An all-over breathing crack on the plate surface having arbitrary depth and location is assumed to be nonpropagating and parallel to one side of the plate. Based on a piecewise model, the nonlinear dynamic behaviors of thin plate with the all-over breathing crack are studied to analyze the effect of external excitation amplitudes and frequencies on cracked plate with different crack parameters (crack depth and crack location). Firstly, the mode shape functions of cracked thin plate are obtained by using the simply supported boundary conditions and the boundary conditions along the crack line. Then, natural frequencies and mode functions of the cracked plate are calculated, which are assessed with FEM results. The stress functions of thin plate with large deflection are obtained by the equations of compatibility in the status of opening and closing of crack, respectively. To compare with the effect of breathing crack on the plate, the nonlinear dynamic responses of open-crack plate and intact plate are analyzed too. Lastly, the waveforms, bifurcation diagrams, and phase portraits of the model are gained by the Runge-Kutta method. It is found that complex nonlinear dynamic behaviors, such as quasi-periodic motion, bifurcation, and chaotic motion, appear in the breathing crack plate.

The plate is widely used in engineering practices such as the aircraft structure and body, concrete floor slabs, and marine industry, to name a few. The existence of a crack in a plate affects its stiffness, mass, and damping properties and then changes its vibration characteristics. For a vibration plate, a crack not only effects the free vibration of plate, including natural frequency and mode shape, but also changes the forced vibration, especially the response of nonlinear dynamics. The decrease of stiffness increases the vibration amplitude of the cracked plate with large deflection. Furthermore, the crack alternately opens and closes during vibration cycle, which forms the cracked structure with nonsmooth characteristics. Then, complex nonlinear dynamics behaviors will appear, which can seriously affect the safety of structures.

An earlier extensive literature review on the vibration of cracked structures could be found in the paper of Dimarogonas [

For the nonlinear dynamic investigation of crack structures, many researchers considered cracks as open-crack models. In 2005, Fu et al. [

In addition to the open crack, the surface crack breathing is a more practical and common situation during the vibration of the cracked structures [

In 2000, Pugno et al. [

The mentioned researches show that breathing crack model could reveal the real nonlinear dynamic behaviors of surface crack structures. Most of the researches on the crack of breathing are focused on the beam, and the nonlinear dynamics study of plate with surface crack employed the open-crack model.

The aim of this paper is to study the bifurcations of thin plate with an all-over breathing crack. The piecewise model is used to describe the opening and closing of breathing crack during the vibration: the crack is taken as open crack when crack opens; otherwise, the cracked plate is regarded as intact plate when crack closes. Based on the von Kármán large deflection theory, Hamilton’s principle is used to establish the nonlinear governing equations of motion for the cracked plate. The mode shape functions are derived using the geometric boundary conditions and the boundary conditions along the crack line of thin plate. The effect of location and depth of crack on mode shapes and free vibration frequencies of cracked plate are analyzed, which is assessed with FEM results. The stress functions of cracked and intact plate with large deflection are obtained by the equations of compatibility. The partial differential equation is discretized via the Galerkin method. The Runge-Kutta method is utilized to investigate the bifurcations and chaotic motions of cracked plate. Nonlinear dynamic behaviors of cracked plate are studied to analyze the effect of external excitation amplitudes and frequencies on plate with different crack parameters which consist of crack depth and crack location.

In this section, the mode shape functions of cracked thin plate are obtained by using the simply supported boundary conditions and internal boundary conditions along the crack.

The simply supported rectangular plate with an all-over breathing crack can be illustrated by Figure

A simply supported rectangular thin plate with and all-over breathing crack.

For the convenience of solving dynamic equations, a stress function

As to the thin plate with an all-over crack, the authors divide the thin plate into two parts, plate I and plate II, at the location

Breathing crack is continuously open-closed during vibration. When the crack is open, at the crack location, the continuity and discontinuity conditions can be written along the crack line as follows:

The mode shape functions for

According to boundary conditions (

Substituting (

The first three-order modes and natural frequencies of the cracked plate can be obtained by the method mentioned above, which is verified by finite element method. We chose parameters of material and size as

Tables

First three-order frequencies of cracked and intact plates.

First-order natural frequency (Hz) | Second-order natural frequency (Hz) | Third-order natural frequency (Hz) | ||||
---|---|---|---|---|---|---|

Theoretical results | FEM | Theoretical results | FEM | Theoretical results | FEM | |

Intact plate | 23.9874 | 25.101 | 38.3798 | 41.853 | 62.3672 | 68.485 |

| 23.9279 | 24.987 | 38.1120 | 41.359 | 62.1117 | 68.071 |

| 23.8698 | 24.941 | 38.3798 | 41.749 | 61.8510 | 67.468 |

First three-order modes frequency of cracked and intact plates.

Theoretical results | FEM | ||
---|---|---|---|

Mode of | Mode of | ||

First order | | | |

| |||

Second order | | | |

| |||

Third order | | | |

Theoretical results and simulation data are in good agreement, and the accuracy of the theoretical method is verified in this section. Comparing theoretical results to simulation data, the first three-order modal figures obtained by theoretical calculation are consistent with FEM simulations. Thus, the modal functions corresponding to the first three-order modal figures are used to construct the solutions of dynamic equations.

