A method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials

Abstract

The present invention belongs to the field of multi-scale mechanical analysis of composite materials, and particularly relates to a method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials. It includes: Step 1: Take a representative volume element, i.e., RVE, from the fiber-reinforced composite material, apply stress in the 2 direction, where the 2 direction is the y direction of the coordinate system, and calculate the fiber strain and matrix strain of the fiber and the matrix in the 2 direction; Step 2: According to the actual deformation of the fiber matrix, introduce a deformation coordination plane and a tensile deformation coordination factor α in the deformation region, calculate the true strain of the fiber matrix when loaded in the 2 direction, and then solve the true macroscopic elastic modulus in the 2 direction; Step 3: Introduce shear deformation coordination factors β and γ to solve the macroscopic shear modulus.

Description

Technical Field

[0001] The present invention belongs to the field of multi-scale mechanical analysis of composite materials, and particularly relates to a method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials. Background Art

[0002] Currently, the methods for predicting the macroscopic engineering elastic constants of composite materials can be divided into two categories: one is the analytical calculation method, represented by the classical mixture method (Rule of Mixture, RoM method), Chamis method, inclusion method, Mori-Tanaka (M-T) method, self-consistent method, and bridging method, etc.; the other is the finite element method, represented by the inclusion method and the asymptotic homogenization method. The analytical calculation method has high calculation efficiency and is fast and convenient to apply in engineering, but the accuracy is relatively low; although the finite element method has a cumbersome calculation process, the accuracy is relatively high. Currently, when predicting the macroscopic mechanical properties of unknown composite materials, the finite element method is usually used to obtain the macroscopic engineering elastic constants.

[0003] In the field of multi-scale calculation and engineering analysis of fiber-reinforced composite materials, it is often necessary to analyze the influence of fiber volume fraction on the macroscopic mechanical properties of composite materials. If the analytical calculation method is adopted, except for the RoM method and the Chamis method, the remaining algorithms such as the M-T method are relatively complex or require obtaining experimental data first, but the calculation results of the RoM method and the Chamis method have large errors. The finite element method is more convenient for a specific structure, but it is very cumbersome when calculating the influence of fiber volume fraction on the macroscopic mechanical properties. Therefore, it is very necessary to develop a simple analytical calculation method. Summary of the Invention

[0004] In view of the above problems existing in the prior art, the present invention proposes a method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials (updated Rule of Mixture, uRoM method). By solving the problem of inconsistent deformation between fibers and the matrix brought about by the classical mixture method and the Chamis method, the prediction accuracy of the mixture method is significantly improved, the prediction of the macroscopic mechanical properties of fiber-reinforced composite materials is realized, and guidance is provided for the multi-scale calculation and engineering analysis of fiber-reinforced composite materials, especially high-performance carbon fiber-reinforced composite materials. Compared with the Chamis method and the RoM method, the uRoM method has the smallest error and the highest accuracy in calculating the results with the M-T method as the benchmark when predicting the macroscopic elastic modulus and shear modulus, and can conveniently realize the prediction of the macroscopic mechanical properties of fiber-reinforced composite materials.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials, comprising:

[0007] Step 1: Take a representative volume element, i.e., RVE, from the fiber-reinforced composite material. Apply stress in the 2-direction, where the 2-direction is the y-direction of the coordinate system. Calculate the fiber strain and matrix strain of the fiber and matrix in the 2-direction.

[0008] Step 2: Based on the theoretical fiber strain and matrix strain of the fiber and matrix in the 2-direction obtained in Step 1, according to the actual deformation of the fiber matrix, introduce a deformation coordination plane and a tensile deformation coordination factor α in the deformation region. Calculate the true strain of the fiber and matrix when loaded in the 2-direction, and then solve the true macroscopic elastic modulus in the 2-direction.

[0009] Step 3: Based on the formula for solving the true macroscopic elastic modulus in Step 2, introduce shear deformation coordination factors β and γ, and solve the true macroscopic engineering elastic constants of the fiber-reinforced composite material in the 1-2 direction, 1-3 direction, and 2-3 direction.

