Method for determining critical surface of smooth test piece during fatigue life prediction

Abstract

The invention provides a method for determining a critical plane during fatigue life prediction of a smooth test piece, and the method comprises the steps: carrying out the high-cycle fatigue test of the test piece, obtaining all stress components at a dangerous point O, and building an original coordinate system O-xyz and a transformed new coordinate system O-abn at the dangerous point O; taking a vector q on the plane aob of the obtained new coordinate system O-abn, wherein the included angle between the vector q and the coordinate axis a is alpha, and obtaining the normal stress of the plane aob and the slitting shear stress along the direction q; a Powell method is used for determining the maximum value of the slitting shear stress variance and determining three angles theta max, alpha max and phi max of a critical plane where the maximum value of the slitting shear stress variance is located; a plane having the maximum normal stress and experiencing the maximum shear stress variance is found from the three resulting angles. According to the method, the problem of low calculation efficiency caused by traversing the whole angle range by using the small increment in the maximum variance method is solved, the maximum normal stress on the plane is considered, and prediction defects in the fatigue life are reduced.

Description

Technical Field

[0001] The 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 fail due to fatigue during use and are often subjected to multiaxial cyclic loads. In order to predict the 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 is the most promising method in the field of multiaxial fatigue life prediction methods. It studies fatigue crack initiation by specifying the plane with the maximum fatigue damage variable inside the material unit.

[0003] The critical surface method selects different damage variables according to the different types of critical surfaces, and then calculates the damage parameters of all planes. The plane with the largest calculation result is defined as the critical surface. It is common to use the equivalent stress amplitude plane as the critical surface. The main methods for calculating the equivalent shear stress amplitude under multiaxial non-proportional loads include the longest projection method, the longest chord method, the minimum circumscribed circle method, the minimum circumscribed ellipse method and the maximum variance method. Except for the maximum variance method, other methods require the calculation of complex shear stress trajectories one by one, and then calculate the equivalent shear stress amplitude and mean, which has low calculation efficiency. The maximum variance method can comprehensively consider the influence of the shear stress value at each moment under the random loading path on the calculation results, and does not require a clear shear stress trajectory. It is not only suitable for constant amplitude fatigue loads, but also can quickly solve the critical surface direction under variable amplitude fatigue loads. For the cycle counting of multiaxial random fatigue loads, it is only necessary to count the shear stress on the critical surface and the normal stress on the critical surface.

[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 three angles θ, φ, and α on the stress decomposition plane and 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 calculating 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 maximum shear stress amplitude variance on the plane and the maximum normal stress variance on the plane. Summary of the invention

[0005] In view of 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 normal stress of plane aob and the shear stress along direction q, as well as the variance and covariance corresponding to the normal stress and the variance corresponding to the shear stress are obtained by the maximum variance method, 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 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 , Powell's method will dynamically adjust the increment size according to the current gradient information to ensure that each step reduces the value of the objective function as quickly as possible, getting rid of the problem of low computational efficiency caused by the traditional method of pre-setting small increments for incremental increase; according to the three angles θ max , α max and φ max , find the region 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 a critical surface of a smooth test piece when predicting fatigue life, and 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. Establish a vector q with an angle α with the coordinate axis a vector on the plane aob of the new coordinate system O-abn obtained in step S1, and 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) get the corresponding variance;

[0009] S3. Use Powell's method 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 :

[0010] S31, respectively set the initial values ​​θ0, α0 and φ0 of θ, α and φ, 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 -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 on the ab-plane and the a-axis, 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 perform one-dimensional search along r1, r2 and r3 in the initial direction R = [r1, r2, r3] respectively, and according to the angle θ after the current iteration k , α k ,φ k and step length a k , the expression for obtaining the new angle value and objective function value is:

[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] In the formula, 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 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, a 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, 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, three angles θ obtained according to step S3 max , α max and φ max , find the maximum normal stress and the maximum shear stress variance Var[τ q (t)] max The 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 obtained as follows:

[0024]

[0025] In the formula, 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 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 normal vector of the new plane, 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 the a-axis 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] In the formula, q x ,q y and q z are the projections of vector q on 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] In the formula, 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 k, i, 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] Assuming that three directions r1, r2, and r3 are currently in use, define a new direction r new ;

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

[0060]

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

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

[0066]

[0067] Update the direction matrix R[i] 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 problem that the traditional method needs to calculate a large number of planes at different angles, and the stress-strain components on each plane need to calculate the entire multi-axis load history, which leads to low calculation efficiency and huge calculation amount; the shear stress used in the maximum variance method is a one-dimensional vector, which avoids 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 variance of the shear stress, getting rid of the problem of low calculation 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 each step reduces the value of the objective function as quickly as possible.

