CN116205109A - Multi-crack coupling propagation track calculation method, equipment and storage medium - Google Patents

Multi-crack coupling propagation track calculation method, equipment and storage medium Download PDF

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CN116205109A
CN116205109A CN202310221761.9A CN202310221761A CN116205109A CN 116205109 A CN116205109 A CN 116205109A CN 202310221761 A CN202310221761 A CN 202310221761A CN 116205109 A CN116205109 A CN 116205109A
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crack
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鲁乃唯
崔健
王鸿浩
罗媛
肖新辉
雷尧
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Changsha University of Science and Technology
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Abstract

The invention discloses a multi-crack coupling propagation track calculation method, equipment and a storage medium, wherein the method comprises the steps of constructing a finite element model, splitting a local sub-model, inserting initial fatigue cracks into welding seams of the local sub-model, and carrying out grid repartition on the local sub-model; readjusting the crack tip grid; calculating stress, displacement and strain of the crack tip node; calculating stress intensity factors according to stress, displacement and strain; determining a median node according to the stress intensity factor, and setting a crack expansion step length of the median node; determining the stress circulation times according to the fatigue crack propagation model, and determining crack propagation step sizes of other nodes according to the crack propagation step sizes of the median nodes; fitting the nodes after expansion according to the nodes corresponding to the expansion step length of the crack, and obtaining a fitting curve, wherein the fitting curve is used as the fatigue crack to continue expansion. The method can accurately calculate the expansion increment of the crack tip node, and improves the track fitting accuracy.

Description

Multi-crack coupling propagation track calculation method, equipment and storage medium
Technical Field
The invention belongs to the technical field of steel bridge fatigue crack analysis, and particularly relates to a multi-crack coupling expansion track calculation method, equipment and a storage medium at a welding seam of a top plate and a longitudinal rib of a steel bridge.
Background
The orthotropic steel bridge deck has the advantages of light weight, high strength, high construction speed, attractive structure, easy maintenance and the like, thereby being widely applied to large-span bridge systems at home and abroad. The orthotropic steel bridge deck slab has milestone significance in the field of bridge engineering, but has a large number of welding lines and obvious stress concentration degree in the self structure, and directly bears the repeated action of vehicle wheel load for a long time, the welding construction details of the top plate and the longitudinal ribs are easy to initiate fatigue cracks and rapidly expand, so that a large number of steel bridge deck slabs at home and abroad have a large number of fatigue cracks in a short service period, wherein the cracks initiated at the welding lines of the top plate and the longitudinal ribs are least easy to observe and most dangerous. With the increase of the service time of the bridge, the number and the size of fatigue cracks of the steel bridge deck plate are greatly increased, and the interaction among dense cracks can accelerate crack initiation and propagation. The fusion phenomenon occurs after the contact of the plurality of crack tips, and large-size cracks are generated. Fatigue crack growth is a very complex process, the conventional fatigue crack growth simulation is often aimed at a growth path, and the crack growth track is rarely researched, so that the fatigue crack growth track is researched by utilizing a theoretical model in order to reveal the morphological evolution rule of the crack growth process.
Fatigue analysis results based on fracture mechanics are widely applied to engineering fields such as bridges, ships, aerospace and the like due to safer conservation. In the aspect of steel bridge deck fatigue analysis theory, a crack propagation method based on Linear Elastic Fracture Mechanics (LEFM) considers factors such as low-amplitude stress, loading sequence, initial crack, plastic deformation and the like, and overcomes various defects of an S-N curve (namely, a stress-life curve, a curve representing the relationship between the fatigue strength and the fatigue life of a standard test piece under a certain cycle characteristic by taking the fatigue strength of a material standard test piece as an ordinate and taking the logarithmic value lgN of the fatigue life as an abscissa). The crack propagation analysis by utilizing fracture mechanics becomes an important research means through a finite element numerical simulation method, and the stress intensity factor can be calculated efficiently and accurately. The crack tip stress intensity factor based on the Paris formula calculates the crack expansion rate, and the Paris fatigue crack expansion model is combined with a crack expansion increment method to realize crack expansion, predict crack expansion generation tracks, characterize crack expansion layer-by-layer transmission and multi-crack fusion phenomena, and reveal the shape change trend of single cracks and multiple cracks.
However, as the crack propagation simulation results calculated based on fracture mechanics criteria cause discontinuous problems due to a stress intensity factor calculation method, the crack propagation simulation results are not subjected to multiple curve fitting crack front edge increment, often show a saw-tooth-shaped characteristic, do not conform to the actual smooth propagation track of a crack in a homogeneous material, and cannot predict the multiple crack fusion propagation evolution of an aging steel bridge deck in service, so that the accuracy of crack propagation track description is low.
