Stress Intensity Factor and Shape Coefficient Correction of Non-Penetrating Three-Dimensional Crack for Brittle Ampoule Bottle with V-Shaped Notch

Abstract

The automatic opening of an ampoule bottle is key to making the operation of the sterility inspection automatic. During the automatic opening, the fracture characteristics on the neck of ampoule bottle need to be deeply understood to avoid the contamination of the samples by preventing the glass fragments from dropping into the ampoule bottle. This paper presents the calculation of fracture characteristic parameters, such as the coefficient K of the stress intensity factor (SIF) and the coefficient F of shape factor (shape coefficient), based on the finite element method (FEM) for a non-penetrating three-dimensional crack. Mechanical and mesh models were built for the special structure of an ampoule bottle with a V-shaped notch, and the influence of mesh size on the coefficient K was evaluated. The mathematical expressions of stress intensity factor and shape coefficient of three types of cracks were established. The results demonstrate that the crack ellipticity (a/c) and crack relative depth (a/t) have significant effects on the K_I, K_II, and K_III of the SIFs. The K_I plays a dominant role, which follows a symmetrical distribution at the symmetrical position on both sides of the deepest crack point, whereas the K_II and K_III can be negligible. The corrected shape coefficient F_I decreases with increasing ellipticity a/c and increases with increasing relative depth a/t under the combined tensile stress and bending loads. The comparison to the literature shows the calculation of the corrected shape coefficient has a high accuracy based on the FEM, which will be applicable and reliable for non-penetrating three-dimensional cracks.

1. Introduction

Sterility inspection plays an important role in controlling food and drug quality for enterprises and regulatory agencies inspecting qualified products. Therefore, the requirements of sterility tests are clearly listed in the latest 2011 GMP (Good Manufacturing Practice for Pharmaceuticals) [1,2,3], the 2015 Chinese Pharmacopoeia and the United States pharmacopeia USP31-NF26 “71” etc. [4,5] editions. The sterility test of the samples can adopt the membrane filtration method and the direct inoculation method. Conventionally, the sterility inspection relies on manual operations, which have low automation and intelligence levels, leading to low efficiency. The key processes of sterility inspection will be greatly affected by personnel. It is possible that there are significant differences in sterility inspection results and efficiency due to the operations of different personnel. Therefore, it is imperative to improve intelligence levels through the automatic design and operation.

For the ampoule bottle filled with the test samples, its automatic opening is key to the operation of the sterility inspection. The process of an automatic opening of an ampoule bottle includes cutting and breaking steps. The cutting aims to produce a non-penetrating crack, avoiding glass fragments breaking and then entering the ampoule bottle to contaminate the samples. The ampoule bottle is made from the glass, which has high compressive strength, hardness and good elasticity at room temperature. In comparison to steel parts, plastics and building materials, the glass has a linear relationship between stress and strain. Griffith first proposed the theory of low stress brittle fracture of glass and established the fracture mechanics of glass [6,7]. Since then, fracture mechanics has been used to study the fracture behavior of the glass, which plays a crucial role in the use of the glass.

Guo et al. [8] studied the phenomenon of crack healing and its relationship with glass transition and characterized the glass transition and explored the glass transition mechanism. Zhao et al. [9] conducted a systematic study on the fracture surface of brittle metallic glass under compressive load, proposed its fracture mechanism and provided a theoretical basis for the development of dynamic fracture of brittle metallic glass. Richard [10] investigated the crack growth rate of the fracture of silicate glass and estimated the kinetic energy at the crack growth by using the strain energy release rate. Pourmoghaddam et al. [11,12] demonstrated the strain energy and fracture pattern morphology of thermally tempered glass and elaborated the relevant parameters that affect glass fracture. For different materials of glass, the fracture characteristics and mechanical properties will be different. The ampoule bottles are fabricated from high borosilicate glass. Compared with ordinary glass, high borosilicate glass has greater thermal stability, mechanical properties, acid and alkali resistance, and no toxic side effects. Therefore, it is widely used in various fields, such as chemical industry, household and medical treatment.

