Additive manufacturing (3D printing) of concrete is rapidly gaining attention as a transformative approach in the construction industry, facilitating the freeform fabrication of structural components through elimination of traditional formwork [1,2,3,4]. While this layered deposition technology presents numerous advantages over conventional construction methods, its widespread implementation is still hindered by several challenges, early-age shrinkage and associated cracking [5,6,7] being an important one. Immediately after extrusion, the printed elements have their entire surface areas exposed, leading to increased rate of moisture loss, resulting in significant plastic shrinkage [8,9]. When this shrinkage is restricted by factors such as overlying layers or non-uniform boundary conditions, tensile stresses develop, which when exceeds the tensile strength of the concrete, lead to cracking [5,8,10]. 3D-printed concrete (3DPC) is also more vulnerable to early-age cracking due to the use of higher binder (especially cement) content, limited bleeding capacity, and absence of coarse aggregates [7,11].
Conventional measurement techniques are often inadequate to accurately capture early-age deformations in 3DPC due to the local heterogeneity introduced by layering variables, overburden effects, and print geometry [10,12]. Each layer undergoes different mechanical constraints, resulting in differential drying, and thus varying shrinkage responses across the printed element. The interaction between substrate and the freshly extruded layers introduces restraint effects, which transmits from the already printed layer to the successive layers that are printed over it. Furthermore, restraint is introduced from adjacently placed printed filaments, leading to additional non-uniform volumetric changes.
Several studies have focused on quantifying shrinkage in 3DPC [9,10,13,14,15,16,17] and most of them consider shrinkage to be a bulk response, neglecting the effect of number and thickness of layers, presence of adjacent filaments, and the evolution of material properties with time and location in the printed structure. Advanced techniques such as digital image correlation (DIC) and embedded sensors, such as fiber-optic sensors, have been used to assess shrinkage strains [18,19,20]. Distributed fiber-optic sensors provide spatially continuous, internally measured early-age strains with appropriate temperature and moisture compensation, offering a complementary view to surface DIC [21]. DIC-and LVDT-based techniques have been shown to highlight the significant early-age deformation caused by rapid evaporation due to increased exposed surface area [7,8,15]. Beyond global deformation mapping, DIC has also been used to detect and quantify plastic shrinkage cracking in substrate-restrained overlays, extracting validated crack properties [22]. Studies on plastic shrinkage cracking in 3DPC have shown that restrained early-age shrinkage leads to rapid crack formation within the first two hours, and promotes interlayer slip that could compromise bond strength between layers [7]. Parallel capillary-pressure tracking has further implicated meniscus-driven suction as a governing driver of early-age strain localization in printed filaments [23]. It has also been shown that increased overburden pressure due to the superimposed layers enhances bleeding and capillary pressure while simultaneously imposing restraint-induced tensile stresses at the bottom layers, which increases the risk of early-age cracking [16].
A predictive framework for total shrinkage in 3DPC that couples internal relative humidity, mass loss, and hydration across specimens with different water-to-cement ratios and surface-to-volume geometries has been reported [24]. The significance of layer build-up rate on early-age cracking in 3DPC has been investigated using embedded strain gauges [14], although layer-dependent deformations were not explicitly addressed. While differential shrinkage in 3DPC has been attributed to heterogeneous drying [7,8] the influence of geometry-induced constraints, such as those arising from layer count (element height) or filament arrangement, have also not been quantified. Shrinkage-induced deformations in 3DPC result from the combined effect of free shrinkage (driven by moisture loss) and mechanical restraint arising from the filament laying process, both in the vertical and horizontal directions. In other words, restraints originate from the confinement imposed by the print bed in the longitudinal (along the layer build-up direction) and lateral (along the member thickness direction) directions, both of which limit deformation. A framework capable of resolving shrinkage behavior along these orthogonal directions is essential for linking moisture-loss kinetics, stiffness development, and the onset of tensile cracking in 3DPC. This study introduces a methodology that disaggregates observed deformation into free shrinkage and geometry-induced restraint components, providing a means to reliably estimate the influence of different mixtures and printing parameters (layer height, overall thickness, and filament geometry) on shrinkage strains and the potential for early-age cracking. Full-field, non-contact strain mapping using DIC [7,18,25] is used to isolate directional restraint-induced deformation across individual printed layers. A simplified mechanics-based approach is used to quantify restraint in the longitudinal and lateral directions, and experimental results are used to elucidate the effect of substrate restraint on shrinkage strains. The methodology proposed here allows for the extraction of shrinkage strains as functions of the material composition and exposure conditions alone, allowing designers to identify and select binder combinations that minimize shrinkage and associated cracking concerns in 3DPC.