Disclosure of Invention
The invention aims to provide a multi-crack coupling expansion track calculation method, equipment and a storage medium, which are used for solving the problems that a crack expansion simulation result calculated based on fracture mechanics criteria often has a saw-tooth-shaped characteristic, is not consistent with an actual smooth expansion track of a crack in a homogeneous material, and cannot predict multi-crack fusion expansion evolution of an aging steel bridge deck in service, so that the accuracy of crack expansion track description is low.
The invention solves the technical problems by the following technical scheme: a method for calculating a multi-crack coupling propagation track at a welding line of a top plate and a longitudinal rib of a steel bridge, comprising the following steps:
step 1: constructing a finite element model of a steel bridge deck, performing grid division on the finite element model, and applying load and boundary conditions;
step 2: splitting a local sub-model comprising a welding line on the finite element model, inserting at least one initial fatigue crack at the welding line of the local sub-model, and performing grid repartition on the local sub-model to obtain a local entity sub-model;
step 3: readjusting the crack tip grid for each fatigue crack to adapt to the singularities of the stress field and the displacement field at the crack tip;
step 4: calculating stress, displacement and strain of each node of the crack tip;
step 5: calculating stress intensity factors of corresponding nodes according to the stress, displacement and strain of each node of the crack tip;
step 6: determining a median node according to the stress intensity factor of each node of the crack tip, and setting the crack expansion step length of the median node; the median node is a node with a median stress intensity factor among all nodes at the tip of the crack;
step 7: selecting a fatigue crack expansion model, determining the stress cycle times according to the fatigue crack expansion model, and further determining crack expansion step sizes of other nodes at the crack tip according to the crack expansion step sizes of the median nodes; wherein the other nodes refer to all nodes except the median node;
step 8: fitting the expanded nodes according to the nodes corresponding to the crack expansion step length expansion to obtain a fitting curve;
step 9: judging whether the crack propagation depth reaches a set threshold value, if so, reaching the fatigue life of the crack, and not performing crack propagation; otherwise, taking the fitting curve as a fatigue crack, and turning to the step 3.
Further, the crack tip grid is represented by a three-layer grid, wherein the inner ring of the crack tip grid template is represented by a quarter 15-node singular wedge-shaped unit, and the outer two layers are represented by a quarter 20-node singular wedge-shaped unit.
Further, the stress intensity factor of each node is calculated by using an M-integration method, and a specific calculation formula is as follows:
Figure BDA0004116969670000021
Figure BDA0004116969670000022
Figure BDA0004116969670000023
wherein M is (1,2) Integrating the interaction of the auxiliary field with the actual field; v is poisson's ratio; e is the elastic modulus;
Figure BDA0004116969670000024
is the splay stress intensity factor of the actual field, +.>
Figure BDA0004116969670000025
For the splay stress intensity factor of the auxiliary field, < ->
Figure BDA0004116969670000026
Is the slip-off stress intensity factor of the actual field, < ->
Figure BDA0004116969670000031
Is the sliding-off stress intensity factor of the auxiliary field, < ->
Figure BDA0004116969670000032
Is the tearing stress intensity factor of the actual field, < ->
Figure BDA0004116969670000033
A tear-off stress intensity factor for the auxiliary field; />
Figure BDA0004116969670000034
The method comprises the steps of respectively obtaining a node unit stress tensor of an actual field and a node unit stress tensor of an auxiliary field, wherein a subscript i represents the external normal direction of a plane where the stress tensor is located, and a subscript j represents the acting direction of the stress tensor; />
Figure BDA0004116969670000035
The strain tensor of the node unit of the actual field and the strain tensor of the node unit of the auxiliary field are respectively shown, wherein a subscript i represents the line element direction, and a subscript j represents the line element deflection direction; a is that q Q is a weight function defined on an integral domain; x is x 1 Representing the coordinate length of the node in the long axis direction of the crack surface and x j The coordinate length of the node in any coordinate axis j of the orthogonal coordinate system is represented; Γ is the integral path around the crack tip and ds is a small increment along the integral path Γ; />
Figure BDA0004116969670000036
The components of the node displacement vector of the actual field in any coordinate axis i of the orthogonal coordinate system and the components of the node displacement vector of the auxiliary field in any coordinate axis i of the orthogonal coordinate system are respectively; delta 1j Representing a kronecker function; w (W) (1,2) Is the interaction strain energy density.
Further, the crack propagation step length of the median node is 15% of the minimum feature size of the crack, wherein the minimum feature size refers to the minimum value of lengths of the crack in all directions.
