A fast calculation method for solving the stress intensity factor at the tip of a double crack in compression and shear closure

Abstract

The present invention discloses a method for calculating the stress intensity factor at the tip of a double crack closed by compression and shear quickly, comprising the following steps: S1: decomposition of the initial double crack model closed by compression and shear; S2: establishment of a multipolar coordinate system of a single crack model closed by compression and shear; S3: solution of the disturbance stress of the mutual influence between cracks; S4: calculation of the stress intensity factor at the crack tip. Compared with the classical theory, the calculation method of the present invention is applicable to the solution of any closed double crack problem under compression and shear, including collinear double cracks, parallel non-collinear double cracks and non-parallel non-collinear double cracks. Compared with the classical elliptic integral method and J-integral finite element method, the calculation method of the present invention has the advantages of higher prediction accuracy, wider application range, simple calculation and low time cost.

Description

Technical Field

[0001] The present invention belongs to the research field of solid mechanics-fracture mechanics, and specifically relates to a calculation method for quickly solving the stress intensity factor at the tip of a compression-shear closed double crack, which can be used for analyzing the failure behavior and fracture mechanism of actual engineering components of elastic-brittle solid materials such as rocks, ceramics, glass and metals. Background Art

[0002] Brittle materials often contain a large number of cracks. Under complex loads, stress concentration at the crack tip can lead to failure of engineering structures. Therefore, the study of crack problems has been a hot topic so far. The classical fracture theory based on Griffith takes cracks under tension and shear as the research object. Rich theoretical research results have been achieved for single cracks and multiple cracks under tension and shear. However, cracks in actual engineering components will also be subjected to compressive shear loads, such as underground rock and concrete. Under compression conditions, cracks close, causing friction between crack surfaces, which will significantly affect the stress intensity factor at the crack tip.

[0003] For a single compressive crack, some scholars directly modify the load direction based on classical fracture mechanics, that is, and (Where: σ ∞ and τ ∞ However, many scholars have pointed out that the contact crack surface is closed everywhere under compression and shear, and based on the view that the material is physically inaccessible, it is believed that K I = 0. Zhu et al. used the boundary collocation method to present analytical and numerical results and analyzed the effects of crack surface friction and confining pressure on the crack stress intensity factor.

[0004] Most of the above studies are for single cracks under compression and shear, while the current research on considering the interaction between multiple cracks is mainly concentrated under tensile shear stress state, and the theoretical analysis under compressive shear stress state is relatively small, and it is mainly based on the simplified pseudo-force method proposed by Kachanov to consider the influence of the interaction between cracks on the stress intensity factor. Basista and Gross improved the Kachanov method, analyzed the interaction between any two cracks under compression and shear stress, and proposed a calculation method for the stress intensity factor at the crack tip considering the friction effect of the crack surface, and compared it with the numerical solution obtained by the boundary element method; in addition, in this paper, Basista and Gross also believed that K I=0. Li et al. decomposed the force along the crack surface into linear and nonlinear components based on the Kachanov method, and the sum of the two equals the total force along the crack surface. At the same time, they further simplified and believed that the interaction of nonlinear components can be ignored. Only the influence of linear components is considered, and it is extended to the closed cracks of compression and shear, and a method for calculating the stress intensity factor of the crack tip considering the interaction of adjacent cracks is proposed. Zhu et al. used the complex potential theory, starting from the far-field stress boundary conditions and the crack surface stress boundary conditions, and derived the full-field stress function of an infinite plate with a single crack under compression and shear. Zhu, Zheng et al. and Fan et al. used the elliptic integral method to solve the crack tip stress intensity factor of collinear equal-length double cracks, collinear equal-length three cracks and collinear unequal-length double cracks under compression and shear, and discussed the influence of the interaction between cracks on the stress intensity factor of the crack tip. However, in their research, the influence of the interaction between cracks on the friction of the crack surface was ignored. In addition, the elliptic integral method can only be used to analyze compression and shear collinear cracks, and it cannot be solved for any non-collinear double crack problem. The existing Fredholm integration method involves singular integral equations, and its solution process is complicated and difficult for engineers to understand, which limits its promotion and application in engineering.

