CN117408011B - A meshless buckling analysis method for variable stiffness plates with delamination based on delamination theory - Google Patents
A meshless buckling analysis method for variable stiffness plates with delamination based on delamination theoryInfo
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Abstract
The invention discloses a delamination-containing stiffness-changing plate grid-free buckling analysis method based on a delamination theory, which comprises the steps of obtaining geometry and material parameter information of a delamination-containing stiffness-changing fiber composite material laminated plate, generating node and integration point information based on numerical value layer number information at a delamination position, calculating and storing node and integration point grid-free shape functions and derivative information thereof, completing grid-free space dispersion of a problem domain, constructing an integral stiffness matrix and a geometric stiffness matrix based on the delamination approximation theory, applying an intrinsic boundary condition through a direct method, calculating buckling characteristic values and buckling characteristic vectors of the delamination-containing stiffness-changing fiber composite material laminated plate, and analyzing the penetration behavior of the delamination-containing stiffness-changing fiber composite material laminated plate based on the buckling characteristic values and the buckling characteristic vectors. The method can be suitable for buckling analysis of the curved fiber composite laminated plate with any delamination, can eliminate mutual invasion of upper and lower parts of a delamination area, and has more reasonable and accurate results.
Description
Technical Field
The invention relates to the technical field of variable-stiffness fiber composite materials, in particular to a grid-free buckling analysis method of a delamination-containing variable-stiffness plate based on a delamination theory.
Background
The resin-based fiber composite material has the advantages of high specific strength, high specific rigidity, corrosion resistance, easiness in processing and forming and the like, and has wide application in various fields such as aerospace, automobiles, ships, wind power and the like. The traditional constant-stiffness fiber composite material laminated board adopts straight fibers, and the variable-stiffness fiber composite material adopts curved fiber layering, so that more design space is obtained, and more excellent mechanical properties can be realized. Moreover, with the development of advanced automatic filament laying technology in recent years, curve fiber laying moves from theoretical assumption to actual preparation, and at present, the laying angle of a composite material can be changed by a method of curve laying filament bundles, so that a more reasonable force transfer path is realized.
Resin-based fiber composite laminates are generally applied in engineering in the form of a panel shell structure, and as a main failure mode of the composite panel shell, the calculation of buckling strength is particularly important in design. Whereas the variable stiffness laminate is more computationally complex due to the curved fiber path. Meanwhile, because of the process and structural characteristics of the fiber composite material laminated plate, the interlayer strength mainly depends on the resin substrate, the adhesion between the substrate and the fibers and the like, and is far lower than the in-layer fiber strength, delamination damage is one of the main damage modes of the fiber composite material laminated plate, and an effective method for calculating the buckling bearing capacity of the fiber composite material laminated plate with delamination rigidity is not seen at present.
Disclosure of Invention
This section is intended to outline some aspects of embodiments of the application and to briefly introduce some preferred embodiments. Some simplifications or omissions may be made in this section as well as in the description of the application and in the title of the application, which may not be used to limit the scope of the application.
The present invention has been made in view of the above-described problems occurring in the prior art.
Therefore, the invention provides a grid-free buckling analysis method for a delamination-containing variable-stiffness plate based on a delamination theory, which solves the problems that the buckling bearing capacity and buckling mode of the existing variable-stiffness fiber composite material laminated plate with delamination cannot be accurately analyzed and predicted.
In order to solve the technical problems, the invention provides the following technical scheme:
the embodiment of the invention provides a delamination-containing stiffness-changing plate grid-free buckling analysis method based on a delamination theory, which comprises the following steps of:
Obtaining the geometry and material parameter information of the fiber composite laminated plate with delamination and variable rigidity;
Generating node and integral point information based on the parameter information and numerical value layer number information at delamination, and calculating and storing node and integral point gridless shape functions and derivative information thereof to finish gridless spatial dispersion of a problem domain;
Based on a layering approximation theory, constructing an overall stiffness matrix and a geometric stiffness matrix, applying an essential boundary condition through a direct method, and calculating a buckling characteristic value and a buckling characteristic vector of the layered variable stiffness fiber composite material laminated plate;
and analyzing the penetration behavior of the delamination-resistant variable-stiffness fiber composite material laminated plate based on the buckling characteristic value and the buckling characteristic vector.