Further, a Paris fatigue crack growth model is adopted as the fatigue crack growth model, and the stress cycle number is determined according to the formula:
Figure BDA0004116969670000037
wherein a is the fatigue crack growth length; n is the number of stress cycles; C. n is a material constant; Δk is the stress intensity factor amplitude,
Figure BDA0004116969670000038
K I to be an open stress intensity factor, K II Is a slip-on stress intensity factor, K III Is a tearing stress intensity factor, v is poisson ratio;
the calculation formula of the crack propagation step length of each node is as follows:
Figure BDA0004116969670000039
wherein Δa i The crack propagation step length of the ith node is i not equal to m; Δa m Crack propagation step length of the median node m; n (N) i The stress cycle number is the i-th node; n (N) m The stress cycle number is the median node m; ΔK i The stress intensity factor magnitude for the i-th node,
Figure BDA00041169696700000310
ΔK m the stress intensity factor magnitude being the median node m,
Figure BDA00041169696700000311
R i 、R m stress ratios of the ith node and the median node m are respectively; da/dN i For the crack growth rate of the ith node, da/dN m Is the crack propagation rate of the median node m.
Further, for single cracks without abrupt change of stress intensity factors, fitting the expanded nodes by adopting a third-order polynomial curve; and for single cracks with abrupt stress intensity factors, fitting the expanded nodes by adopting a cubic spline curve.
Further, when the number of fatigue cracks is more than or equal to 2 and fitting curves corresponding to different fatigue cracks are intersected, turning to the step 3 by taking the curve formed after intersection as a single fatigue crack.
Further, the set threshold is half the thickness of the top plate.
Based on the same idea, the present invention provides an electronic device comprising:
a memory for storing a computer program;
and the processor is used for realizing the multi-crack coupling expansion track calculation method at the welding line of the steel bridge roof and the longitudinal rib when executing the computer program.
Based on the same conception, the invention provides a computer readable storage medium, wherein a computer program is stored on the computer readable storage medium, and the computer program realizes the multi-crack coupling expansion track calculation method at the welding line of the steel bridge roof and the longitudinal rib when being executed by a processor.
Advantageous effects
Compared with the prior art, the invention has the advantages that:
the model constructed by the invention is a numerical model, and can accurately describe the singular terms of the stress field and the displacement field generated by the crack tip under the action of complex load, and accurately calculate the stress intensity factors of all nodes of the crack tip under the action of complex load, thereby accurately calculating the expansion increment of the crack tip node and improving the fitting accuracy of crack expansion tracks; the method can be used for constructing a complex structure, and considering the influence of parameters such as the Poisson ratio of materials and the like on the crack propagation track, the numerical noise can be reduced by using a third-order polynomial curve, and the smooth crack propagation track can be accurately obtained.
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In order to more clearly illustrate the technical solutions of the present invention, the drawings that are needed in the description of the embodiments will be briefly described below, it being obvious that the drawing in the description below is only one embodiment of the present invention, and that other drawings can be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a multi-crack coupling propagation trajectory calculation method in an embodiment of the invention;
fig. 2 is a schematic diagram of a calculation result of a stress intensity factor of a single crack in the embodiment of the present invention, where an analysis step 1 refers to a calculation result of a stress intensity factor of an initial fatigue crack, an analysis step 2 refers to a calculation result of a stress intensity factor of a crack (abbreviated as a first crack) after a first expansion of the initial fatigue crack, an analysis step 3 refers to a calculation result of a stress intensity factor of a crack (abbreviated as a second crack) after a first expansion of the first crack, and an analysis step 4 refers to a calculation result of a stress intensity factor of a crack (abbreviated as a third crack) after a first expansion of the second crack;
FIG. 3 is a schematic diagram of crack growth step size (or crack growth increment) in an embodiment of the invention;
FIG. 4 is a schematic representation of a new crack front prediction in an embodiment of the present invention;
FIG. 5 is a graph showing the crack front propagation results in an embodiment of the present invention;
FIG. 6 is a schematic diagram of single crack propagation trajectories with different initial morphology parameters according to an embodiment of the invention, wherein the plot (a) is a morphology ratio a 0 /c 0 Schematic of single crack propagation trace when=0.2, graph (b) is a morphology ratio of a 0 /c 0 Single crack propagation trace schematic at=1;
fig. 7 is a schematic diagram of a calculation result of a stress intensity factor of a double crack in the embodiment of the present invention, in which, in the analysis step 11, the calculation result of the stress intensity factor of the double crack (abbreviated as a first double crack) obtained after two single cracks intersect, in the analysis step 12, the calculation result of the stress intensity factor of the crack (abbreviated as a second double crack) obtained after the first double crack is propagated once, in the analysis step 13, the calculation result of the stress intensity factor of the crack (abbreviated as a third double crack) obtained after the second double crack is propagated once, in the analysis step 14, the calculation result of the stress intensity factor of the crack (abbreviated as a fourth double crack) obtained after the third double crack is propagated once, in the analysis step 15, the calculation result of the stress intensity factor of the crack (abbreviated as a fifth double crack) obtained after the fourth double crack is propagated once;
FIG. 8 is a schematic representation of a dual crack propagation trajectory in an embodiment of the present invention;
FIG. 9 is a schematic diagram of a single crack propagation trajectory shape evolution law in an embodiment of the present invention;
fig. 10 is a schematic diagram of a shape evolution rule of a double crack propagation track in an embodiment of the invention.
