Estimation method for fatigue crack propagation rate of laminated structure

Abstract

The invention discloses a method for estimating the fatigue crack propagation rate of a laminated structure, and belongs to the technical field of laminated structure fatigue crack prediction. The invention aims to solve the problem that the prediction of the existing fatigue crack growth rate cannot be separated from experimental data support. Comprising the following steps: dividing a laminated structure into an upper sub-beam and a lower sub-beam, calculating tangential displacement and normal displacement of a crack interface, and calculating tangential traction force and normal traction force of the layered crack interface by combining the interface fracture toughness of the laminated structure; calculating the total strain energy of the laminated structure by combining the internal stress resultant force, the bending moment and the shear stress resultant force of the upper sub-beam and the lower sub-beam; establishing a laminated structure balance equation and obtaining a control equation set; and solving the control equation set to obtain the theoretically calculated crack length, cycle index, upper sub-beam applied load theoretical value and upper sub-beam displacement theoretical value of the laminated structure under the fatigue load of the laminated structure, calculating to obtain the cycle energy release rate, and further calculating to obtain the crack growth rate. The method is used for estimating the fatigue crack growth rate.

Description

Technical Field

[0001] The invention relates to a method for estimating a laminate structure fatigue crack growth rate, and belongs to the technical field of laminate structure fatigue crack prediction. Background Art

[0002] During the service life of engineering materials and structures, fatigue failure accounts for more than 80%, and its mechanism research and failure prediction have always been the core challenges of fatigue fracture mechanics. As a classic theory, Paris's law achieves a macroscopic description of the fatigue crack growth law by constructing a single power-law relationship between the fatigue crack growth rate and the stress intensity factor or energy release rate.

[0003] Since fatigue tests are difficult and costly to implement, scholars have developed numerical methods to simulate fatigue crack growth, such as virtual crack closure technology (VCCT) and cohesive force units. Although there are many methods for predicting fatigue crack growth, the prediction of fatigue crack growth behavior still cannot be separated from the support of fatigue experimental data. The classic Paris law is a phenomenological method based on experimental data, which cannot explain the physical nature of fatigue crack growth. As an empirical formula, it requires a large amount of fatigue test data to determine the constants in the formula. In addition, although numerical simulation technology has been developed, it is necessary to obtain basic data through high-cost and long-cycle fatigue tests to calibrate the simulation model parameters, and its prediction accuracy is significantly limited by the support of experimental data. This technical bottleneck seriously restricts the engineering application effectiveness of fatigue life prediction of laminated structures. The huge workload involved in fatigue testing leads to an extension of the experimental cycle, which significantly affects the structural development process. Summary of the invention

[0004] In view of the problem that the existing prediction of fatigue crack growth rate cannot be separated from the support of experimental data, the present invention provides a method for estimating the fatigue crack growth rate of a laminated structure.

[0005] A method for estimating fatigue crack growth rate of a laminated structure of the present invention comprises:

[0006] The laminated structure is divided into an upper sub-beam and a lower sub-beam according to the delamination crack interface, and the displacement field models of the upper sub-beam and the lower sub-beam are established using the Timoshenko beam theory to obtain the mid-plane displacement along the length and thickness directions and the rotation angle formula of the mid-plane relative to the normal direction; the tangential displacement and normal displacement of the crack interface, as well as the resultant internal stress, bending moment and shear stress of the upper sub-beam and the lower sub-beam are obtained by calculation;

[0007] The tangential displacement and normal displacement of the crack interface are used in combination with the interface fracture toughness of the laminated structure to calculate the tangential traction and normal traction of the delamination crack interface; the total strain energy of the laminated structure is then calculated in combination with the resultant internal stress, bending moment and shear stress of the upper sub-beam and the lower sub-beam;

[0008] Based on the total strain energy of the laminated structure and the potential energy of the external load applied to the laminated structure, establish the equilibrium equation of the laminated structure; based on the equilibrium equation of the laminated structure, obtain the control equations;

[0009] Solve the control equations to obtain the crack length, number of cycles, theoretical value of the load applied to the upper sub-beam, and theoretical value of the displacement of the upper sub-beam of the laminated structure under fatigue load by theoretical calculation, calculate the cyclic energy release rate, and further calculate the crack propagation rate.

