#### 4.1. Stress–Strain Relationship and Modulus of Elasticity

Stress–strain curves provide basic data for estimating the modulus of elasticity. Given that concrete is not completely elastic, however, estimating the elastic modulus based on these curves results in various issues. Strictly speaking, the modulus of elasticity only applies to the linear elastic section of the stress–strain curve. In cases where it is difficult to judge the linear elastic section from the curved part, as is the case with concrete, the secant modulus of elasticity, also known as the chord modulus, is alternatively used [

23]. In extrusion-based additive construction, the modulus of elasticity is a critical factor that is used to estimate both the stress caused by drying shrinkage in layered EVA-modified cementitious mortar (

σ =

ε_{sh}E) and the stress caused by thermal expansion (

σ =

αEΔ

T).

In concrete, stress and strain typically have a non-linear relationship, but the relationship can be considered the linear elastic section at a lower stress level. This stress range can reach up to 40% of the ultimate strength, within which concrete can be considered an elastic material.

The stress–strain diagrams of the developed EVA-modified cementitious mortars are presented in

Figure 8. The shapes of these curves resemble those obtained for cementitious paste [

23]. Based on these diagrams, the obtained compressive strengths were 48.3 MPa, 41.8 MPa, 38.2 MPa, 35.7 MPa, and 33.5 MPa when the EVA/cement ratios were 0, 0.05, 0.10, 0.15, and 0.20, respectively. The compressive strength decreased as the EVA/cement ratio increased. The secant modulus was calculated using Equation (1). The results were 21.6 GPa, 19.1 GPa, 18.1 GPa, 17.6 GPa, and 16.7 GPa when the EVA/cement ratios were 0, 0.05, 0.10, 0.15, and 0.20, respectively. The relationship between these elastic modulus measurements and the applied EVA/cement ratios was analyzed, as presented in

Figure 9. A high correlation was found (i.e., the modulus of elasticity tended to decrease as the EVA/cement ratio was increased).

Earlier studies showed that the modulus of elasticity in the compression of unmodified cementitious concrete was 21.1 GPa, while the value ranges were 22.4 GPa to 23.6 GPa, 20.2 GPa to 24.3 GPa, and 10.0 GPa to 19.0 GPa for polyacrylic ester (PAE)–modified cementitious concrete, styrene butadiene rubber (SBR)–modified cementitious concrete, and polyvinyl acetate (PVAC)–modified cementitious concrete, respectively [

18]. The results varied depending on the polymer type and polymer/cement ratio. Overall, the modulus of elasticity tended to decrease as the polymer/cement ratio increased, and the degree of reduction increased when the polymer content was excessive. A decrease in the elastic modulus reduces the stiffness of EVA-modified cementitious mortar but also decreases the level of stress caused by drying shrinkage and temperature change.

It is also known that the modulus of elasticity of polymer-modified cementitious mortar is relatively low, 10 GPa to 30 GPa, because it contains polymer (elastic modulus: 0.1 GPa to 10 GPa) [

24]; the elastic modulus of cementitious concrete in compression generally falls within the range of 14 GPa to 40 GPa [

17]. In the present research, the maximum compressive strains were 0.00256, 0.00267, 0.00272, 0.00278, and 0.00283 when the EVA/cement ratios were 0, 0.05, 0.10, 0.15, and 0.20, respectively. These results indicate that the maximum compressive strain tended to increase as the EVA/cement ratio increased. Given that the figures were 0.002 and 0.0027 for 30 MPa cement concrete and 34 MPa cement paste [

23], respectively, the measured maximum compressive strains were found to be comparable to that of cement paste. Concrete in compression shows some inelastic strain before failure. It is worth noting that the typical level of strain at failure is 0.002 [

17] and the strain of 100 MPa concrete is typically 0.003 to 0.004, while the strain of 20 MPa concrete is 0.002. Each stress corresponds to the ultimate strength. However, under the same stress, regardless of strength, stronger concrete exhibits a lower strain [

23].

In addition, the relationship between the compressive strength and estimated modulus of elasticity was analyzed, as presented in

Figure 10. Here, the modulus of elasticity tended to increase with increases in compressive strength. It is highly certain that in concrete, the compressive strength and modulus of elasticity have a proportional relationship, but an agreement has not been reached on the precise form of that relationship [

23]. As a result, thus far the American Concrete Institute (ACI) Building Code and the Comit Euro-International du B ton and the F d ration International de la Pr contrainte (CEB-FIP) Model Code have proposed different equations [

17]. In the present study, the relationship between the modulus of elasticity and compressive strength of each EVA-modified cementitious mortar was derived as shown in Equation (2):

where

${E}_{c}$ is the modulus of elasticity (GPa) and

${{f}^{\prime}}_{c}$ is the compressive strength in MPa.

#### 4.2. Drying Shrinkage

Drying shrinkage is one of the major causes of cracking in concrete. Once exposed to air, concrete starts to dry out and the dried surface contracts. The internal moisture, however, suppresses shrinkage in the outer part. Accordingly, the surface regions undergo tensile stresses, and when the stress-induced drying shrinkage exceeds the direct tensile strength of the concrete, cracking occurs.

One advantage of 3DCP is that formwork is not necessary. This removes a barrier between the curing concrete and ambient environment. Printed layers often have a greater exposed surface area than cast concrete. However, lower water/cement ratios than those seen in casting concrete are typical in 3DCP mortars. Hence, the likelihood of cracking resulting from autogenous shrinkage is increased. Therefore, mix designs must minimize dimensional changes due to dry and autogenous shrinkage and greater care should be taken when curing [

25].

In typical cementitious mortar and concrete, the degree of drying shrinkage ranges from 200 × 10

^{−6} to 1200 × 10

^{−6}, depending on the aggregate/cement ratio [

26].

