CN115795961B - Method for expanding finite element along grid boundary crack - Google Patents
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Abstract
The application discloses a finite element method for crack propagation along a grid boundary, which comprises the following steps: constructing an extrinsic cohesive force model, determining the relation between separation displacement and cohesive force on a crack surface, calculating the maximum principal stress of the current crack tip node, and judging crack propagation; if the crack is expanded, performing grid topology operation on the current crack tip node to realize the splitting of the current crack tip node and generate a new crack surface; and iterating the new crack surface by adopting a Newton Lafson iteration method until the residual force of the whole new crack surface meets the convergence condition, so that the whole finite element simulation flow is realized, and the simulation of the complex crack is realized.
Description
Technical Field
The application relates to a finite element method for crack propagation along a grid boundary, belonging to the technical field of fracture mechanics.
Background
The material and structure damage can cause accidents and even huge life and property losses, the existence of microcracks often leads to the damage of the material and the structure, the propagation path of the cracks and the deformation process often also need corresponding numerical calculation means, and common fracture mechanics numerical methods comprise a singular unit method, a smooth finite element method, an extended finite element method and the like.
Compared with the hot methods such as the expansion finite element method, the near field dynamics method and the like, the traditional finite element method can also solve the problem of crack expansion simulation, but no detailed crack expansion technology and the defect of how to treat the step of crack expansion problem through the finite element are disclosed.
Disclosure of Invention
The application aims to overcome the defects in the prior art, provides a finite element method for expanding cracks along a grid boundary, and solves the problem that cracks cannot be expanded along a cell boundary directly in the classical finite element method.
In order to achieve the above purpose, the application is realized by adopting the following technical scheme:
the application provides a finite element method for crack propagation along a grid boundary, which comprises the following steps:
constructing an extrinsic cohesive force model, and determining the relationship between separation displacement and cohesive force on a crack surface;
calculating the maximum principal stress of the current crack tip node according to the relation, and judging crack propagation;
if the crack is expanded, performing grid topology operation on the current crack tip node to realize the splitting of the old crack tip node and generate a new crack surface;
and iterating the new crack surface by adopting a Newton Lafson iteration method until the residual force of the whole new crack surface meets the convergence condition, thereby realizing the whole finite element simulation flow.
Further, the relationship between the separation displacement and the cohesion on the crack face is expressed as follows:
wherein ψ (delta) n ,Δ t ) Is a potential function; t (T) n Represents normal cohesive force on crack surface, T t Represents tangential inward focusing forces on the crack face; phi (phi) n Is normal cohesive energy, phi t Delta as tangential cohesive energy n For normal separation displacement, delta t Represents a tangential separation displacement; delta n For normal fail-separate displacement, delta t For tangential failure separation displacement, α and β are parameters controlling the shape of the cohesion-separation displacement curve in the extrinsic cohesion model Γ n And Γ t Representing the energy constant.
Further, the potential function expression is:
the energy constant Γ n And an energy constant Γ t The expression is as follows:
further, calculating the maximum principal stress of the current crack tip node comprises the following steps:
calculating stress data of all unit integral points of the current crack tip node, and calculating the stress data of all crack tip node entity units at the crack tip by a stress reconstruction method;
and (3) averaging stress data of all the crack tip node entity units to obtain average maximum principal stress of the crack tip node, and calculating the direction of the maximum principal stress.
Further, the grid topology operation includes:
establishing a temporary node, temporary units and temporary edges of the old crack tip node;
inserting a new node into the current crack tip node, wherein the edge direction with the smallest included angle with the direction of the maximum principal stress is used as a crack propagation path, and the edge node on the edge is the new crack tip.
Further, the adjacent nodes are nodes around crack tip nodes and are directly connected with the crack tip nodes through unit edges; the temporary units are units which are directly connected with the units around the crack tip nodes; the adjacent edge is a unit edge formed by a crack tip node and an adjacent node; the adjacent edge direction is the direction pointing to the adjacent node by taking the crack tip node as a starting point.
Further, the method includes continuously updating the old crack tip node formation, and constructing cohesive units between the new crack tip and the old crack tip to form a new grid structure.
Further, when the temporary unit of the old crack tip node is located between the new crack upper side and the old crack upper side along the anticlockwise direction, the old crack tip node is not updated;
when the temporary unit of the old crack tip node is located between the new crack underside and the old crack underside in the clockwise direction, the old crack tip node in the temporary unit is replaced with the new node inserted in the previous step.
Further, when the maximum principal stress is greater than the cohesive strength of the extrinsic cohesive model, the crack propagates; otherwise, the crack does not propagate.
