CN114996787A - Stress distribution determination method and device for gradient functional material ball structure - Google Patents

Stress distribution determination method and device for gradient functional material ball structure Download PDF

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CN114996787A
CN114996787A CN202210897801.7A CN202210897801A CN114996787A CN 114996787 A CN114996787 A CN 114996787A CN 202210897801 A CN202210897801 A CN 202210897801A CN 114996787 A CN114996787 A CN 114996787A
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闫林统
赵天宇
叶雨农
李嵬
汪志强
高香珍
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Abstract

The disclosure relates to the field of nondestructive testing of gradient functional materials, and provides a method and a device for determining stress distribution of a gradient functional material ball structure. The method comprises the steps of establishing a gradient functional material ball structure model based on a preset material type, evenly dividing a gradient functional material layer in the model into a plurality of shell layers from inside to outside, establishing a control equation of each shell layer, determining stress general solutions of the shell layers according to an elastic mechanics theory and the control equation, determining stress relations between the shell layers based on the stress general solutions and preset boundary conditions, further determining the stress relations between the innermost shell layer and the outermost shell layer, determining the stress corresponding to the innermost shell layer and the outermost shell layer based on the stress general solutions, the boundary conditions, the number of the shell layers and the material type, further determining the stress corresponding to other shell layers, and obtaining a stress distribution diagram of the model, so that performance indexes of the gradient functional material are tested in a simulation mode, the test cost is reduced, the test efficiency is improved, and resource waste is avoided.

Description

Stress distribution determination method and device for gradient functional material ball structure
Technical Field
The disclosure relates to the technical field of nondestructive testing of gradient functional materials, and in particular relates to a method and a device for determining stress distribution of a ball structure of a gradient functional material.
Background
The gradient functional material is a novel material compounded by two or more than two materials with continuously gradient-changed components and structures, is a novel functional composite material developed to meet the requirement of repeated normal work in extreme environments according to the requirements of high and new technical fields of modern aerospace industry and the like. The design requirement of the gradient functional material changes with the change of the internal position of the machine element, and the design requirement is met by optimizing the overall performance of the component.
From the structural point of view of materials, gradient functional materials are different from homogeneous materials and composite materials. The gradient functional material is formed by selecting two or more materials with different properties, and continuously changing the composition and the structure of the two or more materials to ensure that the interface between the two or more materials disappears, so that the properties of the materials are slowly changed along with the change of the composition and the structure of the materials. From the combination mode of materials, the gradient functional material can be divided into various combination modes such as metal/alloy, metal/nonmetal, nonmetal/ceramic, metal/ceramic, ceramic/ceramic and the like, so that the material with various special functions can be obtained. From the viewpoint of the change of the composition of the material, the gradient functional material can be divided into: gradient functional coating type, i.e., a coating layer formed on a base material in a gradual change; the gradient function connection type is that the material is bonded with the joint between two matrixes and the material composition shows gradient change; gradient functional monolithic type, i.e. a structural material with a gradient gradual change of the material composition from one side to the other.
After the traditional gradient functional material is prepared, indexes such as strength, rigidity, stability, heat resistance and the like of the gradient functional material can be tested only through an experimental mode. However, this method not only requires the use of expensive laboratory equipment, but also is cumbersome and often reaches the physical limit of the test sample, resulting in the destruction of the test sample and the waste of resources.
Disclosure of Invention
The present disclosure is directed to at least one of the problems of the prior art, and provides a method and an apparatus for determining a stress distribution of a ball structure of a gradient functional material.
In one aspect of the present disclosure, a method for determining a stress distribution of a ball structure of a gradient functional material is provided, which includes the following steps:
establishing a gradient functional material ball structure model based on the preset material types and sizes, wherein the gradient functional material ball structure model comprises hollow balls and a gradient functional material layer wrapping the hollow balls;
equally dividing the gradient functional material layer into a plurality of shell layers from inside to outside, and respectively establishing a control equation of each shell layer;
determining the stress general solution of each shell layer according to the theory of elastic mechanics and a control equation;
determining the stress relation among the shell layers based on the stress general solution and the preset boundary conditions;
determining the stress relation between the innermost shell and the outermost shell according to the stress relation among the shells, wherein the innermost shell is the shell closest to the hollow ball, and the outermost shell is the shell farthest from the hollow ball;
respectively determining the stress corresponding to the innermost shell layer and the outermost shell layer based on the stress general solution, the stress relation, the size, the boundary condition, the number of the shell layers and the material type of the innermost shell layer and the outermost shell layer;
and respectively determining the stress corresponding to each shell layer according to the stress corresponding to the innermost shell layer and the outermost shell layer and the stress relation between the shell layers to obtain a stress distribution diagram of the gradient functional material ball structure model.
Optionally, determining a stress general solution of each shell layer according to an elastic mechanics theory and a control equation, including:
and solving a control equation according to a displacement potential function method to obtain a displacement component general solution and a stress component general solution which respectively correspond to each shell.
Alternatively, the general solution for the displacement component is expressed as:
Figure 551316DEST_PATH_IMAGE001
Figure 239786DEST_PATH_IMAGE002
the stress component general solution is expressed as:
Figure 97013DEST_PATH_IMAGE003
Figure 670077DEST_PATH_IMAGE004
Figure 13203DEST_PATH_IMAGE005
Figure 138153DEST_PATH_IMAGE006
wherein the content of the first and second substances,irepresents the number of the shell layers arranged from inside to outside in the structural model of the gradient functional material sphere andiis a positive integer and is a non-zero integer,r、θrespectively representing points in the spherical structural model of the gradient functional material in a spherical coordinate systemrDirection (b),θThe coordinate component of the direction is established by taking the sphere center of the hollow sphere as the origin,
Figure 950252DEST_PATH_IMAGE007
respectively represent shell layersiIn a spherical coordinate systemrDirection (b),θThe displacement component of the direction of the displacement,nthe order of the convergence is represented by,P n2 representing the legendre series of even powers,
Figure 576274DEST_PATH_IMAGE008
representing shell layersiIn a spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 790218DEST_PATH_IMAGE009
representing shell layersiIn a spherical coordinate systemθThe direction of the positive stress is the direction of the positive stress,
Figure 86070DEST_PATH_IMAGE010
outer surface of (2) represents a shell layeriIn a spherical coordinate system
Figure 634732DEST_PATH_IMAGE011
The direction of the positive stress is the direction of the positive stress,
Figure 549598DEST_PATH_IMAGE012
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction ofθThe shear stress in the direction of the direction,
Figure 601736DEST_PATH_IMAGE013
are all shell layersiIs determined by the unknown coefficients of (a) and (b),
Figure 334069DEST_PATH_IMAGE014
Figure 855180DEST_PATH_IMAGE015
are all intermediate variables and are represented as:
Figure 563285DEST_PATH_IMAGE016
wherein the content of the first and second substances,λ i andG i all represent shell layersiAnd is composed of shell layeriThe physical properties of itself.
Optionally, the boundary condition comprises a shell layeriAnd shell layeriA boundary condition of +1, expressed as:
Figure 751821DEST_PATH_IMAGE017
，
Figure 655055DEST_PATH_IMAGE018
wherein, the first and the second end of the pipe are connected with each other,
Figure 178309DEST_PATH_IMAGE019
representing shell layersi+1 outer surface in spherical coordinate systemrA positive stress in a direction of the beam,
Figure 434978DEST_PATH_IMAGE020
representing shell layersi+1 outer surface in spherical coordinate system with normal directionrDirection and direction ofθThe shear stress in the direction of the direction,
Figure 727288DEST_PATH_IMAGE021
respectively represent shell layersi+1 in the spherical coordinate systemrDirection (b),θA displacement component of direction;
the boundary condition further comprises a shell layer 1 and a shell layersIs expressed as:
Figure 535844DEST_PATH_IMAGE022
wherein, the first and the second end of the pipe are connected with each other,
Figure 31548DEST_PATH_IMAGE023
and
Figure 341175DEST_PATH_IMAGE024
respectively represent shell layer 1 and shell layersIn a spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 628937DEST_PATH_IMAGE025
and
Figure 483760DEST_PATH_IMAGE026
respectively represent shell layer 1 and shell layersThe outer surface of the spherical coordinate system has a normal direction ofrDirection and direction ofθThe shear stress in the direction of the steel wire,
Figure 981607DEST_PATH_IMAGE027
、
Figure 314499DEST_PATH_IMAGE028
respectively show the shell 1 in a spherical coordinate systemrDirection (b),θThe displacement component of the direction of the displacement,
Figure 309963DEST_PATH_IMAGE029
respectively represent shell layerssIn a spherical coordinate systemrDirection (b),θA displacement component of direction.