Detailed Description
The following description of the embodiments of the present invention will be made more apparent and fully by reference to the accompanying drawings, in which it is shown, however, only some, but not all embodiments of the invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The technical scheme of the present application is described in detail below with specific examples. The following embodiments may be combined with each other, and some embodiments may not be repeated for the same or similar concepts or processes.
As shown in fig. 1, the method for calculating the multi-crack coupling propagation track at the welding line of the top plate and the longitudinal rib of the steel bridge provided by the embodiment of the invention comprises the following steps:
step 1: and constructing a finite element model of the steel bridge deck, meshing the finite element model, and applying load and boundary conditions.
In this embodiment, an ABAQUS software was used to construct a steel bridge deck segment integral finite element model.
Step 2: and splitting a local sub-model comprising a welding line on the finite element model, inserting at least one initial fatigue crack at the welding line of the local sub-model, and performing grid repartition on the local sub-model to obtain a local entity sub-model.
In order to more accurately calculate parameters such as stress, displacement, strain and the like of each node at the crack tip at the weld joint, a local sub-model is split on the finite element model, and then grid repartition is carried out on the local sub-model, so that the grid density is higher, the calculation precision of the parameters is improved, and the fitting accuracy of crack propagation tracks is further improved. In this embodiment, the mesh density of the local solid submodel is 0.2mm.
Step 3: for each fatigue crack, the crack tip grid is readjusted to accommodate the singularities of the stress and displacement fields at the corresponding location.
In this embodiment, the crack tip grid is represented by quarter node singular units to accommodate the singularities of the stress and displacement fields at the crack tip. Specifically, a three-layer grid is used for representing the crack tip grid, wherein the inner ring of the crack tip grid template is represented by a quarter 15-node singular wedge-shaped unit, and the outer two layers are represented by a quarter 20-node singular wedge-shaped unit. The specific implementation process of readjustment is as follows: binding the crack tip grid template with the crack tip. The crack tip grid template is a three-layer grid, wherein the inner ring unit is a quarter 15-node singular wedge-shaped unit and is placed close to the crack tip, the outer two layers of units are quarter 20-node hexahedral unit rings, and the three rings of unit rings jointly form the crack tip grid template; in order to accurately simulate the singularity of the crack tip stress field-displacement field, the radius of the crack tip grid template cell ring is set to 10% of the minimum feature size of the crack, where the minimum feature size refers to the minimum value of the lengths of the cracks in all directions, and typically the minimum feature size is the crack depth.
Step 4: and calculating the stress, displacement and strain of each node of the crack tip.
Stress, displacement and strain for each node of the crack tip can be obtained using ABAQUS software.
Step 5: and calculating the stress intensity factor of the corresponding node according to the stress, displacement and strain of each node of the crack tip.
