CN114330034A - Calculation method for predicting elastic behavior of compressible-incompressible composite material - Google Patents
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Abstract
The invention discloses a calculation method for predicting elastic behavior of a compressible-incompressible composite material, which belongs to the field of composite material calculation and comprises the following steps: step 1, marking unit material attributes and distributing unit variables to be solved; step 2, obtaining a discrete format of the elastic equation of the compressible-incompressible double-layer composite material; and 3, obtaining the central stress of the unit. The method can avoid shearing self-locking when the FEM solves the problem of the incompressible material and false stress concentration when the composite material is solved, can realize the prediction of the elastic behavior of the incompressible-compressible double-layer composite material, can directly obtain the elastic distribution of different material domains without iterative calculation, has stable calculation result and simple program implementation, and can directly solve the problem of the functional incompressible-compressible material.
Description
Technical Field
The invention relates to the field of composite material calculation, in particular to a calculation method for predicting elastic behavior of a compressible-incompressible composite material.
Background
With the continuous development of rubber-like materials (polymers, living tissues, functional materials) in the engineering field and the biomedical field, and the possibility that incompressible materials may show mechanical properties different from those of compressible materials, the research on the line elasticity problem of incompressible materials has become a hot issue of attention of students in various fields. Incompressible means that the volume of the object remains unchanged (isovolumetric deformation) under any stress condition. Poisson's ratio for isotropic non-crushable elastomers under finite modulusνEqual to 0.5, in the form of an elastomer with an infinite bulk modulus, which can only be deformed volumetrically. Generally speaking, the elasticity problem of the incompressible material cannot be directly calculated by a calculation method of the incompressible material with the Poisson's ratio set to 0.5, and the calculation is carried out by using an equation which introduces hydrostatic pressure and under the incompressible condition.
Currently, many researches on the problem of linear elasticity of incompressible materials are calculated by an analytical method, a Finite Element Method (FEM) and a lattice type/lattice point type finite volume method (CC-FVM/CV-FVM), mainly aiming at isotropic homogeneous/functionally gradient incompressible materials, and the problems mainly exist in the following steps: 1. when the FEM calculates the elasticity problem of the composite material or the incompressible material, the problem of non-physical stress oscillation or unit shearing self-locking near a material interface along the thickness direction can occur; 2. the analytical method is only suitable for simple structure distribution and material distribution, and the real material structure cannot be calculated; 3. when the CC-FVM calculates the elasticity problem of the composite material, false stress jump occurs at the interface of the material. Compared with the method, the CV-FEM has better calculation performance on the calculation functional material and the compressible material, and can effectively avoid the phenomenon of false stress and shear self-locking, however, the method is currently suitable for calculating the elasticity problem under the distribution of the compressible or non-compressible material, and the elasticity problem of a compressible-non-compressible (or compressible-near non-compressible) double-layer composite structure cannot be calculated (see figure 1).
In order to meet the requirements of service environments, a laminated material or a functional gradient material is usually combined by a plurality of materials to play the excellent mechanical and thermal properties of each component material, has very wide application in the fields of aviation/aerospace, biology, medical treatment and the like, and is inevitably related to the problem of elasticity prediction of incompressible-compressible materials, such as the problem of contact between a 3D printed high polymer part (incompressible) and a metal matrix, the problem of heat-force transfer between silicon rubber (incompressible) of a space plane, a rigid heat-proof tile and the metal matrix and the like. Therefore, there is a need to develop a calculation method capable of predicting the elastic behavior of compressible-incompressible composite materials, which provides a necessary solution for the estimation and prediction of the elastic performance of engineering composite structural samples.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides a calculation method for predicting the elastic behavior of a compressible-incompressible composite material, avoids the problems of shear self-locking when an FEM solves the problem of the incompressible material and false stress concentration when the composite material is solved, can realize the elastic behavior prediction of the incompressible-compressible double-layer composite material, can directly obtain the elastic distribution of different material domains, does not need iterative calculation, has stable calculation result and simple program implementation, and can directly solve the problem of the functional incompressible-compressible material.