Type I/II ordinary Portland cement (OPC) conforming to ASTM C150, limestone powder conforming ASTM C568, fly ash conforming to ASTM C618, and a Type IP-equivalent cement conforming to ASTM C595/C595M, containing 35% (by mass) of calcined clay, were used in this study. The particle size distributions of OPC, limestone, fly ash, and Type IP cement are illustrated in Figure 1, with their chemical compositions provided in Table 1. Specific gravities of OPC, limestone, fly ash, and Type IP cement were determined using a gas pycnometer to be 3.24, 2.80, 2.30, and 2.85 respectively. Three different binder compositions, consisting of: (i) OPC and limestone (L) (70% and 30% respectively, by mass) denoted as L30, (ii) OPC, limestone (L), and fly ash (F) (70%, 15%, and 15% respectively) denoted as L15F15, and (iii) Type IPequivalent cement (IP), denoted as IP35 (to indicate that it contains 35% of calcined clay, with a kaolinite content of 50% by mass in the parent clay) were used, as shown in Table 2. In this study, mixture design denotes the full formulation, including binder constituents and chemical admixtures (where used). These mixtures have been previously optimized by the authors [26,27] to satisfy critical requirements for printability, including favorable particle packing density and rheological behavior. Flow-spread values for L30, L15F15, and IP35 were 170-180 mm, determined in accordance with ASTM C1437-15, indicating comparable fresh workability within the printable window. A commercial medium-grade sand (Quikrete) conforming to ASTM C778 with a median particle diameter of 0.2 mm and a maximum particle size of 0.6 mm was used as the fine aggregate. The aggregate volume fraction was maintained as 50% of the total volume of solids across all compositions. A water-to-binder ratio (w/b) of 0.35 was maintained for both cast and printed specimens, to enable sufficient extrudability and buildability. However, a polycarboxylate ether-based superplasticizer was required to achieve desired extrudability and buildability for the mixture made using Type IP-equivalent cement, owing to the presence of calcined clay. The dry materials were premixed for 2 min at a low speed of 136 rpm. The mixing process continued for 3 min at the low speed while gradually adding the premixed water-superplasticizer solution. Final homogenization was achieved through high-speed mixing at 280 rpm for an additional 2 min.
Drying shrinkage was evaluated on prismatic mortar specimens (285 mm × 25 mm × 25 mm), in accordance with ASTM C157 guidelines. The measurement started 24 h after casting, when the samples were demolded. They were placed in a conditioned environment at 25 ± 2 °C and 60 ± 5% relative humidity (RH). Each test result represents the average of three replicates, and shrinkage data were collected for up to 28 days.
The evolution of strain on the prismatic specimens due to drying shrinkage was also monitored using DIC, following the same specimen dimensions, conditioning environment, and demolding time as described above (ASTM C157 procedure). The only modification was the application of black and white acrylic speckles to the surface prior to testing, after demolding. A DIC setup comprising two synchronized cameras was positioned to monitor a predefined region of interest on the speckled surface. The cameras consisted of a stereo pair of Point Grey Grasshopper units (5.0 MP, 2448×2048) with 1/1.8″ sensors and 3.45 µm pixel pitch, with an acquisition limit of up to 5 frames per second. Adequate lighting was provided using high intensity flood lamps, and both cameras were calibrated for optimized aperture and focal clarity to ensure accurate image acquisition throughout testing. Image frames were recorded at 10 min intervals.
Lagrangian strain fields were derived from displacement maps following standard post-processing protocols [28,29]. The subsequent analysis used a subset size of 50 pixels and a step size of 12 pixels, selected to balance spatial resolution and correlation robustness. Results represent the average shrinkage strain obtained from three specimens, monitored over a period of 28 days.
Mortar mixtures (L30, L15F15, and IP35) were extruded using a gantry-based printing system equipped with a screw-driven extrusion mechanism. A 30-mm square nozzle was used to print filaments of square crosssection, 30 mm wide and 30 mm tall. The mixtures enabled prints with smooth surfaces and good interlayer and inter-filament bonding. For the initial shrinkage measurements using DIC and quantification of layer and filament restraint effects on shrinkage, two wall geometries were employed: (i) a single filament thick wall (400 mm long (l) × 30 mm wide (b) × 120 mm tall (d)), composed of single filament in the horizontal plane and four layers in vertical plane (Case A; Figure 2(a)), and (ii) a double-filament wall (400 mm long × 60 mm wide × 120 mm tall), composed of two filaments in the horizontal plane and four layers in vertical plane (Case B; Figure 2(b)). In these figures, the x-y plane represents the print bed, with the xdirection along the print length and the y-direction along the filament width (thickness), while the zdirection corresponds to the layer build-up direction. Case A considers restraint only in the vertical (zdirection), while Case B assesses the influence of lateral (y-direction) restraint as well.
To validate the restraint effects under varying layer, filament numbers and sizes, three additional walls were printed using the L30 mixture: (i) a single-filament wall, similar to Case A but of shorter height (400 mm long × 30 mm wide × 90 mm tall; Figure 2 with same overall size as the previous case, but printed using a 20-mm square nozzle, so as to have more filaments and layers. These additional configurations were designed to validate layer build-up and filament-dependent restraint behavior, including the effect of differing filament widths. The first layer in all configurations corresponds to the filament placed directly on the print bed, followed sequentially by the upper layers. For the prints with multiple filaments, all filaments in the first layer were printed first before overlaying the layers.
