Method for determining critical surface in predicting fatigue life of smooth specimens

Abstract

The present invention provides a method for determining a critical surface when predicting fatigue life of a smooth test piece, comprising the following steps: performing a high-cycle fatigue test on the test piece to obtain all stress components at a dangerous point O, and establishing an original coordinate system O-xyz and a transformed new coordinate system O-abn at the dangerous point O; taking a vector q with an angle α with a coordinate axis a vector on a plane aob of the obtained new coordinate system O-abn to obtain a normal stress on the plane aob and a shear stress along the direction q; and using the Powell method to determine a maximum value of the shear stress variance and three angles θ determining a critical surface where the maximum value of the shear stress variance is located. max , α max and φ max Based on the three obtained angles, the plane with the maximum normal stress and the largest shear stress variance is found. This method solves the computational inefficiency caused by traversing the entire angle range with small increments in the maximum variance method, while also considering the maximum normal stress on the plane, reducing the prediction defects in fatigue life.

Description

Technical Field

[0001] The present invention relates to the technical field of fatigue life prediction of metal materials, and in particular to a method for determining a critical surface of a smooth test piece when predicting fatigue life. Background Art

[0002] Metal materials often experience fatigue failure during use, often under multiaxial cyclic loading. To predict fatigue life under multiaxial loading conditions, a variety of equivalent damage variables and fatigue life prediction models have been proposed. However, the critical plane method (CPM) holds the greatest potential in the field of multiaxial fatigue life prediction. This method studies fatigue crack initiation by specifying the plane within a material element with the maximum fatigue damage variable.

[0003] The critical plane method selects different damage variables depending on the critical plane type. The damage parameters of all planes are then calculated, and the plane with the largest calculated result is defined as the critical plane. It is common to use the equivalent stress amplitude plane as the critical plane. The main methods for calculating equivalent shear stress amplitude under multiaxial non-proportional loading include the longest projection method, the longest chord method, the minimum circumscribed circle method, the minimum circumscribed ellipse method, and the maximum variance method. With the exception of the maximum variance method, all other methods require the individual calculation of complex shear stress trajectories to calculate the equivalent shear stress amplitude and mean, resulting in low computational efficiency. The maximum variance method, on the other hand, comprehensively considers the impact of the shear stress value at each moment under the random loading path on the calculation results, eliminating the need for a clear shear stress trajectory. It is not only applicable to constant amplitude fatigue loading, but can also quickly determine the critical plane direction under variable amplitude fatigue loading. For cycle counting under multiaxial random fatigue loading, only the shear stress and normal stress on the critical plane need to be cycle counted.

[0004] The essence of the maximum variance method is to transform the problem of solving the critical surface into an optimization problem. The traditional way is to traverse the entire angle range of the stress decomposition plane and the three angles θ, φ, and α on the stress decomposition direction in fixed small increments. Due to the limitations of the optimization problem, a local optimal solution may sometimes be obtained, resulting in errors in the calculation of the critical surface. In addition, for multiple global optimal solutions, it is also necessary to screen out the critical surface directions that meet the conditions. Therefore, the present invention proposes a method for determining the critical surface of a smooth test piece when predicting fatigue life by modifying the maximum variance method, taking into account the variance of the maximum shear stress amplitude on the plane and the variance of the maximum normal stress on the plane. Summary of the Invention

[0005] In response to the problems existing in the prior art, the present invention provides a method for determining the critical surface of a smooth test piece when predicting fatigue life. The method uses the maximum variance method to obtain the normal stress of the plane aob and the shear stress along the direction q, as well as the variance and covariance corresponding to the normal stress and the variance corresponding to the shear stress, thereby avoiding the traditional definition of the two-dimensional shear stress amplitude and mean on the critical surface; the Powell method is used to determine the maximum value of the shear stress variance Var[τ q (t)] max and the three angles θ that determine the critical surface where the maximum shear stress variance is located max , α max and φ max The Powell method dynamically adjusts the increment size according to the current gradient information to ensure that the value of the objective function is reduced as quickly as possible at each step, thus getting rid of the problem of low computational efficiency caused by the traditional method of presetting small increments for incremental increase. max , α max and φ max , find the one with the maximum normal stress and experiencing the maximum shear stress variance Var[τ q (t)] max The problem of prediction defects caused by not considering the maximum normal stress on the plane is solved.

[0006] The present invention provides a method for determining the critical surface of a smooth test piece when predicting fatigue life. The specific implementation steps are as follows:

[0007] S1. Perform a high-cycle fatigue test on the test piece to obtain all stress components at the dangerous point O on the test piece, and establish the original coordinate system O-xyz and the transformed new coordinate system O-abn at the dangerous point O on the test piece;

[0008] S2. On the plane aob of the new coordinate system O-abn obtained in step S1, a vector q is established with an angle α with the coordinate axis a vector to obtain the normal stress σ of the plane aob. n (t) and the shear stress τ along direction q q (t), and according to the normal stress σ n (t) to obtain the corresponding variance and covariance, according to the shear stress τ q (t) obtain the corresponding variance;

[0009] S3. Use Powell's method to determine the maximum shear stress variance Var[τ q (t)] max and the three angles θ that determine the critical surface where the maximum shear stress variance is located max , α max and φ max :

[0010] S31. Set the initial values ​​of θ, α and φ respectively, θ0, α0 and φ0, the initial direction R = [r1, r2, r3], the initial step size a0, the constant β (0 < β < 1) and the threshold ε, and obtain the initial value Var[t q (t)]0;

[0011] S32. Define the objective function of the Powell method. The specific expression is:

[0012] g(θ,α,φ)=-Var[t q (t)]=-f(θ,α,φ)

