Method for determining stress of circular film under transversely and uniformly distributed load

Abstract

The invention discloses a method for determining the stress of a circular film under a transversely uniformly distributed load, which comprises the following steps of: applying a transversely uniformly distributed load q to a circular film which is flat before deformation, has the periphery fixedly clamped, the radius a, the thickness h, the Young's modulus E and the Poisson ratio v, and determining the stress of the circular film under the transversely uniformly distributed load q; when the circular thin film is subjected to axial-symmetry deformation, axial-symmetry deflection deformation can be generated in the direction of the load q, and based on static equilibrium analysis of the axial-symmetry deformation problem of the circular thin film, the radial stress sigma r (r) and the circumferential stress sigma t (r) of the circular thin film after axial-symmetry deformation can be determined by utilizing the measured value of the load q.

Description

Technical Field

[0001] The invention relates to the field of engineering technology, and in particular to a method for determining the stress of a circular film under a transversely uniformly distributed load. Background Art

[0002] The analytical solution to the large deflection problem of the axisymmetric deformation of a circular membrane clamped and fixed at the periphery under a uniformly distributed transverse load is used in many engineering and technical fields, for example, in the development of various instruments and sensors, and in the study of the adhesion energy measurement of the membrane / substrate system.

[0003] There are two methods to describe the deformation of continuous media, namely the Lagrangian method and the Euler method. The Lagrangian description method uses the coordinate variables of any point on the continuous medium before deformation as independent variables to describe the deformation problem of the continuous medium, while the Euler description method uses the coordinate variables of any point on the continuous medium after deformation as independent variables to describe the deformation problem of the continuous medium. Therefore, both the Lagrangian coordinate system and the Euler coordinate system can be used to describe the deformation problem of the continuous medium. In elasticity mechanics, the deformation problem of elastic bodies is described by describing the stress, strain, and displacement of any point on the elastic body, which can be described analytically or numerically, that is, the deformation problem of elastic bodies can be described by using continuous functions of stress, strain, and displacement or by giving the digital values ​​of stress, strain, and displacement of a finite number of discrete points on the elastic body. Seeking continuous functions of stress, strain and displacement is to analytically solve the deformation problem of elastic body; seeking digital values ​​of stress, strain and displacement at a finite number of discrete points on elastic body is to numerically solve the deformation problem of elastic body; and the continuous functions and digital values ​​of stress, strain and displacement obtained are what are usually called analytical solutions and numerical solutions of stress, strain and displacement. Therefore, both Lagrangian coordinate system and Euler coordinate system can be used to seek analytical solutions or numerical solutions of stress, strain and displacement. In other words, the deformation problem of elastic body can be solved analytically or numerically in Lagrangian coordinate system and Euler coordinate system, and the (analytical or numerical) solution obtained in Lagrangian (or Euler) coordinate system is usually referred to as Lagrangian (or Euler) solution, and sometimes referred to as pure Lagrangian (or pure Euler) solution, and its independent variable is Lagrangian (or Euler) coordinate variable. However, regarding the large deflection problem of axisymmetric deformation of a circular membrane fixed and clamped at the periphery under a transverse uniformly distributed load, in the existing literature, there is neither a pure Euler analytical solution nor a pure Lagrangian analytical solution to the large deflection problem of axisymmetric deformation, but only a mixed Euler-Lagrangian analytical solution to the large deflection problem of axisymmetric deformation. For example, the analytical solutions used in the invention patent "Method for determining the maximum stress of a circular membrane under a transverse uniformly distributed load" (patent number: 202010168916.3), the invention patent "Method for determining the maximum stress of a circular membrane under a uniformly distributed load" (application number: 202010190003.1), and the invention patent "A method for determining the maximum stress of a circular membrane under a transverse uniformly distributed load" (patent number: 202010218173.6) are all mixed Euler-Lagrangian analytical solutions.

