Inorganically-bound sand cores are used in many light-metal foundries to form cavities in the cast part, which cannot be realised by the mould itself. To enable FEM simulations with core materials, their mechanical properties have to be measured. In this article, we develop methods to determine the Young's and shear modulus, the Poisson number, and the fracture strain. This allows us to fully parametrise an ideal brittle FEM model. We found that the Young's and shear modulus can be obtained acoustically via the impulse excitation technique. The fracture strain was measured with a high- speed camera and a digital image correlation algorithm. Keywords: inorganic sand core materials, elastic properties, fracture strength, fracture strain, water-glass, 3-point-bending, Young's modulus, shear modulus, Poisson's number 1. Introduction A rising number of light-metal foundries are using inorganically-bound core materials instead of organically-bound ones, due to stricter environmen- tal laws and the need for eco-friendly production processes [1]. The cores Corresponding author. tel.: +49 89 289 13984 Email address: philipp.lechner@utg.de, (Philipp Lechner) Preprint submitted to Materials April 29, 2020 consist of sand and a binder agent, which holds the individual particles to- gether and forms a porous material, which can be used for foundry moulds and cores [2]. Inorganic sodium silicate binders (water-glass) do not combust during the casting, while organic binders emit a number of hazardous com- bustion products. However, the absence of a combustion process also means that more binder remains in the cast-in sand core. After a cooling period, the sand cores have to be removed via high-energy hammer impacts to the cast component, which is called de-coring [3]. With inorganic binders, this process is more energy-intensive and strain- ing for the cast components, due to the higher amount of remaining binder. In order to minimise the necessary e ort to remove the core from the cast component, one needs to nd out which de-coring process parameters are energy- and strain- minimal. This directly leads to the goal of generating de-coring simulations which allow to evaluate if the cast part is de-corable. Subsequently, such simulations can be utilised to optimise the process and thus reduce the strain on the component. The rst step of building such a mechanical simulation is to determine the basic material parameters, in order to build a material model. Lechner et al. showed that inorganically-bound core materials behave like a brittle material [4]. For a brittle material, the basic material parameters for a simulation are Young's modulus, shear mod- ulus, fracture stress and fracture strain. In this article, we propose methods to determine those parameters for inorganic core materials. 1.1. Mechanical properties of inorganic sand cores There are numerous articles which study mechanical properties of inorganically-bound core materials. Most focus on the mechanical strength and the in uence of environment and heat treatment on it [5]. However, research also exists, which studies the Young's modulus of core materials. Stauder et al. determined fracture stress and Young's modulus with 3-point- bending (3PB) experiments [6]. Griebel et al. showed that strain should be measured optically, when determining Young's Modulus with a 4-point- bending (4PB) experiment [7]. They further showed that the Young's mod- ulus is dependent on the applied load, and decreases towards the point of fracture. Schneider et al. simulated the Young's modulus by modelling the mi- crostructure of the sand-binder-compound. Furthermore, they validated their simulation with an ultrasound through-transmission test [8]. However, to our 2 knowledge, there is no existing literature on the shear modulus or Poisson number of inorganically-bound core materials. 1.2. Determination of elastic properties with the impulse excitation technique The impulse excitation technique allows the identi cation of natural fre- quencies of the tested specimen [9]. An impact, in our case induced by a hammer, excites the specimen with a short impulse, and causes the subse- quent approximately free oscillation of the specimen measured [10]. There are various possibilities for measuring this oscillation. The sound can be recorded with a microphone or the specimen can be tracked with a contact- less laser vibrometer or a piezo crystal [11]. Furthermore, the induced strains can be measured using strain gauges. Regardless of the method, the data is recorded in the time domain and Fourier-transformed to the frequency domain. In the subsequent spectrum, every excited natural frequency is rep- resented by a peak. With the dimensions of the cuboid specimen and its density, the obtained natural frequency peaks can be used to calculate elastic properties of the material used. There are several analytic solutions for the Young's modulus and the shear modulus. We use the following equations to calculate the Young's modulus E, the shear modulus Gand the Poisson number in this article [12]: E= 0:946mf2 f1l3 bh3 1 + 6:585h l2! (1) E=4mlf2 l1 bh(2) G=4lmf2 t1 bh(h=b) + (b=h) 4(h=b)2:52(h=b)2+ 0:21(h=b)6 (3) =E 2G1 (4) wheremis the mass of the specimen, lthe length, bthe width and hthe height.ff1,fl1andft1are the rst bending, longitudinal and torsional eigenfrequencies, respectively. 