CN117688807A - Structural deformation and damage analysis method based on indirect bond finite element method - Google Patents
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Abstract
The invention discloses a structural deformation and damage analysis method based on an indirect force bond finite element method, which comprises the steps of geometrically modeling a target structure, and generating a traditional finite element method grid by seed distribution subdivision; applying load to the target structure, and calculating a force key weighting coefficient, a force key and the elongation of the force key at any node; judging and finding out broken force bonds according to the elongation; calculating displacement of the target structure after key breaking under the load, and judging whether the number of broken force keys is increased or not; if no increase occurs, the current load step is calculated, and then equivalent damage at any node in the target structure is calculated. The invention utilizes the spatial discrete capability of standard three-node triangle and four-node tetrahedron finite element grids, determines a acting domain through all units connected by the nodes, and constructs indirect force keys connected with the nodes in the acting domain, so that the transmission path of force is defined through the deformation of the force keys, and the damage field of the structure is described by utilizing the fracture of the force keys.
Description
Technical Field
The invention belongs to the technical field of analysis of structural deformation and damage, and particularly relates to a structural deformation and damage analysis method based on an indirect force bond finite element method.
Background
The damage and crack evolution problems of the structure under the action of external load are century difficult problems in the field of solid mechanics, and the structure has important engineering application value in the fields of important engineering such as aerospace, mechanical manufacturing, civil engineering and water conservancy and the like. Classical continuous medium mechanics based on the continuous deformation assumption is no longer applicable to such discontinuous deformation problems because of the absence of spatial derivatives due to discontinuities such as breaks, damages, etc. The last century line elastic fracture mechanics and continuous damage mechanics began to rise, however these methods required the artificial introduction of the initial fracture and did not consider the physical mechanism of crack nucleation, and therefore had certain limitations.
For the damage and the crack evolution problems of the structure, the numerical simulation method can be divided into a discrete crack model and a diffusion crack model according to the crack characterization mode. The former geometrically describes crack discontinuity displacements and strains by introducing additional functions, such as extended finite element methods and embedded finite element methods. Such methods do not determine crack propagation paths and directions well in practical numerical implementations, and when dealing with fracture network problems, the geometric description and subsequent computation of the fracture becomes exceptionally difficult due to multi-fracture intersection, particularly three-dimensional fracture growth and propagation problems. The latter reflects the crack location discontinuity behavior, such as the phase field method, by using field variables to propagate the crack. The fracture mechanical response is described by adopting phase field variable approximation, the material fracture simulation problem is converted into a multi-field coupling calculation solution problem, and the simulation solution problem does not need additional crack initiation criteria and a crack explicit tracking algorithm, so that the complexity of the material fracture damage numerical simulation problem is reduced.
In recent years, the internationally emerging method, the near-field dynamics method, develops a new method, a new solid mechanical framework is established based on the idea of non-local interaction, force bonds for transmitting force are defined between non-contact substance points in a limited distance on an object, and the substance points interact through the force bonds, so that the problem that the spatial derivative does not exist when the traditional continuous medium mechanical treatment breaks is eliminated as the traditional differential equation is converted into the integral-differential equation.
Although the method successfully solves the problem of damage and fracture of part of solid mechanical structures, a large number of problems still exist, so that the method cannot be truly applied to large-scale engineering calculation: 1) The grid dependence is strong, such as near field dynamics, expansion finite element and other methods, and the grid quality has a critical influence on the crack expansion direction and speed; 2) The calculation cost is high, for example, each force key of each object point needs to be solved in near field dynamics, and the calculation cost is far greater than that of the traditional finite element, so that the development of the traditional finite element is limited; 3) The coupling with other methods is inconvenient, the method combined with the finite element is usually used for solving the problem of high calculation cost, and the traditional finite element is used for replacing the non-crack propagation potential area, but the coupling between different methods often needs to be provided with a coupling transition area, and meanwhile, the special boundary problems such as an inner boundary and the like cannot be well solved.
Disclosure of Invention
The invention aims to provide a structural deformation and damage analysis method based on an indirect force bond finite element method, which solves the problems existing in the damage analysis numerical methods such as near field dynamics, a phase field method and an extended finite element method in the prior art.
In order to solve the technical problems, the invention adopts the following technical scheme:
the structural deformation and damage analysis method based on indirect bond finite element method comprises the following steps:
step 1: geometric modeling is carried out on the target structure omega, and after the boundary is distributed according to the target quantity, the boundary is split to generate a traditional finite element method grid, m finite elements and n finite element nodes are obtained, m and n are positive integers, and the quantity is determined by the target structure omega and the distribution condition.
Step 2: applying a load to the target structure Ω, determining e for any node I I With units connected theretoCalculating the force at node IKey weighting coefficient W I Force key->Force key->Elongation of +.>And +.>Ordering from big to small, and recording the sequence as Y.
Step 3: and judging and finding out broken force bonds according to the elongation in the sequence Y.
