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The effects of breathing behaviour on the dynamic response and crack growth are studied through a cracked cantilever beam. The main goal is to reveal the coupling mechanism of dynamic response and crack growth by employing a plain single-degree-of-freedom (SDOF) lumped system with the breathing crack stiffness and friction damping. The friction damping loss factor is derived by using Coulomb friction model and energy principle. Natural frequency, dynamic stress, dynamic stress intensity factor (DSIF), and crack growth are analyzed by case studies in the end. Results indicate that not only does the stiffness oscillates during crack growth corresponding to the physically open and closed states of the crack, but also stiffness and friction damping oscillate nonlinearly with crack growth. This behaviour induces not only nonlinear dynamic response but also nonlinear crack growth. It provides an approximate description of the nonlinearities introduced by the presence of a breathing crack. Therefore, it can be employed to improve the prediction precision of the crack identification and crack growth life of a cracked cantilever beam.

Cantilever beam-like structures such as aircraft wings, engine blades, and rocket bodies are widely used in the aerospace engineering field. Cracks often generate on the surface of a beam when the beam-like structure is subjected to vibratory loading conditions. In order to ensure the safety and reliability of these beam-like structures, study on the dynamic response and crack growth is important to prevent catastrophic failures, now and in the future.

According to whether the crack is in the opening or closing state during working time, the crack model is classified into two types: open crack models or breathing crack models. As the name implies, open crack models are regarded as keeping the crack in the open state during motion. For example, Biswal et al. [

Through these crack models, the vibrational response of the cracked beam had been studied by many authors. Ostachowicz and Krawczuk [

The coupling of the crack growth and the dynamic response is a complicated but important mechanism to the dynamic design and the damage tolerant design of a cracked beam working in vibration environment. However, at present, few works have been done concerning the coupling effects of the fatigue crack growth and the dynamic response. Most of the present works that have been done ignore the fact that the crack severity changes during vibration analysis. In fact, not only will the crack depth increase but also will the microslip of the crack surfaces that are produced when the closure effect is considered for a vibrating beam. In this case, the friction force between the crack surfaces is introduced by the normal pressure forces. This friction force will do work over the microslide distance so that the energy dissipation is produced. When friction energy dissipation happens, damping is introduced. When a cracked beam works in a resonance condition, the friction damping will play a very important role in the dynamic response and crack propagation. This scenario happens frequently in equipment and structures. Therefore, resonance is one of the common reasons of equipment failures. Because the crack alters dynamic properties of the vibrating beam, the dynamic response analysis is often used to calculate the dynamic force applied to the cracked section of beam. These observations imply that the breathing behaviour is a complicated but a real very important factor to the damage tolerant design for a beam. Therefore, it is vital to understand the dynamic response changes caused by the crack growth and its effect upon the crack growth. But the previous research pays little attention to this problem. In order to discover the mechanism of friction damping and its effect on the dynamic response and crack growth, the critical detail is to set up a proper friction damping model for the breathing crack.

The present work is to analyze the first mode frequency of the cracked beam by the mentioned two stiffness models and Galerkin method, to derive a friction damping model for a breathing crack by energy principle and Coulomb friction model, and to discuss the dynamic stress response, stress intensity factor, and crack growth with friction damping loss factor included by case studies in the end.

In this article, an Euler-Bernoulli cantilever beam of 304 stainless steel (304SS) is considered. The material of the beam is assumed to be isotropic and homogenous. And a straight surface crack is assumed to be on the top edge. The coordinate, geometry, and dimensions of the beam are shown in Figure

Model of a cracked cantilever beam, where

Ignoring the structural damping and the crack damping, the transverse dynamic equation of an elastic beam in the

Based on the composition of a forced dynamic response, the first mode does the most contribution. Therefore, only the first vibration mode of the cracked cantilever beam is considered in this article. Suppose that

Based on the boundary condition of a cantilever beam, the vibration shape function of the first mode is given by [

Supposing that the damping of the beam composed of the material damping

Therefore, the cracked cantilever beam is simplified into a SDOF system as shown in Figure

Equivalent SDOF model of the cracked cantilever beam.

