Concrete Failure Modeling Based on Micromechanical Approach Subjected to Static Loading

In this paper, a micromechanical model based on the Mori-Tanaka method and the spring-layer model is developed to study the stress-strain behavior of concrete. The concrete is modeled as a two-phase composite. And the failure of concrete is categorized as mortar failure and interface failure. The research presents a method for estimating the modulus of concrete under its whole loading process. The proposed micromechanical model owns the good capabilities for predicting the entire response of concrete under uniaxial compression. It is suitable that tensile strain is as the criterion of concrete failure and the prediction of crack direction also fits with experimental phenomenon.


I. INTRODUCTION
ecently some researchers studied the elastic properties of concrete using the micromechanical model [1].The interface zone is usually considered as a new elastic phase.However, concrete contains many microcracks, especially in the interface zone, even before any loading is applied (Fig. 1).Some experiments have shown that the matrix just surrounding the aggregate is found to have quite different stiffness properties than that away from the aggregate [2].Here concrete is considered as a kind of two-phase composite (the mortar and coarse aggregates).They are bonded together by the interface zone.Softening of concrete is considered as the appearance and development of microcracks.

II. BASIC FORMULATION
In the Mori-Tanaka method, consider an infinitely extended mortar medium D containing many spherical inclusions (coarse aggregates) with an imperfectly bonded interface (Fig. 2).The elastic stiffness of the mortar and coarse aggregate are L 0 and L 1 .Homogeneous boundary condition in the external surface S of D is employed:  n (S) =  0 .n(1) where  0 is constant stress tensor, and n denotes the external normal to the external surface S. If the solid did not contain any inhomogeneity, the strain field would be  0 = L 0 -1 . 0(2) The stress and strain in mortar differ from  0 and  0 by  ~and  ~.  ~is the perturbed strain due to the presence of all the coarse aggregates.The average stress is The stress and strain of the coarse aggregates experience further perturbations from those of the surrounding mortar by pt  and pt  .The average stress in aggre- gate is here  * is the equivalent transformation strain.
There is an assumption that the two sides of interface always are connected with each other so there is only the displacement jump.The interfacial traction remains continous, while both the normal and the tangential displacements might experience a jump across the interface [3].The interface conditions is In which  T and  N denote the compliance in the tangential and the normal directions.[.] =(out)-(in), n i is the outward unit normal on the interface, and T i =  kj .nj ( ik -n i n k ) and N i = kj .nk .nj .ni represent the shear and the normal tractions. ij is the Kronecker .  T and  N should be positive.
For the coarse aggregates with imperfect interface, we have the relationship between the eigenstrain and perturbation strain in coarse aggregates is Here E ijkl S is Eshelby's solution for uniform eigenstrain problem of inclusion with perfectly bonded interface.For the spherical inclusion, it gives where where Here s i u is the displacement field caused by the interfacial sliding and normal separation.S ij  is the body average of S ij  inside the spherical inclusion: If we separate S ij  into its hydrostatic and deviatoric components, we can write (12) as Furthermore, for spherical inclusion, we have The perturbation stress can also be divided into two parts In this case, Eshelby's solution for spherical inclusion with perfect interface gives where  0 is the shear modulus of cement paste phase,  0 Poisson's ratio of cement paste phase, and  ij '* is the deviatoric part of  ij * .The stress  S inside the inclusion [4] is where a is the radius of the spherical inclusion (aggregate) defined by x i .xi < a 2 for which n i =x i /a.According to Zhong and Meguid's derivation, it is We solve the body average of  S , as According to Equation ( 16), we have * * ' ' where If  denotes the overall average stress tensor and c i (i = 0, 1) represent the volume fractions of cement paste and aggregate, separately, we have If there is not special statements below, hydrostatic and deviatoric components of the stress and strain tensors will be written as  and ', , and '.
When the geometry of coarse aggregates is considered as sphere, the average stresses in the mortar and the coarse aggregates then follow Eqs.( 3), ( 4), (24), and (27), as where i  (i = 0, 1) are the bulk moduli of mortar and coarse aggregates and  i (i = 0, 1) are the shear moduli of mortar and coarse aggregates.The strains are their forms by The body average of the strain can be defined as in which V is the volume of whole body [5].
According to Zhong and Meguid's solution, [u i ] can be written as : Substituting Eqs.(32), (33), (34), (35), and (37) into Eqs.(36), we have Thus we further leads to the effective bulk and shear moduli of the concrete as Due to the isotropy of concrete here,  and  known, the Young's modulus E is