Step 4: the displacement of the target structure omega after key breaking under the load is calculated, and the displacement is specifically as follows: calculating any force keyKey force of->Setting the breaking weight function of all the broken force bonds in the step 3 to 0 to obtain the bonding force q at the node I I The method comprises the steps of carrying out a first treatment on the surface of the Based on the bond force q I The counter force resultant force brought to the node I when the key force is calculated by other node force keys is used for obtaining the node force resultant force of the node I, and the integral rigidity matrix of the structure omega is calculated according to the node force resultant force>Then calculating a force matrix F according to the boundary condition of the target structure omega, calculating to obtain an overall node displacement vector u according to the following formula,
step 5: repeating the steps 2 and 3, and judging whether the number of broken force bonds is increased or not; if no increase exists, the calculation of the current load step is completed, and the step 6 is entered; otherwise, repeating the step 4. Because the force bond fracture will cause the redistribution of the displacement field under the current load step, and the redistributed displacement field may bring new force bond fracture, there are multiple fracture iteration steps under the same load step.
Step 6: calculating equivalent damage d at any node I in target structure omega I The calculation method comprises the following steps:
in the formula e I Representing the number of units connected to node I,indicating the number of force keys in each unit connected with node I, N I Representing all force keys at node I, +.>And (3) breaking the weight function for the force bonds.
Further optimizing, aiming at the problems of deformation and damage of a two-dimensional structure, the force key weighting coefficient W at any node I I Force keyKey force->Rigidity matrix of target structure Ω ≡>The calculation steps of the sum force matrix F are as follows:
a1: calculating a force key weighting coefficient: the mesh generated by the conventional finite element method after seed distribution is a three-node triangle unit; determining e for arbitrary node I I With units connected theretoObtain the area of each cell +.>And the interior angle of the cell at node I +.>Calculating to obtain the weighting coefficient of the force key at the node I>Wherein->A certain value representing the I-th element connected to node I.
A2: acquiring force keys: equiangular generation of a cluster starting from the x-axis and number N at node I I Is intersected with the line segment formed by the node J and the node K at the pointPoint->Is +.>Obtaining all force keys at node IThe total number of force keys at the node I is N I The method comprises the steps of carrying out a first treatment on the surface of the Wherein, is located->The number of force keys in the inner part is +.> Representation and sectionA certain value of the jth force key in the ith unit connected with the point I, wherein the force key intersected with the prefabricated fracture line is a failure force key;
a3: calculating key force: any one force keyThe key force of (2) is calculated as +.>Wherein the method comprises the steps ofFor the force bond stiffness matrix, xi is used in the expression to denote +.>ξ 1 And xi 2 Is force key->Component 1,2 of->Is the relative deformation vector of the force key, +.>Force key->Strain of the located cell>Representation->Mu (ζ, t) is force bond +.>The breaking weight function at time t, c is the force bond constant, for the two-dimensional structure plane stress problem c=3e, plane strain problem c=16e/5,e is the Young's modulus of the target structure Ω.
Force keyThe key force of the (a) brings key force counter force for the node J and the node K>And->Wherein,representing due to calculation +.>Reaction force acting on the joint J during the key force, +.>Representing force key intersection +.>Distance to node K, ">Representing force key intersection +.>Distance to junction J, ">Representing the distance from node J to node K, |·| represents the length of the vector.
A4: integral rigidity matrix for calculating joint resultant force and target structure omegaThe node force resultant force of the node I is composed of the bond force resultant force +.>And the total stiffness matrix of the target structure omega is calculated according to the total composition of the counter force resultant force brought to the node I when the key force is calculated by other node force keys>The formula is as follows:
in the formula e I Representing the number of units connected to node I,represents the number of force keys in each unit connected with the node I, delta theta is a fixed discrete angle, n represents the number of nodes in the target structure omega, and K I Is the cell stiffness matrix of node I.
A5: the force matrix F is calculated according to the boundary conditions of the target structure Ω, with the following formula:
wherein m is the total number of finite units; g i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix, F i Representing finite element Ω i Cell load vector, N i (x) Is a finite element omega i Form function matrix of u i (x)＝N i (x)u i Wherein u is i (x) Is unit omega i Interior any of the twoDisplacement at the intentional point x, u i Is unit omega i The displacement at the node, x is the finite element Ω i Point (N) a (x) Is a finite element omega i A = 1,2,3; b (x) is the external body force vector of the external body force field at the point x, dV x Is an integral infinitesimal.
Further preferably, in the step A2, the node J and the node K are unitsThe 1 st and 2 nd nodes arranged anticlockwise from node I, with relative position +.>Recording force key intersection +.>Position of> Representing force key intersection +.>The distances to nodes J and K, a=j, K; l (L) JK Representing the distance from node J to node K.
In the step A3, since the three-node triangle unit is a constant strain unit, the relative deformation vector of the force key is represented by linear interpolation of the node J and the node K, namelyIn->Representing node displacement, a=i, J, K; meanwhile, the counterforce of the force key can be written as +.>And->
In the step A4, the stiffness matrix K at the node I I The calculation formula of (2) is as follows:
in the formula e I Representing the number of units connected to node I,representing the number of force keys in each unit connected with node I, G i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix>Unit rigidity matrix representing combined action of key force and key force counter force at node I, W I The key force weighting coefficient at the node I is given; for simplicity of expression, the elements in the matrix are +.>
Further optimizing, aiming at the deformation and damage problems of the three-dimensional structure, the force key weighting coefficient W at any node I I Force keyKey force->Rigidity matrix of target structure Ω ≡>The calculation steps of the sum force matrix F are as follows:
b1: calculating a force key weighting coefficient: the mesh of the traditional finite element method generated by subdivision after seed distribution is a four-node tetrahedron unit; determining e for arbitrary node I I With units connected theretoObtaining the volume of each unitAnd the solid angle of the cell at node I +.>Calculating to obtain the weighting coefficient of the force key at the node I>Wherein->A certain value representing the I-th element connected to node I.
