# Study on the Macromesoscopic Yield Criterion of Concrete Material

Authors: Xia XiaoZhou; Zhang Qing; Jiang Qun; Xiao JianQiang
Journal: Advances in Condensed Matter Physics
 (2013)

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## 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.


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## Body

## 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 elementV 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:
(1)σ∈F⟺F(σ)=tI1+J2-h≤0,
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 densityU is defined as follows:
(2)U=σ:ε=[I1J2][εvεd]=I1εv+J2εd=I1εv+(h-tI1)εd,
where εv=tr(ε), χ=ε-εvI/3, and εd=(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:
(3)π(ε)=+∞,εv<tεd,π(ε)=hεd,εv≥tεd.When applying a constant strainE on the representative volume element boundary, the basic equation is as follows:
(4)∇·σ=0(Ω),ε=12(∇⊗u+u⊗∇)(Ω),σ=∂πS∂ε(ΩS),σ=Dg:ε(Ωg),u=E·x(∂Ω),
where ΩS is the mortar domain, and Ωg is the aggregate domain, πS is the energy density support function of the matrix. Let δ=εv-tεd, and consider the nonderivability at δ=0 of π function; a derivable convex function series ψa(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:
(5)t∂ψa∂εv+∂ψa∂εd-h=0.
The solution of (4) can be obtained as follows:
(6)ψa(εv,εd)=fa(δ)+hεd.
As long as when the parameter a tends to be zero, the function fa(δ) reaches positive infinity +∞ as presented in Figure 1, and then the function series ψa(ε) also reaches positive infinity +∞. So, the constructed derivative convex function series ψa(ε) meet the requirement of approaching to π function. Thus, the third subformula of (4) becomes as follows:
(7)σ=∂ψaS∂ε(ΩS).Figure 1
Functionfa(δ).