During the opening of the ampoule bottle, the crack left by cutting the neck of ampoule bottle can be regarded as a semi-elliptical crack through observation and measurement. The initial crack is a non-penetrating three-dimensional crack; its width and depth are different because of the difference in initial incision. According to the geometric characteristics of cracks, the cracks can be divided into three types: penetration cracks, deep buried cracks and surface cracks. According to the mechanical characteristics of cracks, the cracks can be divided into type I cracks (open cracks), type II cracks (slide cracks), type III cracks (tear cracks) and compound cracks. In their actual conditions, most of the cracks are type I cracks [13]. When the SIF of this type of crack reaches a certain critical value, i.e., the material’s fracture toughness, the crack propagates rapidly (called instability propagation) and causes the structure fracture of the component [14]. The early research mainly aimed at the calculation of the SIF of two-dimensional cracks. In practical engineering applications, the forcing components create instability and the fracture is due to the existence of three-dimensional cracks. For the opening of the ampoule bottle, its non-penetrating crack mostly occurs in three-dimensional cracks. Even if the final form of the crack is a through crack, in most cases, its initial state is also in the form of a non-penetrating crack.

At present, the main solutions to the three-dimensional crack’s SIF include: Green’s function method, the finite element method, the dual boundary element method, the multi-zone method, the displacement discontinuity method, etc. [15]. Among them, the FEM has the advantages of high efficiency, low cost, good reliability and is not restricted by the load and geometric shape of the cracked body, so it has been widely used. The literature [16,17,18] demonstrated the applicability and reliability of using the FEM to analyze and evaluate the SIF. Therefore, several important parameters for calculating cracks in fracture mechanics can be easily studied by the FEM, such as the coefficient K of the SIF and the coefficient F of shape factor. As an extremely important parameter to characterize the stress-strain field at the crack tip, the accurate and reliable calculation of the coefficient K is the main criterion for predicting the fracture of cracks in linear elastic materials [19]. Newman et al. [20] demonstrated the empirical SIF equation for surface cracks, which was used to predict crack propagation under tensile or flexural fatigue loading in components of various crack shapes and crack sizes. Miyazaki et al. [21] studied the effect of component geometry and residual stress on the SIF of surface cracks, and the SIF was a good estimate of crack propagation behavior in stress corrosion cracking. Many scholars’ studies have shown that the SIF plays an important role in predicting crack propagation and component fracture. The crack shape coefficient F is a key parameter for the calculation of the coefficient K, which is closely related to the geometry of the cracks and the loading patterns. Predan et al. [22] presented the effect of semi-elliptical cracks on several types of SIFs at different relative depths and at various aspect ratios, suggesting that that crack shape had a significant effect on the fracture. Carpinteri et al. [23] investigated the effects of a notch shape and different cracks on the SIF under tensile and bending loads. Yazdanmehr et al. [24] measured the SIF of the V-shaped mouth of the component under different loads by the focal ray method. Since the body of ampoule bottle is a cylinder and the head presents a convex-shaped surface, the neck can be regarded as a V-shaped notch which is prone to stress concentration, and it makes the analysis of the coefficients K and F become more complicated.

Generally, the SIF calculation method of a typical cracked body model can be approximated by querying the manual. However, under complex loads, for structural components like an ampoule bottle with complex crack shapes or structures, the SIF in many cases is difficult to find from the manual. The main form of load in the opening process of an ampoule bottle is a combined load including both tensile stress and bending moment. However, there are few studies on the crack under the combined loads and few reports on the crack of the neck with a V-shaped notch. Therefore, considering the particularity of the structure of the ampoule bottle and the analytic importance of the SIF and the shape coefficient during its opening, this paper focuses on investigating the calculation of the coefficient K of the SIF and the corrected coefficient F of shape factor based on the FEM, for semi-elliptical cracks on the neck with a V-shaped notch under the combined loads of tensile stress and bending moment. The fracture characteristics of the ampoule bottle can guide its automatic opening design, which plays a vital role in the application of the ampoule bottle for the sterility inspection in the medical and health field.

