CN115828652A - Implantation method of large deformation plastic unit based on FDEM under non-rotation reference system - Google Patents
Implantation method of large deformation plastic unit based on FDEM under non-rotation reference system Download PDFInfo
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Abstract
The invention discloses an implantation method of a large deformation plastic unit based on FDEM under a non-rotating reference system, which comprises the following steps: obtaining a deformation rate and a rotation matrix in the rotation configuration based on the node coordinates of the triangular units; acquiring the deformation rate under the non-rotation configuration based on the deformation rate in the rotation configuration and the rotation matrix; determining elastic test stress, determining the state of the material by combining a yield function to obtain real stress, and performing rotational mapping of an original reference configuration to obtain real Cauchy stress; and calculating contact force and force caused by the cohesive force unit according to FDEM calculation, uniformly converting the contact force and the force into node force, updating node speed and position in the current time step according to Newton's second law, and entering the next time step. The method comprises the steps of calculating the stress tensor under the introduction of a non-rotating reference system, rotating the stress tensor, mapping the stress tensor to an original reference system, and matching with a yield function and a plastic flow rule to realize accurate simulation of plastic behaviors in a wire-discrete element.
Description
Technical Field
The invention belongs to the technical field of calculation of large deformation plasticity of materials, and particularly relates to an implantation method of a large deformation plasticity unit based on FDEM under a non-rotating reference system.
Background
Finite Discrete Element Method (FDEM), which is an emerging numerical method capable of simulating continuous-discontinuous values in recent years, has the basic principle that a four-node cohesive force unit without thickness is inserted between two adjacent triangular Finite elements, and the simulation of cracks is realized through the deformation of the four-node cohesive force unit to final fracture.
However, the existing triangle elements in the finite discrete elements are linear triangle finite element elements, which can simulate static mechanics, but the dynamic problems of high temperature, high pressure, high speed impact and explosion cannot simulate the plasticity and large deformation behavior of the material in the whole process well, and cause over-estimation of stress. For the plastic part of the simulation material, only a few researchers write simple and unverified elastoplasticity constitutive constructs into the four-node cohesive force unit, and cannot add complex and mature elastoplasticity constitutive relational expressions.
Disclosure of Invention
Aiming at the defects or improvement requirements of the prior art, the invention provides the implantation method of the large deformation plastic unit based on the FDEM under the non-rotation reference system, the calculation of the stress tensor under the non-rotation reference system is introduced, then the corresponding stress tensor is rotated to be mapped to the original reference system, and the accurate simulation of plastic behaviors in the wire-discrete elements can be realized by matching with the yield function, different flow laws and the like.
To achieve the above object, according to an aspect of the present invention, there is provided an implantation method of a large deformation plastic unit based on FDEM under a non-rotational reference system, comprising the steps of:
s100, obtaining a deformation rate { D } and a rotation matrix { R } in the rotation configuration based on the node coordinates of the triangular units;
s200, acquiring a deformation rate { D } under a non-rotation configuration based on the deformation rate { D } in the rotation configuration and the rotation matrix { R };
s300, determining elastic trial stress according to the deformation rate { d } in the non-rotation configuration, determining the material state by combining a yield function to obtain real stress in the non-rotation configuration, and performing rotation mapping on the original reference configuration based on the real stress to obtain Cauchy stress in the rotation configuration, namely the real Cauchy stress;
s400, according to FDEM calculation, the contact force and the force caused by the cohesive force unit are calculated and then uniformly converted into node force, the node speed and the position in the current time step are updated according to the Newton' S second law, and the next time step is started.
Further, the S100 specifically includes:
s101, calculating a deformation gradient matrix according to the coordinate information of the existing unit node:
wherein X is the node position in the deformation framework, and X is the node position in the reference configuration;
s102 utilizes the polar decomposition theorem of { F }, so that { F } can be decomposed into:
F＝VR＝RU (16)
wherein { V } and { U } are respectively a bilaterally symmetric positive tensile tensor, { R } representing a rotation tensor; on the other hand, the velocity tensor { L } is represented as:
wherein
Representing the time derivative of { F } -1 The { L } tensor can be decomposed into symmetric parts { D } and anti-symmetric parts { W }:
L＝D+W (18)
{ D } and { W } represent the deformation ratio matrix and rotation matrix, respectively.
