A semi-analytical method for functionally graded materials based on the ritz method

Abstract

A kind of semi-analytical analysis method of functionally graded material based on Ritz method, method includes: S1 obtains the displacement boundary condition of functionally graded material, obtains displacement test function based on the displacement boundary condition;S2 the various energy of the functionally graded material is calculated by calculation module;S3 the various energy calculated in S2 is linked by Hamilton principle, and total energy functional is obtained;S4 the ordinary differential equation set or algebraic equation set of displacement coefficient is solved by the energy functional in S3, and displacement coefficient is obtained;S5 displacement test function is solved based on the displacement coefficient obtained in S4, and all response physical quantities of the functionally graded material are obtained.The modeling of the present application is more simple, and other items can be more conveniently added in original equation in energy mode, and the relationship of thermal load and material property change with temperature is also more easily considered.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of mechanical modeling and analysis, and particularly relates to a semi-analytical analysis method for functionally graded materials based on the Ritz method. BACKGROUND

[0002] With the development of aircraft design technology, the limit of flight speed of aircraft is gradually broken, and many aircrafts have reached the range of hypersonic speed. High-speed flight brings many new challenges to aircraft design. In the process of high-speed flight, the surface of the aircraft and the air will produce violent friction, generating a large amount of aerodynamic heat, so that the surface of the aircraft can reach a high temperature of thousands of degrees. In order to resist such high temperature, it is usually necessary to lay thermal protection materials on the surface, and due to the existence of physical property mutation between different materials, stress concentration, peeling, delamination and other phenomena are easily caused. In view of this phenomenon, the idea of functionally graded materials (FGM) emerges as the times require, that is, two different materials are mixed by a specified continuous function distribution rule, so that the physical properties can be continuously changed. Compared with traditional uniform materials, the new material form of functionally graded materials has great differences:

[0003] Firstly, in the functionally graded material, the material property is no longer a constant, but changes with the spatial position, and the material property of the functionally graded material needs to be expressed by the properties of the two component materials.

[0004] Secondly, due to the non-uniformity of the material property in the thickness direction, the strain of the symmetric surface under load is not 0, that is, the neutral surface is not on the symmetric surface, and the assumption of transverse displacement of 0 in the classical plate theory is not established, so the existing uniform plate theory cannot be directly used.

[0005] In addition, the thermal conductivity coefficient of the functionally graded material also changes with the spatial position, and the thermal conductivity coefficient in the steady-state heat conduction differential equation is no longer a constant. The nature of the differential equation changes from a linear differential equation to a nonlinear differential equation, and the temperature field of the functionally graded material needs to be characterized.

[0006] In addition, the functionally graded material is not uniform in material property, but related to the spatial position, and in the integral, the form is more complex, resulting in a very complex response of many physical quantities related to the functionally graded material.

[0007] At present, the research on the mechanical properties of functionally graded materials at home and abroad can be divided into analytical / semi-analytical method and numerical method. Among them, the numerical method includes finite element method, meshless method, generalized differential quadrature method and the like. In the numerical method, due to the great change of the functionally graded material along the thickness direction, in order to more accurately simulate the properties of the functionally graded material, a large number of nodes need to be discretized in the thickness direction, the grid amount is large, and the calculation efficiency is low. In the analytical / semi-analytical method, most of the current researches are to derive partial differential equations through the form of equilibrium equations, and then to obtain ordinary differential equations or algebraic equations through Galerkin method for solving, and the establishment of the equilibrium equation needs to be more familiar with the mechanical system, and it is not easy to quickly expand, and the Galerkin method belongs to the weighted residual method, and the physical meaning of the unknown coefficient obtained is not clear. SUMMARY

[0008] In order to solve the above problems, the present application provides a semi-analytical analysis method for functionally graded materials based on Ritz method.

[0009] The technical scheme of the present application is as follows:

[0010] A semi-analytical analysis method for functionally graded materials based on Ritz method, the method comprising the following steps:

[0011] S1 obtains the displacement boundary condition of the functionally graded material, and obtains the displacement trial function based on the displacement boundary condition; wherein the displacement trial function is composed of the product of the displacement coefficient and the displacement base function;

[0012] S2 calculates various energies of the functionally graded material through a calculation module; different kinds of energies are processed by the module method in the present application, various energies can be calculated separately, common physical quantity information is transmitted in each energy calculation module, each kind of energy form is specially processed according to the characteristics of the functionally graded material, and each energy calculation module is organically combined together to form information interaction. In the mechanical system, the strain energy is most closely related to the structure itself, and for the functionally graded material, the influence of heat on the structure is also very important due to the working environment of the functionally graded material in high temperature.