[0004] As a problem of elastic mechanics, the large deflection problem of axisymmetric deformation is solved analytically by establishing equilibrium equations (describing the static equilibrium of external forces and membrane response forces), geometric equations (describing the geometric relationship between strain and displacement), and physical equations (describing the physical relationship between stress and strain), and these equations should be established in the same coordinate system, that is, either the Euler coordinate system is used to establish these equations (abbreviated as the equilibrium equations, geometric equations, and physical equations described by Euler), or the Lagrangian coordinate system is used to establish these equations (abbreviated as the equilibrium equations, geometric equations, and physical equations described by Lagrangian), where the mathematical expressions of the physical equations in the two coordinate systems are the same (that is, the mathematical expressions of the physical equations described by Euler and the physical equations described by Lagrangian are the same).

[0005] However, perhaps because "when discussing the static equilibrium problem under the action of external elastic forces, the Euler description method is more intuitive, concise and easy to understand than the Lagrangian description method; and when discussing the geometric relationship between strain and displacement, the Lagrangian description method is more intuitive, concise and easy to understand than the Euler description method", in the existing literature, the equilibrium equations of the large deflection problem of axisymmetric deformation are generally established using the Euler coordinate system, and the geometric equations are generally established using the Lagrangian coordinate system. The analytical solution obtained in this way is the so-called mixed Euler-Lagrangian analytical solution mentioned above. Obviously, the method of using the Euler coordinate system to establish equilibrium equations and the Lagrangian coordinate system to establish geometric equations is wrong in theory. In particular, it will cause confusion in the definition of the independent variables of the obtained mixed Euler-Lagrangian analytical solution, because the Euler coordinate variable refers to any point on the deformed circular membrane, and the Lagrangian coordinate variable refers to any point on the circular membrane before deformation. Therefore, it is unclear whether the independent variable of the obtained mixed Euler-Lagrangian analytical solution refers to any point on the circular membrane after deformation or before deformation, which will cause confusion in the definition of the independent variables. Summary of the invention

[0006] In view of the above problems, the present invention provides a method for determining the stress of a circular film under a uniformly distributed transverse load, and the technical solution adopted comprises the following steps:

[0007] S1. Establish a circular film structure and determine the geometric and physical parameters of the circular film;

[0008] S2. Apply a uniformly distributed transverse load to the circular film;

[0009] S3. Using the Lagrangian description method and based on the static equilibrium conditions of a circular membrane with large deflection deformation, establish the Lagrangian description of the out-of-plane and in-plane equilibrium equations;

[0010] S4. The coordination equations are derived from the geometric equations and physical equations described by Lagrangian;

[0011] S5. The power series method is used to solve the out-of-plane equilibrium equation, the in-plane equilibrium equation and the coordination equation, and the analytical solution of the large deflection problem of the axisymmetric deformation of the circular membrane fixed and clamped around the periphery under the action of the transverse uniformly distributed load is obtained. Based on the analytical solution, the radial stress σ after the axisymmetric deformation of the circular membrane is determined. r (r) and circumferential stress σ t (r).

[0012] Optionally, in S1, the circular film is flat before deformation, and its periphery is fixed and clamped, and its geometric and physical parameters include radius a, thickness h, Young's elastic modulus E, and Poisson's ratio ν.

[0013] Optionally, in S2, after a uniformly distributed transverse load q is applied to the circular film, the circular film will produce an axisymmetric flexural deformation along the direction of the load q.

[0014] Optionally, in S3 and S4, the out-of-plane equilibrium equation is used to describe the equilibrium relationship between the external load and the film stress;

[0015] The in-plane equilibrium equation is used to describe the equilibrium relationship between the stresses of each film;

[0016] The geometric equation is used to describe the geometric relationship between film strain and displacement;

[0017] The physical equation is used to describe the physical relationship between film stress and strain;

[0018] Among them, the independent variables of stress, strain and displacement are all Lagrangian coordinate variables, that is, the coordinates of any point on the circular film before deformation.