B2: acquiring force keys: generating a cluster with equal solid angles at the node I and taking the x axis as a starting point, wherein the number is N I Rays of (1) respectively intersect with the triangular face JKL at pointsPoint->Is +.>Nodes J, K, L represent units->Calculating all force keys ++at node I for three nodes except node I>Wherein is located at the unit->The number of force keys in the inner part is +.>And a certain numerical value of the jth force key in the ith unit connected with the node I is represented, and the force key intersected with the prefabricated crack surface is a failure force key.
B3: calculating key force: any one force keyThe key force of (2) is calculated as +.>Wherein the method comprises the steps ofFor the force bond stiffness matrix, xi is used in the expression to denote +.>ξ 1 、ξ 2 And xi 3 Is force key->Component 1,2,3, < ->Is the relative deformation vector of the force key, +.>Force key->Strain epsilon of the cell in which it is located i Representation ofMu (ζ, t) is force bond +.>And c is a force bond constant c=6E, and E is the Young modulus of the target structure omega.
Force keyWill bring key force counter force for other three joints J, K, LWherein (1)>Representing due to calculation +.>Reaction force acting on the joint J during the key force, +.>Representing due to calculation +.>Counter force acting on node K during key force, +.>Representing due to calculation +.>Counter force acting on the node L when the key force is applied; s is S ΔpKL Representing force key intersection +.>Area S of triangle formed by node K and node L ΔpJL Representing force key intersection +.>And a joint J,The area S of the triangle formed by the nodes L ΔpJK Representing force key intersection +.>Area of triangle formed by node J and node K, S ΔJKL The area of the triangle formed by the node J, the node K and the node L is represented, and the I.S. I represents the length of the vector.
B4: integral rigidity matrix for calculating joint resultant force and target structure omegaThe node force resultant force of the node I is composed of the bond force resultant force +.>And the total stiffness matrix of the structure omega is calculated according to the total composition of the counterforce resultant force brought to the node I when the key force is calculated by other node force keys>The formula is as follows:
in the formula e I Representing the number of units connected to node I,represents the number of force keys in each unit connected with the node I, delta omega is a fixed discrete solid angle, n represents the number of nodes in the target structure omega, and K I Is the cell stiffness matrix of node I.
B5: the force matrix F is calculated according to the boundary conditions of the structure Ω, with the formula:
wherein m is the total number of finite units; g i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix, F i Representing finite element Ω i Cell load vector, N i (x) Is a finite element omega i Is a matrix of form functions (u i (x)＝N i (x)u i Wherein u is i (x) Is unit omega i Displacement at arbitrary point x, u i Is unit omega i Displacement at a node), x is the finite element Ω i Point (N) a (x) A=1, 2,3,4 is the finite element Ω i B (x) is the external body force vector of the external body force field at the point x, dV x Is an integral infinitesimal.
Further preferably, in the step B2, the triangle area coordinates are used for representing the force key intersection pointRelative position within triangle JKL +.>Wherein->
In the step B3, since the four-node tetrahedron unit is a constant strain unit, the relative displacement of the force key can be represented by linear interpolation of the displacement of the node JKLWherein->For node displacement, asThe reaction force of the time key can be written as +.>
In the step B4, the stiffness matrix K at the node I I The calculation formula of (2) is as follows:
in the formula e I Representing the number of units connected to node I,representing the number of force keys in each unit connected with node I, G i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix>Unit rigidity matrix representing combined action of key force and key force counter force at node I, W I For the key force weighting coefficient at node I, the element in the matrix is +.>
Further optimizing, the calculation formula of the force bond breaking weight function mu (ζ, t) is as follows:
further preferably, in the step 2, the force keyElongation of +.>
Further preferably, in the step 3, the decision keyThe method for breaking is as follows: if the elongation of the force key +.>Greater than a set critical elongation l c And judging that the force bond is broken, otherwise, judging that the force bond is not broken.
Further optimizing, in order to increase the calculation speed, the first N in the elongation sequence Y in step 2 is taken each time m Judging the broken key of the item, and judging the broken keyAnd (5) judging that the force bonds are broken, uniformly setting the force bond breaking weight functions of the force bonds to 0, and continuing the calculation of the subsequent steps.
Compared with the prior art, the invention has the following beneficial effects:
1) The invention provides a new numerical calculation technology, namely an indirect force bond finite element method, by utilizing the space discrete capability of standard three-node triangle and four-node tetrahedron finite element grids, and is used for realizing the damage analysis of the structure under a general calculation platform. Specifically, the scope is determined through all units connected with the node, and an indirect force bond connected with the node is constructed in the scope, so that the transmission path of force is defined through deformation of the force bond, and the damage field of the structure is described by using the fracture of the force bond.
2) Compared with the traditional near field dynamics method based on the force key, the method has the advantages that the local area is used as the acting area of the force key, so that the overall rigidity matrix bandwidth of the structure is far smaller than that of the near field dynamics method using the non-local area, and the calculation efficiency of the method is far higher than that of the near field dynamics method under the condition that the distribution number of the structures is consistent. Meanwhile, the method introduces a force key weighting coefficient, so that the calculation of the key force is not influenced by the complete condition of the acting domain, and further, the phenomenon of rigidity softening of near-field dynamics on the surface of the structure is avoided.