[0005] Although there have been certain research results on the calculation methods of stress intensity factors at the tips of closed single cracks and collinear multiple cracks under compression and shear, there are still many deficiencies in the research on the fast calculation methods of stress intensity factors at the tips of non-collinear and arbitrary double cracks under compression and shear, which limits the application and development of theory in guiding practical production activities. Summary of the invention

[0006] In order to solve the above problems, the present invention proposes a calculation method for quickly solving the stress intensity factor at the tip of a compression-shear closed double crack.

[0007] The method for calculating the stress intensity factor at the tip of a double crack closed by compression and shear of the present invention comprises the following steps: S1: decomposition of an initial double crack model closed by compression and shear; S2: establishment of a multipolar coordinate system of a single crack model closed by compression and shear; S3: solution of disturbance stresses affecting each other between cracks; and S4: calculation of the stress intensity factor at the crack tip.

[0008] The superposition principle is also called the superposition property, which is also called additivity in mathematics. The decomposition of the initial compression-shear closed double crack model in step S1 means that, according to the superposition principle, the stress field of the compression-shear closed double crack model under the initial linear elastic condition is decomposed into two parts: (1) the disturbance stress field caused by the first crack on the second crack under the compression-shear action; (2) the disturbance stress field caused by the second crack on the first crack under the compression-shear action.

[0009] Establishing a multipolar coordinate system for the compression-shear closed single crack model in step S2 means that, for an existing compression-shear closed single crack model, the stress field expression around the crack in the Cartesian rectangular coordinate system is converted into a form under the multipolar coordinate system, and dimensionless processing is performed to obtain an expression of the dimensionless stress field around the single crack.

[0010] The expression of the dimensionless stress field around the single crack is:

[0011]

[0012] Where: p is the far-field compression load, f is the friction coefficient of the crack surface, is the angle between the crack and the horizontal direction (counterclockwise is positive), a is the half length of the crack, (r0,θ0), (r1,θ1) and (r2,θ2) are the coordinate parameters in the corresponding multipolar coordinate system, subscript 0 represents the polar coordinate established with the crack center as the origin, subscript 1 and 2 represent the polar coordinate established with the left and right endpoints of the crack as the origin, respectively. These three coordinates refer to the distances from the point on the second crack to the center and left and right endpoints of the first crack in the multi-level coordinate system, (σ x ,σ y ,τ xy ) are the three stress components at any point.

[0013] The solution of the disturbance stress caused by the mutual influence between cracks in step S3 refers to establishing a local multipolar coordinate system with the centers of the first and second cracks as the origins, and solving the disturbance stress on the crack surface.

[0014] The solution to the disturbance stress caused by the mutual influence between the cracks refers to establishing a local multi-level coordinate system with the center of the first crack as the origin to obtain the disturbance stress caused by the first crack on the second crack, and establishing a local multi-polar coordinate system with the center origin of the second crack to obtain the disturbance stress caused by the second crack on the first crack.

[0015] Taking the center of the first crack as the origin, establish a local multipolar coordinate system, and let t(x, y) be any point on the second crack, then the value ranges of (r0, θ0), (r1, θ1) and (r2, θ2) are:

[0016]

[0017]

[0018]

[0019] Where: (x 左1 ,y 左1 )、(x 右1 ,y右1 )、(x 左2 ,y 左2 ) and (x 右2 ,y 右2 ) are the coordinates of the left and right endpoints of the first crack and the second crack in the local multipolar coordinate system.

[0020] Finally, according to formula (1), the expression of the compressive shear stress field in the multipolar coordinate system containing only the first crack is given. Substituting the coordinates of any point on the second crack in sequence, the perturbation stress P(t)+iQ(t) generated by the first crack on the second crack can be approximately obtained (where: P and Q are the perturbation normal stress and perturbation shear stress on the second crack, respectively, and t is any point on the second crack).

[0021] Similarly, by establishing a local multipolar coordinate system with the center of the second crack as the origin, the disturbance stress P(t)+iQ(t) generated by the second crack on the first crack can also be approximately calculated.

