Effect of Copper Slag Content and Hybrid Steel Fiber Addition on the Mechanical Response of an Alkali-Activated Geopolymer Composite

Abstract

This study evaluated the effects of copper slag (CS), dosed relative to the mass of fly ash (FA; CS = 0, 7.5, 15, and 22.5%), and the volume fraction of hybrid steel fibers (V_f = 0.0, 0.5, and 1.0%) on the mechanical response of an alkali-activated geopolymer composite. The tests were performed using a two-factor CS × V_f design (4 × 3), with compressive strength (f_c) and splitting tensile strength (f_ct.sp) determined as the response variables. Statistical analysis showed significant effects of CS, V_f, and CS × V_f on f_c, and a significant CS × V_f interaction for f_ct.sp, confirming that the fiber effect depended on the CS content. The greatest increases in f_c relative to fiber-free composites were obtained for CS = 7.5%: +73% (V_f = 0.5%) and +102% (V_f = 1.0%), and for CS = 22.5%: +75% (V_f = 1.0%). For f_ct.sp, a decrease was found at CS = 0% and V_f = 0.5% (−34%), whereas an increase was observed at CS = 22.5% and V_f = 1.0% (+49%). The interpretation of the mechanical response was extended by DIC-based strain analysis in compression and splitting tests, together with σ_ct.sp–ε_x curves, indicating differences in strain/damage localization and post-cracking response.

1. Introduction

Alkali-activated materials are binder materials obtained through the alkaline activation of aluminosilicate precursors. In the literature, composites based on such binders are often referred to as geopolymer composites [1]. The research interest in this type of composite can be explained, among other factors, by their potential to reduce CO₂ emissions compared with Portland cement-based composites, which is associated with the possibility of using industrial by-products as cement substitutes [2,3,4,5].

The precursors commonly used in geopolymer composites include fly ash (FA) [6,7], ground granulated blast-furnace slag [8,9], and metakaolin [10], whereas the activators commonly used in geopolymer formation include NaOH and alkali silicates, such as Na₂SiO₃, as well as other sodium- and potassium-based compounds used as activators [11]. The mechanical properties of geopolymer composites depend on the reactivity of the precursor, which is governed, among other factors, by its oxide composition and SiO₂/Al₂O₃ ratio [12], the content of the amorphous/reactive fraction [13], and the degree of fineness [11], which affects the dissolution kinetics of active components in the solution [14]. Activation parameters, including the activator concentration [15,16] and sodium silicate content [17,18], are also important.

To accelerate setting during geopolymerization, many studies apply curing/heat treatment at elevated temperature, for example, 140 °C for 24 h [16] or 60 °C for 24 h [19], which may be particularly important for fly ashes with a lower calcium content [20].

The increasing demand for alkali-activated materials has led to substantial consumption of typical precursors, i.e., FA and ground granulated blast-furnace slag, which justifies the investigation of alternative precursor materials, such as copper slag [21]. Copper slag (CS) is a by-product of metallurgical processes whose oxide composition and phase characteristics, including the presence of silica and components richer in Ca and Fe, indicate its potential for use in alkali-activated systems after appropriate material preparation [22,23,24]. At the same time, in many studies, CS has mainly been analyzed as an aggregate or filler in geopolymer composites [25,26,27,28,29], whereas the number of studies in which CS has been considered a controlled component of the precursor system in alkali-activated materials remains limited [22,29,30].

One approach to reducing the brittleness of geopolymer composites and controlling crack propagation is the use of steel fibers, primarily due to their bridging action [4,31]. The effectiveness of this action depends, among other factors, on fiber geometry, volume fraction (V_f), spatial distribution, and the conditions of fiber–matrix interaction. The bridging effect may reduce stress concentration in the cracked zone and stabilize crack development [5]. For this reason, steel fibers are widely used in geopolymer composites as dispersed reinforcement, and their effect is also analyzed with respect to properties determined at maximum load, including fc and f_ct.sp. It has been reported that fiber addition (V_f = 1%) increased f_c by 5–23% in the investigated geopolymer composites [4]. The greatest improvement was achieved with hooked-end fibers, whereas for crimped and straight fibers the strengthening effect was less pronounced and did not show a clear trend [32,33,34,35]. A similar relationship was observed for f_ct.sp, for which the increase ranged from 19% to 43% relative to unreinforced composites [4].

Short straight fibers are generally associated with limiting the development of microcracks [36], whereas longer hooked-end fibers are associated with more effective crack bridging due to more favorable mechanical anchorage in the matrix [37]. For this reason, the hybrid approach is often indicated as potentially beneficial [38,39]. However, the effectiveness of fibers as reinforcement may depend on the properties of the matrix, and thus also on the precursor composition and activation parameters, which modify the conditions of fiber–matrix interaction, including bond/interfacial behavior [40]. Consequently, it is justified to conduct studies in a design that enables the effects of matrix composition and fiber content on f_c and f_ct.sp to be separated [41]. In particular, the number of studies that simultaneously analyze the effect of CS content in the FA–CS precursor system and the content of steel fibers using a design that allows the interaction effects of these factors to be identified remains limited.

The aim of this study is to quantitatively evaluate the effects of copper slag (CS) content in the fly ash–copper slag (FA–CS) system and the volume fraction of hybrid steel fibers (V_f) on the f_c and f_ct.sp of an alkali-activated geopolymer composite. The novelty of the study lies in the use of a factorial design enabling the identification of the CS × V_f interaction for both responses (f_c and f_ct.sp) and in demonstrating that the effect of fibers on f_ct.sp depends on the CS content (CS × V_f interaction). Deformation and damage in compression and splitting tests were also analyzed using digital image correlation (DIC).