III. RESULT AND DISCUSSION
Weibull statistical distribution function has been applied broadly in the field of damage mechanics and concrete failure analysis [5][6][7][8].Lambrigger pointed out Weibull function could correctly characterize the strength and failure of macroscopically homogeneous specimens [9].Here it is applied to evaluate the failure volume of mortar and interfaces.
In the framework of Mori-Tanaka method, the modulus of concrete is calculated by the procedure in Fig. 3 if only the effect of interfaces is considered.
We assume the volume ratio of aggregates, whose interfaces have been destroyed, conform to a Weibull distribution function.Its form is where c i is the volume ratio of interface failure,  is the effective strain of concrete.In the state of the pure compressive stress,  is equal to  3 .The '3' means the compressive stress direction.The  th is the strain threshold, in the state of pure compressive stress, it should be equal to the peak strain of concrete, which is corresponding to the strain value when the concrete reaches the ultimate compressive stress (about  th  0.002).If we consider that c i should be equal to 1 when  is equal to  th , we normalize Equation (43) and have The Weibull distribution can model the distribution of microscopic flaws in the material.The following equation is used to decide the failure volume ratio under the current principal tensile strain.
where  1 is the first principal tensile strain in mortar,  tensile is a strain threshold value (when  1 reach the value, the cracks appear in mortar),  u is the maximum tensile value (when  1 reach the value, the mortar is completely destroyed), m m is a shape index.
Here the main failure reason of concrete is considered as the development of microcracks in mortar and interface microcracks.Microcracks in mortar are considered as a series of aligned microcracks.Interface microcracks are considered as a kind of non-thickness spring layer.
In order to calculate the modulus of concrete, firstly, the modulus of mortar is computed.The principal tension causes the initial aligned microcracks arising if the principal tension strain reaches the critical value.These aligned microcracks should be perpendicular to the direction of the principal tension strain.These aligned microcracks are considered as a kind of materials whose modulus is zero.
For the two-phase composite, we have Equation ( 46) to follow to solve the overall modulus of composite according to Mori-Tanaka method [10].
Here subscript 0 represents matrix and subscript 1 is inclusion.
Microcracks in mortar are considered as a kind of aligned crack.They can be modeled as a sort of special constituent in composite: void.So we have Here c m is the volume ratio of cracks in mortar and 0 represents the material properties of mortar.If there is a series of aligned inclusion in a certain composite (Fig. 4), the corresponding Eshelby's tensor can be given as the following Equation ( 11): For the plane stress problem (Fig. 5), Equation (49) can be expressed like below.

   
When the principal tension reaches the critical value for cracking, we can calculate the current modulus for concrete.Using Equation (45) to get the current volume ratio of aligned microcracks for mortar, we can get the current modulus of mortar.
When the principal tension reaches the critical value for cracking, we can calculate the current modulus for concrete.Using Equation (45) to get the current volume ratio of aligned microcracks for mortar, we can get the current modulus of mortar.
After the modulus of mortar is calculated, the modulus of concrete can be solved.The volume ratio of interface failure for concrete is calculated by Equation (44).The modulus of concrete is shown as follows.

. Outline for calculation of concrete modulus
For the plane stress problem, the Eshelby tensor S can be expressed as the following equation.
For perfect interfaces, we can get the modulus of concrete for perfect interfaces: L p .And we can get the modulus of concrete for interfaces destroyed: L d .
For a certain load stage, the overall modulus of concrete is

