# Effect of T-Stress on Crack-Tip Stress Field and Crack Propagation of Concrete

Authors: Shaoqing Zhou; Limin Li
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

In order to predict the fracture toughness and propagation of the mixed mode fracture of concrete, the modified MTS (maximum tangential stress) criterion combined with T-stress was presented. The modified MTS criterion uses a two-parameter model (stress intensity factor K and T-stress) to describe the concrete’s fracture. Through calculation, it is found that T-stress can affect the maximum tangential stress, especially when T-stress is negative. T-stress is not taken into account in the traditional MTS criterion, which will lead to errors in calculating the maximum circumferential stress. The result of the specimen is not consistent with the predicted result of the traditional MTS. The modified criterion can provide accurate predictions for the experimental results. This provides a basis for the simulation of concrete’s fracture.


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## Body

## 1. Introduction

Concrete is a widely used material in engineering, and its mechanical properties and fracture characteristics are an important basis for concrete and its structural design. Some scholars [1–3] believed that concrete fracture is due to a variety of potential defects and that its fracture process is actually the process of microcrack initiation, propagation, and penetration until macroscopic cracks lead to concrete fracture. How to study this complex mechanical problem has always been a difficult and hot spot in the field of mechanics and material science. After the emergence of fracture mechanics, academic circles set off an upsurge in using fracture mechanics to study concrete. Researchers have carried out many fracture tests on concrete to obtain parameters such as fracture toughness. At present, studies [4, 5] have realized many advantages of studying the macroscopic damage and fracture process of concrete from the mesoscale of concrete. Some scholars [6, 7] thought that concrete could be regarded as a three-phase composite material, with cement mortar as the basic phase, coarse aggregate as the second, and the adhesive tape between them as the dispersed phase. In the research on concrete fracture, some researchers [8, 9] studied the problem of crack path selection under the influence of nearby aggregates. They proposed the theoretically and experimentally judgment criterion of crack propagation path. However, the stress field at the tip of the crack is affected by the nearby aggregate. The crack will rechoose the propagation path and show a complex failure process.Ueda [10] performed the experiments by using polymethyl-methacrylate (PMMA) with a slanted internal crack and found that there exists a difference in crack initiation angles between the MTS criterion and their experiments. They observed that the elastic T-stress, which is the nonsingular term in Williams’ series expansion of stresses, has a significant influence on the crack initiation angle. Prokopyev et al. [11] believed that the T-stress had an impact on the plastic zone size and shape of the crack tip under loading. Rice [12] used the finite element method to study the crack tip stress field about compact tension, bending, and crack center under small-scale yielding condition. Rice found there were 10–20% differences between the finite element and the test results, and if the second crack-tip parameter T-stress was introduced, such differences can be eliminated. Wei et al. [13] carried out mode I fracture tests on two kinds of rocks using a new V-notch short rod bending (VNSRB) specimen. He thinks that the mode I fracture of rock can be better assessed by considering the nonsingular stress terms of the Williams’ expansion (i.e., T-stress).Therefore, T-stress is introduced into the study of concrete fracture in this paper, and the model is established by using the methods of mesomechanics and fracture mechanics. The maximum circumferential stress fracture criterion (traditional MTS criterion) is modified, and a fracture criterion (modified MTS criterion) with two parameters (stress intensity factor K and T-stress) is proposed. Based on this, the fracture of concrete is calculated, and studies show the influence of nonsingular stress (T-stress) on the plastic zone at the crack tip, which provides a basis for accurately simulating concrete fracture.