Compared with the prior art, the application has the beneficial effects that:
the application provides a finite element method for crack propagation along a grid boundary, which describes the relationship between separation displacement and cohesion on a crack surface by constructing an extrinsic PPR cohesion model, realizes the splitting of old crack tip nodes and the generation of a new crack surface by utilizing grid topology operation, establishes a concise crack propagation simulation, can truly reflect the propagation of an open crack, and carries out the simulation of the complex crack.
Drawings
FIG. 1 is a flow chart of a finite element method provided by an embodiment of the present application;
FIG. 2 is a schematic illustration of crack propagation results provided by an embodiment of the present application;
Detailed Description
The application is further described below with reference to the accompanying drawings. The following examples are only for more clearly illustrating the technical aspects of the present application, and are not intended to limit the scope of the present application.
Example 1
As shown in fig. 1, a finite element method flowchart provided by an embodiment of the present application is provided, and the present application constructs a method for finely modeling and digitizing a propellant in a plane stress state, where implementation steps include:
1) The construction of the extrinsic PPR cohesion model describes the relationship between separation displacement and cohesion on the crack face.
2) Calculating the maximum main stress of the crack tip, and judging crack propagation;
3) Splitting the old crack tip node and generating a new crack surface by using grid topology operation;
4) And a Newton Laportson iteration method is adopted to realize the whole finite element simulation flow.
The specific process of the implementation steps is as follows:
in step 1), an extrinsic PPR cohesion model is used, whose potential function is expressed as
In phi n And phi t Represents the cohesive energy, delta, normal and tangential respectively n And delta t Represents normal and tangential separation displacements, respectively; alpha and beta are parameters controlling the shape of the cohesion-separation displacement curve, delta, in the PPR cohesion model n And delta t Is a normal and tangential failsafe displacement Γ n And Γ t Is an energy constant.
Wherein the energy constant Γ n And Γ t The following relationship exists:
and establishing an analytical expression between the cohesive force and the separation displacement on the crack surface by utilizing the relation between the potential function and the separation displacement, wherein the analytical expression is as follows:
wherein T is n And T t The normal cohesive force and the tangential cohesive force on the crack surface are respectively expressed.
And 2) judging crack propagation, calculating stress data of all unit integral points of the crack tip node, and then calculating the stress of all units near the crack tip at the crack tip by using a stress reconstruction method. Averaging the stress data at all crack tips to obtain average maximum principal stress of crack tip nodesAnd calculates the direction of the maximum principal stress.
When the maximum principal stress is greater than the cohesive strength of the extrinsic cohesive model, i.eDuring the time, the crack propagates; otherwise, the crack does not propagate.
Step 3) performing grid topology operation, and establishing adjacent nodes, adjacent units and adjacent edges of the old crack tip node. The adjacent nodes are nodes near crack tip nodes and directly connected with the crack tip nodes through unit edges; the temporary unit is a unit which is near the crack tip node and is directly connected with the unit; the critical edge refers to a unit edge formed by the crack tip node and the critical node. In addition, the edge direction refers to the direction pointing to the edge point by taking the crack tip point as a starting point.
When the crack propagates, a new node is inserted at the crack tip, and the coordinates of the node before and after deformation are consistent with the node of the current crack tip. Then, searching the adjacent edge direction with the smallest included angle with the crack propagation direction calculated in the step 2) near the crack tip node, and taking the adjacent edge as a crack propagation path, wherein the adjacent node on the adjacent edge is a new crack tip.
Considering the generation of new crack surfaces, the adjacent units of the old crack tip nodes are checked to determine whether the node composition needs to be updated.
When the temporary unit of the old crack tip node is positioned between the new crack upper side surface and the old crack upper side surface along the anticlockwise direction, the node composition of the temporary unit does not need to be updated; when an old crack tip node's temporary unit is located between the new crack underside and the old crack underside in the clockwise direction, the old crack tip node number in the temporary unit needs to be replaced with the new node number inserted in the previous step.
Wherein the new node number is consistent with the old crack tip node coordinates.
And finally, constructing cohesive units among the new crack tips, the old crack tips and the newly added nodes to form a new grid structure. The cohesive unit will simulate the cohesive force distribution on the crack face in a finite element simulation flow of the new lattice structure.
Further, in step 4), the finite element simulation flow of crack propagation is consistent with a general finite element simulation method as a whole, and a newton-radson iteration method is adopted. After each load step calculation is completed, the operation of step 2) is performed on the crack tip to judge whether the crack is expanded or not. If crack propagation is possible, the operation of step 3) is required. And then continuously iterating on the load step until the residual force of the new grid structure meets the convergence condition, so as to obtain new grid deformation data meeting the balance state after crack expansion.
