Ex and in situ tests of materials: From design to materials parameters via motion estimation

Materials tests are inevitable to investigate how materials behave under load. This also holds for materials that are not manufactured from bulk material but exhibit an interesting microstructure. Unfortunately, in this case classical stress-strain investigations are no longer sufﬁcient to deduce a proper description of failure mechanisms. To overcome this problem, materials testing is usually combined with an imaging modality to gain information on local material behavior. The combination with classical digital cameras is quite common, but only yields surface information. Here, we present a framework to combine mechanical tests with computed tomography. Such in situ tests are not new, however, we also give information on how to perform tests ex situ. Our special focus is on motion estimation. We use an algorithm that is particularly suitable for delicate behavior of materials with interesting microstructures such as foams. The whole pipeline for materials tests, either in situ or ex situ, is outlined using two foam examples.


Introduction
Digital Image Correlation (DIC) is a widely accepted tool to assess local mechanical behavior within a materials test.Part of its success stems from the fact that it can be integrated into nearly every materials test with manageable effort: Large-scale applications, high or low temperature or complex loading devices do not pose a problem as long as there is enough room for one or two high-speed cameras.Moreover, there exist guidelines and strategies to produce valuable and reliable results [11].Much effort has been put into improvement of the method, not only algorithmically, but also by using cameras that record photos at higher speed with higher resolution.However, one cannot belie the fact that DIC yields reliable information only on the surface of the specimen that is tested.No matter how many cameras are used, the material's behavior in the inner layers remains obscured.This may not cause problems when the material is relatively homogeneous.If however a material features a more complex microstructure which governs the material's behavior, then considering only surface strains does not yield a sufficient description of the material.Concrete for example seems to be a dense, homogeneous material on a macroscopic scale, but exhibits a delicate microstructure with various phases and inclusions of different types.The influence is by far not trivial: In compression tests, cracks may form in the interior of test specimens.When they become visible on the outside, it is often too late to observe the phase of crack initiation.Yet the most interesting stage is initiation, as it gives information about the durability of concrete components.Another example where the local behavior has severe influence on the global material is porous, cellular structures.Metal foams belong to that category and in the literature their compressive behavior is described by three macroscopic stages: An initial elastic regime is followed by a plastic collapse of layers, before eventually one observes global material densification.Microscopically, this scheme is often already violated at the very beginning: Though initially the stress-strain curve shows a linear trend and therefore indicates elastic behavior, one can already observe single breaking struts.These failed struts are often a forerunner to the next stage, the plastic collapse of whole layers.It is therefore inevitable to develop materials tests and evaluation methods that unveil this small scale behavior.A commonly accepted tool for gaining insight into materials microstructures is computed tomography (CT).In the classical setup -laboratory tomographs that come with built-in routines for image reconstruction -it is unfortunately crucial that the specimen does not move during the scan.This is due to reconstruction routines not being externally accessible and being based on filtered backprojection which is very sensitive to motion during the scanning procedure.Nevertheless, it is possible to use CT to not only acquire digitized, three-dimensional representations of materials microstructures, but also time series of images displaying the evolution of the material under load.This is achieved by transforming mechanical tests from dynamic procedures to quasi-static ones, meaning that application of load is interrupted to obtain images of the intermediate, static stages of the experiment.Such a procedure is presented schematically in Figure 1.The main task is then to quantify the motion that the sample underwent during such a test.Receiving a digitized full displacement field from the materials test allows for diverse subsequent uses.Quantification of internal damage, crack opening and propagation, but also comparisons to simulations and material parameter identification have been reported [6].In the following, we describe how to extract the motion that a sample underwent during a mechanical test from a sequence 12th Conference on Industrial Computed Tomography, Fürth, Germany (iCT 2023), www.ict2023.orgFigure 1: Schematic representation of an in situ experiment.A foam specimen is tested while CT images are acquired simultaneously.
of CT images.We specify the benefits of using voxel-accurate algorithms instead of subvolume or finite element (FE) based approaches.We present the algorithms within a complete materials testing pipeline, including settings where in situ tomography is not possible.The procedure is illustrated using two examples: a metal-matrix composite (MMC) foam is investigated in situ, and an aluminium foam is loaded ex situ.