Further, the S200 includes:
the S201 velocity gradient matrix { L } is expressed as follows:
wherein the first term is represented by { Ω }
{ omega } is a symmetric matrix,
being the time derivative of the rotation tensor, R T Is the transpose matrix of the rotation tensor,
for the time derivative of the stretching tensor, U -1 As the inverse matrix of the stretching tensor, the deformation gradient matrix { d } under the non-rotating architecture:
where { D } is a symmetric part of the velocity tensor { L },
for the time derivative of the stretching tensor, U -1 Is inverse of the stretching tensorMatrix, R T Is a transposed matrix of the rotation tensor, R is the rotation tensor;
in order to calculate { d }, S202 calculates a { R } tensor at the current time t, based on the inherent orthogonal rotation tensor describing the rigid rotation:
the subscripts t and t- Δ t represent that the time step for calculating the rotation tensor R is the current time step and the previous time step, and Δ t is the time increment of a single step;
by substituting the above equation, one can obtain:
ω＝w+[trace(V)I-V] -1 z (22)
wherein { W } is a rotation matrix under a rotation framework, trace () is a trace-finding symbol in tensor calculation, { V } is a left-symmetric positive stretching tensor, z is an intermediate calculation vector, and each component is i =0,1,2, which has the following expression:
(z) i ＝e ikj (D) jm (V) mk (23)
subscripts i, ikj, jm, and mk are index notation for tensor calculation, { D } is the deformation rate matrix, { V } is the left-symmetric positive stretch tensor, e ijk Is a third order Levi-Civit tensor;
the antisymmetric part of the second order tensor can be represented by the first order tensor, the vector and the third order tensor:
(Ω) ij ＝e ijk (ω) k (24)
(W) ij ＝e ijk (w) k (25)
wherein 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 symmetric matrices, e ijk Is a third order Levi-Civit tensor;
the transformation ratios in the non-rotated configuration and the rotation matrix can be obtained by combining the above equations.
Further, the step S300 of determining the elastic stress according to the deformation ratio { d } in the non-rotated configuration specifically includes:
s301, acquiring the strain increment in the current time step based on the deformation rate { d } in the non-rotating configuration:
Δε＝dΔt (26)
wherein Δ t is a calculation time step, and { d } is a deformation gradient matrix under a non-rotation framework;
s302 obtaining the trial stress of the current time step based on the strain increment
For the trial stress of the previous time step:
wherein, superscript t refers to the current time step, t-delta t refers to the previous time step, trace (-) is a trace-solving symbol in tensor calculation, I is unit tensor, and lambda and mu are Lame constants of materials;
further, the obtaining of the yield function specifically includes obtaining an equivalent stress according to the test stress, and combining the equivalent stress with the material yield strength to obtain the yield function.
Further, the determining the material state in combination with the yield function in S300 specifically includes:
if the yield function is less than 0, the material is proved not to be subjected to yield, namely in the elastic stage, and the calculated test stress is the true stress, namely
If the yield function is less than 0, the material is indicated to enter a plastic flow stage, and the plastic flow calculation is carried out according to the plastic flow rule to obtain
Further, the step S300 of performing a rotation mapping of the original reference configuration based on the true stress to obtain cauchy stress in a rotation configuration includes:
will be provided with
Mapping of the original reference configuration by the following formula yields cauchy stress in the rotational configuration:
wherein R is t Is the { R } tensor at time t, the rotation matrix in the { R } non-rotated configuration, σ is the stress, R T Is the transposed matrix of the rotation tensor.
According to a second aspect of the present invention, there is provided a material plasticity determination apparatus based on FDEM under a non-rotating reference frame, comprising:
the rotation configuration calculation module is used for obtaining a deformation rate and a rotation matrix in the rotation configuration based on the node coordinates of the triangular units;
the non-rotation configuration calculating module is used for acquiring the deformation rate under the non-rotation configuration based on the deformation rate in the rotation configuration and the rotation matrix;
the plasticity calculation module is used for determining elastic test stress according to the deformation rate in the non-rotation configuration, determining the material state by combining a yield function to obtain real stress in the non-rotation configuration, and performing rotation mapping on the original reference configuration based on the real stress to obtain Cauchy stress in the rotation configuration, namely the real Cauchy stress;
and the material state updating module calculates the contact force and the force caused by the cohesive force unit according to FDEM calculation, then uniformly converts the contact force and the force into node force, updates the node speed and the position in the current time step through Newton's second law and enters the next time step.
