A Micropotential Peridynamics Method for Structural Deformation Analysis

Abstract

The present invention discloses a micro-potential peridynamics method for structural deformation analysis, and the method comprises the following steps: establishing a solid model and determining corresponding material properties; discretizing the structure into a series of material points in a spatial domain; using a second-order deformation measure of bond length change to measure the deformation of the material points; adopting a micro-bond potential energy function between the discretized material points of the structure; introducing the micro-bond potential energy function between the material points, and integrating the obtained micro-bond potential energy function within the near-field range of the material points to obtain the non-local elastic strain energy density of the material points; deriving a bond force model between the material points through the variation of the non-local elastic strain energy density; setting calculation initial conditions and applying force or displacement boundary conditions; submitting static and dynamic calculations; and performing strain analysis based on the deformation results. By introducing the micro-bond potential energy function of the material points, the present invention derives a brand-new bond force model of the material points. The bond force model does not contain any unknowns, does not require simplifying the model to check and calibrate the bond parameters, and simplifies the steps.

Description

Technical Field

[0001] The present invention belongs to the technical field of peridynamics constitutive derivation and numerical simulation, and particularly relates to a micro-potential peridynamics method for structural deformation analysis. Background Technique

[0002] Peridynamics (PD for short) is a multi-scale mechanics method developed recently, which can effectively solve discontinuous problems in macro and micro mechanics fields. The concept of near field comes from the action of long-range forces. Different from the local theory, i.e., the continuous medium mechanics theory, interactions occur only between materially adjacent points in geometric space; the non-local theory is like molecular dynamics, where interactions can occur between materially adjacent points within a certain distance range without the need to be in contact in geometric space, that is, there are long-range forces between materially adjacent points. Currently, there are two theoretical branches - the peridynamics bond theory and the peridynamics state theory.

[0003] The peridynamics state theory was proposed in 2007. Currently, there are few related studies on deriving the bond force by establishing the potential function between materially adjacent points for conventional material models. Jincheng Fan introduced the Xu-Needleman potential function in "A micro-potential based Peridynamic method for deformation and fracturing in solids: A two-dimensional formulation." ([J]. Computer Methods in Applied Mechanics and Engineering 360(2020):112751.), but the determination of the bond parameters in this model needs to be obtained through simplification.

[0004] The peridynamic discontinuous Galerkin finite element method for structural deformation analysis in the Chinese Patent Database (application number: 201910679765.5, publication number: CN 110457790A), the method comprises the following steps: Step 1, establish a solid model, determine each material region and assign corresponding material properties; Step 2, divide the element mesh by using the mesh generation algorithm of the standard finite element method; Step 3, adopt the node replication algorithm to generate discontinuous elements of a new method of generating the discontinuous Galerkin finite element matching peridynamics; Step 4, calculate at the initial moment and update the coordinates of element nodes and Gauss points in real time in subsequent calculation steps; Step 5, introduce a weight function to construct the weak form equation of the bond-type peridynamics, introduce the isoparametric element interpolation shape function to represent the variable at any point by using the node variable value, calculate the equivalent load of the element nodes according to the non-local action of other Gauss points within the near field range of the Gauss point and the relative displacement between Gauss points, calculate the equivalent external load of the node of the stress boundary and body force, calculate the node mass matrix, and give the calculation method of the force and deformation of the whole configuration; Step 6, set the initial conditions, and apply the body force, stress and displacement boundary conditions by using the Lagrange multiplier method; Step 7, submit the static and dynamic calculations; Step 8, carry out stress and strain analysis from the deformation results.

[0005] Comparing with the above patent, there are still the following deficiencies: improving the numerical solution of peridynamics by the discontinuous Galerkin static and dynamic finite element method, but the operation steps of the model parameters established by the discontinuous Galerkin static and dynamic finite element method are cumbersome and limited to the original material deformation constitutive model. Summary of the Invention

[0006] The present invention provides a micropotential peridynamics method that does not require simplification of model parameters for the above deficiencies.

