Abstract
On a mesoscopic level, high performance concrete (HPC) was assumed to be a heterogeneous composite material consisting of aggregates, mortar, and pores. The concrete mesoscopic structure model had been established based on CT image reconstruction. By combining this model with continuum mechanics, damage mechanics, and fracture mechanics, a relatively complete system for concrete mesoscopic mechanics analysis was established to simulate the process of early-age shrinkage cracking in HPC. This process was based on the dispersion crack model. The results indicated that the interface between the aggregate and mortar was the crack point caused by shrinkage cracks in HPC. The locations of early-age shrinkage cracks in HPC were associated with the spacing and the size of the aggregate particle. However, the shrinkage deformation size of the mortar was related to the scope of concrete cracking and was independent of the crack position. Whereas lower water to cement ratios can improve the early strength of concrete, this ratio cannot control early-age shrinkage cracks in HPC.
1. Introduction
In bridge engineering, the phenomenon of early-age cracks in high performance concrete (HPC) appeared frequently. One of the main driving forces of early-age cracks in HPC was shrinkage distortion [1–3]. Early-age cracks appeared because of the hydration reaction of HPC and the difference among internal material properties of each phase. Therefore, the concrete would produce internal micro cracks. These cracks were not load-dependent; rather, the cracks were a defect of the concrete [4–6]. Two major factors contributed to internal micro cracks in HPC. On one hand, because of the different physical and mechanical properties of the mortar and aggregates in the concrete, the early-age concrete mortar produced larger shrinkage deformations in the process of hardening compared to the aggregate, producing asynchronous deformations of the mortar and aggregate. Aggregates function as a constraint on the shrinkage deformation of the mortar. Therefore, the concrete would generate an internal tensile stress, and the interface of aggregate and mortar inside the concrete would generate micro cracks when the tensile stress was sufficiently large. On the other hand, before the initial setting time, the surface of the concrete would appear to secrete water because sinking water inside the concrete aggregate was brought to the surface. This process of moisture movement resulted from the influence of the aggregate. Because of the interface effect, the water would be gathered at the lower edge of the aggregate; this water separated the aggregate and mortar. After consuming this section of water inside the concrete, the interface between the aggregate and mortar would generate pore or micro cracks. When the shrinkage stress in the concrete did not exceed the tensile strength of the concrete, the fissure was stable. When the tensile stress caused by the shrinkage deformation of the HPC was greater than the tensile strength of the concrete, cracks would develop, assemble, and eventually become a macroscopic fracture [7–12].
Combining continuum mechanics, fracture mechanics, and damage mechanics theories, this paper analyzed the process of deformation, damage, and failure at the microscopic level caused by the early-age shrinkage deformation of HPC through a mesoscopic numerical simulation. Simultaneously, the main cause of concrete cracks is considered to be tension. On a mesoscopic level, the HPC was assumed to be a heterogeneous composite material consisting of aggregates, mortar, and pores. The concrete mesoscopic structure model is established based on CT image reconstruction. This method was combined with continuum mechanics, damage mechanics, and fracture mechanics, establishing a relatively complete system for the mesoscopic mechanics analysis of HPC. This method simulated the process of early-age shrinkage cracking in HPC, which is based on the theory of the dispersion crack model.
2. Materials
Ordinary Portland cement (strength grade of 42.5 MPa) and fly ash (I-type high calcium ash) produced in China were used as the cementitious materials used to prepare the mortar. The fine and coarse aggregates were washed river sand with a specific gravity of 2.56, water absorption of 1.01%, fineness modulus of 2.66, and crushed granite with a specific gravity of 2.63, with maximum sizes of 5 mm and 20 mm, respectively. In the concrete, the mechanical properties and shrinkage performances of the mortar were shown in Tables 1 and 2, and the mechanical properties of the aggregate were shown in Table 3. The numerical notation of FA20-0.28S mortar corresponds to the fly ash replacement percentage of cement of 20% and water to cementitious materials ratio of 0.28.