According to a third aspect of the present invention, there is provided an electronic apparatus comprising: at least one central processor; and at least one memory communicatively coupled to the central processor, wherein: the memory stores program instructions executable by the central processing unit, which invokes the program instructions to perform the method.
According to a fourth aspect of the present invention, there is provided a non-transitory computer readable storage medium characterized in that it stores computer instructions which cause a computer to perform the method.
In general, compared with the prior art, the above technical solution contemplated by the present invention can achieve the following beneficial effects:
1. according to the implantation method of the large-deformation plastic unit, the calculation of the stress tensor under the non-rotating reference system is introduced, then the corresponding stress tensor is rotated to be mapped to the original reference system, and the simulation of the plastic behavior in the wire-discrete element can be realized by matching with a yield function, different flow laws and the like.
2. According to the implantation method of the large-deformation plastic unit, the plasticity is simulated by utilizing the property of the triangular unit rather than the property of the cohesive force unit, various complex elastic-plastic structures can be introduced into the triangular unit, the cohesive force cannot be introduced into the complex elastic-plastic structures (such as Moire coulomb, deluke-prague structures and the like), and in the field of traditional finite elements, the plasticity representation is realized through the triangular unit.
3. The implantation method of the large deformation plastic unit is suitable for various complex elastic-plastic constitutive structures, including the famous Mi Saisi, mokolun, delukee-prager, JH2, HJC constitutive relations, the switching between different constitutive relations can be realized only by changing a yield function and a plastic flow rule, and the algorithm has strong transportability and is very stable.
4. The implantation method of the large deformation plastic unit is simultaneously suitable for two-dimensional limited discrete elements (based on a triangular unit and a four-node non-thickness cohesive force unit) and three-dimensional limited discrete elements (based on a tetrahedral unit and a six-node non-thickness cohesive force unit).
5. According to the implantation method of the large deformation plastic unit, the result consistent with the analytic solution can be obtained by a famous standard consolidation experiment of Mokolun, and the effectiveness and the accuracy of the method are proved.
Drawings
FIG. 1 is a flowchart of the finite-discrete element large deformation architecture computation of the present invention;
FIG. 2 is a flow chart of the implantation method of the large deformation plastic unit based on FDEM under non-rotation reference frame;
FIG. 3 is a flow chart of the present invention for obtaining a yield function based on the deformation ratio { d } in a non-rotated configuration;
FIG. 4 is a diagram illustrating a model of the Taylor bar problem and boundary conditions;
FIG. 5 is a graph showing results of Taylor bar simulation (FIG. 5 (a) is the original FDEM simulation result; FIG. 5 (b) is the FDEM simulation result using the macro-deformation framework; FIG. 5 (c) is the experiment result from the literature Mechanics of Taylor impact testing of polycarbonate);
FIG. 6 is a detailed view of results of Taylor bar simulation (FIG. 6 (a) is a graph of results of an original FDEM simulation; FIG. 6 (b) is a graph of results of an FDEM simulation using a large deformation framework; FIG. 6 (c) is a detailed graph of results of an experiment);
FIG. 7 is a schematic diagram of an Oedemeter experiment in Mohr-Coulomb.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
The invention provides an implantation method of a large deformation plastic unit based on FDEM under a non-rotating reference system, which comprises the following steps:
s100, based on the node coordinates of the triangular units, combining with a stress tensor under a non-rotating reference system, and obtaining a deformation rate { d } and a rotation matrix { R } in a non-rotating configuration;
s200, obtaining a yield function based on the deformation rate { d } in the non-rotating configuration;
s300, determining the state of the material according to the yield function to obtain a real stress, and mapping an original reference configuration based on the real stress to obtain a real Cauchy stress;
s400, according to FDEM calculation, the contact force and the force caused by the cohesive force unit are calculated and then uniformly converted into node force, the node speed and the position in the current time step are updated according to the Newton' S second law, and the next time step is started.
In particular, the node coordinates of the triangle elements are three-dimensional coordinates { x, y, z }, and can also be used in two dimensions, which can be regarded as a special case { x, y,0} where z is constantly equal to 0, and for a stress matrix (three-dimensional), it is assumed that the numbers in the matrix represent the corresponding components: the three-dimensional matrix consists of:
there are 9 components in total, 0-8, and two dimensions take only the upper left four terms, i.e.
0,1,3,4.
