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Zirconia grinding fixtures have been widely used in semiconductor industry to improve the quality and precision of the products. For maximizing the service life and minimizing the risks of accidental damage, it is critical to have a better understanding of the fatigue life of zirconia grinding fixtures. To this end, a boundary element method is developed in this paper to investigate their crack growth and fatigue life. To validate the proposed method, the stress intensity factor of a typical plate structure with initial cracks is considered. On this basis, Paris Law is employed in the boundary element model to further study the crack growth and stress distributions in the zirconia fixture under cyclic loads. Numerical results show that stress concentration occurs at the pillar of the fixture, and crack growth is perpendicular to the loading direction.

In semiconductor industry, both grinding and polishing technologies play important roles in the procedures of product manufacturing. In order to improve the quality and precision of products, it is critical to choose an ideal auxiliary fixture to control their surface profiles [

Finite element method (FEM) is considered to be an effective method in investigating the fracture properties [

An alternative approach to study fracture of materials is to use boundary element method (BEM). Compared with the traditional FEM, the BEM provides a higher computational efficiency and accuracy in solving stress concentration problems. BEM has been extensively applied for the solution of stress intensity factor (SIF) [

In this paper, an extension of the BEM is presented to investigate fatigue crack propagations of the zirconia grinding fixture used in semiconductor industry. For each example, the open crack is considered. Stress intensity factor, which is related to the crack size and structural geometric characteristics, is accurately calculated. Moreover, a highlight of the crack evolutions under the fatigue load is studied. The outline of this paper is as follows: Section

Some classic mechanical methods can be employed to solve crack problem [

Singularity problems, such as structural crack, can be easily found in grinding fixtures under multiple cyclic loads. The crack singularity leads to a deterioration of the performance in the procedure of finite element solution due to the singularity property of

Compared with other solution methods based on the partial differential equations, the BEM reduces the dimension of the crack problem. In addition, it is easier for the researchers to discretize the boundary rather than the solution domain. Relatively simple elements are needed to simulate the boundary shape, and the linear algebraic equations with lower order will be acquired. Therefore, the BEM is considered to be a higher effective method to solve the crack singularity problem compared to the FEM. Herein, the BEM is employed to effectively solve the SIF, and the BEM is extended to investigate the mechanical properties and predict their service life.

The boundary integral equation in a 3D elastic domain without consideration of body forces can be written as follows [

In each element, the local coordinate

The relationship between global coordinate system and local coordinate system.

In each element, the traction forces

After discretizing into surface elements, the boundary integral equation can be expressed as follows:

Substituting equation (

By employing the transformation matrix

In fracture mechanics, the open mode or type I crack (as shown in Figure

Open crack mode.

The SIF, which relates to many factors, such as crack size and structural geometric characteristics, reflects the strength of elastic stress field at crack tip. In the modeling procedures by the BEM, longitudinal stresses (

Stress status near the crack tip. (a) Diagram of a three-dimensional crack. (b) Crack coordinate systems.

Displacement components at the crack tip can be written as

If the angle ^{o}, the displacement components at crack region can be simplified as

By employing the displacement components derived from the BEM, the SIF can be expressed as

To validate the proposed BEM method,

A circular hole of

Plate structure with bilateral cracks.

For the open crack mode as shown in Figure

To discuss the relations between SIF and crack length, the boundary element model of the plate as shown in Figure

(a)

Figure

Mises stress distributions of the plate with different crack lengths: (a) 0.5 mm, (b) 1.0 mm, (c) 1.5 mm, (d) 2.0 mm, (e) 2.5 mm, (f) 3.0 mm, (g) 3.5 mm, and (h) 4.0 mm.

A rectangular plate made of zirconia materials with a unilateral crack is also considered here to further validate the established model. The dimension parameters are as follows: length, 10 mm; width, 3 mm; and height, 30 mm. A rectangular precrack with a length of 1 mm is located at the stress concentration region to investigate the crack growth. The constraints and external loadings are applied on the structure as shown in Figure

Plate structure with a unilateral crack.

For the fracture mode of this plate with a unilateral crack as shown in Figure

For investigating the

In order to execute the crack propagation analysis, a precrack with length of 1 mm is also implanted into the boundary element model as shown in Figure

Mises stress distributions of plate with different crack lengths: (a) 1 mm, (b) 2 mm, (c) 3 mm, (d) 4 mm, (e) 5 mm, (f) 6 mm, (g) 7 mm, (h) 8 mm, and (i) 9 mm.

