Study on the Macromesoscopic Yield Criterion of Concrete Material

Abstract

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.

1. 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 [7–11] 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.

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.

2. The Construction of the Function Approximation Series

The concrete sample can be taken as a representative volume element 𝑉 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: 𝜎∈𝐹⟺𝐹(𝜎)=𝑡𝐼₁+𝐽₂−ℎ≤0, (1) where 𝐼₁=tr(𝜎), 𝐽₂=(1/2)𝑠∶𝑠, 𝑡=tan𝜑/9+12tan²𝜑, and ℎ=3𝑐/9+12tan²𝜑. Among them, 𝑠=𝜎−𝐼₁/3, 𝑐 and 𝜑 are the cohesion and the friction angle of the material, respectively.

Energy density 𝑈 is defined as follows: =𝐼𝑈=𝜎∶𝜀₁𝐽₂𝜀_𝑣𝜀_𝑑=𝐼₁𝜀_𝑣+𝐽₂𝜀_𝑑=𝐼₁𝜀_𝑣+ℎ−𝑡𝐼₁𝜀_𝑑, (2) where 𝜀_𝑣=tr(𝜀), 𝜒=𝜀−𝜀_𝑣𝐼/3, and 𝜀_𝑑=(1/2)𝜒∶𝜒. Since 𝐹 is the convex domain, the energy density support function 𝑈 can be taken to the supremum. The support function form can be written as 𝜋(𝜀)=sup{𝜎∶𝜀∣𝜎∈𝐹}. For Drucker-Prager criterion, the 𝜋 expression is as follows: 𝜋(𝜀)=+∞,𝜀_𝑣<𝑡𝜀_𝑑,𝜋(𝜀)=ℎ𝜀_𝑑,𝜀_𝑣≥𝑡𝜀_𝑑. (3)

When applying a constant strain 𝐸 on the representative volume element boundary, the basic equation is as follows: (Ω),1∇⋅𝜎=0𝜀=2(∇⊗𝑢+𝑢⊗∇)(Ω),𝜎=𝜕𝜋^𝑆Ω𝜕𝜀^𝑆,𝜎=𝐷_𝑔∶𝜀Ω^𝑔,𝑢=𝐸⋅𝑥(𝜕Ω), (4) where Ω^𝑆 is the mortar domain, and Ω^𝑔 is the aggregate domain, 𝜋^𝑆 is the energy density support function of the matrix. Let 𝛿=𝜀_𝑣−𝑡𝜀_𝑑, and consider the nonderivability at 𝛿=0 of 𝜋 function; a derivable convex function series 𝜓_𝑎(𝜀) should be constructed. And when parameter 𝑎 tends to be zero, the function series constructed should approach to 𝜋 function. For Drucker-Prager yield function form, it can be written as follows: 𝑡𝜕𝜓_𝑎𝜕𝜀_𝑣+𝜕𝜓_𝑎𝜕𝜀_𝑑−ℎ=0. (5) The solution of (4) can be obtained as follows: 𝜓_𝑎𝜀_𝑣,𝜀_𝑑=𝑓_𝑎(𝛿)+ℎ𝜀_𝑑. (6) As long as when the parameter 𝑎 tends to be zero, the function 𝑓_𝑎(𝛿) reaches positive infinity +∞ as presented in Figure 1, and then the function series 𝜓_𝑎(𝜀) also reaches positive infinity +∞. So, the constructed derivative convex function series 𝜓_𝑎(𝜀) meet the requirement of approaching to 𝜋 function. Thus, the third subformula of (4) becomes as follows: 𝜎=𝜕𝜓^𝑆_𝑎Ω𝜕𝜀^𝑆. (7)

Figure 1 

Function 𝑓𝑎(𝛿).