(a) (
(e) Figure 2: Geometries of 3D printed wall specimens used for early-age shrinkage evaluation: (a) single-filament wall (400 × 30 × 120 mm), (b) double-filament wall (400 × 60 × 120 mm), (c) single-filament wall of reduced height (400 × 30 × 90 mm), (d) four-filament thick wall (400 × 120 × 120 mm), and (e) six-filament thick wall with 20-mm square filaments (400 × 120 × 120 mm).
An in-plane print speed of 30 mm/s and a vertical speed of 5 mm/s was used, with the auger speed calibrated to obtain a consistent extrusion rate of 27 ml/s. Immediately after printing (Figure 3(a)), the exposed surface of the fresh concrete was speckled with white acrylic paint, with the greyish color of the printed mixture serving as the background (Figure 3(b)). Image acquisition commenced soon after and continued for the first two hours to track the very early-age deformation. After this initial phase, image acquisition was suspended for approximately 15 min to allow application of a dual speckle pattern-a white acrylic paint serving as the background and speckles created using black acrylic paint (as shown in x y z 400 mm 120 mm 30 mm 1 st layer 2 nd layer 3 rd layer 4 th layer x y z 30 mm 400 mm 120 mm 1 st layer 2 nd layer 3 rd layer 4 th layer x y z 90 mm 400 mm 30 mm 1 st layer 2 nd layer 3 rd layer x y z 120 mm 30 mm 400 mm 1 st layer 2 nd layer 3 rd layer 4 th layer Longitudinal restraint by substrate (z-direction) Transverse restraint by substrate (y-direction) x y z 20 mm 400 mm 120 mm 1 st layer 2 nd layer 3 rd layer 4 th layer 5 th layer 6 th layer (a) (b) (c) (d) Figure 3: DIC-based setup for early-age shrinkage monitoring in 3D printed walls: (a) freshly printed four-layer wall; (b) white speckle pattern applied on fresh surface; (c) dual-speckle pattern with layer labels (1 st layer nearest to print bed); (d) DIC strain map with virtual extensometers (E0-E3) placed at mid-height of each layer for deformation analysis. E0 E1 E2 E3 Printed specimen Imaging and lighting system Computer system
In addition to the prints described above, single-filament wall elements with four layers (400 mm long × 30 mm wide × 120 mm tall) were printed and subjected to a controlled severe-drying environment in an attempt to induce early-age plastic shrinkage cracking. Immediately after printing, the specimens were placed under a directed air flow of approximately 3 m/s at 25 ± 2 °C and 60 ± 5 % RH to promote rapid surface evaporation. DIC measurements were initiated concurrently to capture the time-dependent deformation and the onset of surface cracking.
To investigate the effect of substrate-induced restraint on early-age shrinkage, a separate set of single-filament, four-layer wall specimens (400 mm long × 30 mm wide × 120 mm tall) was printed over a low-friction interface. Friction effects were minimized by sequentially placing a plastic sheet, applying a thin layer of oil, and covering it with an additional plastic layer before printing. This arrangement minimized interfacial shear transfer from the substrate to the base filament, thereby lowering z-direction restraint at the print bed. Digital image correlation (DIC) measurements were performed under the same severe-drying exposure as mentioned above to monitor the early-age deformation under the reduced restraint condition.
To quantify the time-dependent evolution of stiffness in printed mortar mixtures, Green Compression Tests (GCT) were performed on fresh extruded specimens at predetermined intervals using appropriate modifications to a protocol developed and reported earlier [30]. Mortar was extruded through a 30 mm square nozzle to fabricate a rectangular cuboid (600 × 90 × 120 mm). Immediately after printing, six cylinders (50 mm diameter × 100 mm height) were carefully extracted from the cuboid. The stainless-steel coring tube was internally lubricated to minimize friction and avoid geometric distortion during extraction.
After coring, excess material was trimmed and the specimen was carefully ejected and leveled to ensure flat, parallel loading surfaces. The cylindrical surfaces were then coated with a dual-speckle pattern (white background with black speckles) using acrylic paint, to facilitate DIC-based strain measurement. The cylinders were then subjected to the severe drying exposure condition described earlier, to ensure that the modulus values would be representative of those relevant to the desired exposure conditions. The specimens were tested after 30, 60, 90, 120, 150, and 180 min from printing.
GCT was performed using a displacement-controlled universal testing machine (MTS Exceed Series 40 E42, 5 N-5 kN capacity). Acrylic platens were used at both ends to achieve uniform and aligned contact. A small pre-conditioning displacement (< 0.5 mm) was applied to ensure proper seating. A quasi-static loading rate of 1 mm/min (strain rate of 0.01/min) was used. DIC was used to capture full-field strain distributions and detect strain localizations or heterogeneity during testing. The stress-strain response was derived using machine-recorded load data and DIC-based deformation measurements. The modulus of elasticity at each time interval was computed by applying linear regression to the initial (elastic) portion of the stress-strain curve, prior to the onset of yielding. This procedure enabled reliable and repeatable assessment of the early-age stiffness of the printed mortars under relevant drying and loading conditions, allowing for accurate determination of restraint-induced stress development. Furthermore, the application of an acrylic black-and-white speckle pattern on the specimen did not adversely affect the measured shrinkage response, thus demonstrating the reliability of DIC for capturing deformation behavior in cementitious materials.