Figure

The model of grinding fixture.

Due to the high hardness and brittleness of zirconia materials, machined surfaces of the fixture are prone to produce microcracks, and the cracks will immediately accumulate. In service, periodical cycling loads will be applied on the holes of the adjusting heads to obtain a desired bottom surface profile. It is critical for researchers to grasp crack status and fatigue life to maximize their service life. To this end, the 3D model of the fixture based on the BEM is established. In the fatigue life analysis, the surfaces of the fixed pin holes C and C′ are fully restrained. A cyclic loading is supposed to apply on the hole of force adjusting heads A1–G1. An external load of 0.1 N is applied on the surface

The flow chart of fatigue crack predictions is shown in Figure

The Object Solid Modeler (OSM) is employed to establish the 3D BEM of the grinding fixture.

Meshing the surface and inputting the material properties, boundary conditions. The discretized model based on the BEM is established by employing the Fracture Analysis Code (FRANC3D).

Before the fatigue analysis, a small external load is applied on the fixture, and the stress and deformation cloud diagrams will be acquired. According to the stress distributions, an initial crack is embedded into the maximum stress area of the fixture.

Applying cyclic loading on the fixture, the crack will accumulate and extend. At each crack increment, the SIF will be recalculated.

According to the history data of the SIF and the corresponding crack length, the fatigue life of fixture will be determined.

The process of crack propagation can be summarized as follows:

Based on the boundary element method (BEM), the displacement field of crack surface is solved, and the distribution of SIF along the current crack front is solved by crack opening displacement (COD) fitting method

Based on classical Paris Erdogan Law, the propagation size and position of each point on the crack front edge are calculated

The new crack front is obtained by fitting the new position after propagation by least square method

Repeat the above steps for the specified simulation accuracy

The flow diagram of FRANC3D.

In the mechanical analysis of the fixture, it will be found that the maximum stress is located at the root of the pillar

Mises stress distributions of the fixture. (a) Fixture. (b) Partial enlarged figure.

When calculating the stress intensity factors of three-dimensional cracks by FRANC3D, the cracks are usually simplified to round, ellipse, and rectangle according to the actual situation. According to the experimental observation, the fatigue crack in the pillar of the fixture is basically rectangular, which is continuously expanding forward. Herein, we simplify the crack model into a rectangle in order to further investigate the fatigue crack propagation, and a unilateral crack of 0.1 mm is implanted into the fixture as shown in Figure

Mises stress distributions of the fixture with a unilateral crack. (a) 0.1 mm. (b) 0.5 mm. (c) 1.0 mm. (d) 1.5 mm. (e) 2.0 mm. (f) 2.5 mm.

In order to determine the fatigue life of the fixture under cyclic loads, herein the classical Paris Erdogan Law is employed. The relationship between crack growth rate and SIF range can be written as [

According to the crack propagation law studies mentioned above, an initial crack is supposed and located at the bottom of the pillar. The crack evolution trajectory in the fixture can be confirmed as shown in Figure

Crack in the fixture. (a) Cross section of the crack. (b) Partial enlarged figure of the crack trajectory.

Stress intensity factor (SIF) obtained from each crack growth step (crack length) can be determined by maximum circumferential stress. As shown in Figure

It can be seen from this that the corresponding

Relationships between cycle number and crack length.

In this study, the BEM was developed to predict fatigue life of the zirconia fixture. To validate the proposed method, the stress intensity factors of plate structures with a unilateral precrack and bilateral precracks are investigated. Compared with the theoretical solutions and the FEM, numerical results by the BEM provide a high accuracy in predicting the SIF. The results show that the crack propagation will sharply increase the SIF at crack tip. On this basis, the crack propagation and service life of the zirconia grinding fixture under cyclic load are further studied. The results indicate that the maximum stress is located at the root of the pillar, and the crack length of 1.2 mm is suggested to be the finial failure length of the fixture to avoid a sudden fracture for the fixture.

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical and time limitations.

The authors declare no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (no. 51675397), National Natural Science Foundation of Shaanxi Province (no. 2018JZ5005), China Scholarship Council (no. 201706965037), Fundamental Research Funds for the Central Universities (no. XJS190405), and the 111 Project (no. B14042). The first author is grateful to the Engineering Department, Lancaster University, for the support he has received during the course of his visit.