CN113486496A - Geometric nonlinear analysis method for CFRP (carbon fiber reinforced polymer) laminated plate in damp and hot environment - Google Patents

Geometric nonlinear analysis method for CFRP (carbon fiber reinforced polymer) laminated plate in damp and hot environment Download PDF

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CN113486496A
CN113486496A CN202110652268.3A CN202110652268A CN113486496A CN 113486496 A CN113486496 A CN 113486496A CN 202110652268 A CN202110652268 A CN 202110652268A CN 113486496 A CN113486496 A CN 113486496A
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cfrp
damp
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戴宏亮
唐洪
贺其
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Hunan University
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    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G06F2113/26—Composites
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Abstract

The invention relates to a geometric nonlinear analysis method of a carbon fiber reinforced Composite (CFRP) laminated plate under the action of uniformly distributed load in a damp and hot environment, belonging to the technical field of modern engineering structure analysis and application, namely, the influence of damp and hot effect is considered in an engineering structure of CFRP based on a Reddy high-order shear deformation theory. Aiming at solving the problem of damp-heat deformation in engineering materials, a virtual displacement method is utilized to derive a geometric nonlinear control equation, and the elastic limit rotation is considered as a boundary condition; then, the physical parameters of the CFRP were considered constant in the damp heat effect, and the governing equation was solved using the Galerkin method, in which a clamped rectangular CFRP laminate was used. The invention realizes the prediction of the damp-heat deformation of the engineering structure by using a nonlinear analysis method, provides a new method for predicting the structural damage of the CFRP, and has certain guiding significance in reducing the structural accidents of the composite material.