Optionally, the stress relationship between the shells includes a relationship of unknown coefficients between the shells, which is expressed as:
Figure 725901DEST_PATH_IMAGE030
(ii) a Wherein the content of the first and second substances,
Figure 196196DEST_PATH_IMAGE031
it is indicated that the multiplication is a cumulative multiplication,T 1 a matrix of unknown coefficients representing the innermost shell layer,T i+1 、T i 、T i-1 respectively represent shell layersiShell layer ofi+1, shell layeri-1, and
Figure 847626DEST_PATH_IMAGE032
；R i 、R i-1 individual watchShow shell layeriShell layer ofi-1 distance of the outer surface to the centre of the hollow sphere;M i2 、M i(+1)2 、M i(-1)2 are all intermediate variables, andM i2 expressed as:
Figure 454188DEST_PATH_IMAGE033
；M i3 、M i(+1)3 、M i(-1)3 are all intermediate variables, andM i3 expressed as:
Figure 41027DEST_PATH_IMAGE034
；
Figure 247886DEST_PATH_IMAGE035
is an intermediate variable, expressed as:
Figure 188161DEST_PATH_IMAGE036
。
optionally, the stress relationship between the innermost shell and the outermost shell includes a relationship of unknown coefficients between the innermost shell and the outermost shell, which is expressed as:
Figure 898497DEST_PATH_IMAGE037
wherein the content of the first and second substances,sthe number of the outermost shell layer is shown,T s a matrix representing the unknown coefficient composition of the outermost shell.
Optionally, the lark constant is expressed as:
Figure 531603DEST_PATH_IMAGE038
(ii) a Wherein, the first and the second end of the pipe are connected with each other,E i representing shell layersiIs expressed as
Figure 694600DEST_PATH_IMAGE039
，E 0 The young's modulus of the innermost shell layer is expressed,E s represents the Young's modulus of the outermost shell layer;
Figure 563199DEST_PATH_IMAGE040
representing shell layersiAnd is expressed as
Figure 409932DEST_PATH_IMAGE041
，
Figure 203488DEST_PATH_IMAGE042
The poisson's ratio of the innermost shell layer is expressed,
Figure 870093DEST_PATH_IMAGE043
representing the poisson's ratio of the outermost shell.
In another aspect of the present disclosure, there is provided a stress distribution determining apparatus of a gradient functional material ball structure, the determining apparatus including:
the system comprises an establishing module, a calculating module and a processing module, wherein the establishing module is used for establishing a gradient functional material ball structure model based on the preset material types and sizes, and the gradient functional material ball structure model comprises hollow balls and a gradient functional material layer wrapping the hollow balls;
the dividing module is used for averagely dividing the gradient functional material layer into a plurality of shell layers from inside to outside and respectively establishing a control equation of each shell layer;
the first determining module is used for determining the stress general solution of each shell layer according to the elastomechanics theory and the control equation;
the second determining module is used for determining the stress relation among the shell layers based on the stress general solution and the preset boundary conditions;
the third determining module is used for determining the stress relation between the innermost shell and the outermost shell according to the stress relation among the shells, wherein the innermost shell is the shell closest to the hollow ball, and the outermost shell is the shell farthest from the hollow ball;
the fourth determining module is used for respectively determining the stress corresponding to the innermost shell layer and the outermost shell layer based on the stress general solution, the stress relation, the size, the boundary condition, the number of the shell layers and the material type of the innermost shell layer and the outermost shell layer;
and the fifth determining module is used for determining the stress corresponding to each shell layer according to the stress corresponding to the innermost shell layer and the outermost shell layer respectively and the stress relation between the shell layers respectively to obtain a stress distribution diagram of the gradient functional material ball structure model.
In another aspect of the present disclosure, there is provided an electronic device including:
at least one processor; and a memory communicatively coupled to the at least one processor; wherein the content of the first and second substances,
the memory stores instructions executable by the at least one processor to enable the at least one processor to perform the method for determining a stress distribution of a spherical structure of gradient functional material as described above.
In another aspect of the present disclosure, a computer-readable storage medium is provided, which stores a computer program, and the computer program is executed by a processor to implement the method for determining the stress distribution of the gradient functional material sphere structure described above.
Compared with the prior art, the method firstly establishes a gradient functional material ball structure model based on the preset material types, equally dividing the gradient functional material layer in the model into a plurality of shell layers from inside to outside, respectively establishing a control equation of each shell layer, then determining a stress general solution of each shell layer according to an elastomechanics theory and the control equation, determining a stress relation among the shell layers based on the stress general solution and a preset boundary condition, further determining the stress relation between the innermost shell layer and the outermost shell layer, respectively determining the stress corresponding to the innermost shell layer and the outermost shell layer on the basis of the stress general solution, the boundary condition, the number of the shell layers and the material type of the gradient functional material layer, and determining the stress corresponding to other shell layers, and obtaining a stress distribution diagram of the gradient functional material ball structure model based on the stress corresponding to each shell layer. This openly can be according to the material kind and the size of gradient functional material, combine the environmental condition to determine the inside stress distribution of gradient functional material, thereby can test the performance index of gradient functional material through the emulation mode, and the testing process need not to utilize expensive experimental facilities to carry out the loaded down with trivial details experiment of procedure, also can not lead to the fact destruction to the test sample, not only reduced the test cost, still improved efficiency of software testing, the wasting of resources has been avoided, still can be used to guide the structural design and the material ratio of gradient functional material, thereby produce the more reliable gradient functional material ball of performance according to the environmental condition.
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One or more embodiments are illustrated by way of example in the accompanying drawings, which correspond to the figures in which like reference numerals refer to similar elements and which are not to scale unless otherwise specified.
Fig. 1 is a flowchart of a method for determining a stress distribution of a gradient functional material ball structure according to an embodiment of the present disclosure;
FIG. 2 is a schematic diagram of a structural model of a gradient functional material sphere according to another embodiment of the present disclosure;
fig. 3 is a schematic diagram of a spherical coordinate system according to another embodiment of the disclosure;
FIG. 4 is a schematic force diagram of a structural model of a functionally graded material sphere under the action of a radial point load according to another embodiment of the present disclosure;
FIG. 5 is a schematic diagram of a structural model of a gradient functional material sphere according to another embodiment of the present disclosure and a corresponding stress distribution diagram;
FIG. 6 is a schematic diagram of a structural model of a gradient functional material sphere according to another embodiment of the present disclosure and a corresponding stress distribution diagram;
FIG. 7 is a schematic diagram of a structural model of a gradient functional material sphere according to another embodiment of the present disclosure and a corresponding stress distribution diagram;
FIG. 8 is a schematic diagram of a structural model of a gradient functional material sphere and a corresponding stress distribution map according to another embodiment of the present disclosure;
FIG. 9 is a stress distribution plot of a graded functional material provided in accordance with another embodiment of the present disclosure;
fig. 10 is a schematic structural diagram of a stress distribution determining apparatus of a gradient functional material sphere structure according to another embodiment of the present disclosure;
fig. 11 is a schematic structural diagram of an electronic device according to another embodiment of the present disclosure.
Detailed Description
In general, the dispersed phase is uniformly distributed, and the overall performance of the composite material is the same, but in some cases, it is often desirable that two sides of the same material have different properties or functions, and that two sides with different properties can be perfectly combined, so as to prevent the material from being damaged due to the mismatch of the properties under the severe use conditions. Taking the most representative supersonic combustion ramjet engine in the propulsion system of a space shuttle as an example, the temperature of combustion gas usually exceeds 2000 ℃, which can generate strong thermal shock on the wall of a combustion chamber; while the other side of the combustion chamber wall is subjected to cooling by liquid hydrogen as fuel, typically at a temperature of around-200 c. Thus, the side of the combustion chamber wall that contacts the combustion gases is subjected to extremely high temperatures, while the side that contacts the liquid hydrogen is subjected to extremely low temperatures, which obviously cannot be met by a composite material having the same overall properties. Thus, it is conceivable to use a combination of metal and ceramic, and to use ceramic for coping with high temperature and metal for coping with low temperature. However, when the conventional technique is used to combine metal and ceramic, the interface thermodynamics of the two are not well matched, and the metal and ceramic can still be damaged under the condition of extreme thermal stress. In view of this situation, in 1984, the japanese scientist pingjing stamen first proposed a new concept and concept of a gradient functional material and developed a study. The basic idea of this completely new material design concept is: according to specific requirements, two materials with different properties are selected and used, and the internal interface of the two materials is eliminated by continuously changing the composition and the structure of the two materials, so that the heterogeneous material with the function gradually changed corresponding to the change of the composition and the structure is obtained, and the property mismatching factor of the combining part is reduced and overcome. For example, with the above-described combustion chamber wall, by continuously controlling the change in the internal composition and the fine structure between the ceramic and the metal, no interface occurs between the ceramic and the metal, so that the integrated material in which the ceramic and the metal are combined has a new function of having a good thermal stress resistance strength and a good mechanical strength.
When performance indexes such as strength, rigidity, stability, thermal shock resistance and the like of the gradient functional material are researched, the traditional method is usually tested in an experimental mode. However, this method not only requires the use of expensive laboratory instruments and equipment, but also is cumbersome, and the test sample often reaches the physical limit during the experiment, causing the destruction of the test sample and the waste of resources.
To make the objects, technical solutions and advantages of the embodiments of the present disclosure more apparent, the embodiments of the present disclosure will be described in detail below with reference to the accompanying drawings. However, it will be appreciated by those of ordinary skill in the art that in various embodiments of the disclosure, numerous technical details are set forth in order to provide a better understanding of the disclosure. However, the technical solutions claimed in the present disclosure can be implemented without these technical details and various changes and modifications based on the following embodiments. The following embodiments are divided for convenience of description, and should not constitute any limitation to the specific implementation of the present disclosure, and the embodiments may be mutually incorporated and referred to without contradiction.
