A method for predicting surface crack growth under a rotating bending load condition

Abstract

The application discloses a surface crack propagation prediction method under a rotary bending loading condition and belongs to the mechanical device and transportation technical field.The method comprises the following steps: an initial surface crack propagation arc line and a new surface crack propagation arc line are represented by a five-parameter rotary elliptical model; an axisymmetric point O of the initial surface crack propagation arc line is taken as a center, and 2M+1 crack propagation circles which are uniformly distributed are set, the crack propagation circles are tangent to the initial surface crack propagation arc line and the new surface crack propagation arc line; a crack depth iteration error equation is established by a Paris-Erdogan formula, and iterative calculation is performed until convergence is achieved, so that the prediction of the surface crack propagation shape is completed.The method establishes an elliptical five-parameter model with less shape constraint conditions, can accurately describe the surface crack front shape, and is high in efficiency and accuracy.Meanwhile, the method can be used for representing the asymmetric distribution of the surface crack propagation arc in the propagation process, and is suitable for the multi-axis loading condition.

Description

Technical Field

[0001] This invention relates to the field of mechanical devices and transportation technology, and in particular to a method for predicting surface crack propagation under rotational bending loading conditions. Background Technology

[0002] In engineering, most fatigue cracks in metal structural components initiate on the surface and propagate under fatigue loads, leading to structural failure. During the manufacturing and processing of structural components, defects such as scratches and tool marks easily form on the material surface; these defects can be considered as initial surface cracks. The vast majority of penetrating fatigue cracks discovered in engineering also originate from the gradual propagation of surface cracks. Therefore, surface crack propagation can be considered the initial stage of the structural fatigue fracture process.

[0003] During the propagation of surface cracks, the arc-shaped crack front changes as it expands. Its shape determines the stress field distribution in the vicinity of the crack and the magnitude of the stress intensity factor at the crack front, which in turn affects the accuracy of predicting surface crack depth, residual strength of the material, and fatigue life. Therefore, the evolution law of surface crack shape has attracted much attention.

[0004] Currently, there are two commonly used methods: one is the prediction of surface crack fatigue propagation shape based on the finite element method, and the other is the shape prediction based on the increment of surface crack shape parameters. Lin and Smith proposed a multi-degree-of-freedom numerical simulation program to pre-calculate the stress intensity factor at each point of the surface crack front and simulate the surface crack front curve through cubic spline functions. Song established a functional relationship of the increment of surface crack shape parameters based on the Paris formula and analyzed the fatigue propagation of surface cracks in alloy steel. Hou developed a free-front finite element technique based on the improvement of traditional finite element technology, simulated the surface crack propagation shape, and analyzed the influence of surface crack edge effect, crack tip plastic deformation and other factors on surface crack propagation. Wu Zhixue et al. proposed a new surface crack stress intensity factor calculation formula based on the Newman-Raju surface crack stress intensity factor calculation formula and used this formula to predict the shape change during the surface crack propagation process. Shi proposed a semi-elliptical surface crack propagation shape calculation method based on the Newman-Raju surface crack stress intensity factor calculation formula and calculated the fatigue propagation of surface cracks in 7075-T6 aluminum alloy. Carpinteri proposed a two-parameter variation model to describe circular cracks using the Paris-Erdogan formula. However, most studies describe cracks with a fixed center and a defined shape, and achieve dimensionality reduction by adding constraints, but this inevitably reduces the accuracy of crack propagation shape prediction. Summary of the Invention

[0005] Aiming at the defects of the existing technology, based on the elliptic equation, the present invention establishes an elliptic five-parameter model with an unrestricted center position for accurately describing the front of a surface crack, which is used to predict the shape of the surface crack propagation. Different from the morphological characteristics of the symmetric propagation of the surface crack under a single load, this method can be used to characterize the asymmetric distribution of the surface crack propagation arc during the propagation process, that is, the propagation arc is inclined, and it is applicable to the multi-axial loading situation.