[0010] Preferably, Step 1 includes:

[0011] Step 1.1: Take a representative volume element, i.e., RVE, from the fiber-reinforced composite material. An RVE consists of a matrix wrapping the fiber. The cross-section of the fiber is circular. It is a transversely isotropic material, and the matrix is an isotropic material. Define f to represent the fiber and m to represent the matrix. Use 1, 2, 3 to represent the x, y, z directions of the Cartesian coordinate system, where the 1-direction is the longitudinal direction of the fiber. Then the engineering elastic constants of the fiber are defined as the elastic modulus E of the fiber in the 1-direction f,11 , the elastic modulus E of the fiber in the 2-direction f,22 , the elastic modulus E of the fiber in the 3-direction f,33 , the Poisson's ratio ν of the fiber in the 1-2 direction f,12 , the Poisson's ratio ν of the fiber in the 1-3 direction f,13 , the Poisson's ratio ν of the fiber in the 2-3 direction f,23 , the shear modulus G of the fiber in the 1-2 direction f,12 , the shear modulus G of the fiber in the 1-3 direction f,13 , the shear modulus G of the fiber in the 2-3 direction f,23 , and due to the transverse isotropy of the fiber, E f,22 = E f,33 , ν f,12 = ν f,13 , G f,12 = G f,13 ; The engineering elastic constants of the matrix are defined as the elastic modulus E of the matrix m , the Poisson's ratio ν of the matrix m , the shear modulus G of the matrix m .

[0012] Step 1.2: For the sake of simplicity in modeling, a real RVE plane is intercepted along the 2-3 direction in the RVE, the fiber radius is defined as r, and the fiber volume fraction is c f , the side length of the matrix is l, then c f =πr 2 l / l 3 ;

[0013] Step 1.3: The real RVE plane in Step 1.2 is converted into an equivalent RVE plane with the same fiber volume fraction, and c f =a 2 l / l 3 is calculated, then

[0014] Step 1.4: When a stress σ 2 is applied to the 2 direction of the equivalent RVE plane in Step 1.3, according to the classical mixing method in composite mechanics, the stresses borne by the fiber and the matrix are both σ 2 when, that is, σ f,22 =σ m =σ 2 , σ f,22 is the fiber stress, and the matrix stress is σ m , and the fiber strain ε f,22 and the matrix strain ε m are obtained:

[0015]

[0016] Step 1.5: Comparing the fiber strain ε f,22 and the matrix strain ε m in Step 1.4 gives while E f,22 >E m , so the deformation of the matrix in the length l is much greater than that of the fiber, resulting in deformation incoordination between the fiber and the matrix in the 2 direction. Regarding the fiber as an inclusion phase, the deformation difference Δl between the fiber and the matrix in l is the difference between the actual deformation length l m of the matrix and the actual deformation length a f of the fiber:

[0017]

[0018] The strain difference Δε 22 along the 2 direction in l is:

[0019]

[0020] Preferably, the specific content of Step 2 is:

[0021] Step 2.1: According to the actual deformation of the fiber matrix, for the entire equivalent RVE, based on Equation (3), the sum of the additional strains of the fiber and the matrix along the 2 direction is obtained. It is:

[0022]

[0023] Step 2.2: Assume that the deformation consistent plane of the fiber and the matrix is the deformation coordination plane N of the equivalent RVE plane. 1 -N 2 When loaded, the interaction between the fiber and the matrix is divided into normal stress and shear stress. The shear stress has no influence on the macroscopic mechanical properties, while the normal stress has a greater influence. Assume that the normal stress between the fiber and the matrix is σ s , and the additional strain of the fiber caused by σ s is The additional strain of the matrix is Then there is:

[0024]

[0025] Step 2.3: Because under the action of σ s Then find out and The expression of, and define the tensile deformation coordination factor Then there is:

[0026]

[0027] Step 2.4: Assume that the nominal deformation of the fiber is Δaf and the nominal deformation of the matrix is Δl m , then the nominal deformations of the fiber and the matrix on the RVE are:

[0028]

[0029] Step 2.5: Assume that the nominal strain of the fiber is The nominal strain of the matrix is Then there is:

[0030]

[0031] Step 2.6: On the basis of Step 2.5, if there is an interaction between the fiber and the matrix, define the true strain of the fiber as The true strain of the matrix is By combining Equations (6) and (8), the true macroscopic strain of the equivalent RVE along the 2 direction can be obtained

[0032]

[0033] Furthermore, calculate the true macroscopic elastic modulus in the 2 direction​ is as follows:

[0034]

[0035] Preferably, step 3 specifically includes:

[0036] Step 3.1: According to the RoM method and the Chamis method, G 12 , G 13 , G 23 is similar to the expression of E 22 . Therefore, two shear and stretch deformation coordination factors β and γ are introduced, and their expressions are:

[0037]

