A method for predicting thermal expansion coefficient of composite material in ultra-low temperature and wide temperature range considering temperature nonlinearity of carbon fiber mechanical property

Abstract

This invention relates to a method for predicting the thermal expansion coefficient of composite materials under ultra-low temperature and wide temperature range conditions, considering the temperature nonlinearity of the mechanical properties of carbon fibers. The method includes the following steps: acquiring the low-temperature mechanical properties of carbon fiber reinforced composite materials and their component materials; inverting the explicit functional relationship between the longitudinal thermal expansion coefficient of carbon fibers and temperature from the low-temperature longitudinal thermal expansion coefficient data of the carbon fiber reinforced composite materials; and predicting the thermal expansion coefficient of carbon fiber reinforced composite materials under ultra-low temperature / wide temperature range conditions. This invention solves the problem of poor prediction accuracy of existing methods for predicting the thermal expansion coefficient of carbon fiber reinforced composite materials in low-temperature environments. The method is applicable to various unidirectional fiber reinforced composite materials, has a wide range of applications, the entire calculation process is completely explicit, efficient and convenient, easy to program and develop, and can be applied to the preliminary design of structures such as aircraft and liquid hydrogen storage tanks, demonstrating significant engineering value.

Description

Technical Field

[0001] This invention relates to the field of ultra-low temperature and wide temperature range applications of composite materials, and specifically to a method for predicting the coefficient of thermal expansion of composite materials in ultra-low temperature and wide temperature ranges, taking into account the temperature nonlinearity of the mechanical properties of carbon fibers. Background Technology

[0002] In recent years, the aerospace field has made continuous breakthroughs in lunar exploration, interplanetary exploration, and deep space exploration; in the energy sector, the trend of hydrogen energy moving from demonstration to large-scale production has become apparent. Both of these places stringent requirements on the mechanical properties and stability of materials in ultra-low temperature environments. Carbon fiber reinforced composite materials are known for their extremely high specific strength and specific stiffness, excellent thermal stability and low coefficient of thermal expansion, good fatigue performance, excellent corrosion resistance, and designable anisotropy. They can significantly improve structural performance and reduce structural mass, and are considered the first choice for load-bearing and storage structures in ultra-low temperature environments.

[0003] With varying operating temperatures, the mechanical properties of different components in carbon fiber reinforced composites exhibit different trends. The mismatch in thermal expansion coefficients between the high-stiffness fibers and the flexible matrix is ​​particularly prominent, significantly reshaping the thermal expansion response of the composite and thus affecting structural integrity and sealing performance. Therefore, it is necessary to accurately predict the equivalent thermal expansion coefficient of carbon fiber reinforced composites at low temperatures. Theoretical prediction models for the thermal expansion coefficient of composite materials have evolved over decades, from simple weighted / isostrain assumptions to variational / boundary theories, and further to considering interfaces, shape, and multi-scale numerical homogenization, gradually moving from analytical approximations to multi-scale refined simulations. Due to the high diversity of composite materials, high-precision multi-scale numerical simulations are costly and time-consuming; therefore, the industry still largely uses analytical theoretical models. Studies at room temperature show that among theoretical prediction models for the equivalent thermal expansion coefficient of composite materials, the lower bound of the Rosen & Hashin theory (corresponding to the minimum potential energy principle) best matches experimental values.

[0004] However, current research on the prediction of the low-temperature thermal expansion coefficient of carbon fiber reinforced composites is scarce, with the existing studies mostly focusing on temperatures around -150℃. Predictions for the low-temperature thermal expansion coefficient of carbon fiber reinforced composites within the temperature range of 20K (boiling point of liquid hydrogen at atmospheric pressure) to 77K (boiling point of liquid nitrogen at atmospheric pressure) are extremely rare. Current predictions only consider the functional relationship between the mechanical properties of the matrix material and temperature, and due to the good thermal stability of carbon fibers, their mechanical properties are considered temperature-independent. However, existing research shows that the thermal expansion coefficient of graphite exhibits significant temperature nonlinearity at low temperatures. The assumption that the mechanical properties of carbon fibers are temperature-independent lacks experimental support, and the prediction results cannot capture the temperature nonlinearity exhibited by the longitudinal thermal expansion coefficient of carbon fiber reinforced composites. Summary of the Invention