[0013] Where g(θ,α,φ) is the shear stress variance Var[t q (t)] is the function obtained by multiplying it by -1, θ is the angle between the n-axis of the new coordinate system and the z-axis of the original coordinate system, φ is the angle between the xy-plane of the original coordinate system and the ab-plane of the new coordinate system, α is the angle between the vector q and the a-axis on the ab-plane, and f(θ,α,φ) is the variance of the shear stress Var[t q (t)];

[0014] S33, when the kth iteration has been completed, θ, α and φ are searched in one dimension along r1, r2 and r3 in the initial direction R = [r1, r2, r3], respectively, and the angle θ after the current iteration is obtained. k , α k 、φ k and step length a k , the expressions for obtaining the new angle value and objective function value are:

[0015] θ new =θ k +a k R[i],i=1,2,3,α new =α k +a k R[i],i=1,2,3,

[0016] φ new =φ k +a k R[i],i=1,2,3;

[0017] g(θ,α,φ)=g(θ new ,α new ,φ new );

[0018] Where θ new , α new and φ new are the updated angles θ, α and φ after the k+1th iteration, θk , α k and φ k are the angles θ, α, and φ obtained after the kth iteration, a k is the step size obtained after the kth iteration; g(θ new ,α new ,φ new ) is the function value of the angles θ, α and φ after the k+1th iteration; R[i] is the direction matrix, which initially has three direction vectors;

[0019] S34, if the objective function value Q obtained in step S33 is Q=g(θ ne ,α new ,φ new )-g(θ k ,α k ,φ k )<0, then θ, α and φ are updated, and at this time θ k+1 =θ new , α k+1 =α new ,φ k+1 =φ new , and update the direction, R[k+1]=R[i]; if the objective function value Q=g(θ new ,α new ,φ new )-g(θ k ,α k ,φ k )≥0, the step length is a k+1 =β*a k , and perform directional orthogonalization processing;

[0020] S35, based on step S34, k+1 , α k+1 and φ k+1 As a new round of iteration, k+1 As the new step size, with the new direction r k+1 Evaluate the next objective function;

[0021] S36, if the objective function changes || Q k+1 ||=||g(θ k+1 ,α k+1 ,φ k+1 )-g(θ k ,α k ,φ k )||<ε, the iteration ends and outputs θ max , α max and φ max ; If the objective function changes ||Q k+1 ||=||g(θk+1 ,α k+1 ,φ k+1 )-g(θ k ,α k ,φ k )||≥ε, then continue to update the direction until the objective function changes by ||Q k+1 ||=||g(θ k+1 ,α k+1 ,φ k+1 )-g(θ k ,α k ,φ k )||<ε;

[0022] S4, the three angles θ obtained in step S3 max , α max and φ max , find the maximum normal stress and the maximum shear stress variance Var[τ q (t)] max Plane:

[0023] S41. Screen according to the direction of the critical surface at the dangerous point O on the test piece, and find θ at the dangerous point O on the test piece. max , α max and φ max The angle with the maximum normal stress variance formed by the test piece is the expression of the maximum normal stress on the critical surface at the dangerous point O on the test piece:

[0024]

[0025] Where, σ n,max is the maximum normal stress on the critical surface; σ a is the normal stress amplitude on the critical surface; σ n,m is the mean normal stress on the critical surface; d2 is σ a The matrix of the angle part in s m is σ n,m The matrix of the stress part;

[0026] S42, determining the maximum value among the maximum normal stress at the dangerous point O on the test piece obtained in step S41. And find the three critical angles corresponding to the critical surface at the dangerous point O on the test piece and

[0027] Preferably, in step S1, the stress tensor σ corresponding to time t is ij The expression of (t) is:

[0028]

[0029] Among them, σ x (t), σ y (t), σ z (t) are the normal stresses in the x, y, and z directions respectively; τ xy (t), τ xz (t), τ yz (t) is the shear stress in three directions.

[0030] Preferably, in step S1, the coordinate expressions of the coordinate axes of the new coordinate system O-abn relative to the coordinate axes of the original coordinate system O-xyz are:

[0031]

[0032]

[0033] Where n is the direction of the new plane normal vector, and the angle between it and the original z-axis is θ; a is relative to the x-axis of the old coordinate system, b is relative to the y-axis of the old coordinate system, and the xy plane is rotated around the z-axis by φ to obtain the new ab plane; n x 、n y 、n z are the projections of the n-axis along the x-, y-, and z-axes, respectively. x 、a y 、a z are the projections of axis a along the x, y, and z axes, respectively, and b x 、b y 、b z are the projections of the b-axis along the x-, y-, and z-axes respectively.

[0034] Preferably, in step S2, the unit vector expression of the vector q is:

[0035]

[0036] Where q x ,q y and q z are the projections of vector q in the x, y and z axes respectively.

[0037] Preferably, in step S2, the normal stress σ n The expression of (t) is:

[0038]

[0039] Where n T is the transpose of the n-axis vector;

[0040] The shear stress τ q The expression of (t) is:

[0041]

[0042] Where q T is the transpose of vector q;

[0043] The shear stress τ q (t) is decomposed, and the specific expression is:

[0044]

[0045] Where d1 is the matrix of the angle part of the shear stress τq(t), and s(t) is the matrix of the stress part;

[0046] It can be obtained that the shear stress τ q The variance expression of (t) is:

[0047]

[0048] Where Var[] is the symbol of variance, Cov[] is the symbol of covariance, and d k is the kth, ith, jth item in matrix d1, s i (t), s j (t) are the i-th and j-th entries in the matrix s(t), respectively. The matrix C is a constant matrix;

[0049] The normal stress σ n The two stress components σ of (t) i and σ j The variance expression is:

[0050]

[0051] The normal stress σ n The two stress components σ of (t) i and σ j The covariance expression of is:

[0052]

[0053] Where T is a complete period of a loading stress cycle.