One embodiment of the present disclosure relates to a method for determining stress distribution of a gradient functional material ball structure, the flow of which is shown in fig. 1, and the method includes the following steps:
step 101, establishing a gradient functional material ball structure model based on preset material types and sizes, wherein the gradient functional material ball structure model comprises hollow balls and a gradient functional material layer wrapping the hollow balls.
Specifically, since the gradient functional material ball structure is adapted to different boundary conditions, which is often a hollow ball structure, the size of the gradient functional material ball structure model needs to be preset, and the size may include the inner diameter of the gradient functional material ball structure model, i.e., the radius of the hollow ball, and the outer diameter of the gradient functional material ball structure model, i.e., the distance from the outer surface of the gradient functional material ball structure model to the center of the hollow ball. In addition, since the gradient functional material is composed of two or more materials with different properties and is formed by continuously changing the composition and structure of the two or more materials, the material type of the gradient functional material needs to be preset, and thus, a structural model of the gradient functional material ball including the hollow ball and the gradient functional material layer wrapping the hollow ball is established based on the preset material type and size.
As shown in fig. 2, the structural model of the gradient functional material sphere may include a hollow sphere with a point o as a sphere center and a gradient functional material layer wrapping the hollow sphere. The gradient functional material layer is composed of gradient functional materials, and the included material types are preset material types. For example, when the predetermined material types are ceramic and metal, the gradient functional material layer includes the material types of ceramic and metal.
It should be noted that, in order to obtain different gradient functional material layers, a person skilled in the art may set different types of materials according to actual needs. For example, the predetermined material type may be metal and alloy, metal and nonmetal, nonmetal and ceramic, and the like, which is not limited in the present embodiment.
And 102, averagely dividing the gradient functional material layer into a plurality of shell layers from inside to outside, and respectively establishing a control equation of each shell layer. Specifically, as shown in fig. 2, the gradient functional material layer can be divided into two layers from inside to outsidesA shell layer, wherein,sis an integer of not less than 2. It should be noted that the present embodiment is not limited theretosTo limit the specific values of, for example,smay be 2,3,4,5, etc., and may be selected by those skilled in the art according to actual needs.
The number of the shell layers arranged from the inside to the outside is markedi，iIs a positive integer, theniMay be 1, 2.,s。iand 1, a shell layer 1 in the gradient functional material layer, namely the 1 st shell layer arranged from inside to outside in the gradient functional material layer, is also the shell layer closest to the hollow sphere, namely the innermost shell layer.iIs composed ofsSecond layer arranged from inside to outside in the layer of graded functional materialsAnd the shell layer is also the shell layer farthest from the hollow sphere, namely the outermost shell layer.iIs 1 andsthe value in between indicates the second from inside to outside arrangement in the gradient functional material layeriAnd (4) a shell layer.
The governing equations include physical equations, equilibrium equations, and geometric equations. The physical equations are used to express the relationship between stress and strain, and each shell has 6 physical equations. The equilibrium equation is used to represent the relationship between the stresses, with 3 equilibrium equations per shell. The geometric equations are used to express the relationship between displacement and strain, and each shell has 6 geometric equations.
As shown in fig. 3, a spherical coordinate system is established with the sphere center o of the hollow sphere in the structural model of the sphere made of the gradient functional material as the origin, and then for point (c), (d)r,θ,
Figure 542382DEST_PATH_IMAGE044
)，r、θ、
Figure 492890DEST_PATH_IMAGE044
Respectively representing points in the spherical structural model of the gradient functional material in a spherical coordinate systemrDirection (b),θDirection (b),
Figure 202220DEST_PATH_IMAGE044
The coordinate component of the direction is such that,rthe direction represents from the origin o to the point: (r,θ,
Figure 605388DEST_PATH_IMAGE044
) In the direction of (a) of (b),θdirection represents from the z-axis to a point: (r,θ,
Figure 691156DEST_PATH_IMAGE044
) In the direction of the radius of the shaft,
Figure 371536DEST_PATH_IMAGE044
direction denotes counterclockwise from the x-axis to a point: (r,θ,
Figure 501035DEST_PATH_IMAGE044
) In the direction of the projection of the xy plane.
As shown in FIG. 2, the radius of the hollow sphere is denotedR 0 Shell layer ofiThe distance from the outer surface of (a) to the center o of the hollow sphere is recorded asR i Then, thenR 1 The distance from the outer surface of the shell layer 1 to the center o of the sphere, that is, the distance from the contact surface of the shell layer 1 and the adjacent outer shell layer, that is, the shell layer 2 to the center o of the sphere,R s representing shell layerssTo the center o of the sphere. In conjunction with FIG. 3, for point (A)r,θ,
Figure 142232DEST_PATH_IMAGE044
) When is coming into contact withrThe value of (a) is [0 ", R 0 ) At the time point (r,θ,
Figure 15379DEST_PATH_IMAGE044
) Are points within the hollow sphere. When in userIs taken asR 0 At the time point (r,θ,
Figure 691211DEST_PATH_IMAGE044
) The point on the outer surface of the hollow sphere is the point on the contact surface of the hollow sphere and the innermost shell layer. When in userIs taken from the value ofR i-1 , R i ) At the time point (r,θ,
Figure 726032DEST_PATH_IMAGE044
) Is a shell layeriPoint (2).R i-1 Representing shell layersi-1 distance of the outer surface to the centre of the hollow sphere. When in userIs taken asR i-1 At the time point (r,θ,
Figure 996737DEST_PATH_IMAGE044
) Is a shell layeri1 points on the outer surface. In particular, wheniShell layer of = 1%iShell 0 represents a hollow sphere at-1,R i-1 namely, it isR 0 The distance from the outer surface of the hollow sphere to the center of the hollow sphere, i.e., the radius of the hollow sphere, is shown. When in userIs taken asR i At the time point (r,θ,
Figure 424307DEST_PATH_IMAGE044
) Is a shell layeriOuter ofA point on the surface. For example, whenrIs taken asR 1 Time, point (a)r,θ,
Figure 938334DEST_PATH_IMAGE044
) Is a point on the outer surface of the shell 1 and is also a point on the contact surface of the shell 1 with the shell 2. When in userIs taken asR s At the time point (r,θ,
Figure 691526DEST_PATH_IMAGE044
) Is a shell layersA point on the outer surface of (a).
Based on the structural model of the gradient functional material sphere shown in fig. 2 and the spherical coordinate system shown in fig. 3, the physical equation of each shell layer can be expressed as:
Figure 291004DEST_PATH_IMAGE045
(ii) a Wherein the content of the first and second substances,
Figure 381319DEST_PATH_IMAGE008
representing shell layersiIn a spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 766164DEST_PATH_IMAGE046
representing shell layersiIn a spherical coordinate systemθThe direction of the positive stress is the direction of the positive stress,
Figure 673946DEST_PATH_IMAGE047
representing shell layersiIn a spherical coordinate system
Figure 511452DEST_PATH_IMAGE044
The direction of the positive stress is the direction of the positive stress,
Figure 530093DEST_PATH_IMAGE048
representing shell layersiIn a spherical coordinate systemrA positive strain in the direction of the strain,
Figure 894078DEST_PATH_IMAGE049
representing shell layersiIn a spherical coordinate systemθA positive strain in the direction of the strain,
Figure 723494DEST_PATH_IMAGE050
representing shell layersiIn a spherical coordinate system
Figure 563143DEST_PATH_IMAGE044
A positive strain in the direction of the strain,
Figure 870627DEST_PATH_IMAGE051
representing shell layersiThe outer surface of the spherical coordinate system has a normal direction ofθDirection and direction of
Figure 360557DEST_PATH_IMAGE044
The shear stress in the direction of the direction,
Figure 610142DEST_PATH_IMAGE052
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction ofθThe shear stress in the direction of the steel wire,
Figure 156661DEST_PATH_IMAGE053
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction of
Figure 48263DEST_PATH_IMAGE044
The shear stress in the direction of the direction,
Figure 386840DEST_PATH_IMAGE054
representing shell layersiThe outer surface of the spherical coordinate system has a normal direction ofθDirection and direction of
Figure 558058DEST_PATH_IMAGE044
The shear strain in the direction of the direction,
Figure 106720DEST_PATH_IMAGE055
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction ofθThe shear strain in the direction of the direction,
Figure 490428DEST_PATH_IMAGE056
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction of
Figure 73725DEST_PATH_IMAGE044
Directional shear strain.
λ i AndG i all represent shell layersiAnd is composed of shell layeriThe physical properties of itself. Exemplary, Rad-secret constantλ i AndG i expressed as:
Figure 540479DEST_PATH_IMAGE038
(ii) a Wherein the content of the first and second substances,E i representing shell layersiIs expressed as
Figure 327169DEST_PATH_IMAGE039
，E 0 The young's modulus of the innermost shell layer is expressed,E s the Young's modulus of the outermost shell layer is shown.
Figure 29415DEST_PATH_IMAGE040
Representing shell layersiAnd is expressed as
Figure 217951DEST_PATH_IMAGE041
，
Figure 849746DEST_PATH_IMAGE042
The poisson's ratio of the innermost shell layer is expressed,
Figure 373000DEST_PATH_IMAGE043
representing the poisson's ratio of the outermost shell.