## 2. Effect of T-Stress on the Plastic Yield Zone and the Stress Field at the Crack Tip

### 2.1. Effect of T-Stress on the Plastic Yield Zone at the Crack Tip

In most structural engineering materials, with increasing external stress, there will be a plastic zone near the crack tip. The local stress at the crack tip is exceptionally high, there is a minimal process zone around it, and the material deformation process is very complex. When the plastic zone at the crack tip is small, the first term of Williams’ expansion can be used to describe it. The crack-tip field can be described by a single parameter stress intensity factorK. However, when the plastic zone has a certain size, there is an error in the single parameter [14]. When described together with the second term of Williams’ expansion (i.e., T-stress), it will produce better results. According to the two-parameter theory, regardless of the type of load and the specimen’s shape, as long as the elastic solution has the same KΙ, KΙΙ, and T-stress, it will have the same elastic-plastic field at the crack tip.For most practical engineering materials, when the external load increases, a plastic deformation region will appear near the top of the crack. The shape and size of this area will vary with the stress state. Even in the case of small-scale yield, it is difficult to calculate the shape and size of this plastic zone accurately. If the stress-strain curve is assumed to be a straight line (i.e., the stress and strain become linear) before the yield and the yield condition is satisfied between the stress components obtained by linear elastic analysis during the yield, then an approximate estimation of the shape and size of the yield zone can be obtained.The stress field at the crack tip of the compound fracture is expressed as follows:(1)σxx=KΙ2πrcosθ21−sinθ2sin3θ2−KΙΙ2πrsinθ22+cosθ2cos3θ2+Tσyy=KΙ2πrcosθ21+sinθ2sin3θ2+KΙΙ2πrsinθ2cosθ2cos3θ2σxy=KΙ2πrcosθ2sinθ2cos3θ2+KΙΙ2πrcosθ21−sinθ2sin3θ2.The principal stress at the crack tip can be calculated according to equation (1). For plane problems, the formula for calculating principal stress is as follows:(2)σmax,min=12σxx+σyy±12σxx−σyy2+4σxy2.By substituting equations (1) into (2), a complex expression about the calculation formula of principal stress is obtained, which is abbreviated as(3)σmax,min=fKΙ,KΙΙ,r,θ,T.Using the Mises yield condition,(4)12σ1−σ22+σ2−σ32+σ3−σ12=σs.In the case of small-scale yield under plane strain, it is a good approximation to use the linear elastic theory for fracture analysis. In the case of plane stress, even under a small range of yield conditions, the linear-elastic analysis is quite different from the actual situation because of its plastic zone. Therefore, this paper only considers the plastic’s shape and size in the plane-strain state.In the plane-strain state,(5)σzz=νσxx+σyy=νσmax+σmin=ν2KΙ2πrcosθ2−2KΙΙ2πrsinθ2+T.In the case of plane strain,σxz=σyz=0 and σzz are principal stresses. By substituting equations (3) and (5) into yield condition equation (4), the complex equation of the boundary curve of the yield zone at the leading edge of the crack is obtained, which is abbreviated as(6)r=fKΙ,KΙΙ,ν,θ,T.


### 2.2. Effect of T-Stress on the Stress Field in the Plastic Zone of the Crack Tip

In most structural engineering materials, with increasing external stress, there will be a plastic zone near the crack tip. When the plastic zone of the crack tip is small, the first term of the Williams’ expansion can be used to describe the crack tip field and the crack tip field can be described by a single parameter stress-intensity factorK given the parameter Me (Me=2/πarctanKΙ/KΙΙ. However, when the plastic zone has a certain size, the single parameter has a larger error, and when described together with the second term of Williams’ expansion, more accurate results will be obtained [15]. In this paper, the finite element method is used to calculate, that is, taking the crack tip as the center, the length r as the radius as a circle, and the displacement of equation (7) is applied to the boundary of r. Applying displacement boundary conditions to the plane strain,(7)u=r2π1/212GKΙcosθ2κ−1+2sin2θ2+KΙΙsinθ2κ+1+2cos2θ2+1−ν2ErTcosθv=r2π1/212GKΙsinθ2κ+1−2cos2θ2−KΙΙcosθ2κ−1−2sin2θ2+1+ννErTsinθ.For plane strain,κ=3−4ν.The elastic-plastic field at the crack tip is calculated and analyzed using the finite element model. The Ramberg–Osgood power hardening material is used in the finite element simulation, and the relationship between stress and strain is reproduced as follows:(8)εε0=σσ0+ασσ0n,where n=13 and α=3/7.By fitting the finite element results, the following approximate formulas of T-stress can be obtained:(9)σθθσ0r,T=σθθσ0r,T=0+0.64Tσ0−0.4Tσ02,where σ0 is the yield stress.From the dimensional analysis and the previous formula, it can be deduced that for the two-parameter modified model, the stress at the crack tip can be written in the following more general form:(10)σij=σ0fijrσ0J,θ,Tσ0.In dimensionlessrσ0/J, r is the distance from the crack tip and J is the J integral. σθmax is the maximum circumferential tensile stress, including T-stress, while σθmaxT=0 does not contain T-stress. The parameter Me=2/πarctanKΙ/KΙΙ is a dimensionless parameter describing the stress intensity factor, where KΙ and KΙΙ are types I and II fracture stress intensity factors, respectively. Figure 1 shows the change of σθmax/σmaxT=0 with T/σ0 in the case of rσ0/J=2 and rσ0/J=5, respectively.Figure 1
The variation ofσθmax/σmaxT=0 with T/σ0. (a)rσ0/J=2. (b)rσ0/J=5.
(a)(b)As shown in Figure1, if the T-stress is negative, it can significantly reduce the maximum circumferential stress σθmax. The positive T-stress only slightly increases the value of σθmax. These show that T-stress has an effect on the maximum circumferential stress at the crack tip, especially when T-stress is negative. Taking into account T-stress, the maximum circumferential stress can be calculated more accurately.