[0054] Preferably, in step S2, the shear stress τ q The expression of the matrix d1 for the angle part in (t) is:

[0055]

[0056] Preferably, in step S31, the range of the three angles θ, α and φ is [0, π], and the initial direction is R = [r1, r2, r3], wherein r1 = [1, 0, 0], r2 = [0, 1, 0], and r3 = [0, 0, 1].

[0057] Preferably, in step S34, the specific process of the orthogonalization processing is:

[0058] Assume that three directions r1, r2, and r3 are currently in use, then define a new direction r new ;

[0059] According to the orthogonalization process, each direction r used is obtained i The expression is:

[0060]

[0061] Where i = 1, 2, 3;

[0062] According to the defined direction r new , orthogonal to direction r4, the expression of the orthogonal part is:

[0063]

[0064] Where i = 1, 2, 3;

[0065] Normalizing the orthogonal part of direction r4, the expression of direction r4 is:

[0066]

[0067] The direction matrix R[i] is updated according to the obtained direction r4 to obtain a new direction R = {r1, r2, r3, r4}.

[0068] Preferably, in step S41, the specific calculation process of the maximum normal stress on the critical surface at the dangerous point O on the test piece is:

[0069] The expression for obtaining the normal stress on any plane is:

[0070]

[0071] Where, matrix d2 is the matrix of normal stress with respect to angle;

[0072] According to the normal stress on any plane, the expression of the normal stress variance is:

[0073]

[0074] Normal stress mean σ n,m The expression is:

[0075]

[0076] Normal stress amplitude σ a The expression is:

[0077]

[0078] Through the normal stress variance and normal stress mean σ n,m and normal stress amplitude σ a , and the maximum normal stress on the critical surface at the dangerous point O on the test piece is obtained.

[0079] The present invention has the following advantages:

[0080] 1. The present invention uses the maximum variance method as the basic method, which overcomes the problems of traditional methods that require calculation of a large number of planes at different angles, and the stress and strain components on each plane need to be calculated throughout the multi-axial load history, resulting in low calculation efficiency and huge computational complexity; the shear stress used in the maximum variance method is a one-dimensional vector, avoiding the difficulty of defining the two-dimensional shear stress amplitude and mean on the traditional critical surface.

[0081] 2. The present invention uses the Powell method to optimize the calculation of the shear stress variance, getting rid of the problem of low computational efficiency caused by the traditional method of pre-setting small increments for incremental increase; the size of the increment in the Powell method is determined by a linear search method without the need for pre-setting, which means that the algorithm will dynamically adjust the increment size according to the current gradient information to ensure that the value of the objective function is reduced as quickly as possible at each step.

[0082] 3. In the present invention, among the several global optimal solutions obtained by obtaining the maximum shear stress variance, the three angles θ, α and φ corresponding to each solution are recorded, and then substituted into the maximum normal stress variance in the plane corresponding to these angles, thereby calculating the required critical angle and the critical surface corresponding to the critical angle.

[0083] 4. The method for solving the critical surface proposed in the present invention overcomes the traditional method of needing to calculate the shear stress trajectory line on each plane one by one, and then calculate the equivalent shear stress amplitude and mean, solving the problem of low calculation efficiency and providing a new fatigue analysis auxiliary method for the field of metal fatigue research. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 A flow chart of a method for determining a critical surface when predicting fatigue life of a smooth test piece according to the present invention;

[0085] Figure 2A schematic diagram of coordinate transformation for a method of determining a critical surface when predicting fatigue life of a smooth test piece according to the present invention;

[0086] Figure 3 Schematic diagram of shear stress of the method for determining the critical surface when predicting fatigue life of a smooth test piece according to the present invention;

[0087] Figure 4 A comparison chart of the calculation efficiency of the Powell method and the small increment traversal angle interval method for determining the critical surface when predicting fatigue life of a smooth test piece of the present invention;

[0088] Figure 5 Schematic diagram of the smooth thin-walled circular tube specimen used for comparing the Powell method and the small increment traversal angle interval method in the method of determining the critical surface of the smooth specimen when predicting fatigue life of the present invention. DETAILED DESCRIPTION

[0089] To fully describe the technical content, objectives and effects of the present invention, the following will be described in detail with reference to the accompanying drawings.

[0090] The method of determining the critical surface of a smooth specimen in predicting fatigue life is applicable to any smooth metal material, that is, a smooth metal specimen that is not affected by the notch effect and size effect. The specific implementation process is as follows: Figure 1 As shown:

[0091] S1. The specimen is subjected to cyclic loading to high cycle fatigue in a certain loading mode, and all stress components at the dangerous point O are obtained. The original coordinate system O-xyz and the transformed new coordinate system O-abn are established at the dangerous point O. In a preferred embodiment of the present invention, high cycle fatigue refers to the condition in which a smooth specimen is subjected to cyclic stress below its yield strength and its life is within 10 5 -10 6 Fatigue around.

[0092] S2. On the plane aob of the new coordinate system O-abn obtained in step S1, a vector q is established with an angle α with the coordinate axis a vector, such as Figure 3 As shown, the normal stress σ of plane aob is obtained n (t) and the shear stress τ along direction q q (t), and according to the normal stress σ n (t) to obtain the corresponding variance and covariance, according to the shear stress τ q (t) Get the corresponding variance.

[0093] S3. Use Powell's method to determine the maximum shear stress variance Var[τ q (t)] maxand the three angles θ that determine the critical surface where the maximum shear stress variance is located max , α max and φ max .

