A nonlocal method for heat transfer in cracked functionally graded plates

Abstract

The present invention discloses a non-local method for the heat transfer problem of a cracked functional gradient plate, comprising the following steps: determining the parameters of the functional gradient material and the size of the plate; determining the initial crack position; discretizing the solid model with a suitable grid spacing; determining the control equation of the temperature field of the functional gradient plate and setting the boundary conditions of the model; based on the peridynamic differential operator theory, the control equation and the boundary conditions are converted into a non-local form, discretized and constructed to solve the equation group; solve the equation group to obtain the temperature value of each material point. For cracked heat transfer engineering components, the prediction of the heat flux singularity at the crack tip by the present invention can effectively prevent and reduce the damage to the structure caused by thermal damage and thermal fracture, and increase the service life of the engineering components.

Description

Technical Field

[0001] The invention relates to a modeling method for heat transfer problems of aerospace materials, and in particular to a non-local method for heat transfer problems of functional gradient plates containing cracks. Background Art

[0002] Functionally gradient materials are a new type of composite material whose material properties change smoothly as the structure and composition change in a continuous gradient in the same direction. In the field of aerospace, the use of functionally gradient materials composed of ceramics and metals can give full play to their advantages such as high temperature resistance, shear resistance, and stress concentration elimination, and can significantly improve the service life of aerospace components under high temperature conditions. In this context, it is of great significance to study the heat transfer problem of ceramic-metal-based functionally gradient material structures under thermal loads.

[0003] At present, many scholars have used the grid-based finite element method and boundary element method to solve the heat transfer problem of functional gradient materials. Since these traditional methods have large errors when dealing with problems with continuous changes in material parameters, only by refining the grid can a more accurate solution be obtained, but the corresponding calculation efficiency will be significantly reduced. Peridynamic Differential Operator (PDDO) theory is one of the important representatives of non-local differential operator theory proposed in recent years, which can effectively reduce the error caused by the non-uniformity of functional gradient materials. In addition, similar to solving non-continuous problems such as cracked plates, according to non-local theory, the singularity of the heat flux at the crack tip can be predicted, and the change trend of the heat flux singularity can be analyzed by controlling the change of material parameters. Therefore, using PDDO theory to solve the heat transfer problem of cracked functional gradient plates can effectively improve the calculation accuracy and efficiency of the solution, accurately predict the temperature distribution in the plate and capture the singularity of the heat flux at the crack tip, so as to control, prevent and prevent the occurrence of thermal damage and thermal fracture that endanger the service life of engineering components. Summary of the invention

[0004] Purpose of the invention: In order to overcome the shortcomings of the background technology, the present invention discloses a non-local method for the heat transfer problem of functional gradient plates with cracks.

[0005] Technical solution: The non-local method for heat transfer problem of cracked functional gradient plate disclosed in the present invention comprises the following steps:

[0006] S1. Determine the parameters of the functional gradient material and the size of the plate;

[0007] S2, determine the initial crack location;

[0008] S3, discretize the solid model with appropriate grid spacing;

[0009] S4, determine the control equation of the temperature field of the functional gradient plate and set the boundary conditions of the model;

[0010] S5. Based on the peridynamic differential operator theory, the control equations and boundary conditions are transformed into non-local forms, discretized and the equations to be solved are constructed;

[0011] S6. Solve the equations to obtain the temperature value of each material point.

[0012] The material properties of the functionally gradient plate involved vary gradually from metal to ceramic.

[0013] Furthermore, the functional gradient plate parameters involved include: the thermal conductivity k of the ceramic phase c , thermal conductivity k of the metal phase m , gradient parameter n g , board length W.

[0014] Furthermore, the cracks involved in the solution of this method are adiabatic cracks.

[0015] Furthermore, the grid spacing selected in S3 is Δx=W / 250~W / 100.