2. Theory and Method of Fracture Mechanics Based on the FEM

2.1. Mechanical Model of an Ampoule Bottle with the Crack on the Neck

Ampoule bottles are widely used to hold vaccines, serums and other pharmaceutical solutions, as shown in Figure 1a. The ampoule bottle with the broken neck is found in Figure 1b. The mechanical model of the ampoule bottle established in this paper and the cross section of the neck crack are shown in Figure 2. The fracture characteristics of the semi-elliptical crack with a V-shaped notch on its neck is studied. The material of the ampoule bottle is set as high borosilicate glass, and the physical properties are listed into

Table 1. The material and physical properties of the ampoule bottle.

Items	Material	Density	Elastic Young’s Modulus	Poisson’s Ratio
Ampoule bottle	High borosilicate glass	2.23 g/cm3	67 GPa	0.2

.

The neck of the ampoule bottle can be broken when the loads are applied to the bottle head. The restraint and load conditions (see Figure 2) are set as follows: fully restrained at the bottom of the ampoule bottle, and an upward tensile stress σ on the bottle head. Simultaneously, a bending moment W is applied as a simulated boundary condition. The parameters of the semi-elliptical crack for the FEM calculation include: (1) the outer radius of the neck R, (2) the inner radius r, (3) the length of short semi-axis of the ellipse a, (4) the length of long semi-axis c, and (5) the crack located in the middle of the neck of the ampoule bottle. The maximum angle of the crack tip on both sides from the deepest point of the crack is ${\theta }_{max}$.

2.2. Mesh Model of the FEM in Fracture Mechanics

According to fracture mechanics, there is a stress singularity phenomenon in the area near the crack tip, and the stress asymptotic field can be expressed as:

$$\mathsf{\sigma }=\frac{K}{\sqrt{2\mathsf{\pi }r}}f\left(\theta \right)$$

(1)

where r and θ represent the polar coordinates, and the origin located in the crack tip; f(θ) is a dimensionless parameter and K is the coefficient of the SIF. According to the stress asymptotic field equation, when r approaches zero, the stress at the crack tip approaches infinity, which is called stress singularity [25]. Barry et al. [26] found that the singularity of the stress field near the crack tip can be achieved by moving the middle node of the second-order element near the crack tip along the direction of the crack tip to 1⁄4 node near the crack tip.

In view of the stress singularity at the crack tip, ANSYS provides the mesh division function to deal with it, so that a three-dimensional degenerate singular element can be obtained conveniently. The grid element used in this paper is a three-dimensional degenerate singular isoparametric element generated by SOLID186 element degradation [27]. Figure 3 shows the mesh model of the FEM based on ANSYS, and the local crack meshes were refined to improve calculation accuracy.

2.3. Stress Intensity Factor and Shape Coefficient of the Crack

In general, the SIF, as a physical parameter of linear elastic fracture mechanics, can be quantified by the calculation or finite element analysis. The SIF does not represent the stress at a certain point but the physical quantity of stress field strength. Therefore, it is more acceptable to use the SIF to measure the stress field strength of crack tip for establishing failure conditions [28].

The general expression of the coefficient K of the SIF is:

$$K=F\sigma \sqrt{\mathsf{\pi }a}$$

(2)

where a is the crack depth (the length of short semi-axis of the ellipse crack in Figure 2, σ is the nominal stress, and F is the shape coefficient.

2.4. Calculation of the SIF Based on the FEM

The calculation of the coefficient K of the SIF is based on the interaction integral of finite elements. It can be performed area fraction when dealing with two-dimensional cracking problems and volume fraction for three-dimensional cracks. This dealing is highly accurate and requires fewer units to calculate the SIF, in comparison to the traditional displacement expansion method. The equation of the interaction integral can be written as [29,30]:

$$I=-\int q_{i,j}\left({\sigma }_{ki}{\epsilon }_{ki}^{aux}{\delta }_{ij}-{\sigma }_{kj}^{aux}{u}_{k,i}-{\sigma }_{kj}{u}_{k,i}^{aux}\right)dV/\int \delta q_{n}dS$$

(3)

In which ${\sigma }_{ki}$ and ${\sigma }_{kj}$ are the stress in the real field; $u_{k,i}$ is the displacement in the real field; ${\epsilon }_{ki}^{aux}$ is the strain in the auxiliary field; ${\sigma }_{kj}^{aux}$ is the stress in the auxiliary field; $u_{k,i}^{aux}$ is the displacement in the auxiliary field; $q_{i,j}$ is the crack propagation vector; $q_n$ is the normal direction of the crack propagation; ${\delta }_{ij}$ is the Kronitz symbol.