As a preferable scheme of the delamination-based grid-free buckling analysis method for the delamination-based variable stiffness plate, the invention further comprises the following steps:
if the relative displacement at the delamination interface does not contain elements smaller than the first threshold value, judging that no penetration behavior exists;
If the relative displacement at the delamination interface is smaller than the first threshold value, determining that penetration behavior exists;
When the penetration behavior exists and the penetration displacement is not greater than the allowable penetration threshold, ignoring the penetration condition;
When the penetration behavior exists, if the penetration displacement is larger than the allowable penetration threshold, iteration parameters are determined, an additional stiffness matrix of the spring is calculated, and the overall stiffness matrix is updated until no penetration behavior exists.
As a preferable scheme of the delamination-based grid-free buckling analysis method for the delamination-based variable stiffness plate, the invention comprises the steps of calculating an additional stiffness matrix of a spring, wherein the stiffness of a virtual spring at any node i of a delamination part is expressed as:
Wherein, the For the relative displacement of the interface at delamination interface point x i in the numerical solution for this iteration,Corresponding on diagonal to the overall stiffness matrixIn the item(s) of (c),For the relative displacement of the interface at point x i after stiffness change, r is the iteration parameter and r=0.00001×3 m-1, m is the number of iterations, i is any node with penetration behavior.
As a preferable scheme of the delamination-based grid-free buckling analysis method for the delamination-based variable stiffness plate, the invention comprises the following steps of:
Wherein U is, The displacement parameters of the delamination area and the delamination area of the laminated board are respectively, K b、Ks、Kt is a bending matrix, a shearing matrix and a coupling stiffness matrix which are not delaminated,Bending matrix, shearing matrix and coupling stiffness matrix of delamination and non-delamination coupling respectively,Respectively a bending matrix, a shearing matrix and a coupling stiffness matrix of delamination, wherein K is an overall stiffness matrix.
As a preferable scheme of the delamination-based grid-free buckling analysis method for the delamination-based variable stiffness plate, the method for calculating the geometric stiffness matrix according to the work of NP numerical layer pre-buckling loads comprises the following steps:
the geometric stiffness matrix K G is expressed as:
Wherein, the Is the deflection of the nth numerical layer,The axial compression pre-buckling load is in the horizontal direction of the in-plane coordinates of the nth numerical value layer,The pre-buckling load is axially pressed in the vertical direction of the in-plane coordinates of the nth numerical layer,For in-plane shear pre-buckling load, W 1 is the geometric stiffness matrix of the nth numerical layer, W 4 is the geometric stiffness matrix of the n+1th numerical layer, W 2 is the first coupling stiffness matrix, and W 3 is the second coupling stiffness matrix.
As a preferable scheme of the delamination-based grid-free buckling analysis method for the delamination-based variable stiffness plate, provided by the invention, if delamination exists at both the n-th and n+1-th numerical interfaces, the derivative of the numerical interface deflection can be expressed as follows by a grid-free shape function:
Wherein ne is the number of functions, For the displacement parameters at the non-grid node, W n,x is the derivative of the deflection of the nth numerical layer to the x direction, W n,y is the derivative of the deflection of the nth numerical layer to the y direction, W n+1,x is the derivative of the deflection of the (n+1) th numerical layer to the x direction, W n+1,y is the derivative of the deflection of the (n+1) th numerical layer to the y direction,For the matrix of the shape functions of the nodes at i, N i,x is the derivative of the shape function of the node at i with respect to x, N i,y is the derivative of the shape function of the node at i with respect to y,Is the u-direction displacement parameter of the non-delamination position corresponding to the ith shape function of the n+1th numerical layer,Is the v-direction displacement parameter of the non-delamination position corresponding to the ith shape function of the (n+1) th numerical layer,The w-direction displacement parameter of the non-delamination place corresponding to the ith shape function of the (n+1) -th numerical layer is marked by the subscript "] which indicates delamination.
