The mixed freedom finite element method proposed for contact problems was extended to simulate the fracture mechanics of concrete using the fictitious crack model. Pairs of contact points were set along the potential developing path of the crack. The displacement of structure was chosen as the basic variable, and the nodal contact force in contact region under local coordinate system was selected as the iteration variable to confine the nonlinear iteration process in the potential contact surface which is more numerically efficient. The contact forces and the opening of the crack were obtained explicitly enabling the softening constitutive relation for the concrete to be introduced conveniently by the fictitious crack model. According to the states of the load and the crack, the constitutive relation of concrete under cyclic load is characterized by six contact states with each contact state denoting its own displacement-stress relation. In this paper, the basic idea of the mixed freedom finite element method as well as the constitutive relation of concrete under cyclic load is presented. A numerical method was proposed to simulate crack propagation process in concrete. The accuracy and capability of the proposed method were verified by a numerical example against experiment data.

1. Introduction

Concrete is a highly heterogeneous material with its properties varyong widely from point to point due to the presence of high strength aggregates, medium strength mortar, and weaker mortar-aggregate interfaces. The existence of geometric voids which act as stress raisers can also contribute to the changing property . Due to the low tensile strength of concrete material, cracking is the dominant failure mechanism in concrete structures, and the mechanism of crack initiation and growth has received significant research effort.

In recent years, considerable amount of effort has been devoted to develop numerical models to simulate the fracture behavior of concrete used in civil engineering structures . A number of modeling approaches have been implemented, such as plasticity models [3, 4], microplane models [5, 6], and fracture mechanics models. Generally, in the field of fracture mechanics, the numerical methods based on the finite element method (FEM) are classified into two groups : “smeared crack approach” and “discrete crack approach.” In the smeared crack approach, the fracture is represented in a smeared manner in which an infinite number of parallel cracks with infinitely small opening are (theoretically) distributed (smeared) over the finite element . Those cracks are usually modeled on a fixed finite element mesh and their propagation is simulated by the reduction of the stiffness and strength of the material. Most of the models in this approach treat concrete as a quasi-brittle material that exhibits strain-softening behavior under tensile loading . The constitutive laws, defined by stress-strain relations, are nonlinear and involve a strain softening behavior which can be challenging in the analysis. The system of equations may become ill-posed  and can introduce localization instabilities and spurious mesh sensitivity to the finite element calculations . In the field of fracture mechanics, there are a number of models that can deal with the nonlinear zone ahead of a crack tip in concrete, including the fictitious crack model (FCM) , the crack band model (CBM) , the two-parameter fracture model (TPFM) [13, 14], the size effect model (SZM) , the effective crack model (ECM) [16, 17], and the double-K fracture model (DKFM) .

The discrete approach is preferred when there is one crack, or a finite number of cracks, in the structure . Among them, the cohesive zone model (CZM), which assumes that stresses act across a narrowly open crack, was developed by Hillerborg et al.  under the name FCM. In the literature, it is generally accepted that the FCM is the best simple fracture mechanics model as it provides reasonable approximation of the crack propagation in concrete using only simple calculations. The model is mainly applicable to mode-I fracture, but recently it has been extended to model shear failure mode-II and mode-III cracks . The FCM has been used to investigate the fracture process in concrete under monotonic and cyclic loading. Various constitutive relationships for cyclic tension behaviour of concrete, such as the focal point model  and the continuous function model , have been developed. More recently, linear and bilinear constitutive relationships describing the unloading and reloading scenarios were developed [24, 25]. Although these FE-based models claim to accurately simulate the fracture behaviour of concrete structures, they are usually computationally expensive. This drawback makes them unsuitable for damage detection applications where adaptable and computationally efficient models are needed .

The paper is structured as follows. In the next section, the basic idea of the mixed freedom finite element method for contact problem is given. Section 3 describes the main features of the proposed method, including the improved constitutive relationships for the crack propagation simulation under cyclic load. The numerical implementation and solution algorithms are given in Section 4. Section 5 reports the validation of the model and discusses the findings. Finally, concluding remarks are summarized in Section 6.

