Tunable mechanical behavior of auxetic cementitious cellular composites (CCCs) Experiments and simulations

(cid:1) Cracking-initiated rotation is found as the deformation mechanism of auxetic cementitious cellular composites (CCCs). (cid:1) Compressive behavior of CCCs is found to be crucially inﬂuenced by its geometry features. (cid:1) Re-entrant shape conﬁne zones are discovered for auxetic CCCs under restrained boundary condition. This research presents an investigation of the compressive behavior of auxetic cementitious cellular com- posites (CCCs) using a combination of experiments and ﬁnite element (FE) simulations. Typical auxetic centrosymmetric geometry was used as unit cells for the cellular structure and ﬁber reinforced cemen- titious mortar were used as constituent material. By varying the cellular geometry, three CCCs (P0, P25 and P50) were prepared then experimentally and numerically tested under uniaxial compression with different boundary conditions. Good agreement can be found between experimental and FE simulated results: Only CCCs with chiral section (P25 and P50) exhibited auxetic behavior and a typical compressive stress–strain response with two peaks was found; Under restrained boundary condition, different from the cone conﬁnement zone observed in bulk cementitious materials, re-entrant conﬁnement zone was found in the auxetic CCCs. More importantly, a cracking initiated section rotation mechanism is identiﬁed for the CCCs’ auxetic behavior which is distinct from the elastic instability mechanism of polymeric aux- etic materials with the same cellular structure. In terms of density, energy dissipation ability and Poisson’s ratio, the auxetic CCCs shows excellent properties making them promising in various civil engi- neering applications.


Introduction
The term ''auxetic", first coined by Evans [1] in 1990 s, refers to materials exhibiting negative Poisson's ratio (NPR). The value of Poisson's ratio is defined as the negative of the ratio of transverse strain to the corresponding axial strain resulting from an axial stress. Contrary to conventional materials which show a positive Poisson's ratio, auxetic materials contract laterally when compressed vertically or expand laterally when elongated vertically [2,3]. Auxetic materials have been an extremely hot topic over the past few years, and auxetic behavior has been achieved by different types of materials such as metals [4][5][6], ceramics [7,8], polymers [9][10][11][12][13], and composites [14,15].
Auxetic behavior is usually achieved by introducing special cellular structures [13,16]. By programming the cellular structure [13,17] or modifying the constituent materials [12], mechanical properties of the auxetic materials can be rationally designed. Compared to conventional continuum solids, auxetic materials possess outstanding advantages regarding their mechanical properties in several aspects: on one hand, the cellular structure constructed by periodic unit cells affords them considerably lower weight [18][19][20] comparing to continuum solids. This not only reduces materials usage but expands their application occasions where weight is a crucial factor, for instance in aircrafts [21,22] and protection fabrics [23][24][25]. On the other hand, the unique deformation behavior, i.e. NPR of auxetic materials enables them superior mechanical properties such as high shear resistance, high indentation resistance, high energy absorption, and high damping capability [26]. These mechanical excellences make auxetic materials rather promising in many infrastructure engineering https://doi.org/10.1016/j.conbuildmat.2020.121388 0950-0618/Ó 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). applications such as barriers for impact resistance, offshore breakwaters, dampers for vibration mitigation caused by vehicles or, even in extreme conditions, earthquakes. Compared to other construction materials, such as steel and timbers, cementitious materials, which are the most widely used construction materials, are particularly suitable for these applications because of their capability of obtaining outstanding mechanical properties and durability at relatively low cost. In addition, processing of cementitious materials is quite easy and robust: normally a flowable cementitious mix is directly casted into molds, which ensures the feasibility of adopting cellular structures. Recently, cementitious cellular composites (CCCs) with auxetic behavior have been reported [27,28] by adopting a centrosymmetric chiral (meaning that each unit cell is symmetric however consisted by chiral parts at a centrosymmetric configuration) cellular structure which enables the auxetic CCCs to exhibit excellent deformability and energy dissipation capability.