3) According to the invention, as the standard finite element grid is used for space dispersion, under the condition that the position of the potential occurrence of the crack of the target structure is known, the calculation efficiency can be further improved by using a coupling mode with a finite element method. Compared with near field dynamics, the method can be coupled with a finite element method more naturally, and a coupling transition region is not required to be arranged at the junction of different calculation methods, so that the application scene of the method is further expanded.
Drawings
FIG. 1 is a schematic diagram of a two-dimensional force key and a background grid;
FIG. 2 is a graph of force distribution for a force key at node I and for nodes J and K within a two-dimensional force key;
FIG. 3 is a schematic illustration of geometry and loading conditions of a three-hole precast fracture plate;
FIG. 4 is a schematic diagram of a three-node triangle cell;
FIG. 5 is a flow chart of a multi-stage loading structure damage analysis;
FIG. 6 is a graph of results of a three-hole precast fracture plate structural failure calculation;
FIG. 7 is a graph showing the results of a three-hole precast slit panel X-direction displacement calculation;
FIG. 8 is a graph of calculated Y-direction displacement of a three-hole precast fracture plate;
FIG. 9 is a schematic illustration of geometry and loading conditions of a dual precast slit panel;
FIG. 10 is a graph of the results of a dual precast slit plate structural failure calculation;
FIG. 11 is a graph of the results of a calculation of X-direction displacement of a dual precast slit plate;
FIG. 12 is a graph of the calculated Y-direction displacement of a dual precast slit plate;
FIG. 13 is a schematic illustration of geometry and loading conditions of a precast crack-free L-shaped panel;
FIG. 14 is a graph of the results of a pre-crack free L-shaped panel failure calculation;
FIG. 15 is a graph showing the result of calculation of X-direction displacement of an L-shaped flat plate without precast cracks;
FIG. 16 is a graph showing the results of the calculation of Y-direction displacement of an L-shaped flat plate without precast cracks.
Detailed Description
Specific embodiments are given below in conjunction with specific examples to further illustrate the invention. It should be understood that the examples are only for illustrating the present invention and are not intended to limit the scope of the present invention.
1. An indirect force bond finite element method for deformation analysis of a two-dimensional structure, comprising the steps of:
(A1) Geometric modeling is carried out on a target structure omega, and after the boundary is distributed according to the target quantity, the boundary is split to generate a traditional finite element method grid, specifically a triangular unit with three nodes, m finite elements and n finite element nodes are obtained, and the quantity of m and n is jointly determined by the target structure omega and the distribution condition.
(A2) Determining e for arbitrary node I I With units connected theretoObtain the area of each cell +.>And the interior angle of the cell at node I +.>Calculating to obtain the weighting coefficient of the force key at the node I>Wherein->A certain value representing the I-th element connected to node I.
(A3) Equiangular generation of a cluster starting from the x-axis and number N at node I I Is intersected with the line segment formed by the node J and the node K at the pointAs shown in fig. 1. Node J and node K are units->1 st, 2 nd node arranged anticlockwise from node I, point +.>Is +.>By relative position->Recording force key intersection +.>Is a position of (1), whereinIn->Representing force key intersection +.>Distance to node J and node K, l JK Representing the distance from node J to node K, all force keys +.>The total number of force keys at the node I is N I Wherein is located->The number of force keys in the inner part is +.>Wherein->A certain value of the jth force key in the ith unit connected with the node I is represented, and the force key intersected with the prefabricated fracture line is a failure force key.
(A4) Any one force keyThe key force of (2) is calculated as +.>Wherein->For the force bond stiffness matrix, xi is used in the expression to denote +.>ξ 1 And xi 2 Is force key->Component 1,2 of->Is the relative deformation vector of the force key +.>In->Represents the displacement of the node, mu (ζ, t) is the force key +.>And c is a force bond constant, c=3e is a plane stress problem, c=16e/5 is a plane strain problem, and E is the Young's modulus of the target structure omega. Force key->Will bring a key force counter force to node J and node K +.>And->Wherein->Representing due to calculation +.>The reaction force acting on the joint J at the time of the key force is, |·|| as shown in fig. 2, and indicates the length of the vector.
(A5) The resultant force of node I is composed of the resultant force of bond forceAnd the total stiffness matrix of the structure omega is calculated according to the total composition of the counterforce resultant force brought to the node I when the key force is calculated by other node force keys>The formula is as follows:
in the formula e I Representing the number of units connected to node I,represents the number of force keys in each unit connected with the node I, delta theta is a fixed discrete angle, n represents the number of nodes in the target structure omega, and K I Cell stiffness matrix for node I, G i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix>Unit rigidity matrix representing combined action of key force and key force counter force at node I, W I For the key force weighting coefficient at node I, the element in the matrix is +.>
(A6) The force matrix F is calculated from the boundary conditions of the structure Ω as follows:
wherein m is the total number of finite units; g i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix, F i Representing finite element Ω i Cell load vector, N i (x) Is a finite element omega i Is a matrix of form functions (u i (x)＝N i (x)u i Wherein u is i (x) Is unit omega i Displacement at arbitrary point x, u i Is unit omega i Displacement at a node), x is the finite element Ω i Point (N) a (x) A=1, 2,3 is the finite element Ω i B (x) is the external body force vector of the external body force field at the point x, dV x Is an integral infinitesimal.