Description

Geometric nonlinear analysis method for CFRP (carbon fiber reinforced polymer) laminated plate in damp and hot environment
Technical Field
The invention relates to a geometric nonlinear analysis method of a carbon fiber reinforced composite material laminated plate under the action of uniformly distributed load in a damp and hot environment, belonging to the technical field of modern engineering structure analysis and application.
Background
Various advanced composite materials are largely used in the industrial fields of aerospace and the like in the 21 st century, and the composite materials are used under certain environmental conditions like other engineering materials. Common environmental conditions include humidity, temperature, corrosive media, ultraviolet radiation, load, and the like. For some composite materials, such as aerospace composite materials, the environmental conditions are more complicated, such as high and low temperature alternation and abrupt change of the wet expansion coefficient. The effect of the environmental condition or conditions results in a change in the properties of the composite. Such changes in properties are primarily influenced by environmental factors, resulting in changes or disruptions in the matrix, fibers or fiber-matrix interface. The fiber-matrix interface of the polymer matrix composite material is damaged under the action of load, stress and corrosion, so that the strength and rigidity of the composite material are reduced. Therefore, it is necessary to analyze the composite material in a hot and humid environment to use the composite material reasonably and effectively.
At present, the influence of the damp-heat effect on the mechanical property of the composite material is mainly limited to a small deformation theory. For the theory of geometric non-linearity under hot and humid conditions, most methods are limited to the classical plate-shell theory or Reissner-minidlin theory, although it is also relevant. The existing research results show that the high-order shear deformation theory is closer to the elastic theory solution. However, high order shear deformation of composite materials in hot and humid environments has been rarely observed in prior methods. Therefore, the key point of the invention is the influence of the humid and hot environment on the geometric nonlinearity of the composite material laminated plate under the high-order shear deformation theory, and simultaneously, more factors such as different boundary conditions, change of geometric parameters and the like are considered.
Disclosure of Invention
The invention provides a geometric nonlinear analysis method of a CFRP (carbon fiber reinforced plastic) laminated plate in a damp and hot environment aiming at predicting damp and hot damage deformation of a composite material structure and improving the safety and reliability of the composite material structure in practical engineering application, namely, the influence of damp and hot effects is considered in the CFRP engineering structure based on a Reddy high-order shear deformation theory. Aiming at solving the problem of damp-heat deformation in engineering materials, a virtual displacement method is utilized to derive a geometric nonlinear control equation, and the elastic limit rotation is considered as a boundary condition; then, the physical parameters of the CFRP were considered constant in the damp heat effect, and the governing equation was solved using the Galerkin method, in which a clamped rectangular CFRP laminate was used. The invention realizes the prediction of the damp-heat deformation of the engineering structure by utilizing a nonlinear analysis method, provides a new method for predicting the structural damage of the CFRP, has strong transportability when being applied to different fields, and can combine more abundant engineering experience and the like.
1. A geometrical non-linear analysis method of a CFRP laminated plate in a damp and hot environment is characterized by comprising the following steps:
placing an x-y axis on the middle surface of the CFRP laminated plate to obtain a displacement field of the CFRP laminated plate;
step two, obtaining the constitutive relation of the kth layer in the CFRP laminated plate;
thirdly, obtaining epsilon based on the displacement field of the CFRP laminated plate and based on the Reddy high-order shear deformation theoryx，εy，γyz，γxzAnd gammaxyThe relation between; wherein epsilonxRepresenting strain, ε, in the direction of the x-axisyRepresenting strain in the y-axis direction, gammayzRepresenting the shear strain, gamma, in the y-z planexzRepresenting the shear strain, gamma, in the x-z planexyRepresenting the shear strain in the x-y plane;
step four, representing elastic rotation constraint boundary conditions by using displacement;
and step five, solving a control equation by adopting an iterative method with adjustable parameters to obtain the deflection and the bending stress of the CFRP laminated plate under different temperatures and humidities.
2. The method for analyzing geometric nonlinearity of a CFRP laminate in a hot and humid environment according to claim 1, wherein in the first step, the displacement field of the CFRP laminate is:
Figure BDA0003111452320000021
Figure BDA0003111452320000022
w1(x,y,z)＝w(x,y) (1c)
wherein u is1，v1And w1Is the displacement component of a point outside the middle plane along the x, y and z directions respectively; u, v and w are the displacement components of a point in the mid-plane in the x, y and z directions, respectively;
Figure BDA0003111452320000023
and
Figure BDA0003111452320000024
is the amount of midplane rotation about the x-axis and y-axis, respectively, of the normal;
Figure BDA0003111452320000025
Figure BDA0003111452320000026
and
Figure BDA0003111452320000027
respectively, the displacement field unknown function of the high order shear deformation.
3. The method for analyzing geometric nonlinearity of the CFRP laminate in a hot and humid environment according to claim 2, wherein in the second step, the constitutive relation of the kth layer in the CFRP laminate satisfies:
Figure BDA0003111452320000028
Figure BDA0003111452320000029
wherein sigmaxAnd σyRepresents the positive stress in the x and y directions, respectively; tau isyzAnd τxzRespectively representing the shear stress in the y-z and x-z planes;
Figure BDA00031114523200000210
is the transformed elastic coefficient; Δ c is the change in humidityChanging the gradient; Δ T is the temperature gradient; alpha is alphaxAnd alphayRepresenting the coefficients of thermal expansion in the x and y directions, respectively; beta is axAnd betayThe wet expansion coefficients in the x and y directions are expressed, respectively; alpha is alphaxyAnd betaxyRespectively, the coefficient of thermal expansion and the coefficient of wet expansion in the x-y plane.
4. The method of analyzing geometrical nonlinearity of a CFRP laminate in a hot and humid environment of claim 3, wherein εx，εy，γyz，γxzAnd gammaxyThe relationship between them is as follows:
Figure BDA0003111452320000031
wherein D represents the integral domain; delta represents a variation sign; q represents the uniform load density; h represents the thickness of the CFRP laminate.