[0094] S4, the three angles θ obtained in step S3 max , α max and φ max , find the maximum normal stress and the maximum shear stress variance Var[τ q (t)] max plane.

[0095] Specifically, in step S1, the specific solution process for each coordinate axis of the new coordinate system O-abn is:

[0096] In the original coordinate system O-xyz, the stress tensor σ corresponding to time t is ij The expression of (t) is:

[0097]

[0098] Among them, σ x (t), σ y (t), σ z (t) are the normal stresses in the x, y, and z directions respectively; τ xy (t), τ xz (t), τ yz (t) is the shear stress in three directions. Although there are six shear stresses, according to the shear stress reciprocity law, only three shear stresses are actually needed to describe the stress state at the critical point O.

[0099] Furthermore, in order to obtain the stress and strain components of the dangerous point O on any plane, the new coordinate system O-abn is obtained by transforming the coordinate system, as follows: Figure 2 As shown, the coordinate expressions of each coordinate axis of the new coordinate system O-abn relative to each coordinate axis of the original coordinate system O-xyz are:

[0100]

[0101] Where n is the direction of the new plane normal vector, and the angle between it and the original z-axis is θ; a is relative to the x-axis of the old coordinate system, b is relative to the y-axis of the old coordinate system, and the xy plane is rotated around the z-axis by φ to obtain the new ab plane; n x 、n y 、n z are the projections of the n-axis along the x-, y-, and z-axes, respectively. x 、a y 、a z are the projections of axis a along the x, y, and z axes, respectively, and bx 、b y 、b z are the projections of the b-axis along the x-, y-, and z-axes respectively.

[0102] Specifically, in step S2, the variance and covariance corresponding to the stress and normal stress and the shear stress τ q The specific calculation process of the variance corresponding to (t) is:

[0103] The unit vector expression of vector q is:

[0104]

[0105] Where q x ,q y and q z are the projections of vector q in the x, y and z axes respectively.

[0106] Normal stress σ n The expression of (t) is:

[0107]

[0108] Where n T is the transpose of the n-axis vector.

[0109] Shear stress τ q The expression of (t) is:

[0110]

[0111] Where q T is the transpose of vector q.

[0112] Furthermore, for the sake of simplicity of calculation, the shear stress τ q (t) is decomposed, and the specific expression is:

[0113]

[0114] Where d1 is the shear stress τ q The matrix of the angle part in (t) is related to vector n and vector q, and the expression is:

[0115]

[0116] s(t) is the matrix of stress part, which is expressed as:

[0117]

[0118] From this we can get the shear stress τ q The variance expression of (t) is:

[0119]

[0120] Where Var[] is the symbol of variance, Cov[] is the symbol of covariance, and d k is the kth, ith, jth item in matrix d1, s i (t), s j (t) are the i-th and j-th items in the matrix s(t), respectively. The matrix C is only related to the load history. When the load history is determined, the matrix C is a constant matrix. Because the matrix d1 is related to the three angles θ, α, and φ, The specific expression of matrix C is:

[0121]

[0122] Where V i is the stress σ i The variance of (t), that is, V i =Var[σ i (t)], i = x, y, z, xy, xz, yz; C ij =Cov[σ i (t)]i=x,y,z,xy,xz,yz.

[0123] Normal stress σ n The two stress components σ of (t) i and σ j The expression of the variance of in one period is:

[0124]

[0125] Normal stress σ n The two stress components σ of (t) i and σ j The expression of the covariance of in one period is:

[0126]

[0127] Where T is a complete period of a loading stress cycle.

[0128] Furthermore, in step S3, the Powell method is used to determine the three angles θ of the critical surface where the maximum shear stress variance is located at the dangerous point O on the test piece. max , α max and φ max The specific implementation process is:

[0129] S31. Set the initial values ​​of θ, α and φ respectively as θ0, α0 and φ0, the initial direction R = [r1, r2, r3], the initial step size a0, the constant β (0 < β < 1) and the threshold ε as the termination condition, and obtain the initial value of the shear stress variance at the dangerous point O on the test piece. In the method of the present invention, the range of the three angles θ, α and φ is [0, π], and the initial direction is R = [r1, r2, r3], where r1 = [1, 0, 0], r2 = [0, 1, 0], and r3 = [0, 0, 1].

[0130] S32. Define the objective function of the Powell method. The specific expression is:

[0131] g(θ,α,φ)=-Var[t q (t)]=-f(θ,α,φ)

[0132] Where g(θ,α,φ) is the shear stress variance Var[t q (t)] is the function multiplied by -1. Since Powell's method is often used to obtain the minimum value, the shear stress variance Var[t q (t)] maximum value, at this time the initial objective function value is g(θ0,α0,φ0); θ is the angle between the n-axis of the new coordinate system and the z-axis of the original coordinate system, φ is the angle between the xy-plane of the original coordinate system and the ab-plane of the new coordinate system, α is the angle between the vector q and the a-axis on the ab-plane, and f(θ,α,φ) is the shear stress variance Var[t q (t)].

[0133] S33, when the kth iteration has been completed, θ, α and φ are searched in one dimension along r1, r2 and r3 in the initial direction R = [r1, r2, r3], respectively, and the angle θ after the current iteration is obtained. k , α k 、φ k and step length a k , the expressions for obtaining the new angle value and objective function value are:

[0134] θ new =θ k +a k R[i],i=1,2,3,α new =α k +a k R[i],i=1,2,3,

[0135] φ new =φ k +a k R[i],i=1,2,3;

[0136] g(θ,α,φ)=g(θ new ,α new ,φ new );

[0137] Where θ new , α new and φ new are the updated angles θ, α and φ after the k+1th iteration, θ k , α k and φ k are the angles θ, α, and φ obtained after the kth iteration, a k is the step size obtained after the kth iteration; g(θ new ,α new ,φ new ) is the function value of the angles θ, α and φ after the k+1th iteration; R[i] is the direction matrix, which initially has three direction vectors.