The adopted centrosymmetric chiral structures by [27,28] are usually identified as ''elastic-instability" structures in which the auxetic mechanism is attributed to elastic buckling. In this sense, materials with high deformability, such as polymeric materials, are often used for this type of structures to achieve auxetic behavior. For those commonly used polymers, varying geometrical features of the centrosymmetric chiral structure considerably affects the mechanical properties of the constituted auxetic materials in a large range regarding the stiffness [4,10,29], strength [4,10,29] and deformation localization [30,31] which is the mechanism of NPR behavior. For cementitious materials, similar influence can be identified from the preliminary study by [28]. However, the basic NPR mechanism in CCC's has not yet been fully understood. For a same type of cellular structure, cementitious materials with low deformability and prone to cracking still achieved NPR in [24,25], which indicates an NPR mechanism related to cracking. Usually, boundary restraint condition is believed to significantly impact cracking behavior [32,33] of cementitious materials. For bulk cementitious materials, because of their positive Poisson's ratio (0.15 to 0.22 [34][35][36][37]), the lateral expansion is restrained when vertically compressed by the friction between the specimen and the loading plates. As a result, the compressive behavior such as strength and cracking pattern is different from unrestrained condition. However, for auxetic cementitious materials, because of the NPR lateral contraction instead of expansion is restrained by the loading plates and the frictional restraining force direction is inverted. In this sense, the influence of boundary restraint on the compressive behavior of the auxetic cementitious materials may be rather different.
Programming cellular structures of the auxetic cementitious materials makes a new material developing strategy which, however, requires a rational understanding on the NPR mechanism, the impact of geometrical features and boundary conditions on the mechanical behavior of auxetic materials. This work presents an in-depth investigation of mechanical behavior of auxetic cementitious cellular composites (CCCs) with tuned geometrical features under different boundary conditions. CCCs with a centrosymmetric chiral structure were prepared with the aid of 3D printing technology. By varying geometrical parameters of the unit cell, three types of CCCs were designed, produced and submitted to uniaxial compression under various levels of boundary restraint. In addition to the experiments, finite element numerical models were adopted, calibrated and used to simulate the compression process of the CCCs. With regard to the NPR mechanism, the influence of geometrical features and boundary conditions on the mechanical properties of the auxetic CCCs, analyses coupling experiment and numerical simulation were performed.

Cellular geometry design parameters
Three types of geometry were used in this study. A single unit cell of the cementitious cellular composites (CCCs) is shown in Fig. 1. Geometry parameters are listed in Table 1. The designed CCCs consist of duplicate unit cells: an example of the designed CCCs was shown in Fig. 2. The pattern deformation factor of each group is defined as the increased length percentage of the major axis comparing to P0.

Casting and curing
The same method described in [28] was adopted to prepare CCCs specimens: the designed cellular structures were 3D printed by a commercial fused deposition modelling (FDM) based 3D printer Ultimaker 2 + using acrylonitrile butadiene styrene (ABS) material. The printed structures were glued inside cubic boxes as negative molds. Then, a two-component silicone rubber (Poly-Sil PS 8510) was mixed together and casted into the negative molds. After 2 h, the hardened silicone rubber was taken out of the cubic box and further used as mold for cementitious material casing afterwards. Note that, depending on the printing quality, the actual dimension of the molds may vary within 0.8 mm from the design parameters as 0.8 mm nozzle was used.
A fine-grained fiber reinforced mortar was used as constituent material for the CCCs. CEM I 42.5 N and fly ash were used as binder, PVA fiber was used as reinforcement by 2% of volume. The PVA fiber was produced by Changzhou TianYi Engineering Fiber. Physical and mechanical parameters of the PVA fibers are shown in Table 2. To obtain better fiber distribution, methylcellulose powder produced by Shanghai Ying Jia Industrial Development Co. Ltd. was used as viscosity modifying agent (VA). Mix proportion is listed in Table 3.
Dry components (CEM I 42.5, fly ash, sand, VA) were weighed and mixed for four minutes using a Hobart machine. Then water and superplasticizer were weighted and added to the dry components followed by 2 min mixing. Afterwards fibers were added into the mixed materials and mixed for another 2 min. The mixed fresh grout was casted into the silicone molds, steel molds (100 mm Â 100 mm Â 100 mm) and Styrofoam (180 mm Â 180 mm Â 8 mm) molds respectively. All casted materials were vibrated for 40 s and covered by plastic films to prevent evaporation. After curing under room temperature for 1 day the specimens were demolded and cured in the curing chamber (20℃, 96%RH) until 28 days age. Plate specimens (shown in Fig. 3a) demolded from Styrofoam molds were cut into bars (100 mm Â 30 mm Â 8 mm, shown in Fig. 3b). Cube specimens (Fig. 3c) demolded from steel molds were cut into small cubes (20 mm Â 20 mm Â 20 mm, Fig. 3d).

