^{1}

^{2}

^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

This study used numerical analysis to carry out a large number of numerical model calculations based on a new semicircular bend (SCB) model. Instead of existing numerical computation methods (_{I}, _{II}, and _{III}), investigate the influence of the geometry and material parameters on the fracture behavior under mixed mode I/III loading, and predict the fracture path. The results revealed that the limitations in the scope of the mixity parameter ^{e} in the previous studies can be overcome to a certain extent. The range of ^{e} was established under different Poisson ratios and can be used as a reference for actual material testing. The simulation path is in good agreement with the experimentally obtained fracture path, and the proposed method can be used to simulate the fracture path under mixed mode I/III loading.

Fracture mechanics focuses on the mechanical properties of materials and structures with cracks. The cracks can be inherent in the material or generated during manufacturing, and the existence and growth of these cracks decrease the bearing capacity of the structure or even cause its failure.

The semicircular bend (SCB) is a classic fracture mechanics test that is widely used to investigate solid materials owing to its inherent favorable properties, such as its simplicity, minimal machining requirement, testing convenience, and ease of reaching tensile failure as reported by Mahinda and Kuruppu [_{IC}, and its potential application in the mixed mode fracture test has been investigated [

Because engineering accidents, such as rock fracture and pavement rupture, occur frequently, the investigation of mode III has attracted a great amount of interest. Moreover, observations have shown that flat cracks tend to reorient themselves to oblique planes during propagation and can grow under mixed-mode I/III conditions [

In previous studies [

In the previous numerical studies [_{I}, _{II}, and _{III} values were obtained by the

The schematic diagram of the semicircular geometry is shown in Figure _{I}_{II}_{III}) calculated at the surface contact zone are not reliable, because the local plane strain condition is no longer maintained. The element at the crack tip may be severely distorted, as shown in Figure

Geometry and loading configuration of SCB specimen. (a) Three-dimensional view; (b) front view; (c) top view.

Grid model of SCB specimen. (a) Abaqus mesh. (b) Franc3D mesh. (c) Crack tip mesh.

Element distribution and crack tip types. (a) Hexahedral element. (b) Element rings. (c) Singular wedge-shaped element. (d) Integral domain.

In the created models, radius of SCB specimen (

The fracture parameters (_{I}, _{II}, and _{III}) can be directly extracted using the following expressions. For linear analysis, we can add two valid solutions, and the result is a valid solution, as follows:

Let us consider the corner mark (1) solutions as the Abaqus results and the corner mark (2) solutions as the solutions that we can select. These can be substituted into the expression for the

According to Betti’s reciprocal theorem, the following relationship holds:

By collecting terms, the following relationship can be obtained:

The crack-tip energy release rates can be determined from Irwin’s crack closure integral for small scale yielding assuming plane strain conditions, as follows:

By substituting into the expression for the energy release rate, we obtain the following relationship:

By equating the two definitions for the

We used the Abaqus results for solution (

The asymptotic solution of pure mode crack is evaluated.

_{I} | _{II} | _{III} | |
---|---|---|---|

a | 1.0 | 0.0 | 0.0 |

b | 0.0 | 1.0 | 0.0 |

c | 0.0 | 0.0 | 1.0 |

From the analytical expressions for the crack-front fields, we can obtain the following:

By substituting the value of ^{(2)}, reported in Table ^{(1)}’s, as follows:

_{I}, _{II}, and _{III} can be expressed by the dimensionless parameters _{I}, _{II}, and _{III}, as follows:

To verify that the numerical simulation method can accurately realize mixed I/III mode fracturing, the _{I}, _{II}, and _{III} of the crack tip along the specimen thickness were adopted after normalization processing as shown in Figure _{n} (_{Im} is the maximum value of mode I. The crack length ratio (

Variations of _{n}/_{Im} through crack front of the SCB specimen.

Figure _{I} along the direction of the specimen thickness remained approximately unchanged within a certain range (0.1 < _{I} value of the specimen.

Figures _{I}, _{II}, and _{III} is not reliable, and according to the traditional definition of the singular stress field close to the free surface, it is widely accepted that the crack tip stress field is different, as discussed by Bažant and Estenssoro [_{I}, _{II}, and _{III} close to the free surface should not be considered, and this numerical method is valid and feasible for testing the mixed mode I/III fracture. Notably, in mixed mode I/III loading, the crack tip along the thickness of the specimen completely corresponds to the point where _{I}, _{II}, and _{III} of the specimen.