Based on the structural model of the gradient functional material sphere shown in fig. 2 and the spherical coordinate system shown in fig. 3, the balance equation of each shell layer can be expressed as:
Figure 629669DEST_PATH_IMAGE057
based on the spherical structural model of the gradient functional material shown in fig. 2 and the spherical coordinate system shown in fig. 3, the geometric equation of each shell can be expressed as:
Figure 656400DEST_PATH_IMAGE058
(ii) a Wherein the content of the first and second substances,
Figure 730535DEST_PATH_IMAGE059
respectively represent shell layersiIn a spherical coordinate systemrDirection (b),θDirection (b),
Figure 226238DEST_PATH_IMAGE011
A displacement component of direction.
And 103, determining stress general solutions of the shell layers according to the elastic mechanics theory and the control equation.
For example, in this step, a control equation may be solved according to a displacement potential function method in the theory of elastic mechanics, and a displacement component general solution and a stress component general solution corresponding to each shell are obtained, so that the displacement component general solution and the stress component general solution corresponding to each shell are used as the stress general solution of each shell.
Specifically, in this step, the physical equation, the equilibrium equation, and the geometric equation that are established based on the gradient functional material ball structure model shown in fig. 2 and the ball coordinate system shown in fig. 3 may be solved by a displacement potential function method, so as to obtain a displacement component general solution and a stress component general solution that correspond to each shell layer, respectively.
Wherein, the displacement component general solution can be expressed as:
Figure 535866DEST_PATH_IMAGE060
the stress component general solution can be expressed as:
Figure 167835DEST_PATH_IMAGE061
Figure 271927DEST_PATH_IMAGE004
Figure 645139DEST_PATH_IMAGE005
Figure 243611DEST_PATH_IMAGE062
wherein the content of the first and second substances,nthe order of the convergence is represented by,P n2 representing the legendre series of even powers,
Figure 244934DEST_PATH_IMAGE013
are all shell layersiIs determined by the unknown coefficients of (a) and (b),
Figure 536238DEST_PATH_IMAGE014
Figure 137026DEST_PATH_IMAGE015
are all intermediate variables and are represented as:
Figure 257298DEST_PATH_IMAGE016
。
and 104, determining the stress relation among the shell layers based on the stress general solution and the preset boundary conditions.
Specifically, since the interfaces between different materials in the gradient functional material layer tend to disappear completely, after the gradient functional material layer is divided into a plurality of shell layers on average, the contact between the shell layers can be set to be perfect contact, which means that the outer surfaces of the adjacent shell layers are stressed the same, and further means that the outer surfaces of the shell layers are stressed the same. For example, as shown in FIG. 4, the structural model of the gradient functional material sphere is subjected to radial point loadingpWhile the outer surface of each shell is also exposedpThe function of (1). That is, the boundary condition may appear as a shelliAnd shell layeriA boundary condition of +1, and can be expressed as:
Figure 395018DEST_PATH_IMAGE017
，
Figure 840912DEST_PATH_IMAGE018
wherein the content of the first and second substances,
Figure 532924DEST_PATH_IMAGE063
representing shell layersi+1 outer surface in spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 863412DEST_PATH_IMAGE020
representing shell layersi+1 outer surface in spherical coordinate system with normal directionrDirection and direction ofθThe shear stress in the direction of the direction,
Figure 839327DEST_PATH_IMAGE021
respectively represent shell layersi+1 in spherical coordinate systemrDirection (b),θA displacement component of direction.
In particular, the shell 1 and the shells are stressed the same on the outer surfaces of the shellssIs also equally stressed, i.e. the shell 1 and the shellsThe boundary conditions of (a) can be expressed as:
Figure 206854DEST_PATH_IMAGE064
wherein the content of the first and second substances,
Figure 901010DEST_PATH_IMAGE065
and
Figure 769608DEST_PATH_IMAGE066
respectively represent shell layer 1 and shell layersIn a spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 85183DEST_PATH_IMAGE067
and
Figure 138459DEST_PATH_IMAGE068
respectively represent shell layer 1 and shell layersThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction ofθThe shear stress in the direction of the direction,
Figure 805064DEST_PATH_IMAGE069
、
Figure 432065DEST_PATH_IMAGE070
respectively show the shell 1 in a spherical coordinate systemrDirection (b),θThe displacement component of the direction of the displacement,
Figure 257939DEST_PATH_IMAGE071
respectively represent shell layerssIn a spherical coordinate systemrDirection (b),θA displacement component of direction.
The expressions of the displacement component general solution and the stress component general solution included in the stress general solution and the shell layer in the boundary conditioniAnd shell layeriThe expressions for the boundary condition of +1, taken together, may result in the following expression:
Figure 967269DEST_PATH_IMAGE072
Figure 636016DEST_PATH_IMAGE073
order to
Figure 721784DEST_PATH_IMAGE074
，M i0 Is an intermediate variable, andM i0 and (3) disassembling to obtain:
Figure 402164DEST_PATH_IMAGE075
。
let the intermediate variableM i1 、M i2 、M i3 、
Figure 531663DEST_PATH_IMAGE035
Respectively as follows:
Figure 907281DEST_PATH_IMAGE076
Figure 46007DEST_PATH_IMAGE033
Figure 456260DEST_PATH_IMAGE034
Figure 897605DEST_PATH_IMAGE077
then there is
Figure 275366DEST_PATH_IMAGE078
Let us order
Figure 437357DEST_PATH_IMAGE032
Then, the expression obtained by combining the expression of the displacement component general solution and the expression of the stress component general solution included in the stress general solution with the expression of the boundary condition can be expressed as
Figure 222822DEST_PATH_IMAGE079
Therefore, the relationship of the unknown coefficients among the shell layers in the stress relationship among the shell layers can be obtained and expressed as:
Figure 710436DEST_PATH_IMAGE030
wherein the content of the first and second substances,
Figure 185279DEST_PATH_IMAGE031
it is indicated that the multiplication is a cumulative multiplication,T i+1 、T i 、T i-1 respectively represent shell layersiShell layer ofi+1, shell layeri-1, and
Figure 665808DEST_PATH_IMAGE032
will beT i IniAre respectively replaced byi+1、i-1, i.e. obtainingT i+1 、T i-1 ，T 1 A matrix of unknown coefficient composition representing the innermost shell layer, i.e.
Figure 50653DEST_PATH_IMAGE080
，M i(+1)2 、M i(-1)2 、M i(+1)3 、M i(-1)3 Are all intermediate variables, willM i2 In (1)iAre respectively replaced by (i+1)、(i-1) to obtainM i(+1)2 、M i(-1)2 Will beM i3 IniAre respectively replaced by (i+1)、(i-1) to obtainM i(+1)3 、M i(-1)3 。
And 105, determining the stress relation between the innermost shell layer and the outermost shell layer according to the stress relation among the shell layers.
Specifically, in this step, the relationship of the unknown coefficient between the innermost shell and the outermost shell in the stress relationship between the innermost shell and the outermost shell may be obtained according to the relationship of the unknown coefficient between the shells in the stress relationship between the shells. Therefore, according to the above expression of the relationship of the unknown coefficients between the shells, an expression of the relationship of the unknown coefficients between the innermost shell and the outermost shell can be obtained:
Figure 692856DEST_PATH_IMAGE037
(ii) a Wherein the content of the first and second substances,T s a matrix representing the unknown coefficient composition of the outermost shell, i.e.
Figure 795941DEST_PATH_IMAGE081
。
And 106, respectively determining the stress corresponding to the innermost shell layer and the outermost shell layer based on the stress general solution, the stress relation, the size, the boundary condition, the number of the shell layers and the material type of the innermost shell layer and the outermost shell layer.
Specifically, let an intermediate variable
Figure 689948DEST_PATH_IMAGE082
After the number of the shell layers, the type of the materials and the sizes of the innermost shell layer and the outermost shell layer are determined,R i and the Lambda constantλ i AndG i can be determined, and then
Figure 178567DEST_PATH_IMAGE035
Can be determined and, due to the convergence ordernIt is determined that the user is not in good condition,M i2 、M i3 andM i(+1)2 、M i(+1)3 can also be determined, thereby determining intermediate variablesSSpecific values of (a).
In obtaining intermediate variablesSAfter the specific value of (b), the relationship of the unknown coefficient between the innermost shell and the outermost shell in the stress relationship between the innermost shell and the outermost shell can be expressed asT s =ST 1 I.e. by
Figure 273562DEST_PATH_IMAGE083
. Combining the boundary condition mesolamella 1 with the shellsThe boundary condition of (1) can be solvedsIs unknown, i.e.
Figure 582052DEST_PATH_IMAGE084
. Solving a shell layer 1 and a shell layersIs the unknown coefficient of
Figure 279750DEST_PATH_IMAGE084
Then, the stress flux is respectively substituted into the shell 1 and the shell determined based on the stress fluxsThe stress expression of (1) and (1) can be determinedsThe corresponding stress is the stress corresponding to the innermost shell layer and the outermost shell layer.
And 107, respectively determining the stress corresponding to each shell according to the stress corresponding to the innermost shell and the outermost shell and the stress relation between the shells, and obtaining the stress distribution diagram of the gradient functional material ball structure model.