Specifically, the S100 includes the following processes:
s101, calculating a deformation gradient matrix according to the coordinate information of the existing unit node:
wherein X is a node position in the deformation framework, X is a node position in the reference configuration, and det (F) is a determinant of an F matrix;
s102 utilizes the polar decomposition theorem of { F }, which can decompose { F } into:
F＝VR＝RU (30)
where { V } and { U } are respectively the left-right symmetric positive extension tensors, { R } represents the rotation tensors. On the other hand, the velocity tensor { L } is represented as:
wherein
Representing the time derivative of { F } -1 The { L } tensor can be decomposed into symmetric parts { D } and anti-symmetric parts { W }:
L＝D+W (32)
{ D } and { W } represent the deformation ratio matrix and rotation matrix, respectively.
Specifically, the S200 includes the following processes:
the S201 velocity gradient matrix { L } is expressed as follows:
wherein the first term is represented by { Ω }
Both omega and W are symmetric matrices,
being the time derivative of the rotation tensor, R T Is a transposed matrix of the rotation tensor,
for the time derivative of the stretching tensor, U -1 Is the inverse of the stretch tensor. Deformation gradient matrix under non-rotating architecture { d }:
s202 to calculate { d }, the { R } tensor for the current time t is calculated based on the inherent orthogonal rotation tensor describing the rigid body rotation:
subscripts t and t- Δ t indicate that a time step for calculating the rotation tensor R is a current time step and a previous time step, Δ t is a time increment of a single step, Ω is a symmetric matrix, and I is a unit tensor;
by substituting the above equation, one can obtain:
ω＝w+[trace(V)I-V] -1 z (36)
where { W } is a rotation matrix under a rotation framework, trace (·) is a trace-finding symbol in tensor calculation, { V } is a left-symmetric positive-stretching tensor, z is an intermediate calculation vector, and each component (i =0,1,2) has the following expression:
(z) i ＝e ikj (D) jm (V) mk (37)
the indices i, ikj and jm, mk are the index methods of tensor calculation, e ijk The Levi-Civit tensor is in the third order.
When the expression (5) and the third order unit tensor e in the S201 ikj When the two-dimensional matrix is compressed, the symmetrical part disappears, and three linear equation sets of three independent components of the omega matrix are obtained. The anti-symmetric part of the second-order tensor (index ij) can be represented by a first-order tensor (i.e., vector, index k) and a third-order tensor (index ijk), the following two vectors being defined as the right-hand rotation rate:
(Ω) ij ＝e ijk (ω) k (38)
(W) ij ＝e ijk (w) k (39)
the transformation ratios in the non-rotated configuration and the rotation matrix can be obtained by combining the above equations.
Specifically, the step of determining the elastic stress according to the deformation ratio { d } in the non-rotated configuration in S300 specifically includes:
s301, acquiring the strain increment in the current step based on the deformation rate { d } in the non-rotating configuration:
Δε＝dΔt (40)
wherein, Δ t is a calculation time step, and { d } is a deformation gradient matrix under a non-rotating framework;
s302 obtaining the trial stress of the current time step based on the strain increment
For the trial stress of the previous time step:
wherein, the superscript t refers to the current time step, t-delta t refers to the previous time step, trace (-) is a trace-solving symbol in tensor calculation, I is a unit tensor, and lambda and mu are Lame constants of the material;
the yield function is obtained specifically as follows: and obtaining equivalent stress according to the test stress, and combining the equivalent stress with the yield strength of the material to obtain a yield function.
Specifically, the determining the material state in combination with the yield function in S300 specifically includes:
if the yield function is less than 0, the material is proved not to be subjected to yield, namely in the elastic stage, and the calculated test stress is the true stress, namely
If the yield function is less than 0, the material is indicated to enter a plastic flow stage, and plastic flow calculation is carried out according to a plastic flow rule to obtain
In S300, performing rotation mapping of the original reference configuration based on the true stress to obtain cauchy stress in the rotation configuration includes:
will be provided with
Mapping of the original reference configuration by the following formula yields cauchy stress in the rotational configuration:
wherein R is t Is the { R } tensor at time t, the rotation matrix in the { R } non-rotated configuration, σ is the stress, R T Is the transposed matrix of the rotation tensor.
The Cauchy stress under the rotation configuration is the true Cauchy stress.
Optionally, the method of the present invention may change the yield function and the plastic flow law to realize the switching between different complex elastoplasticity mechanisms, where the complex elastoplasticity mechanisms include Mi Saisi, mokuran, deruke-prague, JH2, HJC constitutive relation.