2. Materials and Methods

2.1. Experimental Program

The experimental program was designed as a two-factor CS × V_f design. Four levels of copper slag content were adopted (CS = 0, 7.5, 15, and 22.5% relative to the mass of FA), together with three levels of the volume fraction of hybrid steel fibers (V_f = 0, 0.5, and 1.0% of the mixture volume). The factor levels were selected as planned comparative levels, enabling the mechanical response of the composites to be assessed with increasing CS content and different steel fiber dosages. Each CS level was combined with each V_f level, which made it possible to verify whether the fiber effect depended on the CS content. The adopted range was not intended to provide full mixture optimization, but rather to analyze the relationships within the predefined CS × V_f design. In total, 12 mixture variants and 72 specimens were prepared. For each variant, three specimens were prepared for compressive strength (f_c) testing and three specimens for splitting tensile strength (f_ct.sp) testing. The obtained results were used to evaluate the effects of CS content and V_f on the analyzed mechanical properties (f_c, f_ct.sp).

2.2. Materials and Specimen Preparation

Commercially supplied class A FA [42] from lignite combustion and CS were used as components of the geopolymer binder for preparing the mixtures (Figure 1a). The specific densities of these materials were ρ = 2200 kg/m³ and ρ = 2900 kg/m³, respectively. CS was introduced as an additional component of the FA–CS precursor system and was dosed at four levels defined relative to the mass of FA, 0%, 7.5%, 15%, and 22.5%, corresponding to m_CS/m_FA ratios of 0, 0.075, 0.15, and 0.225. Sand with a particle size of 0–2 mm and a specific density of ρ = 2650 kg/m³ was used as fine aggregate. The constant binder-to-sand mass ratio was 1.0, with the binder defined as the sum of FA and CS.

An alkaline solution prepared from an aqueous NaOH solution and sodium water glass was used as the activator. The molar concentration of the NaOH solution was 8 M. The NaOH solution was prepared using tap water and NaOH with a purity of >98% (Dragon, Skawina, Poland). The sodium water glass (Dragon, Skawina, Poland) was characterized by a molar modulus Ms = SiO₂/Na₂O > 3 and a density of 1400 kg/m³. The mass ratio of the alkaline solution to the binder was 0.5, whereas the mass ratio of the sodium water glass solution to the NaOH solution was maintained at 2.0. The slight differences in the absolute values of the activator components presented in Table 1 resulted from converting the mixture compositions to 1 m³ while maintaining the specified constant design proportions.

Hybrid dispersed reinforcement consisting of two types of uncoated steel fibers (Figure 1b), dosed at a constant 1:1 mass ratio and with a Young’s modulus of E = 210 GPa, was used (Bekaert, Zwevegem, Belgium). The straight DRAMIX OL fibers had a length of l = 13 mm, a diameter of d = 0.2 mm, an aspect ratio of λ = 65 (λ = l/d), and a tensile strength of f_t = 2000 MPa, whereas the hooked-end DRAMIX 3D fibers had l = 60 mm, d = 0.75 mm, λ = 80, and f_t = 1225 MPa. The total fiber volume fraction was adopted as 0%, 0.5%, or 1.0% relative to the mixture volume (V_f). The mixture compositions are summarized in Table 1.

Table 1. Mixture compositions of the geopolymer composite per 1 m³ (kg/m³) for four CS dosage levels.

Series	FA	CS	Sand	Alkaline Solution			Vf (%)
				Water *	NaOH Pellets	Sodium Silicate
CS-0	850.0	0.0	850.0	107.3	34.3	283.3	0; 0.5; 1
CS-7.5	796.0	59.7	855.7	108.0	34.6	285.2
CS-15	749.0	112.4	861.4	108.8	34.8	287.1
CS-22.5	707.0	159.1	866.1	109.4	35.0	288.7

* Water denotes the mass of water used to prepare the NaOH solution.

Table 1. Mixture compositions of the geopolymer composite per 1 m³ (kg/m³) for four CS dosage levels.

Series	FA	CS	Sand	Alkaline Solution			Vf (%)
				Water *	NaOH Pellets	Sodium Silicate
CS-0	850.0	0.0	850.0	107.3	34.3	283.3	0; 0.5; 1
CS-7.5	796.0	59.7	855.7	108.0	34.6	285.2
CS-15	749.0	112.4	861.4	108.8	34.8	287.1
CS-22.5	707.0	159.1	866.1	109.4	35.0	288.7

* Water denotes the mass of water used to prepare the NaOH solution.

Figure 1. Materials used to prepare the geopolymer composite mixtures: (a) mixture components: 1—fly ash, 2—copper slag, 3—sand, 4—water, 5—NaOH pellets, 6—sodium water glass solution; (b) steel fibers used in the study.

Figure 1. Materials used to prepare the geopolymer composite mixtures: (a) mixture components: 1—fly ash, 2—copper slag, 3—sand, 4—water, 5—NaOH pellets, 6—sodium water glass solution; (b) steel fibers used in the study.

CS was obtained from a commercial supplier in Poland as a waste material from metallurgical/copper-processing operations. In its initial state, CS had a particle size of 100–400 µm. Before milling, the material was preliminarily sieved to remove the coarser fraction and homogenize the feed. Milling was then carried out in a laboratory ball mill (320 rpm, 8 h), using ceramic grinding media at a 2:1 volume ratio relative to the material (Figure 2). The particle size distribution of the powder materials (milled CS and FA) was determined by laser diffraction using a Mastersizer 2000 particle size analyzer with a Hydro wet dispersion unit (Malvern Instruments Ltd., Malvern, UK), with demineralized water used as the dispersing liquid (range: 0.02–2000 µm, ISO 13320 [43]), and is presented in Figure 3.