The purpose of the invention is realized by the following scheme:
a computational method for predicting elastic behavior of a compressible-incompressible composite comprising:
and 3, obtaining the central stress of the unit.
Further, in step 1, the method comprises the sub-steps of: and S1, setting a composite material laying mode, storing the composite material laying mode in the cell center by adopting a multi-grid method, storing the hydrostatic pressure to be solved in the cell center, and marking the discrete cells as composite material interface cells or internal cells.
Further, in step 2, the method comprises the sub-steps of: s2, based on the lattice-point-type FVM discrete elasticity problem control equation, introducing the essential relation of compressible materials or non-compressible materials into the internal control body, introducing the essential relation of the compressible materials or the non-compressible materials into discrete units at the interface in a segmented mode according to unit attribute marks when controlling the volume line integration, and then solving the discrete equation to obtain the displacement at the unit nodes and the hydrostatic pressure distribution at the unit center.
Further, in step 3, the method comprises the sub-steps of: and S3, marking the essential relationship of the compressible material or the incompressible material by the cell attribute to obtain the cell center stress distribution.
Further, in step S1, the method includes the sub-steps of: carrying out material identification on the discrete grid according to the material laying mode of the double-layer composite material, and defining the elastic modulus E and the Poisson ratio in the center of the compressible material unitνCenter-defined modulus of elasticity of incompressible materialEAnd hydrostatic pressure to be relievedP(ii) a And marking the discrete control bodies according to the discrete unit positions and the material attribute distribution to form a non-compressible material area control body, a boundary area control body and a non-compressible area control body.
Further, in step S2, the method includes the sub-steps of: starting from an isotropic steady-state linear elastic equilibrium equation, the control equation of the integral format is as follows:
wherein the content of the first and second substances,
is an integral control line;
is the interface external normal vector;
representing the Cauchy stress tensor;
for isotropic linear elastomers, the constitutive relation is:
wherein the content of the first and second substances,
the strain tensor of Cauchy is expressed,
is a unit tensor, wherein
Expressing the Lame coefficient, when the plane strain assumption is adopted, the expression is:
wherein
In order to be the young's modulus,
is the poisson ratio;
the Cauchy strain tensor expression is:
wherein the content of the first and second substances,u i indicating the displacement in the ith direction,x i is a global coordinate of the ith direction,u j indicating the displacement in the jth direction,x j is the global coordinate of the jth direction.
Poisson's ratio for incompressible materials
Hydrostatic pressure
The variable to be solved is introduced into the equation (2) and is converted into the following equation:
the following non-compressible conditions were introduced for the non-compressible material region:
wherein the content of the first and second substances,Vrepresenting the integrated control volume.
The control body in the compressible material region introduces the formula (2) into the formula (1), the control body in the non-compressible material region introduces the formula (5) into the formula (1), and the control body in the interface is controlled by an integral control lineSDetermining whether the material is compressible or non-compressible material by the grid unit, and introducing the method formula (2) or the formula (5) into the balance equation (1); assuming that the incompressible condition (6) is required to be satisfied in the cell for the incompressible material region, the one at the center of the cell is adoptedxDirection andycalculating the positive strain approximation of the direction;
by the above substeps in step S2, a discrete format of the final equilibrium equation is obtained by performing line integration in the control volume formed by the nodes, see equation (7);
wherein the content of the first and second substances,K xu ，K xv ，K xp is composed ofxDirection to be solved variableu,v,PBy the formula (2), the formula (1), or the formula (5), the formula (1)Obtained for the compressible material coefficientK xp =0；K yu ，K yu ，K yp Is composed ofyDirection to be solved variableu,v,PThe coefficient (c) is obtained by the formula (2), the formula (1) or the formula (5), the formula (1), and the coefficient for the compressible materialK yp =0；K pu ，K pv The coefficients introduced by the incompressible region under the incompressible condition.
