Implantation method of large deformation plastic element based on FDEM in non-rotating reference frame
By introducing stress tensor calculation and rotation mapping in a non-rotating reference frame into the finite discrete element method, and combining it with the yield function, the problem of the inability to simulate material plasticity and large deformation in the existing technology is solved, and accurate and stable simulation of complex elastoplastic constitutive models is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- INST OF ROCK & SOIL MECHANICS CHINESE ACAD OF SCI
- Filing Date
- 2022-06-30
- Publication Date
- 2026-05-19
AI Technical Summary
Existing finite discrete element methods cannot effectively simulate the plasticity and large deformation behavior of materials when simulating dynamic problems such as high temperature, high pressure, high speed impact and explosion, and lead to overestimation of corresponding forces, and cannot incorporate complex elastoplastic constitutive relations.
By introducing stress tensor calculation in a non-rotating reference frame, and combining rotational mapping and yield function with different flow rules, we can achieve accurate simulation of plastic behavior. We can also use the properties of triangular elements to simulate plasticity, which is applicable to complex elastoplastic constitutive relations.
It achieves accurate simulation of plastic behavior, is applicable to two-dimensional and three-dimensional finite discrete element methods, can simulate complex elastoplastic constitutive models, has stable algorithm and strong portability, and the results are consistent with analytical solutions. It is suitable for simulation of various complex materials.
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Figure CN115828652B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of large deformation plasticity calculation technology of materials, and more specifically, relates to a method for implanting large deformation plasticity elements based on FDEM in a non-rotating reference frame. Background Technology
[0002] The Finite Discrete Element Method (FDEM) is a relatively new numerical method that can simulate continuous and discontinuous processes. Its basic principle is to insert a four-node cohesive element with no thickness between two adjacent triangular finite elements, and simulate cracks by deforming the four-node cohesive element until it eventually fractures.
[0003] However, the triangular elements in existing finite element methods are linear triangular finite element elements. While they are acceptable for simulating static mechanics, they cannot accurately simulate the plasticity and large deformation behavior of materials during dynamic problems such as high-temperature, high-pressure, and high-speed impacts and explosions, and may lead to overestimation of corresponding forces. For simulating the plasticity of materials, only a few researchers have incorporated simple and unverified elastoplastic constitutive models into four-node cohesive elements, failing to incorporate complex and mature elastoplastic constitutive relations. Summary of the Invention
[0004] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame. By introducing the calculation of stress tensors in a non-rotating reference frame and then rotating the corresponding stress tensors to map them onto the original reference frame, and in conjunction with yield functions and different flow rules, accurate simulation of plastic behavior in wired-discrete element methods can be achieved.
[0005] To achieve the above objectives, according to one aspect of the present invention, a method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame is provided, comprising the following steps:
[0006] S100 obtains the deformation rate {D} and rotation matrix {R} in the rotation configuration based on the nodal coordinates of the triangular element;
[0007] S200 obtains the deformation rate {d} in the non-rotational configuration based on the deformation rate {D} in the rotational configuration and the rotation matrix {R}.
[0008] S300 determines the elastic test stress based on the deformation rate {d} under the non-rotational configuration, and determines the material state by combining the yield function, thereby obtaining the true stress under the non-rotational configuration. Based on the true stress, a rotational mapping of the original reference configuration is performed to obtain the Cauchy stress under the rotational configuration, which is the true Cauchy stress.
[0009] S400 calculates the contact force and the force caused by the cohesive elements based on FDEM, and then converts them into nodal forces. It then uses Newton's second law to update the nodal velocity and position within the current time step before moving to the next time step.
[0010] Furthermore, S100 specifically includes:
[0011] S101 calculates the deformation gradient matrix based on the existing element node coordinate information:
[0012]
[0013] Where x is the node position in the deformable frame, and X is the node position in the reference configuration;
[0014] Using the polar coordinate decomposition theorem for {F}, S102 can decompose {F} into:
[0015] F = VR = RU (16)
[0016] Where {V} and {U} are left-right symmetrical positive stretching tensors, and {R} represents the rotation tensor; on the other hand, the velocity tensor {L} is expressed as:
[0017]
[0018] in Let {F} be the time derivative. -1} represents the inverse matrix of {F}, and the tensor {L} can be decomposed into a symmetric part {D} and an antisymmetric part {W}:
[0019] L = D + W (18)
[0020] {D} and {W} represent the deformation rate matrix and rotation matrix, respectively.