For CS, the cumulative curve indicated a median particle size of D50 ≈ 20.5 μm, with D10 ≈ 2.6 μm and D90 ≈ 69.5 μm. The cumulative data also show that approximately 97% of the particle volume had a diameter smaller than approximately 105 μm, and that the cumulative curve approached 100% at a diameter of approximately 209 μm, indicating no substantial contribution of coarser fractions within the analyzed measurement range. For FA, the characteristic values of the particle size distribution were shifted toward larger diameters, and the distribution was clearly broader: D50 ≈ 25.1 μm, D10 ≈ 2.8 μm, and D90 ≈ 162.4 μm. The cumulative curve for FA reached approximately 99.8% only at a diameter of approximately 1069 μm, whereas 100% was reached at diameters on the order of 1.4 mm. This indicates that FA contained a higher proportion of coarser fractions, which is reflected in the higher D90 value and the longer “tail” of the cumulative curve. Compared with FA, the milled CS was characterized by a narrower particle size distribution in the fine fraction range. The reduction in particle size, the associated increase in specific surface area, and the mechanical activation of precursors are recognized in the literature as factors that promote increased reactivity, among other mechanisms, by accelerating the dissolution of glassy phases and the progress of binding reactions [22,44].

Figure 2. Preparation of CS for testing: (a) material after preliminary sieving (feed for milling); (b) laboratory ball mill used for grinding; (c) CS powder after milling.

Figure 2. Preparation of CS for testing: (a) material after preliminary sieving (feed for milling); (b) laboratory ball mill used for grinding; (c) CS powder after milling.

Figure 3. Particle size distribution of FA and CS, presented as the cumulative curve (right axis) and the differential curve dV/dlog₁₀(D) (left axis) as a function of particle diameter (logarithmic X-axis), where D10, D50, and D90 denote the diameters corresponding to 10%, 50%, and 90% of the cumulative particle volume, respectively.

Figure 3. Particle size distribution of FA and CS, presented as the cumulative curve (right axis) and the differential curve dV/dlog₁₀(D) (left axis) as a function of particle diameter (logarithmic X-axis), where D10, D50, and D90 denote the diameters corresponding to 10%, 50%, and 90% of the cumulative particle volume, respectively.

The chemical compositions of FA and CS, expressed as oxide contents (wt.%), were determined by wavelength-dispersive X-ray fluorescence (WD-XRF), and the results are summarized in

Table 2. Chemical composition of FA and CS, expressed as oxide contents (wt.%).

	SiO2	Al2O3	Fe2O3	CaO	MgO	K2O	Na2O	TiO2	P2O5	SO3	Cl	CuO	ZnO	PbO	∑MO*
FA	50.54	27.80	8.34	2.98	1.98	4.09	0.94	1.47	0.58	0.38	0.008	0.03	0.04	n.r.	0.81
CS	28.77	9.18	24.32	23.34	4.15	3.32	0.34	0.70	0.15	n.r.	0.010	1.26	1.92	1.08	1.47

n.r.—not reported in the WD-XRF output; ∑MO*—sum of minor oxides not listed separately in the table.

. The materials were ground to a particle size of 50 μm in a disc mill with a zirconia grinding disc and then pressed into pellets. The analysis was performed using a Rigaku Supermini200 spectrometer (Rigaku, Tokyo, Japan).

The WD-XRF results indicate that FA and CS differed substantially in their overall oxide composition. FA was dominated by SiO₂ and Al₂O₃, whose combined content was approximately 78 wt.%, whereas the CaO content was low. This compositional profile is typical of fly ashes used as aluminosilicate precursors in alkali-activated systems. In the case of CS, the contents of SiO₂ and Al₂O₃ were considerably lower, whereas the material was characterized by markedly higher contents of Fe₂O₃ and CaO, amounting to approximately 24 wt.% and 23 wt.%, respectively. This means that increasing the CS content in the CS-0–CS-22.5 series involved introducing into the binder fraction a component with a different oxide composition, particularly one richer in Ca and Fe. At the same time, the overall oxide composition is only one element in assessing the suitability of a precursor for alkaline activation, because its reactivity is also governed by physical and phase characteristics, including particle size distribution, the proportion of amorphous and crystalline phases, and the solubility of components in an alkaline environment [45,46].

The geopolymer mixtures were prepared in the following sequence. The NaOH solution was prepared by dissolving sodium hydroxide in water and then left to cool to approximately 20 °C. The alkaline activator was prepared 60 min before mixing by combining the cooled NaOH solution with sodium water glass and mixing for 3 min. This procedure limited the influence of exothermic effects on the solution temperature at the time of dosing.

FA and CS were mixed in a pan mixer without liquid addition for 3 min. The alkaline activator was then added, and mixing was continued for another 3 min until a homogeneous binder mixture was obtained. In the next step, sand was added and mixing was continued for 5 min. In the steel fiber-reinforced series, the fibers were introduced gradually during mixing, over approximately 3 min, to limit the formation of fiber clusters and ensure the most uniform possible distribution within the mixture volume.

The mixture was placed in molds and compacted on a vibrating table for approximately 20 s. The specimens were heat-cured at 80 °C [47] for 24 h, then cooled to room temperature, demolded, and cured under laboratory conditions until the age of 14 days (T = 20 ± 2 °C; RH = 50 ± 5%).

2.3. Research Methodology

The f_c and f_ct.sp tests were performed on cubes with a side length of a = 150 mm. The f_c and f_ct.sp values were determined based on the maximum load Fmax according to Equations (1) and (2). Before the main loading stage, a preload of 2 kN was applied to stabilize the contact between the specimen and the loading platens. The tests were conducted in accordance with [48,49] using a hydraulic testing machine with a maximum load capacity of 3000 kN (ToniNORM, Berlin, Germany; Figure 4), under piston displacement control at 0.6 mm/min for compression and 0.4 mm/min for splitting. In the splitting test, the specimen was loaded along two opposite generatrices, producing line loading. Along the contact lines, fiberboard strips with a density of ρ > 900 kg/m³, a width of 15 mm, and a thickness of 4 mm were used to reduce local crushing and homogenize the applied pressure. Statistical analyses of the f_c and f_ct.sp results were performed using Statistica 13 software (TIBCO Software Inc., Palo Alto, CA, USA).

$$f_{\mathrm{c}}={\displaystyle \frac{F_{max}}{A_{\mathrm{c}}}}$$

(1)

$$f_{\mathrm{ct}.\mathrm{sp}}={\displaystyle \frac{{2F}_{max}}{\pi Ld}}$$

(2)

where A_c—loaded surface area of the specimen in compression (for cubes, A_c = a²); L—length of the loading line in the splitting test (for cubes, L = a); and d—specimen dimension perpendicular to the loading line (for cubes, d = a).