Figure 5: Comparison of total shrinkage strain over time measured based on ASTM C 157 procedure and DIC for different binder systems. The measurements started after demolding, which occurred 24 h after casting. The average strains shown here represent three replicate samples, with uncertainties < 5% of the average. followed by gradual plateauing (secondary regime). This behavior is primarily attributed to the accelerated evaporation of free water from the freshly extruded surface, which leads to volumetric contraction. As hydration progresses, the setting and hardening process initiates a reduction in evaporation rate, thereby stabilizing the strain profile. Similar observations of early shrinkage profile governed by surface evaporation and setting have been reported elsewhere [7,17]. For all the mixtures and in both configurations, the top-most layer experiences the highest shrinkage strain and bottom-most layer, the lowest, attributable to restraint effects. Restraint refers to the displacement restrictions imposed on the printed element due to boundary conditions, which in turn result in the restriction of the free volumetric contraction of concrete. For the single-filament prints, the shrinkage strain after 24 h is ~0.42% at the 1 st layer for the L30 mixture, increasing to ~1.0% at the 4 th layer.
As expected, for the double-filament prints where the restraint in the lateral direction is also active, the strains are further lower -for the L30 mixture, it varies from ~0.21% at the 1st layer to ~0.65% at the 4th layer. Furthermore, the shrinkage strains in the double-filament prints are noted to exhibit less variation across layers, once again attributed to the enhanced restraint that restricts movement. The shrinkage strains are found to be lower for the L15F15 and the IP35 mixtures as compared to the L30 mixture. The lower shrinkage strains in the IP35 mortar as compared to the limestone-containing mortars until 24 h can be attributed to higher water retention by the calcined clay, and its slower hydration reactions [31,32]. The higher limestone filler content in the L30 mixture increases fines and surface area, suppresses bleeding, and promotes surface moisture loss, resulting in higher shrinkage [33,34]. After 24-48 h, this trend reverses (see Figure 5), where the fine limestone particles (d50 ≈ 2 µm) contribute to nucleation effects and enhanced pore refinement, while the effects of the other supplementary cementing materials are slow to manifest [35,36].
The measured shrinkage strains are influenced not only by material properties but also by geometric confinement in both longitudinal and lateral directions. The print bed restricts deformation at the base, resulting in the highest degree of restraint near the bottom of the printed element, as noticed in the shrinkage strains in Figure 6 and Figure 7. The less stiff (than the print base) layers restrict the movement of the overburden layers resulting in shrinkage in subsequent layers that is higher than that in the layer below. Previous studies have acknowledged the contribution of overburden effects, layer-wise deposition, and inter-filament interactions to anisotropic stress development and early-age cracking in 3DPC systems [37,38]. However, such effects have not been quantified to provide accurate descriptors of the effect of directional restraint on volume change phenomena. This section introduces a simplified framework to correct for geometry-induced restraints in both vertical and lateral directions in 3DPC, in an effort to extract the material's true shrinkage response and elucidate the effects of layer and filament characteristics on shrinkage. While previous experimental approaches [7,9,10,19,37] provide insights into early-age strain evolution, they typically measure total deformation without separating layer-wise deformations or isolating the contribution of the geometry-induced restraints. measurements taken near the mid-height of each printed layer during the initial 24 h after printing. Note that the deformation can be quantified at any location along the print height, and the analysis yields comparable results. In the second stage, the observed strains are adjusted to account for the restraint effects due to the print bed (and underlying layers) on the layers in the build-up direction (in the zdirection). In the third stage, the adjusted strains from the second stage are further corrected to account for the restraint induced by additional filaments in the y-direction. With these corrections (two-stage normalization), the geometry-induced constraints are separated, and the result is the free shrinkage response of the printed material, which is a function of continuing chemical reactions and exposure conditions.
Figure 8: Flow diagram showing the steps involved in evaluating restraint effects to determine free shrinkage.
This step involves measuring the observed deformation (𝜀 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 ; see Figure 6 and Figure 7) at the midheight of each printed layer using DIC during the first 24 h after printing. In lieu of average deformation that is generally recorded, the DIC-based measurement captures layer-wise deformation, thereby enabling a more localized and accurate analysis that accounts for the heterogeneity introduced during the printing process. This approach also allows to predict cracking time and location also, based on observed strain concentrations.