The relationship between the interaction integral and the SIF is [29,30]:

$$I=\frac{2}{E^{*}}\left(K_{\mathrm{I}}{K}_{\mathrm{I}}^{aux}+K_{\mathrm{II}}{K}_{\mathrm{II}}^{aux}\right)+\frac{1}{\mu }{K}_{\mathrm{III}}{K}_{\mathrm{III}}^{aux}$$

(4)

where K_I, K_II, K_III are the coefficients of three types of SIFs, respectively; $K_{\mathrm{I}}^{aux}, K_{\mathrm{II}}^{aux}, K_{\mathrm{III}}^{aux}$ are the coefficients of type I, type II and type III SIFs in the auxiliary field, respectively; If $E^{*}$ = E, it is the plane stress; If $E^{*}$=$E/\left(1-v^2\right)$, it represents the plane strain; E is the elastic Yong’s modulus; v is the Poisson’s ratio; μ is the shear modulus.

2.5. Correction of Crack Shape Coefficient

To simplify the analysis of the cracks, the real cracks can be regarded as the combination of the above three types of SIFs. For practical engineering problems, the correction of crack shape coefficient can help us to quickly calculate the SIF by querying the SIF manual [28]. It can be found that the corrected shape coefficients of three types of SIFs can be expressed as:

$$\left\{\begin{array}{c}{F}_{\mathrm{I}}=K_{\mathrm{I}}/\sigma \sqrt{\mathsf{\pi }a}\\ F_{\mathrm{II}}=K_{\mathrm{II}}/\tau \sqrt{\mathsf{\pi }a}\\ F_{\mathrm{III}}=K_{\mathrm{III}}/\tau \sqrt{\mathsf{\pi }a}\end{array}\right.$$

(5)

2.6. Verification Method of Shape Coefficient Correction

In Equation (5), the coefficient F is the corrected shape coefficient containing the V-shaped notch. To verify its effectiveness, the corrected shape coefficients not containing the V-shaped notch are obtained by converting Equation (5), and a comparison is made to the corrected shape coefficients without V-shaped notches in the literature. The relationship between the shape coefficient for the cracks with the V-shaped notch and without the V-shaped notch can be approximately expressed as:

$$F_{\mathrm{N}}=\left\{\begin{array}{c}F/K_t, \frac{\rho }{R}\to 0\\ F, \frac{\rho }{R}\gg 0\end{array}\right.$$

(6)

where F_N is the corrected shape coefficient of the crack without the V-shaped notch, $K_t$ is the elastic stress concentration coefficient with the V-shaped notch during stretching or bending.

He et al. [31] investigated the variation law of the SIF of semi-elliptical cracks on the cylinder surface under bending and torsion loads with the influence of relative depth of ellipse and relative thickness of cylinder and relative depth of crack and obtained the curve of the corrected shape coefficient at the deepest point of the crack. Meng et al. [32] calculated the three-dimensional stress field at the crack front at the root of the V-shaped notch using a three-dimensional FEM, analyzed the effect of notch restraint on the stress state at the crack front and established an empirical solution for the SIF by considering the effect of notch restraint. However, for special structures such as an ampoule bottle, whose neck is a special V-shaped cut, cracks exist on its surface. As the ampoule bottle is subjected to the loads, fatigue cracks are most likely to occur at the notch, resulting in an unstable fracture. The schematic of the V-shaped notch area is shown in Figure 4, where r₀ represents the depth of the arc at the notch, ρ is the radius of curvature of the arc at the root of the notch, β is the notch angle and p and q are auxiliary parameters to determine the relationship between ρ and r₀.