Digital image correlation (DIC) was used to determine the surface displacement field of the specimens, using the Aramis system (GOM/ZEISS, Braunschweig, Germany). Images were recorded at a frequency of 1 Hz. The camera was positioned approximately 300 mm from the specimen surface, and the image resolution was 4096 × 3000 pixels. The surface was prepared by applying a random high-contrast speckle pattern [50,51]. Virtual extensometers, defined on the specimen surface, were also used to determine average strains over the adopted gauge length.

The displacement field was estimated based on image matching within facets, and the strain fields were then calculated using the engineering strain measure. In this study, maps of the maximum principal strain ε₁ were analyzed and used for qualitative identification of deformation and damage localization zones (Section 3.2). The analyzed area (ROI) was limited using a mask by excluding near-edge zones approximately 5 mm wide, in order to reduce the influence of edge effects and areas with reduced correlation quality.

The DIC analysis parameters were selected separately for compression and splitting: a facet size of 15 × 15 px with a step of 10 px, and a facet size of 21 × 21 px with a step of 7 px, respectively. Here, the facet denotes the subset used for image matching, whereas the step denotes the spacing between measurement points, i.e., the distance between the centers of adjacent facets. These settings were adopted as a compromise between the spatial resolution of the strain maps and correlation stability, including sensitivity to measurement noise and loss of correlation, given the different character of localization and local strain gradients in the two tests.

For the exported ε₁ maps of the compressed specimens selected for the subsequent DIC-based strain analysis, an additional auxiliary image-thresholding analysis was performed using ImageJ.JS. This analysis was used to quantitatively compare the fraction of pixels corresponding to zones of intense ε₁ localization on the observed specimen surface. All four maps were cropped to the same size of 1506 × 1484 px, without image scaling, in order to avoid introducing pixel interpolation. The same thresholding settings were then applied in the HSB color space (Hue–Saturation–Brightness), covering the red range of the common color scale of the ε₁ maps. The adopted indicator was the fraction of pixels satisfying the thresholding criterion relative to the total number of pixels in the analyzed image. This indicator was treated as an auxiliary comparative parameter for ε₁ maps processed according to the same procedure.

Figure 4. Test setup with the DIC system in the compression configuration; in the splitting test, only the testing machine fixture was changed.

Figure 4. Test setup with the DIC system in the compression configuration; in the splitting test, only the testing machine fixture was changed.

3. Results and Discussion

3.1. Compressive and Splitting Tensile Strength

The results of compressive strength f_c and splitting tensile strength f_ct.sp are summarized in Table 3 as individual test results (n = 3 for each CS × V_f combination), whereas Figure 5 presents the mean values with standard deviations (s). Already at the level of mean-value comparison, no consistent trend in the response to increasing fiber volume fraction V_f can be observed across all CS levels. This indicates a possible interaction between the factors, i.e., that the effect of fibers depends on the CS content.

Relative to the fiber-free series (V_f = 0%), the largest increases in the mean f_c values after fiber addition were recorded for the CS-7.5 series, amounting to approximately +73% for V_f = 0.5% and +102% for V_f = 1%, followed by the CS-22.5 series (+24% and +75%, respectively). In the CS-15 series, no clear improvement was observed for V_f = 1% (+5%). For f_ct.sp, a decrease was recorded in the CS-0 series at V_f = 0.5% (−34%), whereas a pronounced increase was observed in the CS-22.5 series at V_f = 1% (+49%). At the intermediate CS levels (CS-7.5 and CS-15), the changes did not show a clear trend within the investigated V_f range (Figure 5).

Table 3. Results of f_c and f_ct.sp (MPa) for composites with different CS contents and V_f values.

	fc			fct.sp
Vf	0%	0.5%	1%	0%	0.5%	1%
	10.44	12.14	15.13	1.69	1.21	1.31
CS-0	13.80	11.18	14.36	1.77	1.31	1.44
	12.02	11.77	13.80	1.61	0.80	1.38
	9.22	14.08	18.46	1.21	1.71	1.49
CS-7.5	8.66	14.79	17.56	1.09	1.63	1.39
	8.94	17.55	18.07	1.34	1.24	1.74
	10.76	13.64	10.95	1.24	1.28	0.76
CS-15	10.81	11.66	11.93	0.80	0.93	0.83
	11.13	13.17	11.35	0.71	0.90	0.82
	10.05	14.86	18.30	1.11	1.17	1.72
CS-22.5	10.66	11.20	17.76	1.28	1.30	1.65
	9.96	11.95	17.56	1.07	1.11	1.76

Table 3. Results of f_c and f_ct.sp (MPa) for composites with different CS contents and V_f values.

	fc			fct.sp
Vf	0%	0.5%	1%	0%	0.5%	1%
	10.44	12.14	15.13	1.69	1.21	1.31
CS-0	13.80	11.18	14.36	1.77	1.31	1.44
	12.02	11.77	13.80	1.61	0.80	1.38
	9.22	14.08	18.46	1.21	1.71	1.49
CS-7.5	8.66	14.79	17.56	1.09	1.63	1.39
	8.94	17.55	18.07	1.34	1.24	1.74
	10.76	13.64	10.95	1.24	1.28	0.76
CS-15	10.81	11.66	11.93	0.80	0.93	0.83
	11.13	13.17	11.35	0.71	0.90	0.82
	10.05	14.86	18.30	1.11	1.17	1.72
CS-22.5	10.66	11.20	17.76	1.28	1.30	1.65
	9.96	11.95	17.56	1.07	1.11	1.76

Figure 5. Mean values (±s, n = 3) of (a) f_c and (b) f_ct.sp for composites with different CS contents and V_f values.