(A7) Solving the equation to obtain the overall node displacement vector u, wherein the equation is as follows:
2. an indirect force bond finite element method for three-dimensional problem structure deformation analysis comprises the following steps:
(B1) Geometric modeling is carried out on a target structure omega, and after the boundary is distributed according to the target quantity, the boundary is split to generate a traditional finite element method grid, specifically four-node tetrahedron units, m finite elements and n finite element nodes are obtained, and the quantity of m and n is jointly determined by the target structure omega and the distribution condition.
(B2) Determining e for arbitrary node I I With units connected theretoObtain the volume of each unit +.>And the solid angle of the cell at node I +.>Calculating to obtain the weighting coefficient of the force key at the node I>Wherein->A certain value representing the I-th element connected to node I.
(B3) Generating a cluster with equal solid angles at the node I and taking the x axis as a starting point, wherein the number is N I Rays of (1) respectively intersect with the triangular face JKL at pointsPoint->Is +.>Representing force key intersection +_using triangle area coordinates>Relative position within triangle JKL +.>Wherein->Nodes J, K, L represent units->Calculating all force keys ++at node I for three nodes except node I>Wherein is located at the unit->The number of force keys in the inner part is +.>A certain value of the jth force key in the ith unit connected with the node I is represented, and the force key intersected with the prefabricated fracture line is a failure force key.
(B4) Any one force keyThe key force of (2) is calculated as +.>Wherein->For the force bond stiffness matrix, xi is used in the expression to denote +.>ξ 1 、ξ 2 And xi 3 Is force key->Component 1,2,3, < ->For the relative deformation vector of the force key, the relative displacement of the force key is represented by linear interpolation of the displacement of the node JKL>Wherein->For node displacement, μ (ζ, t) is force key +.>And c is a force bond constant c=6E, and E is the Young modulus of the target structure omega. Force key->Will bring key force counter force to other three nodesThe term "l" represents the length of the vector.
(B5) The resultant force of node I is composed of the resultant force of bond forceAnd the total stiffness matrix of the structure omega is calculated according to the total composition of the counterforce resultant force brought to the node I when the key force is calculated by other node force keys>The formula is as follows:
in the formula e I Representing the number of units connected to node I,represents the number of force keys in each unit connected with the node I, delta omega is a fixed discrete solid angle, n represents the number of nodes in the target structure omega, and K I Cell stiffness matrix for node I, G i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix>Unit rigidity matrix representing combined action of key force and key force counter force at node I, W I For the key force weighting coefficient at node I, the element in the matrix is +.>
(B6) The force matrix F is calculated from the boundary conditions of the target structure Ω as follows:
wherein m is the total number of finite units; g i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix, F i Representing finite element Ω i Cell load vector, N i (x) Is a finite element omega i Is a matrix of form functions (u i (x)＝N i (x)u i Wherein u is i (x) Is unit omega i Displacement at arbitrary point x, u i Is unit omega i Displacement at a node), x is the finite element Ω i Point (N) a (x) A=1, 2,3,4 is the finite element Ω i B (x) is the external body force vector of the external body force field at the point x, dV x Is an integral infinitesimal;
(B7) Solving the equation to obtain the overall node displacement vector u, wherein the equation is as follows:
3. an indirect bond finite element method for structural damage analysis, comprising the steps of:
(C0) If the research question is two-dimensional, selecting the step (A1-7), if the research question is three-dimensional, selecting the step (B1-7), wherein the step (A1-7) and the step (B1-7) are uniformly described by the step (X1-7) because the content and the symbol system of the step (A1-7) have symmetry, and the step (X1-7) is defaulted to be completed once before the damage analysis is carried out, and all calculation results are reserved;
(C1) Calculate all force keysElongation of +.>The calculation method is->And is about->Ordering from large to small, noting that the sequence is Y, the broken bonds do not enter the sequence.
(C2) Taking out the previous Nm item in the sequence Y, and judging the corresponding force keyWhether the fracture occurs or not, and judging the fracture method comprises the following steps: if the elongation of the force key +.>Greater than a given critical elongation l c The bond breaks, whereas it does not.
(C3) For cleavage in step (C2)And (3) uniformly setting the breaking weight functions of the individual force keys to 0, and executing the steps (X4-7) to calculate the displacement after breaking the keys under the load.
(C4) Executing the step (C1-2) to calculate the number of newly increased broken keysIf no newly added key is broken, the calculation of the current load step is completed, and the step (C5) is entered; otherwise, the process returns to the step (C3).
(C5) Calculating equivalent damage d at any node I in target structure omega I The calculation method comprises the following steps:
in the formula e I Representing the number of units connected to node I,indicating the number of force keys in each unit connected with node I, N I Representing all force keys at node I, +.>And (3) breaking the weight function for the force bonds.
The following is illustrative with reference to specific use procedures:
example 1:
the plane stress problem is set, in particular a flat plate with three round holes and a prefabricated slit, the geometrical dimensions and boundary conditions of which are shown in figure 3. The preset displacement increment is delta u y ＝1×10 -3 mm, maximum load displacement u y =0.1 mm; the Young's modulus of the material of the target structure is E=30GPa and Poisson's ratio v=1/3; critical elongation of force bond l c =0.011, at most allowed number of broken bonds N at a time m =4; the numerical calculation adopts a three-node triangle unit, which is shown in fig. 4, wherein (a) is a target structure unit diagram; (b) is a partial magnified view of the precast fissure.