[0021] Further, S200 includes:
[0022] The velocity gradient matrix {L} of S201 can be expressed in the following form:
[0023]
[0024] Here, {Ω} represents the first term. {Ω} is a symmetric matrix. R is the time derivative of the rotation tensor. T Let be the transpose of the rotation tensor. U is the time derivative of the stretch tensor. -1 Given the inverse of the stretch tensor, the deformation gradient matrix {d} under the non-rotational architecture is:
[0025]
[0026] Where {D} is the symmetric part of the velocity tensor {L}, U is the time derivative of the stretch tensor. -1 R is the inverse matrix of the stretch tensor. T Let R be the transpose of the rotation tensor;
[0027] S202 In order to calculate {d}, based on the intrinsic orthogonal rotation tensor describing the rigid body rotation, calculate the {R} tensor at the current time t:
[0028]
[0029] Wherein, the subscripts t and t-Δt indicate that the time step for calculating the rotation tensor R is the current time step and the previous time step, and Δt is the time increment of a single step;
[0030] By substituting into the formula above, we can obtain:
[0031] ω=w+[trace(V)IV] -1 z (22)
[0032] Where {W} is the rotation matrix under the rotation framework, trace(·) is the trace notation in tensor computation, {V} is the left-symmetric positive stretch tensor, and z is the intermediate computation vector. Each component, i = 0, 1, 2, has the following expression:
[0033] (z) i =e ikj (D) jm (V) mk (twenty three)
[0034] Subscripts i, ikj, jm, and mk are notations used for tensor calculations, {D} is the deformation rate matrix, {V} is the left-symmetric positive tensor, and e ijk It is a third-order Levi-Civit tensor;
[0035] The antisymmetric part of a second-order tensor can be represented by a first-order tensor, i.e., a vector and a third-order tensor:
[0036] (Ω) ij =e ijk (ω) k (twenty four)
[0037] (W) ij =e ijk (w) k (25)
[0038] Where ij is the subscript of the second-order tensor, k is the subscript of the first-order tensor (i.e., the vector), ijk is the subscript of the third-order tensor, {Ω} and {W} are both symmetric matrices, e ijk It is a third-order Levi-Civit tensor;
[0039] Combining the above formulas, we can obtain {d} and {R}, which are the deformation rate and rotation matrix in the non-rotational configuration.
[0040] Furthermore, the determination of the elastic test stress based on the deformation rate {d} under the non-rotational configuration in S300 specifically includes:
[0041] S301 obtains the strain increment in the current time step based on the deformation rate {d} in the non-rotational configuration:
[0042] Δε=dΔt (26)
[0043] Where Δt is the computation time step, and {d} is the deformation gradient matrix under the non-rotational architecture;
[0044] S302 obtains the test stress at this time step based on strain increment. The test stress of the previous time step:
[0045]
[0046] Where the superscript t represents the current time step, t-Δt represents the previous time step, trace(·) is the trace symbol in tensor calculation, I is the unit tensor, λ and μ are the Lamé constants of the material;
[0047] Furthermore, the yield function is obtained by obtaining the equivalent stress based on the test stress, and then combining the equivalent stress with the material yield strength to obtain the yield function.
[0048] Furthermore, the determination of the material state in S300 by combining the yield function specifically includes:
[0049] If the yield function is less than 0, it proves that the material has not yet entered the yielding stage, that is, it is in the elastic stage. At this time, the calculated test stress is the true stress. If the yield function is less than 0, it indicates that the material has entered the plastic flow stage. Plastic flow calculations are performed according to the plastic flow rules to obtain...
[0050] Furthermore, the Cauchy stress in the rotated configuration obtained by performing a rotational mapping of the original reference configuration based on the actual stress in S300 includes:
[0051] Will The Cauchy stress under the rotated configuration is obtained by mapping the original reference configuration using the following formula:
[0052]
[0053] Among them, R t Let R be the {R} tensor at time t, σ be the rotation matrix in the non-rotational configuration of {R}, and R be the stress. T It is the transpose of the rotation tensor.
[0054] According to a second aspect of the present invention, a material plasticity determination apparatus based on FDEM in a non-rotating reference frame is provided, comprising:
[0055] The rotation configuration calculation module is used to obtain the deformation rate and rotation matrix in the rotation configuration based on the nodal coordinates of the triangular elements.