[0006] To achieve the above object, the present invention provides a method for predicting the propagation of a surface crack under a rotating bending load, including:

[0007] (1) Both the initial surface crack propagation arc and the new surface crack propagation arc are represented by a five-parameter rotating ellipse model, which are respectively the long semi-axis a n and a n+1 , the short semi-axis b n and b n+1 , the center coordinates of the rotating ellipse (O nx , O ny ) and (O (n+1)x , O (n+1)y ), the tilt angle t n and t n+1 , where: n is a natural number greater than 0;

[0008] (2) Taking the axisymmetric point O of the initial surface crack propagation arc as the center, 2M + 1 crack propagation circles evenly distributed are set, which are tangent to both the initial surface crack propagation arc and the new surface crack propagation arc. The tangent point coordinates corresponding to the initial surface crack propagation arc are (x jn , y jn ), and the tangent point coordinates corresponding to the new surface crack propagation arc are (x j,(n+1) , y j,(n+1) ); the center coordinates of the crack propagation circle are expressed as (x jc , y jc ); where 0 < j ≤ 2M + 1 and M is a natural number greater than 0;

[0009] (3) Using the 2M + 1 crack propagation circles to establish a crack depth iteration error equation through the Paris-Erdogan formula, and iteratively calculating until convergence to complete the prediction of the surface crack propagation shape.

[0010] Furthermore, the iterative process specifically includes:

[0011] (3.1) Using the Paris-Erdogan to express the surface crack propagation increment:

[0012]

[0013] Where: a represents crack length, N represents fatigue life, da / dN represents crack propagation rate, ΔK represents stress intensity factor range, and C and m are material constants;

[0014] (3.2) Based on the crack propagation step size of the crack propagation circle at point O, calculate the increment of crack propagation at the internal point for each of the crack propagation circles as follows:

[0015]

[0016] Where: △K e This represents the equivalent stress intensity factor related to the stress intensity factors at the initial and new surface crack fronts;

[0017] (3.3) Assuming the stress intensity factor K is a linear function of the crack propagation increment, the deformed equation is used to calculate: da=C(△K) m dN, by adjusting the material constant C, the crack propagation length is increased from a to a+da;

[0018] (3.4) Based on the crack propagation diagram established in step (3.3), the equivalent stress intensity factor ΔK with a step coefficient μ is obtained. e :

[0019]

[0020] Where: 0 < μ < 1;

[0021] (3.5) For each crack propagation circle, calculate the distance from the center of the crack propagation circle to the tangent point of the crack propagation arc on the new surface using geometric relationships;

[0022]

[0023] (3.6) Establish the error equation:

[0024]

[0025] Iterative calculations are performed until the error meets the accuracy requirements. The parameters of the new surface crack propagation arc are then obtained. At this point, it is assumed that the major semi-axis of the rotating ellipse is a. n Extend to the a n+1 The short semi-axis is composed of b n Extend to the b n+1 The coordinates of the center of the rotating ellipse are given by (O... nx O ny ) extended to the (O (n+1)x O (n+1)y The tilt angle of the rotating ellipse is determined by t. n Extend to the t n+1 .

[0026] Furthermore, the coordinate equation of the initial surface crack propagation arc is:

[0027] x = a n (cosθ)cost n -b n (sinθ)sint n +O nx

[0028] y = a n (cosθ)sint n +b n (sinθ)cost n +O ny

[0029] The coordinate equation of the new surface crack propagation arc is:

[0030] x = a n+1 (cosθ)cost n+1 -b n+1 (sinθ)sint n+1 +O (n+1)x

[0031] y = a n+1 (cosθ)sint n+1 +b n+1 (sinθ)cost n+1 +O (n+1)y .