Further, in step S3, the method includes the sub-steps of: the node displacement obtained using equation (7) and the cell static pressure of the material domain divided according to step S1 obtain the stress distribution of the cell center using equation (2) or equation (5).
The invention has the beneficial effects that:
the method provided by the embodiment of the invention can avoid the problems of shear self-locking when the FEM solves the problem of the incompressible material and false stress concentration when the composite material is solved.
The method provided by the embodiment of the invention can realize the prediction of the elastic behavior of the incompressible-compressible double-layer composite material, the elastic distribution of different material domains can be directly obtained, iterative calculation is not needed, the calculation result is stable, the program implementation is simple, and the method can be directly used for solving the functional incompressible-compressible material.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a schematic view of a compressible-incompressible dual layer composite structure under force loading;
FIG. 2 is a flowchart illustrating steps of a method for calculating elastic behavior of a compressible-incompressible dual layer composite according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of discrete cells of a compressible-incompressible bilayer composite at an interface according to an embodiment of the present invention, wherein c1-c4 represents the cell center of 4 cells and CV represents an integral control volume.
Detailed Description
All features disclosed in all embodiments in this specification, or all methods or process steps implicitly disclosed, may be combined and/or expanded, or substituted, in any way, except for mutually exclusive features and/or steps.
As shown in fig. 1 to 3, a calculation method for predicting elastic behavior of a compressible-incompressible composite material includes the following steps:
Carrying out material identification on the discrete grid according to the material laying mode of the double-layer composite material, and defining the elastic modulus E and the Poisson ratio in the center of the compressible material unitνCenter-defined modulus of elasticity of incompressible materialEAnd hydrostatic pressure to be relievedP(ii) a The discrete control bodies are labeled according to the discrete cell locations and material property distributions, as shown in FIG. 3, where non-crushable material zone control bodies (non-interfaces) are formed around nodes 1-3, interface zone control bodies are formed around nodes 4-6, and non-crushable zone control bodies (non-interfaces) are formed around nodes 7-9.
And 2, obtaining a discrete format of the elastic equation of the compressible-incompressible double-layer composite material. In a specific implementation process, based on a lattice-point-type FVM discrete elasticity problem control equation, an intrinsic relation of a compressible material (or a non-compressible material) is adopted for an internal control body (a non-interface discrete unit), the intrinsic relation of the material is introduced according to unit attribute mark segmentation during control of body line integration for the discrete unit at an interface, and displacement at a unit node and hydrostatic pressure distribution at the center of the unit are obtained through solving the discrete equation.
Starting from an isotropic steady-state linear elastic equilibrium equation, the control equation of the integral format is as follows:
wherein the content of the first and second substances,
is an integral control line;
is the interface external normal vector;
representing the Cauchy stress tensor.
For isotropic linear elastomers, the constitutive relation is:
wherein
The strain tensor of Cauchy is expressed,
is a unit tensor, wherein
Expressing the Lame coefficient, when the plane strain assumption is adopted, the expression is:
wherein
In order to be the young's modulus,
is the poisson ratio.
The Cauchy strain tensor expression is:
wherein the content of the first and second substances,u i indicating the displacement in the ith direction,x i is a global coordinate of the ith direction,u j indicating the displacement in the jth direction,x j global coordinates of j-th direction;
poisson's ratio for incompressible or nearly incompressible materials
Hydrostatic pressure
The variable to be solved is introduced into the equation (2) and is converted into the following equation:
for the introduction of incompressible conditions within the incompressible material domain:
wherein the content of the first and second substances,Vrepresenting the integrated control volume.