When there is a periodic force

In order to understand and simulate the open and closed state of the breathing crack, a square wave function is used during analysis. The square wave function is given by

If there is a sine excitation force

Through (

From (

Ignoring the shear force, a cracked cantilever beam can be regarded as two intact beams that are connected by a torsional spring as shown in Figure

Stiffness model of the cracked cantilever beam.

Therefore, the stiffness of the open cracked beam can be calculated by

During crack propagation, the crack tip material will yield and then separate. When the material of the crack tip fractures, the crack will advance. According to the knowledge of materials mechanics, the yield failure of ductile materials often happens related to the maximum shear stress. And the maximum shear stress usually appears on the incline plane which forms a 45° angle to the cross section of the cracked beam under tension stress state for most of steels. Therefore, test results of specimen with smooth surface often yield to some 45° cross inclination strips as shown in Figure

Yield slip line.

Take the Euler-Bernoulli beam having a breathing crack as the research object as shown in Figure

Crack growth road.

Before creating the sliding friction model for the breathing crack, some assumptions should be given at first as follows:

(1) The dynamic response of the cracked beam is stable.

(2) The energy dissipation of the cracked beam is produced just by the friction of the crack surface.

(3) The friction force at the semicrack depth as

(4) The energy dissipation happens only when the crack is in the compressed zone.

(5) The friction force of the crack surfaces satisfies with the Coulomb model law.

(6) The total storage strain energy of the intact beam is equal to that of the breathing cracked beam.

Based on these given assumptions, the surface slide friction model of the breathing crack is set up as shown in Figure

Friction model of crack surface.

According to the geometry of the cracked beam as shown in Figure

Therefore, the normal pressing force is obtained

Because the breathing crack will be opened gradually when it is in intensive zone and will be closed when it is in a compressed zone, the value of pressing force may exist during the half cycle relative to the effect of external harmonic force and the average pressing force

If the frictional coefficient of crack surfaces for 304SS is

In each cycle, the breathing process of the crack from the open state to the close state includes 4 phases. If position ① is the initial position as shown in Figure

It can be seen from Figure

Figure

Therefore, the energy dissipation caused by the friction force within one vibration cycle is obtained

According to the energy principle and the assumptions, the strain energy storage in the cracked beam having rectangular cross section is given by

For a cracked beam, the friction damping loss factor

Therefore, the damping loss factor caused by the friction of the crack surfaces is obtained

If the generalized stiffness

If the breathing behaviour is considered, the stiffness of the cracked beam varies with time within one vibration cycle. It is impossible to plug the stiffness model of a breathing cracked beam into the frequency equation (

For a vibrating cracked beam, the crack propagation speed relies on not only the amplitude of dynamic stress response but also the crack depth. For an edge cracked beam as shown in Figure

Plugging the dynamic displacement response into the stress equation, we obtain

Based on the definition of stress intensity factor (SIF) in fracture mechanics, the SIF of the beam under pure bending can be defined by empirical formula

The shape function is claimed to be of engineering accuracy for any

The parameters such as the crack closure effect, stress ratio, SIF range, critical SIF, and the threshold of SIF are considered for the model of crack growth, which is called the modified Forman Model [

The threshold SIF is given by

In these formulas,

The incremental of crack growth within one vibration cycle is calculated by employing the modified Forman crack growth equation. It is expressed by

The coupling analysis can be done by the above mathematical equations. The stiffness is changed gradually with crack growth first. The damping is also introduced by the friction energy dissipation from crack surfaces. Both of them affect the dynamic response greatly. On the contrary, the crack propagation ratio is also affected by the change of dynamic response.

If the crack depth extends to a certain value or the dynamic stress is too large, alternative failure modes may occur to the beam. Therefore, it is especially important to work out reasonable criteria to indicate the failure. Three failure criteria are considered and shown as follows.

A cantilever beam of 304SS is considered as 1 m in length, 0.12 m in width, 0.15 m in height, and a straight-edge crack with distance

Material properties of 304SS.