[0022] The calculation of the stress intensity factor at the crack tip in step S4 refers to introducing a modified Gauss-Chebyshev quadrature formula to solve the stress intensity factor at the crack tip based on the disturbance stress result.

[0023] Introduce the modified Gauss-Chebyshev quadrature formula:

[0024]

[0025] Where:

[0026]

[0027] Where: a is the half length of the crack, M is the number of discrete crack points, t j is a discrete point on the crack, which is the discrete form of any point t on the crack. The subscript j represents the sequence number of the discrete point.

[0028] The crack tip includes a left endpoint and a right endpoint, and thus, for the left endpoint of the crack:

[0029]

[0030] Similarly, for the right end point of the crack:

[0031]

[0032] The stress intensity factor at the tip of a compression-shear closed double crack considering the interaction between cracks can be determined by equations (7) and (8).

[0033] The expressions of stress intensity factors at the left and right endpoints of a compression-shear closed double crack are:

[0034]

[0035] Where: M is the number of crack discrete points, a is the half length of the crack, P is the perturbation normal stress on the crack, Q is the perturbation shear stress on the crack, t j is a discrete point on the crack, which is the discrete form of any point t on the crack. The subscript j represents the discrete point number, i is an imaginary unit, and K Ⅰ is the mode I stress intensity factor, K Ⅱ is the type II stress intensity factor.

[0036] The beneficial effect of the present invention is that, compared with classical theory, the calculation method of the present invention is suitable for solving any closed double crack problem under compression and shear, including collinear double cracks, parallel non-collinear double cracks and non-parallel non-collinear double crack problems. Compared with the classical elliptic integral method and J-integral finite element method, the calculation method of the present invention has the advantages of higher prediction accuracy, wider application range, simple calculation and low time cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 Schematic diagram of the decomposition of the compression-shear closed double crack model.

[0038] Figure 2 Schematic diagram of the multipolar coordinate system near the first crack.

[0039] Figure 3 Schematic diagram of the collinear closed double crack model under compression and shear.

[0040] Figure 4 Schematic diagram of the compression-shear non-collinear parallel double crack model.

[0041] Figure 5 Distribution curves of disturbance stress on the compression-shear non-collinear parallel double crack surfaces at different inclination angles.

[0042] Figure 6 Change curve of dimensionless stress intensity factor at crack tip. DETAILED DESCRIPTION

[0043] Embodiments of the present invention are described in detail below, and examples of the embodiments are shown in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to be used to explain the present invention, but should not be understood as limiting the present invention.

[0044] The present invention is a calculation method for quickly solving the stress intensity factor of the compression-shear closed double crack tip, and the calculation method comprises the following steps:

[0045] S1. Decomposition of the initial compression-shear closed double crack model

[0046] right Figure 1The closed double crack model under compression and shear shown in the figure, its stress field can be decomposed into: (a) the disturbance stress field P1(t)+iQ1(t) caused by the second crack on the first crack under compression and shear; (b) the disturbance stress field P2(t)+iQ2(t) caused by the first crack on the second crack under compression and shear.

[0047] S2. Establishing a multipolar coordinate system for the compression-shear closed single crack model

[0048] by Figure 1 Take the first crack in the figure as an example, and take its center as the origin. Figure 2 The multipolar coordinate system under compression and shear is shown in Figure 1. Assuming that only the first crack is included, then Figure 1 Under the loading condition shown in (a), the first crack is near Figure 2 The dimensionless stress field expression in the multipolar coordinate system is:

[0049]

[0050] Where: f1 is the friction coefficient of the first crack surface, is the initial inclination angle of the first crack, and a1 is the half length of the first crack.

[0051] S3. Solution of disturbance stress due to mutual influence between cracks

[0052] exist Figure 2 In the multipolar coordinate system shown, t(x, y) is any point on the potential position of the second crack. According to its geometric position, t is known. At this time, according to the corresponding geometric relationship, the (r0, θ0), (r1, θ1) and (r2, θ2) of this point in the multipolar coordinate system are also known. These three coordinates refer to the distances from the point on the second crack to the center and left and right end points of the first crack in the multipolar coordinate system. Substituting their coordinate values ​​into equation (10), the disturbance stress P2(t)+iQ2(t) caused by the first crack on the second crack under compression and shear can be obtained.