[0032] Furthermore, the major semi-axis a of the rotational ellipse is... n+1 The short semi-axis b n+1 The coordinates of the center of the rotating ellipse (O) (n+1)x O (n+1)y The tilt angle t of the rotating ellipse n+1 By repeating steps 1 to 3 as parameters for the new initial crack, the surface crack propagation law can be obtained.

[0033] The beneficial effects of this invention are:

[0034] This method establishes an elliptical five-parameter model with fewer shape constraints, which can accurately describe the shape of the surface crack front and establish an efficient and accurate crack propagation model. Furthermore, unlike the symmetrical morphological characteristics of surface crack propagation under single loading, this method can be used to characterize the asymmetrical distribution of the surface crack propagation arc during the propagation process, i.e., the propagation arc becomes inclined, making it suitable for multi-axis loading scenarios. Attached Figure Description

[0035] Figure 1 It is an axis with a specific transition radius in an embodiment of the present invention.

[0036] Figure 2 This is a schematic diagram of surface cracks in an embodiment of the present invention.

[0037] Figure 3 In this embodiment of the invention, a new surface crack leading edge is defined by a surface crack propagation circle.

[0038] Figure 4 This is a schematic diagram of the model dimensions in an embodiment of the present invention.

[0039] Figure 5 This is a diagram illustrating the application of rotational forces according to an embodiment of the present invention.

[0040] Figure 6 This refers to the mesh division in an embodiment of the present invention.

[0041] Figure 7 This is a comparison chart of experimental data and numerical results from embodiments of the present invention.

[0042] Figure 8 These are the experimental data and numerical results of the left chord length in the embodiments of the present invention.

[0043] Figure 9 These are the experimental data and numerical results of the right chord length in the embodiments of the present invention.

[0044] Figure 10 These are the experimental data and numerical results of the tilt angle in the embodiments of the present invention. Detailed Implementation

[0045] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0046] This embodiment provides a method for predicting surface crack propagation under rotational bending loading conditions, including the following steps:

[0047] S101, the initial surface crack propagation arc, and the new surface crack propagation arc are all represented by a five-parameter rotating ellipse model, where a is the semi-major axis of the rotating ellipse. n and a n+1 short half-axis b n and b n+1 The coordinates of the center of the rotated ellipse (O) nx O ny ) and (O (n+1)x O (n+1)y The tilt angle t of the rotated ellipse n and t n+1 , where n is a natural number greater than 0;

[0048] The surface crack propagation arc of the axle under rotational bending loading is represented by a five-parameter rotating ellipse model, such as... Figure 1 , Figure 2 As shown, the five parameters are the major semi-axis a of the rotating ellipse n , the minor semi-axis b n , the center coordinates of the rotating ellipse (O x , O y ), and the tilt angle t of the rotating ellipse n . The remaining parameters are: the point P moving on the extended arc, the arc length position ratio r, r = s / S (0 ≤ s ≤ S), the crack depth a, the chord length H, the left chord length H L , the right chord length H R and the tilt angle β

[0049] S102. With the axisymmetric point O of the initial surface crack propagation arc as the center, set 2M + 1 crack propagation circles, which are not only tangent to the initial surface crack propagation arc but also tangent to the new surface crack propagation arc. The tangent point coordinates corresponding to the initial surface crack propagation arc are (x jn , y jn ), and the tangent point coordinates corresponding to the new surface crack propagation arc are (x j,(n+1) , y j,(n+1) ); the center coordinates of the crack propagation circle are expressed as (x jc , y jc ), where 0 < j ≤ 2M + 1 and M is a natural number greater than 0

[0050] As Figure 3 shown, in this embodiment, M = 4. With the axisymmetric point O of the initial surface crack propagation arc as the center, set 9 crack propagation circles. They are not only tangent to the initial oblique ellipse crack propagation arc, and the corresponding tangent points are H, G, F, E, O, A, B, C, and D, corresponding to the coordinates (x jn , y jn ); they are also tangent to the new oblique ellipse crack propagation arc, and the corresponding tangent points are H*, G * , F*, E*, O * , A*, B*, C * and D*, corresponding to the coordinates (x j,(n+1) , y j,(n+1) ); the center coordinates of the surface crack propagation circle can be expressed as (x jc , y jc ). In addition, it is assumed that the latest tangent point of the next crack is within the solid surface where the crack is located. Otherwise, the corresponding surface crack propagation circle will disappear in the next expansion stage