As a preferable scheme of the delamination-based grid-free buckling analysis method for the delamination-based variable stiffness plate, the invention further comprises the following steps:
if no delamination exists at the n-th or n+1-th numerical interface, the corresponding relation is that Or (b)Is absent;
applying an intrinsic boundary condition by adopting a direct method, and enabling the boundary node i to be The calculated buckling characteristic value and buckling mode are expressed as:
(K-λKG)U=0
wherein λ is a eigenvalue, U eigenvector.
As a preferable scheme of the grid-free buckling analysis method for the delamination-containing variable stiffness plates based on the delamination theory, the method for obtaining the geometry and material parameter information of the delamination-containing variable stiffness fiber composite material laminated plates at least comprises a plate length L x, a plate width L y, a plate thickness H, a layering number NL, a layering layer thickness t i, a delamination number, delamination positions H/H, delamination widths L dx/Lx、Ldy/Ly, an elastic modulus, a shear modulus and a Poisson ratio.
As a preferable scheme of the delamination-based grid-free buckling analysis method for the delamination-based variable stiffness plate, the invention further comprises the following steps:
input fiber path and ply design
The nodes and integration points are arranged according to the panel length L x and the panel width L y, as well as the number of layers.
The invention relates to a method for analyzing grid-free buckling of a plate with delamination and variable stiffness based on a layering theory, which comprises the steps of calculating and storing node and integral point grid-free shape functions and derivative information thereof, and completing grid-free space dispersion of a problem domain, wherein the method comprises the steps of determining the problem domain according to the geometric dimension of the plate, and regularly arranging a proper amount of nodes in the problem domain;
Each node shape function is calculated by using a radial base point interpolation method, and an approximate displacement field u h (x) is expressed as:
uh(x)＝{r p}G-1U=ΦU
Φ={r p}G-1
wherein phi is a shape function, r, p is a radial basis function and a polynomial basis function, and G is a basis function coefficient.
Compared with the prior art, the method has the beneficial effects that the method can be suitable for buckling analysis of the curved fiber composite material laminated plate with any delamination, can eliminate mutual invasion of upper and lower parts of a delamination area, has more reasonable and accurate results, can more accurately represent the variable stiffness characteristic of the structure, processes any plate shell geometric shape and any delamination geometric shape, does not need to perform secondary approximation on a field function, and has better calculation efficiency. The method of the invention can analyze not only the whole buckling, the mixed buckling and the local buckling, but also eliminate the possible penetration phenomenon of the upper layer and the lower layer of the delamination area in calculation.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings that are needed in the description of the embodiments will be briefly described below, it being obvious that the drawings in the following description are only some embodiments of the present invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art. Wherein:
FIG. 1 is a flow chart of a delamination-based stiffness-changing plate grid-free buckling analysis method according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a model A of a delamination-based stiffness-variable-plate grid-free buckling analysis method based on the delamination theory according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a model B of a delamination-based stiffness-variable-plate grid-free buckling analysis method based on the delamination theory according to an embodiment of the present invention;
FIG. 4 is a graph of a linear variation fiber path for a delamination-based stiffness-variable panel gridless buckling analysis method based on delamination theory according to one embodiment of the present invention;
FIG. 5 is a schematic diagram of a loading mode of a model A of a grid-free buckling analysis method of a delamination-containing variable stiffness plate based on a delamination theory according to an embodiment of the present invention;
FIG. 6 is a schematic diagram of a loading mode of a model B of a grid-free buckling analysis method of a delamination-containing variable stiffness plate based on a delamination theory according to an embodiment of the present invention;
FIG. 7 is a graph showing a buckling mode with a delamination vertical position and a delamination width of 0.1 for a grid-less buckling analysis method for a delamination-based variable stiffness plate according to an embodiment of the present invention;