[0010] According to the estimation method of the fatigue crack propagation rate of the laminated structure of the present invention, the mid-plane displacement and the rotation angle of the mid-plane relative to the normal direction in the length direction and the thickness direction are as follows:

[0011]

[0012] where u r is the displacement of the upper sub-beam and the lower sub-beam along the X-axis in the length direction, r = t, b, t represents the upper sub-beam, and b represents the lower sub-beam; w r is the displacement of the upper sub-beam and the lower sub-beam along the Z-axis in the thickness direction; x is the X-axis coordinate, and z r is the Z-axis coordinate of the upper sub-beam and the lower sub-beam, is the mid-plane displacement of the upper sub-beam and the lower sub-beam in the X-axis direction, is the mid-plane displacement of the upper sub-beam and the lower sub-beam in the Z-axis direction, is the rotation angle of the mid-plane of the upper sub-beam and the lower sub-beam relative to the normal direction.

[0013] According to the estimation method of the fatigue crack propagation rate of the laminated structure of the present invention, the calculation methods for the tangential displacement and the normal displacement of the crack interface are as follows:

[0014]

[0015] where Δu t represents the tangential displacement of the crack interface, and Δu n represents the normal displacement of the crack interface, h b is the thickness of the lower sub-beam, and h t is the thickness of the upper sub-beam.

[0016] According to the estimation method of the fatigue crack propagation rate of the laminated structure of the present invention, the calculation methods for the resultant internal stress, bending moment, and resultant shear stress of the upper sub-beam and the lower sub-beam are as follows:

[0017] According to the strain calculation formulas of the upper sub-beam and the lower sub-beam:

[0018]

[0019] where is the normal strain of the upper sub-beam and the lower sub-beam, are the shear strains of the upper sub - beam and the lower sub - beam. In the variable subscript, x represents the derivative of the corresponding variable;

[0020] Then we get:

[0021]

[0022] In the formula are the resultant internal stresses of the upper sub - beam and the lower sub - beam, are the bending moments of the upper sub - beam and the lower sub - beam, are the resultant shear stresses of the upper sub - beam and the lower sub - beam;

[0023] In the formula, b is the width of the laminated structure, is the normal stress in the X - axis direction, is the shear stress in the XZ plane, are the tensile stiffnesses of the upper sub - beam and the lower sub - beam, are the tensile - bending coupling stiffnesses of the upper sub - beam and the lower sub - beam, are the bending stiffnesses of the upper sub - beam and the lower sub - beam, are the shear stiffnesses of the upper sub - beam and the lower sub - beam.

[0024] According to the method for estimating the fatigue crack growth rate of the laminated structure of the present invention, the calculation method for the interfacial fracture toughness of the laminated structure is:

[0025]

[0026] In the formula, φ n is the interfacial fracture toughness of the laminated structure, P is the test value of the load applied to the upper sub - beam, W is the test value of the displacement of the upper sub - beam, a is the crack length of the laminated structure, and Δ is the crack correction length.

[0027] According to the method for estimating the fatigue crack growth rate of the laminated structure of the present invention, the calculation methods for the tangential traction force and the normal traction force at the delaminated crack interface are:

[0028]

[0029] In the formula, T t is the tangential traction force, T n is the normal traction force, δ0 is the characteristic length of the cohesive zone, and D is the damage variable;

[0030]

[0031] In the formula, k is the correction coefficient, is the cyclic damage increment, is the monotonic damage increment, and t is the time;

[0032]

[0033] In the formula is the cumulative displacement increment in a load cycle period, and δ ∑ is the total displacement required to cause the failure of the cohesive zone, T is the total cohesive tension at the delamination crack interface, and σ f is the cohesive fatigue limit, and σ max is the normal cohesive strength after N load cycles, and σ max,0 is the interface cohesive strength when no damage occurs, and H is the Heaviside function, is the cumulative displacement in a load cycle period, and δ f is the separation displacement corresponding to the cohesive fatigue limit;

[0034]

[0035] Among them, the cohesive fatigue limit σ f is determined by using the conditional yield stress: Define the normal traction force T n - The normal displacement Δu of the crack interface n The parallel line in the elastic response stage of the curve, and shift the parallel line by a specified plastic deformation amount d; The cohesive fatigue limit σ f is obtained through the dimensionless parameters δ n - The normal displacement Δu of the crack interface n at the intersection point of the shifted parallel line and the curve of the normal traction force T f / δ0 and σ f / σ max,0 jointly determined.