[0007] The object of the present invention is achieved as follows: A micropotential peridynamics method for structural deformation analysis, characterized in that: the method comprises the following steps:

[0008] Step 1: Establish a solid model and determine the corresponding material properties;

[0009] Step 2: Discretize the structure into a series of material points in the spatial domain;

[0010] Step 3: Use the second-order deformation measure of bond length change to measure the deformation of material points;

[0011] Step 4: Adopt the microbond potential energy function between the discretized material points of the structure;

[0012] Step 5: Introduce the microbond potential energy function between material points, and obtain the non-local elastic strain energy density of the material point by integrating the obtained microbond potential energy function within the near field range of the material point;

[0013] Step 6: Derive the bond force model between material points through the variation of non-local elastic strain energy density;

[0014] Step 7: Set the calculation initial conditions and apply force or displacement boundary conditions;

[0015] Step 8: Submit the static and dynamic calculations;

[0016] Step 9: Conduct strain analysis based on the deformation results.

[0017] Preferably, in step 3, the deformation of the material point is measured by the second-order deformation measure of the bond length. The second-order deformation measure is:

[0018] Δ = |Y<ξ>| 2 - |ξ| 2 ,

[0019] where ξ is the length of the bond in the reference configuration, Y<ξ> is the deformation state of the bond in the current configuration. The deformation state is related to the undeformed bond length and follows the Cauchy-Born rule Y<ξ> = Fξ, where F is the deformation gradient;

[0020] The second-order deformation measure of the material point bond length can be decomposed into Δ = Δ v + Δ d , and Δ d = 2e ij ξ i ξ j

[0021] where the subscripts i and j represent the material point numbers, and

[0022] Preferably, in step 4, a micro-bond potential energy function is adopted between the discretized material points of the structure. The micro-bond potential energy function is:

[0023]

[0024] where Δ v represents the volume deformation, Δ d represents the tangential deformation, ω1<ξ> and ω2<ξ> represent the influence functions, and ω3<ξ> = ω1<ξ> + ω2<ξ>.

[0025] Preferably, in step 5, the micro-bond potential energy function between material points is introduced, and the non-local elastic strain energy density of the material point is obtained by integrating the micro-bond potential energy function in the near-field range of the material point. The non-local elastic strain energy density is:

[0026] where H represents the near-field of the material point, and V ξ represents the volume of the material point.

[0027] Preferably, in step 6, the bond force model between material points is derived through the variation of the non-local elastic strain energy density, and the bond force model is as follows:

[0028] δW = ∫ H [(2ω1<ξ>Δ v + ω3<ξ>Δ d )δΔ v +(2ω2<ξ>Δ d + ω3<ξ>Δ v )δΔ d dV ξ , and the bond force model of the microbond potential function can be obtained by exchanging the integration variables

[0029] where F1 = (ω1<β> - ω2<β>)Δ, F2 = ω2<β>Δ + ω1<β>Δ v + ω2<β>Δ d , A is the area of the material point before deformation, S β is the area of the material point after deformation, and the influence function is specifically where v is the Poisson's ratio of the material, and δ is the near-field radius scale of the material point.

[0030] Preferably, in step 8, the static and dynamic calculations are submitted by updating the particle displacement, and the updated particle displacement is:

[0031]

[0032]

[0033] where n represents the number of calculation steps, n = 0 represents the initial moment, c is the damping coefficient b is the body force, Λ is the diagonal density matrix, and the value is ξ min is the minimum bond length between material points, V δ is the volume of the near-field range of the material point, α ≥ 1 is the safety factor, a = 2 is taken during calculation, and the equivalent stiffness matrix K is obtained by taking the partial derivative of the mutual force between material points with respect to the relative displacement:

[0034]

[0035] where, V j is the volume of the particle numbered j, and the displacement of the updated particle is

[0036] The beneficial effects of the present invention are as follows: 1. By introducing the microbond potential function of the material point, a new bond force model of the material point is derived. The bond force model does not contain any unknowns, and there is no need to simplify the model to proofread the core bond parameters, thus simplifying the steps.

[0037] 2. Static and dynamic calculations update the particle velocity, define the updated particle velocity, calculate the deformation result, further simplify the steps of static and dynamic calculations, reduce the amount of calculation, and improve technical accuracy and efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a flowchart of the method of the present invention.