Figure 5. Mean values (±s, n = 3) of (a) f_c and (b) f_ct.sp for composites with different CS contents and V_f values.

To quantitatively assess the effects of CS content and V_f, a two-factor model with interaction was used (Equation (3)), with both factors treated as categorical variables. The model parameters were estimated using ordinary least squares (OLS), whereas the significance of the effects was evaluated using two-way analysis of variance with type II sums of squares (

Table 4. Results of the two-way analysis of variance (ANOVA) for f_c and f_ct.sp.

fc
Effect	df	SS	MS	F	p
CS	3	30.47	10.16	9.73	2.20 × 10−4
Vf	2	144.22	72.11	69.05	1.11 × 10−10
CS × Vf	6	98.28	16.38	15.69	2.97 × 10−7
Residual error	24	25.06	1.04	–	–
fct.sp
CS	3	1.523	0.51	17.6	3.0 × 10−6
Vf	2	0.135	0.07	2.35	0.117
CS × Vf	6	1.241	0.21	7.16	1.83 × 10−4
Residual error	24	0.693	0.03	–	–

d_f—degrees of freedom; SS—sum of squares; MS—mean square (SS/d_f); F—test statistic (effect MS/residual MS); p—significance level.

). For f_c, significant effects of CS and V_f were found, together with a significant CS × V_f interaction, confirming that the fiber effect depended on the CS level. For f_ct.sp, the effects of CS and the CS × V_f interaction were significant, whereas the main effect of V_f was not significant. This means that the fiber effect on f_ct.sp changed with the CS level, which justifies including the CS × V_f interaction in the model (Table 4).

$$y=\mu +CS+V_f+CS\times V_f+\epsilon$$

(3)

where y is the analyzed response variable (f_c or f_ct.sp); μ is the intercept; CS and V_f are the factor effects; CS × V_f is the interaction effect; and ε is the random error term with an expected value equal to zero.

The model fits were assessed using basic goodness-of-fit measures and residual diagnostics (

Table 5. Model fit measures (Equation (3)) and residual diagnostics (OLS) for f_c and f_ct.sp.

Response	R2	R2adj	RMSE	Standardized Residuals (min-max)	Cook’s D
fc	0.92	0.88	1.022	−1.97 to 2.62	0.287
fct.sp	0.81	0.72	0.17	−2.21 to 2.33	0.226

R², R²_adj—coefficients of determination; RMSE—root mean square error (MPa); max Cook’s D—maximum value of the measure of the influence of an individual observation on the model fit.

). For f_c, a high goodness of fit was obtained (R² ≈ 0.92) with RMSE ≈ 1.02 MPa, indicating that the model explained a substantial part of the response variability within the investigated design. For f_ct.sp, the fit was weaker (R² ≈ 0.81); however, RMSE ≈ 0.17 MPa corresponded to approximately 10% of the mean level of this response in the entire dataset, indicating a moderate fitting error relative to the measurement scale. The maximum Cook’s D values were <1, which did not indicate observations with an excessively dominant influence on the fit and supported the suitability of the applied model for comparative analysis of the factors.

To assess the adequacy of the linear model described by Equation (3), residual diagnostics were performed, including Q–Q (quantile–quantile) plots of raw residuals and plots of raw residuals versus fitted values (Figure 6). In the Q–Q plots, z_i denotes the theoretical quantiles of the normal distribution, whereas e_i denotes the raw residuals for the i-th observation. This diagnostics was used for qualitative verification of the basic model assumptions, i.e., approximate normality of the residual distribution and the absence of clear variance heterogeneity, which are important for interpreting the ANOVA results. In the plots of residuals versus fitted values, no ordered residual structure and no clear dependence of residual scatter (variance) on the level of fitted values were observed. Therefore, the model assumptions were considered sufficiently satisfied for the purpose of interpreting the ANOVA results.

Because a significant CS × V_f interaction was demonstrated, the interpretation of the results was supplemented with planned comparisons within individual CS levels, treated as an analysis of simple effects relative to the reference level V_f = 0% (

Table 6. Planned comparisons relative to V_f = 0% within CS levels for f_c and f_ct.sp.

Response	CS	Contrast (vs. 0%)	Δ (MPa)	CI90 * (MPa)	pHolm
fc	CS-0	0.5–0	−0.39	[−2.11, 1.33]	0.777
	CS-0	1–0	2.34	[0.63, 4.06]	0.151
	CS-7.5	0.5–0	6.53	[4.79, 8.28]	0.002
	CS-7.5	1–0	9.09	[7.34, 10.84]	<0.001
	CS-15	0.5–0	1.92	[0.86, 2.99]	0.064
	CS-15	1–0	0.51	[−0.56, 1.58]	0.777
	CS-22.5	0.5–0	2.45	[0.61, 4.29]	0.151
	CS-22.5	1–0	7.65	[5.81, 9.49]	0.001
fct.sp	CS-0	0.5–0	−0.58	[−0.85, −0.32]	0.037
	CS-0	1–0	−0.31	[−0.58, −0.05]	0.368
	CS-7.5	0.5–0	0.31	[0.01, 0.62]	0.416
	CS-7.5	1–0	0.33	[0.02, 0.63]	0.416
	CS-15	0.5–0	0.12	[−0.21, 0.45]	1
	CS-15	1–0	−0.11	[−0.44, 0.21]	1
	CS-22.5	0.5–0	0.04	[−0.10, 0.18]	1
	CS-22.5	1–0	0.56	[0.41, 0.70]	0.002

* CI90 is reported without multiplicity adjustment; formal significance was assessed using p_Holm.