The calculation flow chart of the problem is shown in fig. 5, a displacement cloud image and an equivalent damage cloud image of the target structure are output according to the calculation result of each load, and as shown in fig. 6, 7 and 8, the field variable of the target structure is drawn on the initial configuration of the structure, and the color represents the value from low to high from blue to yellow. Wherein (a) is the boundary displacement u y When=0.60 mm, the equivalent damage cloud is shown in fig. 6 (a), the X-direction displacement cloud is shown in fig. 7 (a), and the Y-direction displacement cloud is shown in fig. 8 (a). When the boundary is displaced u y When=0.70 mm, the equivalent damage cloud is shown in fig. 6 (b), the X-direction displacement cloud is shown in fig. 7 (b), and the Y-direction displacement cloud is shown in fig. 8 (b).
Example 2:
the plane stress problem is set, specifically a flat plate structure with double prefabricated cracks, and the geometric dimensions and boundary conditions are shown in fig. 9. The preset displacement increment is delta u y ＝2Δu x ＝6.25×10 -4 mm, maximum load displacement u y ＝0.5mm，u x =0.25 mm; the basic mechanical parameters of the target structural material are consistent with those in example 1; critical elongation of force bond l c =0.012, at most allowed number of broken bonds N at a time m ＝12。
The calculation flow of this example and in example 1According to the calculation result of each load, outputting a displacement cloud image and an equivalent damage cloud image of the target structure, and as shown in fig. 10, 11 and 12, the field variable of the target structure is drawn on the initial configuration of the structure, and the color represents that the numerical value is from low to high from blue to yellow. Wherein when the boundary displacement is u x ＝0.1438mm，u y When= 0.2875mm, the equivalent damage cloud is shown in fig. 10 (a), the X-direction displacement cloud is shown in fig. 11 (a), and the Y-direction displacement cloud is shown in fig. 12 (a). When the boundary is displaced u x ＝0.1875mm，u y When= 0.3750mm, the equivalent damage cloud is shown in fig. 10 (b), the X-direction displacement cloud is shown in fig. 11 (b), and the Y-direction displacement cloud is shown in fig. 12 (b).
Example 3:
the plane stress problem is set, specifically an L-shaped flat plate structure without precast cracks, and the geometric dimension and boundary conditions are shown in figure 13. The preset displacement increment is delta u y =0.01 mm, maximum loading displacement u y =1mm; the basic material parameters of the target structure are consistent with those of example 1, and the maximum number of allowed fracture bonds N is at each time m ＝12。
The calculation flow of this example is identical to the flow chart 5 in example 1, and a displacement cloud image and an equivalent damage cloud image of the target structure are output according to the calculation result of each load, as shown in fig. 14, 15 and 16, the field variable of the target structure is drawn on the initial configuration of the structure, and the color represents a value from low to high from blue to yellow. Wherein when the boundary displacement is u y When=0.19 mm, the equivalent damage cloud is shown in fig. 14 (a), the X-direction displacement cloud is shown in fig. 15 (a), and the Y-direction displacement cloud is shown in fig. 16 (a). When the boundary is displaced u y When=0.29 mm, the equivalent damage cloud is shown in fig. 14 (b), the X-direction displacement cloud is shown in fig. 15 (b), and the Y-direction displacement cloud is shown in fig. 16 (b).
The three examples can show that the method can better predict the tension crack generated by the quasi-brittle material under the action of external load, and the crack form is basically consistent with the actual situation. Meanwhile, the method uses the background grid nodes as basic material points, so that the whole rigidity matrix bandwidth of the structure is equal to that of a standard finite element method. Compared with a near-field dynamics method which is the key-based method but has obviously increased bandwidth and an extended finite element method which uses finite element grids and needs to introduce redundant degrees of freedom, the method can obviously improve the calculation efficiency under the condition that the calculation results are consistent, and therefore has important application value in the engineering fields of civil water conservancy, energy exploitation, aerospace and the like which relate to calculation damage mechanics.
The above examples are given for clarity of illustration only and are not limiting of the embodiments. Other variations or modifications of the above teachings will be apparent to those of ordinary skill in the art. It is not necessary here nor is it exhaustive of all embodiments. While still being apparent from variations or modifications that may be made by those skilled in the art are within the scope of the invention.