[0019] Optionally, in S5, it is obtained by the following equation:

[0020]

[0021]

[0022]

[0023] β0=α0;

[0024]

[0025]

[0026]

[0027]

[0028] Determine α0 and α2, α4, α6, α8, α 10 , α 12 ,β0,β2,β4,β6,β8,β 10 , β 12 The value of , finally, is given by the equation:

[0029]

[0030] Determine the radial stress σ of the circular film at the radial coordinate r r (r), circumferential stress σ t (r), where r represents the radial coordinate of any point on the initially flat circular film, a, h, and r are all in millimeters (mm), and E, q, σ r (r), σ t (r) is in Newton per square millimeter (N / mm 2 ), while ν, α0, α2, α4, α6, α8, α 10 , α 12 ,β0,β2,β4,β6,β8,β 10 , β 12 and Q are dimensionless quantities.

[0031] The beneficial effects of the present invention are:

[0032] Based on the pure Lagrangian analytical solution of the large deflection problem of the axisymmetric deformation of a circular film fixed and clamped around the periphery under a transverse uniformly distributed load, a method for determining the stress of a circular film under a transverse uniformly distributed load is provided. Compared with the analytical solutions in the existing literature, on the one hand, it avoids the difference between the Euler coordinates and the Lagrangian coordinates of the same physical particle point increasing with the increase of the deformation of the circular film, thereby introducing a calculation error that increases with the increase of the deformation of the circular film. Since the larger the deformation of the circular film, the stronger the nonlinearity of its deformation problem, the increase of this calculation error is a strong nonlinear problem; on the other hand, it also avoids the theoretical errors in the aforementioned existing methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 Schematic diagram of the axisymmetric deformation of a circular film with fixed clamping around the periphery under the action of a uniformly distributed transverse load;

[0034] Figure 2 It is a schematic diagram of the static equilibrium of the center part of the deformed circular membrane (0≤r≤a) detached from the body;

[0035] Figure 3 is the microelement ABCD on the deformed circular membrane and its coordinates on the polar coordinate plane Schematic diagram of the initial position A'B'C'D' on the

[0036] Figure 4 is the radial membrane force σ on the microelement ABCD on the deformed circular membrane r (r)h,σ r (r+△r)h and circumferential membrane force σ t (r)h,σ t (r+△r)h in the polar plane Schematic diagram of the projection,

[0037] Among them, 1 is the circular film after axisymmetric deformation, 2 is the clamping device, and 3 is the circular film before deformation. DETAILED DESCRIPTION

[0038] The following is an explanation of the embodiments of the present invention through specific examples. Those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention.

[0039] like Figures 1 to 4 As shown, the method for determining the stress of a circular film under a transversely uniformly distributed load comprises the following steps:

[0040] S1. Establish a circular film structure and determine the geometric and physical parameters of the circular film;

[0041] S2. Apply a uniformly distributed transverse load to the circular film;

[0042] S3. Using the Lagrangian description method and based on the static equilibrium conditions of a circular membrane with large deflection deformation, establish the Lagrangian description of the out-of-plane and in-plane equilibrium equations;

[0043] S4. The coordination equations are derived from the geometric equations and physical equations described by Lagrangian;

[0044] S5. The power series method is used to solve the out-of-plane equilibrium equation, the in-plane equilibrium equation and the coordination equation, and the analytical solution of the large deflection problem of the axisymmetric deformation of the circular membrane fixed and clamped around the periphery under the action of the transverse uniformly distributed load is obtained. Based on the analytical solution, the radial stress σ after the axisymmetric deformation of the circular membrane is determined. r (r) and circumferential stress σ t (r).

