^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

Applying the built dynamic crack model of fibre concrete, bridging fiber segment is substituted by loads. When a crack propagates its fiber continues to break. By the approaches of the theory of complex functions, the problems dealt with can be translated into Riemann-Hilbert problems. Analytical solutions of the displacements, stresses, and dynamic stress intensity factors under the action of of moving variable loads

Regarding the enhancing mechanism of fiber-reinforced concrete, there exist two popular theories [

According to the theory of composite materials, fiber-reinforced concrete is regarded as a multiphase system, that is, the fiber and the concrete separately. The properties of the composite materials are the sum of each ones, and basic assumptions are as follows.

Fibers were distributed continuously and uniformly, and the directions were consistent with the force.

Fiber and matrix bond well, that is, they have the same strain, and no relative sliding occurred.

Fiber and matrix are elastic deformation, and transverse deformation is the same.

In the light of disorder status of the short steel fiber-reinforced concrete, theory of fibrous space is put forward. This theory derives from the theorem of linear elastic fracture mechanics, and fiber space is referred to as the enhancing mechanism of the basic parameters. When fibers are mixed in concrete, they can restrict the crack propagation effectively. The smaller the fiber space is, the greater the stress concentration abates; therefore the strength and toughness of concrete increase easily.

Because fiber mixed into concrete is capable of resisting crack formation and expansion, there is obvious difference in fracture behavior and crack resistance between fiber-reinforced concrete and general concrete. Therefore, fracture characteristic of fiber-reinforced concrete was studied and its fracture model was built, which processes the great significance not only for understanding fiber-reinforced concrete material itself, but also for analyzing structural performance [

Fiber reinforced concrete with fibrous disorder distribution can be regarded as isotropic materials [

When fiber-reinforced concrete was subjected to external loads, a fracture process zone appeared in the front of the main crack tips, and properties in fracture zone have a significant impact on fracture and toughness of fiber-reinforced concrete. After fiber-reinforced concrete starts to crack, crack tip opening was prevented by bridging fibers [

In order to analyze fracture process of fiber-reinforced concrete, a crack model of bridging fibers in concrete was put forward [

Sketch of fracture of fiber reinforced concrete.

This crack model will be divided into three regions. Region A is the fracture section without stress transfer, and the fibers present pullout or break. Region B is regarded as bridging fiber zone which is also known as pseudoplastic zone and begins to fracture, while the fibers can arrest crack propagation; the length of the crack increases slowly and reaches critical state finally. Region C is called the microcrack zone or transition zone, and when the loads aggrandize gradually, region C will transform into region B.

At bridging fiber segment, some fibers across the crack were regarded as closed impact forces which act at the crack surfaces. Bridging fibers reduced the stress intensity factors of the crack tips, and under the conditions of the same fracture toughness of the matrix material, the greater loads needed can induce crack propagation and cause material damage [

Bridging fibers play a vital role in the course of crack arrest, enhancement, and increasing toughness of concrete, especially when the concrete occurs crack, the role of bridging fiber cannot be ignored. Therefore, bridging fiber problems of fiber reinforced concrete are not only an important research task but also a frontline in the field.

During the form and propagation of the crack, bridging fibers appears. That is to say, when the crack propagates, the case of bridge still exists. Accordingly, it is an important meaning to establish an appropriate dynamic model of bridging fibers in studies on fracture problem of fiber reinforced concrete.

If the fiber failure is governed by maximum tensile stress, which occurs at the crack plane, the fiber breaks and hence the crack extension should appear in a self-similar fashion. The fiber breaks lie along a transverse line and therefore present a “V” notch. The crack is supposed to nucleate an infinitesimally small microcrack situated along the

Sketch of crack extension of bridging fibers in fiber reinforced concrete.

Obviously, the dynamic model of crack propagation problem of concrete in Figure

The dynamic model of the crack of bridging fibers in fiber reinforced concrete.

In order to solve efficiently fracture dynamics problems of composite materials, solutions will be obtained under the action of variable loads for mode I moving crack. According to the theorem of generalized functions, the different boundary condition problems considered will be translated into Keldysh-Sedov mixed boundary value problem by means of self-similar functions, and the corresponding solutions will be attained.

Suppose at

Here

An arbitrary continuous function of two variables

Here

Utilizing correlative representations of elastodynamics equations of motion for an orthotropic anisotropic body [

For the case when functions

For the case when functions

The relative self-similar functions are as follows [

Fracture dynamics problems will be studied for an infinite orthotropic anisotropic body. Assume at the initial moment

In order to resolve efficaciously symmetrical dynamics problems with bridging fibers of fiber reinforced concrete, solutions will be found under the conditions of unlike loads for mode I running crack. In the light of the theorem of generalized functions, the different boundary condition problems investigated will be changed into Keldysh-Sedov mixed boundary value problem by the methods of self-similar functions, and the corresponding solutions will be acquired. The problems studied are under the plane strain states.

(1) Postulate, at the initial moment

In this case the displacement will evidently be homogeneous functions, in which,

In terms of (

Deducting from the above formulas, the solution of

In the formula

According to symmetry and the conditions of the infinite point of the plane corresponding to the origin of coordinates of the physical plane as well as singularities of the crack tip [

Substituting (

Then substituting (

Then putting (

The first of (

Integrating (

The displacement

By means of the solution of (

(2) Presume that the rest conditions are the same as those in the above ensample except that the applied loads become an increasing load

In this case the stress will apparently be homogeneous functions, in which

At

In the light of (

From the above formulas, the unique solution of

In the formulae,

In terms of symmetry and the conditions of the infinite point of the plane corresponding to the origin of coordinates of the physical plane as well as singularities of the crack tip [

Then putting (

Substituting (

In an orthotropic isotropic body, the disturbance scope of elastic wave can be illuminated by the circular area of radius

Now inserting (

The limit of the above belongs to the modality

The first of (

After integrating the first term of (

The crack propagates along the

After integrating the second nape of (

The crack moves along the

After integrating the third nape of (

The crack expands along the

The displacement

Then substituting

Analytical solutions need transforming into numerical solutions in the light of real situation of idiographic problems, hence variable law of dynamic stress intensity factor can be denoted better. The corresponding parameters [

Known from (

Relevant numerical values between dynamic stress intensity factor

2 | 4 | 6 | 8 | 10 | |

7.4141 | 5.2426 | 4.2806 | 3.7071 | 3.3157 | |

12 | 14 | 16 | 18 | 20 | |

3.0268 | 2.8023 | 2.6213 | 2.4714 | 2.3446 |

Dynamic stress intensity factors versus time.

In terms of (

Relevant numerical values between dynamic stress intensity factor

2 | 4 | 6 | 8 | 10 | |

0.9380 | 1.3265 | 1.6246 | 1.8759 | 2.0973 | |

12 | 14 | 16 | 18 | 20 | |

2.2975 | 2.4816 | 2.6529 | 2.8139 | 2.9661 |

Dynamic stress intensity factors versus time.

At

At

Analytic solutions of the dynamic crack model of fiber reinforced concrete were found by the measures of complex variable theory. The approach developed in this paper based on the methods of the self-similar functions makes it conceivable to acquire the concrete solution of fiber reinforced concrete and bridging fibrous fracture speed

Utilizing the representation