Mechanical tests
All specimens were taken out of the curing chamber an hour before testing. In order to calibrate the numerical model (ABA-QUS/Explicit), six bar specimens were tested under uniaxial tension, and six cubic specimens were tested under uniaxial compression. The results of these tests are presented in section 3, where the numerical modelling is discussed.
The CCCs were tested under uniaxial compression, four duplicate specimens were tested for each design. One surface of the specimens was painted white and sprayed with red dots in order to perform DIC (digital image correlation). Images were acquired every second during testing by a camera which was placed in front of the specimen. By adopting DIC analysis, local translational shift of the sprayed dots on the specimen was computed. Thus, the displacement of the sprayed dots was obtained and full-filed deformation of the specimen was able to be obtained during the entire compression process. In order to calculate the Poisson's ratio, the

Numerical modelling
In order to provide further understanding of the compressive behavior and NPR mechanism of the CCCs. Numerical models were adopted to simulate the compression process of the CCCs. The simulations were performed by a commercial numerical package (ABAQUS/Explicit). Geometric nonlinearity (Nlgeom) was used and concrete damage plasticity model (CDPM) which is commonly used for simulating fiber reinforced cementitious materials [38][39][40] in ABAQUS was used as material model for the constituent material in this study. According to ABAQUS guide documents, the constitutive equation of CDPM can be described as follows: Where is r Cauchy stress; e and e pl are total strain and equivalent plastic strain respectively; E 0 is initial elastic modulus. In the presence of modulus degradation, d is damage variable and can be defined from 0 to 1. In our study, stiffness degradation in not considered.
The yield criterion beyond which the plastic strain initiates of CDPM is given as: With Where, b r max is maximum principal effective stress; r b0 =r c0 is the ratio of initial equi-biaxial compressive yield stress to initial uniaxial compressive yield stress which has an ABAQUS default value of 1.16 and adopted as input normally. K c describes the tensile meridian second stress invariant to that on compressive meridian for a given invariant p, 0:5 < K c 1 must be satisfied and as an ABAQUS default value K c ¼ 0:667 is also adopted by many researches. Non-associated potential plastic flow is assumed for CDPM and the flow potential G is described as:  Where w is the dilation angle, r t0 is the uniaxial tensile failure strength, is the eccentricity which determines the rate at which the function approaches the asymptote. ¼ 0:1 is adopted as a default value. According to CDPM, uniaxial compression and tension response are characterized by plasticity, the evolution of the failure is controlled by compressive plastic strain e pl c and tensile plastic strain e pl t , respectively. These two variables can be determined by material uniaxial loading behavior as shown in Fig. 4. Under compression, the failure stress (r c0 ) corresponds to onset of microcracking of the cementitious material after elastic stage. Under tension, the failure stress (r tu ) corresponds to the ultimate tensile strength. The compressive inelastic strain (e in ) and tensile crack strain (e ck ) is defined as the total strain minus the elastic strain corresponding to the undamaged material, respectively. In this study, E-modulus degradation is not defined, in this sense compressive plastic strain e pl c ¼ e in and tensile plastic strain e pl t ¼ e in .

Model calibration
Prior to simulating the uniaxial compression behavior of the CCCs, the models were calibrated to obtain proper material input parameters. By fitting uniaxial compression tests on the cubic specimens and uniaxial tension tests on the bar specimens, compressive behavior parameters and tensile behavior parameters of the constituent material can be obtained as model inputs. Balancing model precision and computing cost, two-dimensional plane stress elements with a mesh size of 0.5 mm were chosen and kept constant for all numerical simulations in this study. As described previously, in the CDPM inelastic and cracking strains are used to model the material damage in compression and tension, respectively. Therefore, besides the elastic modulus, density (shown in Table 4) and plasticity parameters (adopted from literature [40], where a rather similar mix was used), inelastic regime (compression) and post peak cracking regime (tension) of the stress-strain response need to be specified to simulate the damage process. Fig. 5a shows the stress-strain curves (blue dashed lines) of the cubic specimens obtained from uniaxial compression experiments. As the constituent material is fiber reinforced, rather ductile compressive damage behavior can be identified from the experimental results: stress increased as compression initiated and kept increasing until compressive strength was reached. Afterwards, the stress started to drop slowly which is normally described as a softening branch [33,41]. Comparing to the typical quasi-brittle damage of unreinforced cementitious material [33,41], the softening branch of fiber reinforced cementitious material is considerably longer as the fibers significantly increase the material ductility. In the stress-strain curves (see Fig. 5a), a long plateau stage was observed from experiments. Several points were chosen by averaging strain and stress values based on the experiment curves (plotted as black dots connected by green line in Fig. 5a) and then used as material inputs for simulation (listed in Table 5 and Table 6, note that in ABAQUS, tension input strain need to be converted to displacement accordingly), the material compressive curve is plotted as green line in Fig. 5a). A numerical compression test was performed using the constitutive curve. The simulated result is also plotted Fig. 5a, marked as ''Calibration" (red line). The simulated compression curve shows good agreement with experiment curves, indicating that the material compressive parameters are calibrated.