3 1.3. Strain determination via digital image correlation For simplicity, reliability and exibility reasons, digital image correlation (DIC) has been established as the standard for optical deformation analy- sis in experimental mechanics of solids. Digital image correlation rests on registration algorithms based on correlation functions. Based on the merg- ing, relative displacements between subsequent images are calculated. The method has been extensively investigated and various task-speci c correla- tion criteria are proposed and used in di erent elds [13]. Integer-pixel accuracy can be achieved using simple correlation criteria. For sub-pixel resolution data, interpolation is necessary, for example coarse- to- ne searching [14], peak- nding algorithms [15], genetic algorithms [16], gradient-based algorithms [13], B-splines, arti cial neural networks [17] and nite elements [18]. Given adequate image acquisition, DIC features scale- invariance leading to a large eld of applications. 2. Materials and experimental methods 2.1. Specimens The specimens used for this work were produced on a Loramendi SLC2 25L core-blowing machine (Loramendi S.Coop., Vitoria-Gasteiz, Spain) with a heated core box and a hot-gas drying device. A H 32 quartz sand (Quarzw- erke GmbH, Frechen, Germany) was bound with an inorganic Inotec binder system (ASK Chemicals GmbH, Hilden Germany). The binder system con- sists of a liquid component EP 4158 (2 wt%) and a powder additive TC 4500 (1.6 wt%), which are measured relative to the sand mass. The temperatures were set to 155C core box and 220C gas temperature. The specimens were stored for 16 hours at 20C and 10 % humidity after production. The dimensions are 173 :5 mm 22:8 mm 22:8 mm. 2.2. Test setup for acoustic determination of elastic properties The three main components of the acoustic test bench are the automatic hammer, the control unit which controls the microphone (S241 from Super- lux, New Taipei, Taiwan) and the hammer, and the support structure with two parallel steel cords. The automatic hammer is actuated by a stepper motor and has a steel tip. One challenge with an automatic hammer is to avoid multiple hits of the specimen. Our hammer is made of ABS poly- mer, which means that it is exible. The stepper motor starts retracting 4 FrameMicrophone Specimen Automatic hammer Figure 1: Test-bench for determining elastic properties via impulse excitation technique. The main components are a microphone, a specimen, an automatic hammer system and the frame of the test-bench. Acoustic measurementFEM simulation Parameter optimisation Elastic material modelPeaks in spectrumInterpolation of eigenfrequenciesFEM support points Figure 2: Algorithm for determining elastic parameters via impulse excitation technique and FEM simulation. The measured and calculated eigenfrequencies are compared and the Young's modulus and Poisson number are optimised such, that the error between the measured and the calculated eigenfrequencies is minimal. 5 the hammer before it hits the specimen. This leads to a de ection of the hammer and a subsequent impact on the specimen, while the stepper motor is already moving backwards. This allows to retract in good time after the rst hit, thus avoiding multiple hits of the specimen. The specimen rests on two parallel steel wires. The control unit consists of a Raspberry Pi, which controls the microphone and the stepper motor via custom python software. The sampling frequency of the system is 96 kHz which leads to a resolution in the frequency domain of 6 Hz. Figure 1 shows the test bench. The impact position was chosen in one corner of the beam in order to excite both exural and torsional eigenmodes. Figure 2 shows the algorithm used in this article to determine elastic prop- erties via impulse excitation technique and FEM simulation. The measured and calculated eigenfrequencies are compared and the Young's modulus and Poisson number are optimised such, that the error between the measured and the calculated eigenfrequencies is minimal. 2.3. Digital image correlation algorithm and high-speed imaging For the DIC analysis in this article, the ICGN algorithm presented by Pan et al. has been adapted [19]. No arti cial speckle pattern has to be ap- plied, since the natural texture of the specimen suces for adequate analysis. First-order shape functions for warping, a subset size of 15 15 pixels and a discrete cosine transform lter proposed by Garcia are used [20]. Further details on the precision, accuracy and present implementation of the algo- rithm are presented by Hartmann et al. [21]. A high-speed camera (Os3-S2 from IDT, Pasadena CA, USA) with magni cation optics (Ultrasonic from Canon, Tokyo, Japan) is used for the acquisition of the deformation (8-bit grey-scale images). The images are recorded at a frequency of 8500 Hz. A macro cold light source (Xenon Nova 300 W, Storz, Tuttlingen, Germany) is used for illumination. 