Specifically, the shell layer 1 and the shell layer are obtainedsIs the unknown coefficient of
Figure 373608DEST_PATH_IMAGE084
Then, by using the relation of the unknown coefficients among the shells in the stress relation among the shells, the unknown coefficients of other shells can be calculated one by one and are respectively substituted into the shell removing layer 1 and the shell determined based on the stress general solutionsThe stress expressions of other shells except the shell 1 can be obtainedsAnd obtaining the stress corresponding to each shell layer by the stress corresponding to other shell layers, and establishing a stress distribution diagram of the gradient functional material layer on the basis, wherein the stress distribution diagram can be used as the stress distribution diagram of the gradient functional material ball structure model.
It should be noted that after the unknown coefficients of each shell are obtained, the unknown coefficients are combined with a stress general solution, so that the stress corresponding to each shell can be quickly obtained according to the material type of the gradient functional material layer, the number of the divided shells, and the boundary condition determined according to the test index of the gradient functional material.
Due to the fact thatsThe value of (a) represents the number of shell layers in the graded functional material layer, and therefore,sthe larger the value of (a) indicates that the gradient functional material layer is more finely divided, the larger the number of shell layers is, so that the stress distribution of the gradient functional material layer determined based on the number of shell layers is more accurate and closer to the internal stress distribution of the gradient functional material constituting the gradient functional material layer. When in usesAnd when the stress distribution tends to be infinite, the stress distribution of the gradient functional material layer can be used as the internal stress distribution of the gradient functional material composing the gradient functional material layer.
Compared with the prior art, the method for constructing the gradient functional material sphere structure model firstly establishes the gradient functional material sphere structure model based on the preset material types, equally dividing the gradient functional material layer in the model into a plurality of shell layers from inside to outside, respectively establishing a control equation of each shell layer, then determining the stress general solution of each shell layer according to the elasto-mechanical theory and the control equation, determining the stress relation among the shell layers based on the stress general solution and preset boundary conditions, further determining the stress relation between the innermost shell layer and the outermost shell layer, and respectively determining the stress corresponding to the innermost shell layer and the outermost shell layer based on the stress general solution, the boundary condition, the number of the shell layers and the material type of the gradient functional material layer, and determining the stress corresponding to other shell layers, and obtaining a stress distribution diagram of the gradient functional material ball structure model based on the stress corresponding to each shell layer. This disclosed embodiment can be according to the material kind and the size of gradient functional material, combine the environmental condition to determine the internal stress distribution of gradient functional material, thereby can test the performance index of gradient functional material through the emulation mode, and the testing process need not to utilize expensive experimental facilities to carry out the loaded down with trivial details experiment of procedure, also can not lead to the fact destruction to the test sample, not only reduced the test cost, still improved efficiency of software testing, the wasting of resources has been avoided, still can be used to guide the structural design and the material ratio of gradient functional material, thereby produce the gradient functional material ball that the performance is more reliable according to the environmental condition.
In order to enable those skilled in the art to better understand the above-described embodiments, a specific example is described below.
A stress distribution determination method for a gradient functional material ball structure comprises the following steps:
the method comprises the following steps: determining the structure and the size of a spherical structure model of the gradient functional material: the gradient functional material ball structure model comprises hollow balls and a gradient functional material layer wrapping the hollow balls; and determining the numerical values of the inner diameter and the outer diameter, namely the radius of the hollow sphere and the distance from the outer surface of the structural model of the gradient functional material sphere to the sphere center of the hollow sphere. Determining the material type and the material elastic constant of the gradient functional material layer, wherein the material elastic constant comprises Young modulus and Poisson ratio. From the Young's module and Poisson's ratio, the Czochralski of a material can be derived using equation 1Number:
Figure 617333DEST_PATH_IMAGE085
（1）
wherein the content of the first and second substances,λandGin order to obtain the constant of the drawing density,Ein order to be the young's modulus,νis the poisson ratio. Determining the test condition of the gradient functional material: equally dividing the gradient functional material layer in the gradient functional material sphere structure model from inside to outsidesAnd (4) a shell layer. Shell layeriIs expressed as the Rad-secret constant of
Figure 429431DEST_PATH_IMAGE038
。iIs the serial number of the shell layers arranged from inside to outside and is a positive integer.E i Representing shell layersiIs expressed as
Figure 789874DEST_PATH_IMAGE039
，E 0 The young's modulus of the innermost shell layer is expressed,E s the Young's modulus of the outermost shell layer is shown.
Figure 738239DEST_PATH_IMAGE040
Representing shell layersiAnd is expressed as
Figure 34091DEST_PATH_IMAGE041
，
Figure 582753DEST_PATH_IMAGE042
The poisson's ratio of the innermost shell layer is expressed,
Figure 497619DEST_PATH_IMAGE043
representing the poisson ratio of the outermost shell. And establishing a structural model of the gradient functional material sphere as shown in figure 2. Wherein, the radius of the hollow ball, namely the inner diameter of the structural model of the gradient functional material ball isR 0 Shell layer ofiHas a distance to the center o of the hollow sphere ofR i Shell layer ofsThe distance from the outer surface to the center o, that is, the outer diameter of the structural model of the gradient functional material ball isR s 。
Step two: establishing a spherical coordinate system as shown in FIG. 3 for points (A), (B), and (C)r,θ,
Figure 80916DEST_PATH_IMAGE044
)，r、θ、
Figure 813249DEST_PATH_IMAGE044
Respectively representing points in the spherical structural model of the gradient functional material in a spherical coordinate systemrDirection (b),θDirection (b),
Figure 334360DEST_PATH_IMAGE044
The coordinate component of the direction is such that,rthe direction represents from the origin o to the point: (r,θ,
Figure 302185DEST_PATH_IMAGE044
) In the direction of (a) of (b),θdirection represents from the z-axis to a point: (r,θ,
Figure 615354DEST_PATH_IMAGE044
) In the direction of the radius of the roller,
Figure 862796DEST_PATH_IMAGE044
direction denotes counterclockwise from the x-axis to a point: (r,θ,
Figure 391909DEST_PATH_IMAGE044
) In the direction of the projection of the xy plane. Based on the theory of elastic mechanicssGoverning equations for each of the individual shells, the governing equations including physical equations, equilibrium equations, and geometric equations, i is 1, 2.,s。
the physical equation reflects the relationship between stress and strain and is expressed as the following formula (2):
Figure 382999DEST_PATH_IMAGE045
(2) (ii) a Wherein the content of the first and second substances,
Figure 675309DEST_PATH_IMAGE008
representing shell layersiIn a spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 483865DEST_PATH_IMAGE046
representing shell layersiIn a spherical coordinate systemθThe direction of the positive stress is the direction of the positive stress,
Figure 245148DEST_PATH_IMAGE047
representing shell layersiIn a spherical coordinate system
Figure 289196DEST_PATH_IMAGE044
The direction of the positive stress is the direction of the positive stress,
Figure 452324DEST_PATH_IMAGE048
representing shell layersiIn a spherical coordinate systemrA positive strain in the direction of the strain,
Figure 290836DEST_PATH_IMAGE049
representing shell layersiIn a spherical coordinate systemθA positive strain in the direction of the strain,
Figure 398469DEST_PATH_IMAGE050
representing shell layersiIn a spherical coordinate system
Figure 996941DEST_PATH_IMAGE044
A positive strain in the direction of the strain,
Figure 263843DEST_PATH_IMAGE051
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofθDirection and direction of
Figure 289568DEST_PATH_IMAGE044
The shear stress in the direction of the direction,
Figure 884497DEST_PATH_IMAGE086
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction ofθThe shear stress in the direction of the direction,
Figure 819085DEST_PATH_IMAGE087
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction of
Figure 425646DEST_PATH_IMAGE044
The shear stress in the direction of the direction,
Figure 871540DEST_PATH_IMAGE088
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofθDirection and direction of
Figure 953766DEST_PATH_IMAGE044
The shear strain in the direction of the direction,
Figure 159619DEST_PATH_IMAGE089
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction ofθThe shear strain in the direction of the direction,
Figure 135534DEST_PATH_IMAGE090
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction of
Figure 237482DEST_PATH_IMAGE044
Directional shear strain.
The equilibrium equation reflects the relationship between the stresses and is expressed as the following equation (3):
Figure 72583DEST_PATH_IMAGE057
（3）。
the geometric equation reflects the relationship between displacement and strain and is expressed as the following formula (4):
Figure 800237DEST_PATH_IMAGE058
（4）
wherein the content of the first and second substances,
Figure 646970DEST_PATH_IMAGE059
respectively represent shell layersiIn a spherical coordinate systemrDirection (b),θDirection (b),
Figure 434666DEST_PATH_IMAGE044
A displacement component of direction.
Step three: boundary conditions for the gradient functional material are established. Because the interface between different materials in the gradient functional material layer tends to disappear completely, when the inner surface of the gradient functional material ball structure model, namely the outer surface of the hollow ball in the gradient functional material ball structure model, is stressed byF(R 0 ,θ,
Figure 101271DEST_PATH_IMAGE044
) When the gradient functional material is used, the outer surface of the ball structure model of the gradient functional material, namely the outer surface of the outermost shell layer, is stressedF(R 0 ,θ,
Figure 773561DEST_PATH_IMAGE044
). Because the stress comprises normal stress and shear stress, the inner surface and the outer surface of the gradient functional material ball structure model correspond to the same normal stress and the same shear stress, and the boundary condition can be expressed as a shell layer 1 and a shell layersBoundary condition of (1), i.e. shell 1 and shellsAre equally stressed.