Combining Computed Tomography and Materials Testing
Testing materials and acquiring CT images simultaneously has been performed for roughly 35 years [5].However, the power of such tests has been exploited much later with the introduction of so called global digital volume correlation (DVC) methods.In situ CT tests are executed quasi static.That means, a load is applied until a certain value and then kept steady as long as the CT image is acquired.As soon as all the projections for image reconstruction are collected, more load can be applied and another image can be taken.This procedure is repeated until further loading steps will not result in new information, for example when a sample is completely compressed or torn apart.This way a time series of CT images is generated.Many classical mechanical tests can be performed quasi static without any loss of information.Common examples include tests for tension and compression, but also torsion.This holds true as long as one is only interested in load induced changes in the material and not in velocity or acceleration.A sequence of images produced in such an in situ test on an MMC foam can be seen in Figure 2. To deduce motion fields from CT images, in situ investigations are considered irreplaceable.That means that in the literature almost all manuscripts dealing with motion estimation investigate in situ tests.However, there are occasions when testing within a tomograph is nt possible.When procedures take very long as in corrosion or fatigue tests or instruments are too large and complex to include them in a tomograph, images have to be acquired after testing.In such ex situ setups, motion induced damage is typically characterized by comparing statistical quantities of the sample before and after loading.For instance, in ex situ concrete tests, measures like crack lengths and widths are compared.Similar methods have been used in composites [7].Yet it is often still possible to estimate motion in ex situ tests.Generally speaking, the most important factor for successful motion estimation between CT images is that both images contain enough similar content.In case of corrosion of concrete this is fulfilled -one observes a thin corroded layer around rebars in reinforced concrete, or initiation of cracks, so there are only minor changes in the image content.Nevertheless, larger deformations allow for the application of a motion estimation algorithm, too.The ex situ compression experiment on an aluminium foam considered here left only half of the foam intact (see Figure 3), yet the full displacement field can be estimated.These examples also show another condition that is ideally fulfilled to generate meaningful tests: The investigated sample should maintain its load induced shape during the test.If this is not the case, ex situ investigations may lead to underestimation of the damage [7].