[0045] In the specific implementation, it is assumed that the circular film is flat before deformation, the periphery is fixed and clamped, its radius is denoted as a, thickness is denoted as h, Young's elastic modulus is denoted as E, and Poisson's ratio is denoted as ν. Then, after a transverse uniformly distributed load q is applied to it, it will produce an axisymmetric flexural deformation along the direction of the load q. If the deflection w at the center point of the circular film is m (i.e. the maximum deflection of the circular film) is greater than 1 / 5 of its thickness h, then this axially symmetrical deflection deformation problem of the circular film is called the large deflection problem of axially symmetrical deformation of the circular film in elastic mechanics. Figure 1 This is a schematic cross-sectional view of the large deflection problem of the axisymmetric deformation of a circular membrane along the diameter of the circular membrane, where the dotted line represents the geometric mid-surface of the initially flat circular membrane, o, r, w represents the introduced cylindrical coordinate system The coordinate origin (located at the centroid of the geometric mid-surface), radial coordinates, circumferential coordinates (due to the cross-sectional view Figure 1 Not shown in ) and transverse coordinates (w also represents the deflection of the circular membrane), polar coordinate plane The plane of the geometric mid-face.

[0046] Next, the out-of-plane equilibrium equation of the Lagrangian description of the large deflection problem of the axisymmetric deformation is established. Figure 1 A detached body is taken from the central area of ​​the large deflection circular film. In the Lagrangian coordinate system, the spatial position of the detached body before and after deformation is shown in the figure of the specification. Figure 2 As shown in the figure, A′A′ represents the position of the detached body in the Lagrangian coordinate system before deformation (since the radius of the detached body before deformation is r, r is the Lagrangian radial coordinate variable describing the detached body before deformation), AA represents the position of the detached body in the Lagrangian coordinate system after deformation (so the radius of the detached body after deformation is r+u, u represents the displacement at the edge of the detached body), σ r , u, θ represent the radial stress, radial displacement and rotation angle of the deformed membrane respectively, and they are all functions of the independent variable r. The total external force π(r+u) of the detached body with a uniformly distributed load q in the transverse direction within r+u is 2 q and membrane force σ at the boundary r+u r The vertical total membrane force of h is 2π(r+u)σ r hsinθ is in static equilibrium under the joint action, so the out-of-plane equilibrium equation established using Lagrangian description is

[0047] 2π(r+u)σ r hsinθ=π(r+u) 2 q, where

[0048]

[0049] Substituting sinθ into the out-of-plane equilibrium equation, the equation is rewritten as

[0050]

[0051] Next, the Lagrangian description of the in-plane equilibrium equation for the large deflection problem of the axisymmetric deformation is established. Figure 1 A microelement ABCD is randomly selected from the large deflection circular film. Before deformation, the microelement is located at A′B′C′D′. The spatial position of the microelement before and after deformation in the Lagrangian coordinate system is shown in the figure of the specification. Figure 3 As shown, the radial coordinates of points A′, D′ and B′, C′ are r and r+Δr respectively, and the coordinates of points A′, B′ and C′, D′ on the circumference are and is the angle bisector of ∠A′oD′, σ r (r)h,σ r (r+△r)h is the radial membrane force, σ t (r)h,σ t (r+△r)h is the circumferential film force. Microelement ABCD is parallel to the polar coordinate plane under the combined action of radial and circumferential film forces. The direction is in a state of static equilibrium, as shown in the attached figure of the specification. Figure 4 As shown, θ(r) and θ(r+△r) represent the boundary and The film rotation angle at r+u represents the boundary The radial coordinates at and The parallel distance between and The radial distance between the two is △r+(du / dr)△r. In the horizontal direction, there are four horizontal forces, namely, The radial membrane force σ r The horizontal component σ of (r+△r)h r (r+△r)hθ(r+△r), acting on the boundary The radial membrane force σ r (r) The horizontal component σ of h r (r)hθ(r), acting on the boundary and The horizontal circumferential membrane force on t (r)h+σ t (r+△r)h] / 2. Therefore, according to the bisector The condition that the resultant force in the direction is equal to zero, the equilibrium equation in the plane described by Lagrange is listed as:

[0052]

[0053] When △r→0 and When , we can use the approximate relationship θ(r+△r)≈θ(r) and σ r (r+△r) and σ t (r+△r) are expanded according to Taylor series and the terms containing △r quadratic and above are omitted, that is, and The above formula is simplified to:

[0054]