[0036] According to the method for estimating the fatigue crack growth rate of the laminated structure of the present invention, the calculation method of the total strain energy of the laminated structure is as follows:

[0037]

[0038] In the formula, U is the total strain energy of the laminated structure, and V t is the volume of the upper sub-beam, and V b is the volume of the lower sub-beam, and A is the interlayer area of the crack interface;

[0039] The potential energy of the external load applied to the laminated structure is expressed as V:

[0040]

[0041] In the formula, L is the length of the laminated structure, and P t is the theoretical value of the load applied by the upper sub-beam, and δ D is the Dirac function, and x e is the X-axis coordinate of the load application position;

[0042]

[0043] According to the method for estimating the fatigue crack growth rate of a laminated structure of the present invention, a balance equation for the laminated structure is established based on the variational method as follows:

[0044] δ(U - V) = 0 (14),

[0045] where δ is the variational symbol;

[0046] Based on the structural balance equation, the following control equation set is obtained:

[0047]

[0048] According to the method for estimating the fatigue crack growth rate of a laminated structure of the present invention, by solving the control equation set, the crack length a, the number of cycles N, the theoretical value P of the load applied to the upper sub-beam, t and the theoretical value w of the displacement of the upper sub-beam t ;;

[0049] The cyclic energy release rate is expressed as ΔG as:

[0050]

[0051] where is the maximum value of the theoretical value of the load applied to the upper sub-beam, is the minimum value of the theoretical value of the load applied to the upper sub-beam, C is the flexibility of the laminated structure; C = w t / P t ;

[0052] The crack length a and the number of cycles N of the laminated structure are fitted with a power function to obtain a functional relationship, and then the functional relationship is differentiated to obtain the crack growth rate da / dN.

[0053] According to the method for estimating the fatigue crack growth rate of a laminated structure of the present invention, the crack growth rate da / dN and the cyclic energy release rate ΔG are plotted on a double logarithmic coordinate system to obtain the fatigue crack growth rate curve of the laminated structure; then the Paris law is used to fit the fatigue crack growth rate curve to obtain a fitted curve; the actual fatigue crack growth rate is determined according to the fitted curve:

[0054]

[0055] where B is a constant related to the material, and m is the exponential constant of the Paris law.

[0056] Advantages of the present invention: Aiming at the technical problem that the prediction of the fatigue crack growth rate of a laminated structure highly depends on the support of fatigue test data to calibrate model parameters, the method of the present invention realizes the estimation of the fatigue crack growth rate of the laminated structure by obtaining load-displacement response data through simple static tests without relying on fatigue test data.

[0057] The method of the present invention breaks through the dependence of traditional models on fatigue experimental data. Its significant advantage is that it can effectively predict the fatigue crack growth rate without calibrating model parameters with fatigue data, relying only on the strain energy release rate obtained from quasi-static tests. The method of the present invention provides an efficient and convenient analysis tool for evaluating the fatigue crack growth behavior of laminated structures, significantly improving the calculation efficiency and engineering applicability of fatigue damage tolerance analysis.

[0058] The method of the present invention is applicable to predicting the fatigue crack growth rate of various types of laminated structures, including but not limited to single metal material laminated plate structures, fiber-reinforced composite material laminated plate structures; hybrid fiber-reinforced and metal material laminated plate structures; foam sandwich structures and honeycomb sandwich structures of metal materials or composite materials, etc. Its wide applicability provides a general prediction method for the fatigue failure problems of laminated structures in engineering applications. Description of the Drawings

[0059] Figure 1 is a schematic diagram of the method for determining the crack correction length Δ in the method for estimating the fatigue crack growth rate of the laminated structure according to the present invention;

[0060] Figure 2 is a schematic diagram of the model of the laminated structure; in the figure, x t is the X-axis coordinate of the upper sub-beam, and x b is the X-axis coordinate of the lower sub-beam;

[0061] Figure 3 is a schematic diagram of the cohesive force model between the upper sub-beam and the lower sub-beam;