In order to verify the effectiveness and accuracy of the method, the following experiments are carried out:
example 1:
taylor bar impact test for large deformation:
the Taylor bar impact problem is a very well-known benchmark test to study plasticity mechanics, using a large deformation architecture and using a Von Mises elastoplasticity model. The property parameters are shown in table 1, fig. 4 depicts the model and boundary conditions of the Taylor bar problem, fig. 5 and fig. 6 show the comparison between the FDEM using the original FDEM and the large deformation framework and the prior experiment, respectively, and it can be seen that the Taylor rod of the original FDEM cannot generate large deformation due to the assumption of small deformation, and the FDEM (the method of the present invention) introducing the large deformation framework can well capture the plastic properties of the material.
TABLE 1 Taylor Bar impact test Property parameters
Example 2:
oedemeter experiment of Mohr-Coulomb elasto-plastic model:
taking the Mohr-Coulomb elastoplasticity constitution as an example, a famous Oedemeter experiment is now performed to establish a model having a length and a width of 1m, respectively, as shown in (a) and (b) of FIG. 7 below, and the parameters of the model are as shown in Table 2. Fig. 7 (a) represents model dimensions and boundary conditions, fig. 7 (b) represents node numbers and grid information, fig. 7 (c) shows the final y-displacement field of the whole process, and fig. 7 (D) shows a comparison graph of the relationship between the y-stress-displacement obtained by FDEM simulation and FLAC3D using the method of the present application.
As can be seen from (a) and (b) in fig. 7, the velocity of y negative direction is applied to the node 2 and the node 3 at 1e-6 m/time step, the whole process lasts for 1000 steps, and finally the y-direction displacement of the node No. 2 and the node No. 3 is-0.01 m, which is shown in (c) in fig. 7.
Fig. 7 (D) can show that the method provided in the present application can obtain the same results in FDEM as the Oedometer experiment in FLAC 3D.
TABLE 2 Oedemeter Experimental parameters
The above-described implementation basis of the present invention is implemented by performing programmed processing by an apparatus having a function of a central processing unit. Therefore, in engineering practice, the technical solutions and functions thereof of the embodiments of the present invention can be packaged into various modules. Based on this reality, on the basis of the above embodiments, the embodiments of the present invention provide a material plasticity determination apparatus based on FDEM under non-rotational reference frame, which is used to perform the implantation method based on large deformation plasticity unit of FDEM under non-rotational reference frame in the above method embodiments. The method comprises the following steps:
the deformation rate and rotation matrix determining module is used for obtaining a deformation rate { d } and a rotation matrix { R } in the non-rotation configuration based on the node coordinates of the triangular units and by combining the stress tensor under the non-rotation reference system;
a yield function determination module for obtaining a yield function based on a deformation rate { d } in the non-rotated configuration;
the real Couchy stress determining module is used for determining the material state according to the yield function to obtain real stress, and mapping the original reference configuration based on the real stress to obtain the real Couchy stress;
and the material state updating module is used for calculating the contact force and the force caused by the cohesive force unit according to FDEM calculation, uniformly converting the contact force and the force into node force, updating the node speed and the position in the current time step through Newton's second law and entering the next time step.
It should be noted that, the apparatus in the apparatus embodiment provided by the present invention may be used for implementing methods in other method embodiments provided by the present invention, except that corresponding function modules are provided, and the principle of the apparatus embodiment provided by the present invention is basically the same as that of the apparatus embodiment provided by the present invention, so long as a person skilled in the art obtains corresponding technical means by combining technical features on the basis of the apparatus embodiment described above, and obtains a technical solution formed by these technical means, on the premise of ensuring that the technical solution has practicability, the apparatus in the apparatus embodiment described above may be modified, so as to obtain a corresponding apparatus class embodiment, which is used for implementing methods in other method class embodiments.
The method of the embodiment of the invention is realized by depending on the electronic equipment, so that the related electronic equipment is necessarily introduced. With this object in mind, an embodiment of the present invention provides an electronic device, as shown in fig. 3, including: the system comprises at least one Central processor (Central processor), a communication Interface (communication Interface), at least one Memory (Memory) and a communication bus, wherein the at least one Central processor, the communication Interface and the at least one Memory are communicated with each other through the communication bus. The at least one central processor may invoke logic instructions in the at least one memory to perform all or a portion of the steps of the methods provided by the various method embodiments described above.
