Possible Application of Computed Tomography for Numerical Simulation of the Damage Mechanism of Cementitious Materials—A Method Review

Abstract

Modeling the meso-scale behavior of a material allows one to answer many problems in an inductive reasoning approach. By simulating the behavior of the sample, researchers can correct the overall behavior of the material. The purpose of this article was to gather the main information linking the issue of tomographic imaging to the modeling of damage mechanisms. This paper addresses the topic of meso-scale modeling of cementitious materials. The authors have mapped the interconnectedness of the issues of cementitious materials and X-ray computed tomography (µCT) according to the Web of Science database. The main interconnecting threads are indicated. The authors focused on the use of µCT. The theoretical basis for conducting tomographic measurements is presented. Reference was made to the basics of physics and mathematics in X-ray projection and data reconstruction. Tools and analyses for data processing are indicated. The benefits of in situ µCT are indicated with reference to the practical application of the method. In addition, the main developments of recent years in the most widely used computational methods for meso-scale models are presented.

1. Introduction

Cementitious materials are heterogeneous, cohesive-frictional materials and exhibit complex non-linear inelastic behavior under multiaxial stress states. Concrete is one of the most common materials, which influences the need for analyses of not only strength [1,2,3], but also experimental [4,5,6], environmental [7,8,9] or thermal [10,11,12] properties. Despite rising costs and the search for new technological solutions [13], interest in cementitious materials as a construction material is not waning. Figure 1a shows the number of articles published on the topic of cementitious materials in the period from January 1990 to November 2022. The data were obtained from the Web of Science database. A dynamic upward trend in research interest is evident. In the last five years alone, more than 10,000 papers have been written on the topic of cementitious materials. Research interest is not declining, and with the development of modern research technologies and techniques, papers are presenting more practical results to push science ahead.

In parallel, advanced testing methods are developing that allow for the description of the internal structure of the material under examination in a non-destructive manner. X-ray computed tomography (µCT, xCT) is a non-invasive imaging technique that has gained popularity in the field of research on cementitious materials. CT scans allow for the creation of three-dimensional images of the internal structure of a material, providing valuable information about its microstructure, porosity and segmentation. This information is critical for understanding the behavior and performance of cement-based materials, such as concrete, and for evaluating factors that affect their durability. For example, CT scans can provide insight into the pore structure of concrete and track the movement of water and air through the material, which can have a significant impact on its overall strength and durability. In addition to its use in the research of cement-based materials, X-ray CT is also used in construction and engineering to assess the quality of concrete and other cement-based materials. CT scans can detect early signs of cracking, deterioration, and other damage after the sample is loaded. This allows one to detect the effects, introduce countermeasures and prevent further damage. This information can also be used to develop new and more durable materials and construction methods. X-ray CT is also useful in the study of cement-based composites, as it allows for a detailed analysis of the behavior and performance of these materials under various loads. Typically, µCT does not leave a mark on the sample, although in the case of electronics, damage or loss of data can occur. The paper [14] widely referes to tomographic studies on cementitious materials. CT imaging provides quantitative information on pores morphology, but unlike microscopy, it gives 3D information [15]. The accuracy of the acquired µCT data makes it highly suitable for the development of meso-scale models. Due to the high cost of the device, xCT is not available in every research facility. Figure 1b shows a graph of articles published on xCT in recent years. It should be noted that the data do not include items related to medicine. The data refer to publications in the Web of Science catalog in categories directly related to construction: materials science multidisciplinary, engineering manufacturing, engineering civil and construction building technology. As with the material itself, xCT technology is of increasing interest. In the last five years, more than 380 articles have been added to the Web of Science catalog on research using CT in materials science.

Generated models based on µCT can be entered into numerical software. The history of numerical calculation in civil engineering dates back to the 1940s, when engineers began using electronic computers to solve complex structural analysis problems. The early methods were limited to solving linear problems, but with advances in computer technology and mathematical techniques, engineers were able to tackle non-linear problems as well. This led to the development of the finite element method (FEM) in the 1960s; FEM became a standard tool for structural analysis and design in civil engineering. The finite element method is a numerical method that allows engineers to analyze and design complex structures. It divides a structure into small elements, representing them mathematically, and solves equations to determine their behavior under different loading conditions. This method has become a staple in modern civil engineering, enabling engineers to optimize designs for safety, cost and sustainability.

In contrast to FEM stands the discrete element method (DEM). DEM is a numerical technique for simulating the behavior of granular materials, such as soil, rock, concrete and debris. It is based on the principle of modeling individual particles and their interactions, rather than treating the material as a continuous mass. This method has become an important tool for civil engineers to analyze and design certain structures, such as retaining walls, slopes and embankments, where the behavior of granular materials is critical. Unlike FEM, which focuses on the behavior of the structure as a whole, DEM provides a more detailed understanding of the behavior of individual particles and their interactions, making it likely to be extremely useful for meso-scale applications.