[0062] Figure 4 is a schematic diagram of the method for determining the cohesive fatigue limit of the cohesive force model by using the conditional yield stress;

[0063] Figure 5 is a schematic diagram of the constitutive relationship of the cohesive force model in the method of the present invention;

[0064] Figure 6 is a schematic diagram of the function obtained by fitting the crack length and compliance data with a power function; in the figure, C = i*a^j represents the form of the fitting equation adopted; i and j are parameters to be determined by fitting, and the i and j in the table in the figure are the parameter values determined by fitting;

[0065] Figure 7It is a schematic diagram of the function obtained by fitting the cycle number and crack length data using a power function; in the figure, a = I + J * N^K; I, J, and K are parameters to be determined by fitting, and the parameter values are determined by fitting.

[0066] Figure 8 It is a comparison chart of the fatigue crack growth rate curves of the composite laminate structure obtained by theoretical calculation and experimental test. Specific embodiments

[0067] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative work belong to the scope of protection of the present invention.

[0068] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0069] The present invention will be further described below in conjunction with the accompanying drawings, but it is not a limitation of the present invention.

[0070] Combined with Figures 1 to 8 As shown, the present invention provides an estimation method for the fatigue crack growth rate of a laminated structure, including

[0071] The laminated structure is divided into an upper sub-beam and a lower sub-beam according to the delamination crack interface, and the Timoshenko beam theory is used to establish the displacement field models of the upper sub-beam and the lower sub-beam, and the mid-plane displacements along the length direction and the thickness direction and the formula for the rotation angle of the mid-plane relative to the normal direction are obtained; through calculation, the tangential displacement, normal displacement of the crack interface, and the resultant internal stress, bending moment, and shear stress resultant of the upper sub-beam and the lower sub-beam are obtained.

[0072] Using the tangential displacement and normal displacement of the crack interface, combined with the interface fracture toughness of the laminated structure, the tangential traction force and normal traction force of the delamination crack interface are calculated; then, combined with the resultant internal stress, bending moment, and shear stress resultant of the upper sub-beam and the lower sub-beam, the total strain energy of the laminated structure is calculated.

[0073] Based on the total strain energy of the laminated structure and the potential energy of the external load applied to the laminated structure, the equilibrium equation of the laminated structure is established; based on the equilibrium equation of the laminated structure, a control equation set is obtained.

[0074] Solve the control equation set to obtain the crack length, cycle number, theoretical value of the load applied to the upper sub-beam, and theoretical value of the displacement of the upper sub-beam of the laminated structure under the fatigue load of the laminated structure obtained by theoretical calculation, calculate the cyclic energy release rate, and further calculate the crack growth rate.

[0075] Furthermore, a mechanical model of the laminated structure is established to calculate the interlaminar crack propagation under monotonic or fatigue loads. The laminated structure is divided into upper and lower sub-beams by a delamination interface. The upper and lower sub-beams can also be composed of single or multiple sub-structures made of the same or different materials. The model in the local coordinate system is as shown in Figure 2 . The displacement field models of the upper and lower sub-beams are established using the Timoshenko beam theory. The formulas for the mid-plane displacements along the length and thickness directions and the rotation angle of the mid-plane relative to the normal direction are:

[0076]

[0077] where u r is the displacement of the upper and lower sub-beams along the X-axis in the length direction, r = t, b, t represents the upper sub-beam, and b represents the lower sub-beam; w r is the displacement of the upper and lower sub-beams along the Z-axis in the thickness direction; x is the X-axis coordinate, z r is the Z-axis coordinate of the upper and lower sub-beams, is the mid-plane displacement of the upper and lower sub-beams in the X-axis direction, is the mid-plane displacement of the upper and lower sub-beams in the Z-axis direction, is the rotation angle of the mid-plane of the upper and lower sub-beams relative to the normal direction.

[0078] The cohesive model is used to represent the constitutive relationship between the upper and lower surfaces of the delamination interface of the laminated structure, as shown in Figure 3 . The calculation methods for the tangential and normal displacements of the crack interface are:

[0079]

[0080] where Δu t represents the tangential displacement of the crack interface, Δu n represents the normal displacement of the crack interface, h b is the thickness of the lower sub-beam, and h t is the thickness of the upper sub-beam.