[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 brought into the maximum normal stress variance in the plane corresponding to these angles, so as to calculate 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 need to calculate the shear stress trajectory on each plane one by one in the traditional method, and then calculate the equivalent shear stress amplitude and mean, solves the problem of low calculation efficiency, and provides 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 of a method for determining a critical surface when predicting fatigue life of a smooth test piece according to the present invention;

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

[0087] Figure 4 A comparison diagram of the calculation efficiency of the method for determining the critical surface of the smooth test piece in predicting fatigue life using the Powell method and the small increment traversal angle interval method;

[0088] Figure 5 A schematic diagram of a smooth thin-walled circular tube specimen used in the present invention to compare the method of determining the critical surface of a smooth test piece when predicting fatigue life with the Powell method and the method of traversing the angle interval with small increments. DETAILED DESCRIPTION

[0089] In order 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 test piece when predicting fatigue life is applicable to any smooth metal material, that is, a smooth metal test piece that is not affected by the notch effect and the size effect. The specific implementation process is as follows: Figure 1 As shown:

[0091] S1. The test piece is subjected to high cycle fatigue cyclic loading according to a certain loading method, and all stress components at the dangerous point O are obtained, and 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 that the smooth test piece is subjected to cyclic stress below its yield strength, and the 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, establish a vector q 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) to obtain the corresponding variance.

[0093] S3. Use Powell's method to determine the maximum value of the 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, three angles θ obtained according to step S3 max , α max and φ max , find the point with the maximum normal stress and the maximum shear stress variance Var[τ q (t)] max plane.

[0095] Specifically, the specific solution process of each coordinate axis of the new coordinate system O-abn in step S1 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 shear stresses in six directions, according to the shear stress reciprocity law, only three shear stresses are actually needed to describe the stress state of the dangerous 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 normal vector of the new plane, 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 the a-axis 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, normal stress, and 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] In the formula, q x ,q y and q z are the projections of vector q on 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] In the formula, 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, expressed as:

[0117]

[0118] It can be obtained that 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 k, i, 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), namely 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 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 conditions, 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)] multiplied by -1. Since the Powell 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 of rotation from the xy-plane of the original coordinate system to the ab-plane of the new coordinate system, α is the angle between the vector q on the ab-plane and the a-axis, 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 perform one-dimensional search along r1, r2 and r3 in the initial direction R = [r1, r2, r3] respectively, and according to the angle θ after the current iteration k , α k ,φ k and step length a k , the expression for obtaining the new angle value and objective function value is:

[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] In the formula, 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 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 at 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 orthogonal processing. The specific process of orthogonal processing includes:

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

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

[0141]

[0142] In the formula, 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] In the formula, i=1, 2, 3.

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

[0147]

[0148] Update the direction matrix R[i] 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, a k+1 As the new step size, with the new direction r k+1 Proceed to evaluate the next objective function.

[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, 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] Further, 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] In the formula, 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] In the formula, σ n,max is the maximum normal stress on the critical surface at the dangerous point O on the test piece; σa 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 n,m The matrix for the stress part.

[0167] S42, determining 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 that can comprehensively consider the influence of the shear stress value at each moment under the random loading path on the calculation result and effectively improve the calculation efficiency. It avoids the cumbersomeness 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 is a further description of 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 in a certain loading mode 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 shear stresses in six directions, according to the shear stress reciprocity law, only three shear stresses are actually needed to describe the stress state of the dangerous 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 normal vector of the new plane, 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 the a-axis 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, establish a vector q 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) to obtain the corresponding variance.

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

[0180]

[0181] In the formula, q x ,q y and q z are the projections of vector q on 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] In the formula, 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, expressed as:

[0193]

[0194] It can be obtained that 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 k, i, 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), namely 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 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 .

[0205] S31, set the initial values ​​of θ, α and φ 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 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. Since 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 of the original coordinate system xy plane rotated to the ab plane of the new coordinate system, α is the angle between the vector q on the ab plane and the a-axis, and f(θ,α,φ) is the shear stress variance function.

[0209] S33, when the kth iteration has been completed, θ, α and φ perform one-dimensional search along r1, r2 and r3 in the initial direction R = [r1, r2, r3] respectively, and according to the angle θ after the current iteration k , α k ,φ k and step length a k , the expression for obtaining the new angle value and objective function value is:

[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] In the formula, 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 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 at 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 orthogonal processing. The specific process of orthogonal processing includes:

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

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

[0217]

[0218] In the formula, 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] In the formula, 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, a k+1 As the new step size, with the new direction r k+1 Proceed to evaluate the next objective function.