Figure 2 shows a linkage map of the issues of cementitious materials raised in articles from the last five years, based on the Web of Science catalog. Visually, four main threads can be distinguished, which can be reduced to three main trends when the content is analyzed. Researchers describe their work in terms of the material used and the research strand (red group), where attention can be drawn to the cementitious composite itself, types of concrete, additives such as fiber and issues (failure, load, uniaxial tests). The second group (light green) includes phenomena and properties such as cracking, self-healing and simulation. The third group comprises entries in the dark green and blue groups. Here, one can distinguish issues concerning bonds, chemical reactions and environmental issues—including ecological aspects (waste placement in the material). In the context of this issue, attention can be drawn to “failure mode”, directly related to “crack”, “model”, “reinforcement” or “load”.

A similar linkage map is shown in Figure 3. The subject of the analysis was CT in the categories materials science multidisciplinary, engineering manufacturing, civil engineering or construction building technology. ‘Concrete’ was mainly linked to ‘microstructure’, ‘model’, ‘pore structure’, ‘evaluation’ and ‘volume’. The main trends in cementitious materials research not only indicate the use of tomographic studies from evaluation but also indicate the search for the use of microstructure or pore network data to define models.

In reviewing the available literature, the authors did not see any papers referring to the use of µCT to develop a path to model the behavior of a sample with numerical software. The aim of this paper is to bring together crucial information about µCT and the conduct of numerical calculations. The authors seek to identify possible directions for the integration of data acquired by µCT scanning together with analysis and testing equipment. Specialized equipment can be used to carry out investigations that best reflect the behavior of cementitious materials with respect to their microstructure.

2. µCT Basics

X-ray tomographs as used in industry rely on X-ray spectroscopy. A complete description is provided in a paper [16] published more than 20 years ago. It still correctly describes the principles of operation. The sample is tested in a special chamber of the device. The sample is positioned between a radiation emitter and the detector. The test is characterized by the position of the sample in the chamber, the radiation power used and the detector operating parameters. Proper selection of the required parameters makes it possible to receive results enabling further 3D reconstruction of the object. Unlike medical tomography (where the patient’s position must be stable), during an industrial scan, the detector and tube remain in a fixed position, while the object under examination is rotated along a vertical axis. Thousands of measurements of the attenuation of the beam of radiation passing through the object under examination are taken during a single scan.

An X-ray tube (commonly called a tube) is an X-ray emitter. In the vacuum tube, electrons are emitted from the heated filament. The electrons are accelerated toward the anode by the potential difference U_ACC. The electrons pass through a hole in the anode to a magnetic coil, which concentrates the electrons over an area of a few µm on the target. The target is covered by a thin layer of tungsten on a diamond. At the target, electrons are abruptly slowed down, resulting in the generation of an X-ray beam. The energy of the electrons is converted 99% into heat, 1% is converted into X-rays [17].

A sample is placed between the radiation emitter and the detector. The sample is mounted on a high-precision manipulator. By convention, the manipulator moves in three perpendicular directions and allows rotation in the vertical axis. The x-ray tube–sample–detector distance affects the magnification of the sample. A sample positioned closer to the tube will give a higher magnification than one positioned next to the detector, affecting the voxel size (vx) measurement accuracy. The voxel size is the size of the cubes from which the whole scan is constructed. The voxel is associated with a gray value and has a size of $\mathrm{V}=\frac{\mathrm{pixel} \mathrm{size}}{\mathrm{magnification}}$. The authors [18] pointed out that specifying the required resolution of a µCT image of particles depends on the targeted measurement task. The experimentally adopted ratio of particle size to voxel size should be greater than 10. However, when transferring these results to other measurement tasks, it is also necessary to take into account the variability in the measurement of the µCT system used and the specific shapes, textures and mineral composition of the studied particles. In [19], the problem of selecting the appropriate resolution and the influence of the voxel size on the results is described. It is necessary to emphasize the important distinction between the resolution of physical objects and the size of pixels (voxels). The scale of object resolution length corresponds to the smallest characteristic that can be resolved in an image and varies with the size of the sample and the internal attenuation contrast of the material in relation to the background. Pixel (voxel) size is the element size of the information in the image.

The shadow of the X-rays is converted into visible light through the scintillation screen, and the visible light is converted into the digital image by the optical sensor. The main advantage of this approach to the detection problem is to obtain a high-contrast image without distortion. Detailed information on the detectors is available in publications [20,21]. ISO 15708 part 1 splits X-ray detectors into two groups: ionization detectors and scintillation detectors. The second common division of modern xCT detectors is a line detector and a flat panel detector. The interior of the tomography chamber equipped with a flat panel detector is shown in Figure 4a.