2. Basic Idea of the Mixed Freedom Finite Element Method

Pairs of contact points were set along the potential developing path of the crack. The mixed freedom FEM proposed for contact problem was extended to simulate the fracture mechanics of concrete using the FCM. The system of forces acting on the structure was divided into two parts: external forces and contact forces. The displacement of structure was chosen as the basic variable, and the nodal contact force in potential contact region under local coordinate system was chosen as the iteration variable, so that the nonlinear iteration process was only limited in the contact surface and much more efficient. The contact forces and the opening of the crack can be obtained explicitly enabling the softening constitutive relation for the concrete to be introduced conveniently by the fictitious crack model.

In Figure 1, the body is divided into two parts (Ω1,Ω2) by the potential developing path of crack which was set in advance. Now we focus our attention on an arbitrary node pair constituted by points 1 and 2. uI is the displacement of the node pair, and fI is the contact force. Here, the superscript I=1,2 denotes the two body parts. ξηζ is the local coordinate system on the interface with ξ being the normal direction and η, ζ the tangent directions. Then, the crack opening displacement (COD) in normal direction between point 1 and 2 can be computed as w=(u1-u2)·ξ.

Mechanical model.

Assuming that the analysis of nth step has been finished, the incremental static equilibrium equation for (n+1)th step is stated as follows by performing a finite element discretization of each body Ω1 and Ω2: (1)KΔun=(Fn+1+fn-BTσndΩ)+Δfn, where K is the global stiffness matrix; Δun is the vector of displacement increment at step n; Fn+1 is the vector of total external load at step n+1; fn is the vector of total contact force at step n; B is the strain-displacement matrix; σn is the Cauchy stress tensor; Δfn is the vector of incremental contact force. Clearly, the first term of the right-hand side (RHS) of the above equation, that is, the content in the bracket, stands for the external load increment of current load step since the contribution of fn to the system is already included into σn.

If a typical decomposition of global stiffness matrix K was performed at the beginning of the analysis for one and only one time, (1) could be rewritten as (2)Δun=K-1(Fn+1+fn-BTσndΩ)+K-1Δfn.

If Δu-n is defined as (3)Δu-n=K-1(Fn+1+fn-BTσndΩ), and the matrix C is introduced into the above equation, (3) is rewritten as (4)Δun=Δu-n+CΔfn, where the matrix C is the flexibility matrix which is defined on the possible contact boundary ΓcI. An arbitrary component of C, cij, represents the flexibility coefficient corresponding to the displacement at the freedom i due to a unit force at the freedom j. Here, i or j is only limited in the freedom of the region where contact is likely to take place.

It is important to be pointed out that (3) and (4) and the following equations in this section are only performed for the degree of freedoms (DOFs) on the potential contact surface, not for all the DOFs of the whole system.

Applying (4) to points 1 and 2 of a given node pair, respectively, gives (5)Δun1=Δu-n1+C1Δfn1,(6)Δun2=Δu-n2+C2Δfn2.

According to Newton’s third law, it is obvious that Δfn1=-Δfn2=Δfn, and, moreover, incorporating flexibility matrix C1 and C2 into C=C1+C2, so that if we subtract (5) from (6), we obtain (7)CΔfn=(Δun1-Δun2)-(Δu-n1-Δu-n2).

Equation (7) is the finite element compliance equation of the mixed finite element method for the static contact problems with friction and initial gaps proposed in this paper. In this equation, the second term in RHS, (Δu-n1-Δu-n2), stands for the difference of incremental displacement only due to the external load increment, as can be seen easily from (3); therefore, it has nothing to do with the current contact state and can be obtained by back substitution directly. However for the first term in RHS, (Δun1-Δun2), it denotes the difference of incremental displacement induced by both the external load increment and the contact force increment. From this point of view, we can see that both the RHS and the left-hand side (LHS) of the above equation are associated with the contact state. An iterative method taking into account different contact states is necessary to solve (7), and this will be given in detail in the next section.