3. Method
3.1. The Injury Tolerance of HPC
Concrete was a multiphase composite material, consisting of aggregates, gelled materials, sand, and water. Because of the different nature of the aggregate and mortar, initial damage was noted in the concrete from the effect of early-age hydration. During early-age shrinkage deformation, the initial damage area of the internal concrete would further extend and become a micro crack area. With increasing shrinkage stresses, the internal micro cracks in the concrete would develop, collect, and finally form a macroscopic crack. With further increases in the shrinkage stress, the macro cracks widen and elongate until the concrete fractures in full. Therefore, the process of early-age shrinkage cracking in concrete would progress through the initial damage, damage accumulation, macroscopic cracking, and macroscopic crack extension.
According to the law of the conservation of energy, material failure consumes energy. The formation process of early-age shrinkage cracks in HPC can be explained by the conservation of energy. In the initial setting of the concrete when the shrinkage deformation initiates, energy is released in the damaged area around each micro crack (according to the energy theorem). This description assumes that the internal micro cracks are uniformly distributed and each micro crack is independent. In addition to the increase in the shrinkage stress, the energy release area around each micro crack gradually expands, interconnecting micro cracks and producing a macroscopic crack. This process results from the heterogeneity inside the concrete, producing a nonlinear stress-strain curve. The nonlinearity results from the internal weak area in the concrete being damaged after stress. This behavior of concrete can be considered as a type of stiffness degradation behavior.
Given the total volume of concrete () and the fact that the damage inside the concrete is caused by shrinkage stress, the volume of the undamaged area is and the volume of the energy release area around the cracks is . These three volumes can be expressed as (1). The shrinkage stress in concrete accumulates gradually with time. When the energy release area is gradually increasing, the undamaged area is gradually decreasing. Assuming that the undamaged area in concrete satisfies a linearly elastic constitutive relation, the corresponding shrinkage stress is the effective stress, , and can be expressed by (3). Introducing the damage variable , (1) and (2) can be written as follows:
The effective shrinkage stress can be defined as follows:where represents the nominal stress (MPa).
On a mesoscopic level, the damage model in continuum mechanics is used to characterize the mechanical properties of the various raw materials in concrete. The Lemaitre strain equivalence principle [13] states that the concrete strain caused by the shrinkage stress after damage is equivalent to the strain caused by the effective shrinkage stress before the damage. The method of calculating the effective stress is displayed in (3). According to this principle, the constitutive relation for the concrete after damage can be represented by the relationship between the nominal stress and strain before the damage (see the following two equations).where is the initial elastic modulus (MPa); is the modulus of elasticity after the damage (MPa); and is the damage variable.
To display the different degrees of damage in concrete, the damage variable D is introduced. The damage variable is equal to zero and one in the intact and destroyed states, respectively. The variable for concrete corresponding to different degrees of damage ranges between zero and one.
3.2. The Constitutive Model of Early-Age Shrinkage Cracking in HPC
The stress-strain relationship curve of the concrete tensile test was shown in Figure 1. At the beginning of the concrete experiencing tension, the strain gradually increased with the stress, displaying a basic linear relationship. Damage to the concrete appeared after the peak stress . At this time, the strain increased and the stress decreased. When the stress reached a zero value, the concrete was completely destroyed and had no ability to hold power. Therefore, when the concrete was in tension, the maximum tensile stress was defined as the tensile strength of concrete. The stress-strain relationship after achieving the maximum tensile stress of concrete was commonly called the softening curve of concrete [14].
The softening curve of concrete in Figure 1 reflected the entire tensile process and the nonlinear relationship of the stress and strain after damage. The nonlinear relationship after concrete damage resulted from interlocking aggregates and mortar inside concrete transferring a portion of the internal force after reaching the tensile strength. Therefore, this nonlinear relationship was important in bridge engineering to develop the full ability of concrete. According to fracture mechanics, the cracking of concrete was related to the stress after injury. Generally, the process of concrete cracking and damage was regarded to occur in three main stages: the generation of micro cracks, the development and gathering of micro cracks, and the cracking destruction of concrete. Understanding the three stages of concrete damage can reveal the process for the early-age shrinkage crack formation in concrete and assist the modeling of the process. Through the above analysis, the formation process of early-age shrinkage cracks resulted from internal micro cracks generating, extending, and combining. Micro cracks were present inside the poured concrete because of the hydration reaction and differences in raw material properties. The shrinkage deformation then gradually accumulated, and when the shrinkage stress reached the tensile strength of concrete, the internal micro cracks developed and the aggregate and mortar would be relatively separated, forming additional micro cracks. With further increase in shrinkage stress, the micro cracks would combine and form a macroscopic crack. After concrete cracking, the ability of the concrete to hold stress included two components: the tensile ability in the orthogonal direction of the fracture and the sliding resistance and friction effect caused by aggregates interlocking within the crack. The process curve diagram of early-age shrinkage crack formation in concrete was obtained through a large number of trials. The diagram displays a highly nonlinear curve in Figure 2.
