Application of Base Force Element Method to Mesomechanics Analysis for Recycled Aggregate Concrete

The base force element method (BFEM) on potential energy principle is used to analyze recycled aggregate concrete (RAC) on mesolevel.Themodel of BFEMwith triangular element is derived.The recycled aggregate concrete is taken as five-phase composites consisting of natural coarse aggregate, new mortar, new interfacial transition zone (ITZ), old mortar, and old ITZ on meso-level. The random aggregate model is used to simulate the mesostructure of recycled aggregate concrete. The mechanics properties of uniaxial compression and tension tests for RAC are simulated using the BFEM, respectively. The simulation results agree with the test results. This research method is a new way for investigating fracture mechanism and numerical simulation of mechanics properties for recycled aggregate concrete.


Introduction
"Concrete" is considered as heterogeneous composites whose mechanical performance is much related to the microstructure of material. The mechanical performances are usually obtained using experimental methods. However, testing usually consumes a large amount of manpower and material resources, and test results are usually more discrete. In order to overcome this defect, the concept of numerical concrete was presented by Wittmann et al. [1] based on micromechanics. Subsequently, some scholars did some creative works in this field and made a number of models. Among them, the two important models are the lattice model and the random aggregate model. For example, Schlangen et al. [2,3] applied the lattice model to simulate the failure mechanism of concrete. Liu and Wang [4] adopted the random aggregate model to simulate cracking process of concrete using FEM. Peng et al. [5] adopted the random aggregate model to simulate the mechanics properties of rolled compacted concrete on meso-level using FEM. Du et al. [6] simulated the failure mechanism of beam under impact loading and triangular cyclic loading by using displacement-controlled FEM, stressstrain curves, and dynamic bending strengths of specimens.
"Recycled aggregate concrete" which is used as a green building material has attracted more and more researchers with the shortage of resources and an increasing number of construction wastes. They have carried out a series of experiments and some conclusions have been obtained. An overview of study on recycled aggregate concrete has been given by Xiao et al. [7]. However, because of the complexity of recycled coarse aggregates, conclusions made by different researchers are usually not very accordant, even opposite sometimes. To remove effects of experimental conditions, some numerical researches on meso-level were considered. For example, numerical simulation on stress-strain curve of recycled aggregate concrete was made by Xiao et al. [8] with lattice model under uniaxial compression. A method on mesomechanics analysis was proposed by Peng et al. [9,16] and Zhou et al. [10] using FEM for recycled aggregate concrete based on random aggregate model. However, the numerical researches on the damage mechanism for recycled aggregate concrete material have just begun.
In recent years, a new type of finite element method, the base force element method (BFEM), has been developed by Peng et al. [9,[11][12][13][14][15][16][17] based on the concept of the base forces 2 Mathematical Problems in Engineering by Gao [18]. In this paper, the BFEM on potential energy principle is used to analyze recycled aggregate concrete (RAC) on meso-level. The mechanics properties of recycled aggregate concrete in uniaxial compression and tension tests are simulated using the BFEM, respectively. The simulation results agree with the test results. This research method is a new way for investigating fracture mechanism and numerical simulation of mechanics properties of recycled aggregate concrete.

Basic Equation
Consider a two-dimensional domain of solid medium; let x ( = 1, 2) denote the Lagrangian coordinate system, where P and Q are the position vectors of a material point before and after deformation, respectively. Two triads for original and current configurations can be defined as follows: Let u denote the displacement of a point; then The gradient of displacement u can be written as follows: Then, the Green strain can be written as In order to describe the stress state at a point Q, a parallelogram with the edges d 1 Q 1 , d 2 Q 2 is shown in Figure 1. Define where 3 = 1 for indexes. Quantities T ( = 1, 2) are called the base forces at point Q in the two-dimensional coordinate system . According to the definitions of various stress tensors, the relation between the base forces and various stress tensors can be given. The Cauchy stress is Further, the base forces are given as follows: in which is the strain energy density and 0 is the mass density before deformation. Equation (7) expresses the T by strain energy directly. Thus, u is just the conjugate variable of T . It can be seen that the mechanics problem can be completely established by means of T and u .

Model of BFEM with Triangular Element
We will derive explicit expressions for stiffness matrices of a triangular element now, based on the concept of "base forces. " Consider a triangular element with boundary as shown in Figure 2.
For the small displacement case, the real strain can be replaced by . We can obtain the average strain in element as in which is the area of element. Substituting (4) into (8), we have Using Green's theorem, (9) becomes where n is the current normal of boundary . When the element is small enough, (10) can be written as where is the length of edge ( = 1, 2, 3), n denotes the external normal of edge ( = 1, 2, 3), and u is the displacement of geometric center of edge ( = 1, 2, 3).
Further, we assume that any edge of the triangular element in the deformation process keeps its edges straight lines. Then, we can obtain the following expression for u : where u and u denote the displacements of both ends of edge ( = 1, 2, 3), respectively. Substituting (12) into (11) yields The summation rule is implied in the above equation, and m is where and are the lengths of edges and and n and n denote the external normals of edges and , respectively.
Then, for an isotropic material, the strain energy in the element is reduced to in which is Young's modulus and ] is Poisson's ratio. Substituting (13) into (15) we have From (16), we can obtain the force acting on this element at node where Here, K is a second-order tensor that is called the stiffness matrix.
The characteristics of the stiffness matrix K compared with the traditional FEM are as follows. (1) This expression of stiffness matrix K can easily be extended to apply to arbitrary polygonal elements problem in two dimensions or arbitrary polyhedral element problem in three dimensions. (2) The expression of the stiffness matrix K is a precise expression, and it is not necessary to introduce Gauss's integral for calculating the stiffness coefficient at a point. The model of the base force element method will be used to analyze the damage problem for recycled aggregate concrete and to analyze the relationships between mesostructure and macroscopic mechanical performance of recycled aggregate concrete.