[0039] Figure 2 It is a horizontal direction strain nephogram of an open-hole flat plate.

[0040] Figure 3 It is a vertical direction strain nephogram of an open-hole flat plate. DETAILED DESCRIPTION OF THE INVENTION

[0041] The following further generalizes the present invention with reference to the accompanying drawings.

[0042] Example:

[0043] As Figure 1 shown, a calculation model is established

[0044] In this example, it is a rectangular plate with a central hole, uniformly loaded at both ends. The upper and lower boundaries of the rectangular plate adopt free boundaries, and uniform tensile loads of 1 MPa act on the left and right boundaries of the model respectively.

[0045] Step 1: Establish a solid model. The model uses a rectangular plate with a central hole. The size of the rectangular plate is 100 mm × 50 mm × 10 mm, the radius of the hole is 10 mm, the elastic modulus is 100 GPa, the Poisson's ratio is 0.4, and the material density is 3 2000 kg / m

[0046] Step 2: Discretize the structure of the rectangular plate into a series of material points in the spatial domain;

[0047] Implement model discretization

[0048] In this example, the size of the discrete material points of the model is dx = 0.025 cm, and the near-field radius δ = 4 × dx. The material is uniformly discretized. There are three layers of particles at the boundary where the load is applied. The numerical implementation adopts an explicit time integration scheme of the dynamic relaxation method, and the time step increment is taken as 1.

[0049] Initialize the information of the material points

[0050] Initialize the physical quantities of the discrete material points of the model in this example, including displacement, velocity, acceleration, and force; apply boundary conditions

[0051] In this example, the boundary condition is the uniform tensile load of 1 MPa acting on the left and right ends of the model.

[0052] Step 3: The deformation of the material point bond adopts the second-order deformation measure of the bond length Δ = |Y<ξ>| 2 -|ξ| 2 , where ξ is the bond length in the reference configuration, Y<ξ> is the deformation state of the bond in the current configuration, and the deformation state is related to the undeformed bond length, following the Cauchy-Born rule Y<ξ> = Fξ, where F is the deformation gradient;

[0053] The second-order deformation measure of the material point bond length can be decomposed into Δ = Δ v +Δ d , and Δ d = 2e ij ξ i ξ j

[0054] where the subscripts i and j represent the material point numbers, and

[0055] where dSξ = ξdξdα, α is the angle between material points, δ is the size of the near-field radius.

[0056] Step 4: Introduce the micro-bond potential function between material points

[0057] where, Δ v represents the volume deformation, Δ d represents the tangential deformation, ω1<ξ> and ω2〈ξ〉 represent the influence functions, and ω3〈ξ〉 = ω1<ξ> + ω2<ξ>;

[0058] Step 5: Introduce the micro-bond potential energy function, and the non-local elastic strain energy density of the material point is obtained by integrating the micro-bond potential energy function in the near-field range of the material point.

[0059] Calculate the bond force of the material point

[0060] Step 6: Derive the bond force model between material points through the variation of the non-local elastic strain energy density;

[0061] The bond force of the model in this embodiment is

[0062] Through δW = ∫ H [(2ω1〈ξ)Δ v +ω3〈ξ)Δ d )δΔ v +(2ω2〈ξ>Δ d +ω3〈ξ)Δ v )δΔ d dV ξ, the bond force model of the microbond potential function can be obtained by exchanging the integration variables

[0063] where F1 = (ω1<β> - ω2<β>)Δ, F2 = ω2<β>Δ + ω1<β>Δ v + ω2<β>Δ d , A is the area of the material point before deformation, and S β is the area of the material point after deformation. The influence function is specifically where v is the Poisson's ratio of the material, and δ is the near-field radius scale of the material point.

[0064] Update the material point information

[0065] Update the physical quantities of the discrete material points in the model of this embodiment, including acceleration, velocity, and displacement.

[0066] Step 7: Set the calculation initial conditions and apply force or displacement boundary conditions;

[0067] Output the material point information

[0068] Output the position, velocity, displacement, and strain of the material points in the model of this embodiment.