The control body in the pressable domain (non-interface) introduces the formula (2) into the formula (1), the control body in the non-pressable domain (non-interface) introduces the formula (5) into the formula (1), and the integral control line is used for controlling the control body at the interfaceSThe grid cell where the material is determined to be compressible (or non-compressible), the present method (2) (or equation (5)) is introduced into the equilibrium equation (1). Assuming that the incompressible material region needs to satisfy the incompressible condition (6) in the cell, the center of the cell is adoptedxDirection andydirection positive strain approximatorAnd (4) calculating. The discrete format of the final equilibrium equation can be obtained by performing the line integration in the control volume formed by the nodes through the discrete steps described above (see equation (7)), wherein,K xu ，K xv ，K xp is composed ofxDirection to be solved variableu,v,PThe coefficient (c) is obtained by the formula (2), the formula (1) or the formula (5), the formula (1), and the coefficient for the compressible materialK xp =0；K yu ，K yu ，K yp Is composed ofyDirection to be solved variableu,v,PThe coefficient (c) is obtained by the formula (2), the formula (1) or the formula (5), the formula (1), and the coefficient for the compressible materialK yp =0；K pu ，K pv The coefficients introduced by the incompressible region under the incompressible condition.
And 3, obtaining the central stress of the unit. In a specific implementation, the cell center stress distribution is obtained by marking the essential relationship of using compressible materials (non-compressible materials) by cell attributes.
The node displacement obtained by using equation (7) and the cell static pressure of the material domain divided according to step 1 obtain the stress distribution of the cell center using equation (2) (or equation (5)).
The technical scheme of the embodiment of the invention is based on the method for solving the elastic problem of the compressible-incompressible material established by the lattice-point FVM, can automatically meet the flux conservation characteristic at the interface of the composite material, does not need to additionally process the material parameters at the interface, can realize the introduction of material attributes and the intrinsic relation in different domains by adopting the staggered grid technology, realizes the direct coupling solution in different material domains, and has the following advantages:
1. and performing full-field dispersion of the balance equation of the compressible-incompressible double-layer composite material based on the lattice point type FVM.
2. The material properties and the hydrostatic pressure to be resolved are defined in the center of the cell using a multiple-grid technique. The incompressible condition within the domain of incompressible material is discrete within a grid cell; the balance equation is dispersed in a control body taking a node as a center, and the prediction of the elastic response of the material at the interface of the composite material can be realized through the material property change and the intrinsic relationship introduced by the multi-grid technology.
Example 1: a computational method for predicting elastic behavior of a compressible-incompressible composite comprising:
and 3, obtaining the central stress of the unit.
Example 2: on the basis of the embodiment 1, in the step 1, the method comprises the following substeps: and S1, setting a composite material laying mode, storing the composite material laying mode in the cell center by adopting a multi-grid method, storing the hydrostatic pressure to be solved in the cell center, and marking the discrete cells as composite material interface cells or internal cells.
Example 3: on the basis of the embodiment 1, in the step 2, the method comprises the following substeps: s2, based on the lattice-point-type FVM discrete elasticity problem control equation, introducing the intrinsic-sufficient relation of compressible materials or non-compressible materials into the internal control body, and when the line integration of discrete units at the interface is controlled, introducing the intrinsic-sufficient relation of the materials in a segmented mode according to unit attribute marks, and solving the discrete equation to obtain the displacement at the node of the unit and the hydrostatic pressure distribution at the center of the unit.
Example 4: on the basis of the embodiment 1, in the step 3, the method comprises the following substeps: and S3, marking the essential relationship of the compressible material or the incompressible material by the cell attribute to obtain the cell center stress distribution.
Example 5: on the basis of embodiment 2, in step S1, the method includes the sub-steps of: carrying out material identification on the discrete grid according to the material laying mode of the double-layer composite material, and defining the elastic modulus E and the Poisson ratio in the center of the compressible material unitνCenter-defined modulus of elasticity of incompressible materialEAnd hydrostatic pressure to be relievedP(ii) a And marking the discrete control bodies according to the discrete unit positions and the material attribute distribution to form non-compressible material area control bodies, interface area control bodies and non-compressible area control bodies.