Step four: and solving a physical equation, a balance equation and a geometric equation which are included in the control equation by using a displacement potential function method in the elastic mechanics theory to obtain a displacement component general solution and a stress component general solution of each shell layer in the gradient functional material layer. Wherein the general solution of the displacement component is expressed by the following formulas (5) and (6):
Figure 198769DEST_PATH_IMAGE001
（5）
Figure 908099DEST_PATH_IMAGE091
（6）
the stress component general solution is expressed by the following formulas (7) to (10):
Figure 311267DEST_PATH_IMAGE092
（7）
Figure 397035DEST_PATH_IMAGE004
（8）
Figure 202049DEST_PATH_IMAGE005
（9）
Figure 82280DEST_PATH_IMAGE093
（10）
wherein the content of the first and second substances,nthe order of the convergence is indicated,P n2 representing the legendre series of even powers,
Figure 848111DEST_PATH_IMAGE013
are all shell layersiIs determined by the unknown coefficients of (a) and (b),
Figure 986837DEST_PATH_IMAGE014
Figure 131511DEST_PATH_IMAGE015
are all intermediate variables and are represented by the following formula (11):
Figure 697490DEST_PATH_IMAGE094
（11）
step five: because the interfaces among different materials in the gradient functional material layer tend to disappear completely, after the gradient functional material layer is divided into a plurality of shell layers on average, the contact among the shell layers is set to be perfect contact, namely the stress of the outer surfaces of the shell layers is the same, and the boundary condition shows that the shell layers are the sameiAnd shell layeriA boundary condition of +1, and is represented by the following formula (12):
Figure 216196DEST_PATH_IMAGE017
，
Figure 502821DEST_PATH_IMAGE018
（12）
wherein the content of the first and second substances,
Figure 10988DEST_PATH_IMAGE019
representing shell layersi+1 outer surface in spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 233022DEST_PATH_IMAGE020
representing shell layersi+1 outer surface in spherical coordinate system with normal directionrDirection and direction ofθThe shear stress in the direction of the direction,
Figure 98079DEST_PATH_IMAGE021
respectively represent shell layersi+1 in the spherical coordinate systemrDirection (b),θA displacement component of direction.
Combining the displacement component general solution and the stress component general solution obtained in the fourth step with the boundary condition represented by the above formula (12), obtaining the following formulas (13) to (16):
Figure 63761DEST_PATH_IMAGE095
（13）
Figure 432294DEST_PATH_IMAGE096
（14）
Figure 90809DEST_PATH_IMAGE097
（15）
Figure 52949DEST_PATH_IMAGE098
（16）
the above equations (13) to (16) are converted into a matrix form: order toM i0 Is a middle changeAmount of, andM i0 represented by the following formula (17),T i a matrix representing the unknown coefficient composition of the shell, anT i Represented by the following formula (18):
Figure 806010DEST_PATH_IMAGE099
（17）
Figure 310940DEST_PATH_IMAGE032
(18) (ii) a To pairM i0 (ii) performing a resolution to obtain formula (19):
Figure 124045DEST_PATH_IMAGE100
（19）
let the intermediate variableM i1 、M i2 、M i3 、
Figure 448847DEST_PATH_IMAGE101
Respectively, the following formulas (20), (21), (22) and (23):
Figure 271178DEST_PATH_IMAGE076
（20）
Figure 365036DEST_PATH_IMAGE033
（21）
Figure 489987DEST_PATH_IMAGE034
（22）
Figure 557212DEST_PATH_IMAGE077
（23）
it can thus be found that,M i0 can be represented by the following formula (24):
Figure 668387DEST_PATH_IMAGE078
(24) (ii) a The above formulas (13) to (16) can be represented by the following formula (25):
Figure 866019DEST_PATH_IMAGE079
（25）
due to convergence ordernIs determined, therefore, after the material type of the gradient functional material layer and the number of the shell layers are determined,M i2 、M i3 andM i(+1)2 、M i(+1)3 it is determined so that the relationship of the unknown coefficients of the adjacent shell layers can be obtained from the above equation (25) and expressed as the following equation (26), wherein,
Figure 37238DEST_PATH_IMAGE102
represents a cumulative multiplication:
Figure 320320DEST_PATH_IMAGE030
（26）。
according to the above formula (26), the outermost shell layer, i.e., the first shell layer, can be obtainedsThe relationship between the unknown coefficients of the individual shell layers and the unknown coefficient of the innermost shell layer, i.e., the 1 st shell layer, is expressed by the following formula (27):
Figure 625400DEST_PATH_IMAGE037
(27) (ii) a Order toSIs an intermediate variable and is represented by the following formula (28):
Figure 428271DEST_PATH_IMAGE082
(28) (ii) a The following formula (29) can be obtained:T s =ST 1 (29) by developing the above formula (29), the following formula (30) can be obtained:
Figure 19658DEST_PATH_IMAGE083
(30) (ii) a The mesolamella 1 and the mesolamella according to the above formula (30) and the boundary conditionssThe boundary condition of (2) can be solvedLayer 1 and Shell layersIs unknown, i.e.
Figure 806348DEST_PATH_IMAGE084
. When solving the unknown coefficient of the shell 1, i.e.
Figure 508594DEST_PATH_IMAGE103
Then, the unknown coefficients of the shells are substituted into the formula (26) to obtain the unknown coefficients of the shells, and the unknown coefficients of the shells are substituted into the corresponding displacement component solutions and stress component solutions to obtain the corresponding displacements and stresses of the shells respectively.
Step six: when in usesThe larger the numerical value is, the more finely the gradient functional material layer is divided, the more the number of the shell layers is, the more accurate the stress distribution of the gradient functional material layer determined based on the number of the shell layers is, and the closer the stress distribution is to the internal stress distribution of the gradient functional material composing the gradient functional material layer, so that the nondestructive testing of the gradient functional material is realized according to the internal stress distribution of the gradient functional material.
The method for determining the stress distribution of the ball structure of the gradient functional material according to the above embodiment is further described below by taking the gradient functional material composed of ceramic and steel as an example.
And setting the included angle between the point load acting range and the axis of the gradient functional material ball structure model as 5 degrees. Determining the size of the gradient functional material, and recording the inner diameter of the spherical structure model of the gradient functional material, namely the radius of the hollow sphereR 0 The outer diameter of the gradient functional material ball structure model, i.e. the distance from the outer surface of the gradient functional material ball structure model to the center of the hollow ball, is recorded asR s Let us orderR 0 /R s And = 0.2. The material types of the composition gradient functional material layer are set as ceramic and steel, and the Young modulus and the Poisson ratio of the ceramic are respectivelyE 0 And
Figure 821764DEST_PATH_IMAGE042
young's modulus and Poisson's ratio of steel are respectivelyE s And
Figure 69205DEST_PATH_IMAGE043
wherein, in the step (A),E 0 =80GPa，
Figure 592459DEST_PATH_IMAGE042
=0.3，E s =200GPa，
Figure 849128DEST_PATH_IMAGE043
and = 0.3. The boundary condition is set such that the outer surface of each shell is subjected to a radial point load and the inner surface of each shell is not subjected to any force. Establishing a stress schematic diagram of a gradient functional material ball structure model according to the boundary condition, wherein as shown in fig. 4, the outer surface of each shell layer is subjected to radial point loadpThe function of (1). The distance from a point in the spherical structure model of the gradient functional material to the spherical center of the hollow sphere isrThen, thenrThe value of (a) is [0, R s ]. When in userThe value of (a) is [0, R 0 ) When the point is a point in the hollow sphere; when the temperature is higher than the set temperaturerIs taken asR 0 When the hollow ball is in the hollow ball outer surface, the point is the point on the contact surface of the hollow ball and the innermost shell layer; when in userIs taken asR i When this point is the shell layeriA point on the outer surface of (a) a,iis taken as 1, 2.,s。
as shown in the left diagram of fig. 5, the gradient functional material layer is divided into 2 layers on average, i.e. the order ofs=2, theniThe value of (A) is 1 and 2, the Young modulus and the Poisson ratio of the innermost shell layer are respectively 80GPa and 0.3, the Young modulus and the Poisson ratio of the outermost shell layer are respectively 200GPa and 0.3,R 1 /R s 0.6, the stress distribution of the stress axis of the gradient functional material layer obtained by utilizing the steps from the first step to the fifth step and combining the boundary conditions is shown in the right graph in fig. 5.
As shown in the left diagram of fig. 6, the gradient functional material layer is divided into 3 layers on average, i.e. the order ofs=3, theniThe value of (a) is 1,2 and 3, and the Young modulus and the Poisson ratio of each shell layer are respectively 80GPa and 0.3; 140GPa, 0.3; 200GPa and 0.3 GPa of the total weight of the composite material,R 1 /R s the content of the organic acid was 0.47,R 2 /R s 0.74, the stress distribution of the stress axis of the gradient functional material layer obtained by the steps one to five and combining the boundary conditions is shown in the right graph in fig. 6.