3. FCM for the Crack Propagation under Cyclic Load

The strain softening behaviour can be applied to establish the so-called FCM according to Hillerborg  to simulate crack formation. In Figure 2, a zone with more microcracks called fracture process zone will appear before crack propagation because of the inhomogeneity of concrete. The fictitious cracks in this zone can transfer tensile stress, and the stress value can be obtained from the σ-w softening curve given its crack width. The nonlinear behavior of the propagating crack is described by a fictitious crack with cohesive forces acting between its interfaces . At the crack tip, the maximum stress is equal to the tensile strength ft, and, for all other points along the crack path, the transmitted stress σ depends on the crack opening w and is defined by the strain softening diagram. The area under the σ-w curve is equal to the specific fracture energy GF. The fracture energy is the ratio of the energy necessary for total fracture of a specimen to the projection of the fracture surface area. GF is related to material characteristic constants such as concrete mixture ratio, strength, aggregate type, particle diameters, and cement grade and will be determined by experiment. According to the widely adopted CEB-FIP Model Code 1990, GF can be determined by (8)GF=(0.0204+0.0053dmax0.958)(fcfc0)0.7, where fc0=10 MPa and dmax/mm is the maximum aggregate size. The value of w0 is of certain relation with the aggregate size; for example, in CEB-FIP Model Code 1990, it is recommended that w0=αFGF/ft, αF is the parameter relative to the maximum aggregate size.

Fictitious crack model.

There are different kinds of loading/unloading model proposed by many researchers for crack propagation under cyclic load, such as the simple elastic loading and unloading model, Hordijk-Reinhardt model [28, 29], and the Toumi model ; see Figures 3(a)~3(c).

Based on the Toumi model , an improved constitutive model shown in Figure 4 is proposed for the crack propagation under cyclic load. The main improvement lies in that the loading and reloading curve equation of Toumi model is adjusted to ensure the continuity of displacement and stress during the crack propagation which is convenient for FEM implementation. The equations used in this improved curve are shown as follows.

(1) The softening curve of concrete is based on Cornelissen’s (softening) function, which can be expressed as  (9)σ=f(w)=([1+(c1ww0)3]e-c2w/w0111-ww0(1+c13)e-c2{[1+(c1ww0)3]})ft, where coefficients c1, c2 takes the recommended values 1.0 and 5.64, respectively. The ultimate crack developing width w0 was determined from the assumption that the area under the softening curve is equal to the specific fracture energy GF and can be approximated by w0=5.618Gf/ft.

(2) The equation of the unloading curve at point C shown in Figure 4 is (10)σ=fu(w)=ww(c)[σ(c)-α(σ(c)-ft)]+α(σ(c)-ft), where σ(c), w(c) represent the stress at point C and displacement of interface opening during unloading and α is a material characteristic parameter describing the relationship between the compressive stress required to reclose the crack with unloading stress and the tensile strength in fracture process zone. According to the experimental results of Reinhardt et al. , a recommended value of α=0.35 is used in this paper. In cracked areas, when unloading if w>w0, then fu(w)=0. Otherwise, if 0<w<w0, we have σ(c)=0, and the stress σ can be calculated using (10).

(3) The equation of reloading curve at point D in the proposed model is (11)σ=fl(w)=(w-w(D)w(E)-w(D))γ(σ(E)-σ(D))+σ(D), where γ is the shape parameter of reloading curve, which can be approximated by γ=1/(1+1.2w(E)/w0) according to the experimental data of .

The stress value at point E is governed by the reduction factor together with the stress value at point C and can be determined using the following equation: (12)σ(E)=(1-β)σ(C).

The value of β takes as a constant of 0.05 according to Horri’s suggestion .