Motion Estimation
As indicated earlier, in situ investigations gained momentum with the development of suitable motion estimation algorithms.To formalize the approach, we consider the following setup.We denote images by I 0 , I 1 and assume that they map from an open and bounded domain Ω ⊂ R 3 to the real numbers, so I 0 , I 1 : by a transform φ : R 3 → R 3 that transfers the images such that Ideally, the transformation φ mimics the motion that the real-world-sample underwent.To compute the motion, Equation (1) has to be solved for φ .It is usually impossible to compute the solution directly, therefore often a weak formulation is used.That is, one minimizes with respect to some norm, for example which in the discrete case reduces to the sum of squared differences of both images.In the in situ case φ will resemble displacement that has been induced solemnly by the materials tests, so we have Note that u in fact depends on x, which means we can have a different displacement vector for each position x.In this case the problem is severely underdetermined, as we try to derive a three-dimensional displacement vector from only a scalar correspondence.
State-of-the-art DVC tries to overcome this problem by not estimating an individual displacement vector per voxel, but an average over several tenths of voxels.In local DVC, this is achieved rather trivially by imposing a coarser grid, the so called subvolumes, and estimating one vector per subvolume.In global DVC, the subvolume grid is replaced by a FE mesh, and nodal displacements are calculated.In both ways the number of degrees of freedom is decreased, such that the problem can be solved based on image data.
On the other hand, the mathematical theory of inverse or ill-posed problems proposes regularization to tackle this problem.The use of total variation for motion estimation is very well established in 2D applications and small displacements since the seminal work of Brox et al. [4] from 2004.We recently extended this work to 3D and applied it in materials science [12].The idea is to use a differentiable approximation to the L 1 norm as distance function, i.e. to use a concave function where | • | is the Euclidean norm.It is known from classical 2D image processing, that using such a norm reduces sensitivity towards outliers, promotes sparsity, and preserves sharp edges [8].
Copyright 2022 -by the Authors.Licensed under a Creative Commons Attribution 4.0 International License.The total variation regularization is also approximated by using Ψ, and reads A third penalty is posed on differences in gradients of both images to emphasize features, mainly prominent edges.It reads The benefit of such an additional term is demonstrated in [12].Eventually, the displacement u = (u, v, w), is found by minimizing The regularization parameters λ > 0 and µ > 0 are chosen problem dependent.
In [12], we showed the superior performance of this approach compared to state-of-the-art methods.The averaging nature (also visually observable in [12]) of DVC based methods inhibits representation of small scale behavior in the computed displacement fields.Subvolume based methods will almost always overlook delicate behavior like breaking struts or crack initiation.And another advantage comes with a voxel based method: The intermediate displacement fields can be used to render the whole experiment.That means we get a visual, repeatable 3D impression of the local changes in the microstructure.Note that the sought quantity u in Equation ( 5) is still "hidden" inside the image I 1 .That means our minimization problem features a nonlinearity that is very difficult to resolve.Many motion estimation algorithms use tools like the Taylor approximation to remove this nonlinearity.Unfortunately, no general assumption can be imposed on the image to guarantee a well-defined derivative.One should therefore keep in mind that the optimization process has to be well initialized to prevent the algorithm from being trapped in a local minimum.A very common procedure that we also incorporated in 3D is the so called coarse-to-fine strategy.The scheme is depicted in Figure 4 for a 2D image.A coarse scale image is generated by consecutively smoothing and subsampling the original image.The solution on the coarse scale is then used as initial displacement for the next finer scale.The number n of necessary scales and the resolution reduction factor d can be calculated via Copyright 2022 -by the Authors.Licensed under a Creative Commons Attribution 4.0 International License.
12th Conference on Industrial Computed Tomography, Fürth, Germany (iCT 2023), www.ict2023.orgwhere N is the size of the longest edge of the image.By setting the product equal to 1 we roughly end up on a scale where the largest displacement does not exceed one voxel edge length.The maximal displacement can be roughly estimated in advance, see for example Figures 3c and 3d, where it is chosen as the difference of the sample boundaries.Usually, a resolution reduction factor between 0.7 and 0.95 is recommended.5 shows a heatmap suggesting how to choose the parameters.In the ex situ case the transformation φ does not only consist of the test induced displacement, but also of a rigid transformation due to removing the sample from the CT and inserting it back in.Classically, rigid body motion consists of six degrees of freedom, three for rotation and three for translation.It can be represented by a matrix vector multiplication with the help of homogeneous coordinates, that is, a coordinate (x, y, z, 1) is transformed by a matrix M ∈ R 4×4 to new coordinates (x , y , z , 1), and M is composed of two matrices M = T R such that Copyright Here, T denotes the translation matrix and R = R x R y R z the rotation matrix, where we exemplarily denote the rotation through angle θ about the x-axis by In this case, φ now reads and we minimize for u and M. The two motion steps, loading by u and movement by M, are typically introduced independently.Hence, it is usually possible to separate both computations.That means, M is calculated first and applied to generate a new image.Then the displacement u is calculated based on the new image as in the in situ setting.Thankfully, rigid body motions can be computed rather easily.Usually, algorithms for this task can be found under the name rigid registration, and many implementations are even available open source.In Table 1 we summarize three of the most common ones, including information on programming language, a rough estimate on runtime based on personal experience, and licensing.Nevertheless, none of these methods can forego Taylor approximation: u and M are both evaluated within the argument of the image function I 1 and the resulting nonlinearity has to be removed.So, as before, careful preinitialization is crucial.In the case of the ex situ compressed foam this can be visualized quite vividly: When the sample was removed for loading and inserted again after being loaded, different faces of the sample were put to the front, inducing a missalignment by 90 • .The registration algorithm now only manages to align the outer shape of the specimen, but the overall rotation through 90 • is too large to be computed by any registration algorithm.The algorithm will always be trapped in a locally optimal solution (alignment of the outer shape), but will not recover the correct global result, which includes the large rotation.Experience within our group showed that rigid registration algorithms work reliably if the rotation to be computed is not larger than 10 • .This is therefore the precision which should be reached by the manual prealignment.If manual prealignment has to be applied anyway, one might raise the question of the necessity of the additional rigid registration.However, in the aluminium foam sample, the final computed rotation had an accuracy of a couple of hundredths of a degree, which is impossible to achieve by human eyesight.The consequences of bad prealignment when estimating the translation are visualized in Figure 6a.There is still a significant mismatch between both images, resulting from a global translation that was not removed manually.If one now performs the rigid registration, the algorithm will, due to the large pores in the foam structure, end up in a local minimum.The cells coincide roughly, but in comparison to an image where registration was performed after manual prealignment (Figure 6b), the difference is obvious.