IV. MODEL PARAMETER
Here the main parameters are the shape index: m, m m , and the threshold value of strain: ε u , ε th .These parameters usually can not be measured directly.Based on simulation for concrete, m and ε th , related to the interface failure and compression of concrete, are set to 3 and 0.008.m m and ε u are related to the mortar failure and tension of concrete and the basic assumption of concrete failure is tensile strain.Here ε u is defined as follows [12,13].
where f t is tensile strength for concrete, G F is fracture energy of concrete and l m is eigen-length for mortar.According to CEB-FIP Code (1993), we have the simple empirical formula relating G F (J/m 2 = N/m) to the conventional quality control parameter-namely the mean compressive strength of concrete f c ' (MPa) [7] as G F = α F (f' c ) 0.7 .The empirical coefficient α F depends on the maximum aggregate size g (Table 1).Normally, the compressive strength of concrete is given in the common experiment.If the proposed model is used, the tensile strength should be calculated by some empirical equations.There is an empirical formula, which has been suggested by ACI Committee 209 for computing the direct tensile strength of different weight concrete.For normal weight concrete, f' c , f t are expressed in MPa by the following equation: (55) Then the parameter l m should be defined.According to Bazant's random particle model, there is l m = β F .g and roughly β F is considered to be equal to 1/2.The expression of ε u can be written like the following equation.Step 1: Decide the value of the two important parameters Step 2: Compute the corresponding parameters further Step 3: Get the modulus of concrete  If these parameter values of the certain concrete are given in experiments, the proposed model can predict the stress-strain behavior very well.Studies in this research show the m m is related to the strength of mortar.The stronger the strength of mortar, the larger the value of m m is ε u is related to the ductile properties of concrete.The smaller the strength of mortar/concrete, the better the ductile properties of concrete or mortar is so ε u is larger.When we do not know the exact value of m m and ε u , we should use Equations ( 57) and (58) to calculate them.Here ε u is revised as ε r .For the different strength level, the following equations are recommended to calculate the ultimate tensile strain and m m.

V. DAMAGE EFFECT
Here it is worth nothing that the lateral deformation increases significantly at higher stress level after the peak loading point.And the large cracks appear and crack growth becomes unstable.Therefore, the change from volume decrease to sudden volume increases leads to find a proper way for the description of lateral strain during the descending branch.Here the following equation is assumed to modify the behavior of lateral strain after the peak loading point under uniaxial compression.There are some suitable experimental data available [14].The concrete and mortar are tested.The mortar and concrete were pan-mixed in the laboratory and were cast in steel moulds (100 mm in diameter and 200 mm in height).The mix design is shown in Table 2.The properties of mortar are shown in Table 3.
In Fig. 6, it is shown that the proposed model can predict the mortar behavior very well, especially for the ascending part and the peak point.It is also shown that the aligned microcrack model can properly evaluate the failure of mortar.
The proposed model is further used to predict the compressive behavior of concrete.The volume ratio of mortar and coarse aggregates (Table 4) can be calculated if their densities are known.Here they are assumed.All related data are shown in Table 4.These responses of the predicted compressive behavior are shown in Fig. 7-Fig.12.These results also show that the comparison for lateral strain does not agree very well.The main reason is that the behavior of concrete in the tensile direction is considered in the model to be that of a continuum material no matter how serious the cracks are.
In order to explore the model capabilities of predicting the transition in behavior from low to high strength concrete, the experimental data for different strength level of concrete are collected in Tables 5 and 6.In comparison with that obtained experimentally, these predictions are very accurate shown in Fig. 13 and Fig.

TABLE 1 .
COEFFICIENT α F WITH MAXIMUM AGGREGATE SIZE g

TABLE 2 .
SPECIFIC WEIGHT RATIO FOR MORTAR AND CONCRETE

TABLE 3 .
ELASTIC PROPERTIES OF G40 MORTAR

TABLE 5 .
MATERIAL PROPERTIES AND COEFFICIENTS FOR THE PROPOSED MODEL(NEVILLE, 1996)

TABLE 6 .
MATERIAL PROPERTIES AND COEFFICIENTS FOR THE PROPOSED MODEL (DAHL, 1992) American Society of Mechanical Engineers, Applied Mechanics Division, AMD, Vol.205, pp.21-34VI.CONCLUSIONThe proposed micromechanical model owns the good capabilities for predicting the entire response of concrete under uniaxial compression.It is suitable that tensile strain is as the criterion of concrete failure and the prediction of crack direction also fits with experimental phenomenon.And Weibull distribution function can describe the behavior of crack development for mortar and interface.