A similar calibration approach was also applied for tension model input parameters. The comparison of model input curves and experimental curves are shown in Fig. 5b. After the first crack (witnessed by a sharp drop of the stress-inelastic strain curve), the specimen did not fail completely, owning the so called ''slip hardening" behavior [42][43][44][45]. Slip hardening is often witnessed in PVA fiber reinforced cementitious materials. PVA fibers are flexible and  less hard comparing to its embedding cementitious matrix. When crack opening increase, these PVA fibers are slowly pulled out from the matrix and polymeric branches are peeled off from the PVA fiber by the hard matrix. As a result, a jamming effect generates between the PVA fiber and the matrix which resists fiber pullout and leads to an increasing load after cracking. Owing to this slip hardening behavior, the load still increased again until a second peak was reached. The constituent material does not show strain hardening behavior and as a result only one crack was witnessed.
In this study, as the tension specimens were meshed homogenously (no mesh heterogeneity was introduced comparing to what discrete models [46][47][48][49][50] usually do), inelastic deformation will be easily localized at where it initiates. As a result, the slip-hardening behavior will be rather difficult to be captured by simulation thus the simulated second peak is slightly lower than the experiment (Fig. 5b red line). In any case, the simulated curve still shows good agreement with experiment, in this case the tension model input parameters are also well calibrated regarding the good agreement between simulation and experiment. Normally, for CDPM compression and tension damage parameters can be assigned which is used to capture E-modulus degradation during unloading process. However, in this study as unloading process was not included in the experiment these damage parameters were not specifically assigned. Therefore, according to the CDPM constitutive law, compressive inelastic strain and tensile cracking strain are assumed to be equal to plastic strain which indicates damage, as described previously.

Simulation of compressive tests on CCCs
ABAQUS/Explicit was used to simulate the compression process of the CCCs. As shown in Fig. 6, in order to simulate the real experimental condition of the compression tests, in the numerical model external load was not directly applied on the specimen, instead, two loading plates (160 mm in length and 5 mm in height) were added as rigid body parts and placed at the top and the bottom of the specimen. Vertical displacement was applied downwards on the top plate while the bottom plate was fixed.
In addition, according to the experimental conditions (listed in Table 1), the specimen P25 was tested under three types of boundary restraint conditions: low friction boundary condition which simulates the unrestrained compression test (P25), single side confined boundary condition which simulates the single side glued compression test (P25SG) and both sides confined boundary condition which simulates the double side glued compression test (P25DG) test. Correspondingly, three types of contact rules were applied between the specimen and the steel plates in the simulations, respectively. For low friction boundary conditions, although plastic films were applied between the specimen and loading plates, slight friction still occurred during testing. Under ideal frictionless condition (friction coefficient is 0), the specimen would move horizontally in the simulation when compressed which however, doesn't match practical experiment conditions. So, by a trial and error method, the friction coefficient is increased from 0 until the compressed specimens don't move horizontally meanwhile the simulated deformation process of the unit cells in contact with the plates shows good agreement with experiment. Therefore, a low friction coefficient 0.105 was assumed and used to simulate the slight friction between the specimen and loading plates. Comparatively, for single side glued condition, glued elements at the specimen top were completely tied with the loading plate while the bottom side is the same with low friction boundary condition; for double side glued condition, glued elements at both sides of the specimen were completely tied with the top and bottom loading plates, respectively. Similar to previous approach, the friction coefficient between the constituent materials is decreased from 1 until the deformation pattern matches experiment, which consequently also ensures the stress-strain response in agreement with experiment. A friction coefficient 0.95 is assumed as the selfcontact parameter between the constituent material elements.   Fig. 7 and Fig. 9 show the stress-strain curves and compression process (until 40% strain) obtained from experiments and numerical simulations of the CCC specimens. Note that as the CDPM is used in the simulation, cracks are simulated by plastic deformation. Therefore, highly deformed locations in Fig. 7 and Fig. 9 show high plastic strain which correspond to the cracks observed in experiments. In general, owing to the difference in geometrical features, P0, P25 and P50 behave significantly different during the compression process. With regard to the entire compressive behavior, for all tested three different geometries, a very good agreement was found between experimental results and numerical simulations.