[0053] Similarly, the disturbance stress P1(t)+iQ1(t) caused by the second crack on the first crack under compression and shear can also be obtained.

[0054] S4. Calculation of stress intensity factor at crack tip

[0055] From formula (9), we can know that the calculation formula of the stress intensity factor at the left and right ends of the first crack is:

[0056]

[0057] Where:

[0058]

[0059] Where a1 is the half length of the first crack, subscript 1 indicates the first crack, M is the number of discrete points of the first crack, and in this example, M=15. From S3, the disturbance stress P1(t)+iQ1(t) distributed on the first crack is known. Therefore, for each discrete point, the disturbance stress P1(t j )+iQ1(t j ) is also known. Substituting it into equation (11) can obtain the stress intensity factor at the left and right endpoints of the first crack.

[0060] Similarly, the stress intensity factors at the left and right endpoints of the second crack can also be obtained.

[0061] Example 1

[0062] Taking a compression-shear collinear closed double crack as an example, the model is as follows Figure 3 As shown in the figure, the length of the first crack on the left side of the model is 1 cm, the length of the second crack on the right side is 2 cm, and the distance between the inner end points of the two cracks is 1.75 cm. In this example, the friction between the crack surfaces is not considered, so f = 0. The comparison between the J-integral finite element simulation results and the theoretical results of this method is shown in Tables 1 to 4.

[0063] Table 1. Comparison of the J-integral finite element simulation results at the left end point of the first crack and the theoretical results of this method

[0064]

[0065]

[0066] Table 2. Comparison of the J-integral finite element simulation results at the right end point of the first crack and the theoretical results of this method

[0067]

[0068] Table 3. Comparison of the J-integral finite element simulation results at the left end point of the second crack and the theoretical results of this method

[0069]

[0070] Table 4. Comparison of the J-integral finite element simulation results at the right end point of the second crack and the theoretical results of this method

[0071]

[0072] From the comparative analysis of Tables 1 to 4, it can be seen that the results obtained by this method are almost consistent with the results obtained by J-integral. Compared with the classical theoretical results, the prediction accuracy is higher, which also proves the effectiveness of the prediction results of this method. However, the J-integral method requires numerical calculations through finite element software, and its modeling is relatively complex and the calculation time cost is high. The method of the present invention does not require complex and time-consuming finite element calculations, and the calculation is simple and the cost is low.

[0073] Example 2

[0074] For Figure 4 The compression-shear non-collinear parallel double crack model shown in the figure cannot be solved by the classical method. According to the method of the present invention, the relationship between the disturbance stress distribution on the crack surface and the stress intensity factor at the crack tip and the geometric position can be obtained, as shown in Figure 5 and Figure 6 shown.

[0075] Depend on Figure 5 It can be seen that under the action of compression and shear, although the absolute values ​​of the inclination angles of the forward-angle rock bridge and the reverse-angle rock bridge are the same, the disturbance stresses distributed along the crack surface caused by the interaction of cracks are completely different. The disturbance stress includes normal stress and shear stress. For example, if the disturbance normal stress distributed along the crack surface is tensile stress in the forward-angle rock bridge, it is compressive stress in the reverse-angle rock bridge, and the direction is exactly opposite. When the disturbance normal stress generated along the crack surface is tensile stress, it will offset part of the compressive stress transmitted to the crack surface by the far-field load. However, since the disturbance tensile stress cannot completely offset the far-field compressive stress, the crack surface is still in a closed state. At the same time, since part of the compressive stress is offset, the friction along the crack surface will decrease, and the corresponding shear stress on the crack surface will increase, thereby weakening the suppression effect on the singular stress field at the crack tip, which will promote the stress intensity factor K. II When the disturbance stress generated along the crack surface is compressive stress, the friction between the crack surfaces will be enhanced, and the corresponding shear stress on the crack surface will be reduced, thereby increasing the suppression of the singular stress field at the crack tip, which will lead to the stress intensity factor K II Reduce.