[0056] The non-rotational configuration calculation module is used to obtain the deformation rate in the non-rotational configuration based on the deformation rate in the rotational configuration and the rotation matrix.
[0057] The plasticity calculation module is used to determine the elastic test stress based on the deformation rate under the non-rotational configuration, and to determine the material state by combining the yield function, so as to obtain the true stress under the non-rotational configuration. Based on the true stress, the original reference configuration is rotated to obtain the Cauchy stress under the rotational configuration, which is the true Cauchy stress.
[0058] The material state update module calculates the contact force and the force caused by the cohesive elements based on FDEM calculations, and then converts them into nodal forces. It then uses Newton's second law to update the nodal velocity and position within the current time step before moving to the next time step.
[0059] According to a third aspect of the present invention, an electronic device is provided, comprising: at least one central processing unit (CPU); and at least one memory communicatively connected to the CPU, wherein: the memory stores program instructions executable by the CPU, and the CPU invokes the program instructions to execute the method described herein.
[0060] According to a fourth aspect of the present invention, a non-transitory computer-readable storage medium is provided, characterized in that the non-transitory computer-readable storage medium stores computer instructions that cause a computer to perform the method described thereon.
[0061] In summary, compared with the prior art, the above-described technical solutions conceived by this invention can achieve the following beneficial effects:
[0062] 1. The large deformation plastic element implantation method of the present invention, by introducing the calculation of stress tensor under a non-rotating reference frame, and then rotating the corresponding stress tensor to map it to the original reference frame, can realize the simulation of plastic behavior in wired-discrete element method by combining yield function and different flow rules.
[0063] 2. The large deformation plastic element implantation method of the present invention utilizes the properties of triangular elements rather than the properties of cohesive elements to simulate plasticity. Triangular elements can introduce various complex elastoplastic constitutive models, while cohesive elements cannot introduce complex elastoplastic constitutive models (such as Mohr-Coulomb, Drucker-Plag constitutive models, etc.). In the traditional finite element field, plasticity is characterized by triangular elements.
[0064] 3. The large deformation plastic element implantation method of the present invention is applicable to various complex elastoplastic constitutive relations, including the well-known Mises, Mohr-Coulomb, Drucker-Plag, JH2, and HJC constitutive relations. Switching between different constitutive relations can be achieved simply by changing the yield function and the plastic flow law. The algorithm is highly portable and very stable.
[0065] 4. The large deformation plastic element implantation method of the present invention is applicable to both two-dimensional finite discrete element method (based on triangular elements and four-node thicknessless cohesive elements) and three-dimensional finite discrete element method (based on tetrahedral elements and six-node thicknessless cohesive elements).
[0066] 5. The method for implanting large deformation plastic units of the present invention, taking the famous Mohr-Coulomb benchmark consolidation experiment as an example, can obtain results consistent with the analytical solution, proving the effectiveness and accuracy of the method. Attached Figure Description
[0067] Figure 1 This is a flowchart of the computational process for the finite-discrete element large deformation architecture of this invention.
[0068] Figure 2 This is a flowchart of the implantation method of large deformation plastic elements based on FDEM in a non-rotating reference frame according to the present invention;
[0069] Figure 3 This is a flowchart illustrating the process of obtaining the yield function based on the deformation rate {d} in a non-rotational configuration according to the present invention.
[0070] Figure 4 A schematic diagram illustrating the model and boundary conditions of the Taylor Bar problem;
[0071] Figure 5 Figure 1 shows the results of the Taylor bar simulation experiment. Figure 5 (a) shows the original FDEM simulation results; Figure 5 (b) shows the FDEM simulation results using a large deformation frame; Figure 5 (c) shows the experimental results, from the literature Mechanics of Taylor impact testing of polycarbonate;
[0072] Figure 6A detailed schematic diagram of the Taylor bar simulation results ( Figure 6 (a) shows the original FDEM simulation results; Figure 6 (b) shows the FDEM simulation results using a large deformation framework; Figure 6 (c) shows a detailed diagram of the experimental results.
[0073] Figure 7 This is a schematic diagram of the Oedometer experiment in the Mohr-Coulomb. Detailed Implementation
[0074] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.
[0075] This invention provides a method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame, comprising the following steps:
[0076] S100 uses the nodal coordinates of triangular elements and the stress tensor in a non-rotating reference frame to obtain the deformation rate {d} and rotation matrix {R} in the non-rotating configuration.