Claims (9)
1. The structural deformation and damage analysis method based on the indirect force bond finite element method is characterized by comprising the following steps of:
step 1: geometric modeling is carried out on a target structure omega, and after the boundary is distributed according to the target quantity, the boundary is split to generate a traditional finite element method grid, m finite elements and n finite element nodes are obtained, m and n are positive integers, and the quantity is determined by the target structure omega and the distribution condition;
step 2: applying a load to the target structure Ω, determining e for any node I I With units connected theretoCalculating a force key weighting coefficient W at node I I Force key->Force key->Elongation of +.>And +.>Sequencing from big to small, and recording the sequence as Y;
step 3: judging and finding out broken force bonds according to the elongation in the sequence Y;
step 4: the displacement of the target structure omega after key breaking under the load is calculated, and the displacement is specifically as follows: calculating any force keyKey force of (a)Setting the breaking weight function of all the broken force bonds in the step 3 to 0 to obtain the bonding force q at the node I I The method comprises the steps of carrying out a first treatment on the surface of the Based on the bond force q I The counter force resultant force brought to the node I when the key force is calculated by other node force keys is used for obtaining the node force resultant force of the node I, and the integral rigidity matrix of the structure omega is calculated according to the node force resultant force>Then calculating a force matrix F according to the boundary condition of the target structure omega, calculating to obtain an overall node displacement vector u according to the following formula,
step 5: repeating the steps 2 and 3, and judging whether the number of broken force bonds is increased or not; if no increase exists, the calculation of the current load step is completed, and the step 6 is entered; otherwise, repeating the step 4;
step 6: calculating equivalent damage d at any node I in target structure omega I The calculation method comprises the following steps:
in the formula e I Representing the number of units connected to node I,indicating the number of force keys in each unit connected with node I, N I Representing all force keys at node I, +.>And (3) breaking the weight function for the force bonds.
2. The method for analyzing structural deformation and damage based on indirect force bond finite element method according to claim 1, wherein the force bond weighting coefficient W at any node I is specific to the two-dimensional structural deformation and damage problem I Force keyKey force->Rigidity matrix of target structure Ω ≡>The calculation steps of the sum force matrix F are as follows:
a1: calculating a force key weighting coefficient: the mesh generated by the conventional finite element method after seed distribution is a three-node triangle unit; determining e for arbitrary node I I With units connected theretoObtain the area of each cell +.>And the interior angle of the cell at node I +.>Calculating to obtain the weighting coefficient of the force key at the node I>Wherein->A certain value representing the I-th element connected to node I;
a2: acquiring force keys: equiangular generation of a cluster starting from the x-axis and number N at node I r Is intersected with the line segment formed by the node J and the node K at the pointPoint->Is +.>Obtaining all force keys at node IThe total number of force keys at the node I is N I The method comprises the steps of carrying out a first treatment on the surface of the Wherein, is located->The number of force keys in the inner part is +.> A certain numerical value of a jth force key in an ith unit connected with the node I is represented, and the force key intersected with the prefabricated fracture line is a failure force key;
a3: calculating key force: any one force keyThe key force of (2) is calculated as +.>Wherein->For the force bond stiffness matrix, xi is used in the expression to denote +.>ξ 1 And xi 2 Is force key->Component 1,2 of->Is the relative deformation vector of the force key, +.>Force key->Strain of the located cell>Representation->Mu (ζ, t) is force bond +.>A breaking weight function at the time t, c being a force key constant;
force keyThe key force of the (a) brings key force counter force for the node J and the node K>And->Wherein (1)>Representing due to calculation +.>Reaction force acting on the joint J during the key force, +.>Representing force key intersection +.>Distance to node K, ">Representing force key intersection +.>Distance to junction J, ">Representing the distance from node J to node K, the term "l" represents the length of the vector;
a4: integral rigidity matrix for calculating joint resultant force and target structure omegaThe resultant force of node I is composed of the resultant force of bond forceAnd the total stiffness matrix of the target structure omega is calculated according to the total composition of the counter force resultant force brought to the node I when the key force is calculated by other node force keys>The formula is as follows:
in the formula e I Representing the number of units connected to node I,represents the number of force keys in each unit connected with the node I, delta theta is a fixed discrete angle, n represents the number of nodes in the target structure omega, and K I A cell stiffness matrix for node I;
a5: the force matrix F is calculated according to the boundary conditions of the target structure Ω, with the following formula:
wherein m is the total number of finite units; g i Is a finite element omega i T represents the transpose of the matrix, F i Representing finite element Ω i Cell load vector, N i (x) Is a finite element omega i Form function matrix of u i (x)＝N i (x)u i Wherein u is i (x) Is unit omega i Displacement at arbitrary point x, u i Is unit omega i The displacement at the node, x is the finite element Ω i Point (N) a (x) Is a finite element omega i A = 1,2,3; b (x) is the external body force vector of the external body force field at the point x, dV x Is an integral infinitesimal.