[0005] The technical problem solved by this invention: To overcome the insufficient accuracy of existing predictions for the low-temperature thermal expansion coefficient of carbon fiber reinforced composites, this invention proposes a method for predicting the ultra-low temperature and wide-temperature-range thermal expansion coefficient of carbon fiber reinforced composites, considering the nonlinearity of the longitudinal thermal expansion coefficient of carbon fibers with respect to temperature. Based on existing experimental data on the low-temperature thermal expansion of carbon fiber reinforced composites, an explicit functional relationship between the longitudinal thermal expansion coefficient of carbon fibers and temperature is obtained through inversion. Combined with the Rosen & Hashin theoretical prediction model, this provides an accurate theoretical method for predicting the low-temperature thermal expansion coefficient of carbon fiber reinforced composites.

[0006] The technical solution of this invention: A method for predicting the coefficient of thermal expansion of composite materials at ultra-low temperatures and over a wide temperature range, considering the temperature nonlinearity of the mechanical properties of carbon fibers, comprising the following steps:

[0007] Step A: Obtain the low-temperature mechanical properties of carbon fiber reinforced composite materials and their component materials. The process is as follows:

[0008] (A1) Obtain the longitudinal thermal expansion coefficient of carbon fiber reinforced composite material from low temperature to room temperature through experiments or literature review, requiring no less than 6 data points to be obtained, and the temperature distribution should be as uniform as possible.

[0009] (A2) Obtain the longitudinal and transverse elastic modulus, Poisson's ratio and coefficient of thermal expansion of the fiber phase of carbon fiber reinforced composite material at room temperature through experiments or literature review.

[0010] (A3) The Poisson's ratio of the matrix phase of carbon fiber reinforced composite material at room temperature was obtained through experiments or literature review.

[0011] (A4) The elastic modulus of the matrix phase of carbon fiber reinforced composite materials at room temperature and low temperature was obtained through experiments or literature review, and its functional relationship with temperature was fitted using a linear function, as follows:

[0012]

[0013] In the formula, the superscript m represents the matrix, T is the temperature, and E is the elastic modulus of the material.

[0014] (A5) The coefficients of thermal expansion of the matrix phase of carbon fiber reinforced composites at room temperature and low temperature were obtained through experiments or literature review, and their functional relationship with temperature was fitted using a linear function, as follows:

[0015]

[0016] In the formula, α m is the coefficient of thermal expansion of the matrix material.

[0017] Step B: Using the low-temperature longitudinal thermal expansion coefficient data of carbon fiber reinforced composites, the explicit functional relationship between the longitudinal thermal expansion coefficient of carbon fiber and temperature is obtained through inversion. The process is as follows:

[0018] (B1) Carbon fibers generally possess good thermal stability, and their mechanical properties can be approximated as independent of temperature in most engineering analyses. However, existing research has shown that the longitudinal thermal expansion coefficient of carbon fibers exhibits a significant nonlinear characteristic with temperature variation. To accurately characterize this temperature dependence, this invention uses a rational number function to fit the longitudinal thermal expansion coefficient of carbon fibers. Compared to polynomial fitting, rational number functions simultaneously include numerator and denominator terms, possessing stronger nonlinear expressive power and more effectively describing rapidly changing data with large local curvature and approximate asymptotic characteristics. Furthermore, compared to higher-order polynomials, which are prone to oscillations or divergence at the endpoints of the interval, rational functions typically exhibit better stability at both ends of the fitting interval, thereby improving the stability and reliability of the fitting results. This invention uses a rational number function with a numerator power of 3 and a denominator power of 2, and its functional form is as follows:

[0019]

[0020] In the formula, the superscript f represents carbon fiber, and the subscript L represents longitudinal direction. Therefore This is the longitudinal thermal expansion coefficient of carbon fiber. , , , , , , Let be the undetermined coefficients of a rational function, where Set it to 1.

[0021] (B2) According to Rosen & Hashin theory, the theoretical prediction formula for the longitudinal thermal expansion coefficient of unidirectional fiber-reinforced composites, considering only the temperature dependence of the matrix material's mechanical properties, is as follows:

[0022]

[0023] In the formula, V is the volume fraction of the component.

[0024] By introducing the temperature dependence of the longitudinal thermal expansion coefficient of carbon fiber into formula (4), formula (5) is obtained.