2.1. Absorption of X-rays

For details of the technology and background on the principles of physics, see the [22,23,24]. The theoretical basis of X-ray computed tomography is based on the Beer-Lambert law. A monochromatic radiation beam I₀ after passing through a homogeneous absorbing medium of thickness l is weakened according to Equation (1):

$${\mathrm{I}=\mathrm{I}}_0{\mathrm{e}}^{-\mathrm{kl}}$$

(1)

The number k is the linear attenuation coefficient. Its value is material-specific and depends on the density, atomic number, contribution for photoelectric attenuation, incoherent Compton scattering and coherent Rayleigh scattering. Equation (1) can be developed to the form shown by Equation (2).

$${\mathrm{I}=\mathrm{I}}_0\exp \left[-{{\displaystyle \int }}_0^{\mathrm{l}}\mathrm{k}\left(\overrightarrow{{\mathrm{r}}_0}+\overrightarrow{\mathrm{s}\mathrm{n}}\right) \mathrm{d}\mathrm{s}\right]$$

(2)

In the above equation $\mathrm{k}\overrightarrow{{\mathrm{r}}_0}$ refers to the linear attenuation coefficient at a point (x, y, z) in the local sense. The length s is the length of the path measured along the ray. X-ray radiation has its direction described by a vector $\overrightarrow{\mathrm{n}}=({\mathrm{n}}_{\mathrm{x}}{,\mathrm{n}}_{\mathrm{y}}{,\mathrm{n}}_{\mathrm{z}})$. The integral is determined between the entry point of the radiation into the sample $\overrightarrow{{\mathrm{r}}_0}{=(\mathrm{x}}_0{,\mathrm{y}}_0{,\mathrm{z}}_0)$, and the exit at a distance of $\mathrm{l}$.

2.2. Principles and Assumptions in Industrial µCT Scanners

As for further considerations, the assumption of the absence of imperfections in the arrangement of the device generating the projections must be made. It is also assumed that the radiation falls perpendicular to the detector plane and that sample rotation occurs in the vertical axis. According to the standard axes in tomographs (e.g., GE phoenix M), the radiation propagates in the direction of the $\overrightarrow{{\mathrm{e}}_{\mathrm{z}}}$, direction $\overrightarrow{{\mathrm{e}}_{\mathrm{y}}}$ is the vertical axis (around which rotation takes place) and $\overrightarrow{{\mathrm{e}}_{\mathrm{x}}}$ is the horizontal axis parallel to the detector plane. The axes are drawn in Figure 4a. A single pixel of the (x, y) detector will receive a signal I (x, y) of power according to the Beer-Lambert law:

$${\mathrm{I}(\mathrm{x},\mathrm{y})=\mathrm{I}}_0\exp \left[-{{\displaystyle \int }}_0^{\mathrm{l}}\mathrm{k}\left(\mathrm{x}, \mathrm{y},\mathrm{z}\right) \mathrm{dz}\right]$$

(3)

Author [22] also indicates the practical application of the projection of the sample’s X-ray attenuation field, which is defined as below.

$$\mathrm{P}\left(\mathrm{x}, \mathrm{y}\right)=\int \mathrm{k}\left(\mathrm{x}, \mathrm{y},\mathrm{z}\right) \mathrm{dz}$$

(4)

Value $\mathrm{P}\left(\mathrm{x}, \mathrm{y}\right)$ can be defined as the difference $\mathrm{P}\left(\mathrm{x}, \mathrm{y}\right)=\log {\mathrm{I}}_0-\log \mathrm{I} \left(\mathrm{x},\mathrm{y}\right)$. In addition, the rotation of the sample in the vertical axis by a given angle θ is taken into account. Assuming the system ($\mathrm{O}, \overrightarrow{{\mathrm{e}}_{\mathrm{X}}}, \overrightarrow{{\mathrm{e}}_{\mathrm{Y}}}, \overrightarrow{{\mathrm{e}}_{\mathrm{Z}}})$ on the device’s manipulator, it is given:

$$\overrightarrow{{\mathrm{e}}_{\mathrm{X}}}=\cos \mathsf{\theta }\overrightarrow{{\mathrm{e}}_{\mathrm{x}}}-\sin \mathsf{\theta }\overrightarrow{{\mathrm{e}}_{\mathrm{z}}}, \overrightarrow{{\mathrm{e}}_{\mathrm{Y}}}=\overrightarrow{{\mathrm{e}}_{\mathrm{y}}}, \overrightarrow{{\mathrm{e}}_{\mathrm{Z}}}=\cos \mathsf{\theta }\overrightarrow{{\mathrm{e}}_{\mathrm{z}}}+\sin \mathsf{\theta }\overrightarrow{{\mathrm{e}}_{\mathrm{x}}}$$

(5)

Making the final transition from the local system to (X, Y, Z) the projection P (x, z, θ) with angle θ is described by Equation (6).

$$\mathrm{P}\left(\mathrm{x}, \mathrm{y}, \mathsf{\theta }\right)=\int \mathrm{k}\left(\mathrm{x} \cos \mathsf{\theta }-\mathrm{z} \sin \mathsf{\theta }, \mathrm{y},\mathrm{z} \cos \mathsf{\theta }+\mathrm{z}\sin \mathsf{\theta }\right) \mathrm{dz}$$

(6)

Obtaining k (X, Y, Z) from the set of projections P (x, y, θ) is an inverse problem. In practice, this process is called reconstruction. After the reconstruction process based on a set of 2D projections, the device operator receives a set of 3D data allowing inspection of the material interior. A single projection is shown in Figure 4b.