FIG. 8 is a graph of buckling modes with a delamination vertical position of 0.1 and a width of 0.5 for a delamination-based grid-free buckling analysis method for a delamination-based variable stiffness plate according to one embodiment of the present invention;
FIG. 9 is a graph of buckling modes with a delamination vertical position of 0.1 and a width of 0.9 for a delamination-based grid-free buckling analysis method for a delamination-based variable stiffness plate according to one embodiment of the present invention;
FIG. 10 is a graph of buckling modes with a delamination vertical position of 0.3 and a width of 0.1 for a delamination-based grid-free buckling analysis method for a delamination-based variable stiffness plate according to an embodiment of the present invention;
FIG. 11 is a graph of buckling modes with a delamination vertical position of 0.3 and a width of 0.5 for a delamination-based grid-free buckling analysis method for a delamination-based variable stiffness plate according to one embodiment of the present invention;
FIG. 12 is a graph of buckling modes with a delamination vertical position of 0.3 and a width of 0.9 for a delamination-based grid-free buckling analysis method for a delamination-based variable stiffness plate according to one embodiment of the present invention;
FIG. 13 is a graph of buckling modes with a delamination vertical position of 0.5 and a width of 0.1 for a delamination-based grid-free buckling analysis method for a delamination-based variable stiffness plate according to one embodiment of the present invention;
FIG. 14 is a graph of buckling modes with a delamination vertical position of 0.5 and a width of 0.5 for a delamination-based grid-free buckling analysis method for a delamination-based variable stiffness plate according to one embodiment of the present invention;
FIG. 15 is a graph of buckling modes with a delamination vertical position of 0.5 and a width of 0.9 for a delamination-based grid-free buckling analysis method for a delamination-based variable stiffness plate according to one embodiment of the present invention;
FIG. 16 is a graph showing a buckling mode of model B using additional springs to remove penetration for a delamination-containing stiffness-changing plate grid-free buckling analysis method according to one embodiment of the present invention;
FIG. 17 is a graph showing a buckling mode of model B of a grid-free buckling analysis method for a delamination-containing variable stiffness plate without additional springs according to an embodiment of the present invention.
Detailed Description
So that the manner in which the above recited objects, features and advantages of the present invention can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to the embodiments, some of which are illustrated in the appended drawings. All other embodiments, which can be made by one of ordinary skill in the art based on the embodiments of the present invention without making any inventive effort, shall fall within the scope of the present invention.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, but the present invention may be practiced in other ways other than those described herein, and persons skilled in the art will readily appreciate that the present invention is not limited to the specific embodiments disclosed below.
Further, reference herein to "one embodiment" or "an embodiment" means that a particular feature, structure, or characteristic can be included in at least one implementation of the invention. The appearances of the phrase "in one embodiment" in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments mutually exclusive of other embodiments.
While the embodiments of the present invention have been illustrated and described in detail in the drawings, the cross-sectional view of the device structure is not to scale in the general sense for ease of illustration, and the drawings are merely exemplary and should not be construed as limiting the scope of the invention. In addition, the three-dimensional dimensions of length, width and depth should be included in actual fabrication.
Also in the description of the present invention, it should be noted that the orientation or positional relationship indicated by the terms "upper, lower, inner and outer", etc. are based on the orientation or positional relationship shown in the drawings, are merely for convenience of describing the present invention and simplifying the description, and do not indicate or imply that the apparatus or elements referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus should not be construed as limiting the present invention. Furthermore, the terms "first, second, or third" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance.