; contrasts: 0.5–0 and 1–0). Mean differences are presented as Δ together with 90% confidence intervals (CI90). The significance of the comparisons was assessed based on p_Holm values, i.e., p-values adjusted using Holm’s method within the set of contrasts for a given response, applied to reduce the risk of Type I error in multiple comparisons. The CI90 intervals for Δ were determined within each CS level based on Student’s t-distribution and the residual error (MSE), with d_f = 6.

The planned comparisons confirmed that the effect of V_f depended on the CS level. For f_c, after Holm correction, the most pronounced positive differences were found in the CS-7.5 series for V_f = 0.5% and V_f = 1%, and in the CS-22.5 series for V_f = 1%. In the CS-15 series, no improvement was confirmed for V_f = 1%, whereas for V_f = 0.5% only a borderline effect was observed (p_Holm = 0.064). For f_ct.sp, the differences that remained significant after Holm correction were limited to the CS-0 series, where a decrease was observed for V_f = 0.5%, and the CS-22.5 series, where an increase was observed for V_f = 1%. This indicates that the beneficial fiber effect in the splitting test became clearly evident only at the highest CS content.

Because f_c and f_ct.sp were determined in separate tests, no paired observations were obtained for the two parameters; therefore, their correlation was not analyzed at the level of individual results. The comparison was limited to evaluating the effects of CS and V_f within the same factorial design. The obtained results indicate that, within the investigated range of compositions, fiber addition increased f_c only at selected CS levels, whereas the effect on f_ct.sp was conditional and became significant only at the highest slag content (CS-22.5). In addition, for the 12 points corresponding to the mean values of the series from Table 3, a linear fit between f_ct.sp and f_c yielded R² = 0.44.

The literature emphasizes that steel fibers exert their primary effect in the post-cracking response, i.e., in controlling crack opening and in the response after F_max is reached, whereas their effect on f_c and f_ct.sp may vary and depend on matrix properties and production technology [52,53], including mixture consistency, compaction, and fiber dispersion [54]. In this context, the varied f_ct.sp response is consistent with the fact that, in the splitting test, the contribution of fibers may be reflected not only at the F_max level, but also in the mode of crack localization and development and in the effective number of fibers crossing the cracked zone [55].

Studies on the alkaline activation of CS indicate that the alkali level, expressed as Na₂Oeq relative to the precursor mass, and the silicate modulus control the intensity of dissolution and the formation of reaction products, which translates into the strength development of alkali-activated composites [56]. In the FA + CS system investigated in this study, the alkali level was close to the value of 6% indicated in [22] as favorable for systems containing CS. At the same time, changing the CS content changed the composition of the binder fraction: FA was richer in SiO₂ and Al₂O₃, whereas CS introduced higher amounts of CaO and Fe₂O₃ (Table 2). In fly ash-based systems of this type, activated with NaOH and sodium silicate solution, the reaction products are usually described as being dominated by N-A-S-H gel, whereas greater calcium availability may lead to the coexistence of C-A-S-H or C-(N)-A-S-H type products [57,58,59]. However, the direction of this effect does not follow directly from the CaO content alone, as determined by WD-XRF, because it depends on the phase form of calcium, its solubility, the activator composition, and the Si–Al–Ca proportions in the reacting system [58,59]. In the case of CS, the high Fe₂O₃ content may also be important, because studies on the alkaline activation of copper slag and reviews of Fe-rich precursors indicate a possible contribution of Fe to reaction products, depending on the type of Fe phases, the degree of amorphousness/reactivity of the raw material, and the activation parameters [22,60]. The cited literature background indicates possible factors governing the differentiation of matrix properties in FA–CS systems; however, it does not provide a direct basis for identifying reaction products in the composites investigated in this study. Because no phase or microstructural analyses were performed in the present study, the observed differences in f_c and f_ct.sp are not attributed to specific matrix phases, reaction products, or quantitatively determined CS reactivity.

Within this interpretative scope, the differences in the f_c and f_ct.sp responses after fiber addition between CS levels may be considered in relation to matrix properties, which may have affected both the contribution of fibers to load transfer after cracking and the initiation, localization, and development of cracking. The CS × V_f interaction demonstrated in the statistical analysis therefore means that the effect of fibers on f_c and f_ct.sp differed between CS levels and cannot be interpreted as a constant effect across the entire investigated composition range. Since the number of observations in each CS × V_f combination was n = 3, the conclusions were limited to the investigated ranges of CS and V_f, and the interpretation was based on the direction of the effects and on the interaction revealed by model (3).

3.2. Post-Failure Surface Images and DIC-Based Strain Analysis

To supplement the interpretation of the mechanical results with an analysis of strain fields determined by DIC, maps of the maximum principal strain ε₁ are presented for selected specimens tested in compression (Figure 7) and splitting tension (Figure 8). The ε₁ maps were used to identify and qualitatively compare strain localization zones associated with crack development and damage [50]. The DIC examples were selected deliberately on the basis of planned comparisons relative to the V_f = 0% level within individual CS levels, as presented in Section 3.1 and summarized in Table 6. This selection was intended to enable comparison between specimens without fibers and specimens with fibers within the same CS level for selected cases showing different effects of fiber addition. In compression, one pair corresponding to a pronounced and statistically confirmed increase in fc after fiber addition and one pair for which such an increase was not statistically confirmed were included. In splitting tension, pairs corresponding to statistically confirmed changes in f_ct.sp in opposite directions were selected: a decrease at CS = 0% and an increase at CS = 22.5%. For each specimen, the ε₁ map at the state corresponding to F_max was compared with the post-failure surface image.