The mixity parameter ^{e} can be used to describe the relative contributions of mode I and mode III, as expressed by equation (^{e} is equal to one, and decreases as the contribution of mode III increases:

To quantify the role of mode I and mode III in the numerical simulation, we propose to substitute equation (^{e}| can vary from 0 to 1, corresponding to the pure mode III and mode I loading conditions, respectively:

The crack length ratio (_{I}, _{III}, and |^{e}| were investigated. Figures _{I}, _{III}, and |^{e}|, with a crack inclination angle (

Variations of _{I} and _{III} with

Variations of |^{e}| with

The following conclusions were drawn from image analysis:

_{III} remained at zero, only when _{I}. But for the _{III}, the increase of

For the pure mode I (

With the increase of _{I} gradually decreased, and |_{III}| first increased and then decreased. Additionally, _{III}|_{max} slowly increased with the increase of _{I} to the changes of _{III} was maximum at |_{III}|_{max}.

The increase of ^{e}| values when ^{e}|. Additionally, as

Half of the loading point span ratio (_{I}, _{III}, and |^{e}| were investigated. Figures _{I}, _{III}, and |^{e}| with the crack inclination angle (^{e}|_{min} obtained with the change of

Variations of _{I} and _{III} with

Variations of |^{e}| with

Variations of |^{e}|_{min} for different values of

The following conclusions were drawn from image analysis:

For the _{I} values. But for the _{III}, the increase of

The value of _{III}|_{max} stabilized at approximately 49°.

For the values of |^{e}|, the increase of ^{e}; when the value of ^{e} was smaller under the same

The value of ^{e}|_{min} of all SCB specimens. As ^{e}|_{min} value was obtained and the minimum value was 0.39. In this case, the contribution of model III was obviously higher compared with that of model I. Additionally, for smaller ^{e}|_{min}; however, when ^{e}|_{min} was more sensitive to the variations of

Notably, the rule discussed in the previous section is not described.

Using the _{I}, _{II}, and _{III} values of the crack tip and the maximum tensile stress criterion (MTS), three model groups (

A comparison diagram between the paper [

Entire process of crack growth path. (a) Pure mode I with

Under mixed mode I/III loading conditions, the crack surface growth morphology was different from the in-plane fracture. Figure

Considering the direction of the crack kink angle (

The relationship between the release rate of the mechanical energy and the angle is expressed as follows:

By superposing the contributions of mode III, the corresponding transformed stress intensity factor can be expressed as follows.

By substituting equation (

Nonplanar propagation mode of crack. (a) Crack surface deflection; (b) crack surface torsion.

As can be seen in the graph, in the case of pure mode I fracturing (_{III}/_{I} = 0), _{III}/_{I} = 0.25), as shown in Figure

Variations of kink angle with

0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | |
---|---|---|---|---|---|---|---|

0 | 0.055 | 0.003 | 0.083 | 0.129 | 0.094 | −0.007 | 0.027 |

10 | −5.691 | −3.742 | −2.034 | −0.008 | 2.198 | 4.115 | 6.070 |

20 | −12.993 | −8.229 | −3.891 | 0.0435 | 3.933 | 8.833 | 13.207 |

30 | −20.466 | −13.083 | −6.010 | 0.071 | 6.762 | 13.082 | 20.372 |

40 | −25.093 | −16.241 | −7.939 | 0.055 | 8.163 | 16.276 | 25.144 |

50 | −30.394 | −19.982 | −9.789 | −0.169 | 10.919 | 21.063 | 31.432 |

60 | −39.125 | −24.966 | −12.113 | −0.025 | 13.049 | 26.081 | 40.163 |

The following conclusions were drawn from this study:

Numerical modeling combined with Abaqus and Franc3D analysis was used to investigate the influence of half of the loading point span ratio (

The increase of _{I}. But for the _{III}, only the increase of _{I} decreased with the changes of _{III} became maximum at |_{III}|_{max}.

The obvious approach toward reducing the mixity parameter |^{e}| is to decrease ^{e}| by changing

The |^{e}|_{min} values of the SCB specimens were obtained under different

The numerical parameters obtained by the

All data used to support the findings of this study are included within the article.

There is no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge Senior Engineer Haijun Wang and Dr. Shuyang Yu for their supports in this research program, and they also extend their thanks to the National Key R&D Program of China (2017YFC0404902), the National Natural Science Foundation of China (51739008), the Natural Science Foundation of Jiangsu Province of China (BK20171130), and the basic scientific research operating expenses of the Public Welfare Scientific Research Institutes at the Central Level of China (Y419005).