[0051] The coordinates (x jn , y jnThe coordinates () are uniformly distributed at the initial crack tip, where the subscript j represents H, G, F, E, O, A, B, C, and D, and the corresponding coordinate parametric equations are:

[0052] x = a n (cosθ)cost n -b n (sinθ)sint n +O nx

[0053] y = a n (cosθ)sint n +b n (sinθ)cost n +O ny (1)

[0054] From the recursive equation, we can obtain the elliptic coordinate parameter equation corresponding to the next extended oblique elliptic arc as follows:

[0055] x = a n+1 (cosθ)cost n+1 -b n+1 (sinθ)sint n+1 +O (n+1)x

[0056] y = a n+1 (cosθ)sint n+1 +b n+1 (sinθ)cost n+1 +O (n+1)y (2)

[0057] Points H*, G * ,F*,E*,O * ,A*,B*,C * and the coordinates of D* are (x j,(n+1) ,y j,(n+1) ).

[0058] S103. Using the 2M+1 crack propagation circles, establish the crack depth iterative error equation through the Paris-Erdogan formula, and iterate until convergence to complete the prediction of the surface crack propagation shape.

[0059] The iterative prediction process is as follows:

[0060] (1) The surface crack propagation increment at these nine points is expressed using the Paris-Erdogan formula, as follows:

[0061]

[0062] Where: a represents crack length, N represents fatigue life, da / dN represents crack propagation rate, ΔK represents stress intensity factor range, and C and m are material constants.

[0063] (2) Based on the crack propagation step size of the crack propagation circle at point O, the increment O of the crack propagation at the internal point is given after each calculation of the crack morphology. * All points H * G * ,F * E * O * A * B * C * and D * The crack propagation increment can be given by the following formula.

[0064]

[0065] In the above equation, △K e This represents the equivalent stress intensity factor related to the initial and new crack front stress intensity factors.

[0066] (3) Assume the stress intensity factor K is a linear function of the crack propagation increment. The number of crack propagation steps can be assumed to be arbitrary. The modified equation is used to calculate...

[0067] da=C(△K) m dN (5)

[0068] By adjusting the material constant C, the crack propagation length is increased from a to a+da, that is, it is repeatedly increased from each step to the last step.

[0069] (4) Based on the crack propagation diagram established in step (3), obtain the equivalent stress intensity factor ΔK with a stepping coefficient μ. e :

[0070]

[0071] At the start of the iteration, a larger value μ can be used to avoid divergence.

[0072] (5) The distance from the crack propagation circle center to H was calculated using geometric relationships. * G * ,F * E * O * A * B * C * and D * Distance between points:

[0073]

[0074] (6) Establish the error equation:

[0075]

[0076] Iterative calculations continue until the error meets the accuracy requirements, at which point the new surface crack parameters are obtained. At this point, it is assumed that the major semi-axis of the rotating ellipse is determined by a. n Extend to the a n+1 The minor semi-axis is determined by b n Extend to the b n+1 The coordinates of the center of the rotated ellipse are given by (O) nx O ny ) extended to the (O (n+1)x O (n+1)y The tilt angle of the rotated ellipse is determined by t. n Extend to the t n+1 .