The stress functions of cracked thin plate with large deflection are obtained by the equations of compatibility. Using the first three modes of mode shape function (

During vibration, surface crack will be continuously open-closed. When the crack is open, the modes of crack thin plate are adopted and the stress function of crack thin plate considering large deflection is obtained as (

Having obtained the mode shape functions and stress function expression of crack thin plate and intact plate, the authors use Galerkin method to discrete the dynamic equation (

As to intact plate, the modal function and stress function of intact plate are adopted. Putting expressions (

The opening and closing of breathing crack are described as piecewise model during the vibration. The authors assume that the crack is on the top surface of the plate and the transversal displacement

The Runge-Kutta algorithm is utilized to numerically analyze the nonlinear dynamical behaviors of the cracked plate under different excitation amplitude and frequency. The nonlinear dynamical behaviors of the plate with different crack location and depth are studied by analyzing the bifurcation diagrams. To compare with the effect of breathing crack on the plate, the nonlinear dynamic responses of the plate with an open crack and the intact plate are analyzed too. In numerical simulation, we chose parameters of material and size as

For intact plate, as shown in Figure ^{2} to 4000 N/m^{2}, while the excitation frequency is

The bifurcation diagram of intact plate for transverse displacement

The nonlinear dynamic behaviors of cracked plate are studied to analyze the effect of external excitation amplitudes and frequencies on plate with different crack depth. The rectangular plates have a crack at

Figures

The bifurcation diagram of plate with a breathing crack (

The bifurcation diagram of plate with a breathing crack (

The bifurcation diagram of plate with a breathing crack (

The bifurcation diagram of plate with an open crack (

The bifurcation diagram of plate with an open crack (

The bifurcation diagram of plate with an open crack (

The above bifurcation diagrams show that breathing crack plate has complex nonlinear phenomenon: quasi-periodic motion, bifurcation motion, and chaotic motion. With the change of crack depth, the bifurcation diagrams of breathing crack plates represent different nonlinear dynamic behaviors. However, the motion of open-crack plates is relatively simple, which merely demonstrates the single-cycle movement and the movement of doubling the cycle.

As for bifurcation diagram 4, the corresponding waveform and phase diagrams are gained when the external excitation amplitudes are 2500, 3100, 3300, 3400, 3700, and 3800 N/m^{2}, respectively, as shown in Figure

Waveforms and bifurcation of the plate with breathing crack; relative crack depth

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

For one cycle motion, as shown in Figure

In this section, the effect of crack location on nonlinear dynamic behavior is studied to analyze the bifurcation diagrams of plates subject to external excitation. The simply supported rectangular plates have a crack at depth (

Figures

The bifurcation diagram of plate with a breathing crack (

The bifurcation diagram of plate with a breathing crack (

The bifurcation diagram of plate with a breathing crack (

The bifurcation diagram of plate with an open crack (

The bifurcation diagram of plate with an a open crack (

The bifurcation diagram of plate with an open crack (

The piecewise model is used to describe the opening and closing of crack during the vibration. The dynamic equations are solved by using the stress function and the mode functions derived by this paper. Bifurcation diagrams, wave-shape diagrams, and phase diagrams of simply supported rectangular thin plate with an all-over breathing crack are investigated in this paper. The following conclusions have been obtained:

It is complex to derive the mode functions of in-plane displacements (

The previous study of the nonlinear dynamics of surface crack plate usually uses open-crack model. However, the authors find that the nonlinear dynamic behaviors of open-crack model are similar to those of the intact plate, which does not reveal the effect of crack on nonlinear dynamic behaviors of plate. Breathing crack model reflects the real process of opening-closing of surface plate and presents more complex phenomena. Through three sets of bifurcation diagrams, we find that breathing crack plate has complex nonlinear phenomena such as quasi-periodic motion, bifurcation motion, and chaotic motion, and the nonlinear dynamic behaviors of open-crack model are similar to intact plate model, which merely presents the single-cycle motion and the double periodic motion.

The deeper the crack is, the smaller the critical excitation frequency will be, which means that the increase of the crack depth will make the motion of crack plate more complicated. There are two reasons for this phenomenon. Firstly, the increase of the crack depth reduces the stiffness of plate, resulting in the increase of vibration amplitude for the cracked plate which leads to complex nonlinear dynamic behavior. Secondly, the crack alternately opens and closes during vibrational cycle and the increase of the crack depth forms the cracked structure with strong nonsmooth characteristics.

When the crack is near the center of the plate, complex nonlinear dynamic behavior more easily occurs in the vibration of cracked plate subject to certain excitation amplitude and frequency. This is because the maximum transversal displacement is near the center of the plate, and the farther away the crack is from the locations of zero displacement (

The coefficients given in (

The authors declare that there is no conflict of interests regarding the publication of this article.

The authors gratefully acknowledge the National Natural Science Foundation of China (NNSFC) through Grant nos. 11172011 and 11472019.