Filippi et al. [33] obtained an accurate stress field equation for calculating notch stress by compensating the influence of finite size on the stress field. In this paper, the stress field equation of the Filippi notch under tensile stress load can be expressed as:

$$\begin{array}{cc}\left(\begin{array}{c}{\sigma }_{\theta }\\ \begin{array}{c}{\sigma }_r\\ {\tau }_{r\theta }\end{array}\end{array}\right)=& {\lambda }_1{r}^{{\lambda }_1-1}{a}_1(\left\{\begin{array}{c}\left(1+{\lambda }_1\right)\cos \left(1-{\lambda }_1\right)\theta \\ \left(3-{\lambda }_1\right)\cos \left(1-{\lambda }_1\right)\theta \\ \left(1-{\lambda }_1\right)\sin \left(1-{\lambda }_1\right)\theta \end{array}\right\}+{\chi }_{b_1}\left(1-{\lambda }_1\right)\left\{\begin{array}{c}\cos \left(1+{\lambda }_1\right)\theta \\ -\cos \left(1+{\lambda }_1\right)\\ \sin \left(1+{\lambda }_1\right)\theta \end{array}\right\}\\ & +\frac{q}{4\left(q-1\right)}{\left(\frac{r}{r_0}\right)}^{{\mu }_1-{\lambda }_1}\left[{\chi }_{d_1}\left\{\begin{array}{c}\left(1+{\mu }_1\right)\cos \left(1-{\mu }_1\right)\theta \\ \left(3-{\mu }_1\right)\cos \left(1-{\mu }_1\right)\theta \\ \left(1-{\mu }_1\right)\sin \left(1-{\mu }_1\right)\theta \end{array}\right\}\right]\\ & +{\chi }_{c_1}\left\{\begin{array}{c}\cos \left(1+{\mu }_1\right)\theta \\ -\cos \left(1+{\mu }_1\right)\theta \\ \sin \left(1+{\mu }_1\right)\theta \end{array}\right\})\end{array}$$

(7)

Under shear load, the stress field equation of the Filippi notch can be written as:

$$\begin{array}{cc}\left(\begin{array}{c}{\sigma }_{\theta }\\ \begin{array}{c}{\sigma }_r\\ {\tau }_{r\theta }\end{array}\end{array}\right)=& {\lambda }_2{r}^{{\lambda }_2-1}{a}_2(\left\{\begin{array}{c}\left(1+{\lambda }_2\right)\sin \left(1-{\lambda }_2\right)\theta \\ \left(3-{\lambda }_1\right)\sin \left(1-{\lambda }_2\right)\theta \\ \left(1-{\lambda }_1\right)\cos \left(1-{\lambda }_2\right)\theta \end{array}\right\}+{\chi }_{b_1}\left(1+{\lambda }_1\right)\left\{\begin{array}{c}\sin \left(1+{\lambda }_2\right)\theta \\ -\sin \left(1+{\lambda }_2\right)\\ \cos \left(1+{\lambda }_2\right)\theta \end{array}\right\}\\ & +\frac{1}{4\left({\mu }_2-1\right)}{\left(\frac{r}{r_0}\right)}^{{\mu }_1-{\lambda }_1}\left[{\chi }_{d_2}\left\{\begin{array}{c}\left(1+{\mu }_2\right)\sin \left(1-{\mu }_2\right)\theta \\ \left(3-{\mu }_2\right)\sin \left(1-{\mu }_2\right)\theta \\ \left(1-{\mu }_2\right)\cos \left(1-{\mu }_2\right)\theta \end{array}\right\}\right]\\ & +{\chi }_{c_2}\left\{\begin{array}{c}-\sin \left(1+{\mu }_2\right)\theta \\ \sin \left(1+{\mu }_2\right)\theta \\ -\cos \left(1+{\mu }_2\right)\theta \end{array}\right\})\end{array}$$

(8)

Characteristic parameters such as ${\lambda }_1$, ${\mu }_1$, ${\chi }_{b1}$, ${\chi }_{c1}$, ${\chi }_{d1}$, ${\lambda }_2$, ${\mu }_2$, ${\chi }_{b2}$, ${\chi }_{c2}$, ${\chi }_{d2}$ can be calculated by the Filippi notch stress field equation under different notch angles. In this paper, the notch angle was set to 135°, and the stress concentration factor under tensile stress load is:

$$K_t={\sigma }_{max}\sqrt{2\mathsf{\pi }}\left\{r_0^{{\lambda }_1-1}/\left(1+\frac{\left(1+{\mu }_1\right){\chi }_{d_1}+{\chi }_{c_1}}{\left(1+{\lambda }_1\right)+{\chi }_{b_1}\left(1-{\lambda }_1\right)}·\frac{q}{4\left(q-1\right)}\right)\right\}$$