[0077] This embodiment uses Abaqus software for modeling and simulation, such as... Figure 4 As shown. A bending moment of 120000 N·mm and a rotational force of 523.599 radians / second are applied to the large end face of the model, as follows. Figure 5 As shown, the model is then meshed, see... Figure 6 Finally, the simulation results are analyzed. The five-parameter rotating ellipse model of this invention can be used to predict surface crack propagation under rotating bending loading conditions. Figure 7 The red expanding arc represents the numerical results obtained using this method, while the black expanding arc represents the experimental results obtained by R. Hannemann et al. using experimental methods. A comparison of the numerical results and experimental data shows good agreement between the two. Figures 8-10 The experimental data and numerical results were analyzed from three aspects to verify the feasibility of this method.

[0078] Symbols in the diagram: H eAL Left chord length H under experimental data nAL : Left chord length H under numerical simulation eAR : Right chord length H under experimental data nAR : Right chord length in numerical simulation.

[0079] The above embodiments are preferred embodiments of the present invention and are not intended to limit the patent scope of the present invention. Any equivalent structural transformations made using the contents of the present invention specification and drawings under the concept of the present invention, or direct / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A method for predicting the shape of surface crack propagation under rotational bending loading conditions, characterized in that, include: (1) Both the initial surface crack propagation arc and the new surface crack propagation arc are represented by a five-parameter rotating ellipse model, where are the major semi-axis of the rotating ellipse, respectively. and short half-shaft and The coordinates of the center of the rotated ellipse and The tilt angle of the rotated ellipse and ,in: n It is a natural number greater than 0; (2) The axisymmetric point of the initial surface crack propagation arc Centered on, set a uniformly distributed 2 M+ A crack propagation circle is formed, which is tangent not only to the initial surface crack propagation arc but also to the new surface crack propagation arc. The coordinates of the tangent point to the initial surface crack propagation arc are: The coordinates of the tangent point of the new surface crack propagation arc. The coordinates of the center of the crack propagation circle are represented as follows: ,in ; (3) Using the above 2 M+1 The crack propagation circle is used to establish an iterative error equation for crack depth using the Paris-Erdogan formula. The equation is iteratively calculated until convergence, thus completing the prediction of the surface crack propagation shape. The iterative process specifically includes: (3.1) Using Paris-Erdogan to express the surface crack propagation increment: in: a Represents the crack length. N Represents fatigue life. Represents the crack propagation rate. Represents the range of stress intensity factors. and These are material constants; (3.2) According to The crack propagation step size of the point crack propagation circle is calculated as follows: The increment of crack propagation within each of the stated crack propagation circles at internal points is: in: This represents the equivalent stress intensity factor related to the stress intensity factors at the leading edge of the initial surface crack and the new surface crack; (3.3) Assuming stress intensity factor It is a linear function of the crack propagation increment, calculated using the modified equation: By adjusting the material constants This causes the crack to propagate from... Increase to ; (3.4) Based on the crack propagation diagram established in step (3.3), a crack propagation diagram with step coefficients is obtained. Equivalent stress intensity factor : in: ; (3.5) For each crack propagation circle, calculate the distance from the center of the crack propagation circle to the tangent point of the crack propagation arc on the new surface using geometric relationships; (3.6) Establish the error equation: Iterative calculation until the error is reached. Error To meet the accuracy requirements, the parameters of the new surface crack propagation arc are obtained. At this point, it is assumed that the major semi-axis of the rotational ellipse is... Extend to The short semi-axis is composed of Extend to The coordinates of the center of the rotating ellipse are given by Extend to The tilt angle of the rotating ellipse is determined by... Extend to .

2. The method for predicting the surface crack propagation shape under rotational bending loading conditions according to claim 1, characterized in that: The coordinate equation of the initial surface crack propagation arc is: The coordinate equation of the new surface crack propagation arc is: 。 3. The method for predicting the surface crack propagation shape under rotational bending loading conditions according to claim 1, characterized in that: The major semi-axis of the rotating ellipse The short half-shaft The coordinates of the center of the rotating ellipse The tilt angle of the rotating ellipse By repeating steps (1) to (3) as parameters for the new initial crack, the surface crack propagation law can be obtained.

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