[0077] S200 obtains the yield function based on the deformation rate {d} in the non-rotational configuration;
[0078] S300 determines the material state based on the yield function, obtains the true stress, and then maps the original reference configuration based on the true stress to obtain the true Cauchy stress.
[0079] S400 calculates the contact force and the force caused by the cohesive elements based on FDEM, and then converts them into nodal forces. It then uses Newton's second law to update the nodal velocity and position within the current time step before moving to the next time step.
[0080] Specifically, the node coordinates of the triangular element are three-dimensional coordinates {x, y, z}, but can also be used in two dimensions. Two dimensions can be viewed as a special case where z is always equal to 0 {x, y, 0}. For a stress matrix (three-dimensional), assuming the numbers in the matrix represent the corresponding components: the three-dimensional matrix composition is as follows: There are 9 components from 0 to 8, but only the top four terms in the two-dimensional representation are taken. The four items are 0, 1, 3, and 4.
[0081] Specifically, S100 includes the following process:
[0082] S101 calculates the deformation gradient matrix based on the existing element node coordinate information:
[0083]
[0084] Where x is the node position in the deformable frame, X is the node position in the reference configuration, and det(F) is the determinant of matrix F;
[0085] Using the polar coordinate decomposition theorem for {F}, S102 can decompose {F} into:
[0086] F = VR = RU (30)
[0087] Where {V} and {U} are left-right symmetrical positive stretching tensors, and {R} represents the rotation tensor. On the other hand, the velocity tensor {L} is expressed as:
[0088]
[0089] in Let {F} be the time derivative. -1} represents the inverse matrix of {F}, and the tensor {L} can be decomposed into a symmetric part {D} and an antisymmetric part {W}:
[0090] L=D+W (32)
[0091] {D} and {W} represent the deformation rate matrix and rotation matrix, respectively.
[0092] Specifically, S200 includes the following process:
[0093] The velocity gradient matrix {L} of S201 can be expressed in the following form:
[0094]
[0095] Here, {Ω} represents the first term. Both {Ω} and {W} are symmetric matrices. R is the time derivative of the rotation tensor. T Let be the transpose of the rotation tensor. U is the time derivative of the stretch tensor. -1 Let be the inverse matrix of the stretch tensor. The deformation gradient matrix {d} under the non-rotational architecture:
[0096]
[0097] S202 In order to calculate {d}, based on the intrinsic orthogonal rotation tensor describing the rigid body rotation, calculate the {R} tensor at the current time t:
[0098]
[0099] Where the subscripts t and t-Δt indicate that the time step for calculating the rotation tensor R is the current time step and the previous time step, Δt is the time increment of a single step, {Ω} is a symmetric matrix, and I is a unit tensor;
[0100] By substituting into the formula above, we can obtain:
[0101] ω=w+[trace(V)IV] -1 z (36)
[0102] Where {W} is the rotation matrix under the rotation framework, trace(·) is the trace notation in tensor computation, {V} is the left-symmetric positive stretch tensor, and z is the intermediate computation vector, each of which (i = 0, 1, 2) has the following expression:
[0103] (z) i =e ikj (D) jm (V) mk (37) The subscripts i, ikj, jm, and mk are all notations for tensor computation, e ijk It is a third-order Levi-Civit tensor.
[0104] When expression (5) in S201 is related to the third-order identity tensor e ikj When contracted, its symmetric part disappears, resulting in a system of three linear equations for the three independent components of the Ω matrix. The antisymmetric part of the second-order tensor (subscript ij) can be represented by a first-order tensor (i.e., a vector, subscript k) and a third-order tensor (subscript ijk). The following two vectors are defined as right-hand rotation rates:
[0105] (Ω) ij =e ijk (ω) k (38)
[0106] (W) ij =e ijk (w) k (39)
[0107] Combining the above formulas, we can obtain {d} and {R}, which are the deformation rate and rotation matrix in the non-rotational configuration.
[0108] Specifically, determining the elastic test stress based on the deformation rate {d} under the non-rotational configuration in S300 includes:
[0109] S301 obtains the strain increment in the current time step based on the deformation rate {d} in the non-rotational configuration:
[0110] Δε=dΔt (40)
[0111] Where Δt is the computation time step, and {d} is the deformation gradient matrix under the non-rotational architecture;
[0112] S302 obtains the test stress at this time step based on strain increment. The test stress of the previous time step:
[0113]
[0114] Where the superscript t represents the current time step, t-Δt represents the previous time step, trace(·) is the trace symbol in tensor calculation, I is the unit tensor, λ and μ are the Lamé constants of the material;
[0115] The yield function is obtained by: obtaining the equivalent stress based on the test stress, and combining the equivalent stress with the material yield strength to obtain the yield function.