[0081] Furthermore, the calculation methods for the internal stress resultant, bending moment, and shear stress resultant of the upper and lower sub-beams are:

[0082] According to the strain calculation formulas of the upper and lower sub-beams:

[0083]

[0084] where is the normal strain of the upper and lower sub-beams, is the shear strain of the upper and lower sub-beams. In the variable subscript, x represents the derivative of the corresponding variable;

[0085] Then we get:

[0086]

[0087] In the formula is the resultant internal stress of the upper sub-beam and the lower sub-beam, is the bending moment of the upper sub-beam and the lower sub-beam, is the resultant shear stress of the upper sub-beam and the lower sub-beam;

[0088] In the formula, b is the width of the laminated structure, is the normal stress in the X-axis direction, is the shear stress in the XZ plane, is the tensile stiffness of the upper sub-beam and the lower sub-beam, is the tensile-bending coupling stiffness of the upper sub-beam and the lower sub-beam, is the bending stiffness of the upper sub-beam and the lower sub-beam, is the shear stiffness of the upper sub-beam and the lower sub-beam.

[0089] The double-cantilever beam tensile test under monotonic load is carried out on the laminated structure with prefabricated delamination cracks to obtain the load-displacement response data of the structure, and the interface fracture toughness of the laminated structure is calculated by the modified beam theory. The calculation method of the interface fracture toughness of the laminated structure is:

[0090]

[0091] In the formula, φ n is the interface fracture toughness of the laminated structure, P is the test value of the load applied to the upper sub-beam, W is the test value of the displacement of the upper sub-beam, a is the crack length of the laminated structure, and Δ is the crack correction length.

[0092] Combined with Figure 1 shown, the crack correction length Δ is obtained by extrapolating the linear fitting equation of the cube root of the compliance C 1 / 3 and the crack length a to C 1 / 3 = 0.

[0093] The calculation methods of the tangential traction force and the normal traction force at the delamination crack interface are:

[0094]

[0095] In the formula, T t is the tangential traction force, T n is the normal traction force; is equal to the fracture toughness calculated in the static test; δ0 is the characteristic length of the cohesive zone, and D is the damage variable, indicating the degradation degree of the interface performance;

[0096] Under fatigue load, the expression of the damage variable D is:

[0097]

[0098] where k is the correction coefficient with a value of 0.01; is the cyclic damage increment, is the monotonic damage increment, and t is time;

[0099]

[0100] In the formula is the cumulative displacement increment of one load cycle, and δ ∑ is the total displacement required to cause the cohesive zone to fail, is the total cohesive tension at the delamination crack interface, σ f is the cohesive fatigue limit, and σ max is the normal cohesive strength after N load cycles, and σ max,0 is the interface cohesive strength when no damage occurs, and σ max = σ max，0 (1 - D); H is the Heaviside function, and the function value is 1 only when the value inside the parentheses is greater than or equal to 0, otherwise the function value is 0; is the cumulative displacement of one load cycle, and δ f is the separation displacement corresponding to the cohesive fatigue limit;

[0101]

[0102] Furthermore, the conditional yield stress is adopted to define the cohesive fatigue limit of the cohesive force model. Specifically, by defining a normal traction force T n - the normal displacement Δu of the crack interface n a parallel line in the elastic response stage of the curve, and shifting the parallel line by a specified plastic deformation amount d, the cohesive fatigue limit σ f is jointly determined by two dimensionless parameters (δ f / δ0 and σ f / σ max,0 ) at the intersection point of the shifted line and the traction - displacement curve, as shown in Figure 4 . The total displacement δ Σ for the failure of the cohesive zone is defined as 10δ0, and this displacement point corresponds to the tension value degraded to 0.1% of the initial cohesive strength. The schematic diagram of the constitutive relationship of the finally determined cohesive force model is shown in Figure 5 . When , both monotonic damage and fatigue damage occur simultaneously in the cohesive zone, and the two damage increments coexist. The larger one is the final damage increment of the current cycle, and the unloading path linearly unloads from the current state point back to the traction zero point; when , the cohesive tension exceeds the cohesive fatigue limit but is less than the cohesive strength, and only fatigue damage occurs in the cohesive zone, and the unloading path linearly unloads from the current state point back to the traction zero point; when When the cohesive tension is in the elastic section and no damage occurs, the unloading path returns to the origin along the elastic section.