5. The method for analyzing geometric nonlinearity of a CFRP laminate in a hot and humid environment according to claim 4, wherein in the fourth step, the constraint conditions for elastic rotation are represented by displacement
φη＝0,φξ,ξ＝±θ1φξ(Ifξ＝0,1) (4a)
φξ＝0,φη,η＝±θ2φη(Ifη＝0,1) (4b)
Wherein theta isi(i＝1,2)∈[0,∞)；φηAnd phiξIndicating the deflection angle around the material principal axis in the directions of 0 and 1, wherein 0 is the material matrix direction and 1 is the material fiber direction; η and ξ represent subscripts.
6. The method of analyzing geometric nonlinearity of a CFRP laminate in a hot and humid environment of claim 5, wherein in step five, an iterative method with adjustable parameters is used to solve the control equation:
Figure BDA0003111452320000032
psi is an adjustable integral parameter, and psi is smaller when the load is smaller; when the load is large, psi can take a large value; wherein u ismn,vmn,wmn,xmn,ymnRepresenting a series term.
The invention has the beneficial effects that:
(1) the geometric non-linear analysis method of the CFRP laminated plate in the damp and hot environment uses more representative boundary conditions and more representative constitutive relation, and uses a simpler solving method to replace the prior analysis method. Since the transverse shear modulus of elasticity is much lower than the in-plane modulus of elasticity, shear deformation plate theory must be employed in the analysis of the laminate. The method not only considers the theoretical influence of the high-order shear deformation plate, but also considers the influence of comprehensive environmental factors such as temperature, humidity and elastic foundation.
(2) Aiming at the nonlinear problem of the wet-heat environment and the elastic foundation composite material laminate, the geometric nonlinear analysis method of the CFRP laminated plate in the wet-heat environment adopts a Vlasov method to convert a nonlinear non-homogeneous partial differential equation system into a nonlinear non-homogeneous algebraic equation system, then converts the nonlinear algebraic equation system into a linear algebraic equation system, and simplifies the algebraic equation systems with a large number of unknowns by utilizing a large sparse matrix technology to find an optimal approximation and iteration method.
Description of the drawings:
FIG. 1 is a graph showing the effect of the damp heat environment on the load-center deflection and bending moment curves of CFRP laminates in accordance with example 1 of the present invention;
FIG. 2 is a three-dimensional distribution of deflection of a CFRP panel in accordance with example 2 of the invention;
fig. 3 is a three-dimensional distribution of bending stresses for a CFRP sheet according to example 2 of the present invention.
The specific implementation method comprises the following steps:
the present invention will be described in further detail with reference to specific embodiments, but the scope of the present invention is not limited to the description.
The geometric non-linear analysis method for the CFRP laminated plate in the damp and hot environment provided by the embodiment of the invention comprises the following steps:
(1) placing the x-y axis at the mid-plane of the panel, consider the displacement field of a CFRP laminate as:
Figure BDA0003111452320000041
Figure BDA0003111452320000042
w1(x,y,z)＝w(x,y) (1c)
wherein u is1，v1And w1Is the displacement component of a point outside the middle plane along the x, y and z directions respectively; u, v and w are the displacement components of a point in the mid-plane in the x, y and z directions, respectively;
Figure BDA0003111452320000043
and
Figure BDA0003111452320000044
is the amount of midplane rotation about the y-axis and the x-axis, respectively, of the normal;
Figure BDA0003111452320000045
Figure BDA0003111452320000046
and
Figure BDA0003111452320000047
is a displacement field unknown function of high order shear deformation.
(2) Considering the constitutive relation of the k-th layer in the CFRP laminated plate, the method satisfies the following conditions:
Figure BDA0003111452320000048
Figure BDA0003111452320000049
(3) based on the formula (1), considering the Reddy high-order shear deformation theory and using the formula (2), epsilon can be obtainedx，εy，γyz，γxzAnd gammaxyThe relation satisfied between them. Then, considering the principle of virtual displacement, we can obtain:
Figure BDA00031114523200000410
wherein D represents the integral domain; delta represents a variation sign; q represents the uniform load density.
(4) The elastic rotation constraint boundary conditions expressed by displacement are as follows:
φη＝0,φξ,ξ＝±θ1φξ(Ifξ＝0,1) (4a)
φξ＝0,φη,η＝±θ2φη(Ifη＝0,1)(4b)
wherein theta isi(i＝1,2)∈[0,∞)。
(5) And solving a control equation by adopting an iterative method with adjustable parameters, namely:
Figure BDA0003111452320000051
where Ψ is an adjustable integer parameter. When the load is small, Ψ is small; when the load is large, Ψ may take a large value.
The embodiment adopts the damp and hot deformation prediction method of the engineering structure provided by the invention to predict the deformation of a numerical simulation example and a composite material structure based on geometric nonlinear analysis in the damp and hot effect of CFRP.
Example 1: the length, width and height of the CFRP laminated plate respectively meet 1, 1 and 0.002; the damp and hot environment satisfies that delta T is 0-50C, and delta C is 0-1%: changing the relation of load-center deflection (W) and load-center bending moment in damp and hot environment
Figure BDA0003111452320000052
The relationship of (a) is shown in FIG. 1.
As is clear from fig. 1, the increase in temperature and humidity simultaneously increases the bending deflection and the bending moment (bending stress), and it is clear that the bending behavior of the CFRP is adversely affected by the hot and humid environment. It is worth noting that in a hot and humid environment, the most significant effect is the temperature factor, while the effect of humidity is very small. If the coefficient of thermal expansion α and the coefficient of wet expansion β are of the same order of magnitude, the wet heat and torque are also of the same order of magnitude, but Δ T and Δ c differ too much. In this example, Δ T is 0 to 50C, and Δ C is 0% to 1%, and the influence of humidification on the bending is very small because the increase in Δ C is very limited.
Example 2: considering the damp-heat environment Δ T ═ 25C, Δ C ═ 0.5%; the three-dimensional deflection distribution and the three-dimensional distribution of bending stress of the CFRP sheet are shown in fig. 2 and 3, with corresponding contour lines below.
Fig. 2 and 3 clearly show the deformation of the whole CFRP panel after loading. The maximum deflection is obviously the midpoint, as shown in fig. 2. The distribution position and magnitude of the bending stress can be clearly seen from the three-dimensional graph and contour lines in fig. 3. Since both the CFRP panel structure and the load are symmetric about the centerline, the deflection of fig. 2 and the bending stress of fig. 3 are also symmetric about the centerline.