[0116] Specifically, determining the material state in S300 by incorporating the yield function includes:
[0117] If the yield function is less than 0, it proves that the material has not yet entered the yielding stage, that is, it is in the elastic stage. At this time, the calculated test stress is the true stress. If the yield function is less than 0, it indicates that the material has entered the plastic flow stage. Plastic flow calculations are performed according to the plastic flow rules to obtain...
[0118] The Cauchy stress in the rotated configuration obtained by performing a rotational mapping of the original reference configuration based on the actual stress in S300 includes:
[0119] Will The Cauchy stress under the rotated configuration is obtained by mapping the original reference configuration using the following formula:
[0120]
[0121] Among them, R t Let R be the {R} tensor at time t, σ be the rotation matrix in the non-rotational configuration of {R}, and R be the stress. T It is the transpose of the rotation tensor.
[0122] The Cauchy stress under the rotational configuration is the true Cauchy stress.
[0123] Optionally, the method of the present invention can switch between different complex elastoplastic constitutive relations by changing the yield function and the plastic flow law. The complex elastoplastic constitutive relations include Mises, Moercoulomb, Drucker-Plag, JH2, and HJC constitutive relations.
[0124] To verify the effectiveness and accuracy of this method, the following experiments were conducted:
[0125] Example 1:
[0126] Large deformation Taylor bar impact test:
[0127] The Taylor bar impact problem is a well-known benchmark test in the study of plasticity. This Taylor bar uses a large deformation structure and employs the Von Mises elastoplastic model. Its properties are shown in Table 1. Figure 4 The model and boundary conditions of the Taylor Bar problem are described. Figure 5 and Figure 6 The comparisons between the original FDEM and the FDEM with a large deformation framework and existing experiments are shown respectively. It can be seen that the Taylor bar of the original FDEM cannot produce large deformation due to the small deformation assumption. The FDEM with a large deformation framework (the method of this invention) can capture the plastic properties of the material very well.
[0128] Table 1. Properties of Taylor bar impact test
[0129]
[0130] Example 2:
[0131] Oedometer experiment on the Mohr-Coulomb elastoplastic model:
[0132] Taking the Mohr-Coulomb elastoplastic constitutive model as an example, the famous Oedometer experiment is conducted, and a model with a length and width of 1m is constructed, as follows: Figure 7 As shown in (a) and (b), the model parameters use the parameters listed in Table 2. Figure 7 In this context, (a) represents the model dimensions and boundary conditions. Figure 7 In the diagram, (b) represents the node number and grid information. Figure 7 (c) in the diagram shows the final y-direction displacement field of the entire process. Figure 7 Figure (d) in the figure shows a comparison of the y-direction stress-displacement relationship between the results obtained using the method of this application through FDEM simulation and those obtained in FLAC 3D.
[0133] Combination Figure 7 As shown in (a) and (b), when a negative y-axis velocity is applied to nodes 2 and 3 at 1e-6 m / h steps, and the entire process lasts for 1000 steps, the final y-axis displacement of nodes 2 and 3 is -0.01 m, which is... Figure 7 As shown in (c) in the figure.
[0134] Figure 7(d) in the figure shows that the method provided in this application can obtain the same results in FDEM as the Oedometer experiment in FLAC3D.
[0135] Table 2 Oedometer experimental parameters
[0136]
[0137]
[0138] The above-described implementation of this invention is achieved through programmed processing using a device with a central processing unit (CPU). Therefore, in practical engineering, the technical solutions and functions of the various embodiments of this invention can be encapsulated into various modules. Based on this reality, and building upon the above embodiments, this invention provides a material plasticity determination device based on a non-rotating reference frame FDEM. This device is used to execute the large deformation plasticity element implantation method based on a non-rotating reference frame FDEM in the above method embodiments. It includes:
[0139] The deformation rate and rotation matrix determination module is used to obtain the deformation rate {d} and rotation matrix {R} in the non-rotation configuration based on the nodal coordinates of the triangular element and the stress tensor in the non-rotation reference frame.