(9)

The stress concentration factor under shear load is:

$$K_t={\sigma }_{max}\sqrt{2\mathsf{\pi }}\left\{r_0^{{\lambda }_2-1}/\left(1+\frac{\left(1+{\mu }_2\right){\chi }_{d_2}+{\chi }_{c_2}}{\left(1+{\lambda }_2\right)+{\chi }_{b_2}\left(1-{\lambda }_2\right)}·\frac{1}{4\left({\mu }_2-1\right)}\right)\right\}$$

(10)

The notch stress concentration factor of the ampoule bottle under the corresponding load can be calculated by Equations (7)–(10), and the corrected shape coefficient of the crack without a V-shaped notch $F_{\mathrm{N}}$ can be obtained by substituting $K_t$ into the Equation (6).

3. Results and Discussion

3.1. The Influence of Mesh Size on the Coefficient K

Mesh size of the FEM possibly affects the calculation accuracy of the SIF. Therefore, the influence of mesh sizes ranging from 0.1 mm to 2.0 mm on the SIF was evaluated, as shown in Figure 5. The SIF varies a little with the mesh sizes of 0.1 mm–0.7 mm. However, the SIF significantly fluctuates at 0.7 mm–2.0 mm. To obtain higher accuracy and efficiency, in this paper, the mesh size of 0.1 mm was selected to carry out the calculation of the SIF.

3.2. Results and Analysis of the Coefficient K at the Crack Front

Many parameters influence the coefficient K of the SIF, such as long semi-axial length, short semi-axial length and component thickness [34,35]. To analyze the relationship between the influence of various factors on the SIF, the effects of different crack ellipticity (a/c) and different crack relative depth (a/t) on the SIF were calculated in this paper, where t is the thickness of ampoule bottle body (t = R − r). Thus, the corrected shape coefficients of components under different relative thickness (R/r) were obtained. Supposing that the long half-axis length c was a fixed value, the values of a/c were set to 0.2, 0.4, 0.6, 0.8, respectively, the a/t values were set to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8, respectively. The effects of these parameters on the coefficients K of the SIFs are shown in Figure 6, Figure 7 and Figure 8. The abscissa shown in the figures represents the position ratio of different depth of the semi-elliptical crack front edge, i.e., ξ = $\theta /{\theta }_{max}$, and the origin is at the deepest point of the crack, where θ and ${\theta }_{max}$ are shown in Figure 2.

Figure 6 shows that the type I SIF (K_I) at the symmetrical position on both sides of the deepest semi-elliptical crack follows a symmetrical distribution. When the relative ellipticity ratio a/c is constant, but the relative crack depth a/t varies, that is, the ampoule bottle thickness gradually thickened, the curve changed from a concave and convex M-shape to a slightly protruding shape in the middle and a relatively smooth downward shape on both sides, and the SIF value at its corresponding position gradually decreased. When a/c = 0.2 and a/c = 0.4, the maximum value of SIF is located on both sides of the deepest crack point near the crack tip. When a/c = 0.6 and a/c = 0.8, the coefficient K_I of the SIF is the minimum at the deepest crack point, and the SIF increases as it gets closer to the crack tip. In addition, when a/c = 0.8, the SIF of the deepest crack point at any a/t is smaller than those at a/t = 0.7 and at a/t = 0.6. It means that when the crack ellipticity rate a/c reaches a certain value, the SIF in the deepest part of the crack may decrease as the a/t continuously increases.

It can be inferred from Figure 6 that the depth of the crack has a great impact on the type I SIF at the deepest point, whereas it has a small impact on the type I SIF at the front of the cracks on both sides. The thickness of the ampoule bottle directly affects the SIF of the entire crack front. It suggests that the thicker the bottle thickness is, the smoother the corresponding SIF curve becomes and the smaller the change is in the corresponding coefficient K_I of the SIF. When both the crack depth and the bottle thickness are changed, the effect of the crack depth on the crack SIF is greater than that of the thickness on the SIF.