The terms "mounted, connected, and coupled" as used herein, unless otherwise specifically indicated and defined, shall be construed broadly and include, for example, fixed, removable, or integral, as well as mechanical, electrical, or direct, as well as indirect via intermediaries, or communication between two elements. The specific meaning of the above terms in the present invention will be understood in specific cases by those of ordinary skill in the art.
Example 1
Referring to fig. 1, in one embodiment of the present invention, a delamination-containing variable stiffness plate gridless buckling analysis method based on delamination theory is provided, including:
s1, acquiring the geometry and material parameter information of a fiber composite laminated plate with delamination and variable rigidity;
Further, the geometric and material parameter information of the laminated plate with the delamination variable stiffness fiber composite material at least comprises a plate length L x, a plate width L y, a plate thickness H, a layering number NL, a layering layer thickness t i, the delamination number, a delamination position H/H, a delamination width L dx/Lx、Ldy/Ly, an elastic modulus, a shear modulus and a Poisson ratio;
Further, the fiber paths and ply designs are input, and the fiber paths in the embodiment of the invention are predetermined by a function theta (x, y) to analyze any fiber path and ply design, and the fiber paths are expressed as linear fiber paths:
Wherein x, y are the coordinates of the calculated points, phi is the rotation angle of the coordinate axes, d 0 represents the linear variation period of the fiber, T 0 and T 1 are the angles between the fiber and the x' axis at the beginning and end of the period respectively, and theta is the fiber deflection angle.
Specifically, the fiber path is denoted as [ phi < T 0T1 > ].
S2, generating node and integral point information based on parameter information combined with numerical value layer number information at delamination, calculating and storing a node and integral point grid-free shape function and derivative information thereof, and completing grid-free space dispersion of a problem domain;
furthermore, the embodiment of the invention adopts Gaussian integration according to the plate length L x and the plate width L y, the number of numerical layers, the arrangement nodes and the integration points, and the number of the numerical layers used by the layering approximation theory can be larger than or equal to the number of the layers.
Further, calculating a node and Gaussian point gridless function and a derivative thereof, and storing a spare tone;
Specifically, a problem domain is determined according to the geometric dimension of the plate, a proper amount of nodes are regularly arranged in the problem domain, RPIM is adopted to calculate the node shape function phi, and an approximation function is set as
Where n is the number of nodes in the computation point support domain, m is the number of terms of the polynomial basis function, r i (x) and p j (x) are the radial basis function and the polynomial basis function, respectively, and a i (x) and b j (x) are the corresponding coefficients.
Let the nodes in the computation point support domain satisfy the above approximation formula, it is possible to obtain:
where u is the column vector of the field function and R and P are matrices of the row vector of the radial basis function and the row vector of the polynomial basis function, respectively, whereby:
uh(x)＝{r p}G-1U=ΦU
Φ={r p}G-1
S3, constructing an overall stiffness matrix and a geometric stiffness matrix based on a layering approximation theory, applying an essential boundary condition through a direct method, and calculating a buckling characteristic value and a buckling characteristic vector of the fiber composite laminated board with delamination and rigidity variation;
further, the overall stiffness matrix is calculated from the matrix form of the strain energy variation of the lower laminate, expressed as:
Wherein U is, The displacement parameters of the delamination area and the delamination area of the laminated board are respectively, K b、Ks、Kt is a bending matrix, a shearing matrix and a coupling stiffness matrix which are not delaminated,Bending matrix, shearing matrix and coupling stiffness matrix of delamination and non-delamination coupling respectively,Respectively a bending matrix, a shearing matrix and a coupling stiffness matrix of delamination, wherein K is an overall stiffness matrix.
Specifically, the non-delaminated bending matrix is expressed as:
Wherein T represents the matrix transpose, NP is the number of numerical layers, M is the plane of the numerical layers and m=np+1, Is the ordinate of the lower and upper surfaces of the kth numerical layer,In the form of a non-delaminated elastic matrix,For the non-delaminated bending matrix at the I, J value plane, H I,HJ is the interpolation function at the I, J value plane,For the flexural modulus of the k-value layer, B b is the strain matrix of the non-delaminated bending matrix, D b is the strain matrix, H I (z) is the global interpolation function of the displacement component along the thickness direction, ψ I (z) is the step function representing delamination.