3. Nonlinear Homogenization Technique

The stress-strain relationship can be derived as follows by the potential function (6): 𝜎=𝜕𝜓_𝑎𝜕𝜀_𝑣+𝜕𝜓_𝑎𝜕𝜀_𝑑=𝑓^′_𝑎(𝛿)+𝑓^′_𝑎(𝛿)(−𝑡)+ℎ^𝛿≠0𝑓−−−−→=^′𝑎(𝛿)𝛿𝜀_𝑣+𝑓^′_𝑎(𝛿)(−𝑡)𝜀_𝑑𝜀_𝑑𝑓+ℎ−^′𝑎(𝛿)𝑡𝜀_𝑑𝛿=𝐾𝜀_𝑣+2𝐺𝜀_𝑑+𝐻, (8) where 𝐾=𝑓^′_𝑎(𝛿)/𝛿, 2𝐺=−𝑓^′_𝑎(𝛿)𝑡/𝜀_𝑑, and 𝐻=ℎ−𝑓^′_𝑎(𝛿)𝑡𝜀_𝑑/𝛿, which are the bulk modulus, two times of shear modulus, and the spherical prestressed, respectively. Since 𝛿 is negative and 𝑓^′_𝑎(𝛿)<0 except for the stress status 𝜎=ℎ, 𝐾, 𝐺 are positive and depend on 𝜀_𝑣 and 𝜀_𝑑. In addition, if 𝜎≠ℎ, the equal formula of lim_𝑎→0𝛿=0 can be obtained from the definition of the function 𝑓_𝑎(𝛿). Since the strain fields both in the matrix domain and in the aggregates domain of the question (7) are nonuniform, 𝐾, 𝐺 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 𝜀, and the non-uniform problem (7) can use the classical homogenization method to deal with as follows:𝜎=𝐾^eq𝜀_𝑣+2𝐺^eq𝜀_𝑑+𝐻^eq, (9a) where, 𝐾^eq=𝑓^′_𝑎𝛿𝛿,2𝐺^eq𝑓=−^′𝑎𝛿𝑡𝜀_𝑑,𝐻^eq𝑓=ℎ−^′𝑎𝛿𝑡𝜀_𝑑𝛿. (9b)Among them, 𝛿=𝜀_𝑣−𝑡𝜀_𝑑. Then (7) becomes as follows: 𝜎=𝐾^eq𝜀_𝑣+2𝐺^eq𝜀_𝑑+𝐻^eqΩ^𝑆. (10)

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: 𝜀_𝑣=⟨𝜀_𝑣⟩_Ω,𝜀_𝑑=⟨𝜀_𝑑∶𝜀_𝑑⟩_Ω^𝑆, (11) 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: 1𝑊=2𝐸∶1𝐿∶𝐸=2𝜀∶𝐿∶𝜀_Ω. (12)