## 2.1. Effect of T-Stress on the Plastic Yield Zone at the Crack Tip

In most structural engineering materials, with increasing external stress, there will be a plastic zone near the crack tip. The local stress at the crack tip is exceptionally high, there is a minimal process zone around it, and the material deformation process is very complex. When the plastic zone at the crack tip is small, the first term of Williams’ expansion can be used to describe it. The crack-tip field can be described by a single parameter stress intensity factorK. However, when the plastic zone has a certain size, there is an error in the single parameter [14]. When described together with the second term of Williams’ expansion (i.e., T-stress), it will produce better results. According to the two-parameter theory, regardless of the type of load and the specimen’s shape, as long as the elastic solution has the same KΙ, KΙΙ, and T-stress, it will have the same elastic-plastic field at the crack tip.For most practical engineering materials, when the external load increases, a plastic deformation region will appear near the top of the crack. The shape and size of this area will vary with the stress state. Even in the case of small-scale yield, it is difficult to calculate the shape and size of this plastic zone accurately. If the stress-strain curve is assumed to be a straight line (i.e., the stress and strain become linear) before the yield and the yield condition is satisfied between the stress components obtained by linear elastic analysis during the yield, then an approximate estimation of the shape and size of the yield zone can be obtained.The stress field at the crack tip of the compound fracture is expressed as follows:(1)σxx=KΙ2πrcosθ21−sinθ2sin3θ2−KΙΙ2πrsinθ22+cosθ2cos3θ2+Tσyy=KΙ2πrcosθ21+sinθ2sin3θ2+KΙΙ2πrsinθ2cosθ2cos3θ2σxy=KΙ2πrcosθ2sinθ2cos3θ2+KΙΙ2πrcosθ21−sinθ2sin3θ2.The principal stress at the crack tip can be calculated according to equation (1). For plane problems, the formula for calculating principal stress is as follows:(2)σmax,min=12σxx+σyy±12σxx−σyy2+4σxy2.By substituting equations (1) into (2), a complex expression about the calculation formula of principal stress is obtained, which is abbreviated as(3)σmax,min=fKΙ,KΙΙ,r,θ,T.Using the Mises yield condition,(4)12σ1−σ22+σ2−σ32+σ3−σ12=σs.In the case of small-scale yield under plane strain, it is a good approximation to use the linear elastic theory for fracture analysis. In the case of plane stress, even under a small range of yield conditions, the linear-elastic analysis is quite different from the actual situation because of its plastic zone. Therefore, this paper only considers the plastic’s shape and size in the plane-strain state.In the plane-strain state,(5)σzz=νσxx+σyy=νσmax+σmin=ν2KΙ2πrcosθ2−2KΙΙ2πrsinθ2+T.In the case of plane strain,σxz=σyz=0 and σzz are principal stresses. By substituting equations (3) and (5) into yield condition equation (4), the complex equation of the boundary curve of the yield zone at the leading edge of the crack is obtained, which is abbreviated as(6)r=fKΙ,KΙΙ,ν,θ,T.