In 3DPC components, the base layers experience significant restraint due to the stiffness of the substrate which restricts free shrinkage, resulting in lower observed deformations [8,14]. In this study, longitudinal (z-direction) restraint is treated primarily as a substrate-induced one that transmits upward through the height of the wall, producing a restraint field that is maximum at the base and decays toward the top. The top-most layer can be considered to experience a near-zero restraint. Although this vertical gradient in restraint has been acknowledged [7,10,16], the influence of such differential restraint on early-age deformation behavior has seldom been quantified. Hence, a restraint coefficient in the z-direction ( 𝑧 ) is introduced here. 𝑧 is defined as the ratio of vertical deformation at a given layer height (𝛿 𝑧 ) to the total vertical deformation of the entire printed wall (𝛿 𝑡𝑜𝑡𝑎𝑙 ) as shown in Equation 2.1 which quantifies the degree of restraint experienced at each layer height.
Measurement and quantification of early-age deformation (Stage I) Assessment and correction of restraint effect along z-direction (Stage II) Evaluation and adjustment of restraint effect along y-direction (Stage III) Quantification of early-age deformation Restraint coefficient along z-direction Restraint coefficient along y-direction Modified strain along z-direction Modified strain along z-and y-direction Extraction of free shrinkage Input Output Theory ( ) ( ) Single-filament: Multi-filament:
Since the printed concrete is assessed during the fresh state, deformation comprises both elastic and plastic components due to the material's thixotropic and partially set nature:
However, green compression tests on samples extruded from the printed specimens indicated that the maximum self-weight-induced stress at the bottom filament at different times remained below the corresponding elastic yield stress of the mortar. Consequently, the observed vertical deformation can be considered to be primarily elastic (𝛿 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 ), given by Hooke's law. The effective load acting at height 𝑑 𝑧 is influenced by the weight and build-up of material above, where 𝑑 𝑧 is the vertical distance from the top of the print to the layer of interest. Thus, the axial load (𝑃 𝑧 ) acting at different distances from the top can be expressed in terms of the material unit weight (𝜌) and geometry, as 𝑃 𝑧 = 𝜌𝐴𝑑 𝑧 . Given that the crosssectional area of interest for the self-weight (A = l × b) and the elastic modulus (E) remains the same, the longitudinal restraint coefficient 𝑧 can be expressed as:
This formulation indicates that 𝑧 varies quadratically with normalized height, increasing from zero at the top (where 𝑑 𝑧 = 0) to unity at the base (where 𝑑 𝑧 = 𝑑). By definition, 𝑧 is always bound between 0 and 1 (0 < 𝑧 < 1). This boundary condition is physically governed by the progressively increasing confinement from accumulated self-weight and substrate interaction. To express the restraint variation in a generalized form, it is expressed here as a function of a non-dimensional parameter ( 𝑑 𝑧 𝑑 ⁄ ), which varies from 0 at the top to 1 at the bottom. As illustrated in Figure 9(a), the bottom-most layer, constrained by both substrate interaction and accumulated overburden, experiences maximum restraint, while the top-most layer, being minimally confined, exhibits near-zero restraint.
(a) (b)
) )
Figure 9: Restraint effects in 3DPC elements: (a) as a function of height (z-direction) and (b) as a function of thickness (y-direction). In (b), values are normalized to the minimum section thickness, with a single 30 mm filament case taken as the one with zero lateral restraint.
Normalizing the observed deformation with (1- 𝑧 ), as shown in Equation 2.4, yields the intrinsic shrinkage strain (𝜀 𝑧 ), effectively eliminating the influence of longitudinal restraint along build-up direction. Note that, if there is only one filament being built up, this is effectively the free shrinkage strain, (𝜀 𝑧 = 𝜀 𝑓𝑟𝑒𝑒-3𝐷 ), which is a function of the mixture characteristics and the exposure conditions.
𝜀 𝑧 = 𝜀 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 1 - 𝑧 (Equation 2.4)
The third stage of the model addresses the restraint effects induced by the thickness (y-direction) of the printed element, particularly those associated with increasing wall thickness or filament count. Previous studies have recognized that thicker printed elements tend to exhibit reduced deformation due to enhanced lateral confinement [10,39]. Figures 6 and 7 clearly show that, for all the mixtures tested, even at the top-most layer where longitudinal (z-direction) restraint is minimal, double-filament walls exhibit approximately 35% lower strain compared to single-filament walls, indicative of the presence of lateral (transverse) restraint in multi-filament walls. Here, a restraint coefficient in the y-direction ( 𝑦 ), is introduced, that complements 𝑧 defined earlier for the z-direction.
Since 𝑦 is the restraint factor in the lateral direction, it can be considered to be zero for a single-filament print (unless a single filament wall is printed to thicken an existing wall, as in the case of structural strengthening or for architectural purposes-such cases are not considered here). Thus, the relationship between the shrinkage strains in the z-direction for a single and double filament print can be expressed as:
For each binder system used here, 𝑦 was determined by equating the free shrinkage strain of a singlefilament wall to that of a double-filament wall after removing the z-direction restraint as described above.