2.4. Test setup for measuring fracture stress and strain We used 3-point-bending (3PB) experiments to determine the tensile frac- ture stress and strain. The distance between the supports was 150 mm. The experiments were carried out on a Zwick Z20 (ZwickRoell GmbH & Co. KG, Ulm, Germany) equipped with a 20 kN force sensor. The stress was calcu- lated from the maximum force at fracture, via beam theory according to [22]: 3PB=3F3PBl 2bh2(5) 6 x=173.5 mmy=22.8 mm z=22.8 mm Imprint I Figure 3: Meshed specimen for FEM calculation of natural frequencies and eigenmodes. where3PBis the maximum stress in the beam. F3PBis the applied force andlis the distance between the supports. bis the width and hthe height of the beam. The fracture strain was determined using 3PB as well. The high-speed camera was aligned to the area around the punch, which has the highest ten- sile stresses. The high-speed data was evaluated with the algorithm described in section 2.3. 3. FEM calculation of natural frequencies The natural frequencies and eigenmodes of the specimen were calculated with the FEM-Software Abaqus 2018 (Dassault Systems, Velizy-Villacoublay, France). The model is shown in Figure 3. The specimens were measured and modelled, including minor production inaccuracies: for example the imprint I on the front side of the beam is due to the shooting nozzle. The model was meshed with tetrahedron elements with an approximate global size of 2 mm. The calculation was done with a Lanczos solver. Figure 4 depicts FEM results for eigenmodes 1-6. The simulation shows that each exural eigenmode is split in two very close neighbours, due to the asymmetry on the front side of the beam, which results from the production process. For reasons of simplicity, only the rst one of the neighbour-pairs will be considered in this article. 4. Experimental results 4.1. Acoustical determination of elastic parameters We simulated the exact geometry of the beam to obtain the natural fre- quencies (<20 kHz) based on a purely elastic model, with Young's modulus and the Poisson number as parameters. Comparing the simulation results with the measured spectrum allows the identi cation and matching of the 7 1st exural eigenmode (f1) 1st torsional eigenmode (t1) 2nd exural eigenmode (f2) 1st longitudinal eigenmode (l1) 3rd exural eigenmode (f3) 2nd torsional eigenmode (t2) Figure 4: Eigenmodes 1-6 of the specimen obtained by FEM simulation speci c eigenmodes with the peaks of the spectrum. Figure 5 shows a spec- tral analysis of one exemplary data set. We calculated the Young's and shear moduli via Equations 1 and 3. Subsequently, the natural frequencies were calculated using FEM and the obtained elastic parameters. These natural frequencies are marked in the gure with dotted vertical lines and assigned to the speci c eigenmodes with the sequence of eigenfrequencies from FEM. Until 5:5 kHz, the FEM results t the measured peaks, while for higher fre- quencies they do not, especially for torsional frequencies. The reason for this is that the mechanical reality di ers from the mate- rial model utilised in the FEM-calculation. We studied the distance between successive natural frequencies to show this and calculated ratios of successive natural frequencies. This ratio is independent of the elastic parameters in simulation. For example, the rst and second exural frequency can both be used to determine the Young's modulus. Their quotient is determined by the geometrical dimensions of the specimen. Since the dimensions of the mea- sured beam and the simulation are approximately identical, this can serve as a criterion to show how well the elastic two-parameter model corresponds with the mechanical reality. Table 1 shows a comparison between the simula- tion and the experiment. There are small deviations between the simulation and the experiment, which is to be expected. However, one outlier can be identi ed. The ratio ft1=ft2is signi cantly smaller in the experiment (1.81 compared to 2.00). To exclude the test setup as a reason for this behaviour, we changed the support to a cross, and included steel specimens as well. A 8 ff1=ff2ff2=ff3ff3=ff4ft1=ft2ft2=ft3fl1=fl2 Simulation 2.54 1.78 1.50 2.00 1.50 2.00 Experiment 2.59 1.75 1.50 1.81 1.50 1.98 Table 1: Ratios of successive natural frequencies obtained by simulation and experiment. The ratios are independent of the elastic parameters. variation of the supports had no in uence on this e ect, while a measure- ment using steel showed a ratio close to 2.00. This leads to the conclusion that this e ect is in fact a typical behaviour for the inorganically-bound core materials used and