The distribution curves of type II SIF (K_II) are shown in Figure 7, under both tensile stress and bending moment loads, which have intersection points in the region near the tip of both sides of the semi-elliptical crack. The K_II of near intersection points are approximately equal. The K_II is greatest at the deepest part of the crack and symmetrical at both sides of the deepest point. When a/c = 0.2, the K_II fluctuates greatly. It gradually becomes smaller and smoother as the ampoule bottle thickness increases.

It can be seen from Figure 8 that the K_III at the deepest part of the crack is approximately 0, and the negative value indicates the opposite direction of the symmetric shear force on both sides of the semi-elliptic crack center. When a/c is constant, the larger the a/t value is, and the greater the K_III value becomes. It indicates that when the thickness of the ampoule bottle becomes thinner, the larger the K_III is, and the more likely the ampoule bottle is to break.

According to type I, type II and type III SIFs at the deepest point of the crack, the ratios of $\left|K_{\mathrm{I}}/K_{\mathrm{II}}\right|$ and $|K_{\mathrm{I}}/K_{\mathrm{III}}|$ of different crack sizes can be calculated, and the results are listed in

Table 2. The ratios of $\left|K_{\mathrm{I}}/K_{\mathrm{II}}\right|$ and $|K_{\mathrm{I}}/K_{\mathrm{III}}|$ at the deepest point of the crack.

a/t	a/c
	0.2		0.4		0.6		0.8
	$\left|{\mathit{K}}_{\mathit{I}}/{\mathit{K}}_{\mathit{I}\mathit{I}}\right|$	$|{\mathit{K}}_{\mathit{I}}/{\mathit{K}}_{\mathit{I}\mathit{I}\mathit{I}}|$	$\left|{\mathit{K}}_{\mathit{I}}/{\mathit{K}}_{\mathit{I}\mathit{I}}\right|$	$|{\mathit{K}}_{\mathit{I}}/{\mathit{K}}_{\mathit{I}\mathit{I}\mathit{I}}|$	$\left|{\mathit{K}}_{\mathit{I}}/{\mathit{K}}_{\mathit{I}\mathit{I}}\right|$	$|{\mathit{K}}_{\mathit{I}}/{\mathit{K}}_{\mathit{I}\mathit{I}\mathit{I}}|$	$\left|{\mathit{K}}_{\mathit{I}}/{\mathit{K}}_{\mathit{I}\mathit{I}}\right|$	$|{\mathit{K}}_{\mathit{I}}/{\mathit{K}}_{\mathit{I}\mathit{I}\mathit{I}}|$
0.2	821.02	+∞	1000.00	+∞	35.71	+∞	25.71	+∞
0.3	77.50	+∞	40.00	+∞	1500.00	+∞	50.26	+∞
0.4	51.65	+∞	38.60	+∞	56.25	+∞	1351.35	+∞
0.5	49.29	+∞	34.67	+∞	39.23	+∞	60.52	+∞
0.6	43.88	+∞	29.11	+∞	30.66	+∞	32.96	+∞
0.7	34.67	+∞	30.16	+∞	25.21	+∞	22.73	+∞
0.8	28.57	+∞	29.23	+∞	23.33	+∞	21.53	+∞

. For the three types of cracks at the deepest point of the component containing a V-shaped notch, the coefficient K differs greatly. It is obvious that K_I are much larger than K_II in the same case, and K_III is close to 0. Therefore, type I SIF plays a dominant role, whereas type II and III SIFs can be negligible. It has good agreement with the results of the hole edge cracking of screwed joints studied by Yu et al. [36]. It is known that for the three types of cracks without a V-shaped notch, the difference in the coefficient K is not large, so K_I can be used as the main criterion for the fracture of components containing a V-shaped notch. Considering the actual engineering significance of the K_I, a correction will be made to the crack shape coefficient of K_I in the following discussion.

3.3. Corrected Shape Coefficient with a V-Shaped Notch

The K_I at the deepest point of the crack under different conditions can be obtained by the FEM, and then the corrected shape coefficient F_I of type I SIF with a V-shaped notch was calculated by Equation (5). Figure 9 shows the data calculated according to the crack shape coefficient expression. Under different ratios of R/r, the F_I curve is numerically fitted by the least square method by changing the ellipticity ratio a/c and the relative crack depth a/t.