[0103] The calculation method of the total strain energy of the laminated structure is as follows:

[0104]

[0105] In the formula, U is the total strain energy of the laminated structure, V t is the volume of the upper sub-beam, V b is the volume of the lower sub-beam, and A is the interlayer area of the crack interface;

[0106] The potential energy of the external load applied to the laminated structure is expressed as V:

[0107]

[0108] In the formula, L is the length of the laminated structure, P t is the theoretical value of the load applied to the upper sub-beam, δ D is the Dirac function. The function value is 1 only when the value in the parentheses is 0, otherwise the function value is 0; x e is the X-axis coordinate of the load application position;

[0109]

[0110] Based on the principle of minimum potential energy, the equilibrium equation of the laminated structure is established based on the variational method as follows:

[0111] δ(U - V) = 0(14),

[0112] In the formula, δ is the variational symbol;

[0113] Based on the structural equilibrium equation, the following control equation set is obtained:

[0114]

[0115]

[0116] By solving the above first-order ordinary differential control equation set through numerical calculation software, the crack length a, the number of cycles N, the theoretical value P of the load applied to the upper sub-beam t and the theoretical value w of the displacement of the upper sub-beam t ;

[0117] The cyclic energy release rate calculated by the theoretical method for the load and displacement is expressed as ΔG as follows:

[0118]

[0119] In the formula Apply the maximum theoretical value of the load to the upper sub - beam. Apply the minimum theoretical value of the load to the upper sub - beam. C is the flexibility of the laminated structure; C = w t / P t ; The function relationship can be obtained by fitting the crack length and flexibility data, and the differential operation is performed on the function relationship.

[0120] The function relationship is obtained by fitting the crack length a and the number of cycles N of the laminated structure with a power function, and then the differential operation is performed on the function relationship to obtain the crack growth rate da / dN.

[0121] Furthermore, plot the crack growth rate da / dN and the cyclic energy release rate ΔG in a double - logarithmic coordinate system to obtain the fatigue crack growth rate curve of the laminated structure; then use Paris' law to fit the fatigue crack growth rate curve to obtain the fitted curve; determine the actual fatigue crack growth rate according to the fitted curve:

[0122]

[0123] In the formula, B is a constant related to the material, and m is the exponential constant of Paris' law.

[0124] Example:

[0125] S1. Prepare a composite laminated plate structure with a pre - fabricated delamination crack. As Figure 2 shown, the total length L of the specimen is 200 mm, the width b is 25 mm, the total thickness is 4.5 mm. The upper and lower sub - beams are divided into sub - beams of equal thickness by the pre - fabricated delamination crack, and the pre - fabricated crack length Lc = 30 mm. The basic mechanical properties of the composite material used are shown in Table 1. Apply a tensile load to the composite laminated plate structure with a pre - fabricated delamination crack at the position of x e = 25 mm, conduct a double - cantilever beam tensile test under monotonic load, obtain the load - displacement response data of the structure, and calculate the interfacial fracture toughness of the composite laminated plate structure using formula (5).

[0126] Table 1 Basic mechanical properties of the composite laminated plate

[0127]

[0128] S2. Establish a mechanical model of the composite laminated plate structure according to the geometric parameters and material parameters of the structure in step S1. Input the basic mechanical properties of the composite laminated plate into formula (4), and input the interfacial fracture toughness obtained from the quasi - static tensile test into formula (6).

[0129] S3. Adopt the conditional yield stress to define the cohesive fatigue limit of the cohesive force model. As Figure 3 shown, draw a parallel line to the elastic response stage of the traction-displacement curve, offset by the specified plastic deformation amount d. The value of the conditional yield stress is jointly determined by two dimensionless parameters (δ f / δ0 and σ f / σ max,0 ) at the intersection point of this offset line and the traction-displacement curve. The values of the two dimensionless parameters calculated with different plastic deformation amounts d are shown in Table 2.