The partial derivatives of 𝑊 to 𝐾^eq and 𝐺^eq can be gained respectively, by formula (12) as follows: 121−_𝑖𝑐_𝑖𝜀²_𝑣=12𝜕𝐸∶𝐿𝜕𝐾^eq∶𝐸,1−_𝑖𝑐_𝑖𝜀²_𝑑=12𝜕𝐸∶𝐿𝜕𝐺^eq∶𝐸. (13) It can be further obtained as follows by the assumption of material isotropic characteristic: 121−_𝑖𝑐_𝑖𝜀²_𝑣=12𝜕𝐾𝜕𝐾^eq𝐸²_𝑣+𝜕𝐺𝜕𝐾^eq𝐸²_𝑑,1−_𝑖𝑐_𝑖𝜀²_𝑑=12𝜕𝐾𝜕𝐺^eq𝐸²_𝑣+𝜕𝐺𝜕𝐺^eq𝐸²_𝑑. (14) 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 𝐸 to the average strain of matrix ⟨𝜀⟩_Ω^𝑆 which is equivalent to the action of average strain ⟨𝜀⟩_Ω^𝑆 of matrix applied to far field when each inclusion is put embedded into the infinity matrix. So, 𝐸=1−_𝑖𝑐_𝑖⟨𝜀⟩_Ω^𝑆+_𝑖𝑐_𝑖⟨𝜀⟩_Ω^𝑔=1−_𝑖𝑐_𝑖⟨𝜀⟩_Ω^𝑆+_𝑖𝑐_𝑖𝐴^𝑖_𝑐⟨𝜀⟩_Ω^𝑆, (15) where 𝐴^𝑖_𝑐 is the strain localization tensor, and 𝐴^𝑖_𝑐={𝐼+𝑆^𝑖∶𝑀∶(𝐿^𝑖−𝐿)}⁻¹. Here, 𝑆^𝑖 is the average Eshelby tensor. 𝑀 and 𝐿 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: ⟨𝜀⟩_Ω^𝑆=𝐸1−∑_𝑖𝑐_𝑖𝐼+∑_𝑖𝑐_𝑖𝐴^𝑖_𝑐. (16) And subsequently, the bulk average strain of inclusion is obtained as 𝜀_𝑣Ω𝑆=𝜀_𝑣=𝐸_𝑣1−∑_𝑖𝑐_𝑖𝑠^𝑖₁1−𝜆_𝐾/𝜆_𝐾+𝑠^𝑖₁1−𝜆_𝐾. (17) The mechanical property of all the aggregate is assumed to be the same, and the stiffness ratios are defined as 𝜆_𝐾=𝐾^eq/𝐾^𝑔 and 𝜆_𝐺=𝐺^eq/𝐺^𝑔. The average Eshelby tensor expressions can be given as follows by [14]: 𝑆=𝑠₁,𝑠₂, (18) where 𝑠₁=𝛼^𝐸(𝑘₀+1)−(𝛼^𝐸−1)𝜆₀, 𝑠₂=𝛽^𝐸(1+𝑃₀+(2/5)𝑄₀+(3/5)𝑁₀)−(𝛽^𝐸−1)(𝐴₀+(2/5)𝐵₀).

Among 𝛼^𝐸=3𝐾/(3𝐾+4𝐺), 𝛽^𝐸=6(𝐾+2𝐺)/5(3𝐾+4𝐺) (for spherical inclusion). The determination of the coefficients 𝑘₀, 𝜆₀, 𝑃^𝑖₀, 𝑄^𝑖₀, 𝑁^𝑖₀, 𝐴^𝑖₀, and 𝐵^𝑖₀ 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: 𝑢_𝑗𝛿_𝑖𝑗−𝑛_𝑖𝑛_𝑗=𝛼𝜎_𝑘𝑗𝑛_𝑗𝛿_𝑖𝑘−𝑛_𝑖𝑛_𝑘,𝑢_𝑗𝑛_𝑖𝑛_𝑗=𝛽𝜎_𝑘𝑗𝑛_𝑘𝑛_𝑗𝑛_𝑖, (19) where [⋅]=(out)−(in), 𝑛_𝑖 denotes the external normal direction of the interface, 𝑢_𝑗 the displacement at interface, and 𝜎_𝑖𝑗 the interface stress component. It can be seen from (18) that 𝛼 and 𝛽 denote the interface characterization behaviors. For example, (𝛼=0,𝛽=0) is corresponding to the perfect interface, (𝛼≠0,𝛽=0) is corresponding to the slip in tangential direction and perfect in normal direction, and (𝛼→∞,𝛽=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 𝐾=𝐾^eq1+_𝑖𝑐_𝑖1−𝑠^𝑖₁1−𝜆_𝐾𝜆_𝐾+𝑠^𝑖₁1−𝜆_𝐾×1−_𝑖𝑐_𝑖𝑠^𝑖₁1−𝜆_𝐾𝜆_𝐾+𝑠^𝑖₁1−𝜆_𝐾⁻¹,𝐺=𝐺^eq1+_𝑖𝑐_𝑖1−𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺×1−_𝑖𝑐_𝑖𝑠^𝑖₂(1−𝜆_𝐺)𝜆_𝐺+𝑠^𝑖₂(1−𝜆_𝐺)⁻¹, (20) where 𝑐_𝑖 is the volume occupancy of the 𝑖th phase aggregate.

4. 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 𝐸⁽¹⁾=0 and 𝐻^eq(1)=𝐻^eq, respectively, the other is that the representative volume element of concrete is subjected by 𝐸⁽²⁾=𝐸 and 𝐻^eq(2)=0, respectively.