As shown in Figure 9, the corrected shape coefficient F_I decreases with increasing ellipticity a/c and increases with increasing crack relative depth a/t under combined tensile stress and bending loads. The law of the corrected shape coefficient of the crack in this paper agrees well with that of the root crack with the T-shaped welded joint studied by Song et al. [37]. For the ampoule bottle with a V-shaped notch, its corrected shape coefficient increases significantly after a/t = 0.6. By binomial fitting of the curves shown in Figure 9, the expressions between the corrected shape coefficient of type I SIF and the crack relative depth can be obtained and are listed in

Table 3. The binomial fitting results of the corrected shape coefficient F_I and the crack relative depth.

a/c	R/r
	1.2	1.4	1.6	1.8
0.2	1.90357(a/t)2+ 0.02693(a/t)+ 3.65907	2.52857(a/t)2− 0.63657(a/t)+ 3.84057	2.47716(a/t)2− 0.35966(a/t)+ 3.66329	2.73214(a/t)2− 0.49364(a/t)+ 3.69364
0.4	1.6(a/t)2− 0.291(a/t)+ 3.3385	1.03571(a/t)2+ 0.28429(a/t)+ 3.26621	1.92857(a/t)2− 0.49857(a/t)+ 3.38957	2.20714(a/t)2− 0.39214(a/t)+ 3.36914
0.6	1.44286(a/t)2− 0.430(a/t)+ 2.95586	0.175(a/t)2+ 0.5125(a/t)+ 2.926	0.2(a/t)2+ 0.46(a/t)+ 2.915	1.15357(a/t)2+ 0.09493(a/t)+ 2.97157
0.8	−0.125(a/t)2+ 0.5925(a/t)+ 2.475	0.52143(a/t)2 0.09243(a/t)+ 2.68243	0.35(a/t)2+ 0.189(a/t)+ 2.618	0.55714(a/t)2+ 0.27886(a/t)+ 2.63214

.

3.4. Verification of the Corrected Shape Coefficient

Corrected shape coefficient of type I SIF with a V-shaped notch has been obtained, and a comparison of the results between this paper and the literature needs to be made in order to verify its effectiveness. In Section 2.6, the verification method has been introduced, thus the corrected shape coefficients not containing a V-shaped notch F_N can be calculated by Equations (6)–(10). The comparison of the corrected shape coefficients obtained in this paper and those in reference [31] is shown in Figure 10. It shows that the fitting curves of the two are approximate, suggesting that the corrected shape coefficient containing a V-shaped notch obtained by the FEM is acceptable and reliable.

In practical engineering applications, when calculating the crack-related parameters of components with a V-shaped notch, such as in ampoule bottles, we can directly query the required stress intensity factor and shape coefficient according to the graph or fitting binomial expression in this paper. It can also be utilized to calculate the crack-related parameters of the component without a V-shaped notch based on the FEM.

4. Conclusions

The fracture characteristics of a non-penetrating three-dimensional crack for an ampoule bottle with a V-shaped notch were investigated by using the FEM, and the characteristic parameters such as the coefficient K of the SIF and the corrected shape coefficient F were calculated. The influence of mesh sizes ranging from 0.1 mm to 0.7 mm on the calculation of the coefficient K is relatively weak, but the coefficient K significantly fluctuates at 0.7 mm–2.0 mm. It is suggested that the mesh size of 0.1 mm is acceptable to execute the calculation of the coefficient K to obtain high accuracy and efficiency.

The parameters of crack ellipticity (a/c) and crack relative depth (a/t) play significant roles in the SIF coefficients K_I, K_II and K_III of three types of cracks and the corresponding corrected shape coefficients. The coefficient K_I follows a symmetrical distribution at both sides of the deepest semi-elliptical crack. The depth of the crack has a great impact on the K_I at the deepest point, whereas it has a small impact at the crack front on both sides. The relationship between the corrected shape coefficient F_I and the crack relative depth (a/t) follows a binominal fitting function.

The investigation of fracture characteristics of an ampoule bottle can push for the automatic design of its opening, which will be beneficial to promote the application of ampoule bottles for sterility inspections in the medical and health fields.