2.3. Reconstruction

Once the projections giving information about the cross sections have been obtained, the algorithm proceeds to the logarithm. Projections are processed by inversion and calculation of the logarithm of the grey values performing $-{{\displaystyle \int }}_{\mathrm{L}}^{}\mathrm{k}\left(\mathrm{x},\mathrm{y}\right) \mathrm{dl}=-\ln \frac{{\mathrm{I}}_{\mathrm{Out}}}{{\mathrm{I}}_{\mathrm{O}}}$. Then ramp filtering is carried out [25], which enhances edges or higher frequencies. To reconstruct the 3D image, the 2D projections are reprojected into a virtual sample volume. Subsequently, reconstruction takes place.

The reconstruction method called the reverse radon transform [26] is usually based on the Fourier transform. The formula is valid for the case where the source of the beams describes a bounded curve satisfying a set of weak conditions. A reduction in the dimension of the problem (from three to two) can be made by mapping the 3D space as a set of cross sections. A complex mathematical description was proposed in the 1980s [27] and has been developed over the years [28]. The method is based on the inverse Fourier transform of $\hat{\mathrm{f}}$, the support of which is contained in the compact space Ω. We will denote the inner product of two vectors α, β ∈ R₃ by <α, β>. We will denote the integral of the function over the domain D in R₃ by ${{\displaystyle \int }}_{\mathrm{D}}^{}$. For the following formula, f is a real integrable function on R₃.

$$\mathrm{f}\left(\mathrm{x}\right)={{\displaystyle \int }}_{{\mathrm{R}}^{\mathrm{3}}}^{} \hat{\mathrm{f}}\left(\mathsf{\xi }\right){\mathrm{e}}^{\mathrm{2}\mathrm{i}\mathsf{\pi }<\mathsf{\xi },\mathrm{x}>}\mathrm{d}\mathsf{\xi }$$

(7)

In most cases, the radiation used has a polychromatic characteristic [29]. Most reconstruction methods cannot correctly check the non-linearity of the absorption of polychromatic X-rays [30]. As a result of beam hardening, the images may contain artifacts that significantly reduce the quality of the data. An extensive description of the problems associated with reconstruction is given in [22]. The author described the issues of geometric considerations, calibration, artifacts and software. Currently, the offered software has a built-in reconstruction module dedicated to data from a specific manufacturer’s device.

The second method of data reconstruction that is less commonly used is algebraic reconstruction techniques (ART) [31]. It has been developed into a simultaneous algebraic reconstruction technique. The basics of ART algorithms are simple and intuitive. Each measured (via absorbance) density is projected back through the reconstruction space in which densities are iteratively adjusted to bring each reconstructed projection into correspondence with the measured projection [32]. It is also worth mentioning the development of deep learning-based reconstruction methods [33,34,35,36]. These methods are still being developed and are far less common.

2.4. Use of Computed Tomography

The resolution obtained in modern µCT research is very well suited to the study of the structure of heterogeneous cementitious materials whose fine fractions are 100 µm (or more) in size [37]. Because of its accuracy and detail, µCT has applications in the study of precision materials used, for example, in aerospace [38]. Numerous publications point to the usefulness of µCT in the study of various types of cementitious materials. Looking at the material itself, several groups can be distinguished. Research on cement paste has been carried out in [39,40,41,42]. The authors [43] propose a method to quantify the degree of hydration of slag-blended cement paste with the application of µCT. Applications for µCT in plain concretes were found in publications [44,45], lightweight concretes [46,47], foamed concretes [48,49], high porosity concretes [47], magnesium oxychloride cement concrete [50] and reinforced concrete [51,52].

The µCT data were used to assess the compactness of the concrete [53]. The technique has also been used to characterize the surface of concretes [54]. Studies of the effect of static loading on concrete specimens using tomographic imaging are known [55]. The usefulness of the tomographic image method in the evaluation of shear tests is indicated in [56]. There are studies that show how to determine the change in pore inertness of a concrete sample subjected to freezing cycles by tomography using a medical device [57]. Basic tomographic studies of steel fiber-reinforced concretes are presented in the publication [58]. Tomographic analysis mainly focused on the arrangement of the fibers in volume and the cracks created during the strength tests. The effectiveness of neutron tomography is indicated for the analysis of samples with high moisture content. Neutrons are very strongly absorbed by water, making it possible to image the water distribution with high contrast. Another advantage is that neutrons have high penetrating power, making it possible to image large samples. The disadvantage is the complexity of such systems [59]. In [60], the effectiveness of neutron tomography in tracking water movement in samples of cementitious materials is indicated. The use of neutron tomography has been well demonstrated in [61]. The researchers conducted an analysis of cement-based materials for the conditioning of low- and intermediate-level radioactive waste. A large amount of water inside hardened cement samples deteriorates the quality of neutron radiographs due to a large number of incoherent scattering effects. There are also known applications of dual-energy X-ray computed tomography for the detection of voids in fiber-reinforced composites [62].