## 2.2. Effect of T-Stress on the Stress Field in the Plastic Zone of the Crack Tip

In most structural engineering materials, with increasing external stress, there will be a plastic zone near the crack tip. When the plastic zone of the crack tip is small, the first term of the Williams’ expansion can be used to describe the crack tip field and the crack tip field can be described by a single parameter stress-intensity factorK given the parameter Me (Me=2/πarctanKΙ/KΙΙ. However, when the plastic zone has a certain size, the single parameter has a larger error, and when described together with the second term of Williams’ expansion, more accurate results will be obtained [15]. In this paper, the finite element method is used to calculate, that is, taking the crack tip as the center, the length r as the radius as a circle, and the displacement of equation (7) is applied to the boundary of r. Applying displacement boundary conditions to the plane strain,(7)u=r2π1/212GKΙcosθ2κ−1+2sin2θ2+KΙΙsinθ2κ+1+2cos2θ2+1−ν2ErTcosθv=r2π1/212GKΙsinθ2κ+1−2cos2θ2−KΙΙcosθ2κ−1−2sin2θ2+1+ννErTsinθ.For plane strain,κ=3−4ν.The elastic-plastic field at the crack tip is calculated and analyzed using the finite element model. The Ramberg–Osgood power hardening material is used in the finite element simulation, and the relationship between stress and strain is reproduced as follows:(8)εε0=σσ0+ασσ0n,where n=13 and α=3/7.By fitting the finite element results, the following approximate formulas of T-stress can be obtained:(9)σθθσ0r,T=σθθσ0r,T=0+0.64Tσ0−0.4Tσ02,where σ0 is the yield stress.From the dimensional analysis and the previous formula, it can be deduced that for the two-parameter modified model, the stress at the crack tip can be written in the following more general form:(10)σij=σ0fijrσ0J,θ,Tσ0.In dimensionlessrσ0/J, r is the distance from the crack tip and J is the J integral. σθmax is the maximum circumferential tensile stress, including T-stress, while σθmaxT=0 does not contain T-stress. The parameter Me=2/πarctanKΙ/KΙΙ is a dimensionless parameter describing the stress intensity factor, where KΙ and KΙΙ are types I and II fracture stress intensity factors, respectively. Figure 1 shows the change of σθmax/σmaxT=0 with T/σ0 in the case of rσ0/J=2 and rσ0/J=5, respectively.Figure 1
The variation ofσθmax/σmaxT=0 with T/σ0. (a)rσ0/J=2. (b)rσ0/J=5.
(a)(b)As shown in Figure1, if the T-stress is negative, it can significantly reduce the maximum circumferential stress σθmax. The positive T-stress only slightly increases the value of σθmax. These show that T-stress has an effect on the maximum circumferential stress at the crack tip, especially when T-stress is negative. Taking into account T-stress, the maximum circumferential stress can be calculated more accurately.


## 3. Finite Element Simulation of Concrete Crack Propagation

### 3.1. Monte Carlo Method and Random Number Generation

At the mesoscopic level, concrete is regarded as a three-phase composite composed of coarse aggregate, cement mortar, and a transition interface. In practical engineering, the manufacture of concrete components is very random, so in the process of concrete numerical simulation, to get the aggregate model consistent with reality, Li et al. [16] regard concrete as a composite material formed by the random dispersion of coarse aggregate into the cement mortar matrix and then use the Monte Carlo method to generate a series of random numbers on the computer to determine the geometric position of the random aggregate.There are many ways to generate random numbers using mathematical methods, and random numbers can be obtained from the transformation of uniformly distributed random numbers. There are also many methods to generate uniformly distributed random numbers, among which the most commonly used is the mixed congruence method, whose formula is as follows:(11)Rn+1=λRn+CmodMrn=XnM−1modM.Equation (11) represents the congruence equation modulo M, where r is the nth random number and λ,M, and C are positive integers.


### 3.2. Generation and Delivery of Random Aggregates

Assuming that the aggregate particle is spherical, its cross-sectional particle diameter can be determined using the Fuller curve, and its gradation curve is expressed as follows:(12)γ=DDmax1/2,where γ is the percentage by mass of aggregate D passing through the sieve hole. Dmax is the maximum aggregate particle size.Based on the Fuller formula, the probabilityPcD<D0 of aggregate diameter D<D0 at any point in the inner section plane of the specimen is expressed as follows:(13)PcD<D0=Pk1.065D0Dmax0.5−0.053D0Dmax4−0.012D0Dmax6−0.045D0Dmax8+0.0025D0Dmax10,where Pk is the percentage of aggregate volume to the total volume.So, the probability distribution curvePcD<D0 can be drawn according to different D0 values.


### 3.3. Material Performance Distribution

After randomly generating aggregate particles, the random aggregate model of a concrete specimen is divided automatically by using the Delaunay triangulation principle [17]. When the triangular element is in the aggregate particle, it is defined as the aggregate unit; when the triangular element is in the mortar area, it is defined as the mortar unit; and when the triangular element is in the connecting area, it is defined as the bonding unit.


## 3.1. Monte Carlo Method and Random Number Generation

At the mesoscopic level, concrete is regarded as a three-phase composite composed of coarse aggregate, cement mortar, and a transition interface. In practical engineering, the manufacture of concrete components is very random, so in the process of concrete numerical simulation, to get the aggregate model consistent with reality, Li et al. [16] regard concrete as a composite material formed by the random dispersion of coarse aggregate into the cement mortar matrix and then use the Monte Carlo method to generate a series of random numbers on the computer to determine the geometric position of the random aggregate.There are many ways to generate random numbers using mathematical methods, and random numbers can be obtained from the transformation of uniformly distributed random numbers. There are also many methods to generate uniformly distributed random numbers, among which the most commonly used is the mixed congruence method, whose formula is as follows:(11)Rn+1=λRn+CmodMrn=XnM−1modM.Equation (11) represents the congruence equation modulo M, where r is the nth random number and λ,M, and C are positive integers.