[0025]

[0026] Since the experiment to determine the longitudinal thermal expansion coefficient of carbon fiber is difficult to carry out, the publicly available experimental results are very limited. Therefore, this invention obtains the longitudinal thermal expansion coefficient of carbon fiber at the corresponding temperature point by inverting the experimental value of the longitudinal thermal expansion coefficient of unidirectional fiber reinforced composite material at different temperatures. Formula (5) can be rewritten to obtain formula (6).

[0027]

[0028] (B3) Substitute the longitudinal thermal expansion coefficient data of the carbon fiber reinforced composite material obtained in step (A1) into formula (6) to invert and obtain the longitudinal thermal expansion coefficient of carbon fiber at the corresponding temperature point. Then, use formula (3) to fit the data and determine the undetermined coefficients of the rational number function. Thus, the explicit nonlinear relationship between the longitudinal thermal expansion coefficient of carbon fiber and temperature is obtained.

[0029] Step C: Predict the coefficient of thermal expansion of carbon fiber reinforced composites at ultra-low temperatures / wide temperature range. The process is as follows:

[0030] (C1) Combining formula (1), formula (2) and formula (3) after the undetermined coefficients are determined, formula (5) is used to predict the longitudinal thermal expansion coefficient of carbon fiber reinforced composites under ultra-low temperature / wide temperature range.

[0031] (C2) According to Rosen & Hashin theory, the theoretical prediction formula for the coefficient of thermal expansion of unidirectional composite materials, considering only the temperature dependence of the mechanical properties of the matrix material, is as follows:

[0032]

[0033] In the formula, the superscript "—" indicates volume average, no superscript indicates the equivalent quantity of composite materials, and the subscript T indicates transverse direction. For the flexibility tensor Quantity, For a fourth-order unit tensor Quantity.

[0034] By introducing the temperature dependence of the longitudinal thermal expansion coefficient of carbon fiber into formula (7), formula (8) is obtained.

[0035]

[0036] Combining formulas (1), (2) and (3) after the undetermined coefficients are determined, formula (8) is used to predict the transverse thermal expansion coefficient of carbon fiber reinforced composites under ultra-low temperature / wide temperature range.

[0037] The advantages of this invention compared to the prior art are:

[0038] 1. To address the problem of poor prediction accuracy of existing methods for predicting the thermal expansion coefficient of carbon fiber reinforced composites in low-temperature environments, a new method for predicting the thermal expansion coefficient of carbon fiber reinforced composites at ultra-low temperatures and over a wide temperature range, which considers the temperature nonlinearity of the longitudinal thermal expansion coefficient of carbon fibers, is proposed.

[0039] 2. The method proposed in this invention is applicable to various unidirectional fiber reinforced composite materials, is easy to program and develop, and has a wide range of applications.

[0040] 3. This invention uses a simple rational number function to capture the temperature nonlinearity of the longitudinal thermal expansion coefficient of carbon fiber. The entire calculation process is completely explicit, efficient, convenient, and highly readable. It can be applied to the preliminary design of structures such as aircraft and liquid hydrogen storage tanks, and has significant engineering value. Attached Figure Description

[0041] Figure 1 This is a flowchart illustrating a method for predicting the thermal expansion coefficient of composite materials under ultra-low temperature and wide temperature range, taking into account the temperature nonlinearity of the mechanical properties of carbon fibers.

[0042] Figure 2 This is the inversion result of the longitudinal thermal expansion coefficient of carbon fiber in Example 1 and the fitting curve of the rational number function;

[0043] Figure 3 This is a comparison chart of the predicted and experimental results of the longitudinal thermal expansion coefficient of carbon fiber reinforced composite material under ultra-low temperature and wide temperature range in Example 1.

[0044] Figure 4 This is a comparison chart of the predicted and experimental results of the transverse thermal expansion coefficient of carbon fiber reinforced composite material at ultra-low temperature and over a wide temperature range in Example 1. Detailed Implementation

[0045] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0046] like Figure 1 As shown, the specific implementation steps of the present invention for predicting the coefficient of thermal expansion of composite materials under ultra-low temperature and wide temperature range considering the temperature nonlinearity of carbon fiber mechanical properties are as follows:

[0047] Example 1: Prediction of thermal expansion coefficient of M40J carbon fiber cyanate resin composite material system under ultra-low temperature and wide temperature range

[0048] 1. Obtain the low-temperature mechanical properties of carbon fiber reinforced composite materials and their component materials.