The compressive behavior of P0 resembles the compressive damage process of conventional fiber reinforced cementitious materials [39,51]: a single peak can be found during the entire compression process. In the pre-peak regime, an ascending branch can be found in the stress-strain curves (Fig. 7a), specimen deformed elastically as soon as external load was applied. When the peak load was reached, cracks started to appear and propagate through the cellular structure. It can be seen from both experiment and simulation in Fig. 7b, cracks (high strain area) propagated through the cellular structure by connecting hollow parts of the specimen forming shear crack planes which is commonly found in the compressive failure process of conventional cementitious materials [33]. At the macroscale, although the diagonal fracture plane looks very similar to shear bands caused by plastic slip (which is the failure mode of ductile materials), the microscopic mechanism for the CCCs is very different. Indeed, the PVA fiber reinforced material shows some ductility after the peak load (see tensile stress-strain curve in Fig. 5b), but this is caused by the fiber bridging after cracking. So, the microscopic mechanism of the failure of CCCs is still brittle cracking. Cementitious materials are very heterogenous, and their compressive failure is a process of tensile microcracking (Mode I fracture at microscale) inclined to the shear direction [52]. Depending on the boundary conditions, inclination angle of the shear fracture plane varies: under frictionless condition, an inclination around 21 degrees has been observed [33]. For cellular materials, the fracture plane must follow the configuration of the cellular structure/elements, as voids cannot transfer stresses. A similar phenomenon is commonly witnessed in discrete element fracture modeling. For example, realistic diagonal cracks can be simulated only when diagonal orientated elements are present. In our study, the shear fracture plane must follow the ''section" part in each single unit which has the highest stress as the ''section" are the load bearing part of the cellular structure. Actually, the diagonal shear fracture plane can be also witnessed from the experimental result. A symmetric diagonal plane from top left to bottom right direction can be observed in the experiment. However, since the top and the bottom surface of the experimental sample are not perfectly flat, the shear plane is not exactly diagonal direction but somewhat deflected. Eventually, cracks generally propagated through the entire structure leading to the failure of the compressed specimen. Correspondingly, from stress-strain  curves (Fig. 7a), a long descending branch can be witnessed in the post peak regime. Within the entire compression process P0 did not show auxetic behavior, which is rather different from the behavior of similar P0 structure made using highly elastic polymers [53]. The Poisson's ratio development of P0 can be seen later from Fig. 10. The compressive behavior of P25 and P50 is significantly different: obvious lateral contraction can be witnessed when P25 (see Fig. 9a) and P50 (see Fig. 9b) was compressed. Namely, the NPR or the so-called auxetic behavior was achieved. Therefore, P25 and P50 can be categorized as ACCCs (auxetic CCCs). As previously defined in [27,28], the experimental compression process of these ACCCs can be divided into two stages until 40% of strain (marked in Fig. 8). In the first stage, both from experiment and the simulated curves, the stress-strain response of the ACCCs looks quite similar to that of P0: a peak load can be found which consisted by an ascending branch and a long descending branch. The compression process of the ACCCs, however, is rather different. As can be seen from Fig. 9, the ACCCs exhibit obvious lateral contraction during the entire first stage, the cellular structure of the ACCCs was generally damaged and eventually disappeared until the ACCCs were compacted to be ''solid" material. Even more detailed information can be found from the Poisson ratio curves in Fig. 10. For all three cellular structures, the experimental measured Poisson's ratio shows good consistence with the simulated results: during the entire compression process P0 shows lateral expansion, a positive Poisson's ratio was obtained as a result. Comparatively P25 and P50 show lateral contraction as soon as the compression initiated. For the ACCCs, the division of compression stages is based on their stress-strain response as well as the corresponding development of the Poisson's ratio. Throughout the first stage, the Poisson's ratio kept decreasing until a minimum Poisson's was reached at a certain strain indicating that the cellular structure was completely compacted. The first stage ended at this strain, afterwards the ACCCs started to expand under compression although the calculated Poisson's ratio is still negative.