Previous studies had experimentally tested the concrete tensile damage process and fit softening curves to the tensile damage. As the most representative softening curve model for concrete, Reinhardt et al. (1984) established the expression for the concrete softening curve (see (6)) [15]. Cornelissen et al. (1986) established (7) [16]. Gopalartnam and Shah studied the influence of the strain softening of material of the stress distribution of the virtual crack area and established (8) [17]. where is the crack opening width (μm); and is the maximum strain softening curve (με).
By comparing the experiments, the above expressions can reflect the cracking process of concrete. However, the structure of the above expressions was complex and inconvenient in practical applications. Therefore, additional studies simplified the concrete softening curve according to the characteristics of the curve. The curve was expressed as a piecewise linear function, producing a simple expression that can satisfy the precision required in engineering applications. This process had been widely applied. In 1976, Hillerborg recommended expressing the concrete tensile softening curve as a straight line. In 1985, Rots et al. [18] expressed the concrete cracking process caused by a tensile load with a bilinear curve. Figure 3 displays the bilinear model diagram of the strain softening curve when the concrete is in tension.
When under a uniaxial tensile stress, the concrete specimen would fracture along the direction of maximum tensile stress. The fault fracture was the only fracture and occurred perpendicular to the loading direction. When under unidirectional compression, the sides of the concrete specimen expand outwards, and the concrete lateral surface would be damaged under a tensile stress. The compression damage of concrete presented multiple parallel cracks, and the fracture direction and loading were in the identical direction. Through the above analysis, tension fractures were the main reason for concrete damage. Therefore, the constitutive model for the early-age shrinkage cracking of concrete only considered the concrete tensile damage and the fact that concrete in compression presents a linearly elastic constitutive relation. The concrete constitutive model for early-age shrinkage cracking was shown in Figure 3. A double dogleg was noted in the softening curve, in which indicates the maximum tensile strength; indicates the principal tensile strain; indicates the residual tensile strength of inflection point M after concrete damage; indicates the principal tensile strain of point M; indicates the ultimate tensile strain when the concrete specimens completely lose resistance to tension. From Figure 3, the relationship between the maximum tensile strength of the concrete and the residual tensile strength can be shown by the following equation: where indicates the residual strength coefficient of the concrete after damage and indicates the ultimate strain coefficient of the concrete cracking. The definition of the variable stiffness degradation, , can be determined using the following equation:
3.3. Dispersion Crack Model of HPC
In the early-age shrinkage cracking process of HPC, the tensile cracking of the interface area between the mortar and aggregate was the main cause of damage. In the finite element analysis, a concrete smeared cracking model was suitable for the study mainly through the open crack problem. It is assumed that the constitutive relation of concrete was linearly elastic before cracking in this model. In the calculation, we can remove the unit satisfying the damage condition according to the concrete crack criterion. In the finite element simulation, the dispersion crack model was mainly based on the model established by Lubliner et al. (1989) [19] and by Lee and Fenves (1998) [20]. The total strain rate of the model was divided into the elastic strain rate and the plastic strain rate (see the following equation).
The plastic strain rate increment in the finite element simulation can be expressed by the following equation: where denotes the plastic strain rate increment of concrete within time increment ; denotes the initial strain rate increment; denotes the final strain rate increment; and denotes the coefficient of the strain rate related to the development of strain rate of materials.
This model assumes that was the initial modulus of elasticity (no damage) of concrete, and in the case of uniaxial tension or compression, the stress-strain relationship of concrete was shown as follows:
According to (13), effective tensile stresses and effective compressive stresses can be obtained using the following equations: where and are the tensile stress and compressive stress of concrete, respectively; and are the effective tensile stress and compressive stress of concrete, respectively; and are the total tensile strain and total compressive strain of concrete, respectively; and are the equivalent plastic tensile strain and compressive strain of concrete, respectively; and and are the tensile injury tolerance and compressive injury tolerance of concrete, respectively.