4. Numerical Implementation

When using mixed freedom FEM to simulate fracture process, the initial condition was taken from the converged result (contact state and contact force between crack surfaces) of former FEM step. Using the overall displacement field obtained from (1), the joint opening displacement can be determined and employed to solve the new contact force on joint interface iteratively. Assuming that the ith contact iteration has been finished, the displacement and stress expression for (i+1)th iteration is stated as follows: (13)wn+1i+1=wn+1i+(Δun1,i-Δun2,i)·ξ,(σn+1i+1)ξ=σn+1i+1·ξ. Here, the superscript denotes the contact iteration step while the subscript stands for the loading increment step. The displacement and stress from the above equation are prediction results and must be checked against the contact state after (i+1)th iteration. Based on the calculated result from the former loading step and ith contact iteration, as well as the joint opening displacement and contact stress after (i+1)th iteration, the contact states were processed and updated which will be explained in detail as follows.

If a contact state is at the initial closed state, where the initial condition is w=0, fη2+fζ2<-μfξ+Ac,  f1=-f2, ftr=ft. After (i+1)th iteration, if the normal stress of the crack surfaces satisfied (σn+1i+1)ξ>ft, the contact state is updated to fictitious crack extension state, and fn+1i+1=ftA is set for the next iteration; otherwise, if fη2+fζ2-μfξ+Ac was satisfied, the current state is changed to sliding state, and a new condition of fη2+fζ2=-μfξ+Ac is set accordingly. fξ is the normal contact force component; fη, fζ are two tangential components of the contact forces; μ is the friction coefficient, A is the controlled area of pair nodes; c is cohesive strength; ft is the tensile strength of the crack surface; ftr is the residual tensile strength of the crack surface.

If the current state is at the fictitious crack extension state, where the condition is 0<w<w0, the normal stress must satisfy (9). If the opening width reaches the condition of wn+1i+1wn, the current state is adjusted to fictitious crack unloading state, and contact force is set as fn+1i+1=(σn)ξA, stress at the unloading point is set as σ(C)=(σn)ξ, and the residual tensile strength of the contact pair is changed to ftr=(1-β)σ(C). If (wn+1i+1w0) is met, the current state is changed to opening state, and new values are set to make fn+1i+1=0, ftr=0. (σn)ξ is the normal stress of crack at nth loading step.

If the current state is at the fictitious crack unloading state, where the condition is 0<ww(C), the normal stress must satisfy (10). If the opening width of crack after iteration meet wn+1i+1wn, the current state is changed to fictitious crack reloading state, and contact force is set as fn+1i+1=(σn)ξA, and stress at the reloading point is set as σ(D)=(σn)ξ, σ(E)=ftr. If (wn+1i+1<0) is satisfied, the current state is changed to crack closed state, and the width is set to  w=0  to avoid crack surface embedding.

If current state is at the crack closed state, where the condition is w=0, ftr<ft, and σn+1i+1<fu(0). If the normal stress after iteration satisfies σn+1i+1>fu(0), the contact state is changed to fictitious crack reloading state and is set as fn+1i+1=fu(0)A, where fu(0) is the stress when the crack is fully closed.

If the current state is at the fictitious crack reloading state, where the condition is 0<ww(E), the normal stress is satisfied in (11) σ=fl(w). If the normal stress after iteration satisfies σn+1i+1=ftr, and the opening width of crack surface after iteration meets the condition wn+1i+1>w(E), then the current state is changed to the fictitious crack extension state.

If the current state is at the opening state, where the condition is ww0, f1=f2=0. If the opening degree of joints after iteration meets w<w0, the current state is changed to the fictitious crack unloading state, and the stress at loading point is set σ(C)=0.

If the current state is at the sliding state, where the condition is fη2+fζ2=-μfξ+Ac. If fη2+fζ2<-μfξ+Ac is met after iteration, the current state is changed to the initial closed state.

Having updated the new contact state of one loading increment step, the RHS of (7) is obtained and can be used to get the overall displacement field from (1). Then, the width of the crack surface can be updated, and the new contact force vector can be determined from the stress-displacement curve. The updated contact force and state are taken into the next iteration, and the above process is repeated until both the state and force have converged which will lead to the start of the next loading step.