[0138] S34, if the objective function value Q obtained in step S33 is Q=g(θ ne ,α new ,φ new )-g(θ k ,α k ,φ k )<0, then θ, α and φ are updated, and at this time θ k+1 =θ new , α k+1 =α new ,φ k+1 =φ new , and update the direction, at this time R[k+1]=R[i], that is, the k+1th update is performed on r1, r2 and r3; if the objective function value Q=g(θ new ,α new ,φ new )-g(θ k ,α k ,φ k )≥0, the step size is reduced to a k+1 =β*a k , and perform directional orthogonalization processing. The specific process of orthogonalization processing includes:

[0139] Assume that three directions r1, r2, and r3 are currently in use, then define a new direction r new .

[0140] According to the orthogonalization process, each direction r used is obtained i The expression is:

[0141]

[0142] Wherein, i=1, 2, 3.

[0143] According to the defined direction r new , orthogonal to direction r4, the expression of the orthogonal part is:

[0144]

[0145] Wherein, i=1, 2, 3.

[0146] Normalizing the orthogonal part of direction r4, the expression of direction r4 is:

[0147]

[0148] The direction matrix R[i] is updated according to the obtained direction r4 to obtain a new direction R = {r1, r2, r3, r4}.

[0149] S35, based on step S34, regression iteration, with θ k+1 , α k+1 and φ k+1 As a new round of iteration, k+1 As the new step size, with the new direction r k+1 Proceed to the next objective function evaluation.

[0150] S36, if the objective function changes || Q k+1 ||=||g(θ k+1 ,α k+1 ,φ k+1 )-g(θ k ,α k ,φ k )||<ε, the iteration ends and outputs θ max , α max and φ max ; If the objective function changes ||Q k+1 ||=||g(θ k+1 ,α k+1 ,φ k+1 )-g(θ k ,α k ,φ k )||≥ε, then continue to update the direction until the objective function changes by ||Q k+1 ||=||g(θ k+1 ,α k+1 ,φ k+1 )-g(θ k ,α k ,φ k )||<ε.

[0151] Furthermore, in step S4, it is determined that the dangerous point O on the test piece has the maximum normal stress and experiences the maximum shear stress variance Var[τ q (t)] max The specific implementation steps of the plane are:

[0152] S41. Due to the limitations of the optimization problem, multiple global optimal solutions may be obtained, which will lead to finding multiple θ at the same time. max , α max 、φ max Therefore, it is necessary to screen the direction of the critical surface at the dangerous point O on the test piece and find θ max , α max and φ max The angle with the maximum normal stress variance is formed by:

[0153] The expression for obtaining the normal stress on any plane is:

[0154]

[0155] Wherein, matrix d2 is the matrix of normal stress with respect to angle, and its expression is:

[0156]

[0157] According to the normal stress on any plane, the expression of the normal stress variance is:

[0158]

[0159] Normal stress mean σ n,m The expression is:

[0160]

[0161] Where, σ i,m is the stress component σ i The mean of .

[0162] Normal stress amplitude σ a The expression is:

[0163]

[0164] Through the normal stress variance and normal stress mean σ n,m and normal stress amplitude σ a , the maximum normal stress on the critical surface is obtained, and the expression is:

[0165]

[0166] Where, σ n,max is the maximum normal stress on the critical surface at the dangerous point O on the test piece;a is the normal stress amplitude on the critical surface at the dangerous point O on the test piece; n,m is the mean normal stress on the critical surface at the dangerous point O on the test piece; d2 is σ a The matrix of the angle part in s m is σ n,m The matrix for the stress part.

[0167] S42. Determine the maximum value among the maximum normal stresses obtained in step S41. And find the three critical angles corresponding to the critical surface at the dangerous point O on the test piece and and That is, the angle corresponding to the critical surface at the dangerous point O on the test piece that needs to be found, that is, the critical angle, so that the required critical surface can be obtained.

[0168] The method of the present invention relies on the maximum variance method as a basic method, which can comprehensively consider the influence of the shear stress value at each moment under the random loading path on the calculation results and effectively improve the calculation efficiency. It avoids the tediousness and omissions of other methods for solving the critical surface based on the shear stress trajectory line, can effectively reduce a large amount of calculation time, and has good engineering practice significance.

[0169] The following further describes a method for determining a critical surface when predicting fatigue life of a smooth test piece according to the present invention in conjunction with an embodiment:

[0170] S1. Cyclic load the test piece to high cycle fatigue according to a certain loading method to obtain all stress components at the dangerous point O, and establish the original coordinate system O-xyz and the transformed new coordinate system O-abn at the dangerous point O.

[0171] Specifically, in step S1, the stress tensor σ corresponding to time t is ij The expression of (t) is:

[0172]

[0173] Among them, σ x (t), σ y (t), σ z (t) are the normal stresses in the x, y, and z directions respectively; τ xy (t), τ xz (t), τ yz (t) is the shear stress in three directions. Although there are six shear stresses, according to the shear stress reciprocity law, only three shear stresses are actually needed to describe the stress state at the critical point O.