Equations (13) represent the stress-strain relationship of concrete under uniaxial stress. In this paper, the calculation model for the shrinkage cracking of concrete assumes that the damaged concrete remains isotropous in the plane stress state. Simultaneously, we reference the definition of damage in the Fenves model, and the constitutive model of concrete shrinkage cracking under the plane stress state is shown in the following equation: where represents the initial elastic matrix of concrete in the plane, which can be determined by the following equation:where represented the damage variable when the concrete was under uniaxial tension. The definition of was shown in (5)–(10). represented the weight factor for the multiaxial stress state, defined as shown in the following equation: where (for two dimensions, ; for three dimensions, ) is the main stress component. The Macaulay operator, , is defined in the following equation:
3.4. Crack Criterion
The crack criterion for early-age shrinkage cracking used the maximum tensile stress criterion (Rankine strength criterion). According to the strength criterion, concrete was in brittle failure when the shrinkage stress of any point inside the concrete had reached the uniaxial tensile strength of the concrete, regardless of whether other normal or shearing stresses were noted. Therefore, the strength perpendicular to the plane, plane, and plane can be shown (see the following equation). Equations (5)–(19) showed the concrete strength in different directions, which was expressed by the spherical stress tensor and the second stress invariants (see the following equation). When and , the failure criterion was , and we can write the following two equations: Combining (21) with (22), we can write equation (23). Combining (24) into (23), we can write the following equation: In the plane, the following equations can be written:When , the above equations become the following:
According to the above formula, we can draw compressive and tensile meridians of the strength criterion for the maximum tensile stress. The meridian was shown in Figure 4, and the plane was shown in Figure 5. When I1 is identical, this projection of the failure surface in the plane is a regular triangle. When °, the failure surface is a straight line on the meridian. The failure surface is therefore a regular triangle in space. When remains unchanged, such as ° or °, the failure surface is a straight line on the meridian. The shape of the failure surface is a triangle cone. The test surface of the crack of the maximum tensile stress criterion in the deviatoric stress plane, meridian plan, and state of plane stress is shown in Figures 4–6. The direction perpendicular to the existing crack surface can also form the second crack. In this calculation, each material point allows a crack.
3.5. Mesoscopic Numerical Model of HPC
During the mesoscopic simulation, a portion of the concrete specimens in early-age shrinkage deformation were selected for simulation because of the large number of concrete units. Figure 7 displays the mesostructure of the concrete obtained by industrial CT scans and an image reconstruction technique. Considering the computational time and resources, the size of the numerical model size was 90 mm × 100 mm. This section investigates the cracking process of the early-age shrinkage deformation under external constraints shown in Figure 8. The outer parts of left and right sides of the model were restrained.
The stress and strain displayed a nonlinear relationship in the damaged concrete because the interlocking between the aggregate and mortar in HPC can transfer internal energy even after exceeding the tensile strength of the concrete. From the results, the early-age shrinkage cracking process of the HPC can be reasonably represented by a double dogleg stress-strain constitutive model. The maximum tensile stress failure criterion was selected as the failure criterion of concrete. In the process of mesoscopic simulation, the shrinkage deformation of the concrete accumulated as the ages of concrete. When the maximum tensile strain of the concrete mesoscopic unit satisfies , the unit was considered uninjured and linearly elastic. With further accumulation of shrinkage deformation and the maximum tensile strain at , the concrete mesoscopic unit was considered in the first damaged stage. With accumulation of shrinkage deformation, , the concrete mesoscopic unit was considered in the second damaged stage. These two stages were a progressive process. When the accumulation of shrinkage deformation was sufficiently large, the maximum tensile strain, , of the concrete mesoscopic unit was greater than , and at this time, the concrete mesoscopic units were destroyed. When concrete mesoscopic units were destroyed, we can break the unit by multiplying a small coefficient for the stiffness of the unit. The process simulating the early-age shrinkage cracking in HPC using a finite element mesoscopic method repeated the above process until the loading was completed.
At each stage in the simulation, the location of the damage in the concrete model was different. In the process of numerical simulation, once the mesoscopic unit of concrete was damaged, a new elasticity modulus would be automatically assigned to the unit. The injured concrete mesoscopic unit would then intuitively be reflected in the simulation of the next step.