Claims (6)

1. A geometrical non-linear analysis method of a CFRP laminated plate in a damp and hot environment is characterized by comprising the following steps:
placing an x-y axis on the middle surface of the CFRP laminated plate to obtain a displacement field of the CFRP laminated plate;
step two, obtaining the constitutive relation of the kth layer in the CFRP laminated plate;
thirdly, obtaining epsilon based on the displacement field of the CFRP laminated plate and based on the Reddy high-order shear deformation theoryx，εy，γyz，γxzAnd gammaxyThe relation between; wherein epsilonxRepresenting strain, ε, in the direction of the x-axisyRepresenting strain in the y-axis direction, gammayzRepresenting the shear strain, gamma, in the y-z planexzRepresenting the shear strain, gamma, in the x-z planexyRepresenting the shear strain in the x-y plane;
step four, representing elastic rotation constraint boundary conditions by using displacement;
and step five, solving a control equation by adopting an iterative method with adjustable parameters to obtain the deflection and the bending stress of the CFRP laminated plate under different temperatures and humidities.
2. The method for analyzing geometric nonlinearity of a CFRP laminate in a hot and humid environment according to claim 1, wherein in the first step, the displacement field of the CFRP laminate is:
Figure FDA0003111452310000011
Figure FDA0003111452310000012
w1(x，y，z)＝w(x，y) (1c)
wherein u is1(x，y，z)，v1(x, y, z) and w1(x, y, z) are the displacement components of a point outside the median plane (x, y) in the x, y and z directions, respectively; u (x, y), v (x, y) and w (x, y) are the displacement components of a point in the midplane (x, y) in the x, y and z directions, respectively;
Figure FDA0003111452310000013
and
Figure FDA0003111452310000014
the amount of midplane rotation about the x-axis and the y-axis, respectively, of the normal;
Figure FDA0003111452310000015
and
Figure FDA0003111452310000016
respectively, the displacement x field unknown function of the high order shear deformation.
3. The method for analyzing geometric nonlinearity of the CFRP laminate in a hot and humid environment according to claim 2, wherein in the second step, the constitutive relation of the kth layer in the CFRP laminate satisfies:
Figure FDA0003111452310000017
Figure FDA0003111452310000021
wherein sigmaxAnd σyRepresents the positive stress in the x and y directions, respectively; tau isxy、τyzAnd τxzRespectively representing the shear stress in the X-Y, Y-Z and X-Z planes;
Figure FDA0003111452310000022
is the transformed elastic coefficient; Δ c is the humidity change gradient; Δ T is the temperature gradient; alpha is alphaxAnd alphayRepresenting the coefficients of thermal expansion in the x and y directions, respectively; beta is axAnd betayThe wet expansion coefficients in the x and y directions are expressed, respectively; alpha is alphaxyAnd betaxyRespectively, the coefficient of thermal expansion and the coefficient of wet expansion in the x-y plane.
4. The method of analyzing geometrical nonlinearity of a CFRP laminate in a hot and humid environment of claim 3, wherein εx，εy，γyz，γxzAnd gammaxyThe relationship between them is as follows:
Figure FDA0003111452310000023
wherein D represents the integral domain; delta represents a variation sign; q represents the uniform load density; h represents the thickness of the CFRP laminate.
5. The method for analyzing geometric nonlinearity of a CFRP laminate in a hot and humid environment according to claim 4, wherein in the fourth step, the constraint conditions for elastic rotation are represented by displacement
Figure FDA0003111452310000024
Figure FDA0003111452310000025
Wherein
Figure FDA0003111452310000026
φηAnd phiξRespectively representing deflection angles around the directions of 0 and 1 of a main shaft of the material, wherein 0 is the direction of a material matrix, and 1 is the direction of a material fiber; η and ξ represent subscripts.
6. The method of analyzing geometric nonlinearity of a CFRP laminate in a hot and humid environment of claim 5, wherein in step five, an iterative method with adjustable parameters is used to solve the control equation:
Figure FDA0003111452310000027
where Ψ is an adjustable integer, where umn，vmn，wmn，xmn，ymnRespectively representing the series terms.
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Publication number Priority date Publication date Assignee Title
US20200276771A1 (en) * 2019-02-28 2020-09-03 Zhejiang University Carbon fiber reinforced plastic electrofusion fitting and a self-monitoring method of strain
CN110133101A (en) * 2019-05-27 2019-08-16 东北大学 A kind of fiber-reinforced composite material board kinetics of high temperature performance degradation analysis method
CN112613205A (en) * 2020-12-12 2021-04-06 天津理工大学 Modeling method for fiber reinforced composite material low-speed cutting technological process simulation

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