Study on the Macromesoscopic Yield Criterion of Concrete Material

In mesoscopy, the concrete material is a kind of composite consisting of mortar matrix, aggregate, and the interface between them. And the shape of aggregate is assumed to be spherical and the mortar matrix is supposed to satisfy the Drucker-Prager yield criterion. The energy density support function is introduced to reflect the yield surface of mortar matrix. In order to solve the nonderivability on the yield boundary, the function approximation series has been constructed to substitute for the energy density support function. Finally, based on the function series, a macromesoscopic yield criterion of concrete material has been derived by the nonlinear homogenized technique.Through the macromesoscopic yield criterion established, the influence law of the bonding status of interface and the fraction of aggregate on the macrofriction coefficient is explored.


Introduction
In mesoscopy, the concrete can be taken as a multiphase composite material which is made of aggregate, mortar matrix, and the interface between them.And the macroscopic mechanical properties of concrete depend on the mechanical characteristic of the mesocomponents and their connection status.Compared with the well-rounded estimation theory of the effective modulus, the estimation of composite strength which is very sensitive to the mesostructures is still not mature enough.For all that, there are still many scholars developing this research.Huaxiang et al. [1] used the homogenization method of mesomechanics to estimate the ultimate loads applied and the macroyield criterion of composites is further obtained through the corresponding relationship between the macromechanical property and the meso structures.Barthelemy and Dormieux [2] studied the relationship of macro friction coefficient with the matrix friction coefficient and aggregates volume occupancy under the perfect interface case and the non-friction sliding interface case by using the meso-inclusion theory and obtained the macroconstitutive model which can reflect the mesoparameters.For the composite strength of cement matrix, by many components experiment, Maalej et al. [3] obtained the conclusion that the cement matrix occupy contributes the cohesive strength as proportional pattern and has almost no effect on the macrofriction coefficient and made explanation of mechanical mechanism for this experimental results.For the estimation of effective property of composite material considering the nonlinear characteristic of matrix, Willis [4,5] studied the effective property estimation problem of composite material in which the nonlinear matrix includes the rigid inclusion and void inclusion by using Hashin-Shtrikman variational structure method [6,7].Ponte Castañeda [7][8][9][10][11] studied the estimation of effective property of composite material in which the matrix is nonlinear as power form and the bonds between matrix and inclusions are supposed perfect by using his proposed variational structure method.Zhang and Xia [12] studied the nonlinear composite properties of concrete and explored the influence of interface connect status on the concrete effective properties considering the nonlinear soft characteristic of cement matrix by variational structure method proposed by Ponte Castañeda.

Advances in Condensed Matter Physics
In this paper, the aggregate component is supposed to never yield and its shape is assumed to be spherical.For the mortar matrix as friction type of material, the Drucker-Prager yield criterion is applied to describe its yield behavior.The macroscopic yield criterion can be established by introducing the potential function method and combining the estimation of the effective modulus.And the influence law of aggregate content and bond characteristic to the inner friction coefficient of concrete can be explored by the established macroscopic yield criterion.The potential function will use a kind of support function, the energy density support function.
The average Eshelby tensor which can reflect the connecting status of interface between the inclusion and the matrix including imperfect connecting status [13,14] is introduced to explore the influence law of interface connecting status on macroyield characteristic of concrete.
The energy density support function [14] has been used to reflect the yield surface of the mortar matrix successfully.And through this function, the influence of the air-entrained water reduce agent on concrete inner friction coefficient was explored subsequently [15].This paper will still adopt the energy density support function and its function approximation series to describe the yield surface of concrete, and the macromesoscopic yield criterion is expected to be established.

The Construction of the Function Approximation Series
The concrete sample can be taken as a representative volume element  which is a multiphase composite material made of cement mortar (matrix), aggregate, and the interface between them.For the mortar and interface phase, the Drucker-Prager yield criterion is applied as follows: where  1 = tr(), √ 2 = √(1/2) : ,  = tan /√9 + 12tan 2 , and ℎ = 3/√9 + 12tan 2 .Among them,  =  −  1 /3,  and  are the cohesion and the friction angle of the material, respectively.Energy density  is defined as follows: where  V = tr(),  =  −  V /3, and   = √(1/2) : .Since  is the convex domain, the energy density support function  can be taken to the supremum.The support function form can be written as () = sup{ :  |  ∈ }.For Drucker-Prager criterion, the  expression is as follows: When applying a constant strain  on the representative volume element boundary, the basic equation is as follows: where Ω  is the mortar domain, and Ω  is the aggregate domain,   is the energy density support function of the matrix.Let  =  V −   , and consider the nonderivability at  = 0 of  function; a derivable convex function series   () should be constructed.And when parameter  tends to be zero, the function series constructed should approach to  function.For Drucker-Prager yield function form, it can be written as follows: The solution of (4) can be obtained as follows: As long as when the parameter  tends to be zero, the function   () reaches positive infinity +∞ as presented in Figure 1, and then the function series   () also reaches positive infinity +∞.So, the constructed derivative convex function series   () meet the requirement of approaching to  function.Thus, the third subformula of (4) becomes as follows:

Nonlinear Homogenization Technique
The stress-strain relationship can be derived as follows by the potential function ( 6): where  =    ()/, 2 = −   ()/  , and  = ℎ −    ()  / , which are the bulk modulus, two times of shear modulus, and the spherical prestressed, respectively.Since  is negative and    () < 0 except for the stress status  = ℎ, ,  are positive and depend on  V and   .In addition, if  ̸ = ℎ, the equal formula of lim  → 0  = 0 can be obtained from the definition of the function   ().Since the strain fields both in the matrix domain and in the aggregates domain of the question (7) are nonuniform, ,  take on nonuniform distribution.Based on the classical homogenization method [16] which is only to deal with the heterogeneous material consisted of homogeneous phases, the real strain field is substituted by an equivalent alternative strain , and the nonuniform problem (7) can use the classical homogenization method to deal with as follows: where, Among them,  =  V −   .Then (7) becomes as follows: In order to ensure that the effective strain is not rougher compared with true strain fields, the first-order form is adopted to estimate the bulk strain and the second-order form to estimate deviator strain invariants as follows: where ⟨⋅⟩ is an averaged operator.The relationship between the average strain of inclusion and the applied strain at far field can be deduced by the following energy equivalent principle: The partial derivatives of  to  eq and  eq can be gained respectively, by formula (12) as follows: It can be further obtained as follows by the assumption of material isotropic characteristic: The Mori-Tanaka method [17] is adopted to estimate the average strain of the inclusions.This method can consider the interaction between inclusions by changing the far field strain  to the average strain of matrix ⟨⟩ Ω  which is equivalent to the action of average strain ⟨⟩ Ω  of matrix applied to far field when each inclusion is put embedded into the infinity matrix.So, where    is the strain localization tensor, and Here,   is the average Eshelby tensor. and  are softness and stiffness tensor of the matrix, respectively.Because of the application of the average Eshelby tensor in the strain localization tensor, the inclusion average strain includes the influence of relative displacement generated by the imperfect interface.So the formula (15) can be further trimmed as follows: And subsequently, the bulk average strain of inclusion is obtained as The mechanical property of all the aggregate is assumed to be the same, and the stiffness ratios are defined as   =  eq /  and   =  eq /  .The average Eshelby tensor expressions can be given as follows by [14]: where Among   = 3/(3 + 4),   = 6( + 2)/5(3 + 4) (for spherical inclusion).The determination of the coefficients  0 ,  0 ,   0 ,   0 ,   0 ,   0 , and   0 are related with the interface parameters.Assuming that the interface is of elastic and isotropic characteristic, the interface constitutive relation equation can be decomposed into two parts presented as follows when the interface embedding effect and the couple effect of constitutive relationship between the normal direction and tangential direction is not considered: where [⋅] = (out) − (in),   denotes the external normal direction of the interface,   the displacement at interface, and   the interface stress component.It can be seen from ( 18) that  and  denote the interface characterization behaviors.For example, ( = 0,  = 0) is corresponding to the perfect interface, ( ̸ = 0,  = 0) is corresponding to the slip in tangential direction and perfect in normal direction, and ( → ∞,  = 0) is corresponding to the free sliding status of interface.The linear interface condition is essentially equivalent to using a nonthickness linear spring layer to simulate the connection behavior of the imperfect interface.The details can be found in [14].The estimation formula of the effective bulk modulus and shear modulus for the isotropic case [17,18] can be easily derived by the formula (18).Consider where   is the volume occupancy of the th phase aggregate.

