD is the stiffest composite, followed by CFC and finally MoGr.While this is an important result, it is clearly one of the many parameters to consider in determining the best material for the application.The figures of merit described earlier are reproduced in Table VI, with the inclusion of the values obtained from the data presented in this work.This shows how the data presented here aid in the selection of the most suitable material.
The results of this study confirm that MoGr satisfies the requirements for HL-LHC primary and secondary collimators, while CuCD meets the HL-LHC targets for tertiary collimators.The two materials are proposed for replacement of CFC and W-alloy, respectively.This has an impact on the design of the collimator jaw: MoGr, differently from CFC, cannot be produced as a monolithic structure 1 m long, and CuCD cannot be threaded and screwed to the Glidcop support, as was done in the case of the W-alloy.A new modular design, Fig. 25, universal to primary, secondary, and tertiary collimator jaws, was studied, allowing to change the function of the collimator by simply replacing the absorber blocks in the housing [38].The jaw can host CuCD or MoGr blocks interchangeably.

FIG. 10 .
FIG. 10.Relative change in length and CTE against temperature with initial thermal reference of 49 °C (H) and 50.095 °C (C) for CuCD.

FIG. 12 .
FIG. 12. Relative change in length and CTE against temperature with initial thermal reference of 48 °C (H) and 50.1 °C (C) for MoGr along (T).

C
FIG. 15.Plot of flexural stress against strain for CuCD.

FIG. 17 .
FIG. 17. Plot of flexural stress against strain for CFC.Longitudinal data and relevant axes marked with (L).Transverse data and relevant axes marked with (T).Perpendicular data and relevant axes marked (P).

TABLE I .
Figures of merit for CFC and tungsten alloy currently adopted in LHC collimators, and the requirements for HL-LHC.

TABLE III .
Flexural and compressive elastic moduli [GPa] along the respective orientations.

TABLE VI .
Figures of merit for current materials adopted in LHC collimators (CFC and tungsten alloy), requirements for HL-LHC, and figures for the materials under investigation.