Full-scale testing and multiphysics modeling of a reinforced shot-earth concrete vault with self-sensing properties

Measurement Science and Technology PAPER • OPEN ACCESS Full-scale testing and multiphysics modeling of a reinforced shot-earth concrete vault with selfsensing properties To cite this article: Antonella D’Alessandro et al 2024 Meas. Sci. Technol. 35 105116 View the article online for updates and enhancements. You may also like The direction vector assisted INS/spectral redshift integrated navigation system for spacecraft Xu Peng, Yan Li, Gaoge Hu et al. - Transfer learning-based channel attention enhancement network combined with Gramian angular domain field for fault diagnosis Dongxiao Hou, Jintao Mu, Bo Zhang et al. - Measurement of air-water counter-current flow rates in vertical annulus using multiple differential pressure signals and machine learning Feng Cao, Ruirong Dang, Bo Dang et al. - Measurement Science and Technology Meas. Sci. Technol. 35 (2024) 105116 (13pp) https://doi.org/10.1088/1361-6501/ad6173 Full-scale testing and multiphysics modeling of a reinforced shot-earth concrete vault with self-sensing properties Antonella D’Alessandro1,∗, Andrea Meoni1, Rubén Rodríguez Romero2, Enrique García-Macías3, Marco Viviani4and Filippo Ubertini1 1Department of Civil and Environmental Engineering, University of Perugia, via Goffredo Duranti 93, 06125 Perugia, PG, Italy 2Department of Continuum Mechanics and Structural Analysis, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain 3Department of Structural Mechanics and Hydraulic Engineering, University of Granada, Campus Universitario de Fuentenueva s/n, 18071 Granada, Spain 4HEIG-VD/HES-SO—Haute Ecole d’Ingénierie et de Gestion du Canton de Vaud, Route de Cheseaux 1, CH-1401 Yverdon, Switzerland E-mail: [email protected] Received 4 May 2024, revised 23 June 2024 Accepted for publication 10 July 2024 Published 30 July 2024 Abstract Civil constructions significantly contribute to greenhouse gas emissions and entail extensive energy and resource consumption, leading to a substantial ecological footprint. Research into eco-friendly engineering solutions is therefore currently imperative, particularly to mitigate the impact of concrete technology. Among potential alternatives, shot-earth-concrete, which combines cement and earth as a binder matrix and is applied via spraying, emerges as a promising option. Furthermore, this composite material allows for the incorporation of nano and micro-fillers, thereby providing room for enhancing mechanical properties and providing multifunctional capabilities. This paper investigates the damage detection capabilities of a novel smart shot-earth concrete with carbon microfibers, by investigating the strain sensing performance of a full-scale vault with a span of 4 m, mechanically tested until failure. The material’s strain and damage sensing capabilities involve its capacity to produce an electrical response (manifested as a relative change in resistance) corresponding to the applied strain in its uncracked state, as well as to exhibit a significant alteration in electrical resistance upon cracking. A detailed multiphysics numerical (i.e. mechanical and electrical) model is also developed to aid the interpretation of the experimental results. The experimental test was conducted by the application of an increasing vertical load at a quarter of the span, while ∗Author to whom any correspondence should be addressed. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1 © 2024 The Author(s). Published by IOP Publishing Ltd Meas. Sci. Technol. 35 (2024) 105116 A D’Alessandro et al modelling of the element was carried out by considering a piezoresistive material, with coupled mechanical and electrical constitutive properties, including a new law to reproduce the degradation of the electrical conductivity with tensile cracking. Another notable aspect of the simulation was the consideration of the effects of the electrical conduction through the rebars, which was found critical to accurately reproduce the full-scale electromechanical response of the vault. By correlating the outcomes from external displacement transducers with the self-monitoring features inherent in the proposed material, significant insights were gleaned. The findings indicated that the proposed smart-earth composite, besides being well suited for structural applications, also exhibits a distinctive electromechanical behavior that enables the early detection of damage initiation. The results of the paper represent an important step toward the real application of smart earth-concrete in the construction field, demonstrating the effectiveness and feasibility of full-scale strain and damage monitoring even in the presence of steel reinforcement. Keywords: smart shot-earth concrete, self-sensing materials, damage detection, full-scale experimental tests, numerical multiphysics modelling, carbon microfibers, mechanical and electrical properties 1. Introduction Cementitious materials and concrete industries play a pivotal role in the construction sector. These materials offer exceptional versatility for various building types, owing