The Macroscopic Yield Criterion
Since the constitutive relation of the matrix and aggregates in ( 11) is linear, it is decomposed into two basic problems, one is that the representative volume element of concrete is subjected by  (1) = 0 and  eq (1) =  eq , respectively, the other is that the representative volume element of concrete is subjected by  (2) =  and  eq(2) = 0, respectively.Here, we use Σ  = tr(Σ)/3 and Σ V = √Σ  : Σ  to denote the macroscopic stress invariant.Among them, Σ  = Σ−Σ  .
Obviously the macroscopic stress generated by subproblem one is  eq ; For subproblem two, the macroscopic stress generated depended on the effective modulus of the representative volume element of concrete.Thus, the macrostressstrain relationship can be gained as follows after superposition: Here,  and  are given by formula (17) and  eq and  eq are given by formula (9b).Since lim  → 0  = 0, the following formula can be gained: It can be further gained as follows: lim {( 6 ( eq + 2 eq ) 5 (3 eq + 4 eq ) ) Now, we will derive the relationship between   and   .It can be analyzed from expression ( 14) that the solution of expression / eq and / eq is the key of gaining the relationship between   and   .Since the matrix modulus involved in   and   is constant, the two partial derivative functions can be solved as follows from formula (14).Consider where Substituting the formulas ( 16) and ( 24) into the second formula of the formula ( 13) and further simplification, the relation between   and   can be gained as follows: Substituting the formulas ( 17), (20), and (27) into formula (21), the macrobulk stress and macrodeviator stress can be gained as follows: Substituting the formulas ( 17) and (9b) into the formulas ( 27) and (28) respectively, and further simplification, the macrobulk stress and macrodeviator stress become as follows: The macro-micro constitutive relation of the concrete can be obtained as the follows by eliminating    () of formula (30) and formula (31): The above formula is further abbreviated as follows: where ) . ).It is worth to explain that the macroscopic friction coefficient could be of singularity because of the possibility that its denominator is zero.So it has to regulate the following expression to avoid the occurrence of singularity: where err is a small tolerant threshold.

Results Analysis
The relationship between the macroinner friction coefficient  and microstructure can be explored by the macromicroconstitutive relations (33).And the laws of macroscopic inner friction coefficient with aggregate occupancy, stiffness ratio of mortar matrix to aggregate, and softness of interface for different matrix inner friction coefficient are presented as Figures 2-6.It can be seen from Figures 2 and 3 that the interface connection status is a key factor obviously influencing the overall inner friction coefficient , and the effective inner friction coefficient even decreased with the increment of the imperfect interface occupancy.What is more, the effective inner friction coefficient increases more rapidly with the increment of aggregate occupancy for infinite aggregate stiffness case, as presented Figure 3 than that of common aggregate stiffness case, as shown Figure 2. When all of the interface connection statuses are imperfect, the overall inner friction coefficient decreases with the increment of aggregate occupancy, as presented Figures 2-4.And it can be seen from Figure 4 that the effective inner friction coefficient decreases with the increment of stiffness ratio of mortar matrix to aggregate.Compared with Figure 5 reflecting the influence of tangential softness  of interfaces on the macroinner friction coefficient, the influence of the normal softness  of interface on the macroinner friction coefficient is more violent, as presented Figure 6.All the changing trends presented in the Figures 2-6 agree well with the experience intuition.

Summary
In this paper, the concrete material is taken as a composite material consisting of aggregate, mortar matrix, and interface between them in the mesoscopic.Based on the assumption that the shape of aggregate is spherical, the aggregate phase is never yield and the mortar matrix satisfies Drucker-Prager yield criterion, and the macromesoscopic yield criterion of concrete material is derived by introducing the energy density supporting function to reflect the yield surface of the mortar matrix and the nonlinear homogenization technique.An approximation function series can be constructed to solve the nonderivable problem at the yield boundary.Finally, the influence laws of the interface connection status, aggregate content, and the rigidity ratio of matrix to aggregate on macroscopic inner friction coefficient are explored by the macromesoscopic yield criterion derived.
The results show that the influence laws gained agree well with the real case.Compared with the [2], which can only consider the estimation of the macroinner friction coefficient under the nonfriction interface case, this paper can not only deal with that of arbitrary interface connection status but also is not limited by the assumption of the aggregate rigid infinity.So this theory proposed takes a big step forward, but there is still some unestimated or estimation failure problem because of the singularity.

2 )Figure 2 :
Figure 2: The change curve of the macroscopic inner friction coefficient with aggregate occupancy for different matrix inner friction coefficient (  = 0.45).

Figure 3 :Figure 4 :
Figure 3: The change curve of the macroscopic inner friction coefficient with aggregate occupancy for different matrix inner friction coefficient (  = 0.0).

Figure 5 :Figure 6 :
Figure 5: The change curve of the macroscopic inner friction coefficient with aggregate occupancy for different tangential softness  of interfaces (  = 0.0).