[0013] S3 connects various energies calculated in S2 through Hamilton principle to obtain the total energy functional, and the variation of the energy functional of the actual existing physical system is 0;

[0014] S4 according to the principle of variation method, the variation of the total energy functional is 0, which is equivalent to that the partial derivative obtained for each displacement coefficient is 0, so that the ordinary differential equation set or algebraic equation set about the displacement coefficient can be obtained, and the displacement coefficient can be obtained by directly solving the equation set;

[0015] S5 obtains a displacement trial function based on the displacement coefficient obtained in S4, and obtains all response physical quantities of the functionally graded material.

[0016] Further, the calculation module in S2 comprises a thermal calculation module and a strain energy calculation module.

[0017] Further, the thermal calculation module specifically comprises:

[0018] Under given temperature boundary conditions, the expression of the internal temperature field and the spatial position is obtained by solving a nonlinear ordinary differential equation;

[0019] The thermal stress is calculated through the new elastic modulus and the thermal expansion coefficient;

[0020] The thermal stress is regarded as a compression load, and the work done by the compression load caused by the thermal stress, i.e. the energy influence of the thermal stress on the functionally graded material, is obtained by integrating the total calculation domain of the functionally graded material.

[0021] Further, the thermal calculation module further comprises:

[0022] If the temperature-dependent characteristics of the material properties need to be considered, the obtained expression of the temperature field is substituted into the function relationship of the material properties changing with temperature to obtain new material properties, and the new temperature-dependent material properties are output.

[0023] Further, the strain energy calculation module specifically comprises:

[0024] First, the geometric parameters of the functionally graded material are calculated according to the displacement trial function, wherein the geometric parameters include a slope and a curvature;

[0025] The position of the physical mid-plane is obtained according to the relationship between the material properties inside the functionally graded material and the spatial position;

[0026] The strain inside the functionally graded material is obtained according to the geometric parameters and the position of the physical mid-plane;

[0027] The stress inside the functionally graded material is obtained according to the physical properties of the functionally graded material;

[0028] The product of the stress and the strain obtained in the above two steps is integrated in the calculation domain of the functionally graded material to obtain the strain energy of the functionally graded material.

[0029] Further, the position of the physical mid-plane is the position of the plane where the strain is 0.

[0030] Technical effects of the present application:

[0031] (1) The process is streamlined and modularized, and the influence of various factors on the mechanical system is represented in the form of energy, which can facilitate the acquisition of functionally graded materials;

[0032] (2) The modeling is simple. The energy expression can be established through basic physical concepts, and then the ordinary differential equation system or algebraic equation system can be obtained to complete the solution.

[0033] (3) The results are simple in form and fast in solution, and it is easier to obtain the influence of each physical quantity on mechanical properties;

[0034] (4) The influence of the thermal environment on functionally graded materials can be considered, including the load caused by thermal stress and the influence of temperature on material properties. Attached Figure Description

[0035] The accompanying drawings illustrate various embodiments generally by way of example rather than limitation, and are used, together with the specification and claims, to explain embodiments of the invention. Where appropriate, the same reference numerals are used in all drawings to refer to the same or similar parts. Such embodiments are illustrative and are not intended to be exhaustive or exclusive embodiments of the apparatus or method.

[0036] Figure 1 A schematic diagram of the overall calculation process of the present invention is shown;

[0037] Figure 2 A schematic diagram of the thermal calculation module of the present invention is shown;

[0038] Figure 3 A schematic diagram of the strain energy calculation module of the present invention is shown. Detailed Implementation

[0039] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0040] This invention takes a functionally graded material thin plate with a power function distribution that is simply supported on four sides as an example.

[0041] The geometric dimensions of the functionally graded material plate are length a, width b, and thickness h, respectively, and the properties of the component materials can be represented by P. c and P m Given a gradient exponent of k, the relationship between the material properties and spatial location of a functionally graded material can be expressed as follows: In the formula, P(z)FGM is a spatially related attribute, a is the plate length, b is the plate width, h is the plate thickness, k is the gradient exponent, and P c and P m For ceramic and metallic materials, z represents the z-axis coordinate.

[0042] Step one, in the displacement boundary condition, the displacement trial function can be expressed as In the formula, w is the displacement trial function, c mn is the displacement coefficient, m, n are the order of x, y direction respectively, M, N are the maximum order of x, y direction selected respectively.

[0043] Step two, run each energy calculation module

[0044] 2.1 Enter the thermal calculation module.

[0045] (1) Since the thermal conductivity of the functionally graded material is related to the spatial position, the heat conduction control equation becomes In the formula, T is the temperature, λ is the thermal conductivity. Substitute the expression of the thermal conductivity and the spatial position In the formula, λ c is the thermal conductivity of the ceramic, λ m is the thermal conductivity of the metal, the general form of the nonlinear temperature field expression can be obtained which changes with the gradient index k.