4. Results and Discussion
Through the numerical simulation of the cracking process, the early cracking mechanism and trends in the HPC on the mesoscopic level were obtained. The data were shown in Figure 9, which was the simulated result of the shrinkage cracking process of FA20-0.28 concrete under external constraints. The position of the interior cracks in the concrete was at the interface between the aggregate, mortar, and interspace because of the constraining effect from the surface of the aggregate to the mortar. At the interface, a stress concentration phenomenon appeared, contributing to the interface cracking. The mesoscopic shrinkage crack of HPC gradually extends to the weak areas as the specimen ages. The location of the crack point was generally around the aggregate with a larger particle size and a large interface area with the mortar. The constraining effect on the mortar was higher, producing a larger internal stress in the mortar and becoming the cracking point. Along with age, micro cracks gradually expand, extend, and combine to form macro cracks.
(a) Four hours
(b) The first day
(c) The second day
(d) The third day
(e) The eighth day
(f) The tenth day
4.1. Age of the Concrete
From the simulation, the first step loading was the initial setting time. During this stage, the early-age shrinkage of HPC was small () and no damage was noted inside the concrete. After the first day, the shrinkage deformation increased and a small amount of cell damage was noted (the first damaged stage: ), and the location of damage was mainly at the interface between the aggregates and mortar. This position was considered as the cracking point. In the second day, additional mortar units were damaged, but all were in the first damaged stage. The locations of these units were at the interface between the aggregates and mortar. In the third day, the shrinkage deformation of the mortar and the number of injured mortar units were gradually increasing. A wide range of injured units were noted in the second damaged stage (), whereas no damage was noted on the aggregate units. In the fourth day, the constraining shrinkage stress reaches a maximum, and the mortar reaches a maximum number of damaged cells. The number of mortar units in the second damaged stage was also gradually increasing. A small amount of aggregate unit damage was noted. After the fourth day, the stage was in an unloading phase. The number of injured mortar units was decreasing because the number of units that have been destroyed () was increasing. During unloading, the number of damaged units of mortar was constantly increasing. The damaged units form micro cracks which expand and combine to form macroscopic cracks. After eight days, the early shrinkage crack was basically stable, whereas the aggregate unit displays only a few damaged units.
At the mesoscopic level, the mechanism of the formation of early shrinkage cracks in HPC resulted from the shrinkage deformation of mortar being externally restrained and the aggregate. These characteristics produced stress concentrations in the mortar on the aggregate surface. The shrinkage deformation of mortar gradually increased with age. When the restraint stress in the mortar was greater than the tensile strength of the mortar, mesoscopic microfractures were formed in the interface of the aggregate and mortar. These microfractures expand, extend, and combine to form macro cracks with age of HPC. The interface between the aggregate and mortar was the cracking point.
4.2. Aggregate
From Figure 10, the early-age shrinkage cracking location was related to the distribution of aggregate. The areas with a small aggregate spacing were the cracking point. In these areas, the shrinkage of mortar was remarkably influenced by the aggregate, and the mortar was restrained by the maximum tensile stress. The area affected by the aggregate surface effect and the poor compactness of mortar produced a lower tensile strength than in other regions. Therefore, the mortar in these areas easily achieved the tensile strength of the mortar and the area of the cracking point.
From the simulation results, the position of early-age shrinkage cracks of concrete was also linked to the size of the aggregate particle. Larger particle sizes and higher restraining stresses on the surrounding mortar resulted in a higher probability of shrinkage cracking in the mortar around the aggregate.
The distribution in the mortar and the particle size of the aggregate in the concrete both display a substantial effect on the early-age shrinkage crack of concrete. Therefore the aggregate spacing, average mortar thickness, and the particle size were the factors that should be considered in the mix ratio design of concrete to control the early shrinkage cracking of concrete.
4.3. Mortar
Through the laboratory experiment, it obtained the tendency of change of the shrinkage deformation in the mortar of the different water to binder ratio and the effects of shrinkage deformations in mortar on shrinkage deformations and early-age cracking in the concrete by numerical simulation. Using a numerical simulation, results for the shrinkage cracking of concrete with different water to binder ratios and temporal distribution of restraining stresses were obtained in Figures 11–15.