In addition, the logic instructions in the at least one memory may be implemented in software functional units and stored in a computer readable storage medium when sold or used as a stand-alone product. Based on such understanding, the technical solution of the present invention may be embodied in the form of a software product, which is stored in a storage medium and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device) to execute all or part of the steps of the method according to the method embodiments of the present invention. And the aforementioned storage medium includes: a U-disk, a removable hard disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a magnetic disk or an optical disk, and other various media capable of storing program codes.
The above-described embodiments of the apparatus are merely illustrative, and the units described as separate parts may or may not be physically separate, and parts displayed as units may or may not be physical units, may be located in one place, or may be distributed on a plurality of network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the solution of the present embodiment. One of ordinary skill in the art can understand and implement it without inventive effort.
Through the above description of the embodiments, those skilled in the art will clearly understand that each embodiment can be implemented by software plus a necessary general hardware platform, and certainly can also be implemented by hardware. With this understanding in mind, the above-described technical solutions may be embodied in the form of a software product, which can be stored in a computer-readable storage medium such as ROM/RAM, magnetic disk, optical disk, etc., and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute the methods described in the embodiments or some parts of the embodiments.
The flowchart and block diagrams in the figures 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 recognition, each block in the flowchart or block diagrams may represent a module, a program segment, or a portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems which perform the specified functions or acts, or combinations of special purpose hardware and computer instructions.
In this patent, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising … …" does not exclude the presence of another like element in a process, method, article, or apparatus that comprises the element.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (10)
1. A method for implanting a large deformation plastic unit based on FDEM under a non-rotation reference frame is characterized by comprising the following steps:
s100, obtaining a deformation rate { D } and a rotation matrix { R } in the rotation configuration based on the node coordinates of the triangular units;
s200, acquiring a deformation rate { D } under a non-rotation configuration based on the deformation rate { D } in the rotation configuration and the rotation matrix { R };
s300, determining elastic trial stress according to the deformation rate { d } in the non-rotation configuration, determining the material state by combining a yield function to obtain real stress in the non-rotation configuration, and performing rotation mapping on the original reference configuration based on the real stress to obtain Cauchy stress in the rotation configuration, namely the real Cauchy stress;
s400, according to FDEM calculation, the contact force and the force caused by the cohesive force unit are calculated and then uniformly converted into node force, the node speed and the position in the current time step are updated according to the Newton' S second law, and the next time step is started.
2. The method for implanting a large deformation plastic unit based on FDEM under non-rotational reference frame as claimed in claim 1, wherein said S100 specifically comprises:
s101, calculating a deformation gradient matrix according to the coordinate information of the existing unit nodes:
wherein X is the node position in the deformation framework, and X is the node position in the reference configuration;
s102 utilizes the polar decomposition theorem of { F }, which can decompose { F } into:
F＝VR＝RU (2)
wherein { V } and { U } are respectively a bilaterally symmetric positive tensile tensor, { R } representing a rotation tensor; on the other hand, the velocity tensor { L } is expressed as:
wherein
Representing the time derivative of { F } -1 The { L } tensor can be decomposed into symmetric parts { D } and anti-symmetric parts { W }:
L＝D+W (4)
{ D } and { W } represent the deformation ratio matrix and rotation matrix, respectively.
3. The method for implanting a large deformation plastic unit for FDEM under non-rotational reference frame as claimed in claim 2, wherein said S200 comprises:
the S201 velocity gradient matrix { L } is expressed as follows:
wherein the first term is represented by { Ω }
{ omega } is a symmetric matrix,
being the time derivative of the rotation tensor, R T Is a transposed matrix of the rotation tensor,
for the time derivative of the stretching tensor, U -1 As the inverse matrix of the stretching tensor, the deformation gradient matrix { d } under the non-rotating architecture:
where { D } is a symmetric part of the velocity tensor { L },
for the time derivative of the stretching tensor, U -1 As inverse matrix of the extension tensor, R T Is a transposed matrix of the rotation tensor, R is the rotation tensor;
in order to calculate { d }, S202 calculates a { R } tensor at the current time t, based on the inherent orthogonal rotation tensor describing the rigid rotation:
the subscripts t and t- Δ t represent that the time step for calculating the rotation tensor R is the current time step and the previous time step, and Δ t is the time increment of a single step;
by substituting the above equation, one can obtain:
ω＝w+[trace(V)I-V] -1 z (8)
wherein { W } is a rotation matrix under a rotation framework, trace () is a trace-finding symbol in tensor calculation, { V } is a left-symmetric positive tensile tensor, z is an intermediate calculation vector, i =0,1,2, and has the following expression:
(z) i ＝e ikj (D) jm (V) mk (9)
subscripts i, ikj, jm, and mk are index notation for tensor calculation, { D } is the deformation rate matrix, { V } is the left-symmetric positive stretch tensor, e ijk Is a third order Levi-Civit tensor;
the antisymmetric part of the second order tensor can be represented by the first order tensor, the vector and the third order tensor:
(Ω) ij ＝e ijk (ω) k (10)
(W) ij ＝e ijk (w) k (11)
wherein 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 symmetric matrices, e ijk Is a third order Levi-Civit tensor;
the above equations are combined to obtain { d } and { R }, i.e., the deformation ratio and the rotation matrix in the non-rotated configuration.