Here, we use Σ_𝑚=tr(Σ)/3 and Σ_𝑣=Σ_𝑑∶Σ_𝑑 to denote the macroscopic stress invariant. Among them, Σ_𝑑=Σ−Σ_𝑚𝐼.

Obviously the macroscopic stress generated by subproblem one is 𝐻^eq; For subproblem two, the macroscopic stress generated depended on the effective modulus of the representative volume element of concrete. Thus, the macrostress-strain relationship can be gained as follows after superposition: Σ_𝑚=𝐾𝐸_𝑣+𝐻^eq,Σ_𝑑=2𝐺𝐸_𝑑. (21) Here, 𝐾 and 𝐺 are given by formula (17) and 𝐾^eq and 𝐺^eq are given by formula (9b). Since lim_𝑎→0𝛿=0, the following formula can be gained: lim_𝑎→0𝐺^eq𝐾^eq=lim_𝑎→0−𝑡𝛿2𝜀_𝑑=0. (22) It can be further gained as follows: lim_𝑎→0𝑠₁=lim_𝑎→0𝛼^𝐸𝑘₀+1−𝛼^𝐸−1𝜆₀=lim_𝑎→03𝐾^eq3𝐾^eq+4𝐺^eq𝑘₀+1−𝜆₀+𝜆₀≈𝑘₀+1lim_𝑎→0𝑠₂=lim_𝑎→0𝛽^𝐸1+𝑃₀+25𝑄₀+35𝑁₀−𝛽−1𝐴₀+25𝐵₀=lim_𝑎→06𝐾^eq+2𝐺^eq5(3𝐾^eq+4𝐺^eq)×1+𝑃₀+25𝑄₀+35𝑁₀−𝐴₀−25𝐵₀+𝐴₀+25𝐵₀≈251+𝑃₀+25𝑄₀+35𝑁₀+35𝐴₀+25𝐵₀. (23)(24) Now, we will derive the relationship between 𝜀_𝑑 and 𝐸_𝑑. It can be analyzed from expression (14) that the solution of expression 𝜕𝐾/𝜕𝐺^eq and 𝜕𝐺/𝜕𝐺^eq is the key of gaining the relationship between 𝜀_𝑑 and 𝐸_𝑑. Since the matrix modulus involved in 𝜆_𝐾 and 𝜆_𝐺 is constant, the two partial derivative functions can be solved as follows from formula (14). Consider 𝜕𝐾𝜕𝐺^eq𝜕≈0𝐺𝜕𝐺^eq≈1+_𝑖𝑐_𝑖1−𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺×1−_𝑖𝑐_𝑖𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺⁻¹, (25) where 𝑠^𝑖₁=𝑘^𝑖₀𝑠+1,_𝑖2=251+𝑃^𝑖₀+25𝑄^𝑖₀+35𝑁^𝑖₀+35𝐴^𝑖₀+25𝐵^𝑖₀. (26) Substituting the formulas (16) and (24) into the second formula of the formula (13) and further simplification, the relation between 𝜀_𝑑 and 𝐸_𝑑 can be gained as follows: 𝜀_𝑑=𝐸_𝑑×1+_𝑖𝑐_𝑖1−𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺×1−_𝑖𝑐_𝑖𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺⁻¹×1−_𝑖𝑐_𝑖⁻¹12. (27) Substituting the formulas (17), (20), and (27) into formula (21), the macrobulk stress and macrodeviator stress can be gained as follows: Σ_𝑚=𝐾𝐸_𝑣+𝐻^eq=𝐾^eq𝐸_𝑣×1+_𝑖𝑐_𝑖1−𝑠^𝑖₁1−𝜆_𝐾𝜆_𝐾+𝑠^𝑖₁1−𝜆_𝐾×1−_𝑖𝑐_𝑖𝑠^𝑖₁1−𝜆_𝐾𝜆_𝐾+𝑠^𝑖₁1−𝜆_𝐾⁻¹+𝐻^eq=𝐾^eq1+_𝑖𝑐_𝑖1−𝑠^𝑖₁1−𝜆_𝐾𝜆_𝐾+𝑠^𝑖₁1−𝜆_𝐾𝜀_𝑣+ℎ−𝐻^eq,Σ_𝑑=2𝐺𝐸_𝑑=2𝐺^eq𝐸_𝑑×1+_𝑖𝑐_𝑖1−𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺×1−_𝑖𝑐_𝑖𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺⁻¹. (28)(29) 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: Σ_𝑚−ℎ≈𝑓^′_𝑎𝛿21−3×_𝑖𝑐_𝑖1−𝜆₀1−𝜆_𝐾𝜆_𝐾+𝑘^𝑖₀+1−2𝜆^𝑖₀1−𝜆_𝐾𝛼²,Σ_𝑑=−𝑓^′_𝑎𝛿𝑡×1+_𝑖𝑐_𝑖1−𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺×1−_𝑖𝑐_𝑖×1−_𝑖𝑐_𝑖𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺^−11/2. (30)(31) The macro-micro constitutive relation of the concrete can be obtained as the follows by eliminating 𝑓^′_𝑎(𝛿) of formula (30) and formula (31): Σ_𝑑=−Σ_𝑚𝑡×−ℎ1+_𝑖𝑐_𝑖1−𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺×1−_𝑖𝑐_𝑖×1−_𝑖𝑐_𝑖𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺^−11/2×21−3_𝑖𝑐_𝑖1−𝜆₀1−𝜆_𝐾𝜆_𝐾+𝑘^𝑖₀+1−2𝜆^𝑖₀1−𝜆_𝐾𝑡²⁻¹. (32) The above formula is further abbreviated as follows: Σ_𝑑=−𝑇Σ_𝑚−ℎ, (33) where 𝑇=1+_𝑖𝑐_𝑖1−𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺×1−_𝑖𝑐_𝑖×1−_𝑖𝑐_𝑖𝑠^𝑖₂1−𝜆_𝐺𝜆_𝐺+𝑠^𝑖₂1−𝜆_𝐺^−11/2×21−3_𝑖𝑐_𝑖(1−𝜆₀)(1−𝜆_𝐾)𝜆_𝐾+(𝑘^𝑖₀+1−2𝜆^𝑖₀)(1−𝜆_𝐾)⁻¹𝑡. (34) Among them, 𝑠^𝑖₁=𝑘^𝑖₀+1, 𝑠^𝑖₂=(2/5)(1+𝑃^𝑖₀+(2/5)𝑄^𝑖₀+(3/5)𝑁^𝑖₀)+(3/5)(𝐴^𝑖₀+(2/5)𝐵^𝑖₀).