[0046] (2) Then, if the material properties related to temperature are considered, i.e. P(T) = P0(P -1 T -1 +1+P1T+P2T 2 +P3T 3 ), in the formula, P0 is the material property at the reference temperature, P -1 is the temperature-1 order coefficient, P1 is the temperature 1 order coefficient, P2 is the temperature 2 order coefficient, P3 is the temperature 3 order coefficient. Substitute the obtained nonlinear temperature field T(z) to obtain the material property P(z, T) related to both temperature and spatial position in the functionally graded material.

[0047] (3) Further, the thermal stress can be obtained as σ T = E(z, T) α(z, T) T(z), in the formula, E is the elastic modulus, α is the thermal expansion coefficient, T is the temperature, σ T is the thermal stress.

[0048] (4) The influence of the thermal stress on the functionally graded material plate can be obtained by the compression load work brought by the thermal stress In the formula, v is the Poisson's ratio.

[0049] 2.2 Then, through the strain energy calculation module, the plate theory uses the classical plate theory, i.e. ignoring the influence of transverse shear stress, the temperature-dependent material properties obtained by the thermal calculation module can be transmitted.

[0050] (1) The slope and curvature of the functionally graded material can be calculated as

[0051] where i, j are x or y, where β is the geometric slope, and k is the geometric curvature.

[0052] (2) Introducing the concept of physical mid-plane, because the material properties of functionally graded materials are not symmetric about the geometric mid-plane, so the strain of the geometric mid-plane is not 0 when under load. By calculating the geometric mid-plane, the position of strain 0, z0 is the physical mid-plane,

[0053]

[0054] (3-4) According to the definition of the physical mid-plane, the strain and stress (take x direction as an example) can be expressed as

[0055] where ε is the strain, and σ is the stress.

[0056] (5) Then the strain energy of the functionally graded material can be calculated, and output.

[0057] Step three, according to Hamilton's principle, the actual physical system δ (U-W) =0, where U is the strain energy, and W is the work done by external force.

[0058] Step four, by substituting various energies, the F (c mn ) =0 about displacement coefficient can be obtained, and the value of c mn is solved.

[0059] Step five, by substituting c mn , the expression of each physical quantity of the functionally graded material can be obtained.

[0060] The above is only the preferred specific embodiment of the present application, but the protection scope of the present application is not limited to this, any skilled person in the art, according to the technical solution and the inventive concept of the present application, within the technical range disclosed by the present application, makes equivalent replacement or change, should be covered in the protection scope of the present application.

Claims

1. A semi-analytical analysis method for functionally graded materials based on the Ritz method, characterized in that, The method includes the following steps: S1 Obtain the displacement boundary conditions of the functionally graded material, and obtain the displacement trial function based on the displacement boundary conditions; S2 calculates the various energies of the functionally graded material using the calculation module; S3 connects the various energies calculated in S2 using Hamilton's principle to obtain the overall energy functional; S4 Based on the principle of variational method, the displacement coefficient is obtained by solving the ordinary differential equations or algebraic equations of the displacement coefficient through the energy functional described in S3. S5. Based on the displacement coefficients obtained in S4, solve the displacement trial function to obtain all the response physical quantities of the functionally graded material; The calculation module in S2 includes a thermal calculation module and a strain energy calculation module; The strain energy calculation module is specifically as follows: First, the geometric parameters of the functionally graded material are calculated based on the displacement trial function, and the geometric parameters include slope and curvature; Based on the calculated relationship between the internal material properties and spatial position of the functionally graded material, the position of the physical mid-surface is obtained, which is the position of the plane where the strain is 0. The strain inside the functionally graded material is obtained based on the geometric parameters and the physical mid-surface position. Based on the physical properties of the functionally graded material, the stress inside the functionally graded material is obtained; Based on the product of stress and strain obtained in the previous two steps, the product is integrated within the computational domain of the functionally graded material to obtain the strain energy of the functionally graded material.

2. The semi-analytical analysis method for functionally graded materials based on the Ritz method according to claim 1, characterized in that, The thermal calculation module is specifically as follows: Under given temperature boundary conditions, the expression for the internal temperature field and spatial location is obtained by solving nonlinear ordinary differential equations; the obtained expression for the temperature field is substituted into the functional relationship between material properties and temperature to obtain new material properties, and the new temperature-related material properties are output. Thermal stress is calculated using the new elastic modulus and coefficient of thermal expansion; Treating thermal stress as a compressive load and integrating it within the overall computational domain of the functionally graded material, we obtain the work done by the compressive load caused by thermal stress, which is the energy effect of thermal stress on the functionally graded material.

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