In the first stage, the differences found between P0, P25 and P50 can be attributed to their geometrical features. The peak load values of P0, P25 and P50 is normalized by their relative density, both in the experiments and the simulations the first peak (see Fig. 11a) of P0 is always the highest and P25 is lower while the P50 is the lowest because of the stress concentration induced by the cellular structure under compression. More in-depth discussion related to the stress concentration induced by geometrical features is presented later. On the other hand, as the sections start to get in contact near the minor axis end of the ellipse and the ellipse minor axis of P25 (6 mm) was higher than P50 (4 mm), P25 had more space in each unit cell to be compressed during the first stage. Therefore, the first stage of P25 ended at 13.5% of strain while P50 ended at 10%.
Normally, elastic response of the auxetic Normalized Emodulus of the CCCs is shown in Fig. 11c, obviously the cellular structure drastically decreased E-modulus of all the CCCs comparing to the used constituent material (8.23 GPa) which can be explained by the fact that CCCs have a significantly lower density. However, for the CCCs as the relative density increases (from P0 to P50), the elastic modulus, on the contrary, decreases. This indicates that the deformation pattern or the shape of the cellular structure, and not the density, dominates the elastic response of the CCCs.
While the first stage indicated the damage and compacting process of the cellular structure, the second stage indicated the damage process of the constituent material. In the second stage, the ''sections" of each unit cell started to contact and cracks started to propagate in the sections generally leading to the failure of the compacted ''solid" constituent material. Accordingly, in the stress-strain curves, a second peak which consisted of another ascending and descending branch can be identified. In this sense, this process is also rather similar to a compressive damage process of conventional fiber reinforced materials. It can be seen from Fig. 11b, because of higher specimen volume, P50 (79.75 cm 3 ) has a higher second peak than P25 (67.68 cm 3 ). Meanwhile the simulated second peak load is slightly lower than the experiment which may be caused by the slight inconsistence in cellular geometry and the boundary friction between the numerical models and the real experiments. This two-stage compressive behavior was also observed in many other auxetic materials [9].

Mechanism of auxetic behavior
With regard to the auxetic mechanism, a rod-spring model was proposed in [16] for a similar centrosymmetric chiral structure however constituted by elastomers in which localized deformation is a crucial factor for the auxetic behavior. For cementitious materials, because of their low deformability, the auxetic behavior is more related to cracking rather than only elastic deformation. Regarding to the auxetic mechanism of the ACCCs in this study, more specifically, a hypothesis was raised by [27]. There, the origin of this behavior was assumed to be both the crack bridging ability due to the fibers in the constituent material and the chirality of the cellular structure. In this study, because a same mixture as in [27] was used, the influence of constituent material was eliminated. With the aid of numerical simulations, more proofs can be identified supporting the hypothesis raised in [27].  According to [27], chirality of the ''sections" in each unit cell introduces stress concentration when external compressive load is applied. Fig. 12 shows the distribution of maximum principle stress (tensile stress) and minimum principal stress (compressive stress) on three cellular structures in the elastic regime subjected to the same load (250 N). It can be seen from Fig. 12a, for P0 the highest tensile stress distributes in the ''section" region of the cellular structure which will eventually result in cracking when the principal tensile stress reaches the tensile strength of the constituent material. However, for P25 and P50, the tensile principal stress concentrates in the ''joint" region of each unit cell. More importantly, as assumed by [27,30], when this cellular structure is loaded, because of the section chirality, the ''joint" of each unit cell is submitted to a condition similar to four point bending scheme (shown in Fig. 13), under which the minor and major axis side of the ellipse is in tension and compression, respectively. The simulation results are in good agreement with this assumption: it can be seen from Fig. 12b and Fig. 12c that for both P25 and P50 tensile principal stress mainly concentrates at the ''joint" near the minor axis side, meanwhile compressive stress mainly concentrates near the major axis side. Owing to the stress concentration, a crack initiates at the tension stress concentrated area as soon as the tensile stress reaches the material tensile strength. Comparing to P25, higher concentrated tensile stress can be found in P50 under the same external load (250 N): the highest tensile stress of P50 at the ''joint" is almost exceeding the tensile strength (1.74 MPa) of the constituent material, while the highest tensile stress of P25 is lower than 1 MPa. This explains the fact that the first peak load of P50 is lower than P25 (see Fig. 11a).