[0130] Table 2 Dimensionless parameter values corresponding to different plastic deformation amounts

[0131]

[0132] S4. Solve the control equation groups of formulas (15)-(26) to obtain the interlaminar crack length a, the number of cycles N, the theoretical value P of the load applied to the upper sub-beam, t and the theoretical value w t of the displacement of the upper sub-beam under the fatigue load of the composite laminate structure obtained by theoretical calculation.

[0133] S5. Adopt the load and displacement data calculated in step S4, and calculate the cyclic energy release rate ΔG of the structure under the fatigue load using formula (27). dC / da in the formula is obtained by fitting the crack length and flexibility data to obtain a functional relationship and performing a differential operation. As Figure 6 shown.

[0134] S6. Adopt the crack propagation length and number of cycles data calculated in step S4, fit the calculated data using a power function to obtain a functional relationship, and the crack propagation rate da / dN can be obtained through a differential operation. As Figure 7 shown.

[0135] S7. Plot the theoretically calculated crack propagation rate da / dN and the cyclic energy release rate ΔG in a double logarithmic coordinate system to obtain the fatigue crack propagation rate curve of the composite laminate structure. As Figure 8 shown. Fit the fatigue crack propagation rate curves of theoretical calculation and experimental test using Paris' law to obtain the exponential constant of Paris' law, as shown in Table 3.

[0136] Table 3 Exponential constant m of Paris' law obtained from theoretical calculation and experimental test

[0137]

[0138] Although the present invention has been described herein with reference to particular embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the present invention. Accordingly, it should be understood that numerous modifications may be made to the exemplary embodiments, and other arrangements may be devised, without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that the different dependent claims and the features described herein may be combined in a manner different from that described in the original claims. It should also be understood that the features described in connection with separate embodiments may be used in other described embodiments.

Claims

1. A method for estimating the fatigue crack growth rate of a laminated structure, characterized in that including, divide the laminated structure into an upper sub - beam and a lower sub - beam according to the delamination crack interface, and establish displacement field models of the upper sub - beam and the lower sub - beam using the Timoshenko beam theory to obtain the mid - plane displacements along the length and thickness directions and the formula for the rotation angle of the mid - plane relative to the normal direction; through calculation, obtain the tangential displacement, normal displacement of the crack interface, as well as the resultant internal stress, bending moment and resultant shear stress of the upper sub - beam and the lower sub - beam; use the tangential displacement and normal displacement of the crack interface, combine with the interface fracture toughness of the laminated structure to calculate the tangential traction and normal traction of the delamination crack interface; then combine with the resultant internal stress, bending moment and resultant shear stress of the upper sub - beam and the lower sub - beam to calculate the total strain energy of the laminated structure; establish the equilibrium equation of the laminated structure based on the total strain energy of the laminated structure and the potential energy of the external load applied to the laminated structure; obtain the control equation set based on the equilibrium equation of the laminated structure; solve the control equation set to obtain the crack length, number of cycles, theoretical value of the load applied to the upper sub - beam and theoretical value of the displacement of the upper sub - beam of the laminated structure under fatigue load by theoretical calculation, calculate the cyclic energy release rate, and further calculate the crack growth rate.

2. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 1, wherein, the formulas for the mid - plane displacements along the length and thickness directions and the rotation angle of the mid - plane relative to the normal direction are: where u r is the displacement of the upper and lower sub - beams along the X - axis in the length direction, r = t, b, t represents the upper sub - beam, b represents the lower sub - beam; w r is the displacement of the upper and lower sub - beams along the Z - axis in the thickness direction; x is the X - axis coordinate, z r is the Z - axis coordinate of the upper and lower sub - beams, is the mid - surface displacement of the upper and lower sub - beams in the X - axis direction, is the mid - surface displacement of the upper and lower sub - beams in the Z - axis direction, is the rotation angle of the mid - surface of the upper and lower sub - beams relative to the normal direction.

3. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 2, wherein the calculation methods for the tangential displacement and normal displacement of the crack interface are: where Δu t represents the tangential displacement of the crack interface, and Δu n represents the normal displacement of the crack interface, h b is the thickness of the lower sub-beam, and h t is the thickness of the upper sub-beam.

4. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 3, wherein, the calculation methods for the resultant internal stress, bending moment and resultant shear stress of the upper sub - beam and the lower sub - beam are: According to the strain calculation formulas of the upper sub - beam and the lower sub - beam: where are the normal strains of the upper and lower sub - beams, are the shear strains of the upper and lower sub - beams. In the variable subscripts, x represents the derivative of the corresponding variable; then obtain: In the formula is the resultant internal stress of the upper sub-beam and the lower sub-beam, is the bending moment of the upper sub-beam and the lower sub-beam, is the resultant shear stress of the upper sub-beam and the lower sub-beam; where b is the width of the laminated structure, is the normal stress in the positive X-axis direction, is the shear stress in the XZ plane, are the tensile stiffnesses of the upper and lower sub-beams, is the tensile-bending coupling stiffness of the upper and lower sub-beams, is the bending stiffness of the upper and lower sub-beams, is the shear stiffness of the upper and lower sub-beams.

5. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 4, wherein, the calculation method for the interface fracture toughness of the laminated structure is: where φ n is the interfacial fracture toughness of the laminated structure, P is the test value of the load applied to the upper sub-beam, W is the test value of the displacement of the upper sub-beam, a is the crack length of the laminated structure, and Δ is the crack correction length.

6. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 5, wherein, the calculation methods for the tangential traction and normal traction of the delamination crack interface are: where T t is the tangential traction force, T n is the normal traction force, δ0 is the characteristic length of the cohesive zone, and D is the damage variable; where k is the correction coefficient, is the cyclic damage increment, is the monotonic damage increment, and t is time; where is the cumulative displacement increment for one load cycle period, δ ∑ is the total displacement required to cause failure of the cohesive zone, is the total cohesive tension of the delamination crack interface, σ f is the cohesive fatigue limit, σ max is the normal cohesive strength after N load cycles, σ max,0 is the interface cohesive strength without damage, H is the Heaviside function, is the cumulative displacement for one load cycle period, δ f is the separation displacement corresponding to the cohesive fatigue limit; Among them, the cohesive fatigue limit σ f is determined by the conditional yield stress: Define the normal traction force T n - The normal displacement Δu of the crack interface n Parallel line in the elastic response stage of the curve, offset the parallel line by a specified plastic deformation amount d; the cohesive fatigue limit σ f Through the intersection point of the offset parallel line and the normal traction force T n - The normal displacement Δu of the crack interface n The dimensionless parameter δ at the intersection of the curve f / δ0 and σ f / σ max,0 are jointly determined.

7. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 6, wherein, the calculation method for the total strain energy of the laminated structure is: where \(U\) is the total strain energy of the laminated structure, \(V\) t is the volume of the upper sub-beam, \(V\) b is the volume of the lower sub-beam, and \(A\) is the interfacial area between layers of the crack interface; The potential energy of the external load applied to the laminated structure is expressed as V: where L is the length of the laminated structure, P t is the theoretical value of the load applied to the upper sub-beam, δ D is the Dirac function, and x e is the X-axis coordinate of the load application position; 8. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 7, wherein, establish the equilibrium equation of the laminated structure based on the variational method as: δ(U - V)=0(14), where δ is the variational symbol; obtain the following control equation set based on the structure equilibrium equation:

9. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 8, wherein, Solving the control equations to obtain the crack length a, the number of cycles N, and the theoretical value P of the load applied to the upper sub-beam of the laminated structure under the fatigue load of the laminated structure obtained by theoretical calculation t and the theoretical value w of the displacement of the upper sub-beam t ; express the cyclic energy release rate as ΔG as: In the formula is the maximum theoretical value of the load applied to the upper sub-beam, is the minimum theoretical value of the load applied to the upper sub-beam, and C is the flexibility of the laminated structure; C = w t / P t ; Use a power function to fit the crack length a and the number of cycles N of the laminated structure to obtain a functional relationship, and then perform a differential operation on the functional relationship to obtain the crack growth rate da / dN.

10. The method for estimating the fatigue crack growth rate of the laminated structure according to claim 9, wherein, The crack growth rate da / dN and the cyclic energy release rate ΔG are plotted on a double logarithmic coordinate system to obtain the fatigue crack growth rate curve of the laminated structure; then the Paris law is used to fit the fatigue crack growth rate curve to obtain the fitted curve; the actual fatigue crack growth rate is determined according to the fitted curve: where B is a constant related to the material, and m is the exponential constant of the Paris law.

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