4. The method for implanting a large deformation plastic unit based on FDEM under non-rotational reference frame as claimed in claim 1, wherein the determining the elastic trial stress according to the deformation ratio { d } under non-rotational configuration in S300 specifically comprises:
s301, acquiring the strain increment in the current step based on the deformation rate { d } in the non-rotating configuration:
Δε＝dΔt (12)
wherein, Δ t is a calculation time step, and { d } is a deformation gradient matrix under a non-rotating framework;
s302 obtaining the trial stress of the current time step based on the strain increment
For the trial stress of the previous time step:
wherein the superscript t refers to the time step, t-delta t refers to the time step, trace (-) is the trace-solving symbol in tensor calculation, I is unit tensor, and lambda and mu are Lame constants of the material.
5. The method for implanting the FDEM-based large deformation plastic unit under the non-rotating reference frame as claimed in claim 4, wherein the yield function is obtained by obtaining an equivalent stress according to a test stress and combining the equivalent stress and a material yield strength to obtain the yield function.
6. The method for implanting a large deformation plastic unit based on FDEM under non-rotational reference frame as claimed in claim 1, wherein the determining the material state in combination with the yield function in S300 comprises:
if the yield function is less than 0, the material is proved not to be subjected to yield, namely in the elastic stage, and the calculated test stress is the true stress, namely
If the yield function is less than 0, the material is indicated to enter a plastic flow stage, and the plastic flow calculation is carried out according to the plastic flow rule to obtain
7. The method for implanting a large deformation plastic unit based on FDEM under non-rotational reference frame as claimed in claim 1, wherein the step S300 of performing rotational mapping of the original reference configuration based on the true stress to obtain Cauchy stress under rotational configuration comprises:
will be provided with
Mapping of the original reference configuration by the following formula yields cauchy stress in the rotational configuration:
wherein R is t Is the { R } tensor at time t, the rotation matrix in the { R } non-rotated configuration, σ is the stress, R T Is the transposed matrix of the rotation tensor.
8. A material plasticity determination device based on FDEM under a non-rotating reference frame, comprising:
the rotation configuration calculation module is used for obtaining a deformation rate and a rotation matrix in the rotation configuration based on the node coordinates of the triangular units;
the non-rotation configuration calculating module is used for acquiring the deformation rate under the non-rotation configuration based on the deformation rate in the rotation configuration and the rotation matrix;
the plasticity calculation module is used for determining elastic test stress according to the deformation rate in the non-rotation configuration, determining the material state by combining a yield function to obtain real stress in the non-rotation configuration, and performing rotation mapping on the original reference configuration based on the real stress to obtain Cauchy stress in the rotation configuration, namely the real Cauchy stress;
and the material state updating module calculates the contact force and the force caused by the cohesive force unit according to FDEM calculation, then uniformly converts the contact force and the force into node force, updates the node speed and the position in the current time step through Newton's second law and enters the next time step.
9. An electronic device, comprising: at least one central processor; and at least one memory communicatively coupled to the central processor, wherein: the memory stores program instructions executable by a central processing unit, the central processing unit calling program instructions capable of performing the method of any one of claims 1 to 7.
10. A non-transitory computer-readable storage medium storing computer instructions for causing a computer to perform the method of any one of claims 1-7.
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| CN117307176B (en) * | 2023-11-29 | 2024-02-20 | 中铁三局集团广东建设工程有限公司 | Transverse channel construction method based on formed pilot tunnel |
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