3. The method for analyzing structural deformation and damage based on indirect force bond finite element method according to claim 2, wherein in the step A2, the nodes J and K are unitsThe 1 st and 2 nd nodes arranged anticlockwise from node I, with relative position +.>Recording force key intersection +.>Position of> Representing force key intersectionThe distances to nodes J and K, a=j, K; l (L) JK Representing the distance from node J to node K;
in the step A3, since the three-node triangle unit is a constant strain unit, the relative deformation vector of the force key is represented by linear interpolation of the node J and the node K, namelyIn->Representing node displacement, a=i, J, K; meanwhile, the counterforce of the force key can be written as +.>And->
In the step A4, the stiffness matrix K at the node I I The calculation formula of (2) is as follows:
in the formula e I Representing the number of units connected to node I,representing the number of force keys in each unit connected with node I, G i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix>Unit rigidity matrix representing combined action of key force and key force counter force at node I, W I The key force weighting coefficient at the node I is given; for simplicity of expression, the elements in the matrix are +.>
4. Root of Chinese characterThe method for analyzing structural deformation and damage based on indirect force bond finite element method according to claim 1, wherein the force bond weighting coefficient W at any node I is specific to the three-dimensional structural deformation and damage problem I Force keyKey force->Rigidity matrix of target structure Ω ≡>The calculation steps of the sum force matrix F are as follows:
b1: calculating a force key weighting coefficient: the mesh of the traditional finite element method generated by subdivision after seed distribution is a four-node tetrahedron unit; determining e for arbitrary node I I With units connected theretoObtaining the volume of each unitAnd the solid angle of the cell at node I +.>Calculating to obtain the weighting coefficient of the force key at the node I>Wherein->A certain value representing the I-th element connected to node I;
b2: acquiring force keys: generating a cluster with equal solid angles at the node I and taking the x axis as a starting point, wherein the number is N I Rays of (1) respectively intersect with the triangular face JKL at pointsPoint->Is +.>Nodes J, K, L represent units->Calculating all force keys ++at node I for three nodes except node I>Wherein is located at the unit->The number of force keys in the inner part is +.> A certain numerical value of a jth force key in an ith unit connected with the node I is represented, and the force key intersected with the prefabricated crack surface is a failure force key;
b3: calculating key force: any one force keyThe key force of (2) is calculated as +.>Wherein the method comprises the steps ofFor the force bond stiffness matrix, xi is used in the expression to simplify the expression/>ξ 1 、ξ 2 、ξ 2 And xi 3 Is force key->Component 1,2,3, < ->Is the relative deformation vector of the force key, +.>Force key->Strain epsilon of the cell in which it is located i Representation ofMu (ζ, t) is force bond +.>A breaking weight function at the time t, c is a force bond constant c=6E, and E is the Young's modulus of the target structure omega;
force keyWill bring key force counter force for other three joints J, K, LWherein (1)>Representing due to calculation +.>Reaction force acting on the joint J during the key force, +.>Representing due to calculation +.>Counter force acting on node K during key force, +.>Representing due to calculation +.>Counter force acting on the node L when the key force is applied; s is S ΔpKL Representing force key intersection +.>Area S of triangle formed by node K and node L ΔpJL Representing force key intersection +.>Area of triangle formed by the node J and the node L, S ΔpJK Representing force key intersection +.>Area of triangle formed by node J and node K, S ΔJKL The area of a triangle formed by the node J, the node K and the node L is represented, and I.S. I represents the length of the vector;
b4: integral rigidity matrix for calculating joint resultant force and target structure omegaThe resultant force of node I is composed of the resultant force of bond forceAnd other node force keys to give node I when calculating key forceThe resultant forces of the opposing forces together form a global stiffness matrix for the structure Ω calculated from this>The formula is as follows:
in the formula e I Representing the number of units connected to node I,represents the number of force keys in each unit connected with the node I, delta omega is a fixed discrete solid angle, n represents the number of nodes in the target structure omega, and K I A cell stiffness matrix for node I;
b5, calculating a force matrix F according to the boundary condition of the structure omega, wherein the formula is as follows:
wherein m is the total number of finite units; g i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix, F i Representing finite element Ω i Cell load vector, N i (x) Is a finite element omega i Is a matrix of form functions (u i (x)＝N i (x)u i Wherein u is i (x) Is unit omega i Displacement at arbitrary point x, u i Is unit omega i Displacement at a node), x is the finite element Ω i Point (N) a (x) A=1, 2,3,4 is the finite element Ω i B (x) is the external body force vector of the external body force field at the point x, dV x Is an integral infinitesimal.
5. The method for analyzing structural deformation and damage based on indirect force key finite element method according to claim 4, wherein in the step B2, the force key intersection point is represented by using triangle area coordinatesRelative position within triangle JKLWherein->
In the step B3, since the four-node tetrahedron unit is a constant strain unit, the relative displacement of the force key can be represented by linear interpolation of the displacement of the node JKLWherein->For the displacement of the node, the reaction force of the force key can be written as +.>
In the step B4, the stiffness matrix K at the node I I The calculation formula of (2) is as follows:
in the formula e I Representing the number of units connected to node I,representing the number of force keys in each unit connected with node I, G i Is a finite element omega i Is a transformation matrix of node degrees of freedom, · T Representing the transpose of the matrix>Unit rigidity matrix representing combined action of key force and key force counter force at node I, W I For the key force weighting coefficient at node I, the element in the matrix is +.>
6. The method for analyzing structural deformation and damage based on indirect force bond finite element method according to any one of claims 1 to 5, wherein the calculation formula of the force bond breaking weight function μ (ζ, t) is:
7. the method for analyzing structural deformation and damage based on indirect force key finite element method according to claim 6, wherein in the step 2, force keys are usedElongation of +.>
8. The method for analyzing structural deformation and damage based on indirect force key finite element method according to claim 7, wherein in the step 3, the force key is determinedThe method for breaking is as follows: if the elongation of the force key +.>Greater than a set critical elongation l c And judging that the force bond is broken, otherwise, judging that the force bond is not broken.
9. The method for analyzing structural deformation and damage based on indirect force bond finite element method according to claim 8, wherein the first N in the elongation sequence Y in step 2 is taken every time in order to increase the calculation speed m Judging the broken key of the item, and judging the broken keyAnd (5) judging that the force bonds are broken, uniformly setting the force bond breaking weight functions of the force bonds to 0, and continuing the calculation of the subsequent steps.
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