[0016] Furthermore, the control equation of the temperature field in S4 is derived as follows:

[0017] The steady-state heat conduction equation is:

[0018]

[0019] In the formula, T is temperature, k is thermal conductivity, and Q is heat source;

[0020] For a two-dimensional problem, it is assumed that the material properties of the functionally gradient plate vary with x, that is:

[0021] k=k(x)

[0022] In the absence of a heat source, the temperature field control equation of the functionally gradient plate is:

[0023]

[0024] Furthermore, there are three types of boundary conditions set in S4:

[0025] Dirichlet boundary conditions:

[0026] Neumann boundary conditions:

[0027] The third type of boundary conditions:

[0028] In the formula, is a known temperature, is the known heat flux, h is the convective heat transfer coefficient, and T0 is the ambient temperature.

[0029] Furthermore, the specific calculation method for discretizing the control equation in S5 is:

[0030] For a field function f(x)=f(x,y) in a two-dimensional space, perform an N-order Taylor expansion at point x:

[0031]

[0032] Where R(N,x) refers to the negligible remainder, n1 and n2 represent the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 represent the unit vectors in the x and y directions, ξ1 and ξ2 represent the projection of the length between two material points in the x and y directions;

[0033] Using the properties of orthogonal functions, use the PD function to convert partial differentials into integral form

[0034]

[0035] Where 1≤p1+p2≤N, H x Represents the near field range of point x, and the PD function is:

[0036]

[0037] in is the weight function, is the unknown coefficient matrix, which can be obtained by the following formula:

[0038]

[0039]

[0040]

[0041] in is a matrix of shape, is a known coefficient matrix;

[0042] The control equations in S4 and the Neumann boundary conditions and the third-order boundary conditions involving differentials are rewritten in their differential form into integral form as follows:

[0043]

[0044]

[0045]

[0046] where T′=T(x′,y′),

[0047] Discretize the above formula to get:

[0048]

[0049]

[0050]

[0051] The symbols with subscripts (k) and (j) represent the functions at two different material points (x (k) ,y (k) ) and (x (j) ,y (j) ), N (k) Represents a material point (x (k) ,y (k) )The total number of material points within the near field.

[0052] Furthermore, the method for solving the temperature in S6 is:

[0053] The discrete control equations and boundary conditions are linear and can be composed of the following linear equations:

[0054] AT=0,BT=c

[0055] Where the matrices A and B represent the coefficients generated by the differential equation, the vector c represents the known values ​​of the material points at the boundary, and the vector T is the temperature to be determined, which can be written as:

[0056]

[0057] Where N tol Represents the total number of discrete points,

[0058] By introducing the Lagrangian operator ε and using variational analysis, the above equations can be expressed as follows:

[0059]

[0060] The temperature value at each material point can be obtained by programming.

[0061] Beneficial effects: Compared with the prior art: The present invention controls the properties of functional gradient materials by adjusting parameters and uses the peridynamic differential operator theory to model the functional gradient plates under thermal loads. Different from the traditional mesh-based finite element and boundary element methods, the peridynamic differential operator theory builds a bridge between the differential form and the integral form of the temperature field control equation of the functional gradient plate, avoiding the singularity caused by derivation in discontinuous problems, effectively reducing the large errors caused by local differential operators when solving heat transfer problems with non-uniform functional gradient materials, improving the calculation accuracy of the solution, and being able to accurately predict the temperature distribution of the functional gradient plate under temperature loads;

[0062] For heat transfer engineering components containing cracks, the prediction of the heat flux singularity at the crack tip by the present invention can effectively prevent and reduce the damage to the structure caused by thermal damage and thermal fracture, thereby increasing the service life of the engineering components. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 This is a schematic diagram of a cracked functional gradient plate model of the present invention;

[0064] Figure 2 is the gradient parameter n in the embodiment of the present invention g =1, the calculated temperature distribution diagram;

[0065] Figure 3 is the gradient parameter n in the embodiment of the present invention g =1, the calculated heat flux distribution in the y direction;

[0066] Figure 4 is the gradient parameter n in the embodiment of the present invention g =100, the calculated temperature distribution diagram;

[0067] Figure 5 is the gradient parameter n in the embodiment of the present invention g =100, the calculated heat flux distribution in the y direction;