[0049] According to literature review, M40J carbon fiber has a longitudinal elastic modulus of 392 GPa, a transverse elastic modulus of 23.1 GPa, a longitudinal Poisson's ratio of 0.2, a transverse Poisson's ratio of 0.4, and a longitudinal coefficient of thermal expansion of -0.8 × 10⁻⁶. -6 ·K -1 The lateral thermal expansion coefficient is 8.7 × 10⁻⁶. -6 ·K -1 The linear relationship between the elastic modulus and thermal expansion coefficient of cyanate ester resin and temperature in the low-temperature to room-temperature range is as follows:

[0050]

[0051] The cyanate ester resin has a Poisson's ratio of 0.36 at room temperature.

[0052] Based on the low-temperature thermal expansion test results of M40J carbon fiber cyanate resin composite material, its longitudinal thermal expansion coefficient at 20K, 77K, 135K, 185K, 240K, and 280K is -0.7999×10⁻⁶. -6 ·K -1 -0.8036×10 -6 ·K -1 -1.2100×10 -6 ·K -1 -0.7353×10 -6 ·K -1 -0.2922×10 -6 ·K -1 -0.1746×10 -6 ·K -1 .

[0053] 2. The nonlinear relationship between the longitudinal thermal expansion coefficient of carbon fiber and temperature was obtained by inversion.

[0054] Based on the experimental values ​​of the longitudinal thermal expansion coefficient of the composite material and the mechanical properties of each component, the longitudinal thermal expansion coefficient of the carbon fiber at each test temperature is obtained by inversion using equation (6). The undetermined coefficients in equation (3) are obtained by least squares fitting, as shown in the table below.

[0055] 1.7598 -15.4551 36.1046 -2.8304 1.0000 -3.1671 2.6050

[0056] The longitudinal thermal expansion coefficient of carbon fiber at each test temperature point obtained by inversion from equation (5) and the data curves obtained by fitting rational number functions are shown below. Figure 2 As shown.

[0057] 3. Conduct theoretical predictions of the thermal expansion coefficients of composite materials at ultra-low temperatures and over a wide temperature range.

[0058] Substituting equations (1), (2), and (3) after determining the undetermined coefficients into equations (5) and (8), we can predict the longitudinal and transverse thermal expansion coefficients of carbon fiber reinforced composite materials at ultra-low temperatures and over a wide temperature range. The prediction results are as follows: Figure 3 , Figure 4 As shown, the prediction results considering the temperature nonlinearity of the longitudinal thermal expansion coefficient of carbon fiber successfully reproduce the temperature nonlinearity relationship of the longitudinal thermal expansion coefficient of carbon fiber reinforced composites. Within the temperature range of 20K to 300K, the prediction error of the longitudinal thermal expansion coefficient does not exceed 20%, and the prediction error of the transverse thermal expansion coefficient does not exceed 10%.

Claims

1. A method for predicting the coefficient of thermal expansion of composite materials at ultra-low temperatures and over a wide temperature range, considering the temperature nonlinearity of the mechanical properties of carbon fibers, characterized in that... Includes the following steps: Step A: Obtain the low-temperature mechanical properties of carbon fiber reinforced composite materials and their component materials; Step B involves using the low-temperature longitudinal thermal expansion coefficient data of carbon fiber reinforced composite materials to invert and obtain the explicit functional relationship between the longitudinal thermal expansion coefficient of carbon fiber and temperature. Step C: Predict the coefficient of thermal expansion of carbon fiber reinforced composites under ultra-low temperature / wide temperature range conditions.