In this embodiment, the stress intensity factor of each node is calculated by using an M-integration method, and a specific calculation formula is as follows:
Figure BDA0004116969670000061
Figure BDA0004116969670000062
Figure BDA0004116969670000063
wherein M is (1,2) Integrating the interaction of the auxiliary field with the actual field; v is poisson's ratio; e is the elastic modulus;
Figure BDA0004116969670000064
is the splay stress intensity factor of the actual field, +.>
Figure BDA0004116969670000065
For the splay stress intensity factor of the auxiliary field, < ->
Figure BDA0004116969670000066
Is the slip-off stress intensity factor of the actual field, < ->
Figure BDA0004116969670000067
Is the sliding-off stress intensity factor of the auxiliary field, < ->
Figure BDA0004116969670000068
Is the tearing stress intensity factor of the actual field, < ->
Figure BDA0004116969670000069
A tear-off stress intensity factor for the auxiliary field; />
Figure BDA00041169696700000610
The method comprises the steps of respectively obtaining a node unit stress tensor of an actual field and a node unit stress tensor of an auxiliary field, wherein a subscript i represents the external normal direction of a plane where the stress tensor is located, and a subscript j represents the acting direction of the stress tensor; />
Figure BDA00041169696700000611
The strain tensor of the node unit of the actual field and the strain tensor of the node unit of the auxiliary field are respectively shown, wherein a subscript i represents the line element direction, and a subscript j represents the line element deflection direction; a is that q Q is a weight function defined on an integral domain; x is x 1 Representing the coordinate length of the node in the long axis direction of the crack surface and x j The coordinate length of the node in any coordinate axis j (namely X/Y/Z axis) of an orthogonal coordinate system is represented; Γ is the surrounding crackAn integration path of the striation tip, ds, is a small increment along the integration path Γ; />
Figure BDA00041169696700000612
The components of the node displacement vector of the actual field in any coordinate axis i (namely X/Y/Z axis) of the orthogonal coordinate system and the components of the node displacement vector of the auxiliary field in any coordinate axis i of the orthogonal coordinate system are respectively; delta 1j The clerk function is represented, and the subscript 1j has no practical meaning; w (W) (1,2) Is the interaction strain energy density. In this embodiment, an orthogonal coordinate system is constructed at the crack tip, the crack depth direction is the X axis, the long axis direction of the crack surface is the Y axis, and the direction perpendicular to the crack surface is the Z axis.
In the online elastic fracture mechanics, cracks generated in consideration of different stress forms can be also classified into different types. Cracks that lead to fracture of the structure mainly include three forms: open type cracks (type I), slide-open type cracks (type II), tear-open type cracks (type III).
Considering two independent equilibrium states of the elastomer, the superscript (1) and the superscript (2) represent the actual field and the auxiliary field, respectively, and an effective solution, M, is obtained by de-superposition of these two fields (1,2) Is the interaction integral of the auxiliary field with the actual field, i.e. M-integral. Formula (1) describes the correlation between M-integral, material properties and stress intensity factors, formula (2) is an expression of M-integral, and formula (3) is an interaction strain energy density W (1,2) Is defined in (a). The stress intensity factors (open stress intensity factor K) of different types of fatigue cracks can be treated by the combined type (1) - (3) I Slip-on stress intensity factor K II Tear-open stress intensity factor K III ) Solving, as shown in fig. 2, the calculation result of the single crack stress intensity factor is shown, wherein the crack front distance normalization refers to the normalization processing of the crack front length to be a number between (0, 1), and the normalization processing is used for representing the stress intensity factor of the corresponding node of the crack tip.
Step 6: and determining a median node according to the stress intensity factor of each node of the crack tip, and setting the crack expansion step length of the median node.
And arranging stress intensity factors of all nodes at the tip of the crack according to the size, wherein the node with the stress intensity factor being the median is the median node. For example, the crack tip is provided with 5 nodes, and after stress intensity factors of the 5 nodes are arranged according to the size, the 3 rd node is the median node. When the number of nodes at the tip of the crack is even, the two nodes in the middle after arrangement are all median nodes, and the stress intensity factor of the median node becomes the average value of the stress intensity factors of the two nodes. For example, the crack tip is provided with 6 nodes, after stress intensity factors of the 6 nodes are arranged according to the size, the 3 rd node and the 4 th node are median nodes, and the stress intensity factors of the 3 rd node and the 4 th node are changed into the average value of original stress intensity factors of the 3 rd node and the 4 th node.
In this embodiment, the crack growth step (or growth increment) of the median node is set to 15% of the minimum feature size of the crack, where the minimum feature size is the minimum value of the lengths of the cracks in each direction, and the minimum feature size is typically the crack depth.
Step 7: and selecting a fatigue crack extension model, determining the stress circulation times according to the fatigue crack extension model, and further determining the crack extension step length of other nodes at the crack tip according to the crack extension step length of the median node.
Based on fracture mechanics theory, selecting a Paris fatigue crack growth model as a fatigue crack growth model, wherein the specific expression of the Paris fatigue crack growth model is as follows:
Figure BDA0004116969670000071
/>
wherein a is the fatigue crack growth length; n is the number of stress cycles; C. n is a material constant; Δk is the stress intensity factor amplitude,
Figure BDA0004116969670000081
K I to be an open stress intensity factor, K II Is a slip-on stress intensity factor, K III Is a tearing stress intensity factorV is poisson's ratio. Taking c=5.21×10 according to the recommended value proposed by BS7910 - 13 N·mm -3/2 ，n＝3。
The stress cycle number N can be determined according to the formula (4), and the crack expansion step length of other nodes at the crack tip is calculated according to the crack expansion step length of the median node and combining with the Paris fatigue crack expansion model, wherein the specific calculation formula is as follows:
Figure BDA0004116969670000082
wherein Δa i The crack propagation step length of the ith node is i not equal to m; Δa m Crack propagation step length of the median node m; n (N) i The stress cycle number is the i-th node; n (N) m The stress cycle number is the median node m; ΔK i The stress intensity factor magnitude for the i-th node,
Figure BDA0004116969670000083
ΔK m the stress intensity factor magnitude being the median node m,
Figure BDA0004116969670000084
R i 、R m stress ratios of the ith node and the median node m are respectively; da/dN i For the crack growth rate of the ith node, da/dN m Is the crack propagation rate of the median node m. Wherein the other nodes refer to all nodes except the median node. FIG. 3 shows crack propagation steps of different nodes, corresponding nodes are propagated according to the crack propagation steps, and curve fitting is performed on the propagated nodes to obtain predicted crack leading edges (namely new fatigue cracks), so that a crack propagation method with specified propagation steps is realized.