[0140] The yield function determination module is used to obtain the yield function based on the deformation rate {d} in the non-rotational configuration;
[0141] The module for determining the true Cauchy stress is used to determine the material state based on the yield function, obtain the true stress, and then map the original reference configuration based on the true stress to obtain the true Cauchy stress.
[0142] The material state update module is used to calculate the contact force and the force caused by the cohesive elements based on FDEM, and then convert them into nodal forces. The nodal velocity and position within the current time step are updated using Newton's second law before proceeding to the next time step.
[0143] It should be noted that the apparatus in the device embodiments provided by the present invention can be used not only to implement the methods in the above method embodiments, but also to implement the methods in other method embodiments provided by the present invention. The only difference is that corresponding functional modules are set. The principle is basically the same as that of the above device embodiments provided by the present invention. As long as those skilled in the art can improve the apparatus in the above device embodiments by referring to the specific technical solutions in other method embodiments and combining technical features to obtain corresponding technical means and technical solutions composed of these technical means, on the basis of the above device embodiments, under the premise of ensuring the practicality of the technical solutions, so as to obtain corresponding device-type embodiments for implementing the methods in other method-type embodiments.
[0144] The method in this embodiment of the invention is implemented using an electronic device; therefore, it is necessary to introduce the relevant electronic device. For this purpose, this embodiment of the invention provides an electronic device, such as... Figure 3 As shown, the electronic device includes: at least one central processor, a communications interface, at least one memory, and a communications bus, wherein the at least one central processor, the communications interface, and the at least one memory communicate with each other via the communications bus. The at least one central processor can invoke logical instructions in the at least one memory to execute all or part of the steps of the methods provided in the foregoing method embodiments.
[0145] Furthermore, when the logical instructions in at least one of the aforementioned memories can be implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various method embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.
[0146] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.
[0147] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.
[0148] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. Based on this understanding, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions marked in the blocks may occur in a different order than those shown in the drawings. For example, two consecutive blocks may actually be executed substantially in parallel, or sometimes in reverse order, depending on the functions involved. It should also be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, can be implemented using a dedicated hardware-based system that performs the specified function or action, or using a combination of dedicated hardware and computer instructions.
[0149] In this patent, the terms "comprising," "including," or any other variations thereof are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising..." does not exclude the presence of additional identical elements in the process, method, article, or apparatus that includes said element.
[0150] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame, characterized in that, Includes the following steps: S100 obtains the deformation rate {D} and rotation matrix {R} in the rotation configuration based on the nodal coordinates of the triangular element; Specifically, S100 includes: S101 calculates the deformation gradient matrix based on the existing element node coordinate information: Where x is the node position in the deformable frame, and X is the node position in the reference configuration; Using the polar coordinate decomposition theorem of {F}, S102 can decompose {F} into: Where {V} and {U} are left-right symmetrical positive stretching tensors, and {R} represents the rotation tensor; on the other hand, the velocity tensor {L} is expressed as: in express Time derivative, represent The inverse matrix, the {L} tensor, can be decomposed into a symmetric part {D} and an antisymmetric part {W}: {D} and {W} represent the deformation rate matrix and rotation matrix, respectively; S200 obtains the deformation rate {d} in the non-rotational configuration based on the deformation rate {D} in the rotational configuration and the rotation matrix {R}; S200 includes: The S201 velocity gradient matrix {L}, i.e., the velocity tensor, is expressed in the following form: Here, {Ω} represents the first term. {Ω} is a symmetric matrix. The time derivative of the rotation tensor. Let be the transpose of the rotation tensor. This is the time derivative of the stretch tensor. Let {d} be the inverse matrix of the stretch tensor, and the deformation gradient matrix under the non-rotational architecture, i.e., the deformation rate under the non-rotational configuration: Where {D} is the symmetric part of the velocity tensor {L}, This is the time derivative of the stretch tensor. For the stretch tensor inverse matrix, Let be the transpose of the rotation tensor. It is a rotation tensor; S202 To calculate {d}, based on the intrinsic orthogonal rotation tensor describing the rigid body rotation, the {R} tensor at the current time t is calculated: Where the subscripts t and t-Δt denote the calculation of the rotation tensor. The time step is the current time step and the previous time step, and Δt is the time increment of a single step. S300 determines the elastic test stress based on the deformation rate {d} under the non-rotational configuration, and determines the material state by combining the yield function to obtain the true stress under the non-rotational configuration. Based on the true stress, a rotational mapping of the original reference configuration is performed to obtain the Cauchy stress under the rotational configuration, which is the true Cauchy stress. S400 calculates the contact force and the force caused by the cohesive elements based on FDEM, and then converts them into nodal forces. It then uses Newton's second law to update the nodal velocity and position within the current time step before moving to the next time step.