Porosity analyzes are often used when looking for defects in concrete [59,63] or changes in the material caused by external influences. Leading software that works with CT data allows automated porosity/inclusion analysis. Pointing out the usefulness of tomography, one should also mention the emerging trend of 3D printing of cementitious materials. Research [64] looks at the use of a halloysite nanotube as a mineral additive to enhance the properties of 3D printable geopolymer mortar. Authors used μCT imaging technology to examine both mold-cast and 3D-printed specimens to assess their internal structures. Tomography was utilized to visualize the material and measure the level of porosity. By analyzing the data, the study was able to compare the porosity of different samples, taking into account not only the overall porosity but also the number and size of pores. Another example of research that can be mentioned is described in [65]. The microstructure of 3DP-ECC was studied using µCT scanner. The pores’ features and distribution were captured and analyzed. The CT scans showed that the porosity within the layers and between the inter- and intra-layer did not have significant differences, and it was only possible to differentiate the layers based on their volumes with the help of CT scans.

Tomography is also used to indicate geometric changes resulting from external impacts. In [66] authors worked on a method to assess the wear of shaft guides with the application of nondestructive and semi-destructive testing methods. Among the tools used was µCT. The authors took samples of wooden guides and analyzed their wear under laboratory conditions. µCT allowed internal structural damage to be indicated and a nominal/actual analysis was carried out. Analysis was used to map the structure destruction of the guide surface. A broad overview of existing methods of working with CT-acquired data is given in the paper [53].

2.5. Dynamic CT and Damage Mechanisms of Cementitious Materials

Dynamic CT (4DCT, CT-4D) is a technique widely used in imaging during medical examinations [67,68,69]. Procedures have been developed to enable the tracking of changes in the tested object under the influence of an external phenomenon. It involves recording multiple images over time with the ability to play back the processes as video. In X-ray tomography, it is possible to obtain measurements of the specimen during external influence. In the context of analyzing the mechanism of damage, the behavior of the concrete structure during each loading stage is important. The crack formation process is shown in Figure 5. A natural phenomenon resulting from the difference in the properties of the aggregate and the matrix is the occurrence of tensile stresses. In the initial phases of loading, the formation of microcracks occurs. These microcracks generate bridging stresses as they move. When the tensile strength is exceeded, a decrease occurs as the load increases, and the friction between the concrete elements decreases due to the enlargement of the cracks. A detailed description of the phenomenon is presented in [70]. Due to the material characteristics, the available sources refer to specific mixtures. However, detailed descriptions affecting the possibility of 3D modeling of the damage mechanism of cementitious materials are still lacking.

Computed tomography has been used to assess the mechanism of damage [72]. Authors [72] highlighted that high-resolution computer tomography can be used to identify microstructures in composite materials in three dimensions, such as fiber distribution, particle volume fraction and damage mechanisms such as fiber cracks and breaks. There are devices such as the in situ loadcell tensile stage for X-ray CT applications [73]. It allows for static or thermal effects inside the tomography chamber. Due to their low popularity and high cost, there are still few publications indicating the use of such tests. The procedure involves a gradual increase in the forces acting on the sample. After an increment of force, the piston is stopped, and a 360° scan is performed. The measurement is followed by another force increase and another scan. Once the specimen is destroyed (or the set force is reached), the operator completes the test by pulling the entire loadcell with the sample out of the tomograph chamber. A scan performed in this form should be as short as possible to reduce the effects of relaxation [74]. There are known applications of the method for the assessment of damage of composite materials [75]. It has been found that a reduction in total porosity due to increased load can only be predicted by in situ CT measurements, which confirms the usefulness of the proposed tests. Examples of cross-sections obtained in the in situ compression test are shown in Figure 6.

Authors [76] used X-ray µCT in situ loadcell to investigate the internal structure of a 3D-printed sample of alloy. In the paper, a tensile test was used. Measurements were carried out at six load levels. The authors used porosity analysis to quantify the changes that occur in the sample. According to the paper, the volume of this pore did not change significantly in the tensile process. In addition, the data obtained allows a visual assessment of the formation of the failure process, the determination of the angle of cracking in relation to the direction of force or the development of plastic deformation. Similar studies have been carried out for carbon fiber/epoxy composites [77], CFRP laminates [78], asphalts [79] and shales [80].

In situ X-ray studies on cement paste were carried out almost 10 years ago [81]. The researchers indicated the possibility of determining the deformation of the specimens under external loading. The sample was subjected to further load thresholds in a compression test inside the CT chamber. After reaching the stage, CT scan was performed. Researchers used the average gray scale values and standard deviations of the grayscale values as a measure of the change in the sample. Once the results of the proposed analysis are visualized, the change in sample structure is easily discernible, which will greatly facilitate the calibration of the numerical models. A similar research process was presented by the authors in [74]. The researchers subjected cement-based composites to strain-hardening tests. Tension and compression experiments were carried out on specially prepared miniature specimens in a testing rig integrated in an industrial computed microtomography. The paper raises several technically relevant issues. The duration of the scan should be as short as possible. This affects relaxation effects and measurement reliability. The duration of a single scan was approximately one hour, so the elongation (force) plots show significant drops at the time of scanning. A single voxel had a dimension of (13 µm)³. This resolution allowed the authors to detect microcracks. The authors further indicated the usefulness of 2D radiography as a primary tool for describing changes within a structure. Bending tests are also used to assess the damage of concrete [82]. Measurements of crack mouth opening displacement was performed in two steps. Beam was scanned under the existing residual force and the second one was scanned after beam unloading. The maximum width of the crack during continuous µCT scanning was approximately 20% to 30% higher than the beam being unloaded to the scanner. In situ studies to determine the differences in meso-scale damage evolution of concrete under static and dynamic tensile loading using µCT are presented in [83]. It has been confirmed that the failure mechanism is specific to the type of loading. The nature of the failure can be determined by evaluating the tomographic data.