[0068] Figure 6 Graph showing heat flux distribution when x=0.25 m for different gradient parameters in an embodiment of the present invention;

[0069] Figure 7 Graph showing the temperature distribution when x=0.25 m with different gradient parameters in the embodiment of the present invention. DETAILED DESCRIPTION

[0070] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0071] S1, such as Figure 1As shown, a two-dimensional model of a square plate of functionally gradient material with cracks under thermal load is constructed. In this embodiment, the functionally gradient plate is made of a mixture of ceramic zirconium oxide (ZrO2) and metal monel (70Ni-30Cu). For this two-phase composite material, the Mori-Tanaka method is selected to estimate the effective performance, and the metal and ceramic are used as the particle phase and the matrix phase respectively, and their parameters are shown in the following table.

[0072] Table 1 Functionally graded plate parameters

[0073]

[0074] S2. Select the crack length as 2a=0.5m, and the coordinates of the two ends of the crack as (0,0.25) and (0,-0.25).

[0075] S3. Select a grid spacing of Δx = W / 200 = 5 mm to divide the solid model into a discrete model of material points.

[0076] S4, determine the control equation of the temperature field and set the boundary conditions of the model;

[0077] In this embodiment, the derivation process of the control equation of the temperature field is:

[0078] Steady-state heat conduction equation:

[0079] Where T is temperature, k is thermal conductivity, and Q is heat source.

[0080] For a two-dimensional problem, the thermal conductivity k of the functionally gradient material is related as follows:

[0081]

[0082] V c represents the volume fraction of the ceramic phase, assuming it is a power function, with the following relationship:

[0083]

[0084] where n g is the gradient parameter. The volume fraction of the metal phase is V m =1-V c .

[0085] In the absence of a heat source, the temperature field control equation of the functionally gradient plate is:

[0086]

[0087] The boundary conditions of this embodiment are:

[0088] Dirichlet boundary conditions:

[0089] Neumann boundary conditions:

[0090] The given temperature T R =300K, T L =1000K.

[0091] S5. Based on the peridynamic differential operator theory, the control equations and boundary conditions are discretized and a set of equations is constructed for solution.

[0092] The field function f(x)=f(x,y) in two-dimensional space is expanded by Nth order Taylor at point x:

[0093]

[0094] Where R(N,x) refers to the negligible remainder, n1 and n2 represent the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 represent the unit vectors in the x and y directions, ξ1 and ξ2 represent the projection of the length between two material points in the x and y directions;

[0095] Using the properties of orthogonal functions, use the PD function to convert partial differentials into integral form

[0096]

[0097] Where 1≤p1+p2≤N, H x Represents the near field range of point x, and the PD function is:

[0098]

[0099] in is the unknown coefficient matrix, which can be obtained by the following formula:

[0100]

[0101]

[0102]

[0103] in is a matrix of shape, is a known coefficient matrix;

[0104] The control equation and Neumann boundary condition in this embodiment can be rewritten from their differential form to integral form as follows:

[0105]

[0106]

[0107] Discretize the above control equations, Neumann boundary conditions and Dirichlet boundary conditions to obtain:

[0108]

[0109] T ,2(k) =0,

[0110] T (k) =T L ,

[0111] T (k) =T R ,

[0112] The discrete control equations and boundary conditions are linear and can be composed of the following linear equations:

[0113] AT=0,BT=c

[0114] Where the matrices A and B represent the coefficients resulting from the PD differentiation of the equation. The vector c represents the known values ​​of the material points at the boundary. The vector T is the temperature to be determined, which can be written as:

[0115]

[0116] Where N tol Represents the total number of discrete points.

[0117] By introducing the Lagrangian operator ε and using variational analysis, the above equations can be expressed as follows:

[0118]

[0119] The temperature value at each material point can be obtained by programming.

[0120] like Figures 2 to 5 As shown, they are n g =1 and 100 when the temperature distribution and heat flux distribution in the y direction. Figure 6 and Figure 7 The n given g =1, 5, 10, 50, 100, the curve of the heat flux in the y direction and the temperature distribution along the x direction at y = 0.25m.