From the simulation results, increasing water to binder ratios caused the contraction deformation of the mortar and caused the area of early shrinkage cracking to decrease. When the water to binder ratio reached 0.28, the contraction deformation of the mortar, the damaged area, and the width of the cracks were at a maximum. When the water to binder ratio reached 0.45, the damaged area was at a minimum and was rather distributed. In certain aggregate distributions and particle sizes, the regional locations for the shrinkage cracking of HPC with different water to binder ratios were consistent; only the size of the cracks varies. In general, the shrinkage deformation size of the mortar was related to the concrete crack area and was independent of the crack position.
From Figures 11–15 that displayed the temporal distribution of the restraining stress using numerical simulations of early-age shrinkage cracking, the restraining stress also increased with age. The time required to reach the maximum restraining stress was related to the size of the mortar contraction deformation. Damage would appear when the restraining stress of the concrete reaches a maximum value. With further aging, the damaged area gradually expanded, whereas the restraining stress gradually decreased to zero. Thereafter, the damaged area caused by early-age shrinkage deformation tends to be stable.
During the second day after casting a concrete with water to binder ratio of 0.28, the internal restraining stress of HPC reaches a maximum of 1.94 MPa as shown in Figure 11. The damaged area gradually expanded with age, whereas the restraining stress gradually decreased. This result highlighted the fact that, within two days after pouring, damage appeared inside the concrete because of strain increases from mortar shrinkage. On the following day, macro cracks appear, and the cracking load was 1.94 MPa. On the fourth day, a shrinkage crack had been formed and stabilized. Therefore, the first four days after concrete pouring were important for concrete crack formation.
From Figures 12–15, the maximum restraining stress of the concrete was the load of the early-age shrinkage cracking. Increasing the water to binder ratio delayed the occurrence of the biggest constraining stress and increased the maximum constraining stress. A concrete with a water to binder ratio of 0.32, 0.36, 0.40, and 0.45 displayed a maximum constraining stress of 2.86, 3.03, 3.28, and 3.57 MPa, respectively, on the third, fourth, fifth, and seventh days after pouring, respectively. By comparing the simulation results, reducing the water to binder ratio from 0.45 to 0.28 caused the early-age cracking loaded to reduce from 3.57 MPa to 1.94 MPa. Reducing the water to binder ratio also caused the time of early-age shrinkage cracking of HPC to advance from the seventh day to the second day after pouring. Therefore, the cracking performance of HPC reduced with lower water to binder ratios. These results showed that whereas the lower water to binder ratio can improve the concrete strength, the early-age shrinkage deformation of HPC would also increase and the size of the early-age shrinkage deformation was the main cause of early-age shrinkage cracking. In engineering applications, more early-age shrinkage cracks were noted in HPC with low water to binder ratios than in normal concrete. Therefore, when preparing the mix proportion of HPC, the water to binder ratio cannot be arbitrarily reduced to increase the strength.
5. Conclusions
A mesoscopic model of HPC was built based on CT image reconstruction. A double dogleg stress-strain constitutive model for HPC was also introduced. This process adopted a maximum tensile stress criterion under a planar state and established a relatively complete system for mesoscopic mechanical analysis of HPC. This method was based on a dispersion crack model. Overall, this model simulated the early-age shrinkage cracking of HPC. The following conclusions are obtained.
(1) On the mesoscopic level, the interface between the aggregate and mortar was the cracking point in HPC. These microfractures stretch, expand, and combine, forming macro cracks with age.
(2) The location of the early-age shrinkage cracks was associated with the spacing and particle size of the aggregate. Smaller distances between the aggregates and larger particle sizes produced higher restraining stresses on the surrounding mortar.
(3) For certain aggregate distributions and particle sizes, the location of shrinkage cracking was consistent for concrete with different water to binder ratio; only the size of the cracking area was different. In general, the size of the mortar contraction deformation was related to the concrete crack area and was independent of the crack position.
(4) By comparing the simulation results, the lower water to binder ratio can improve the concrete strength, but this ratio can also decrease the early-age shrinkage deformation of HPC. Therefore, when preparing the mix proportion for HPC, the water to binder ratio cannot be arbitrarily reduced when attempting to improve the strength.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was financially supported by the National Natural Science Foundation of China (51568053) and Natural Science Foundation of the Inner Mongolia autonomous region of China (2015BS0507).