[0069] Step 8: Submit the static and dynamic calculations

[0070] Update the particle displacement to where n represents the number of calculation steps, n = 0 represents the initial moment, c is the damping coefficient b is the body force, Λ is the diagonal density matrix, and its value is ξ min is the minimum bond length between material points, V δ is the volume of the near-field range of the material point, α ≥ 1 is the safety factor, take a = 2, and the equivalent stiffness matrix K is obtained by taking the partial derivative of the mutual force between material points with respect to the relative displacement: V j is the volume of the particle numbered j, and update the displacement of the particle to

[0071] Step 9: Conduct strain analysis based on the deformation results

[0072] Judge whether the calculation time step reaches the set value

[0073] If the set time step is reached, terminate the calculation; otherwise, repeat the steps of applying boundary conditions, calculating the bond force of the material points, updating the material point information, outputting the material point information, and judging whether the calculation time step reaches the set value.

[0074] From Figure 2 and 3It can be seen that in this embodiment, stress concentration phenomena occur at the upper and lower edges of the circular hole as the model is stretched, and positive and negative pressure zones alternate along the opening, which is consistent with the observed phenomena in reality.

[0075] Working principle: First, the structure is discretized into a series of material points in the spatial domain, and corresponding material properties are assigned. The deformation of the material point bonds uses the second-order deformation measure of bond length change. A microbond potential function is used between the discretized material points of the structure. The non-local elastic strain energy density of the material points is obtained by integrating the microbond potential energy function within its near-field range. The bond force model between the material points is derived through the variation of the non-local elastic strain energy density. Set the initial calculation conditions, apply force or displacement boundary conditions, and submit the static and dynamic calculations to obtain the deformation and strain analysis of the structure.

[0076] The above are only the embodiments of the present invention and are not used to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the scope of the claims of the present invention.

Claims

1. A micropotential peridynamics method for structural deformation analysis, characterized in that: The method includes the following steps: Step 1: Establish an entity model and determine the corresponding material properties; Step 2: Discretize the structure into a series of material points in the spatial domain; Step 3: Use the second-order deformation measure of bond length change to measure the deformation of material points; Step 4: Adopt the microbond potential energy function between the discretized material points of the structure; The microbond potential energy function is: ； Among them, represents volumetric deformation, represents tangential deformation, and represents the influence function, ; Step 5: Introduce the microbond potential energy function between material points, and obtain the non-local elastic strain energy density of the material point by integrating the obtained microbond potential energy function in the near-field range of the material point; The non-local elastic strain energy density is as follows: ; Among them, represents the near field of the material point, represents the volume of the material point; Step 6: Derive the bond force model between material points through the variation of the non-local elastic strain energy density; Step 7: Set the calculation initial conditions and apply force or displacement boundary conditions; Step 8: Submit the static and dynamic calculations; Update the particle velocity to: ， ； ， ； where n represents the number of calculation steps, n = 0 represents the initial moment, c is the damping coefficient, ; b is the body force, is the diagonal density matrix, and its value is ; is the minimum bond length between material points, is the volume of the near-field range of material points, is the safety factor, taking , and the equivalent stiffness matrix is obtained by taking the partial derivative of the relative displacement with respect to the mutual force between material points: ； Among them, is the volume of the particle with particle number j, and the displacement of the updated particle is ; Step 9: Conduct strain analysis based on the deformation results.

2. The micropotential peridynamics method according to claim 1, wherein: In Step 3, the second-order deformation measure of bond length change is used to measure the deformation of material points, and the second-order deformation measure is: ； Among them, is the length of the bond in the reference configuration, is the deformed state of the bond in the current configuration. The deformed state is related to the undeformed bond length and follows the Cauchy-Born rule , where is the deformation gradient; The second-order deformation measure of the particle bond length is decomposed into , ; Among them, the subscript and represent the material point numbers, and .

3. The micropotential peridynamics method according to claim 1, wherein: In Step 6, the bond force model between material points is derived through the variation of the non-local elastic strain energy density, and the bond force model is: ； The bond force model of the microbond potential function can be obtained by exchanging the integration variables ; Among them, , , is the area of the material point before deformation, is the area of the material point after deformation. The influence function is specifically , where, is the Poisson's ratio of the material, is the near-field radius scale of the material point.

Images