[0174] Furthermore, in order to obtain the stress and strain components of the dangerous point O on any plane, the new coordinate system O-abn is obtained by transforming the coordinate system, as follows: Figure 2 As shown, the coordinate expressions of each coordinate axis of the new coordinate system O-abn relative to each coordinate axis of the original coordinate system O-xyz are:

[0175]

[0176]

[0177] Where n is the direction of the new plane normal vector, and the angle between it and the original z-axis is θ; a is relative to the x-axis of the old coordinate system, b is relative to the y-axis of the old coordinate system, and the xy plane is rotated around the z-axis by φ to obtain the new ab plane; n x 、n y 、n z are the projections of the n-axis along the x-, y-, and z-axes, respectively. x 、a y 、a z are the projections of axis a along the x, y, and z axes, respectively, and b x 、b y 、b z are the projections of the b-axis along the x-, y-, and z-axes respectively.

[0178] S2. On the plane aob of the new coordinate system O-abn obtained in step S1, a vector q is established with an angle α with the coordinate axis a vector, such as Figure 3 As shown, the normal stress σ of plane aob is obtained n (t) and the shear stress τ along direction q q (t), and according to the normal stress σ n (t) to obtain the corresponding variance and covariance, according to the shear stress τ q (t) Get the corresponding variance.

[0179] Specifically, in step S2, the unit vector expression of vector q is:

[0180]

[0181] Where q x ,q y and q z are the projections of vector q in the x, y and z axes respectively.

[0182] Normal stress σ n The expression of (t) is:

[0183]

[0184] Where n T is the transpose of the n-axis vector.

[0185] Shear stress τ q The expression of (t) is:

[0186]

[0187] Where q T is the transpose of vector q.

[0188] Furthermore, for the sake of simplicity of calculation, the shear stress τ q (t) is decomposed, and the specific expression is:

[0189]

[0190] Where d1 is the shear stress τ q The matrix of the angle part in (t) is related to vector n and vector q, and the expression is:

[0191]

[0192] s(t) is the matrix of stress part, which is expressed as:

[0193]

[0194] From this we can get the shear stress τ q The variance expression of (t) is:

[0195]

[0196] Where Var[] is the symbol of variance, Cov[] is the symbol of covariance, and d k is the kth, ith, jth item in matrix d1, s i (t), s j (t) are the i-th and j-th items in the matrix s(t), respectively. The matrix C is only related to the load history. When the load history is determined, the matrix C is a constant matrix. Because the matrix d1 is related to the three angles θ, α, and φ, The specific expression of matrix C is:

[0197]

[0198] Where V i is the stress σ i The variance of (t), that is, V i =Var[σ i (t)], i = x, y, z, xy, xz, yz; C ij =Cov[σ i (t)]i=x,y,z,xy,xz,yz.

[0199] Normal stress σ n The two stress components σ of (t) i and σ j The expression of the variance of in one period is:

[0200]

[0201] Normal stress σ n The two stress components σ of (t) i and σ j The expression of the covariance of in one period is:

[0202]

[0203] Where T is a complete period of a loading stress cycle.

[0204] S3. Use Powell's method to determine the maximum shear stress variance Var[τ q (t)] max and the three angles θ that determine the critical surface where the maximum shear stress variance is located max , α max and φ max .

[0205] S31, set the initial values ​​of θ, α and φ as θ0, α0 and φ0 respectively, the initial direction R = [r1, r2, r3], the initial step size a0, the constant β (0 < β < 1) and the threshold ε as the termination condition, and obtain the initial value of the shear stress variance In the method of the present invention, the range of the three angles θ, α and φ is [0, π], and the initial direction is R = [r1, r2, r3], where r1 = [1, 0, 0], r2 = [0, 1, 0], and r3 = [0, 0, 1].

[0206] S32. Define the objective function of the Powell method. The specific expression is:

[0207]

[0208] Where g(θ,α,φ) is the variance of the shear stress The function multiplied by -1, because the Powell method is often used to obtain the minimum value, the function g(θ,α,φ) multiplied by -1 is taken as the objective function and its minimum value is obtained to obtain the shear stress variance. The maximum value, at this time the initial objective function value is g(θ0,α0,φ0); θ is the angle between the n-axis of the new coordinate system and the z-axis of the original coordinate system, φ is the angle between the xy plane of the original coordinate system and the ab plane of the new coordinate system, α is the angle between the vector q and the a-axis on the ab plane, and f(θ,α,φ) is the variance of the shear stress function.

[0209] S33, when the kth iteration has been completed, θ, α and φ are searched in one dimension along r1, r2 and r3 in the initial direction R = [r1, r2, r3], respectively, and the angle θ after the current iteration is obtained. k , α k 、φ k and step length a k , the expressions for obtaining the new angle value and objective function value are:

[0210] θ new =θ k +a k R[i],i=1,2,3,α new =α k +a k R[i],i=1,2,3,

[0211] φ new =φ k +a k R[i],i=1,2,3;

[0212] g(θ,α,φ)=g(θ new ,α new ,φ new );

[0213] Where θ new , α new and φ new are the updated angles θ, α and φ after the k+1th iteration, θ k , α k and φ k are the angles θ, α, and φ obtained after the kth iteration, a k is the step size obtained after the kth iteration; g(θ new ,α new ,φ new ) is the function value of the angles θ, α and φ after the k+1th iteration; R[i] is the direction matrix, which initially has three direction vectors.

[0214] S34, if the objective function value Q obtained in step S33 is Q=g(θ ne ,α new ,φ new )-g(θ k ,α k ,φ k )<0, then θ, α and φ are updated, and at this time θ k+1 =θ new , α k+1 =α new ,φ k+1=φ new , and update the direction, at this time R[k+1]=R[i], that is, the k+1th update is performed on r1, r2 and r3; if the objective function value Q=g(θ new ,α new ,φ new )-g(θ k ,α k ,φ k )≥0, the step size is reduced to a k+1 =β*a k , and perform directional orthogonalization processing. The specific process of orthogonalization processing includes:

[0215] Assume that three directions r1, r2, and r3 are currently in use, then define a new direction r new .