Step 8: and fitting the expanded nodes according to the nodes corresponding to the crack expansion step length expansion to obtain a fitting curve.
In this embodiment, the third-order polynomial curve is used to fit the expanded node, so that numerical noise can be reduced, a new crack front extends to the surface of the structure, the position of the new crack tip is determined, a smooth fitted curve is obtained, the corresponding crack expansion front fitted curve is shown in fig. 4, and the crack front expansion result is shown in fig. 5.
Step 9: judging whether the crack propagation depth reaches a set threshold value, if so, reaching the fatigue life of the crack, and not performing crack propagation; otherwise, taking the fitting curve as a fatigue crack, and turning to the step 3.
In this embodiment, the set threshold is set to be half of the top plate thickness, and when the crack propagation depth reaches half of the top plate thickness, the crack is considered to have penetrated the top plate, and the fatigue life of the crack is reached, so the number of load actions when the crack propagates to half of the top plate thickness is used as the fatigue crack propagation life of the structure, and the crack propagation is no longer performed. And when the crack propagation depth is less than half of the thickness of the top plate, taking the fitted curve in the step 8 as a new fatigue crack, repeating the steps 3-9 to continue crack propagation, and recovering the singular units at the crack tip of the original fatigue crack into normal hexahedral units.
FIG. 6 shows single crack propagation trajectories for different initial morphology parameters, wherein a 0 Initial depth for fatigue crack insertion, c 0 Initial semi-Long Length for fatigue crack, a 0 /c 0 Represents the initial morphology ratio of the crack, morphology ratio a as the crack propagates 0 /c 0 Gradual change, always tending to a fixed value. When a is 0 When=2 mm, fig. 6 (a) shows that the aspect ratio is a 0 /c 0 Single crack propagation trace when=0.2, fig. 6 (b) shows a morphology ratio of a 0 /c 0 Single crack propagation trace at=1.
When the initial fatigue crack in the step 3 is 2 (for example, crack 1 and crack 2), the single crack is expanded respectively, and when the fitted curve after the crack 1 is expanded for multiple times and the fitted curve after the crack 2 is expanded for multiple times are intersected, a single crack (namely, the coupled single crack) is formed by the fitted curve after the crack 1 is expanded for multiple times and the fitted curve after the crack 2 is expanded for multiple times, and then the single crack is expanded until the crack expansion depth reaches half of the thickness of the top plate. Therefore, the method can be suitable for multi-crack coupling expansion track calculation.
Fig. 7 shows the calculation result of the single crack stress intensity factor after coupling, wherein the crack front distance normalization refers to the normalization of the crack front length to a number between (0, 1), and the normalization is used to represent the stress intensity factor of the crack tip corresponding node. FIG. 8 shows a double crack propagation trace, forming a single crack (as shown by the dashed line in FIG. 8) when two fitted curves after single crack propagation intersect, FIG. 8, a 0 Crack spacing s=12mm, morphology ratio a 0 /c 0 ＝0.5。
As shown in fig. 7, the geometrical jump of the coupled single crack generally causes abrupt changes in the stress intensity factor along the crack tip, and a third order polynomial curve fitting is no longer applied, but a cubic spline curve is used to capture the discrete points of the propagation increment or step size, and a concave crack is formed by the integration of the cubic spline curves (as shown by the dashed line in fig. 8). For crack fronts with recessed segments, new crack fronts obtained according to the crack propagation step size of the set median node may have penetrations or interferences, which are found to be reversed at the crack fronts using a cubic spline curve fit, points of penetration or interference will be identified as being discarded.