As shown in fig. 2, a schematic diagram of crack growth results achieved by the method is given, and the mesh deformation results in the schematic diagram can be known: the finite element method for crack propagation along the grid boundary provided by the application can realize the simulation process of crack propagation along the grid boundary.
It will be appreciated by those skilled in the art that embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, systems, and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
The scope of the application is obviously not limited to these specific embodiments. Equivalent modifications and substitutions for related technical features may be made by those skilled in the art without departing from the principles of the present application, and such modifications and substitutions will fall within the scope of the present application.
Claims (6)
1. A method of propagating finite elements along a grid boundary, comprising the steps of:
constructing an extrinsic cohesive force model, and determining the relationship between separation displacement and cohesive force on a crack surface;
calculating the maximum principal stress of the current crack tip node according to the relation, and judging crack propagation;
if the crack is expanded, performing grid topology operation on the current crack tip node to realize the splitting of the old crack tip node and generate a new crack surface;
iteration is carried out on the new crack surface by adopting a Newton Lafson iteration method until the residual force of the whole new crack surface meets the convergence condition, so that the whole finite element simulation flow is realized;
the method for calculating the maximum main stress of the current crack tip node comprises the following steps:
calculating stress data of all unit integral points of the current crack tip node, and calculating the stress data of all crack tip node entity units at the crack tip by a stress reconstruction method;
averaging stress data of all the crack tip node entity units at the crack tip to obtain the maximum principal stress of the crack tip node, and calculating the direction of the maximum principal stress;
the relationship between the separation displacement and the cohesion on the crack face is expressed as follows:
，
in the method, in the process of the application,is a potential function; />Represents the normal cohesion on the crack face, +.>Represents tangential inward focusing forces on the crack face;is normal cohesive energy->Is cohesive energy in tangential direction +.>For normal separation displacement->Represents a tangential separation displacement;for normal fail-separate displacement->For tangential fail-separate displacement, +.>And->Is a parameter for controlling the shape of cohesion-separation displacement curve in extrinsic cohesion model, +.>And->Representing an energy constant;
the potential function expression is:
；
the energy constantAnd energy constant->The expression is as follows:
。
2. a method of propagating finite elements along a grid boundary as claimed in claim 1, wherein said grid topology operation comprises:
establishing a temporary node, temporary units and temporary edges of the old crack tip node;
inserting a new node into the current crack tip node, wherein the edge direction with the smallest included angle with the direction of the maximum principal stress is used as a crack propagation path, and the edge node on the edge is the new crack tip.
3. A method of finite element propagation along a grid boundary as claimed in claim 2, wherein the said adjacent nodes are nodes around the crack tip node directly connected to the crack tip node by cell edges; the temporary units are units which are directly connected with the units around the crack tip nodes; the adjacent edge is a unit edge formed by a crack tip node and an adjacent node; the adjacent edge direction is the direction pointing to the adjacent node by taking the crack tip node as a starting point.
4. A method of propagating finite elements along a grid boundary according to claim 2, comprising continuously updating old crack tip node formations, constructing cohesive units between new and old crack tips, forming a new grid structure.
5. The method of claim 4, wherein the old crack tip node configuration is not updated when the temporary elements of the old crack tip node are located between the new crack upper side and the old crack upper side in a counter-clockwise direction;
when the temporary unit of the old crack tip node is located between the new crack underside and the old crack underside in the clockwise direction, the old crack tip node in the temporary unit is replaced with the new node inserted in the previous step.
6. A method of finite element propagation along a grid boundary as claimed in claim 1, wherein the crack propagates when the maximum principal stress is greater than the cohesive strength of the extrinsic cohesive model; otherwise, the crack does not propagate.
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| US10458937B2 (en) * | 2018-03-08 | 2019-10-29 | Shijiazhuang Tiedao University | Electromagnetic detector for detection of interface cracks in a piezoelectric-piezomagnetic laminated structure |
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| CN113033038A (en) * | 2021-02-23 | 2021-06-25 | 中国石油天然气股份有限公司 | Method for simulating random expansion of hydraulic fractures in fractured shale reservoir |
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Inventor after: Cui Huiru Inventor after: Ding Jian Inventor after: Cheng Zijian Inventor after: Wang Daqing Inventor before: Cui Huiru Inventor before: Ding Jian Inventor before: Cheng Zijian Inventor before: Wang Daqing |