to their ease of use and manufacturing. Cement-based materials can be significantly enhanced through the incorporation of additives and Supplementary Cementitious Materials (SCMs), thereby improving their physical, mechanical, and technological properties [1,2]. However, factors such as production processes, component degradation, modifications, restoration, and maintenance of concrete structures throughout their service lives demand careful consideration for their impact on costs and environment [3,4]. As a matter of fact, the environmental impact of cementitious materials, including resource waste, CO2emissions, and other greenhouse gas releases, can be significant and necessitates mitigation or redesigning [5–7]. Over the years, there has been a growing awareness of the environmental impact of the concrete industry, which has motivated the development of several eco-friendly strategies to mitigate the phenomenon [8]. Approaches to reducing environmental impact include using less harmful components, optimizing/modifying production processes, enhancing durability and resistance of the final product to prolong structural service life, and minimizing maintenance interventions [9–11]. Incorporating natural or recycled components can also diminish the environmental footprint of cementitious composites [12,13]. Among possible natural components, earth emerges as a promising alternative to reduce the amount of cement in concrete production. As one of the earliest construction materials, earth can be used in dried or fired form, the former often being recycled from excavation or construction sites [14–16]. However, the inclusion of earth in concrete may affect its strength and durability, necessitating the addition of further cement binders, reinforcement additives, or tailored fillers for enhanced performance [17,18]. Specific nano- or micro-fillers, with electrically conductive and piezoresistive capabilities, can enhance the performance of earth-concrete providing smart multifunctional properties, such as selfsensing capabilities [19–21]. The ability to self-monitor the integrity and consistency of structural elements increases safety thanks to the possibility of early detecting cracks and other damage scenarios [22,23]. Nevertheless, although smart earth-concrete has been already proposed by the authors for small samples [24], it has been never applied and tested in fullscale structures. This paper presents the development of a strain- and damage-sensing fiber-reinforced shot-earth composite and its pioneering application to a full-scale smart structural specimen consisting of a shallow steel reinforced barrel vault with a span of 4 m entirely constructed from the novel composite via spraying. An experimental test is carried out where the vault is pushed down to failure, and its electrical response is measured. Additionally, a multiphysics numerical model is developed to reproduce and interpret the experimental results. It is found that the proposed material is well suited for structural applications, as it outputs an easy-to-measure electrical response that is correlated to mechanical strain in elastic conditions and provides evidence of crack formations in the postelastic stage, thus being well apt for self-monitoring structures. 2. Smart shot-earth concrete The current research work endeavors to develop and assess a pioneering multifunctional construction material blending earth and cement, enriched with carbon micro-fillers, to bolster the sustainability and performance attributes of structural elements. While the mix design of the binder is based on previous studies [25,26], the application of the smart material through spraying and the full-scale validation are novelties of the present work. The interest in utilizing earth as a binder for construction materials is primarily motivated by its sustainability, ease of accessibility, and the enhanced natural comfort it brings 2 Meas. Sci. Technol. 35 (2024) 105116 A D’Alessandro et al [27,28]. Various methods exist for incorporating earth into buildings, depending on the specific element being considered. Usually, raw earth is adopted for non-structural components, coatings, or internal interventions [29–31] because of its low mechanical strength and low resistance to environmental conditions, such as rain, and variations in temperature and humidity. For load-bearing structures, fired earth elements, in the form of bricks and tiles, are diffusely organized in more complex assemblies able to resist dead and service loads [32,33]. However, this type of application possesses a high environmental footprint due to the energy consumption and emissions during the preparation process [34,35]. Raw earth constructions offer environmental advantages over conventional materials and techniques. Current research focuses on investigating the seismic response of earth buildings, the mechanical properties of earth masonry, as well as its thermal and hygrothermal performance [36]. However, earth requires stabilization, achievable through deep compaction or the use of binders. Emerging technologies in stabilized soilconcrete involve high-speed projection of a