3. Numerical Analysis

Concrete structures are becoming increasingly analyzed in detail using numerical software. Computed tomography allows the structure of the scanned element to be accurately represented on a meso-scale. The resulting geometry is the basis for further calculations. Numerical analysis is based on the solution of a set of equilibrium equations and kinematic admittance of the displacement field. The models are based on boundary and initial conditions, which are assumptions of the designer. The finite element method is one of the most popular and widespread computational methods used in construction, mechanics, fluid, air or heat flow simulations. In its basic form, it is based on Equation (8), where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, [u] is the displacement matrix and [F] is the force matrix.

$$\left[M\right]\left[\ddot{u}\right]+\left[C\right]\left[\dot{u}\right]+\left[K\right]\left[u\right]=\left[F\right]$$

(8)

It is generally assumed that concrete is treated as an initially isotropic material with anisotropies caused by a change in stresses [84]. The combination of static and kinematic admittances is the essence of constitutive models. The material constitutive law usually arises as a result of complex experiments [85]. A complex review of the literature on constitutive models is presented in [86]. The publication highlights the main constitutive models and their division. The models were divided into linear and non-linear models. In addition, the general applicability of both groups is pointed out. Linear elastic models are the most basic type of constitutive models used to describe the behavior of concrete. In these models, concrete is treated as having linear elastic properties until it reaches its ultimate strength, at which point it fails in a brittle manner. For concrete under tension, the linear elastic model is accurate and sufficient in predicting behavior up to failure due to the low failure strength. However, concrete under multiaxial compressive stress exhibits nonlinear behavior and linear elastic models are not effective in these cases. Nonlinear constitutive models can provide better results and there are two main approaches used for nonlinear modeling: secant formulation (total stress-strain) and tangential stress-strain (incremental) formulation. There is still a need to find new models for special applications [87,88]. The correlation of constitutive relationships allows for a more accurate representation of the behavior of the simulated phenomenon. The most advanced models for complex concrete failure simulations are shown in

Table 1. Examples of complex constitutive models used for cementitious materials.

Model	Applications	References
Localizing gradient damage model	investigate the quasi-brittle fracture behavior of concrete at meso-scale; simulation of static tensile cracking on double-edge-notched specimen, direct tension test under impact loading and l-shaped specimen under static and dynamic tensile cracking; fiber reinforced ultra-high performance concrete bending simulation;	[89,90,91]
Willam-Warnke failure criterion	describing behavior of Ca-Si hollow blocks (masonry); triaxial behavior of lightweight concrete; modeling of reinforced concrete; ultra-high performance concrete nonlinear analysis	[92,93,94,95]
Menetrey and Willam	plain and reinforced concrete structures; combined fracture-plastic model, peak load evaluation; analysis of steel-reinforced concrete walls and slabs	[96,97,98,99]
Onate et al.	modified version of the Mohr–Coulomb yield surface	[86]

.

The numerical approach and X-ray CT technologies are widely used for simulations that have higher accuracy, but each numerical model should be created on the basis of a section and scanning of physical specimens, which is called experimental dependence [100]. Applications of reconstructed models and analyses to develop models of carbon fiber-reinforced textile composites are known [101,102]. Tomographic data can be represented with varying degrees of accuracy as grids fed into numerical environments generating similar behavior with real samples [45]. In addition, the development of methods to extract CAD models from µCT data makes it easier to work with scans of poor quality or full of artifacts [26]. Voronoi cells are also known to be used to describe a sample based on µCT scan data [103]. The 3D Voronoi diagram of the sample is created by using the Voronoi cells, which represent the aggregates and mortar in the meso-scale. The Voronoi algorithm generates a new synthetic dataset representing the sample based on statistics determined by measurements. The generation of 3D meso-scale models on the basis of µCT tests allows accurate simulations and calibration of numerical parameters with in situ experimental results. Studies linking experiments and numerical analyses to a given load set are summarized in

Table 2. Sources describing the type of effect.

Loading	References
Uniaxial	[104,105,106,107,108]
Biaxial	[109,110,111,112,113]
Triaxial	[114,115,116,117]
Cyclic	[117,118,119,120]
Thermal	[121,122,123]
Hydro-mechanical behavior	[117,124,125]

.