A Pipeline for Materials Testing
To generate meaningful images to apply motion estimation in the first place, the materials test has to be designed carefully.Therefore, the first step in performing mechanical tests with CT image acquisition is to decide which loading stages are of interest and qualify as candidates for the quasi-static interruption.This means, that any experiment is interrupted at certain, predefined stages and that one expects the specimen to be in equilibrium at this stage.Only this assumption guarantees that a clear CT image can be acquired.If the loading procedure involves for example impact loading, the test cannot be interrupted.
One could nevertheless still try to acquire images before and after the test -however, if the specimen is deformed too severely, displacements can not be calculated.The MMC foam in Figure 2 serves as a very good example.Similar samples have been tested macroscopically quite extensively [2,3].It was found that such foams behave more or less like metallic foams, and -as indicated earlier -one can identify the regimes of elasticity, plastic collapse, and densification quite well.Based on the stress-strain curves from measurements, interruptions of the compression were fixed at 2%, 4%, 6%, 10%, and 20% engineering strain.Figure 2 shows one corresponding slice in all of the 3D images during the compression test.The test was well designed -we can see all stages of materials behavior that we expected beforehand.
Copyright 2022 -by the Authors.Licensed under a Creative Commons Attribution 4.0 International License.In the ex situ setup, exactly the same considerations have to be made.However, when placing the samples in the tomograph after applying the load, a lot of computational effort can be saved.Cuboidal samples should be positioned such that the top face always remains the same.Ideally, also the front facing the X-ray source should coincide.That is unfortunately often a very tedious task.If one decides to attach any kind of marker to help with positioning, it has to be ensured that the contrast is not influenced.If a material or liquid with high attenuation coefficient is chosen, it may cause low contrast in parts with similar attenuation coefficients.Also markers may be lost during the testing procedure.For instance, when the aluminium foam was loaded, cleaning of the sample after the testing removed the previously applied marker.Figures 3a and 3b show the consequence.By observing the highlighted pores bottom left one can see that the samples are rotated approximately through 90 • .Nevertheless, ensuring that at least some face looks at the detector, preinitialization as described in the previous section needs only rotations through 90 • .