mixture of earth, cement, aggregates, and water (known as shot-earth). The preparation involves dry mixing of components, and the spraying process reduces water content, yielding a denser and more efficient material [37]. This technique, based on both the use of stabilizer and high-speed spraying, is considered in this work and extended to include carbon microfiber reinforcements, allowing to produce a construction material with strength comparable to light concrete [38]. In particular, the matrix for the spray process must include several key features to enhance the mechanical performance of the material. First, the water amount must be sufficient for the earth to stick together while avoiding any excess that could compromise the overall compaction of the material. The aggregate must be sufficiently large in size to compact the material upon impact due to its kinetic energy. Finally, the dry components must be graded so that the packing density is optimal, resulting in a compact material. For other types of applications, such as thermal ones, the matrix of shot-earth and the process must be optimized to avoid creating a too-dense material. Furthermore, the use of an even rate of stabilization is typical for elements such as vaults and beams. In elements such as bearing walls, it is preferred to reduce the rate of stabilization gradually from the bottom to the top of the wall, when compatible with the specifics of the site, as is commonly proposed for non-bearing walls. The composite nature of shot-earth allows for modifications in the mix design through the addition of fillers and fibers. Notably, incorporating carbon fillers can confer multifunctional capabilities to binders, as demonstrated in various studies [39,40]. In particular, the piezoresistive properties of the composites when electrically conductive fibers are in the optimal amount close to the percolation threshold, enhance the monitoring capability of the compound [41]. By correlating variations in electrical features with changes in strain or stress, it becomes possible to detect load redistributions or excessive loading conditions. Similarly, detecting significant changes in electrical conductivity can allow to directly detect cracks and other types of local damages [42,43]. Various suitable fillers are available in both the literature and in the market for this purpose [44–46]. However, for specific real-scale applications, fibers that do not require difficult dispersion procedures are desirable. The authors opted for carbon microfibers due to their unique ability to be added without the use of surfactants or complex procedures like sonication, as well as their optimal aspect ratio for enhancing electrical properties in large-scale samples [24,47]. The fibers were incorporated into the shotearth through a tailored preparation process, as outlined in the following sections. 3. Experimental methods The experimental campaign undertaken in this study involved constructing a full-scale vault using sprayed shot-earth doped with carbon microfibers and subjecting it to compressive eccentric loading until failure. Key aspects of the experiments included selecting the components for the mix design, fabricating the full-scale specimen, and establishing the mechanical and electrical setups, as detailed in the following paragraphs. 3.1. Materials The developed smart earth-concrete comprises a binder composed of both earth (providing sustainable characteristics) and cement (enhancing material strength and durability). The specific formulation involved 7 parts of excavated earth and 2 parts of cement; aggregates were added in the amount of 7 parts (figure 1(a)). The aggregate used was a mix of 4– 8 mm and 8–16 mm alluvial aggregate, dredged from Lake Neuchˆ atel and commonly used in the area for producing concrete in local plants. The cement was type CEM I 42.5N, bought in bags from a local supplier (Jura Cement). The earth came from an excavation site in Ursy, Switzerland, and had a low clay content [38,48]. The full-scale circular shallow barrel vault, having a span of 400 cm, a rise of 50 cm, a thickness of 40 cm, and a width of 200 cm was produced using a high-speed spraying process, as outlined in Birgin et al [24] (figure 2(c)). The mixture was pressurized and propelled onto the vault-shaped mold at high velocity, adding the minimum amount of water for obtaining a workable compound assuring a certain compactness. Just before the spraying of the material, carbon microfibers, in the amount of 0.05% with respect to the weight of the binder, i.e. earth and cement, were scattered, to enhance their dispersion (figure 2(b)). Carbon microfibers were sourced from SGL Carbon, specifically type SIGRAFIL®, with a diameter of 5 µm and a length of 6 mm, and were incorporated into the dry components. This particular amount of inclusions has been selected after previous research carried out by the authors on the specific construction material [24,49]. It is noteworthy that this marks the first application in the literature of microfibers in sprayed shot-earth, selected for their appropriate characteristics to prevent agglomeration and blockage of the spraying tube. Figure 1represents the step production of the smart fullscale vault. The vault was reinforced with an electro-welded mesh with a diameter of 6 mm and a spacing of 20 cm, applied 3 Meas. Sci. Technol. 