[0216] According to the orthogonalization process, each direction r used is obtained i The expression is:

[0217]

[0218] Wherein, i=1, 2, 3.

[0219] According to the defined direction r new , orthogonal to direction r4, the expression of the orthogonal part is:

[0220]

[0221] Wherein, i=1, 2, 3.

[0222] Normalizing the orthogonal part of direction r4, the expression of direction r4 is:

[0223]

[0224] The direction matrix R[i] is updated according to the obtained direction r4 to obtain a new direction R = {r1, r2, r3, r4}.

[0225] S35, based on step S34, regression iteration, with θ k+1 , α k+1 and φ k+1 As a new round of iteration, k+1 As the new step size, with the new direction r k+1 Proceed to the next objective function evaluation.

[0226] S36, if the objective function changes || Q k+1 ||=||g(θ k+1 ,α k+1,φ k+1 )-g(θ k ,α k ,φ k )||<ε, the iteration ends and outputs θ max , α max and φ max ; If the objective function changes ||Q k+1 ||=||g(θ k+1 ,α k+1 ,φ k+1 )-g(θ k ,α k ,φ k )||≥ε, then continue to update the direction until the objective function changes by ||Q k+1 ||=||g(θ k+1 ,α k+1 ,φ k+1 )-g(θ k ,α k ,φ k )||<ε.

[0227] S4, the three angles θ obtained in step S3 max , α max and φ max , find the maximum normal stress and the maximum shear stress variance Var[τ q (t)] max plane.

[0228] S41. Due to the limitations of the optimization problem, multiple global optimal solutions may be obtained, which will lead to finding multiple θ at the same time. max , α max 、φ max Therefore, it is necessary to screen the direction of the critical surface at the dangerous point O on the test piece and find θ max , α max and φ max The angle with the maximum normal stress variance is formed by:

[0229] The expression for obtaining the normal stress on any plane is:

[0230]

[0231] Wherein, matrix d2 is the matrix of normal stress with respect to angle, and its expression is:

[0232]

[0233] According to the normal stress on any plane, the expression of the normal stress variance is:

[0234]

[0235] Normal stress mean σ n,m The expression is:

[0236]

[0237] Where, σ i,m is the stress component σ i The mean of .

[0238] Normal stress amplitude σ a The expression is:

[0239]

[0240] Through the normal stress variance and normal stress mean σ n,m and normal stress amplitude σ a , the maximum normal stress on the critical surface at the dangerous point O on the test piece is obtained, and the expression is:

[0241]

[0242] Where, σ n,max is the maximum normal stress on the critical surface at the dangerous point O on the test piece; a is the normal stress amplitude on the critical surface at the dangerous point O on the test piece; n,m is the mean normal stress on the critical surface at the dangerous point O on the test piece; d2 is σ a The matrix of the angle part in s m is σ n,m The matrix for the stress part.

[0243] S42. Determine the maximum value among the maximum normal stresses obtained in step S41. And find the three critical angles corresponding to the critical surface at the dangerous point O on the test piece and and That is, the angle corresponding to the critical surface at the dangerous point O on the test piece that needs to be found, that is, the critical angle, so that the required critical surface can be obtained.

[0244] In this specific embodiment, the Powell method is used to optimize the calculation in the search for the critical angle, and compared with the traditional method of increasing the calculation by every 1°, as shown in FIG. Figure 4 As shown, it can be seen that the method of the present invention can greatly improve the calculation efficiency, thereby being able to obtain the required critical surface angle, and laying a good foundation for obtaining the fatigue life of the specimen.

[0245] like Figure 5As shown in the figure, the smooth thin-walled circular tube specimen used for comparison, F1 in the figure is tension and compression, F2 is torsion, and the figure indicates the situation where tension, compression and torsion occur simultaneously, while Figure 4 The “phase difference x°” indicates that the phase difference of the loading waveform is x° when two loading methods are carried out simultaneously. “Tension and compression” and “torsion” refer to loading methods with only “tension and compression” or “torsion”.

[0246] The embodiments described above are merely descriptions of preferred implementations of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should fall within the scope of protection determined by the claims of the present invention.