Processing of the Motion Estimation
Having fixed the parameters we are able to calculate the displacement vector fields.We now have several options to deduce information from them.The most obvious one is just information gain to develop new hypotheses concerning materials behavior.The original work regarding the in situ tests of the MMC foams for example states that plastic deformation is only visible from 10% compressive strain on, and even there the authors struggle to find real fracture of struts.Our method however was able to find a fracture already at 2% compressive strain -a behavior that fits much better to the assumption of pseudo elasticity of foams [10].Figure 7 shows the corresponding slices, where the fracture can be identified by the jump in the displacement field.It is also visible in the ε 33 component of the strain.Though compressed quite severely, even the aluminium foam sample allows for some conclusions.The displacement field for example exhibits highest strain rates at those areas where more material from the already damaged areas can be found.Figure 8 shows this behavior.Another possibility of using the full field displacement measurements is to derive constitutive parameters from them.Assume for the moment that we want to derive constitutive parameters for elasticity.The material is mainly governed by a (possibly heterogeneous) fourth order Hooke tensor [1].Having measured full-field displacements at hand, there are several ways of using them to deduce the tensor components.The "classical" method is by finite element model updating.Here, the distance between simulated and measured displacements is minimized by varying over the materials parameters.Another example is the virtual fields method which applies the principle of virtual work to derive a linear system whose solution then represents the desired materials parameters.For a detailed derivation and comparison of these two methods, we refer to the work of Avril et al. [1].

Conclusion
In this contribution we presented how to combine materials test with CT not only in situ but also ex situ.Special emphasis is put on motion estimation.Hence, by combining CT and testing, we do not only obtain a digitized version of the interior of our samples, but also compute full-field, voxel accurate displacement fields.We used a novel sophisticated motion estimation algorithm which allows to gain more information than with classical state-of-the-art approaches.We investigated two examples of a similar material, outlined the relevant (mechanical) issues and presented the associated tests.We showed how to carry out such tests in situ, that means we perform a mechanical test and in parallel acquire volume images by CT.We are aware that the more complicated and delicate the material is, the more complicated materials tests can become.Often these tests have to be performed by highly specialized machines which do not allow for an integration into a CT device.We therefore also presented examples where specimens have been loaded ex situ.We showed that such tests can be put into an in situ like context, including the computation of accurate local deformation fields.We showed that our setup has the great advantage over commonly applied DIC of being able to unveil deformation inside the sample.Moreover, we proved that our motion field estimation algorithm captures local sudden changes better than conventional DVC.

Figure 2 :
Figure 2: 2D slices of CT images of an MMC foam, unloaded and loading stages according to the applied quasi-static loading scheme.The sample is compressed from the right.Fracture can be observed already at 2%, collapsing layers and densification further on.Sample, in situ testing, and imaging: TU Bergakademie Freiberg.

Figure 3 :
Figure 3: Scanning setup of ex situ test of aluminium foam.Top: Pictures of positioning in tomograph.Bottom: Corresponding CT slice view of the aluminium foam sample before and after loading.Sample and testing by TU Kaiserslautern, CT imaging by Fraunhofer ITWM.

Figure 4 :
Figure 4: Image pyramid in 2D.Gaussian blurring and subsampling are performed to get a coarse, low resolution version of the original image.Image taken from [9].

Figure 5 :
Figure 5: Heat map indicating how parameters for the image pyramid should be chosen.X-axis: Number of levels in the pyramid.Y-axis: Resolution reduction factor.Color: Maximal displacement value in voxels that can be captured by the whole coarse-tofine scheme.NaN values indicate parameter combinations that exceed image limits, as this scheme condenses the image edge length below 1.

Figure 6 :
Figure 6: Overlay images of an aluminium foam.Left: Erroneous rigid registration due to missing preapplied translation.One can see clearly that large pores roughly coincide, whereas registration does not fit in small pores.Right: Correct registration after manual application of a translation.

Figure 7 :
Figure 7: 2D slices of CT images of an MMC foam, unloaded and loaded, with corresponding slice from the 3D displacement field.Sample, in situ testing, and imaging: TU Bergakademie Freiberg.

Figure 8 :
Figure 8: 2D slices of CT images of aluminium foam, unloaded and loaded ex situ, with corresponding slices of the 3D displacement field magnitudes.In (c) the full range of magnitude is displayed.The large deformation of the sample is resembled quite accurately.In (d) the magnitude is scaled such that the local behavior in the deformed middle layer can be observed.

Table 1 :
Algorithms for rigid registration.Higher number of stars indicates higher computation time.
2022 -by the Authors.Licensed under a Creative Commons Attribution 4.0 International License.