As described in section 4.1, the cellular structure of the ACCCs was destroyed in the first stage. This process was initiated by cracking which subsequently led to compacting of the unit cell until the cellular structure generally disappeared. Auxetic behavior was observed during this process. For P25 (Fig. 14a) and P50 (Fig. 14b), cracks first appear at the tensile stress concentrated area. The simulation results and DIC results (see Fig. 15) show good agreement with regard to the cracking location. After cracking, similar to the rod-spring model described by [30], in each unit cell the joints act as hinges which allows the sections to rotate and eventually the sections get in contact with each other. As the horizontal length of the unit cell decreases with rotation, globally the CCCs exhibit auxetic behavior. In this sense, the behavior can be Fig. 10. Poisson's ratio of P0, P25 and P50 obtained from experiment and simulation, standard deviation is indicated; the experimental result of P25 is adopted from [27]. Fig. 11. Comparison of the a) the normalized first peak load, b) the normalized second peak load and c) normalized E-modulus between experiment and simulation, experiment standard deviation is indicated.  seen as similar to the rotating square geometries [54][55][56] where the joints act as nodes and the sections act as rotating bodies. In addition, because the ellipse minor axis length of P25 (6 mm, see Table 1) is 30% higher than P50 (4 mm, see Table 1), larger strain (3.5% higher than P50) is needed for P25 to be compacted. In summary, the mechanism of the auxetic behavior of ACCCs can be described as cracking initiated section rotation.

Influence of boundary conditions
Boundary restraint conditions have a significant influence on the cracking behavior of cementitious materials under uniaxial compression as the confinement provided by the loading plates affects the development of the fracture localization [32]. For bulk cementitious materials, due to the positive Poisson's ratio, uniaxial vertically compressed specimen exhibits horizontal expansion. Under high friction boundary condition, horizontal expansion near the loading plate is restrained which creates local confinement: consequently, cracking will occur in the unconfined area. A cone shape can be observed (schematics is shown in Fig. 16), with triaxial compressive stress arising locally within the specimen which leads to a higher overall peak load [32,33]. This phenomenon is comparable with the triaxial compressed concrete. For the auxetic CCCs, the confined zone can be also witnessed: however, unlike bulk cementitious materials, the compressed specimen tries to contract laterally under compressive vertical load, owing to the NPR effect. Therefore, the direction of confinement is inverted, which leads to re-entrant angle shape confined zone instead of a cone. As can be seen from Fig. 17a, for P25SG (the P25 specimen with top side glued on the loading plate) at 5% strain obvious difference can be seen between the glued side (top) and low friction side (bottom). The undeformed cells show a re-entrant shape confined zone. At the low friction side (bottom), the cellular structure is compacted by a previously described cracking initiated rotation mechanism, while at the glued side (top), the cells are barely deformed because of the lateral confinement from the upper loading plate. Similarly, for P25DG (the P25 specimen with both top and bottom side glued on the loading plates) at 5% strain, both sides show re-entrant confined zones consisting of undeformed cells near the two loading plates (see Fig. 17b), similar deformation pattern can be also seen on elastomers [46], however in elastic stage caused by elastic instability.
The stress-strain response of P25DG and P25SG is also obviously affected by the boundary restraint. As already discussed in the previous section that void closure causes the typical twostage stress-strain response of P25. Under low friction condition, the cells are not confined and are free to crack and deform. Consequently, the first peak from the P25 stress-strain curve depicts cracking of the cellular structure, while subsequently the second load increase after section rotation depicts compacting of the constituent material. Under glued boundary conditions, the typical second load increase still can be found both from the experiment and simulated stress-strain curves (see Fig. 18). However, the crack initiation as well as the section rotation were restrained. As shown in Fig. 19, due to the highest applied restraint, the normalized first peak load of P25DG (3.48 MPa from the experiment and 3.18 MPa from the simulation) is also the highest. The first peak load of P25SG is lower (2.75 MPa from the experiment and 2.56 MPa from the simulation) because of less restraint. Finally, the first peak load of P25 is the lowest (2.18 MPa from experiment  and 2.47 MPa from simulation) because of the low friction boundary condition. This phenomenon can be attributed to the confined fracture localization which is similar to the fact that bulk cementitious materials show higher compressive strength under restraint boundary condition [32]. As mentioned previously, cementitious materials are very heterogenous, their compressive failure is a process of tensile microcracking (Mode I fracture at microscale) and crack localization (crack shearing). For bulk cementitious materials, compressive strength is reached around the onset of crack localization. When a macroscopic compressive load is present and the specimen is laterally confinement, lateral tensile stresses introduced at the micro heterogeneities are reduced by the confinement. Higher compressive load is then required to generate microcracks as a result. Also, when microcracks are present, lower shear stress will be present due to lateral confinement which slows down shearing of the microcracks due to an increase of the frictional stresses on the crack surface and compressive strength is increased as a result. For P25, the first peak strength is mainly determined by the tensile cracking at the joints. Higher compressive load is required to generate microcracks at the joints due to lateral confinement, as a result first peak strength of confined P25 is higher. After the first peak, the second load increase of P25DG and P25SG is still visible both in experiment and simulated curves. However, as the cell sections rotation near the loading plates was restrained, less cells can deform and be compacted which led to slower load increase and is seen in the stress-strain curves by much longer and milder ascending branch under glued boundary conditions.