[0121] It can be seen from the simulation results that this method can efficiently and accurately simulate the temperature field and heat flux distribution of the functional gradient plate with cracks. At the same time, this method can realize the accurate prediction of the heat flux singularity at the crack tip, improve the ability of artificial intervention, protect engineering components, and increase the service life of components, and promote the further development of thermal protection of heat transfer structures.

Claims

1. A non-local method for heat transfer problems in cracked functionally gradient plates, characterized in that: The following steps are involved: S1. Determine the parameters of the functional gradient material and the size of the plate; S2, determine the initial crack location; S3, discretize the solid model with appropriate grid spacing; S4, determine the control equation of the temperature field of the functional gradient plate and set the boundary conditions of the model; S5. Based on the peridynamic differential operator theory, the control equations and boundary conditions are transformed into non-local forms, discretized and the equations to be solved are constructed; S6. Solve the equations to obtain the temperature value of each material point; Among them, the derivation process of the control equation of the temperature field in S4 is: The steady-state heat conduction equation is: In the formula, T is temperature, k is thermal conductivity, and Q is heat source; For a two-dimensional problem, it is assumed that the material properties of the functionally gradient plate vary with x, that is: k=k(x) In the absence of a heat source, the temperature field control equation of the functionally gradient plate is: There are three types of boundary conditions in heat conduction: Dirichlet boundary conditions: Neumann boundary conditions: The third type of boundary conditions: In the formula, is a known temperature, is the known heat flux, h is the convection heat transfer coefficient, and T0 is the ambient temperature; The specific calculation method for discretization of the control equation in S5 is: For a field function f(x)=f(x,y) in a two-dimensional space, perform an N-order Taylor expansion at point x: Where R(N,x) refers to the negligible remainder, n1 and n2 represent the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 represent the unit vectors in the x and y directions, ξ1 and ξ2 represent the projection of the length between two material points in the x and y directions; Using the properties of orthogonal functions, use the PD function to convert partial differentials into integral form Where 1≤p1+p2≤N, H x Represents the near field range of point x, and the PD function is: in is the weight function, is the unknown coefficient matrix, which can be obtained by the following formula: in is a matrix of shape, is a known coefficient matrix; The control equations in S4 and the Neumann boundary conditions and the third-type boundary conditions involving differentials can be rewritten from their differential forms into integral forms as follows: where T′=T(x′,y′), Discretize the above formula to get: The symbols with subscripts (k) and (j) represent the functions at two different material points (x (k) ,y (k) ) and (x (j) ,y (j) ), N (k) Represents a material point (x (k) ,y (k) )The total number of material points within the near field.

2. The non-local method for heat transfer problem of functionally gradient plate with cracks according to claim 1, characterized in that: The material properties of the functionally gradient plate involved change gradually from metal to ceramic.

3. The non-local method for heat transfer problem of functionally gradient plate with cracks according to claim 1, characterized in that: The functional gradient plate parameters involved include: thermal conductivity k of the ceramic phase c , thermal conductivity k of the metal phase m , gradient parameter n g , board length W.

4. The non-local method for heat transfer problem of functionally gradient plate with cracks according to claim 1, characterized in that: The cracks involved in the solution of this method are adiabatic cracks.

5. The non-local method for heat transfer problem of functionally gradient plate with cracks according to claim 1, characterized in that: The grid spacing selected in S3 is Δx=W / 250~W / 100.

6. The non-local method for heat transfer problem of functionally gradient plate with cracks according to claim 1, characterized in that: The method for solving temperature in S6 is: The discrete control equations and boundary conditions are linear and can be composed of the following linear equations: AT=0,BT=c Where the matrices A and B represent the coefficients generated by the differential equation, the vector c represents the known values ​​of the material points at the boundary, and the vector T is the temperature to be determined, which can be written as: Where N tol Represents the total number of discrete points, By introducing the Lagrangian operator ε and using variational analysis, the above equations can be expressed as follows: The temperature value at each material point can be obtained by programming.

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