[0055] in,

[0056]

[0057] Then the in-plane equilibrium equation can be written as:

[0058]

[0059] At the same time, the geometric equation of the Lagrangian description of the large deflection problem of axisymmetric deformation can be written as:

[0060]

[0061] The physical equation described by Lagrange can be written as:

[0062]

[0063] Among them, e r 、e t Represents the radial and circumferential strains of the deformed membrane. The coordination equation is derived using the geometric equation and the physical equation:

[0064]

[0065] Since the periphery of the circular film is fixed and clamped, at the position r = a of the circular film, the circumferential strain is zero and the deflection is also zero, that is, when r = a,

[0066] When r=a, w=0.

[0067] The out-of-plane balance equation, in-plane balance equation, and coordination equation contain σ r , σ t , w, use the power series method to solve these three equations. r , σ t, w is expanded into a power series about r / a

[0068]

[0069] σ r (r), σ t Substitute the power series expressions of w(r) and w(r) into the out-of-plane equilibrium equation, in-plane equilibrium equation, and coordination equation, and let r 2i The sum of the coefficients of (i=0,1,2,…,6) is zero, and we get a series of power series coefficients α 2i (i=0,1,2,…,6),β 2i (i=0,1,2,…,6),γ 2i (i=0,1,2,…,6) equations, through which α 2i (i=0,1,2,…,6),β 2i (i=0,1,2,…,6),γ 2i (i=0,1,2,…,6) is expressed as a polynomial with an undetermined coefficient α0. γ0 is also an undetermined coefficient. The two undetermined coefficients α0 and γ0 are obtained by the boundary conditions. The value of the undetermined coefficient α0 is determined by the boundary condition r=a when e t (r)=[σ t (r)-νσ r (r)] / E=0, that is, by the equation:

[0070]

[0071] The value of the unknown coefficient γ0 is determined by the boundary condition w(r) = 0 when r = a, that is, by the equation:

[0072]

[0073] So far, the analytical solution of the large deflection problem of the axisymmetric deformation of a circular membrane with fixed clamping around it under transverse uniform load is obtained. Based on this analytical solution, a method for determining the stress of a circular membrane under transverse uniform load is given.

[0074] At the same time, Figures 1 to 4 In the equation, a represents the radius of the circular film and the inner radius of the clamping device, q represents the lateral uniform load, and w represents the m represents the maximum deflection of the circular membrane after axisymmetric deformation, o, r, w represents the origin, radial coordinate, circumferential coordinate, and transverse coordinate of the cylindrical coordinate system, respectively. Δr, Respectively represent r, The radial and circumferential increments of points A′, D′ and B′, C′ are r and r+Δr respectively, and the circumferential coordinates of points A′, B′ and C′, D′ are and r+u means The radial coordinate at △r+(du / dr)△r represents and The radial distance between is the angle bisector of ∠A′oD′, σ r (r), σ t (r), u(r), w(r), θ(r) represent the radial stress, circumferential stress, radial displacement, lateral displacement, and film rotation angle of the deformed circular film at the radial coordinate r, respectively. r (r+△r),σ t (r+△r), u(r+△r), w(r+△r), and θ(r+△r) represent the radial stress, circumferential stress, radial displacement, lateral displacement, and film rotation angle of the deformed circular film at the radial coordinate r+Δr, respectively.

[0075] Combined with specific numerical values, the technical solution of the present invention is further explained: there is a circular film that is initially flat before deformation, the periphery is fixed and clamped, the radius a = 70mm, the thickness h = 0.3mm, and the Young's elastic modulus E = 3.01N / mm 2 , Poisson's ratio ν = 0.45, when a uniformly distributed transverse load q is applied to it, it will produce axisymmetric flexural deformation along the direction of the load q, and the measured load q = 0.005N / mm 2 , using the method provided by the present invention, from the equation:

[0076]

[0077]

[0078]

[0079] β0=α0;

[0080]

[0081]

[0082]

[0083]