## 3. Nonlinear Homogenization Technique

The stress-strain relationship can be derived as follows by the potential function (6):
(8)σ=∂ψa∂εv+∂ψa∂εd→δ≠0=fa′(δ)+fa′(δ)(-t)+h→δ≠0=fa′(δ)δεv+fa′(δ)(-t)εdεd+h-fa′(δ)tεdδ→δ≠0=Kεv+2Gεd+H,
where K=fa′(δ)/δ, 2G=-fa′(δ)t/εd, and H=h-fa′(δ)tεd/δ, which are the bulk modulus, two times of shear modulus, and the spherical prestressed, respectively. Since δ is negative and fa′(δ)<0 except for the stress status σ=h, K, G are positive and depend on εv and εd. In addition, if σ≠h, the equal formula of lima→0δ=0 can be obtained from the definition of the function fa(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:(9a)σ=Keqε-v+2Geqε-d+Heq,
where,
(9b)Keq=fa′(δ-)δ-,2Geq=-fa′(δ-)tε-d,Heq=h-fa′(δ-)tε-dδ-.Among them, δ-=ε-v-tε-d. Then (7) becomes as follows:
(10)σ=Keqε-v+2Geqε-d+Heq(ΩS).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:(11)ε-v=〈εv〉Ω,ε-d=〈εd:εd〉ΩS,
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:
(12)W=12E:L-:E=〈12ε-:L:ε-〉Ω.The partial derivatives ofW to Keq and Geq can be gained respectively, by formula (12) as follows:
(13)12(1-∑ici)ε-v2=12E:∂L-∂Keq:E,(1-∑ici)ε-d2=12E:∂L-∂Geq:E.
It can be further obtained as follows by the assumption of material isotropic characteristic:
(14)12(1-∑ici)ε-v2=12∂K-∂KeqEv2+∂G-∂KeqEd2,(1-∑ici)ε-d2=12∂K-∂GeqEv2+∂G-∂GeqEd2.
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〉ΩS which is equivalent to the action of average strain 〈ε〉ΩS of matrix applied to far field when each inclusion is put embedded into the infinity matrix. So,
(15)E=(1-∑ici)〈ε〉ΩS+∑ici〈ε〉Ωg=(1-∑ici)〈ε〉ΩS+∑iciAci〈ε〉ΩS,
where Aci is the strain localization tensor, and Aci={I+S-i:M:(Li-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:
(16)〈ε〉ΩS=E(1-∑ici)I+∑iciAci.
And subsequently, the bulk average strain of inclusion is obtained as
(17)〈εv〉ΩS=ε-v=Ev1-∑i(cis1i(1-λK)/(λK+s1i(1-λK))).
The mechanical property of all the aggregate is assumed to be the same, and the stiffness ratios are defined as λK=Keq/Kg and λG=Geq/Gg. The average Eshelby tensor expressions can be given as follows by [14]:
(18)S-=(s1,s2),
where s1=αE(k0+1)-(αE-1)λ0, s2=βE(1+P0+(2/5)Q0+(3/5)N0)-(βE-1)(A0+(2/5)B0).AmongαE=3K/(3K+4G), βE=6(K+2G)/5(3K+4G) (for spherical inclusion). The determination of the coefficients k0, λ0, P0i, Q0i, N0i, A0i, and B0i 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:
(19)[uj](δij-ninj)=ασkjnj(δik-nink),[uj]ninj=βσkjnknjni,
where [·]=(out)-(in), ni denotes the external normal direction of the interface, uj the displacement at interface, and σij 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
(20)K-=Keq[1+∑ici(1-s1i)(1-λK)λK+s1i(1-λK)]×[1-∑icis1i(1-λK)λK+s1i(1-λK)]-1,G-=Geq[1+∑ici(1-s2i)(1-λG)λG+s2i(1-λG)]×[1-∑icis2i(1-λG)λG+s2i(1-λG)]-1,
where ci 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 Heq(1)=Heq, respectively, the other is that the representative volume element of concrete is subjected by E(2)=E and Heq(2)=0, respectively.Here, we useΣm=tr(Σ)/3 and Σv=Σd:Σd to denote the macroscopic stress invariant. Among them, Σd=Σ-ΣmI.Obviously the macroscopic stress generated by subproblem one isHeq; 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:
(21)Σm=K-Ev+Heq,Σd=2G-Ed.
Here, K- and G- are given by formula (17) and Keq and Geq are given by formula (9b). Since lima→0δ-=0, the following formula can be gained:
(22)lima→0GeqKeq=lima→0-tδ-2ε-d=0.
It can be further gained as follows:
(23)lima→0s1=lima→0{αE(k0+1)-(αE-1)λ0}=lima→0{(3Keq3Keq+4Geq)(k0+1-λ0)+λ0}≈k0+1(24)lima→0s2=lima→0{βE(1+P0+25Q0+35N0)-(β-1)(A0+25B0)}=lima→0{(6(Keq+2Geq)5(3Keq+4Geq))×(1+P0+25Q0+35N0-A0-25B0)+A0+25B0(6(Keq+2Geq)5(3Keq+4Geq))}≈25(1+P0+25Q0+35N0)+35(A0+25B0).
Now, we will derive the relationship between ε-d and Ed. It can be analyzed from expression (14) that the solution of expression ∂K-/∂Geq and ∂G-/∂Geq is the key of gaining the relationship between ε-d and Ed. Since the matrix modulus involved in λK and λG is constant, the two partial derivative functions can be solved as follows from formula (14). Consider
(25)∂K-∂Geq≈0∂G-∂Geq≈[1+∑ici(1-s2i)(1-λG)λG+s2i(1-λG)]×[1-∑icis2i(1-λG)λG+s2i(1-λG)]-1,
where
(26)s1i=k0i+1,s2i=25(1+P0i+25Q0i+35N0i)+35(A0i+25B0i).
Substituting the formulas (16) and (24) into the second formula of the formula (13) and further simplification, the relation between ε-d and Ed can be gained as follows:
(27)ε-d=Ed×((1-∑ici)-1[1+∑ici(1-s2i)(1-λG)λG+s2i(1-λG)]×[1-∑icis2i(1-λG)λG+s2i(1-λG)]-1×(1-∑ici)-1)12.
Substituting the formulas (17), (20), and (27) into formula (21), the macrobulk stress and macrodeviator stress can be gained as follows:
(28)Σm=K-Ev+Heq=KeqEv×[1+∑ici(1-s1i)(1-λK)λK+s1i(1-λK)]×[1-∑icis1i(1-λK)λK+s1i(1-λK)]-1+Heq=Keq[1+∑ici(1-s1i)(1-λK)λK+s1i(1-λK)]ε-v+h-Heq,(29)Σd=2G-Ed=2GeqEd×[1+∑ici(1-s2i)(1-λG)λG+s2i(1-λG)]×[1-∑icis2i(1-λG)λG+s2i(1-λG)]-1.
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:
(30)Σm-h≈fa′(δ-)(∑ici(1-λ0)(1-λK)λK+(k0i+1-2λ0i)(1-λK)1-23×∑ici(1-λ0)(1-λK)λK+(k0i+1-2λ0i)(1-λK)α2),(31)Σd=-fa′(δ-)t×([1-∑icis2i(1-λG)λG+s2i(1-λG)]-1[1+∑ici(1-s2i)(1-λG)λG+s2i(1-λG)]×(1-∑ici)×[1-∑icis2i(1-λG)λG+s2i(1-λG)]-1)1/2.
The macro-micro constitutive relation of the concrete can be obtained as the follows by eliminating fa′(δ-) of formula (30) and formula (31):
(32)Σd=-(Σm-h)t×([1-∑icis2i(1-λG)λG+s2i(1-λG)]-1[1+∑ici(1-s2i)(1-λG)λG+s2i(1-λG)]×(1-∑ici)×[1-∑icis2i(1-λG)λG+s2i(1-λG)]-1)1/2×(1-23∑ici(1-λ0)(1-λK)λK+(k0i+1-2λ0i)(1-λK)t2)-1.
The above formula is further abbreviated as follows:
(33)Σd=-T(Σm-h),
where
(34)T=(×(1-23∑ici(1-λ0)(1-λK)λK+(k0i+1-2λ0i)(1-λK))-1([1-∑icis2i(1-λG)λG+s2i(1-λG)]-1[1+∑ici(1-s2i)(1-λG)λG+s2i(1-λG)]×(1-∑ici)×[1-∑icis2i(1-λG)λG+s2i(1-λG)]-1)1/2×(1-23∑ici(1-λ0)(1-λK)λK+(k0i+1-2λ0i)(1-λK))-1)t.
Among them, s1i=k0i+1, s2i=(2/5)(1+P0i+(2/5)Q0i+(3/5)N0i)+(3/5)(A0i+(2/5)B0i).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:(35)|1-23∑ici(1-λ0)(1-λK)λK+(k0i+1-2λ0i)(1-λK)|<err,
where err is a small tolerant threshold.


## 5. Results Analysis

The relationship between the macroinner friction coefficientT 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 softnessα of interfaces (λK=0.0).Figure 6
The change curve of the macroscopic inner friction coefficient with aggregate occupancy for different normal softnessβ 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.



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