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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.

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 [

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 [

The discrete approach is preferred when there is one crack, or a finite number of cracks, in the structure [

This paper presents an efficient approach of modeling the monotonic and cyclic flexural cracking behaviour of concrete using a FCM. Under the cyclic load, the constitutive relation of concrete fracture is very complex, the same displacement may correspond to three different kinds of stress states, and numerical conditions of the transformation between each stress state are not easy to define. In numerical analysis, it will lead to instability problems during the overlapping process of positive and negative stiffness. In the analysis of concrete crack propagation under cyclic load, the states of load (loading, unloading, and reloading) at joint interface should be defined first because different states correspond to different displacement-stress curves. And the softening load that fictitious crack plane transfer is a function of opening displacement on this plane based on fictitious crack model, but the opening displacement is an unknown quantity still to be determined, which means that the propagation calculation of concrete cracks under cyclic load is a nonlinear iterative process. This situation is very similar to contact problems in practical engineering in which the state of load process, contact forces, and displacements on interfaces cannot be known in advance. The solution is to make the equation fulfill the equilibrium condition in general and satisfy the constitutive relation between opening displacement and interface tension on interface from the determined states of load and crack.

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

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

Mechanical model.

Assuming that the analysis of

If a typical decomposition of global stiffness matrix

If

It is important to be pointed out that (

Applying (

According to Newton’s third law, it is obvious that

Equation (

The strain softening behaviour can be applied to establish the so-called FCM according to Hillerborg [

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 [

Loading/unloading model for crack propagation under cyclic load. (a) Elastic loading and unloading model, (b) Hordijk-Reinhardt model, and (c) Toumi model.

Based on the Toumi model [

The improved loading/unloading for crack propagation under cyclic load.

(1) The softening curve of concrete is based on Cornelissen’s (softening) function, which can be expressed as [

(2) The equation of the unloading curve at point

(3) The equation of reloading curve at point

The stress value at point

The value of

In the propagation of the crack in concrete, there are three loading process states, that is, loading, unloading, and reloading. For model I crack, there are three interface gap states, that is, closure, opening, and virtual opening. In order to describe the whole loading/unloading process of model I crack in a physical model, six contact states are employed. In Figure

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 (

Having updated the new contact state of one loading increment step, the RHS of (

The simulation of crack propagation of Toumi’s [

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

Experimental and FEM results for monotonous load.

Experimental and FEM results for cyclic load.

Figure

Normal stress distribution along the crack. (a) Loading,

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

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

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.

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

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).