To accomplish this, experimental results from one-, two-, and four-filament walls were used, corresponding to wall thicknesses of 30, 60, and 120 mm respectively. These correspond to normalized thicknesses of 1, 2, and 4 in Figure 9(b). The first few points in Figure 9(b) are thus derived from experimental results and averaged over multiple trials (with standard deviations found to be less than 1.5%), and similar values were obtained across different binder systems. Beyond a normalized thickness value of 4, 𝑦 was extended by scaling the experimentally determined ratio of reduction in free shrinkage with each doubling of thickness, which was determined to 0.65, irrespective of the material type. In other words, 𝜀 𝑓𝑟𝑒𝑒-2𝑡 = 0.65 𝜀 𝑓𝑟𝑒𝑒-𝑡 ; 𝜀 𝑓𝑟𝑒𝑒-4𝑡 = 0.65 𝜀 𝑓𝑟𝑒𝑒-2𝑡 (where 𝜀 𝑓𝑟𝑒𝑒-𝑡 , 𝜀 𝑓𝑟𝑒𝑒-2𝑡 , and 𝜀 𝑓𝑟𝑒𝑒-4𝑡 are free shrinkages at wall thicknesses of t, 2t, and 4t respectively). It is seen (based on limited experiments shown in a later section of this paper) that this ratio remains effectively invariant across reasonable changes in layer width and height when the overall thickness is matched. The subsequent points were therefore extrapolated to illustrate the asymptotic approach of 𝑦 towards unity with increasing thickness (e.g., increasing number of filaments).
This approach ensures that 𝑦 captures the additional restraint introduced by increasing thickness. As the number of filaments increases, the lateral restraint becomes more pronounced, and 𝑦 increases, slowly approaching unity in significantly thicker cross-sections (as illustrated in Figure 9(b)). Figure 9(b) also
shows the diminishing marginal enhancement in restraint as thickness is increased. In other words, this parameter theoretically remains lower than 1.0 even in very thick sections (even at a 𝑏 𝑦 𝑏 ⁄ of 16, it is only ~0.90). This coefficient represents the fraction of deformation prevented by lateral confinement relative to an unconstrained single-filament and is also theoretically bounded between 0 and 1 (0 < 𝑦 < 1). A value of zero indicates no lateral restraint, while a value of 1.0 represents a fully restrained cross-section.
The previously height-adjusted strains (𝜀 𝑧 ) are further normalized using (1 - 𝑦 ), yielding 𝜀 𝑧𝑦 (Equation 2.6), which represents the case where the experimental shrinkage of a 3D printed wall is adjusted for both the z-and y-direction geometric effects, yielding the intrinsic free strain that is representative of the material and exposure effects.
The validity and generality of the experimentally derived 𝑦 values are further assessed using multifilament (≥ 4) configurations, as discussed in the following sections.
Figure (a) and (b) illustrate the shrinkage strain normalized by the z-direction restraint coefficient (as described in Equation 2.4) for both single-and double-filament prints for the L30 mixture. The observed strains were shown in Figures 6(a) and 7(a) respectively. The layer height 𝑑 𝑧 corresponding to each virtual extensometer (E0 to E3), as captured in the DIC-based strain maps (see Figure 3(d)), was used to compute the restraint coefficient 𝑧 at each layer using Equation 2.3.
Upon applying the z-direction correction, the observed strain 𝜀 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 for different layers converge, indicating that the previously observed variation in shrinkage between layers can primarily be attributed to the restraint imposed by the substrate and to a lesser degree, the underlying layers (since the extrusion time interval between layers is minimal, the restraint imposed by the underlying layers can be considered to be minor. If the overlying layer is laid after one immediately below has set, this will still be captured in 𝑧 ). The z-direction normalized strain values obtained from this step are referred to as 𝜀 𝑧 (as shown in geometries. This assumption allows 𝜀 𝑧 to serve as the baseline for evaluating additional restraint effects in multi-filament configurations. For single filament prints, Figure (a) shows that the normalized shrinkage strain (𝜀 𝑧 = 𝜀 𝑓𝑟𝑒𝑒-3𝐷 ) for the L30 mixture is about 1.0%. This is equal to the strain in the top-most layer observed from the DIC experiments for this mixture, which is expected because the calculation scheme for restraint assumes a 𝑧 of zero for the top-most layer. Similar trends were observed for L15F15 and IP35 mixtures; their plots are not shown here since no additional insights are gleaned beyond the differences in peak magnitude (~0.9% and ~0.7%, respectively). For double filament walls, Figure (b) similarly shows the results for the L30 mixture, with the shrinkage adjusted for z-direction restraint (𝜀 𝑧 ). The peak shrinkage value for the L30 mixture was ~0.65%, whereas for the L15F15 and IP35 mixtures (not shown here), the values were ~0.58% and ~0.55%, respectively. effects can be attributed to interfacial bonding between adjacent filaments, which restricts movement perpendicular to the print direction. Thus, while accounting for substrate and layer restraint alone is sufficient in slender geometries, wider or multi-filament elements require consideration of lateral restraint effects also, as shown below.
Studies have shown that lateral confinement becomes increasingly significant with increasing element width [9,40]. These effects are particularly pronounced during the initial setting period, when the material retains sufficient plasticity to undergo volumetric changes but begins to experience restraint from adjacent filaments.