Paper [37] refers to the application of models of concrete reinforced with polymer short fibers obtained by µCT to finite element calculations. The authors did not transfer the sample model into the numerical environment. The µCT data were used to quantify the pore distribution in the sample and to determine the geometry of the fibers. The statistical data obtained allowed for the determination of general material characteristics capable of addressing a wide range of applications. The analytical and numerical calculations are in good agreement, providing a maximum discrepancy of ~2.5%. The paper [126] refers to the study of the adhesion of steel fibers to concrete. The authors focused on an accurate representation of the fiber geometry in order to perform specimen-specific numerical calculations. The research indicated a high correlation between the change in porosity (appearance of defects) in the matrix and the occurrence of boundary stresses in the FEM calculations. The graphs, together with the FEM solution, are shown in Figure 7. In the study [127], micro-CT was employed to obtain the non-randomly oriented distribution of ellipsoidal pores in cement paste with an AE-agent added. The authors proved that µCT provides accurate data about non-randomly oriented pores, and thereby aids in the utilization of the numerical and Mori–Tanaka method. In [128], the authors presented a method to accurately generate a 3D mesh based on µCT gray value. This study proposes a FEM model to investigate the deterioration of concrete subjected to multiple physical forces, based on its 3D microstructure obtained through X-ray CT imaging. The CT images are converted into a mathematical matrix and transformed into 3D voxels with varying gray values, which serve as the basis for the FEM analysis. A threshold value of 200 is set to separate the mortar and aggregate components in concrete. The generated data were used for mechanical, thermal and moisture simulations.

Cementitious materials are prone to cracking under extreme events and rough environments, leading to extensive inelastic behavior. On the other hand, nucleation of cracks in heterogeneous materials such as concrete usually occurs at the meso-scale around the weakest areas—the interfacial transition zones (ITZs)—between aggregates and the surrounding cementitious mass or matrix [129]. It is reasonable to implement such zones in the models being built. The real thickness of ITZ in concrete has been found to be around 10–50 μm [103,130]. Due to the grid size, an ITZ thickness of 100–800 µm is assumed for meso-scale summation [131,132], although the studies indicate the introduction of ITZ with a thickness up to 1 mm [133]. It has been shown that changing the thickness of the ITZ has a negligible effect on the results obtained [134]. Given the size of the ITZ, meshing models in FEM can be problematic [135]. A good description of the meshing ITZ method is presented in [136]. The numerical tests demonstrated that using solid wedge elements to depict the ITZ in concrete effectively captures the distinctive behavior of concrete under tension, including the nonlinear response before reaching the maximum stress and the decrease in strength caused by the formation and progression of microcracks in the ITZ and mortar. In [137], two meso-scale 2D models were generated based on µCT studies. Two approaches were utilized: a continuous approach and a discrete approach. Attention was paid to the shape of the fracture zone between the aggregate grains. In addition, the influence of the properties of interfacial transitional zones on fracture was investigated. DEM offered valuable insights into the fracture progression, including both micro and macro cracks. The research revealed that the mechanical properties of ITZs have a significant impact on the overall strength and macrocracking of the material, highlighting the importance of conducting experimental studies, creating 3D models, and considering small grain size in DEM calculations. The evaluation of poorly characterized ITZ influences the strength of concrete [138], but the elastic modulus of concrete is insensitive to the strength of ITZ/aggregate [139]. In [139] it was pointed out that heterogeneity in the ITZ/mortar has no obvious effect on the mechanical behavior of the concrete and elastic modulus of concrete is insensitive to the strength of ITZ/ aggregate. The ITZ zone is shown in Figure 7e.

Another method that is being developed is differential quadrature method (DQM). DQM was first proposed by Bellman and his associates in the early 1970s. The method is mainly based on the idea of Taylor series expansion. The temperature-dependent nonlinear dynamic stability of the functionally graded CNT reinforced visco-plate was present in [140]. The method was used for buckling analysis [141]. The results obtained through DQM have been found to match well with those generated by the popular finite element method (FEM) and the ANSYS software. This suggests that DQM can be used as a substitute for the commonly used FEM.

Simulations of damage mechanisms were also performed using the finite difference method (FDM) [142]. The authors analyzed the effect of loads on a coal sample. The authors noted the impossibility of introducing all discontinuities into the model. Another method practiced for foamed concrete was the discrete lattice model (DLM). In the lattice model, the continuum is schematized as a set of beam elements that can transfer axial and shear forces and bending moment [143]. The discrete element method (DEM) also offers a practical approach to the issue. DEM is a set of numerical methods and algorithms aimed at calculating the physical properties of a large number of objects in free motion. DEM allows for determining the trajectories of particle motion and interactions between elements of discrete media. The application of DEM for concretes is presented together with the algorithms in [144]. The paper [145] presented a method in which laser scanning was used to map the geometry of the aggregate. Using DEM, it was shown that the failure behavior of the degree of concrete and the damage of different components under dynamic tensile loading are influenced by the ratio of aggregate strength to mortar strength. Other studies (FEM) [146] showed that the morphology of the aggregate has a non-negligible influence on post-peak mechanical behavior, the orientation of the aggregate affects the initiation and propagation and self-restraint stresses reduce the strength of concrete. Paper [147] presents meso-scale thermomechanical analyses of plain concrete based on DEM. Very high accuracy with the experiment was obtained at temperatures up to 550°C. The authors proved that the developed 2D meso-scale DEM model is reasonably accurate in predicting the transient thermo-mechanical behavior of concrete. More implementations with listed constitutive models are shown in

Table 3. Application of numerical methods with a focus on constitutive models.