[0076] Depend on Figure 6 It can be seen that when the absolute value of the rock bridge inclination angle |β| is small, the parallel non-collinear cracks show an enhancement effect; when the absolute value of the rock bridge inclination angle |β| is large, the cracks show a shielding effect, especially the inner end point enters the shielding interval earlier. In addition, it can be seen that the corresponding crack tip singularities in the rock bridge with a positive inclination angle and the rock bridge with a negative inclination angle do not show a symmetrical relationship.

[0077] The results obtained by this method can be used to explain why rock bridge crack specimens exhibit different mechanical properties under compression and shear.

[0078] From the above implementation cases, it can be seen that the present invention provides a calculation method for quickly solving the stress intensity factor at the tip of a double crack closed by compression and shear. This method has the advantages of being simple, fast and accurate, and can provide a theoretical basis for the fracture analysis of components containing double cracks under compression and shear.

[0079] Any matters not described in the present invention are applicable to the prior art.

[0080] Although the above embodiments have been shown and described, it is to be understood that the above embodiments are exemplary and are not to be construed as limitations on the present invention. Changes, modifications, substitutions and variations of the above embodiments by those of ordinary skill in the art are all within the scope of protection of the present invention.

Claims

1. A method for quickly calculating the stress intensity factor at the tip of a double crack in compression and shear closure, characterized in that: The following steps are involved: S1: Decomposition of the initial compression-shear closed double crack model: According to the superposition principle, the stress field of the compression-shear closed double crack model is decomposed into two parts: the disturbance stress field caused by the first crack on the second crack under compression-shear action and the disturbance stress field caused by the second crack on the first crack under compression-shear action; S2: Establishing a multipolar coordinate system for a compression-shear closed single crack model: Converting the stress field expression around the crack of the compression-shear closed single crack model in the Cartesian rectangular coordinate system into a multipolar coordinate system, and performing dimensionless processing to obtain the expression of the dimensionless stress field around the single crack; S3: Solution of disturbance stress caused by mutual influence between cracks: With the centers of the first and second cracks as the origins, local multipolar coordinate systems are established to solve the disturbance stress on the crack surface; specifically, with the center of the first crack as the origin, a local multipolar coordinate system is established to obtain the disturbance stress caused by the first crack on the second crack, and with the center of the second crack as the origin, a local multipolar coordinate system is established to obtain the disturbance stress caused by the second crack on the first crack; S4: Calculation of stress intensity factor at crack tip: The modified Gauss-Chebyshev quadrature formula is introduced to solve the stress intensity factor at the crack tip based on the perturbation stress results.

2. According to the calculation method for quickly solving the stress intensity factor of the double crack tip of compression and shear closure according to claim 1, the expression of the dimensionless stress field around the single crack is: Where: p is the far-field compression load, f is the friction coefficient of the crack surface, is the angle between the crack and the horizontal direction, with the counterclockwise direction as positive, a is the half-length of the crack, (r0,θ0), (r1,θ1) and (r2,θ2) are the coordinate parameters in the corresponding multipolar coordinate system, subscript 0 represents the polar coordinate established with the crack center as the origin, subscript 1 and 2 represent the polar coordinate established with the left and right endpoints of the crack as the origin, respectively. These three coordinates refer to the distances from the point on the second crack to the center and left and right endpoints of the first crack in the multi-level coordinate system, (σ x ,σ y ,τ xy ) are the three stress components at any point.

3. The method for calculating the stress intensity factor at the tip of a compression-shear closed double crack according to claim 1 is characterized in that: The crack tip includes a left endpoint and a right endpoint, and the expressions of the stress intensity factors of the left endpoint and the right endpoint are: Where: M is the number of discrete crack points, a is the half length of the crack, P is the perturbation normal stress on the crack, Q is the perturbation shear stress on the crack, t is any point on the crack, i is an imaginary unit, K Ⅰ is the mode I stress intensity factor, K Ⅱ is the type II stress intensity factor.

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