Figure (c) presents the shrinkage strain profiles for the double-filament L30 mixture prints after accounting for the lateral (y-direction) restraint effects on shrinkage strains on which longitudinal restraint corrections have already been implemented. The resulting strain (𝜀 𝑧𝑦 ), corrected for both vertical (buildup direction) and lateral (additional thickness) geometric effects, is the free shrinkage strain (𝜀 𝑓𝑟𝑒𝑒-3𝐷 ), as it reflects the material's deformation response independent of external restraint. It can be seen that the strain profiles of all the layers converge, and exhibit strong similarity with the trends and values observed after longitudinal correction alone in the single-filament specimens (Figure (a)). This similarity confirms that once both z-and y-direction geometric effects are systematically corrected, the underlying shrinkage behavior is independent of filament configuration. For the double-filament configuration, Figure (c) presents the normalized shrinkage strain (𝜀 𝑧𝑦 = 𝜀 𝑓𝑟𝑒𝑒-3𝐷 ) for the L30 mixture, which reached a peak of approximately 1.0%. The corresponding peak strains for L15F15 and IP35 mixtures, which also followed the same trends as shown in the figures, were around 0.9% and 0.7%, respectively. Thus, the approach proposed here not only enables direct comparison of shrinkage strain across different print geometries but also provides a consistent framework to evaluate the material's deformation capacity free from boundary-induced artifacts.
All the validation exercises presented in this section utilize the L30 mixture. (a) (b) (c) Figure 12: Shrinkage strain profiles for a four-filament wall made using L30 mixture: (a) at various layer heights, (b) after z-direction restraint correction, and (c) after both z-and y-direction restraint correction). Replicate measurements within 8% of each other.
Figure 13 shows an extended validation of the restraint correction framework through a six-filament, sixlayer wall printed with the L30 mixture using a 20-mm square nozzle. Strain evolution at the mid-thickness of each individual layer over a period of 24 h is shown in Figure 13(a), which shows even lower strains than a four-filament wall section, further reinforcing the effect of restraint. It also needs to be noted that the slope of the primary regime is much lower in this case than a two-or four-filament wall section, indicating the effect of restraint in the shrinkage rate (a potential effect of surface area-to-volume ratio, a parameter not addressed in detail in this paper), in addition to the total strains. Similar correction approaches to account for the longitudinal and lateral restraints were carried out and the resultant strains are shown in Figure 13 (b) and (c) respectively. Once again, the maximum free strains converged to a value of ~1.0%, reflecting the intrinsic shrinkage characteristics observed in experiments involving other geometries described earlier. This approach shows that the corrected shrinkage strain reflects the intrinsic material behavior governed primarily by binder composition and water loss kinetics driven by environmental exposure and chemical reactions. The successful validation of shrinkage strains in a sixfilament, six-layer wall (with different filament width than the earlier cases) also substantiates the determination of 𝑦 values where a constant multiplier was used to reduce the free shrinkage with each doubling of thickness.
(a) (b) (c) Figure 13: Shrinkage strain profiles for a six filament wall made using L30 mixture: (a) at various layer heights, (b) after z-direction restraint correction, and (c) after both z-and y-direction restraint correction). Quantitative profiles are based on three specimens; measurement variation remained below 6% of the mean.
Figure 14(a) shows the DIC-measured shrinkage strain profile for a single-filament, four-layer wall (400 × 30 × 120 mm) for the L30 mixture, subjected to forced convection drying. As noted earlier, the height-dependent gradient in shrinkage strain arises from longitudinal restraint, and severe drying further exacerbates shrinkage strains due to accelerated evaporation (e.g., maximum strains in the layers range from ~1.0% to ~2.3% under this drying regime, as opposed to ~0.45% to 1.0% under the exposure condition used in the other experiments reported before in the paper for the single-filament wall of the same dimension, made using the L30 mixture). After applying the longitudinal correction (using Equation 2.4), the strains converged to 𝜀 𝑧 , which in this single-filament case directly represents 𝜀 𝑓𝑟𝑒𝑒-3𝐷 , with free shrinkage strain of ~2.3% (Figure 14 The tensile stresses induced by the accumulated restraint were calculated by multiplying the layer-wise restraint strains with the corresponding modulus values from GCT (𝜀 restrained × E), and are shown in Figure 15(c). Under the imposed severe drying conditions, the first crack was observed in the bottommost layer (1 st layer) at approximately 1.45 h after printing, corresponding to a tensile stress of 0.17 MPa.
Since the average DIC-determined strains in the layer increases with time (even after local strain at the crack zone drops to zero), increase in average tensile stress with time in the cracked layer is noticed.
Following the first crack, second and third cracks were noticed in the bottom-most layer. The onset of each crack causes a partial local stress release within the cracked zone, along with strain redistribution to the adjacent intact zones. While this redistribution mechanism is well established, the DIC analysis focused on a localized region in the bottom-most layer along the crack path to resolve the immediate post-cracking strain relaxation, as shown in Figure 14(d). Over time, as drying persisted and material stiffness increased, the balance between restraint-induced stress and evolving tensile capacity was repeatedly surpassed in successive layers, leading to progressive upward crack propagation. This sequential failure mechanism reflects the combined influence of rapid moisture loss, stiffness gains due to hydration, and geometric restraint from the printed configuration.