35 (2024) 105116 A D’Alessandro et al Figure 1. Preparation of the smart-earth vault: (a) scheme of the production, (b) picture during the projection process of the element. Figure 2. Positioning of the linear variable displacement transducers (a), and electrical setup (b) adopted to perform the experimental tests. at a distance of 5 cm from the intrados. Figure 1(d) indicates that internal electrodes were applied after the production of half thickness, by fixing them on the external opposite sides of the vault. It is therefore important to remark that electrodes and steel rebars were placed at a distance of 5 cm. Electrodes consisted of copper wires, 1 mm in diameter, embedded in the middle of the vault thickness (equal to 20 cm), and applied transversally to the span of the vault, along the entire width of 200 cm. The diameter of the electrodes and their placement have been selected by the authors based on previous experimental campaigns and field tests conducted on bidimensional elements made of similar construction materials [50]. 3.2. Preparation of the samples As already mentioned, the vault, with dimensions of 40 cm in thickness, a span of 400 cm, and a length of 200 cm, was reinforced with steel rebars. During the projection process, five copper wires, each with a diameter of 1 mm, were positioned on the right-hand side of the vault at mid-height, spaced 40 cm apart. Figure 1illustrates the scheme of the preparation process of the smart shot-earth vault (figure 1(a)) and a picture of the element production (figure 1(b)). The appearance of the vault after curing and instrumentation is represented in figure 2. As shown in the figure, Linear Variable Displacement Transducers (LVDTs) were placed at both sides of the vault to measure its deflections, while electrical measurements were carried out on the smart material through connections applied at one side. 3.3. Experimental setup LVDTs recorded the vertical displacements of the vault at the lateral sides during the whole failure test. Electrical measurements involved applying a square wave voltage input to the smart material and recording the resulting voltage drop output between single couples of contiguous electrodes (figure 2). Specifically, the power supply voltage, a 20 V peak-to-peak signal input with a frequency of 1 Hz and a duty cycle of 50%, was provided to electrode e2by using a function generator (RIGOL DG1022), while electrode e4was connected to the ground. The choice of these two active electrodes allowed to obtain good measurements with a limited external voltage. In particular, voltage drops were simultaneously measured by using a multichannel analog input module (NI PXIe-4302) mounted on an NI PXIe-1073 chassis with a sample rate of 10 Hz at sections 1–4of the vault (corresponding to Channels 1, 2, 3, and 4 of the acquisition system), as well as across a shunt resistor (Channel 0) connected in series with the circuit to retrieve the electrical current Icrossing the specimen. Electrical resistance corresponding to each section of the vault was computed using Ohm’s law as Ri=Vi(t)/I, with Vi(t) representing the voltage drop measured at the ith section of the structural element. Note that, for the chosen electrodes’ configuration, current crossing channels 2 and 3 is not equal to the one crossing channel 1 and channel 4. However, due to the conductivity of the rebars, voltage drops across all channels were very similar therefore indicating similar current values across all pairs of electrodes. 4 Meas. Sci. Technol. 35 (2024) 105116 A D’Alessandro et al Figure 3. Failure test configuration: (a) general setup of acquisition systems, (b) detail of the instrumented zone closed to the applied load. Figure 4. Images of cracks during the test and of the failure mechanism. It follows that the calculated values of electrical resistance, although approximated, can be considered indicative of the actual electrical resistance across each pair of electrodes and, as shown later in the paper, suited for damage detection. Values of the voltage drops considered in the computations were sampled at 80% of the constant part of their output signal. Accordingly, the electrical resistance of each section of the vault was sampled at 1 Hz [51]. The relative change in electrical resistance was then determined for each section of the vault as follows: ∆R R0=R−R0 R0,(1) where R0is the electrical resistance of each section of the vault at the beginning of the test, while Rrepresents its subsequent values. The initial (unstrained) values of electrical resistance in the analyzed sections (i.e. from Channel 1 to Channel 4) were 15.8, 15.2, 12.9, and 12.8 Ω, respectively, proving both the homogeneity of the material and its electrical conductivity. It is noted however that the rebars contributed to electrical conduction from one electrode to the other, resulting in higher conductivity with respect to the shot-earth material. The load configuration