In the ε₁ maps for the specimens subjected to compression (Figure 7a), strain localization bands corresponding to zones of increased ε₁ values are visible. In the specimens without fibers (CS-7.5/V_f = 0%, CS-15/V_f = 0%), localization takes the form of a smaller number of distinct, continuous or partially continuous bands. In the specimens with fibers (CS-7.5/V_f = 1%, CS-15/V_f = 1%), the localization pattern is more dispersed and includes a larger number of shorter bands with different orientations, visible in different parts of the analyzed surface. To supplement this observation with a quantitative element, the relative area fraction of pixels assigned to zones of intense ε₁ localization was determined for the ε₁ maps presented in Figure 7a. This fraction was 1.25% for CS-7.5/V_f = 0% and 1.62% for CS-7.5/V_f = 1%, whereas for the CS-15 series it was 0.90% for V_f = 0% and 2.48% for V_f = 1%. These results support the qualitative observation that, after fiber addition, strain localization on the observed specimen surface had a more dispersed character. This indicator was treated as an auxiliary comparative parameter determined in ImageJ.JS for ε₁ maps processed according to the same thresholding procedure. The distribution of bands with increased ε₁ values remains qualitatively consistent with the damage morphology observed in the post-failure surface images (Figure 7b). In addition, for the same pairs of specimens, the compressive strain ε_y at the state corresponding to F_max was analyzed based on the readings from a virtual extensometer with a 60 mm gauge length, located centrally on the specimen surface. For the CS-7.5 series, ε_y increased from 2.11‰ to 3.24‰ after the introduction of fibers, whereas in the CS-15 series it increased from 2.88‰ to 3.50‰. This value was treated only as a supplement to the interpretation of the ε₁ maps, because the compressive strain of the composite determined in DIC measurements may strongly depend on the location and length of the adopted gauge length, as well as on the boundary conditions of the test, which affect the local distribution of displacements and strains [61].

In the ε₁ maps for the specimens subjected to splitting tension (Figure 8a), the localization of increased ε₁ values takes the form of a narrow zone with a generally vertical course, consistent with the typical splitting line. The differences between the variants mainly concern the continuity and local branching of this zone. In the post-failure surface images (Figure 8b), complete separation of the specimen is visible in the fiber-free variants, whereas in the fiber-reinforced variants crack opening is limited, which is consistent with the expected effect of fibers on load transfer after cracking, i.e., the bridging mechanism.

It should be emphasized that, in the DIC analysis used in this study, the strain field was determined from the gradient of the displacement field in the measurement plane, according to the adopted strain measure. In DIC, the displacement field is estimated discretely on a grid of points based on image matching within facets, i.e., subsets, and the strains are then determined numerically from the derivatives of this field. Consequently, the ε₁ field was interpreted as a description of deformation and damage localization at the scale defined by the DIC calculation parameters, including facet size and grid step, and not as a direct measurement of crack geometry in the sense of a discrete crack opening.

Figure 7. f_c test: (a) ε₁ maps at the state corresponding to F_max, presented using a common color scale; (b) post-failure surface images of the specimens.

Figure 7. f_c test: (a) ε₁ maps at the state corresponding to F_max, presented using a common color scale; (b) post-failure surface images of the specimens.

Figure 8. f_ct.sp test: (a) ε₁ maps at the state corresponding to F_max, presented using a common color scale; (b) post-failure surface images.

Figure 8. f_ct.sp test: (a) ε₁ maps at the state corresponding to F_max, presented using a common color scale; (b) post-failure surface images.

To supplement the qualitative analysis of the ε₁ maps for the f_ct.sp test, an additional analysis of ε_x strains, i.e., strains transverse to the loading direction of the specimen, was performed using virtual extensometers (VEs). The engineering strain definition ε_x = Δl/L was adopted. Nine VEs with a gauge length of 80 mm were used to determine ε_x, arranged at different height levels of the specimen (Figure 9). A detailed presentation of the VE arrangement and the variation in ε_x along the specimen height is provided for the CS-0/V_f = 0.5% specimen, treated as an example of using virtual extensometers to describe the development of transverse strains in the splitting test.

The adopted VE arrangement enabled ε_x to be recorded at different height levels of the specimen. The purpose of this arrangement was to capture changes in the sign and magnitude of ε_x along the specimen height, because in the splitting test tensile strains, i.e., positive strains, develop in the central part, whereas the influence of compressive strains, i.e., negative strains, becomes evident near the load application zones [62].

The strain–time histories ε_x(t) recorded by the individual VEs are shown in Figure 10. Up to the moment immediately preceding the attainment of F_max by the specimen, the histories showed a similar, approximately linear character. Positive ε_x values were recorded by the extensometers located in the central part of the specimen height, whereas near the upper and lower edges the influence of strains with the opposite sign was evident. After F_max was reached, at approximately t = 110 s, a rapid increase in ε_x was observed, associated with crack propagation in the specimen. In the subsequent part of the response, ε_x continued to increase; however, after the initial jump, a period of less intensive growth followed, and at approximately t = 160 s a renewed increase in the intensity of changes was visible. In the final range of the test, the rate of ε_x increase decreased again.

The variations in ε_x along the specimen height at selected test times are shown in Figure 11. Before cracking, the ε_x curves had an approximately symmetric arrangement, with tension in the central part and compression near the upper and lower edges, as observed for t = 50 and 100 s. As F_max was approached, ε_x values increased in the central region. This is particularly evident for t = 110 s, when a rapid increase in ε_x was observed in the central part of the specimen after cracking. This indicates that, after cracking, further deformation was concentrated primarily in the central zone, consistent with the course of the splitting line. This observation is consistent with the ε₁ maps in Figure 8a, where the localization of increased ε₁ values takes the form of a narrow, generally vertical zone.