(a) (b) (c) Figure 15: Evolution of: (a) time-dependent modulus of elasticity; measurements at 1.5, 2, 2.5, and 3 h, with the trend extended backward to the mixture's initial setting time, (b) layer-wise average restrained strain evolution over time, and (c) estimated average tensile stress profiles across the layers, for the L30 mixture.
Figure 16(a) shows the shrinkage behavior of an identical single-filament, four-layer wall (400 × 30 × 120 mm) printed over a low-friction interface consisting of oiled plastic sheets. For this series, the uncertainty of the DIC-derived strain estimates was ~ 7%. Unlike the previous cases, the strain distribution across the layers is confined in a narrower range, confirming that reducing substrate friction effectively lowers z-direction restraint. The strain in the 1 st layer is significantly higher than in the more restrained case presented in the earlier experiments, indicating that the wall could deform more freely in the early plastic stage. While complete elimination of restraint is practically unachievable, the observed shrinkage profiles approach an ideal free-shrinkage response, with only minor differences between layers due to reduced interfacial resistance. The smaller band of shrinkage profiles suggests that the primary contributor to early-age restraint in slender 3D-printed walls is the frictional effects at the print bed or at the completely hardened layer when walls are built up over multiple days.
The corresponding DIC-derived horizontal strain field is shown in Figure 16(b), which reveals no pronounced strain localization near the base filament, in contrast to the severe drying case.
Early-age shrinkage cracking is a concern for 3D printed concrete structures and is influenced by the geometry of the printed element (particularly the number of layers and filaments and their dimensions) and surface roughness of the substrate. The interaction between moisture loss, structural build-up, and geometric restraint directly affects the time-dependent shrinkage. This study introduces a framework to characterize early-age shrinkage in 3D printed concrete elements by integrating digital image correlationbased, layer-specific surface strain measurements and analytical quantifications of geometry-based restraints. DIC-based measurements of strain starting 24 h from casting were found to be consistent with measurements obtained using standardized ASTM tests, and thus DIC was used to measure shrinkage strains starting immediately after printing. The speckling patterns were adjusted before and after the setting of the mortar to allow for image contrasts but did not influence the measurements. The following conclusions are obtained from this work:
1. Early-age surface strains measured by DIC on single-and double filament walls demonstrated the influence of geometry and surface roughness of the substrate. A rapid increase in strain in the first few hours (primary regime) was followed by gradual plateauing (secondary regime). The lowest -0.0153 -0.0261 lagrange strains were observed on the bottom-most layer in contact with the substrate and the highest strains in the top-most layer, due to the restraint induced by the substrate. The magnitude of strains was lower in the multi-filament ( 2) walls because of the restraint in the lateral direction (y-direction) also, while the single-filament walls were subjected to only the longitudinal restraint (z-direction). Surface strains measured using DIC were much lower, and their appearance delayed in thicker, multifilament sections because geometric restraint masked the material response, even when moisture loss conditions were similar.
2. An analytical framework was developed to quantify the restraint effects in the longitudinal and lateral directions as a function of the geometry. A two-stage methodology-first determining the longitudinal restraint and correcting the measured strains using this parameter and then implementing the lateral restraint correction-was used. For single-filament walls, only the first stage is necessary. The time-dependent free shrinkage strains thus obtained are only dependent on the mixture design (binder components, w/b, and any other additives) and exposure conditions.
3. The framework was validated using walls printed using different materials, multiple layers and filaments, and different element dimensions. The validation effort also enabled the verification of a few simplifying, yet mechanistically consistent assumptions that were used in the formulations of the restraint coefficients.
4. The influence of evaporation rate and substrate friction were also evaluated. The tensile stress development was calculated using residual strains (the difference between DIC-measured strains and analytically calculated free strains) and time-dependent elastic modulus determined using green compression tests on cylinders cored from the printed walls. Cracks initiated at the layer in contact with the substrate and propagated along the height as drying progressed. Reducing the friction at the substrate significantly reduced the vertical restraint, producing nearly uniform layer-wise strains and enabling the base-layer response to approach the free shrinkage. This suggested that substrate friction is the primary source of early restraint in slender prints and that its mitigation suppresses restraint-induced tensile stress and cracking.
The framework proposed here helps accurately interpret differential shrinkage in 3D printed concrete elements. This allows for reliable assessment of binder performance and supports the development of mixture designs that mitigate the risk of shrinkage-induced cracking. This approach can also be used to quantify the restraint effects as a function of geometry. The methodology can also support the development of models for restraint-induced tensile stresses and the potential for shrinkage cracking in 3D-printed structures. Note that the current model assumes near-perfect interfacial bonding and does not capture deformation occurring during deposition/printing, especially when printing under severe exposure conditions that impose significant deformations. Accordingly, the derived restraint coefficients may need to be revised where bonding is deficient or printing-induced strains are significant.