## 3.2. Generation and Delivery of Random Aggregates

Assuming that the aggregate particle is spherical, its cross-sectional particle diameter can be determined using the Fuller curve, and its gradation curve is expressed as follows:(12)γ=DDmax1/2,where γ is the percentage by mass of aggregate D passing through the sieve hole. Dmax is the maximum aggregate particle size.Based on the Fuller formula, the probabilityPcD<D0 of aggregate diameter D<D0 at any point in the inner section plane of the specimen is expressed as follows:(13)PcD<D0=Pk1.065D0Dmax0.5−0.053D0Dmax4−0.012D0Dmax6−0.045D0Dmax8+0.0025D0Dmax10,where Pk is the percentage of aggregate volume to the total volume.So, the probability distribution curvePcD<D0 can be drawn according to different D0 values.


## 3.3. Material Performance Distribution

After randomly generating aggregate particles, the random aggregate model of a concrete specimen is divided automatically by using the Delaunay triangulation principle [17]. When the triangular element is in the aggregate particle, it is defined as the aggregate unit; when the triangular element is in the mortar area, it is defined as the mortar unit; and when the triangular element is in the connecting area, it is defined as the bonding unit.


## 4. Simulation Calculation of Mixed Mode Fracture Process of the Concrete Beam

With reference to Galvez et al.’s study [18], three-point bending beam test was conducted. The geometry, force, and boundary conditions of the notched beam are shown in Figure 2. The material properties of aggregate, mortar, and adhesive tape are given in Table 1.Figure 2
Geometric dimensions, forces, and boundary conditions of specimens.Table 1
Material properties of aggregate, mortar, and adhesive tape.
Material typeElastic modulus (GPa)Poisson’s ratioUltimate crack width (mm)Tensile strength (MPa)Concrete45.40.21——Mortar30.850.180.0534.5Aggregate700.150.0534.5Interface layer300.210.034.2It can be seen from Figure3 that there is an error between the prediction results of the traditional maximum circumferential stress fracture criterion (without T-stress) and the experimental results, but the results obtained by the modified maximum circumferential stress fracture criterion (including T-stress) in this paper are consistent with the experimental results. When the cracked specimen is loaded by the mixed model, although the stress intensity factor describes the stress singularity at the crack tip, the positive or negative numerical magnitude of T-stress can also affect the fracture toughness, cracking angle, and the stability in the crack extension. Zhou and Guo [19] showed that under the I fracture model, crack extension is stable when the T-stress is negative, but when the T-stress is positive, the crack extension will be deflected. Therefore, it is more accurate to consider the effect of T-stress on the crack when studying the crack expansion path. A modified MTS criterion is used for predicting the crack trajectories of concrete. In the modified MTS fracture criterion, the effect of nonsingular stress term (T-stress) is considered in addition to the stress intensity factor (K). The T-stress can influence the direction of the crack’s propagation and fracture toughness.Figure 3
Envelope of experimental crack propagation trajectory and simulation results.Figure4 shows the experimental data envelope diagram made by Galvez et al. [18] and the numerical simulation of concrete fracture. From Figure 4, it can be obtained that the numerical results (include T-stress) and the experimental results are in good agreement. This is because T-stress can affect the plastic zone in front of the crack, so the crack opening value, the displacement of the loading point, and the displacement of the B point are affected. At the same time, the maximum circumferential stress is also affected by T-stress, so the two-parameter model is used to calculate the crack opening slip displacement more accurately.Figure 4
Experimental envelope data and numerical simulation curves. (a) Displacement change curve of load point. (b) CMOD (crack mouth opening displacement) curve. (c) Displacement change curve of point B.
(a)(b)(c)


## 5. Conclusions

(1)
T-stress has a significant influence on the plastic yield zone and the stress field at the crack tip. Negative T-stress can significantly reduce the maximum circumferential stress, while positive T-stress only slightly increases the maximum circumferential stress.(2)
The traditional criterion of maximum circumferential stress only considers the singular term. That is, only the first term is retained in the Williams’ characteristic expansion of the stress field at the crack tip. To analyze the fracture of concrete more accurately, a modified maximum circumferential stress criterion is proposed. The criterion takes into account not only the singular term in the expansion but also the influence of a constant term (i.e., T-stress). The modified maximum circumferential stress criterion is the two-parameter K-T model. Taking into account the effect of T-stress on crack, the results show that the theoretical prediction results agree well with the experimental results.



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