Figure 11 presents the drying shrinkage test results for up to 28 days in relation to the EVA/cement ratio.

Figure 11a shows the results for up to 24 h, while

Figure 11b indicates the results for up to 28 days. Here, the results are presented in two separate figures to make the initial-stage strain caused by drying shrinkage more distinct.

One hour after placement, autogenous shrinkage was initiated; substantial shrinkage occurred from 4 to 11 h. After demolding, drying shrinkage continued to increase until 28 days. At 28 days, the drying shrinkage was 331 × 10

^{−6}, 349 × 10

^{−6}, 379 × 10

^{−6}, 418 × 10

^{−6}, and 461 × 10

^{−6} when the EVA/cement ratios were 0, 0.05, 0.10, 0.15, and 0.20, respectively, indicating that the shrinkage increased as the EVA/cement ratio increased (see

Figure 12). Based on these results, an increased rate was calculated when the EVA/cement ratio of zero was set as a reference. The rates were 5%, 14%, 26%, and 39% when the EVA/cement ratios were 0.05, 0.10, 0.15, and 0.20, respectively. This drying shrinkage development trend was related to an increase in the water/cement ratio from 0.45 to 0.55 with an increase in the EVA/cement ratio, as shown in

Table 8. This is considered a disadvantage of the redispersible EVA powder.

In a previous study by Weng et al. [

27], at 28 days and a water/cement ratio of 0.5, the drying shrinkages were 0.0128%, 0.0217%, 0.0222%, and 0.0224% when the EVA/cement ratios were 0, 0.03, 0.05, and 0.08, respectively. When the water/cement ratio was 0.6, the drying shrinkages were 0.0380%, 0.0527%, 0.0538%, and 0.0546% and the EVA/cement ratios were 0, 0.03, 0.05, and 0.08, respectively. This indicates that the drying shrinkage increased as both the EVA/cement ratio and water/cement ratio increased.

In contrast, in SBR latex-modified cementitious mixtures, drying shrinkage was reported to decrease with an increase in polymer content [

19,

24,

28]. Likewise, when modified with latex, cementitious mortar exhibited less drying shrinkage; this is ascribed to the effects of the surfactants and antifoamers contained in the latex [

29]. The significant drying shrinkage seen in EVA-modified cementitious mixtures can be significantly reduced by using shrinkage reducing agents such as polyethylene glycol [

30] and ethylene [

18], but attention must be paid when following this course because adverse effects, such as strength degradation, may occur.

#### 4.3. Coefficient of Thermal Expansion

The coefficient of thermal expansion is defined as the change in the unit length of a material for a unit change in temperature. Thermal shrinkage strain is determined by the degree of temperature drop in concrete and its coefficient of linear thermal expansion [

17]. The coefficient of the thermal expansion of concrete is determined by the combined values of the dissimilar thermal coefficients of its two main constituents (i.e., cement paste and aggregates) [

23]. Concrete structures are deformed by temperature variations resulting from the hydration reaction of cement or atmospheric temperature changes. When this temperature variation causes the tensile stress of concrete to exceed its tensile stress, cracking is initiated. Layered cementitious materials built through extrusion-based additive construction are expected to undergo cracking and delamination due to temperature change.

In the present study, thermal strain tests were conducted, and the results are presented in

Figure 13. The thermal strains were 437 × 10

^{−6}, 530 × 10

^{−6}, 643 × 10

^{−6}, 812 × 10

^{−6}, and 997 × 10

^{−6} when the EVA/cement ratios were 0, 0.05, 0.10, 0.015, and 0.020, respectively. It was found that in all cases, the thermal strain was reached within about two hours. The coefficient of thermal expansion was estimated by dividing each thermal strain measurement by the corresponding temperature rise. Once calculated, its relationship with the EVA/cement ratio was analyzed, as shown in

Figure 14. Here, the coefficients of thermal expansion were 7.9 × 10

^{−6}/°C, 9.6 × 10

^{−6}/°C, 11.7 × 10

^{−6}/°C, 14.8 × 10

^{−6}/°C, and 18.1 × 10

^{−6}/°C when the EVA/cement ratios were 0, 0.05, 0.10, 0.15, and 0.20, respectively. This indicates that the thermal expansion coefficient tended to increase as the EVA/cement ratio increased, and the correlation between the two factors was high.

Previous studies have reported that the coefficient of thermal expansion of polymer-modified cementitious mortar ranged from 9 × 10

^{−6}/°C to 10 × 10

^{−6}/°C [

24]. Notably, however, when the polymer/cement ratio ranged from 0.10 to 0.20, the coefficient of thermal expansion of SBR latex-modified cementitious mortar was reported to be between 7.7 × 10

^{−6}/°C and 8.6×10

^{−6}/°C, which was not significantly different from that of unmodified mortar at 7.9 × 10

^{−6}/°C [

18]. For ordinary cementitious concrete, Neville [

23] reported that the coefficient of linear thermal expansion of hydrated cementitious paste varied from 11 × 10

^{−6}/°C to 20 × 10

^{−6}/°C. Mehta et al. [

17] presented that the coefficient of linear thermal expansion of cementitious mortars was approximately 18 × 10

^{−6}/°C for cementitious paste, 12 × 10

^{−6}/°C for mortar, and between 6 × 10

^{−6}/°C and 12 × 10

^{−6}/°C for concrete. As explained above, the measured coefficient of thermal expansion was determined to be significantly higher in the present study when compared to the results reported by previous studies on polymer-modified cementitious mortars. At the same time, the coefficient of thermal expansion of ordinary cementitious concrete and mortars measured in the present study was largely comparable to those reported in similar previous research.