As shown in the left diagram of fig. 7, the gradient functional material layer is divided into 4 layers on average, i.e. the order ofs=4, theniThe values of (a) and (b) are 1,2,3 and 4, and the Young modulus and the Poisson ratio of each shell layer are respectively 80GPa and 0.3; 120GPa, 0.3; 160GPa, 0.3; 200GPa and 0.3 GPa of the total weight of the composite material,R 1 /R s the content of the organic acid is 0.4,R 2 /R s the content of the organic acid is 0.6,R 3 /R s and (5) 0.8, obtaining the stress distribution of the stress axis of the gradient functional material layer by utilizing the first step to the fifth step and combining the boundary conditions, wherein the stress distribution is shown in the right graph in fig. 7.
As shown in the left diagram of fig. 8, the gradient functional material layer is divided into 5 layers on average, i.e. lets=5, theniThe value of (a) is 1,2,3,4 and 5, and the Young modulus and the Poisson ratio of each shell layer are respectively 80GPa and 0.3; 110GPa, 0.3; 140GPa, 0.3; 170GPa, 0.3; 200GPa and 0.3 GPa of the total weight of the composite material,R 1 /R s the content of the carbon dioxide is 0.36,R 2 /R s is a content of at least 0.52,R 3 /R s the content of the acid was 0.68,R 4 /R s 0.84, and combining the boundary conditions by the steps one to five, the stress distribution of the stress axis of the gradient functional material layer is obtained as shown in the right graph in fig. 8.
With the increasing number of the shells, the stress distribution of the gradient functional material layer obtained by the first step to the fifth step and combining the boundary conditions gradually approaches to the internal stress distribution of the gradient functional material forming the gradient functional material layer, and the stress distribution diagrams in fig. 5 to 8 are compared and fitted to obtain the internal stress distribution of the gradient functional material formed by ceramic and steel as shown in fig. 9, so that the nondestructive testing of the gradient functional material formed by ceramic and steel can be realized based on fig. 9.
Another embodiment of the present disclosure relates to a stress distribution determining apparatus of a gradient functional material ball structure, as shown in fig. 10, including:
the building module 1001 is used for building a gradient functional material ball structure model based on preset material types and sizes, wherein the gradient functional material ball structure model comprises a hollow ball and a gradient functional material layer wrapping the hollow ball; the dividing module 1002 is configured to averagely divide the gradient functional material layer into multiple shell layers from inside to outside, and respectively establish a control equation of each shell layer; the first determining module 1003 is used for determining stress general solutions of all shell layers according to an elastic mechanics theory and a control equation; a second determining module 1004, configured to determine a stress relationship between the shells based on a stress general solution and a preset boundary condition; a third determining module 1005, configured to determine a stress relationship between an innermost shell and an outermost shell according to the stress relationship between the shells, where the innermost shell is a shell closest to the hollow sphere, and the outermost shell is a shell farthest from the hollow sphere; a fourth determining module 1006, configured to determine, based on a stress general solution, a stress relationship, a size, a boundary condition, a number of shells, and a material type of the innermost shell and the outermost shell, stresses corresponding to the innermost shell and the outermost shell, respectively; a fifth determining module 1007, configured to determine the stress corresponding to each shell according to the stress corresponding to each innermost shell and the stress corresponding to each outermost shell, and the stress relationship between the shells, so as to obtain a stress distribution map of the gradient functional material ball structure model.
For a specific implementation method of the device for determining stress distribution of a ball structure of a gradient functional material provided in the embodiments of the present disclosure, reference may be made to the method for determining stress distribution of a ball structure of a gradient functional material provided in the embodiments of the present disclosure, and details are not repeated here.
Compared with the prior art, the method for constructing the gradient functional material sphere structure model firstly establishes the gradient functional material sphere structure model based on the preset material types, equally dividing the gradient functional material layer in the model into a plurality of shell layers from inside to outside, respectively establishing a control equation of each shell layer, then determining the stress general solution of each shell layer according to the elasto-mechanical theory and the control equation, determining the stress relation among the shell layers based on the stress general solution and preset boundary conditions, further determining the stress relation between the innermost shell layer and the outermost shell layer, respectively determining the stress corresponding to the innermost shell layer and the outermost shell layer on the basis of the stress general solution, the boundary condition, the number of the shell layers and the material type of the gradient functional material layer, and determining the stress corresponding to other shell layers, and obtaining a stress distribution diagram of the gradient functional material ball structure model based on the stress corresponding to each shell layer. This disclosed embodiment can be according to the material kind and the size of gradient functional material, combine the environmental condition to determine the internal stress distribution of gradient functional material, thereby can test the performance index of gradient functional material through the emulation mode, and the testing process need not to utilize expensive experimental facilities to carry out the loaded down with trivial details experiment of procedure, also can not lead to the fact destruction to the test sample, not only reduced the test cost, still improved efficiency of software testing, the wasting of resources has been avoided, still can be used to guide the structural design and the material ratio of gradient functional material, thereby produce the more reliable gradient functional material ball of performance according to the environmental condition.
Another embodiment of the present disclosure relates to an electronic device, as shown in fig. 11, including: at least one processor 1101; and a memory 1102 communicatively coupled to the at least one processor 1101; the memory 1102 stores instructions executable by the at least one processor 1101, and the instructions are executed by the at least one processor 1101, so that the at least one processor 1101 can execute the method for determining the stress distribution of the spherical structure of the gradient functional material according to the above embodiment.
Where the memory and processor are connected by a bus, the bus may comprise any number of interconnected buses and bridges, the bus connecting together various circuits of the memory and the processor or processors. The bus may also connect various other circuits such as peripherals, voltage regulators, power management circuits, and the like, which are well known in the art, and therefore, will not be described any further herein. A bus interface provides an interface between the bus and the transceiver. The transceiver may be one element or a plurality of elements, such as a plurality of receivers and transmitters, providing a means for communicating with various other apparatus over a transmission medium. The data processed by the processor is transmitted over a wireless medium via an antenna, which further receives the data and transmits the data to the processor. The processor is responsible for managing the bus and general processing and may also provide various functions including timing, peripheral interfaces, voltage regulation, power management, and other control functions. And the memory may be used to store data used by the processor in performing operations.
Another embodiment of the present disclosure relates to a computer-readable storage medium, which stores a computer program, and when the computer program is executed by a processor, the computer program implements the method for determining the stress distribution of the gradient functional material sphere structure according to the above embodiment.
That is, as can be understood by those skilled in the art, all or part of the steps in the method according to the foregoing embodiments may be implemented by a program instructing related hardware, where the program is stored in a storage medium and includes several instructions to enable a device (which may be a single chip, a chip, or the like) or a processor (processor) to execute all or part of the steps in the method according to each embodiment of the present disclosure. And the aforementioned storage medium includes: various media capable of storing program codes, such as a usb disk, a removable hard disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a magnetic disk, or an optical disk.
It will be understood by those of ordinary skill in the art that the foregoing embodiments are specific embodiments for practicing the present disclosure, and that various changes in form and details may be made therein without departing from the spirit and scope of the present disclosure in practice.

Claims (10)

1. A stress distribution determination method of a gradient functional material sphere structure is characterized by comprising the following steps:
establishing a gradient functional material ball structure model based on the preset material types and sizes, wherein the gradient functional material ball structure model comprises a hollow ball and a gradient functional material layer wrapping the hollow ball;
equally dividing the gradient functional material layer into a plurality of shell layers from inside to outside, and respectively establishing a control equation of each shell layer;
determining the stress general solution of each shell layer according to an elastic mechanics theory and the control equation;
determining a stress relation between the shell layers based on the stress general solution and a preset boundary condition;
determining the stress relation between the innermost shell and the outermost shell according to the stress relation between the shells, wherein the innermost shell is the shell closest to the hollow sphere, and the outermost shell is the shell farthest from the hollow sphere;
respectively determining the stress corresponding to the innermost shell layer and the outermost shell layer based on the stress general solution, the stress relation between the innermost shell layer and the outermost shell layer, the size, the boundary condition, the number of the shell layers and the material type;
and respectively determining the stress corresponding to each shell layer according to the stress corresponding to the innermost shell layer and the outermost shell layer and the stress relation between the shell layers to obtain the stress distribution diagram of the gradient functional material ball structure model.
2. The method of claim 1, wherein determining the stress solution for each shell layer based on the theory of elastic mechanics and the governing equation comprises:
and solving the control equation according to a displacement potential function method to obtain a displacement component general solution and a stress component general solution which respectively correspond to each shell layer.