that, contrary to the initial impression, the physics in the simulation corresponds better to the experiment at higher torsional frequencies. In order to minimise this model error, more than the rst exural and torsional frequency can be used to calculate the Young's modulus and the Poisson number. This was performed via linear optimisation in Matlab (MathWorks Inc., Natick, USA). We simulated natural frequencies at sup- port points for the following tupel of Young's modulus and Poisson num- ber [(5500,0);(5500,0.28);(6000,0.14);(6500,0);(6500,0.28)]. The results were used to t a linear model, which calculates natural frequencies in the two- dimensional space of the support points. Additionally, we introduced an error function which calculates the root mean square error (rms) for multi- ple peaks between experiment and simulation. This model can be optimised such that the rms is minimal and thus ts the elastic model optimally to the obtained experimental data. Figure 6 shows the result of this optimisation (eigenmodes used: f1,f2,f3,t1,t2,t3,l1,l2). The simulated eigenmodes match signi cantly better to the measured spectrum, while t1remains an outlier. In Table 2 the elastic parameters for seven specimens S1-S7 are shown. The mean Young's modulus is 5 :75 GPa with standard deviation of 0:391 GPa. The mean shear modulus is 2 :43 GPa with a standard deviation of 0:203 GPa. With Equation 4 this leads to a mean Poisson number of 0.183. 4.2. Optical determination of fracture strain with high-speed images In this section, we determine the fracture strain via digital image corre- lation and compare the result to a strain calculation based on the Young's modulus and on the fracture force calculated using Equation 5 and Hooke's law. Figure 7 shows the strain xxin the beam in the area under the punch for nine di erent points in time, taken from one exemplary high-speed data set. 9 0 2000 4000 6000 8000 10000 12000 Frequency in Hz00.20.40.60.811.2Normalised amplitudef1t1f2l1f3t2f4t3l2f5t4Figure 5: Spectral analysis of the impulse excitation data. The doted lines mark natural frequencies based on FEM calculations, with Young's modulus and the shear modulus calculated analytically using f1 and t1, respectively. 0 2000 4000 6000 8000 10000 12000 Frequency in Hz00.20.40.60.811.2Normalised amplitudef1t1f2l1f3t2f4t3l2f5t4 Figure 6: Spectral analysis of the impulse excitation data. The doted lines mark natural frequencies based on FEM calculations. The Young's and shear modulus of the FEM material model result from a error minimisation between the peaks in the spectrum and the calculated eigenfrequencies ( <12 kHz) 10 Specimen no. S1 S2 S3 S4 S5 S6 S7 Young's modulus in GPa 6.18 5.92 5.62 5.15 5.89 - 5.89 Shear modulus in GPa 2.70 2.59 2.36 2.24 2.27 - - Strain in m/m 597 - - 578 - 580 576 Stress in MPa 2.92 2.95 2.84 2.78 2.51 2.70 2.21 Table 2: Results for elastic parameters of inorganically-bound core materials. The eval- uation of the acoustical data was performed with an error minimisation for the Young's modulus and the shear modulus. It should be noted that positive stress translates to tension, and negative stress to compression. As can be expected from a 3PB setup, there are tensile stresses in the lower half of the beam. Furthermore, the fracture is initiated under the punch at the lower beam surface, where analytically the highest tensile stresses are located. Starting from time t2, the tensile stresses start to localise. In t9, the fracture is complete. However, on the original images the human eye is not yet able to detect a fracture. A fracture can be seen as soon as the beam starts to fall and the fracture opens up at time t10. This image cannot be evaluated with the DIC-algorithm any more, since the fracture opening is too wide. For the fracture strain calculation, we used the last image without localising tensile strain. In our exemplary data set, this is t1. The strain was calculated with a mean over a region of interest, which is placed at the location of the subsequent stress localisation, as marked with a rectangle in t1. Please note, that only four specimens were recorded with the high-speed camera. The average of the evaluated results is 583 µm=m, and the results are shown in Table 2. 