Numerical Method	Matrix Model	Aggregates Model	ITZ Model	Reference
FEM	Holmquist–Johnson–Cook (HJC) model	Holmquist–Johnson–Cook (HJC) model	Not modeled	[148]
	Holmquist–Johnson–Cook (HJC) model	Holmquist–Johnson–Cook (HJC) model	Holmquist–Johnson–Cook (HJC) model	[149]
	K&C Concrete Damage Model	K&C Concrete Damage Model	K&C Concrete Damage Model	[150]
	Plastic damage model (Lee and Fenves)	Linearly elastic	Not modeled	[151]
	Concrete Damage Model #71 LSDYNA	Linearly elastic	Concrete Damage Model #71 LSDYNA	[152]
	K&C Concrete Damage Model	Nonlinear plastic	K&C Concrete Damage Model	[153]
	Mat_72R3 LsDyna	Pseudo Tensor (Mat_16 in LSDYNA)	Not Modeled	[154]
	Drucker Prager model Ottosen model	Drucker Prager model	Not Modeled	[155]
	Concrete damaged plasticity model (CDPM)	Linearly elastic	CDPM	[156]
	CDPM	Linearly elastic	CDPM	[132]
	Plasticity model with isotropic damage model	Ottosen model	Cohesive model	[157]
	CDPM	Linearly elastic	CDPM	[158]
	CDPM	Linearly elastic	Not Modeled	[159]
	Elasto-viscoplastic damage model	Linearly elastic	Elasto-viscoplastic damage model	[160]
	Gurson model	Linearly elastic	Not Modeled	[161]
	Isotropic damage model	Linearly elastic	Not modeled	[162]
	CDPM	Linearly elastic	CDPM	[163]
FDM	Mohr-Coulomb model	Non-cementitious material		[142]
DLM	Discrete lattice fracture model	Application for foamed concrete		[143]
Numerical Method	Contact Model			Reference
DEM	Flat-Joint model			[146,145]
	Burger model			[164]
	Temperature-dependent Mohr–Coulomb failure envelope			[147]
	Multi-level stress corrosion			[119]

.

4. Summary

This paper analyzes the state of the art in data acquisition and modeling of mechanisms occurring in cementitious material samples. The main focus of the discussion is on the use of µCT in the acquisition of geometry data. In situ µCT studies were indicated as a potential source of information on the failure mechanism. Next, the main currents in computational methods that allow meso-scale analyses of cementitious materials were indicated. On the basis of the available studies, the following summary can be formulated.

X-ray projections provide basic information about the structure of a material. X-rays are the main and leading tool in non-destructive 3D imaging methods. X-ray projections are based on basic principles of physics and can easily indicate potential damage in a sample. Measured absorbance is a function mainly of the density and atomic number of the material. Additionally, projections made correctly can be used as an input for 3D reconstruction, where CT scans are involved. Modern µCT is a high-precision method. Today’s devices can image cores 5 cm in diameter with an accuracy of 25–50 µm. The data obtained provide accurate information about the structure of cementitious materials. The use of high-contrast detectors influences the ability to distinguish between materials of different densities. The reconstructed data sets allow for analysis of porosity or distribution. Data acquired by µCT allow for precise mapping of the sample. Geometric data were very detailed. The cited studies indicate paths that allow the generation of finite element meshes or the use of Voronoi cells. Geometric models and their numerical counterparts allow simulation of phenomena occurring in the material.

In situ testing in CT can indicate additional relationships in the destruction processes of many materials, including cementitious materials. The innovative research path allows for the calibration of models with real data. The correlation is made between the results of a simulation based on the geometry of a specific sample together with experimental data. This action affects the maximum model match. Meso-scale modeling techniques can be divided into continuous and discrete methods. Numerical methods have developed a large number of constitutive models. The literature indicates high accuracy in the calculations. Meso-scale FEM models require the use of ITZ. Models developed with high accuracy require significant computational resources. The number of elements increases abruptly as the analyzed volume increases. Meso-scale models allow simulation of phenomena occurring in the structure of cementitious materials. In the near future, it is not expected to be possible to simulate full meso-scale structures. With the development of hardware, it will be possible to use tomographic analyses to determine porosity conditions and material distribution for simulations using meso-scale elements. At present, in special cases, it is possible to make the overall parameter in the material model dependent on the actual value of, for example, porosity.

The model calibration should start with the correct acquisition of geometric data. Given the purpose factor and voxel size information, the first scan of the sample can be done outside the in situ loadcell device. A direct scan will improve data quality and accuracy. Subsequent CT scans can take place at next force levels. After gaining information about a given sample, two approaches are possible: a) mapping the sample as a mesh, taking into account the geometry of the whole sample—selected aggregate and pores, or b) compiling statistical data and further generating models (e.g., Voronoi tessellation). The first approach will provide the ability to compare numerical results with in situ µCT studies. Statistically generated models can also be used, and calibration will refer to maximum strengths. Mesh-based numerical models (FEM, FDM, DQM) allow the generation of 3D elements with a high representation of the actual geometry. These methods allow accurate calibration with the obtained in situ µCT results. However, when using statistical data, it is worth using the advantages of discrete methods (DEM). However, there is still a lack of studies that directly compare the two approaches in tests.