The bending matrix of delamination versus non-delamination coupling is expressed as:
Wherein, the For a strain matrix that couples bending matrices with delamination and non-delamination,The J-layer values plane the bending stiffness matrix for the I-th layer coupled with delamination and non-delamination.
The bending matrix for delamination is expressed as:
Wherein, the The J-layer value is the bending stiffness moment of the planar layer for delamination I.
The non-delaminated shear matrix is expressed as:
Wherein B s is a strain matrix of a non-delamination moment of bending shear coupling stiffness, The J-layer value is the bending shear coupling stiffness moment of the planar layer, i.e., non-delaminated.
The shear stiffness matrix coupled with delamination and non-delamination is expressed as:
Wherein, the A strain matrix of bending shear coupled stiffness moments coupled with delamination and non-delamination,The bending shear coupling stiffness moment of the planar layer is the I/J layer number value of the non-delamination and delamination coupling,The strain matrices are respectively the ith shape function.
The shear stiffness matrix for delamination is expressed as:
Wherein, the The J-layer value is the moment of bending shear coupling stiffness of the planar layer for delamination I.
The non-delaminated shear stiffness matrix is expressed as:
Wherein B t is the strain matrix of the non-delaminated shear stiffness matrix,
The shear stiffness matrix coupled with delamination and non-delamination is expressed as:
Wherein, the For a strain matrix of a shear stiffness matrix coupled with delamination and non-delamination,
The shear stiffness matrix for delamination is expressed as:
Wherein, the The strain matrix is a function of the ith shape.
Further, if the global interpolation function H I (z) is a Lagrange interpolation function, it can be expressed as follows
Wherein h k is the thickness of the kth numerical layer, The thickness coordinate of the lower surface of the kth numerical layer is z, which is a vertical coordinate.
Further, the step function of delamination is expressed as:
Wherein z I represents the ordinate of the ith delamination.
The strain matrix, which is the ith shape function, is expressed as follows:
Wherein phi i、φi,x and phi i,y are respectively gridless functions and their derivatives in the x, y directions.
AndThe elastic coefficients of the in-plane bending, transverse shearing, in-plane bending and transverse expansion coupling parts of the kth numerical layer under the global coordinate system are respectively expressed as follows
The stiffness coefficient matrix of the k-th layer of the laminated plate under the global coordinate system can be obtained by converting the stiffness matrix of the material coordinate system by the following coordinates, and is expressed as follows:
Wherein θ is the fiber angle.
Further, calculating the geometric stiffness matrix based on the work performed by the NP number of numerical layer pre-buckling loads includes:
The geometric stiffness matrix K G is expressed as:
Wherein, the Is the deflection of the nth numerical layer,The axial compression pre-buckling load is in the horizontal direction of the in-plane coordinates of the nth numerical value layer,The pre-buckling load is axially pressed in the vertical direction of the in-plane coordinates of the nth numerical layer,For in-plane shear pre-buckling load, W 1 is the geometric stiffness matrix of the nth numerical layer, W 4 is the geometric stiffness matrix of the n+1th numerical layer, W 2 is the first coupling stiffness matrix, and W 3 is the second coupling stiffness matrix.
Specifically, the geometric stiffness matrix of the nth numerical layer is expressed as:
wherein W n,x,Wn,y is the derivative of deflection of the nth numerical layer with respect to the x, y directions.