Figure 9. Arrangement of VE-1–VE-9 used to record ε_x strains in the f_ct.sp test of the CS-0/V_f = 0.5% specimen; arrows indicate the force direction; grid spacing is given in mm; ε_x values for VE-1–VE-9 are shown at the state immediately before F_max was reached.

Figure 9. Arrangement of VE-1–VE-9 used to record ε_x strains in the f_ct.sp test of the CS-0/V_f = 0.5% specimen; arrows indicate the force direction; grid spacing is given in mm; ε_x values for VE-1–VE-9 are shown at the state immediately before F_max was reached.

Figure 10. Changes in ε_x strains recorded by VE-1–VE-9 as a function of time during the f_ct.sp test of the CS-0/V_f = 0.5% specimen.

Figure 10. Changes in ε_x strains recorded by VE-1–VE-9 as a function of time during the f_ct.sp test of the CS-0/V_f = 0.5% specimen.

Figure 11. Changes in ε_x strains along the height of the CS-0/V_f = 0.5% specimen at selected times during the f_ct.sp test, determined from VE readings.

Figure 11. Changes in ε_x strains along the height of the CS-0/V_f = 0.5% specimen at selected times during the f_ct.sp test, determined from VE readings.

Figure 12 presents the relationship between the splitting tensile stress σ_ct.sp, determined according to Equation (2), and ε_x for the specimens for which the ε₁ maps were analyzed previously (Figure 8), with the location and gauge length being the same as for VE-5 in Figure 9. After cracking, ε_x no longer describes material strain in a continuum sense; therefore, these curves were treated as a supplementary record of the specimen response over the adopted gauge length. In the pre-cracking range (Figure 12a), the curves showed a similar, approximately linear character, while differing in slope and in the ε_x value corresponding to the maximum stress σ_ct.sp.max. In the specimens without fibers, the ε_x values corresponding to σ_ct.sp.max were clearly lower than in the fiber-reinforced variants: they were 0.28‰ for CS-0/V_f = 0% and 0.33‰ for CS-22.5/V_f = 0%, whereas for CS-0/V_f = 0.5% and CS-22.5/V_f = 1% they were 3.76‰ and 0.81‰, respectively. For the fiber-reinforced specimens, σ_ct.sp values were then read at ε_x = 4‰, 8‰, and 10‰. For CS-0/V_f = 0.5%, these values were 1.20, 0.75, and 0.70 MPa, corresponding to 98.2%, 61.2%, and 57.6% of σ_ct.sp.max, respectively. For CS-22.5/V_f = 1%, the corresponding values were 1.43, 1.07, and 0.90 MPa, i.e., 81.1%, 60.5%, and 51.4% of σ_ct.sp.max. These readings indicate that, in the analyzed examples, the fiber-reinforced specimens retained part of their load after reaching F_max over the further ε_x range, which is consistent with the failure morphology shown in Figure 8.

4. Conclusions

Based on the presented test results and the conducted analyses, the following conclusions were formulated:

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The applied 4 × 3 factorial design (CS × V_f) revealed a significant CS × V_f interaction for both compressive strength f_c and splitting tensile strength f_ct.sp. This means that the effect of steel fiber dosage depends on the content of copper slag (CS) in the fly ash–copper slag (FA–CS) precursor system.

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For f_c, the effects of CS, V_f, and CS × V_f were significant. The most clearly confirmed increases relative to V_f = 0, after Holm correction, were obtained for CS = 7.5%—+73% (V_f = 0.5%; Δ = +6.53 MPa) and +102% (V_f = 1.0%; Δ = +9.09 MPa)—and for CS = 22.5%—+75% (V_f = 1.0%; Δ = +7.65 MPa).

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The use of steel fibers did not lead to an increase in f_ct.sp across the entire investigated CS range. The direction and magnitude of the changes depended on the CS level, as indicated by the CS × V_f interaction. After Holm correction, the statistically confirmed effects were limited to a 34% decrease for CS = 0% at V_f = 0.5% (Δ = −0.58 MPa) and a 49% increase for CS = 22.5% at V_f = 1.0% (Δ = +0.56 MPa).

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Due to the significant CS × V_f interaction, the conclusions regarding the effect of fibers on the mechanical properties of the composite were formulated separately for each CS level, i.e., by comparing V_f levels relative to V_f = 0 at constant CS. This approach avoids generalizations that would disregard the dependence of the fiber effect on CS content.

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The DIC analysis and post-failure images supplemented the interpretation of the mechanical results by providing a qualitative assessment of deformation/damage localization in selected specimens. The ε₁ maps at Fmax indicated that, in compression, the fiber-reinforced specimens exhibited a more dispersed pattern of increased-strain bands, whereas in splitting, a narrow localization zone consistent with the course of the splitting line dominated. The ε_x analysis based on virtual extensometers showed variation in transverse strains along the specimen height, with positive values in the central part and strains of the opposite sign near the load application zones. The σ_ct.sp–ε_x relationships for the central VE-5 gauge showed that the fiber-reinforced specimens maintained the ability to carry load at higher ε_x values after F_max was reached, which was consistent with the greater specimen integrity visible in the post-failure images.

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Within the investigated range of CS content relative to the mass of FA, alkali-activated composites with different mechanical responses in compression and splitting were obtained. The results define the mechanical effects of using CS in the FA–CS system, but they do not allow its contribution to the formation of reaction products to be unequivocally assessed. A full evaluation of the function of CS as a component of the precursor system requires further phase and microstructural studies, as well as analysis of the fiber–matrix interfacial zone. The obtained relationships, including the CS × V_f interaction, should also be interpreted with reference to the adopted specimen preparation and curing conditions, which comprised exposure at 80 °C for 24 h and testing after 14 days. Under different curing conditions or at a different testing age, the course of these relationships may differ; therefore, their generalization requires separate experimental verification.