Random Aggregate Model for RAC
Based on the Fuller grading curve, Walraven and Reinhardt [19] put the three-dimensional grading curve into the probability of any point located in the sectional plane of specimens, and its expression is as follows: where is the volume percentage of aggregate volume among the specimens, in general = 0.75, 0 is the diameter of sieve pore, and max is the maximum aggregate size.
According to (20), the numbers of coarse aggregate particles with various sizes can be obtained. By Monte Carlo method, random to create the centroid coordinates of all kinds of coarse aggregate particles, namely, to generate random aggregate model as Figure 3. According to the projection method, we dissect the specimens of RAC with different phases of materials. Then, the phase of recycled coarse aggregate, the phase of new hardened cement, the phase of old hardened cement, and the phase of new and old interfacial transition zone (ITZ) can be judged by a computer code.

Damage Model of Materials
Components of RAC such as recycled coarse aggregate, new mortar, old mortar, new interfacial transition zone (new ITZ), and old interfacial transition zone (old ITZ) are basically quasi-brittle material, whose failure patterns are mainly brittle failure. In this paper, according to the characteristics of recycled aggregate concrete on meso-structure, the damage degradation of recycled aggregate concrete is described by the bilinear damage model, and the failure principal is the criterion of maximum tensile strain. Damage constitutive model is defined as̃= (1 − ) as shown in Figure 4, where the damage factor can be expressed as follows: where is the tensile strength of material, the residual tensile strength is defined as = , the residual strength coefficient ranges from 0 to 1, the residual strain is = 0 , is the residual strain coefficient, the ultimate strain is defined as = 0 , where is ultimate strain coefficient, and is principal tensile strain of element.

Numerical Examples
According to the test results achieved from the experiments, material parameters of recycled aggregate concrete are  Table 1.

Simulation on Uniaxial Compressive Strength of RAC.
Based on the uniaxial compression experiments, uniaxial compressive strengths for RAC specimens are investigated using the BFEM as shown in Figure 5. The dimension of compression specimen is 100 mm × 100 mm × 100 mm. The numbers of large aggregates (representative size is 17.5 mm), middle aggregates (representative size is 12.5 mm), and small aggregates (representative size is 7.5 mm) are 3, 9, and 37, respectively. In order to simplify the numerical calculation, the three-dimensional structure is schematized as a plane stress problem and the geometry of the analytical specimen is 100 mm × 100 mm. The random aggregate models of a RAC specimen are shown in Figure 6. Three specimens with different random aggregate distribution are simulated and the mean compressive strength of these specimens is obtained.
The uniaxial compressive stress-strain curve of recycled aggregate concrete is as shown in Figure 7. The result of BFEM on mesodamage analysis for RAC is consistent with the test results [20].
The propagation process of cracks of the RAC specimen by uniaxial compression is shown in Figure 8.

Simulation on Uniaxial Tensile Strength of RAC.
Based on the uniaxial tension experiments, uniaxial tensile strengths for RAC specimens are investigated using the BFEM as shown in Figure 9.
The dimension of tension specimen is 100 mm × 100 mm × 100 mm. The numbers of large aggregates (representative size is 17.5 mm), middle aggregates (representative size is 12.5 mm), and small aggregates (representative size is 7.5 mm) are 3, 9, and 37, respectively. In order to simplify the numerical calculation, the three-dimensional structure is schematized as a plane stress problem and the geometry of the analytical specimen is 100 mm × 100 mm. The random aggregate model of a RAC specimen is shown in Figure 6. Three specimens with different random aggregate distribution are simulated and the mean tensile strength of these specimens is obtained.
The uniaxial tensile stress-strain curve of recycled aggregate concrete is as shown in Figure 10.
The tensile strengths of the three specimens are 2.64 MPa, 2.44 MPa, and 2.63 MPa. The average uniaxial tensile strength of the specimen group is 2.57 MPa. The result of BFEM on meso-damage analysis for RAC is consistent with the test results [21].
The propagation process of cracks of the RAC specimen by uniaxial tension is shown in Figure 11.

Conclusions
(1) In this paper, a model of the base force element method (BFEM) is proposed for the damage analysis problem and is used to simulate the relations of mesostructure and macro-strength of recycled aggregate concrete (RAC). The characteristics of the BFEM are that the expression of the stiffness matrix K is a precise expression and it is not necessary to introduce the interpolation function and Gauss's integral for calculating the stiffness coefficient at a point. The numerical results show that this method can be used for damage analysis of the RAC.
(2) In order to simulate the meso-structure of recycled aggregate concrete material, recycled aggregate concrete is taken as five-phase composites consisting of natural coarse aggregate, new mortar, new interfacial transition zone (ITZ), old mortar, and old ITZ on meso-level in this paper. The random aggregate models are used for the numerical simulations of uniaxial compressive and tensile performances of recycled aggregate concrete, respectively. The results of the BFEM on meso-damage analysis for RAC are consistent with the test results.

Conflict of Interests
The authors declare that there is no conflict of interests.