for the failure test is depicted in figure 3. The experimentation was performed under displacement control, maintaining a constant displacement rate of 0.005 mm s−1. Hydraulic jacks were employed to apply the vertical load on section 3of the structural element. A steel beam positioned atop a concrete curb was utilized for load redistribution across the whole width of the specimen. Figure 4depicts the initial formation of cracks on the vault during the test. Specifically, thin cracks emerged at the intrados of the structural element near the load application area, as well as at the extrados in proximity to the right support. Failure was characterized by a classic four-hinge mechanism. In figure 5, the time histories of the relative change in electrical resistance for the monitored sections of the vault are presented. The trends for sections 3and 4exhibited a steady increase as the applied load intensified, particularly until approximately 15 min into the test. A noticeable deviation in these trends occurred shortly before the 15 minute mark, coinciding with the onset of the first cracks on the vault. Conversely, the trends observed for sections 1and 2indic- ated a smaller sensitivity to load increments, at least until the initiation of cracking. However, even these sections exhibited 5 Meas. Sci. Technol. 35 (2024) 105116 A D’Alessandro et al Figure 5. Time histories of the normalized change in the electrical resistance obtained for the monitored channels of the vault, and scheme of the considered channels. a discernible but moderate change in the monitored electrical output trends. As plastic hinges formed, internal stress redistribution likely occurred within the vault, leading to the detection of additional load increments in sections 1and 2of the structural element. Overall, the trends in the relative change of electrical resistance across monitored channels consistently mirrored the applied load and the pattern of crack development. 4. Multiphysics numerical modelling The reinforced earth concrete vault was modelled using ABAQUS 6.10, a commercial software for Finite Element (FE) analysis [52]. The adopted model parameters are summarized in table 1and taken from [25,49,53]. Hereinafter, the details of the constitutive model adopted for the composite material and the electromechanical simulation of the vault are reported in sections 4.1 and 4.2, respectively. 4.1. Mechanical behavior The mechanical behavior of the vault is simulated by assuming an elastic, perfectly plastic model for the steel rebars, and an elastoplastic model with stiffness degradation for the smart earth. Specifically, the Concrete Damage Plasticity model (CDP), originally proposed by García-Macías et al [53] and implemented in ABAQUS, has been adopted. This model is wellsuited for the modelling of brittle materials, allowing the consideration of cracking in tension and crushing in compression. On this basis, inelastic strains in compression εin cand tension εin tare estimated as: εin c=εc− σc E0, εin t=εt− σt E0,(2) where εcand εtstand for the total strain under compression and tension, respectively, and E0is the initial modulus of elasticity for the undamaged material. Hereinafter, subscripts ‘c’ and ‘t’ are used to relate the corresponding quantities to compression and tensile stresses. Table 1. Electromechanical constitutive properties adopted for modelling of smart earth concrete (∗Term σb0/σc0 denotes the ratio of the second stress invariant on the tensile meridian). References: [24,54,56]. Smart earth concrete Mechanical properties Density ρ2070 kg m−3 Elastic modulus E010 990 MPa Poisson’s ratio ν0.16 Compressive strength fcm 9.50 MPa Direct tensile strength fct 1.47 MPa Rate of weakening nw1.3 Ultimate compressive strain εcu 2‰ CDP Eccentricity 0.1 Viscosity parameter 0.001 Dilation angle 15◦ Kca 0.66 σb0/σc0 ∗1.16 Electrical properties Gauge factor λ172.73 Unstrained electrical conductivity σE,0 2.5 ×10−05 S m−1 Permeability parameters (n,k) (1,20) Steel reinforcements Mechanical properties Density ρ7850 kg m−3 Elastic modulus E0200 GPa Poisson’s ratio ν0.3 Yield strength fy380 MPa Electrical properties Electrical conductivity σE8.0 ×1006 S m−1 According to the CDP formulation by [53], the plastic strains (εpl c,εpl t) are determined through a modified Drucker– Prager criterion with five constitutive parameters as indicated in table 1(for further details, readers may refer to [55,57]). On this basis, it is possible to compute the uniaxial compression and tensile stresses considering damage-induced stiffness degradation as: σc= (1−dc)E0(εc−εpl c), σt= (1−dt)E0(εt−εpl t)(3) where dcand dtdenote degradation variables for compression and tensile stresses spanning from 0 (undamaged) to 1 (complete loss of stiffness). For the definition of the post-elastic stress–strain curves, given the lack of specific models for the smart earth, the formulation for unreinforced concrete reported in EN 1992-1-1 has been adopted [54]: σc=fcm kη−η2 1+(k−2)η, k1=1.05Ecm εc1 fcm , η =εc εc1 (4) 6 Meas. Sci. Technol. 