5. Numerical Experiments

The simulation of crack propagation of Toumi’s  beam problem under cyclic load was carried out to verify the proposed scheme. The numerical setup is present in Figure 5 along with the mesh configurations. Similarly, an initial crack is set in BC section, and AB is the pair of contact points. The mesh size near the crack is 1 mm, and other parameters are

length = 320 mm,

depth = 80 mm,

thickness = 50 mm,

initial crack length = 40 mm,

tensile strength = 5.2 MPa,

fracture energy = 34.2 N/m,

elastic modulus = 31.6 GPa,

poisson’s ratio = 0.2.

Finite element model for Toumi’s beam.

The load-displacement curve under monotonous load was shown in Figure 6 and compared against the experimental data . The simulated result lies between the two experimental curves and followed the trend closely. In Figure 7, the load-displacement curve under cyclic load condition was plotted and matches well the experimental data for each loading cycle.

Experimental and FEM results for monotonous load.

Experimental and FEM results for cyclic load.

Figure 8 shows the distributions of normal stress along the height of crack under three different loading conditions when the vertical displacement is 0.04 mm and 0.07 mm respectively. No abrupt stress change has been observed as the stress curve is in a continuous and smooth way, and the distribution of normal tensile stress of crack fits nicely with the concrete softening curve adopted. When unloading, the crack surface below the tip of fracture process zone transited gradually from tensile state to compressive state, and the compressive stress increases gradually towards the end. However, if the crack zone is long (Dy=0.07 mm), and the opening width of the bottom crack surface is too large, the compressive stress tends to decrease. When reloading, the crack surface below the tip is mainly in the tensile state. Both findings match well general understanding and demonstrate great accuracy and capability of the current scheme.

For simulating the crack propagation, the size of mesh is a factor of great importance. To fully understand the influence of the mesh size on calculated results and investigate the mesh dependence of the suggested method, the load-displacement curves near the crack were plotted using different meshing sizes varying from 1 mm to 8 mm (about 1/40~1/5 of the crack height) in Figure 9. It can be seen that for a meshing size less than 1/10 of the crack height, the result of different meshes shows little change, while if the meshing size and crack height ratio is bigger than 1/5, the load-displacement curve is not smooth anymore. In this case, the trend of the curve in general agreed well with the result of fine mesh despite the big error in the far end.

Load displacement curve for different mesh sizes. (a) Monotonous load, and (b) cyclic load.

6. Conclusion

The mixed freedom finite element method proposed for contact problems is extended to investigate the crack propagation under cyclic load using the constitutive relation of model I crack. According to the states of the load and the crack, the constitutive relation is characterized by six contact states and solved iteratively. The current scheme is verified against a number of experimental cases and proved to be accurate and reliable.

Using the mixed freedom finite element method to simulate the crack propagation, the softening constitutive relation of concrete is implemented directly into the contact iteration. It will not lead to numerical instability during the overlapping process of positive and negative stiffness, which is convenient to adopt the softening constitutive relation of random shape. Moreover, the dependence of the result on the size of mesh is not obvious.

As there are limited resources available about the softening relation of mixed type crack, this paper only showed simulations on the propagation of model I type crack. However, the authors believe the proposed method is applicable for other type crack as well. Due to the fact that mixed freedom finite element method needs to preset pairs of contact points in the potential contact area, the application of the current method is confined to problems which the paths of the crack propagation are already known. Future work involves continuing the current work on different types of cracks, and a more detailed analysis of the calculation parameters of the constitutive relations is presented in this paper.

Conflict of Interests

The authors do not have any conflict of interests with the content of the paper.

Acknowledgments

The present work is supported by the National Natural Science Foundation of China (no. 51279050), the National High-tech R&D Program of China (863 Program) (no. 2012BAK10B04), the National Key Technology R&D Program in 12th Five-Year Plan (no. SS2012AA112507), and the Non-Profit Industry Financial Program of MWR (Ministry of Water Resources of China) (no. 201301058).