5. Discussion Comparing the results of the eigenmode simulation and the acoustically obtained data shows that it is necessary to calculate the Young's modulus and the shear modulus with multiple natural frequencies. This is in contrast to EN 843-2, which standardises the determination of elastic parameters of ceramics. There, the recommendation is to use the rst exural and the rst torsional natural frequency to calculate the Young's and shear moduli, re- spectively. Our results show that especially the rst torsional eigenfrequency di ers from the theoretical one obtained by mechanical calculations based on 11 t1 t2 t3 t4 t5 t6 t7 t8 t9 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 Strainxxin mm/m1 0 -1 Figure 7: Strain calculation via digital image correlation for a 3PB experiment and 10 points in time ( t1-t10). Only the area under the punch is analysed. Starting at t2, the strain starts to localise until the fracture is complete at t9, which is still undetectable by the human eye. 12 an elastic material model with two parameters (Young's modulus and Pois- son number). This e ect is due to the complex material behaviour, which results from a inhomogenous micro-structure of the sand-binder compound. However, taking more eigenmodes into account minimises this model error, since the simulation model ts quite well for higher frequencies and the ex- ural eigenmodes in general. Minimising the error between multiple peaks in the measured spectrum and the simulation leads to a stable evaluation of the acoustical data obtained with the impulse excitation technique. In literature, only the Young's modulus for inorganically-bound core ma- terials can be found, while the shear modulus and Poisson number are yet to be determined. Furthermore, the Young's modulus is highly in uenced by the production parameters, environmental conditions, the type of sand and the binder used for the core blowing. This makes it dicult to validate the results obtained in the relevant literature. Griebel et al. determined the Young's modulus of a di erent type of inorganic core to be approximately 7 GPa. They further showed that the Young's modulus is in uenced by the applied load, which leads to a Young's modulus which is 20% lower upon fracture stress [7]. This is a known behaviour for various materials including metals, which has led to the practice of using only low-stress parts of the strain-stress curve for the calculation of the Young's modulus [23]. Here, acoustical determination of the elastic parameters has two advantages over tensile tests. First, the impulse excitation technique only applies a low load on the specimen. Second, with a tensile test, the shear modulus cannot be determined without an additional torsion test. Further validation of the acoustic results can be achieved by comparing them to the DIC data through Hook's law. =E (6) whereis the stress, Ethe Young's modulus and the strain. Calculating the theoretical strain at fracture with the mean Young's modulus and the mean fracture stress of our experiments yields a fracture strain of 468 µm=m, which is 20 % lower than the actual measured fracture strain. Since the Young's modulus is obtained with an acoustical low-load measurement, this is in close agreement with Griebel et al., and validates both, the strain and the Young's modulus measurement. [7] Analysing the scatter of the results for fracture strain and fracture stress shows that the results of the fracture strain have signi cantly less scatter 13 than the fracture stress. This is due to the calculation of the fracture stress, which assumes that the applied load is supported by a homogeneous specimen with spatially constant material behaviour. However, core-blowing leads to a complex micro-structure which does not always ful l this assumption. Taking specimen S7 as example: The fracture stress is signi cantly lower, while the fracture strain is in close agreement with the other specimens. This leads to the assumption that the specimen had a local defect which was the origin of the premature fracture, and shows that the local measurement leads to a more precise and stable determination of the fracture behaviour. However, the numerical requirements, the manual evaluation e ort and the high data amount, which has to be stored and moved by the high-speed camera (10 GB per measurement) makes it a time-expensive method. For this reason, we propose to parametrise future FEM simulations with a fracture strain calculated from acoustical determined elastic parameters and a fracture stress, which is corrected by the factor 0.8 determined in this article. 6. Conclusion In this article, we developed methods to determine all necessary physical properties for an ideal brittle material model of inorganically-bound core materials. The Young's and shear moduli were determined via the impulse excitation technique. The fracture stress and fracture strain were determined using an universal testing machine and a high-speed camera. With these properties, inorganically-bound core materials can be simulated with a fully parametrised ideal brittle model for the rst time. This enables a simulation of complex stress states with FEM, and a comparison of the elastic simulation results with physical experiments. Our future work will include a study regarding the yield surface of inorganically-bound core materials, in order to characterise the plastic material behaviour in addition to the elastic model investigated in this article. Acknowledgements This research was supported by Deutsche Forschungsgemeinschaft DFG (VO 1487/37-1). We would like to thank Quarzwerke GmbH and ASK Chem- icals GmbH for providing the materials needed for conducting this research. 14