Example 6: on the basis of embodiment 3, in step S2, the method includes the sub-steps of: starting from an isotropic steady-state linear elastic equilibrium equation, the control equation of the integral format is as follows:
wherein the content of the first and second substances,
is an integral control line;nis the interface external normal vector;
representing the Cauchy stress tensor;
for isotropic linear elastomers, the constitutive relation is:
wherein
The strain tensor of Cauchy is expressed,
is a unit tensor, wherein
Expressing the Lame coefficient, when the plane strain assumption is adopted, the expression is:
wherein
In order to be the young's modulus,
is the poisson ratio;
the Cauchy strain tensor expression is:
wherein the content of the first and second substances,u i indicating the displacement in the ith direction,x i is a global coordinate of the ith direction,u j indicating the displacement in the jth direction,x j global coordinates of j-th direction;
poisson's ratio for incompressible materials
Hydrostatic pressure
The variable to be solved is introduced into the equation (2) and is converted into the following equation:
for the introduction of incompressible conditions within the incompressible material domain:
wherein the content of the first and second substances,Vrepresenting the integrated control volume.
The control body in the compressible region introduces the formula (2) into the formula (1), the control body in the non-compressible region introduces the formula (5) into the formula (1), and the control body in the interface is controlled by an integral control lineSThe grid cells determine whether the material is compressible or non-compressible, and the method formula (2) or formula (5) is introduced into the balance equation (1)(ii) a Assuming that the incompressible material region needs to satisfy the incompressible condition (6) in the cell, the center of the cell is adoptedxDirection andycalculating the positive strain approximation of the direction;
performing line integration in a control body formed by nodes through the discrete steps to obtain a discrete format of a final equilibrium equation, which is shown in an expression (7); wherein the content of the first and second substances,K xu ，K xv ，K xp is composed ofxDirection to be solved variableu,v,PThe coefficient (c) is obtained by the formula (2), the formula (1) or the formula (5), the formula (1), and the coefficient for the compressible materialK xp =0；K yu ，K yu ，K yp Is composed ofyDirection to be solved variableu,v,PThe coefficient (c) is obtained by the formula (2), the formula (1) or the formula (5), the formula (1), and the coefficient for the compressible materialK yp =0；K pu ，K pv The coefficients introduced for the non-compressible region under non-compressible conditions;
example 7: on the basis of embodiment 6, in step S3, the method includes the sub-steps of: the node displacement obtained using equation (7) and the cell static pressure of the material domain divided according to step S1 obtain the stress distribution of the cell center using equation (2) or equation (5).
The functionality of the present invention, if implemented in the form of software functional units and sold or used as a stand-alone product, may be stored in a computer readable storage medium. 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 all or part of the steps of the method according to the embodiments of the present invention are executed in a computer device (which may be a personal computer, a server, or a network device) and corresponding software. And the aforementioned storage medium includes: various media capable of storing program codes, such as a usb disk, a removable hard disk, or an optical disk, exist in a read-only Memory (RAM), a Random Access Memory (RAM), and the like, for performing a test or actual data in a program implementation.
Claims (7)
1. A computational method for predicting elastic behavior of a compressible-incompressible composite, comprising:
step 1, marking unit material attributes and variables to be solved of a distribution unit;
step 2, obtaining a discrete format of the elastic equation of the compressible-incompressible double-layer composite material;
and 3, obtaining the central stress of the unit.
2. A calculation method for predicting the elastic behaviour of a compressible-incompressible composite according to claim 1, characterised in that in step 1 it comprises the sub-steps of: and S1, setting a composite material laying mode, storing the composite material laying mode in the cell center by adopting a multi-grid method, storing the hydrostatic pressure to be solved in the cell center, and marking the discrete cells as composite material interface cells or internal cells.
3. A calculation method for predicting the elastic behaviour of a compressible-incompressible composite according to claim 1, characterised in that in step 2 it comprises the sub-steps of: s2, based on the lattice-point-type FVM discrete elasticity problem control equation, introducing the essential relation of compressible materials or non-compressible materials into the internal control body, introducing the essential relation of the compressible materials or the non-compressible materials into discrete units at the interface in a segmented mode according to unit attribute marks when controlling the volume line integration, and then solving the discrete equation to obtain the displacement at the unit nodes and the hydrostatic pressure distribution at the unit center.