[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, 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, three angles θ obtained according to 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] In the formula, 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] In the formula, σ n,max is the maximum normal stress on the critical surface at the dangerous point O on the test piece; σ 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 n,m The matrix for the stress part.

[0243] S42, determining 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° increment, 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, laying a good foundation for obtaining the fatigue life of the specimen.

[0245] like Figure 5As shown in the figure, the smooth thin-walled round 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, and “tension and compression” and “torsion” refer to loading methods with only “tension and compression” or “torsion”.

[0246] The embodiments described above are only descriptions of the preferred implementation modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design 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 all fall within the protection scope determined by the claims of the present invention.

Claims

1. A method for determining a critical surface when predicting fatigue life of a smooth test piece, characterized in that: The 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. Establish a vector q with an angle α with the coordinate axis a vector on the plane aob of the new coordinate system O-abn obtained in step S1, and 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) get the corresponding variance; S3. Use Powell's method 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 : S31, respectively set the initial values ​​θ0, α0 and φ0 of θ, α and φ, 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; S32. Define the objective function of the Powell method. The specific expression is: g(θ,α,φ)=-Var[t q (t)]=-f(θ,α,φ) Where g(θ,α,φ) is the shear stress variance Var[t q (t)] is the function obtained by multiplying -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 on the ab-plane and the a-axis, and f(θ,α,φ) is the variance of the shear stress Var[t q (t)]; S33, when the kth iteration has been completed, θ, α and φ perform one-dimensional search along r1, r2 and r3 in the initial direction R = [r1, r2, r3] respectively, and according to the angle θ after the current iteration k , α k ,φ k and step length a k , the expression for obtaining the new angle value and objective function value is: i new =θ k +a k R[i],i=1,2,3,a new =a k +a k R[i],i=1,2,3, f new =φ k +a k R[i],i=1,2,3; g(θ,α,φ)=g(θ new ,a new ,f new ); In the formula, θ new , α new and φ new are the angles θ, α and φ updated 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; 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 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 orthogonal processing; S35, based on step S34, k+1 , α k+1 and φ k+1 As a new round of iteration, a k+1 As the new step size, with the new direction r k+1 Evaluate the next objective function; S36. If the objective function changes || Q k+1 ||=||g(θ k+1 ,α k+1 ,φk +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, 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; S4, three angles θ obtained according to step S3 max , α max and φ max , find the point with the maximum normal stress and the maximum shear stress variance Var[τ q (t)] max The plane: 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 obtained as follows: In the formula, 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 n,m 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 of a smooth test piece when predicting fatigue life 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 normal vector of the new plane, 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 the a-axis 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 of a smooth test piece when predicting fatigue life according to claim 1, characterized in that: In step S2, the unit vector expression of the vector q is: In the formula, q x ,q y and q z are the projections of vector q on 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: In the formula, q T is the transpose of vector q; The shear stress τ q (t) is decomposed, and the specific expression is: 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; 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 k, i, 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 σ of (t) i and σ j The variance expression is: The normal stress σ n The two stress components σ of (t) i and σ j The covariance expression is: Where T is a complete period of a loading stress cycle.

6. The method for determining the critical surface of a smooth test piece when predicting fatigue life according to claim 5, characterized in that: In step S2, the shear stress τ q The expression of the matrix d1 for the angle part in (t) is:

7. The method for determining the critical surface of a smooth test piece when predicting fatigue life according to claim 1, characterized in that: 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], where r1 = [1, 0, 0], r2 = [0, 1, 0], and r3 = [0, 0, 1].

8. The method for determining the critical surface of a smooth test piece when predicting fatigue life according to claim 1, characterized in that: In step S34, the specific process of the orthogonalization process is: Assuming that three directions r1, r2, and r3 are currently in use, define a new direction r new ; According to the orthogonalization process, each direction r used is obtained i The expression is: Wherein, i = 1, 2, 3; According to the defined direction r new , orthogonal to direction r4, the expression of the orthogonal part is: Wherein, i = 1, 2, 3; Normalizing the orthogonal part of direction r4, the expression of direction r4 is: Update the direction matrix R[i] according to the obtained direction r4 to obtain a new direction R = {r1, r2, r3, r4}.

9. The method for determining the critical surface of a smooth test piece when predicting fatigue life 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: Where, matrix d2 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 nm The expression is: Normal stress amplitude σ a The expression is: 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.

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