3. The determination method according to claim 2, wherein the displacement component general solution is expressed as:
Figure 17773DEST_PATH_IMAGE001
the stress component general solution is expressed as:
Figure 138176DEST_PATH_IMAGE002
Figure 245809DEST_PATH_IMAGE003
Figure 93548DEST_PATH_IMAGE004
Figure 845603DEST_PATH_IMAGE005
wherein the content of the first and second substances,irepresents the serial number of the shell layers arranged from inside to outside in the structural model of the gradient functional material sphere andiis a positive integer and is a non-zero integer,r、θrespectively representing points in the spherical structural model of the gradient functional material in a spherical coordinate systemrDirection (b),θA coordinate component of the direction, the spherical coordinate system is established by taking the spherical center of the hollow sphere as an origin,
Figure 120596DEST_PATH_IMAGE006
respectively represent shell layersiIn a spherical coordinate systemrDirection (b),θThe displacement component of the direction of the displacement,nthe order of the convergence is represented by,P n2 representing the legendre series of even powers,
Figure 981104DEST_PATH_IMAGE007
representing shell layersiIn a spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 383267DEST_PATH_IMAGE008
representing shell layersiIn a spherical coordinate systemθThe direction of the positive stress is the direction of the positive stress,
Figure 239096DEST_PATH_IMAGE009
representing shell layersiIn a spherical coordinate system
Figure 435722DEST_PATH_IMAGE010
The direction of the positive stress is the direction of the positive stress,
Figure 642582DEST_PATH_IMAGE011
representing shell layersiThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction ofθThe shear stress in the direction of the direction,
Figure 707490DEST_PATH_IMAGE012
are all shell layersiIs determined by the unknown coefficients of (a) and (b),
Figure 168558DEST_PATH_IMAGE013
are all intermediate variables and are represented as:
Figure 45073DEST_PATH_IMAGE014
wherein the content of the first and second substances,λ i andG i all represent shell layersiAnd is composed of shell layeriThe physical properties of itself.
4. The determination method according to claim 3,
the boundary condition comprises a shell layeriAnd shell layeriA boundary condition of +1, expressed as:
Figure 489961DEST_PATH_IMAGE015
wherein the content of the first and second substances,
Figure 217614DEST_PATH_IMAGE016
representing shell layersi+1 outer surfaceIn a spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 798768DEST_PATH_IMAGE017
representing shell layersi+1 outer surface in spherical coordinate system with normal directionrDirection and direction ofθThe shear stress in the direction of the steel wire,
Figure 461831DEST_PATH_IMAGE018
respectively represent shell layersi+1 in the spherical coordinate systemrDirection (b),θA displacement component of direction;
the boundary condition further comprises a shell layer 1 and a shell layersIs expressed as:
Figure 643282DEST_PATH_IMAGE019
wherein the content of the first and second substances,
Figure 190938DEST_PATH_IMAGE020
and
Figure 16812DEST_PATH_IMAGE021
respectively represent shell layer 1 and shell layersIn a spherical coordinate systemrThe direction of the positive stress is the direction of the positive stress,
Figure 240989DEST_PATH_IMAGE022
and
Figure 394889DEST_PATH_IMAGE023
respectively represent shell layer 1 and shell layersThe outer surface has a normal direction in a spherical coordinate system ofrDirection and direction ofθThe shear stress in the direction of the direction,
Figure 464345DEST_PATH_IMAGE024
、
Figure 410305DEST_PATH_IMAGE025
respectively representThe shell layer 1 is in a spherical coordinate systemrDirection (b),θThe displacement component of the direction of the displacement,
Figure 556115DEST_PATH_IMAGE026
respectively represent shell layerssIn a spherical coordinate systemrDirection (b),θA displacement component of direction.
5. The method of claim 4, wherein the stress relationship between the shells comprises an unknown coefficient relationship between the shells, expressed as:
Figure 452439DEST_PATH_IMAGE027
wherein the content of the first and second substances,
Figure 76319DEST_PATH_IMAGE028
it is indicated that the multiplication is a cumulative multiplication,T 1 a matrix of unknown coefficients representing the innermost shell layer,T i+1 、T i 、T i-1 respectively represent shell layersiShell layer ofi+1, shell layeri-1 matrix of unknown coefficients, and
Figure 876784DEST_PATH_IMAGE029
；
R i 、R i-1 respectively represent shell layersiShell layer ofi-1 distance of the outer surface of the hollow sphere to the centre of the sphere;
M i 2 、M i(+1)2 、M i(-1)2 are all intermediate variables, andM i2 represents:
Figure 177184DEST_PATH_IMAGE030
M i 3 、M i(+1)3 、M i(-1)3 are all intermediate variables, andM i3 expressed as:
Figure 305677DEST_PATH_IMAGE031
Figure 982515DEST_PATH_IMAGE032
is an intermediate variable, expressed as:
Figure 247275DEST_PATH_IMAGE033
。
6. the method of determining as defined in claim 5, wherein the stress relationship of the innermost and outermost shells comprises a relationship of unknown coefficients between the innermost and outermost shells expressed as:
Figure 859522DEST_PATH_IMAGE034
wherein the content of the first and second substances,srepresents the number of the outermost shell layer,T s a matrix representing the unknown coefficient composition of the outermost shell.
7. The determination method according to any one of claims 3 to 6, characterized in that the Larger constant is expressed as:
Figure 724578DEST_PATH_IMAGE035
wherein the content of the first and second substances,E i representing shell layersiOf (2)Modulus and is expressed as
Figure 690260DEST_PATH_IMAGE036
，E 0 Represents the Young's modulus of the innermost shell layer,E s represents the Young modulus of the outermost shell layer;
Figure 324373DEST_PATH_IMAGE037
representing shell layersiAnd is expressed as
Figure 717308DEST_PATH_IMAGE038
，
Figure 945027DEST_PATH_IMAGE039
Representing the poisson's ratio of the innermost shell layer,
Figure 981246DEST_PATH_IMAGE040
representing the poisson's ratio of the outermost shell.
8. A stress distribution determining apparatus for a gradient functional material sphere structure, the determining apparatus comprising:
the system comprises an establishing module, a calculating module and a processing module, wherein the establishing module is used for establishing a gradient functional material ball structure model based on preset material types and sizes, and the gradient functional material ball structure model comprises a hollow ball and a gradient functional material layer wrapping the hollow ball;
the dividing module is used for averagely dividing the gradient functional material layer into a plurality of shell layers from inside to outside and respectively establishing a control equation of each shell layer;
the first determining module is used for determining the stress general solution of each shell layer according to an elastic mechanics theory and the control equation;
the second determining module is used for determining the stress relation among the shell layers based on the stress general solution and a preset boundary condition;
the third determining module is used for determining the stress relation between the innermost shell and the outermost shell according to the stress relation between the shells, wherein the innermost shell is the shell closest to the hollow ball, and the outermost shell is the shell farthest from the hollow ball;
a fourth determining module, configured to determine stresses corresponding to the innermost shell and the outermost shell, respectively, based on the stress general solution, the stress relationship between the innermost shell and the outermost shell, the size, the boundary condition, the number of shells, and the material type;
and the fifth determining module is used for respectively determining the stress corresponding to each shell layer according to the stress corresponding to the innermost shell layer and the outermost shell layer and the stress relation between the shell layers to obtain a stress distribution map of the gradient functional material ball structure model.
9. An electronic device, comprising:
at least one processor; and the number of the first and second groups,
a memory communicatively coupled to the at least one processor; wherein the content of the first and second substances,
the memory stores instructions executable by the at least one processor to enable the at least one processor to perform the method of determining a stress distribution of a gradient functional material sphere structure of any one of claims 1 to 7.
10. A computer-readable storage medium, in which a computer program is stored, which, when being executed by a processor, implements the method for determining a stress distribution of a spherical structure of a gradient functional material according to any one of claims 1 to 7.
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Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101477030A (en) * 2008-01-02 2009-07-08 中国人民解放军装甲兵工程学院 Measuring method for residual stress in coating
CN105740601A (en) * 2016-01-18 2016-07-06 北京理工大学 Analytic analysis method for stress field of single-layer or multi-shell hollow sphere or solid sphere under action of point load
CN110442945A (en) * 2019-07-26 2019-11-12 浙江工业大学 A kind of analytic method for gradient elasticity polishing disk distribution of contact
CN111783319A (en) * 2020-08-03 2020-10-16 内蒙古工业大学 Bending deformation analysis method and device for functional gradient multilayer magnetoelectric elastic nano plate
CN112380745A (en) * 2020-11-12 2021-02-19 山东大学 Physical neutral surface-based SPH (particle beam resonance) analysis model without grids for functionally graded composite material
CN114441590A (en) * 2021-12-22 2022-05-06 中国航天空气动力技术研究院 Method and system for determining heat transfer and mechanical properties of gradient heat-proof material
WO2022121203A1 (en) * 2020-12-08 2022-06-16 江苏科技大学 Method for calculating spherical shell surface three-dimensional crack propagation fatigue life

Patent Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101477030A (en) * 2008-01-02 2009-07-08 中国人民解放军装甲兵工程学院 Measuring method for residual stress in coating
CN105740601A (en) * 2016-01-18 2016-07-06 北京理工大学 Analytic analysis method for stress field of single-layer or multi-shell hollow sphere or solid sphere under action of point load
CN110442945A (en) * 2019-07-26 2019-11-12 浙江工业大学 A kind of analytic method for gradient elasticity polishing disk distribution of contact
CN111783319A (en) * 2020-08-03 2020-10-16 内蒙古工业大学 Bending deformation analysis method and device for functional gradient multilayer magnetoelectric elastic nano plate
CN112380745A (en) * 2020-11-12 2021-02-19 山东大学 Physical neutral surface-based SPH (particle beam resonance) analysis model without grids for functionally graded composite material
WO2022121203A1 (en) * 2020-12-08 2022-06-16 江苏科技大学 Method for calculating spherical shell surface three-dimensional crack propagation fatigue life
CN114441590A (en) * 2021-12-22 2022-05-06 中国航天空气动力技术研究院 Method and system for determining heat transfer and mechanical properties of gradient heat-proof material

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