The first coupling stiffness matrix and the second coupling stiffness matrix are expressed as:
the geometric stiffness matrix of the n+1th numerical layer is expressed as:
Further, assuming delamination at both the n-th and n+1-th numerical interfaces, the derivative of the numerical interface deflection can be expressed by a gridless function as:
Wherein ne is the number of shape functions, For the corresponding displacement parameters, W n,x is the derivative of the deflection of the nth value layer with respect to the x-direction, W n,y is the derivative of the deflection of the nth value layer with respect to the y-direction, W n+1,x is the derivative of the deflection of the n+1th value layer with respect to the x-direction, W n+1,y is the derivative of the deflection of the n+1th value layer with respect to the y-direction,For the matrix of the shape functions of the nodes at i, N i,x is the derivative of the shape function of the node at i with respect to x, N i,y is the derivative of the shape function of the node at i with respect to y,Is the u-direction displacement parameter of the non-delamination position corresponding to the ith shape function of the n+1th numerical layer,Is the v-direction displacement parameter of the non-delamination position corresponding to the ith shape function of the (n+1) th numerical layer,The w-direction displacement parameter of the non-delamination place corresponding to the ith shape function of the (n+1) -th numerical layer is marked by the subscript "] which indicates delamination.
Further, if there is no delamination at the n-th or n+1-th numerical interface, the corresponding relation isOr (b)Is absent.
Applying an intrinsic boundary condition by adopting a direct method, and enabling the boundary node i to beAnd solving a system equation (K-lambda K G) U=0 to obtain a buckling characteristic value and a buckling mode.
And S4, analyzing the penetration behavior of the delamination stiffness-variable fiber composite material laminated plate based on the buckling characteristic value and the buckling characteristic vector.
Still further, still include:
if the relative displacement at the delamination interface does not contain elements smaller than the first threshold value, judging that no penetration behavior exists;
If the relative displacement at the delamination interface is smaller than the first threshold value, determining that penetration behavior exists;
When the penetration behavior exists and the penetration displacement is not greater than the allowable penetration threshold, ignoring the penetration condition;
When the penetration behavior exists, if the penetration displacement is larger than the allowable penetration threshold, iteration parameters are determined, an additional stiffness matrix of the spring is calculated, and the overall stiffness matrix is updated until no penetration behavior exists.
It should be noted that, in the embodiment of the present invention, the value of the first threshold is 0, the allowable penetration threshold range is [10 -4,10-6 ], and the specific value of the allowable penetration threshold can be selected according to the model size and the specific problem.
Specifically, calculating the additional stiffness matrix for the spring includes expressing the stiffness of the virtual spring at any node of the delamination site as:
Wherein, the For the relative displacement of the interface at delamination interface point x i in the numerical solution for this iteration,Corresponding on diagonal to the overall stiffness matrixIn the item(s) of (c),For the relative displacement of the interface at point x i after stiffness change, r is the iteration parameter and r=0.00001×3 m-1, m is the number of iterations, i is any node with penetration behavior.
It should be noted that setting r to a parameter that increases exponentially with the number of iterations is to minimize the effect of additional stiffness on node displacement at other locations when the number of iterations is low, and to converge as quickly as possible when the number of iterations is high. The choice of the parameter r determines the speed of iteration convergence and the accuracy of the iteration result.
Example 2
Referring to fig. 2-17, an embodiment of the present invention, which is different from the first embodiment, provides a specific example of a delamination-containing stiffness-changing plate grid-free buckling analysis method based on a delamination theory, and verifies the beneficial effects of my invention.
In this example, 2 different delamination conditions were provided, model a and model B, respectively.
As shown in fig. 2, model a has a longitudinal through delamination, where L x＝4m,Ly =4m, h=0.01 m, for the geometry of model a. The boundary condition is simple on opposite sides and free on opposite sides, and the thickness of each layer is t i =0.001 m. Delamination position H/h=0.1, 0.3, 0.5, delamination width L d/Lx =0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9;
As shown in fig. 3, which is the geometry of model B, there is a rectangular delamination area in the middle of model B, where L x＝Ly =0.81408 m, h=1.016 mm. Its delamination position H/h=0.25, delamination width L dx/Lx＝Ldy/Ly =0.6. Boundary conditions are simple supports on opposite sides and free opposite sides;
In the material coordinate system, the elastic modulus of the laminated plate is E 11＝181Gpa,E22＝E33 = 10.273Gpa, the shear modulus G 12＝G23＝G13 = 7.170Gpa, and the Poisson ratio v 12 =0.28. The fiber path definition function θ (x, y) is input such that the fiber lay state of the model A laminate is [ + -0 <0|0> ] 10 as shown in FIG. 4, and the fiber orientation of the model B is [ + -90 <0|75> ] 2s as shown in FIG. 5.