35 (2024) 105116 A D’Alessandro et al where Ecm stands for the secant modulus of elasticity determined following the empirical expression by EN 1992-1-1 as Ecm =22(0.1fcm)0.3, with fcm the average compressive strength. The term εc1 in equation (4) denotes the corresponding strain at the compressive strength, also estimated following the Eurocode as εc1 =0.7(fcm)0.31. For the definition of the post-elastic tensile stress–strain curve, the formulation proposed by Wang and Hsu [56] has been adopted as: σt=fct(εct εt)nw ,(5) where εct =fct/E0is the tensile strain corresponding to tensile strength and nw denotes the rate of weakening. From the previous formulation, it is possible to define completely the post-elastic stress–strain curves of the material considering the material parameters furnished in table 1. 4.2. Electromechanical coupling For the electrical behavior of the smart earth, a linear piezoresistivity model is assumed in the elastic uncracked stage. Following the theoretical formulation presented in [57], similar to cement-based materials doped with carbon nanotubes, it is assumed that the smart earth behaves as a volumetric strain sensor. Such an assumption allows the definition of one single piezoresistive coefficient (λ) or gauge factor to characterize the strain-induced variations in the electrical conductivity of the material as: ∆ρE ρE,0=λ·trace(ε) = λ(ε11 +ε22 +ε33),(6) where ρE,0and ρE= (σE)−1denote the unstrained and straindependent electrical resistivity values of the material, εthe strain tensor, and ε11, ε22, ε33 the dilation strain values along the diagonal of ε. In addition, to account for the electrical permeability properties of the material when tensional cracks develop, an innovative degradation model is proposed in this work. In particular, a two-parameter sigmoid model is adopted as a function of the degradation parameter dtto describe the electrical permeability, of the material, P, as: P=1−e−k(1−dt)n 1−e−k(7) which directly affects the electrical conductivity of the material, i.e. PσE. The degradation parameters nand kallow for defining a comprehensive family of degradation curves as illustrated in the parametric analysis in figure 6(b). These profiles allow for giving some consideration to fiber bridging effects across the crack surfaces, coupling the Figure 6. Stress and degradation parameters versus inelastic strains (a), and parametric analysis of the proposed electrical permeability Pfunction (b). non-linear mechanical behavior with the electrical response. The overall mechanical and electrical responses are therefore coupled both in the uncracked elastic stage (piezoresistivity, equation (1)) and in the cracked one (degradation model, equation (3)). Finally, it is worth mentioning that the rebars are modeled as electrically conductive materials with no piezoresistive effects. 4.3. Macroscale multiphysics simulation The previous formulation has been implemented in ABAQUS through custom-made Python scripts and a user Material (UMAT) Fortran Subroutine. The main body of the vault was then meshed using 3D solid tetrahedral elements (Q3D4) with 4 nodes, incorporating thermal, electrical, and structural coupling (Q3D4) (figure 7), resulting in a total of 481 778 nodes and 2791 370 elements. The simulations were conducted on an Intel®Core™ i7- 7700 CPU @ 3.60 GHz server computer with 32 GB RAM, taking approximately 6 d per simulation. To reproduce the rigid body rotations of the reinforced concrete supports, two master nodes (MN1 and MN2) have been defined at the centroids of the supports. These nodes are linked to the contact surfaces of the vault through rigid body kinematic conditions. On this basis, the translational and Rz (torsional) and Rx (transversal) rotational degrees of freedom of the master nodes are constrained. Finally, vertical displacements Uz are 7 Meas. Sci. Technol. 35 (2024) 105116 A D’Alessandro et al Figure 7. Finite element meshing of the smart earth vault. Figure 8. Electromechanical boundary conditions of the smart earth vault. imposed on the contact surface between the steel profile and the upper surface of the vault (see figure 8). For simplicity, the steel reinforcements are coupled with the surrounding smart earth by coupling the degrees of freedom of the coincident nodes, assuming perfectly bonded conditions. With regard to the electrical boundary conditions, the electrodes are not explicitly modeled. Instead, assuming the electrodes are perfectly conductive, they are simulated by coupling the potential degrees of freedom of the nodes along the electrodes. Then, similarly to the experimental setting, a voltage of 20 V is applied to the numerical model. 5. Results and discussion Figure 9furnishes a comparison between the experimental and numerical crack patterns. To shed light on the effects of the conductive steel reinforcements, the results obtained with the previously introduced FE model with and without steel reinforcements are reported hereinafter. In particular, the mechanical and electrical responses of the sample are depicted in figures 10–12, respectively. First of all, figures 10(a) and (b) show the force–displacement curves obtained experimentally against the numerical predictions without and with steel reinforcements, respectively. 8