4. A calculation method for predicting the elastic behaviour of a compressible-incompressible composite according to claim 3, characterised in that in step 3 it comprises the sub-steps of: and S3, marking the essential relationship of the compressible material or the incompressible material by the cell attribute to obtain the cell center stress distribution.
5. A calculation method for predicting the elastic behavior of a compressible-incompressible composite according to claim 2, wherein in step S1, it comprises the sub-steps of: carrying out material identification on the discrete grid according to the material laying mode of the double-layer composite material, and defining the elastic modulus E and the Poisson ratio in the center of the compressible material unitνCenter-defined modulus of elasticity of incompressible materialEAnd hydrostatic pressure to be relievedP(ii) a And marking the discrete control bodies according to the discrete unit positions and the material attribute distribution to form a non-compressible material area control body, a boundary area control body and a non-compressible area control body.
6. A calculation method for predicting the elastic behavior of a compressible-incompressible composite according to claim 3, wherein in step S2, it comprises the sub-steps of: starting from an isotropic steady-state linear elastic equilibrium equation, the control equation of the integral format is as follows:
wherein the content of the first and second substances,
is an integral control line;
is the interface external normal vector;
representing the Cauchy stress tensor;
for isotropic linear elastomers, the constitutive relation is:
wherein the content of the first and second substances,
the strain tensor of Cauchy is expressed,
is a unit tensor, wherein
Expressing the Lame coefficient, when the plane strain assumption is adopted, the expression is:
wherein
In order to be the young's modulus,
is the poisson ratio;
the Cauchy strain tensor expression is:
wherein the content of the first and second substances,u i indicating the displacement in the ith direction,x i is a global coordinate of the ith direction,u j indicating the displacement in the jth direction,x j global coordinates of j-th direction;
poisson's ratio for incompressible materials
Hydrostatic pressure
The variable to be solved is introduced into the equation (2) and is converted into the following equation:
the following non-compressible conditions were introduced for the non-compressible material region:
wherein the content of the first and second substances,Vrepresenting an integral control volume;
the control body in the compressible material region introduces the formula (2) into the formula (1), the control body in the non-compressible material region introduces the formula (5) into the formula (1), and the control body in the interface is controlled by an integral control lineSDetermining whether the material is compressible or non-compressible material by the grid unit, and introducing the method formula (2) or the formula (5) into the balance equation (1); assuming that the incompressible condition (6) is required to be satisfied in the cell for the incompressible material region, the one at the center of the cell is adoptedxDirection andycalculating the positive strain approximation of the direction;
by the above substeps in step S2, a discrete format of the final equilibrium equation is obtained by performing line integration in the control volume formed by the nodes, see equation (7);
wherein the content of the first and second substances,K xu ，K xv ，K xp is composed ofxDirection to be solved variableu,v,PThe coefficient (c) is obtained by the formula (2), the formula (1) or the formula (5), the formula (1), and the coefficient for the compressible materialK xp =0；K yu ，K yu ，K yp Is composed ofyDirection to be solved variableu,v,PThe coefficient (c) is obtained by the formula (2), the formula (1) or the formula (5), the formula (1), and the coefficient for the compressible materialK yp =0；K pu ，K pv The coefficients introduced by the incompressible region under the incompressible condition.
7. The calculation method for predicting the elastic behavior of a compressible-incompressible composite according to claim 6, wherein the step S3 comprises the following sub-steps: the node displacement obtained using equation (7) and the cell static pressure of the material domain divided according to step S1 obtain the stress distribution of the cell center using equation (2) or equation (5).
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| CN115995277A (en) * | 2023-03-22 | 2023-04-21 | 中国空气动力研究与发展中心计算空气动力研究所 | Method, device, equipment and medium for evaluating material dynamic characteristics |
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