Young’s modulus | 2.04 × 10^{5} |

Poisson’s ratio | 0.3 |

Mass density ^{3}) | 7860 |

Yield stress | 275.8 |

Ultimate stress | 620 |

Fracture toughness ^{0.5}) | 7645 |

Threshold value ^{0.5}) | 121.6 |

Crack growth properties of 304SS.

| 6 × 10^{−10} |

| 0.25 |

| 0.25 |

| 3.0 |

| 0.3 |

| 2.5 |

Because the damping properties in the stainless steel subjected to deformation and temperature, the maximum damping loss factor is not bigger than 0.01 [

Figure

Frequencies change with crack severity.

Figure

Friction damping loss factor with crack severity change under different friction coefficients.

Figure

The maximum dynamic stress with crack severity change (

Figure

The maximum dynamic stress with crack severity change of breathing cracked beam.

Figure

The maximum dynamic SIF with crack severity change for different frequency (

Figure

The maximum dynamic SIF with crack severity change of breathing cracked beam.

Figures

Crack depth with vibration cycles for different frequency (

Breathing crack depth with vibration cycles for different friction coefficient and frequency.

Figure

Frequency ratio with vibration cycles (

In this article, the stiffness model and the friction damping model of a breathing cracked beam are derived. The first mode frequency is analyzed by using a SDOF model which is simplified by the Galerkin method. The dynamic stress is derived and the crack growth is studied. The effects of the breathing behaviour on the dynamic response and crack growth are revealed so as to analyze the coupling mechanism of the crack growth and dynamic response approximately. The main conclusions are obtained:

(1) During vibration analysis of a cracked beam, the crack closure effect between opening and closing states must be considered. A breathing crack model is better than the open crack model for dynamic response and crack growth analysis. A sinusoidal breathing function is useful to simulate the transition state between the fully open state and the fully closed state. It meets the object facts and the physical natural of crack. It may be closer to the real motion of the breathing crack than that of the square wave function. Using a breathing crack model is more realistic than using an opening crack model.

(2) Based on energy principle and Coulomb friction model, the friction model is set up and friction damping loss factor is derived. Friction damping not only plays important role in the maximum dynamic response, but also plays important role in the crack growth. Especially to resonance response and crack growth, friction damping effect is very obvious.

(3) The Galerkin method is used to simplify the breathing cracked beam into a SDOF system. The first mode frequency equation is derived approximately. The first mode frequency changes with the crack severity. The frequency decrease ratio of the breathing cracked beam is slower than that of the open cracked beam. If the closure behaviour of the breathing crack is not considered, the degree of crack growth may be underestimated.

(4) The modified Forman equation is employed to simulate the crack growth. A crack growth life analytical method is proposed with the coupling effect of vibration and fatigue crack growth included. Breathing behaviour of the crack is important to crack growth. The dynamical properties play an important role in crack growth. When the exciting frequency is near to the first mode frequency, the crack growth ratio is very fast. The crack growth life of the breathing cracked beam is smaller than that of the open cracked beam.

Crack depth, initial crack depth

The

Area of cross section

Width of beam cross section

Fatigue properties of materials

Coefficient of modal damping

Flexibility of the intact beam

Flexibility due to the presence of the crack

Total flexibility of the open cracked beam

Sliding distance between crack surfaces

Young’s modulus

Closure function of breathing crack

Harmonic load

The generalized force, loading amplitude

Normal pressing force, friction force

Average pressing force

Crack shape function

Height of the beam cross section

Area moment of inertial of the beam cross section

The generalized stiffness

Stiffness of the breathing cracked beam

Stiffness of the close cracked beam

Stiffness of the open cracked beam

Stiffness of the torsional spring

Dynamic stress intensity factor

The maximum dynamic stress intensity factor

Fracture toughness

The threshold of stress intensity factor

The range of stress intensity factor

Distance of crack from the fixed end

Length of the beam

The generalized mass

Moment

Fatigue properties of materials

Numbers of vibration cycle

Shear force

Stress ratio

Time

Amplitude-time function

The strain energy

Energy dissipation

Lateral deflection in

Modal function

The coordinate

Static displacement

Density of material, Poisson’s ratio

Friction coefficient

Excitation frequency

Frequency of the close cracked beam

Frequency of the open cracked beam

Material damping

Frictional damping

Amplify ratio

Frequency ratio

The breathing function

The Dirac function

Nominal stress, normal bulk dynamic stress

Ultimate stress, yield stress.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research work is supported by the National Natural Science Foundation of China (51565039).