By means of the morphology ratio a 0 /c 0 The course of change of (c) explains the single crack propagation trajectory shape evolution law, as shown in fig. 9. By calculating the fatigue single crack propagation trajectory of any initial shape, it is possible to obtain: for a 0 /c 0 Fatigue crack of < 0.25, crack morphology ratio a as crack growth depth increases 0 /c 0 Remains unchanged after increasing to 0.26; for a 0 /c 0 Fatigue crack of > 0.25, crack morphology ratio a as the crack continues to propagate 0 /c 0 The gradual decrease to 0.26 remains unchanged, indicating that the final shape tends to a fixed shape as the fatigue crack propagates.
By means of the morphology ratio a 0 /c 0 The change in (c) explains the shape evolution law of the double crack propagation trajectory, as shown in fig. 10. By calculating the fatigue double crack propagation track of any initial shape, the following can be obtained: fatigue double crack along with expansionThe crack spacing gradually decreases, when a pair of crack front fitting curves contact each other, a concave large crack is formed, namely, a crack fusion phenomenon occurs, at the moment, the stress intensity factor of the merging point is sharply increased, as shown in fig. 7, the form ratio of the crack is suddenly changed, the concave crack is rapidly expanded into a convex crack, continuously expands according to a single crack expansion mode, and finally tends to a fixed shape a 0 /c 0 ＝0.26。
The foregoing disclosure is merely illustrative of specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art will readily recognize that changes and modifications are possible within the scope of the present invention.

Claims (10)

1. The method for calculating the multi-crack coupling propagation track at the welding line of the top plate and the longitudinal rib of the steel bridge is characterized by comprising the following steps:
step 1: constructing a finite element model of a steel bridge deck, performing grid division on the finite element model, and applying load and boundary conditions;
step 2: splitting a local sub-model comprising a welding line on the finite element model, inserting at least one initial fatigue crack at the welding line of the local sub-model, and performing grid repartition on the local sub-model to obtain a local entity sub-model;
step 3: readjusting the crack tip grid for each fatigue crack to adapt to the singularities of the stress field and the displacement field at the crack tip;
step 4: calculating stress, displacement and strain of each node of the crack tip;
step 5: calculating stress intensity factors of corresponding nodes according to the stress, displacement and strain of each node of the crack tip;
step 6: determining a median node according to the stress intensity factor of each node of the crack tip, and setting the crack expansion step length of the median node; the median node is a node with a median stress intensity factor among all nodes at the tip of the crack;
step 7: selecting a fatigue crack expansion model, determining the stress cycle times according to the fatigue crack expansion model, and further determining crack expansion step sizes of other nodes at the crack tip according to the crack expansion step sizes of the median nodes; wherein the other nodes refer to all nodes except the median node;
step 8: fitting the expanded nodes according to the nodes corresponding to the crack expansion step length expansion to obtain a fitting curve;
step 9: judging whether the crack propagation depth reaches a set threshold value, if so, reaching the fatigue life of the crack, and not performing crack propagation; otherwise, taking the fitting curve as a fatigue crack, and turning to the step 3.
2. The method for calculating the multi-crack coupling propagation track at the weld joint of the steel bridge roof and the longitudinal rib according to claim 1, wherein the crack tip grid is represented by a three-layer grid, wherein the inner ring of the crack tip grid template is represented by a quarter 15-node singular wedge-shaped unit, and the outer two layers of the crack tip grid template are represented by a quarter 20-node singular wedge-shaped unit.
3. The method for calculating the multi-crack coupling propagation track at the weld joint of the steel bridge roof and the longitudinal rib according to claim 1, wherein the stress intensity factor of each node is calculated by using an M-integration method, and a specific calculation formula is as follows:
Figure FDA0004116969660000011
Figure FDA0004116969660000012
Figure FDA0004116969660000013
wherein M is (1,2) Integrating the interaction of the auxiliary field with the actual field; v is poisson's ratio; e is the elastic modulus;
Figure FDA0004116969660000014
is the splay stress intensity factor of the actual field, +.>
Figure FDA0004116969660000021
For the splay stress intensity factor of the auxiliary field, < ->
Figure FDA0004116969660000022
Is the slip-off stress intensity factor of the actual field, < ->
Figure FDA0004116969660000023
Is the sliding-off stress intensity factor of the auxiliary field, < ->
Figure FDA0004116969660000024
Is the tearing stress intensity factor of the actual field,
Figure FDA0004116969660000025
a tear-off stress intensity factor for the auxiliary field; />
Figure FDA0004116969660000026
The method comprises the steps of respectively obtaining a node unit stress tensor of an actual field and a node unit stress tensor of an auxiliary field, wherein a subscript i represents the external normal direction of a plane where the stress tensor is located, and a subscript j represents the acting direction of the stress tensor; />
Figure FDA0004116969660000027
The strain tensor of the node unit of the actual field and the strain tensor of the node unit of the auxiliary field are respectively shown, wherein a subscript i represents the line element direction, and a subscript j represents the line element deflection direction; a is that q Q is a weight function defined on an integral domain; x is x 1 Representing nodesLength of coordinates in long axis direction of crack surface, x j The coordinate length of the node in any coordinate axis j of the orthogonal coordinate system is represented; Γ is the integral path around the crack tip and ds is a small increment along the integral path Γ;
Figure FDA0004116969660000028
the components of the node displacement vector of the actual field in any coordinate axis i of the orthogonal coordinate system and the components of the node displacement vector of the auxiliary field in any coordinate axis i of the orthogonal coordinate system are respectively; delta 1j Representing a kronecker function; w (W) (1,2) Is the interaction strain energy density.