Potential application of the CCCs
As a cellular material, low unit weight is an important feature of the CCCs. In our study, the density of the CCCs are 928 kg/m 3 (P0), 987 kg/m 3 (P25) and 1163 kg/m 3 (P50) respectively, which are considerably lower than bulk cementitious materials (1870 kg/ m 3 for the mix we used). In this aspect, the CCCs are rather similar to a widely used lightweight civil engineering material: foam concrete. Regarding the compressive strength, the CCCs are rather comparable to the foam concrete with a density from 900 kg/m 3 to 1200 kg/m 3 , see Fig. 20. On the other hand, as shown in Fig. 21, the compressibility (defined as the strain at compressive strength by percentage) of the ACCCs (22.4% for P25 and 18.0% for P50) is distinctively higher than normal foam concrete (around 1.0%). The large compressibility of ACCCs ensures their excellent energy dissipation capacity (shown in Fig. 22, defined as the area under load-displacement curves divided by volume, due to the compressed displacement varies in different existing researches [57][58][59][60][61][62], the energy dissipation properties of foam concrete has rather large deviation). Comparing to energy dissipation property of foam concrete (0.01 J/cm 3 to 1.70 J/cm 3 with an average value of 0.70 J/cm 3 ), the energy dissipation properties of the ACCCs is good enough to be applied as supplement or substitute. In addition, in the construction of roads or bridge embankments, foam concrete (density from 350 kg/m 3 to 1000 kg/m 3 , compressive strength from 0.5 MPa to 3.5 MPa [63,64]) has been used as filler of the embankment under the road pavement to control settlements of the underlying soil foundation. Using as filler for embank- ment could be a potential application for the ACCCs, not only for their low density and strength (which are similar to those of foam concrete), but, more importantly, the negative Poisson's ratio. The ACCCs do not laterally expand when loaded, thereby avoiding added pressure on adjacent structures which is an excellent feature especially when existing embankment need to be widened.

Conclusions
In the present work, a study coupling experiments and numerical simulations was performed to investigate the compressive behavior of cementitious cellular composites (CCCs) with negative Poisson's ration (NPR). Based on the obtained experimental and numerical results and corresponding analysis, the main conclusions can be drawn as follows: Concrete damage plasticity model (CDPM) proves to be an adequate tool to simulate the uniaxial compressive behavior of this type of cementitious cellular composites (CCCs) regarding to their deformation pattern, Poisson's ratio and stress-strain response. Compressive deformation patterns of the CCCs is crucially influenced by the geometrical features. Only CCCs with chiral sections (P25 and P50) show negative Poisson's ratio and can be defined as auxetic CCCs (ACCCs). Physical and mechanical properties of the ACCCs can be tuned and modified by varying the pattern deformation factor. Increasing the pattern deformation factor increases the relative density compressive strength while decreasing the E-modulus. A cracking initiated section rotation mechanism is identified as a source of the ACCCs auxetic behavior, in contrast to the elastic instability mechanism of the elastomers with similar cellular structures.     21. Compressibility of the CCCs and foam concrete, standard deviation is indicated; data of the foam concrete is adopted from [59][60][61][62]67] Fig. 22. Energy dissipation capacity of the CCCs; data of the foam concrete is adopted from [59][60][61][62]67], due to the compressed strain varies in these studies [59][60][61][62]67], large deviation of foam concrete can be witnessed.
Under boundary confinement, compressed ACCCs undergo lateral restraint. Different from the commonly found cone shape fracture zone contrary to that of bulk cementitious materials which leads to a re-entrant shape fracture zone. The ACCCs have similar physical and mechanical properties to foam concrete used in engineering practice, while their deformability is significantly higher. This makes ACCCs a potential supplement or an alternative of foam concrete in applications such as energy absorption and settlement mitigation.
According to the obtained results, the properties of the CCCs are tunable and can be improved by modifying the cellular structure. Meanwhile, combing the modification of cellular structure and tailoring the properties of the constituent materials may also improve mechanical performance of the CCCs and this can be a future research direction.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.