2. The method for predicting the thermal expansion coefficient of composite materials at ultra-low temperatures and over a wide temperature range, considering the temperature nonlinearity of carbon fiber mechanical properties, as described in claim 1, is characterized in that: The specific implementation process for obtaining the low-temperature mechanical properties of carbon fiber reinforced composite materials and their component materials in step A is as follows: (A1) Obtain the longitudinal thermal expansion coefficient of carbon fiber reinforced composite material from low temperature to room temperature through experiments or literature review, requiring no less than 6 data points to be obtained, and the temperature distribution should be as uniform as possible. (A2) Obtain the longitudinal and transverse elastic modulus, Poisson's ratio and coefficient of thermal expansion of the fiber phase of carbon fiber reinforced composite material at room temperature through experiments or literature review. (A3) The Poisson's ratio of the matrix phase of carbon fiber reinforced composite material at room temperature was obtained through experiments or literature review. (A4) The elastic modulus of the matrix phase of carbon fiber reinforced composite materials at room temperature and low temperature was obtained through experiments or literature review, and its functional relationship with temperature was fitted using a linear function, as follows: In the formula, the superscript m represents the matrix, T is the temperature, and E is the elastic modulus of the material. (A5) The coefficients of thermal expansion of the matrix phase of carbon fiber reinforced composites at room temperature and low temperature were obtained through experiments or literature review, and their functional relationship with temperature was fitted using a linear function, as follows: In the formula, α m is the coefficient of thermal expansion of the material. The process of inverting the low-temperature longitudinal thermal expansion coefficient data of carbon fiber reinforced composite materials to obtain the explicit functional relationship of the longitudinal thermal expansion coefficient of carbon fiber with respect to temperature in step B is as follows: (B1) Carbon fibers generally possess good thermal stability, and their mechanical properties can be approximated as independent of temperature in most engineering analyses. However, existing research has shown that the longitudinal thermal expansion coefficient of carbon fibers exhibits a significant nonlinear characteristic with temperature variation. To accurately characterize this temperature dependence, this invention uses a rational number function to fit the longitudinal thermal expansion coefficient of carbon fibers. Compared to polynomial fitting, rational number functions simultaneously include numerator and denominator terms, possessing stronger nonlinear expressive power and more effectively describing rapidly changing data with large local curvature and approximate asymptotic characteristics. Furthermore, compared to higher-order polynomials, which are prone to oscillations or divergence at the endpoints of the interval, rational functions typically exhibit better stability at both ends of the fitting interval, thereby improving the stability and reliability of the fitting results. This invention uses a rational number function with a numerator power of 3 and a denominator power of 2, and its functional form is as follows: In the formula, the superscript f represents carbon fiber, and the subscript L represents longitudinal direction. Therefore This is the longitudinal thermal expansion coefficient of carbon fiber. , , , , , , Let be the undetermined coefficients of a rational function, where Set it to 1. (B2) According to Rosen & Hashin theory, the theoretical prediction formula for the longitudinal thermal expansion coefficient of unidirectional fiber-reinforced composites, considering only the temperature dependence of the matrix material's mechanical properties, is as follows: In the formula, V is the volume fraction of the component. By introducing the temperature dependence of the longitudinal thermal expansion coefficient of carbon fiber into formula (4), formula (5) is obtained. Since the experiment to determine the longitudinal thermal expansion coefficient of carbon fiber is difficult to carry out, the publicly available experimental results are very limited. Therefore, this invention obtains the longitudinal thermal expansion coefficient of carbon fiber at the corresponding temperature point by inverting the experimental value of the longitudinal thermal expansion coefficient of unidirectional fiber reinforced composite material at different temperatures. Formula (5) can be rewritten to obtain formula (6). (B3) Substitute the longitudinal thermal expansion coefficient data of the carbon fiber reinforced composite material obtained in step (A1) into formula (6) to invert and obtain the longitudinal thermal expansion coefficient of carbon fiber at the corresponding temperature point. Then, use formula (3) to fit the data and determine the undetermined coefficients of the rational number function. Thus, the explicit nonlinear relationship between the longitudinal thermal expansion coefficient of carbon fiber and temperature is obtained. The process of predicting the coefficient of thermal expansion of carbon fiber reinforced composites at ultra-low temperatures / wide temperature range in step C is as follows: (C1) Combining formula (1), formula (2) and formula (3) after the undetermined coefficients are determined, formula (5) is used to predict the longitudinal thermal expansion coefficient of carbon fiber reinforced composites under ultra-low temperature / wide temperature range. (C2) According to Rosen & Hashin theory, the theoretical prediction formula for the coefficient of thermal expansion of unidirectional composite materials, considering only the temperature dependence of the mechanical properties of the matrix material, is as follows: In the formula, the superscript "—" indicates volume average, no superscript indicates the equivalent quantity of composite materials, and the subscript T indicates transverse direction. For the flexibility tensor Quantity, For a fourth-order unit tensor Quantity. By introducing the temperature dependence of the longitudinal thermal expansion coefficient of carbon fiber into formula (7), formula (8) is obtained. Combining formulas (1), (2) and (3) after the undetermined coefficients are determined, formula (8) is used to predict the transverse thermal expansion coefficient of carbon fiber reinforced composites under ultra-low temperature / wide temperature range.

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