4. The method for calculating the multi-crack coupling propagation trajectory at the weld joint of the top plate and the longitudinal rib of the steel bridge according to claim 1, wherein the crack propagation step length of the median node is 15% of the minimum characteristic dimension of the crack, and the minimum characteristic dimension is the minimum value of lengths of the crack in all directions.
5. The method for calculating the multi-crack coupling propagation track at the welding line of the top plate and the longitudinal rib of the steel bridge according to claim 1, wherein a Paris fatigue crack propagation model is adopted as the fatigue crack propagation model, and the stress cycle number is determined according to the formula:
Figure FDA0004116969660000029
wherein a is the fatigue crack growth length; n is the number of stress cycles; C. n is a material constant; Δk is the stress intensity factor amplitude,
Figure FDA00041169696600000210
K I to be an open stress intensity factor, K II Is a slip-on stress intensity factor, K III Is a tearing stress intensity factor, v is poisson ratio;
the calculation formula of the crack propagation step length of each node is as follows:
Figure FDA00041169696600000211
wherein Δa i The crack propagation step length of the ith node is i not equal to m; Δa m Crack propagation step length of the median node m; n (N) i The stress cycle number is the i-th node; n (N) m The stress cycle number is the median node m; ΔK i The stress intensity factor magnitude for the i-th node,
Figure FDA00041169696600000212
ΔK m the stress intensity factor magnitude being the median node m,
Figure FDA0004116969660000031
R i 、R m stress ratios of the ith node and the median node m are respectively; da/dN i For the crack growth rate of the ith node, da/dN m Is the crack propagation rate of the median node m.
6. The method for calculating the multi-crack coupling propagation track at the weld joint of the top plate and the longitudinal rib of the steel bridge according to any one of claims 1 to 5, wherein for single cracks without abrupt change of stress intensity factors, a third-order polynomial curve is adopted to fit the expanded nodes; and for single cracks with abrupt stress intensity factors, fitting the expanded nodes by adopting a cubic spline curve.
7. The method for calculating the multi-crack coupling propagation track at the welding seam of the top plate and the longitudinal rib of the steel bridge according to claim 1, wherein when the number of fatigue cracks is more than or equal to 2 and fitting curves corresponding to different fatigue cracks are intersected, the curve formed after intersection is a single fatigue crack, and the step 3 is shifted.
8. The method for calculating the multi-crack coupling propagation track at the weld joint of the steel bridge roof and the longitudinal rib according to claim 1, wherein the set threshold value is half of the thickness of the roof.
9. An electronic device, the device comprising:
a memory for storing a computer program;
a processor for implementing the multi-crack coupling propagation trajectory calculation method at the weld joint of the steel bridge roof and the longitudinal ribs according to any one of claims 1 to 8 when executing the computer program.
10. A computer readable storage medium, wherein a computer program is stored on the computer readable storage medium, and when the computer program is executed by a processor, the method for calculating the multi-crack coupling propagation track at the welding line of the steel bridge roof and the longitudinal rib is realized.
CN202310221761.9A 2023-03-09 2023-03-09 Multi-crack coupling propagation track calculation method, equipment and storage medium Pending CN116205109A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116522695A (en) * 2023-07-05 2023-08-01 矿冶科技集团有限公司 Rock mass crack growth track construction method, device, equipment and storage medium
CN118171545A (en) * 2024-05-16 2024-06-11 湖南工程学院 Collinear double-crack re-characterization method, device, storage medium and product

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116522695A (en) * 2023-07-05 2023-08-01 矿冶科技集团有限公司 Rock mass crack growth track construction method, device, equipment and storage medium
CN116522695B (en) * 2023-07-05 2023-09-05 矿冶科技集团有限公司 Rock mass crack growth track construction method, device, equipment and storage medium
CN118171545A (en) * 2024-05-16 2024-06-11 湖南工程学院 Collinear double-crack re-characterization method, device, storage medium and product

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