2. The method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame according to claim 1, characterized in that, Substituting formula (7) into formula (6), we get: Where {W} is the rotation matrix under the rotational architecture, trace(∙) is the trace notation in tensor computation, and {V} is the left-symmetric positive stretch tensor. For the intermediate computation vector, each component i=0,1,2 has the following expression: The subscripts i, ikj, jm, and mk are all notations used for tensor calculations, {D} is the deformation rate matrix, and {V} is the left-symmetric positive tensor. It is a third-order Levi-Civit tensor; The antisymmetric part of a second-order tensor can be represented by a first-order tensor, i.e., a vector and a third-order tensor: Where ij is the subscript of the second-order tensor, k is the subscript of the first-order tensor (i.e., the vector), ijk is the subscript of the third-order tensor, and {Ω} and {W} are both symmetric matrices. It is a third-order Levi-Civit tensor; Combining the above formulas (5) to (11), we can obtain {d} and {R}, that is, the deformation rate and rotation matrix in the non-rotational configuration.
3. The method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame according to claim 1, characterized in that, The determination of the elastic test stress in S300 based on the deformation rate {d} under the non-rotational configuration specifically includes: S301 obtains the strain increment in the current time step based on the deformation rate {d} in the non-rotational configuration: in To calculate the time step, {d} is the deformation gradient matrix under the non-rotational architecture; S302 obtains the test stress at this time step based on strain increment. , The test stress of the previous time step: The superscript 't' refers to the current time step. The symbol refers to the previous time step; trace(∙) is the trace notation in tensor computation; I represents the unit tensor. and Let be the Lamé constant of the material.
4. The method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame according to claim 3, characterized in that, Specifically, the yield function is obtained by obtaining the equivalent stress based on the test stress, and then combining the equivalent stress with the material's yield strength to obtain the yield function.
5. The method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame according to claim 1, characterized in that, The determination of the material state in S300 by combining the yield function specifically includes: If the yield function is less than 0, it proves that the material has not yet entered the yielding stage, that is, it is in the elastic stage. At this time, the calculated test stress is the true stress. If the yield function is less than 0, it indicates that the material has entered the plastic flow stage. Plastic flow calculations are then performed according to the plastic flow rules to obtain... .
6. The method for implanting large deformation plastic elements based on FDEM in a non-rotating reference frame according to claim 1, characterized in that, The Cauchy stress in the rotated configuration obtained by performing a rotational mapping of the original reference configuration based on the actual stress in S300 includes: Will The Cauchy stress under the rotated configuration is obtained by mapping the original reference configuration using the following formula: in, Let {R} be the {R} tensor at time t, and let {R} be the rotation matrix in the non-rotational configuration. For stress, It is the transpose of the rotation tensor.
7. A material plasticity determination apparatus based on FDEM in a non-rotating reference frame, the apparatus operating as described in claim 1, characterized in that, include: The rotation configuration calculation module is used to obtain the deformation rate and rotation matrix in the rotation configuration based on the nodal coordinates of the triangular elements. The non-rotational configuration calculation module is used to obtain the deformation rate in the non-rotational configuration based on the deformation rate in the rotational configuration and the rotation matrix. The plasticity calculation module is used to determine the elastic test stress based on the deformation rate under the non-rotational configuration, and to determine the material state by combining the yield function, so as to obtain the true stress under the non-rotational configuration. Based on the true stress, the original reference configuration is rotated to obtain the Cauchy stress under the rotational configuration, which is the true Cauchy stress. The material state update module calculates the contact force and the force caused by the cohesive elements based on FDEM calculations, and then converts them into nodal forces. It then uses Newton's second law to update the nodal velocity and position within the current time step before moving to the next time step.
8. An electronic device, characterized in that, include: At least one central processing unit; And at least one memory communicatively connected to a central processing unit, wherein: the memory stores program instructions executable by the central processing unit, and the central processing unit invokes the program instructions to perform the method according to any one of claims 1-6.
9. A non-transitory computer-readable storage medium, characterized in that, The non-transitory computer-readable storage medium stores computer instructions that cause the computer to perform the method described in any one of claims 1-6.