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: 21−3_𝑖𝑐_𝑖1−𝜆₀1−𝜆_𝐾𝜆_𝐾+𝑘^𝑖₀+1−2𝜆^𝑖₀1−𝜆_𝐾<err, (35) where err is a small tolerant threshold.

5. Results Analysis

The relationship between the macroinner friction coefficient 𝑇 and microstructure can be explored by the macro-microconstitutive 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 𝑇, 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 𝛼 of interfaces on the macroinner friction coefficient, the influence of the normal softness 𝛽 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.

Figure 2 

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

Figure 3 

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

Figure 4 

The change curve of the macroscopic inner friction coefficient with aggregate stiffness ratio for different matrix inner friction coefficient.

Figure 5 

The change curve of the macroscopic inner friction coefficient with aggregate occupancy for different tangential softness 𝛼 of interfaces (𝜆𝐾=0.0).

Figure 6 

The change curve of the macroscopic inner friction coefficient with aggregate occupancy for different normal softness 𝛽 of interface (𝜆𝐾=0.0).

6. 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.

Conflict of Interests

The authors declared that they have no conflict of interests to this work.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (nos. 51179064, 11372099, 11132003, and 51108231) and the state Key Laboratory Open Foundation of Hydrology-Water Resources and Hydraulic Engineering at Hohai University (no. 2011490911). Their financial support is gratefully acknowledged.