[0084] Obtain α0 = 0.2315164063 and γ0 = 0.4948187539 and α2 = 0.02616613067, α4 = -0.005536636959, α6 = -0.001174620580, α8 = -0.00003801122937, α 10=-0.00003344273442, α 12 =-0.00001941054465, β0=0.2315164063, β2=-0.1276709349, β4=0.01296618665, β6=-0.003648057532, β8=-0.0001897028399, β 10 =-0.00004477243687, β 12 =-0.00003293858133, and then from the equation The uniformly distributed load q on the circular film in the lateral direction is determined to be 0.005 N / mm. 2 The radial stress at any point under the action is:

[0085] σ r (r) = -4.221120375 × 10 -27 r 12 -3.563591181×10 -23 r 10 -1.984696443×10 -19 r 8

[0086] -3.005217168×10 -14 r 6 -6.940973447×10 -10 r 4 +1.607348027×10 -5 r 2 +0.6968643830;

[0087] Circumferential stress:

[0088]

[0089] Finally, it should be noted that the above are only preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, it is still possible for those skilled in the art to modify the technical solutions described in the aforementioned embodiments or to make equivalent substitutions for some of the technical features therein. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for determining the stress of a circular film under a uniformly distributed transverse load, characterized in that: The following steps are involved: S1. Establish a circular film structure and determine the geometric and physical parameters of the circular film; S2. Apply a uniformly distributed transverse load to the circular film; S3. Using the Lagrangian description method and based on the static equilibrium conditions of a circular membrane with large deflection deformation, establish the Lagrangian description of the out-of-plane and in-plane equilibrium equations; S4. The coordination equations are derived from the geometric equations and physical equations described by Lagrangian; S5. The power series method is used to solve the out-of-plane equilibrium equation, the in-plane equilibrium equation and the coordination equation, and the analytical solution of the large deflection problem of the axisymmetric deformation of the circular membrane fixed and clamped around the periphery under the action of the transverse uniformly distributed load is obtained. Based on the analytical solution, the radial stress σ after the axisymmetric deformation of the circular membrane is determined. r (r) and circumferential stress σ t (r).

2. The method for determining the stress of a circular film under a uniformly distributed transverse load according to claim 1, characterized in that: In S1, the circular film is flat before deformation, and its periphery is fixed and clamped, and its geometric and physical parameters include radius a, thickness h, Young's elastic modulus E, and Poisson's ratio ν.

3. The method for determining the stress of a circular film under a uniformly distributed transverse load according to claim 1, characterized in that: In S2, after a uniformly distributed transverse load q is applied to the circular film, the circular film will produce an axisymmetric flexural deformation along the direction of the load q.

4. The method for determining the stress of a circular film under a uniformly distributed transverse load according to claim 1, characterized in that: In S3 and S4, the out-of-plane equilibrium equation is used to describe the equilibrium relationship between the external load and the membrane stress; The in-plane equilibrium equation is used to describe the equilibrium relationship between the stresses of each film; The geometric equation is used to describe the geometric relationship between film strain and displacement; The physical equation is used to describe the physical relationship between film stress and strain; Among them, the independent variables of stress, strain and displacement are all Lagrangian coordinate variables, that is, the coordinates of any point on the circular film before deformation.

5. The method for determining the stress of a circular film under a uniformly distributed transverse load according to claim 1, characterized in that: In S5, the following equation is used: β0＝α0； Determine α0 and α2, α4, α6, α8, α 10 ,a 12 ,β0,β2,β4,β6,β8,β 10 、b 12 the value of,finally,by the equation: Determine the radial stress σ of the circular film at the radial coordinate r r (r), circumferential stress σ t (r), where r represents the radial coordinate of any point on the initially flat circular film, a, h, and r are all in millimeters (mm), and E, q, σ r (r), σ t The unit of (r) is Newton per square millimeter (N / mm 2 ), while ν, α0, α2, α4, α6, α8, α 10 , α 12 , β0, β2, β4, β6, β8, β 10 , β 12 and Q are dimensionless quantities.

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