According to the plate length L x and the plate width L y, the model A is divided into 40 equal-sized integrating units, and the model B is divided into 64 integrating units. Both were calculated using 2 number layers. And determining RPIM to calculate a shape function parameter q=1.5, alpha c =1, and a dimensionless support domain size d max =5, regularly arranging 101×11 nodes in a problem domain, and calculating and storing the shape function of the nodes.
Calculating rigidity matrix K of model A and B according to matrix form of strain energy variation of laminated plate, respectively integrating and calculating work of longitudinal pre-buckling load on 2 number layers to obtain model A and B geometric rigidity matrix K G, applying essential boundary condition by adopting direct method, making at boundary node iAnd solving a system equation (K-lambda K G) U=0 to obtain a buckling characteristic value and a buckling mode.
Judging whether penetration behavior exists, namely relative displacement at delamination interfaceJudging whether elements smaller than a preset allowable penetration displacement epsilon exist or not, if not, ending, wherein the allowable penetration displacement epsilon is 0.00001, if so, calculating the spring stiffness K *, updating the integral stiffness matrix K, and solving the buckling characteristic value and the buckling mode until ending after no penetration exists. And drawing a buckling mode diagram according to the buckling characteristic vector.
In this embodiment, the nodes are in a regular arrangement of 101×11, and the result of FEM is obtained by modeling the software Abaqus with continuous shell elements, and the model uses 100×100 elements. Under the condition of a model A, carrying out normalized buckling load comparison on the basis of a 1-dimensional model established by a radial basis gridless layering method (RPIM-LW) and GJ Simitses et al, and adopting a defined dimensionless buckling load to represent buckling performance of the laminated plate, wherein the buckling performance is expressed as follows:
Wherein, the For buckling load, L x is the plate length, E 11 is the x-direction elastic coefficient, and H is the plate thickness.
The comparison results are shown in Table 1:
table 1 model a normalized buckling load for laminated plates with delamination
Referring to fig. 7-15, it can be seen from table 1 that the buckling loads calculated for the three methods of each model are substantially identical, demonstrating the effectiveness of the method of the present invention. For example, as shown in fig. 7, delamination at a vertical position H/h=0.1 is shown, and delamination width is L d/Lx =0.1, i.e., delamination width is 0.4 m.
Based on the method of the present invention under the conditions of model B, the normalized buckling load of VSC laminates with fiber orientation [ ±90<0|75> ] 2s were calculated with and without additional springs to remove breakthrough, respectively, as shown in table 2:
table 2 model B normalized buckling load for layered laminates with delamination
| Whether or not to restrict | Fiber orientation | RPIM-LW |
| Unconstrained buckling | [±90<0|75>]2s | 0.8038 |
| Buckling with contact being taken into account | [±90<0|75>]2s | 1.1454 |
Referring to fig. 16-17, it can be seen from table 2 that the present invention is suitable for buckling analysis of curved fiber composite laminates with arbitrary delamination, and can eliminate the mutual invasion of upper and lower parts of the delamination area, and the result is more reasonable and accurate.
It should be noted that the above embodiments are only for illustrating the technical solution of the present invention and not for limiting the same, and although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that the technical solution of the present invention may be modified or substituted without departing from the spirit and scope of the technical solution of the present invention, which is intended to be covered in the scope of the claims of the present invention.
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