Claims

1. A method for determining the critical surface when predicting fatigue life of a smooth test piece, characterized in that: Specific implementation steps include: S1. Perform a high-cycle fatigue test on the test piece to obtain all stress components at the dangerous point O on the test piece, and establish the original coordinate system O-xyz and the transformed new coordinate system O-abn at the dangerous point O on the test piece; S2. On the plane aob of the new coordinate system O-abn obtained in step S1, a vector q is established with an angle α with the coordinate axis a vector to obtain the normal stress σ of the plane aob. n (t) and the shear stress τ along direction q q (t), and according to the normal stress σ n (t) to obtain the corresponding variance and covariance, according to the shear stress τ q (t) obtain the corresponding variance; S3. Use Powell's method to determine the maximum shear stress variance Var[τ q (t)] max and the three angles that determine the critical surface where the maximum shear stress variance is located 、 and : S31, respectively set 、 and Initial value of 、 and , initial direction , initial step length , constant β and threshold , 0<β<1, the initial value of the shear stress variance at the dangerous point O on the test piece is obtained ; S32. Define the objective function of the Powell method. The specific expression is: ； Where, is the shear stress variance The function after multiplying with -1, is the angle between the n-axis of the new coordinate system and the z-axis of the original coordinate system, It is the angle of rotation from the original coordinate system xy plane to the new coordinate system ab plane. is the angle between vector q and axis a on plane ab, is the shear stress variance function; S33. When the kth iteration has been completed, 、 and Along the initial direction in 、 and Perform a one-dimensional search based on the angle after the current iteration 、 、 and step length , the expressions for obtaining the new angle value and objective function value are: , , ； ； Where, 、 and are updated after the k+1th iteration respectively. 、 and horn, 、 and is obtained after the kth iteration 、 and horn, is the step size obtained after the kth iteration; yes 、 and The function value of the angle is taken after the k+1th iteration; is the direction matrix, initially there are three direction vectors; S34, if the objective function value obtained in step S33 When 、 and Accept the update, now , , , and update the direction, ; If the objective function value obtained in step S33 When , the step length is , and perform directional orthogonalization processing; S35. Based on step S34, 、 and As a new round of iteration, As a new step length, in a new direction Evaluate the next objective function; S36, if the objective function changes When , the iteration ends and outputs 、 and ; If the objective function changes , then continue to update the direction until the objective function changes ; S4, according to the three angles obtained in step S3 、 and , find the maximum normal stress and the maximum shear stress variance Var[τ q (t)] max Plane: S41. Screen according to the direction of the critical surface at the dangerous point O on the test piece, and find the critical surface at the dangerous point O on the test piece. 、 and The angle with the maximum normal stress variance formed by the test piece is the expression of the maximum normal stress on the critical surface at the dangerous point O on the test piece: ； Where, is the maximum normal stress on the critical surface; is the normal stress amplitude on the critical surface; is the mean normal stress on the critical surface; yes The matrix of the angle part; yes The matrix of the stress part; S42, determining the maximum value among the maximum normal stress at the dangerous point O on the test piece obtained in step S41. , and find the three critical angles corresponding to the critical surface at the dangerous point O on the test piece 、 and .

2. The method for determining the critical surface when predicting fatigue life of a smooth test piece according to claim 1, characterized in that: In step S1, the stress tensor σ corresponding to time t is ij The expression of (t) is: ； Among them, σ x (t), σ y (t), σ z (t) are the normal stresses in the x, y, and z directions respectively; τ xy (t), τ xz (t), τ yz (t) is the shear stress in three directions.

3. The method for determining the critical surface when predicting fatigue life of a smooth test piece according to claim 1 or 2, characterized in that: In step S1, the coordinate expressions of the coordinate axes of the new coordinate system O-abn relative to the coordinate axes of the original coordinate system O-xyz are: ； ； ； Where n is the direction of the new plane normal vector, and the angle between it and the original z-axis is θ; a is relative to the x-axis of the old coordinate system, b is relative to the y-axis of the old coordinate system, and the xy plane is rotated around the z-axis by ϕ to obtain the new ab plane; n x 、n y 、n z are the projections of the n-axis along the x-, y-, and z-axes, respectively. x 、a y 、a z are the projections of axis a along the x, y, and z axes, respectively, and b x 、b y 、b z are the projections of the b-axis along the x-, y-, and z-axes respectively.

4. The method for determining the critical surface when predicting fatigue life of a smooth test piece according to claim 1, characterized in that: In step S2, the unit vector expression of the vector q is: ； Where q x ,q y and q z are the projections of vector q in the x, y and z axes respectively.

5. The method for determining the critical surface when predicting fatigue life of a smooth test piece according to claim 1 or 4, characterized in that: In step S2, the normal stress σ n The expression of (t) is: ； Where n T is the transpose of the n-axis vector; The shear stress τ q The expression of (t) is: ； Where q T is the transpose of vector q; The shear stress τ q (t) is decomposed, and the specific expression is: ； Where, is the shear stress τ q The matrix of the angle part in (t) is It is the matrix about stress part; It can be obtained that the shear stress τ q The variance expression of (t) is: ； Where Var[] is the symbol of variance, Cov[] is the symbol of covariance, and d k is the kth, ith, jth item in matrix d1, s i (t), s j (t) are the i-th and j-th entries in the matrix s(t), respectively. The matrix C is a constant matrix; The normal stress σ n The two stress components of (t) and The variance expression is: ； The normal stress σ n The two stress components of (t) and The covariance expression of is: ； Where, is a complete cycle of loading stress.

6. The method for determining the critical surface when predicting fatigue life of a smooth test piece according to claim 5, characterized in that: In step S2, the shear stress τ q (t) The matrix of the angle part The expression is: 。 7. The method for determining the critical surface when predicting fatigue life of a smooth test piece according to claim 1, characterized in that: In step S31, the 、 and The range of the three angles is , the initial direction is ,in, , , .

8. The method for determining the critical surface when predicting fatigue life of a smooth test piece according to claim 1, characterized in that: In step S34, the specific process of the orthogonalization process is: Assume there are currently 3 directions If both are used, define a new direction ; According to the orthogonalization process, each direction used is obtained The expression is: ； Where i=1, 2, 3; According to the direction of the definition , for the direction Orthogonal, the expression of the orthogonal part is: ； Where i=1, 2, 3; Direction Normalize the orthogonal part of The expression is: ； According to the direction obtained Update the direction matrix , get a new direction .

9. The method for determining the critical surface when predicting fatigue life of a smooth test piece according to claim 1, characterized in that: In step S41, the specific calculation process of the maximum normal stress on the critical surface at the dangerous point O on the test piece is: The expression for obtaining the normal stress on any plane is: ； In the formula, the matrix is the matrix of normal stress with respect to angle; According to the normal stress on any plane, the expression of the normal stress variance is: ； Normal stress mean σ n,m The expression is: ； Normal stress amplitude The expression is: ； Through the normal stress variance and normal stress mean σ n,m and normal stress amplitude , and the maximum normal stress on the critical surface at the dangerous point O on the test piece is obtained.

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