[0038] Step 3.2: Based on the formula for solving the true macroscopic elastic modulus in step 2, define the true macroscopic elastic modulus in the 1 direction the true macroscopic elastic modulus in the 3 direction the true macroscopic shear modulus in the 1-2 direction the true macroscopic elastic modulus in the 1-3 direction the true macroscopic elastic modulus in the 2-3 direction the true Poisson's ratio in the 1-2 direction the true Poisson's ratio in the 1-3 direction the true Poisson's ratio in the 2-3 direction Then the true macroscopic engineering elastic constants of the fiber-reinforced composite material are obtained as:

[0039]

[0040] Compared with the prior art, the beneficial effects of the present invention are:

[0041] The uRoM method proposed by the present invention considers the deformation coordination of fibers and matrix in steps 2 and 3, solves the true macroscopic strain of the fiber-reinforced composite material on the equivalent RVE, and solves the problem of low accuracy in predicting the macroscopic elastic modulus and shear modulus by the Chamis method and the RoM method. Compared with the M-T method used as a benchmark, the uRoM method has the smallest error and the highest accuracy in predicting the macroscopic elastic modulus and shear modulus, which is convenient for engineering calculations and can conveniently realize the prediction of the macroscopic mechanical properties of fiber-reinforced composite materials. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The drawings are used to provide a further understanding of the present invention, and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation to the present invention.

[0043] In the drawings:

[0044] Figure 1 It is the flowchart of the method of the present invention;

[0045] Figure 2 It is the schematic diagram of the RVE of the fiber-reinforced composite material of the present invention;

[0046] Figure 3 It is the equivalent conversion diagram of the RVE of the present invention, (a) real RVE, (b) equivalent RVE;

[0047] Figure 4 is E 22 It is the schematic diagram for calculating the principle, (a) real RVE plane, (b) equivalent RVE plane, (c) deformation-compatible equivalent RVE plane;

[0048] Figure 5 shows the predicted results of the macroscopic mechanical properties of IM7 / 8552 of the present invention: (a) (b) (c) (d) (e) (f)

[0049] Figure 6 shows the predicted results of the macroscopic mechanical properties of T700S / Epoxy of the present invention: (a) (b) (c) (d) (e) (f)

[0050] Figure 7 shows the predicted results of the macroscopic mechanical properties of T800H / Epoxy of the present invention: (a) (b) (c) (d) (e) (f)

[0051] Figure 8 shows the predicted results of the macroscopic mechanical properties of E-Glass / PMR-15 of the present invention: (a) (b) (c) (d) (e) (f) Detailed implementation manners

[0052] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are only used to illustrate and explain the present invention, and are not used to limit the present invention.

[0053] Embodiment:

[0054] Fiber-reinforced composites are usually composed of two parts: fibers and matrix. However, due to the differences in mechanical properties of these two parts, it is necessary to predict the mechanical properties of unidirectional fiber-reinforced composite laminates at the macroscopic scale.

[0055] Currently, the commonly used hybrid prediction method models are:

[0056] RoM method:

[0057] The RoM method is the earliest proposed method for predicting the macroscopic mechanical properties of fiber-reinforced composites. Define the true macroscopic elastic modulus E in the 1 direction 11 , the true macroscopic elastic modulus E in the 2 direction 22 , the true macroscopic elastic modulus E in the 3 direction 33 , the true macroscopic shear modulus G in the 1-2 direction 12 , the true macroscopic elastic modulus G in the 1-3 direction 13 , the true macroscopic elastic modulus G in the 2-3 direction 23 , the true Poisson's ratio ν in the 1-2 direction 12 , the true Poisson's ratio ν in the 1-3 direction 12 , the true Poisson's ratio ν in the 2-3 direction 32 , and its model for calculating the macroscopic engineering elastic constants of fiber-reinforced composites is:

[0058]

[0059] In Equation (1), f represents the fiber, m represents the matrix, and the comma does not represent differentiation.

[0060] Chamis method:

[0061] The Chamis method is an improvement based on the RoM method. It converts the true RVE into an equivalent RVE with a square fiber cross-section but unchanged volume, and obtains its model for calculating the macroscopic engineering elastic constants of fiber-reinforced composites as:

[0062]

[0063] The comparison benchmark is the M-T method:

[0064] The calculation model of the M-T method is:

[0065]

[0066] In the formula, Z 1 and Z 2 are two intermediate variables, which are respectively:

[0067]

[0068] The RoM method is the earliest proposed hybrid computational method for predicting the macroscopic mechanical properties of fiber-reinforced composites, with high computational efficiency. However, when predicting the macroscopic elastic modulus and shear modulus, the predicted values of this method are on the low side and the error is too large. The Chaims method improves the prediction accuracy compared with the RoM method by converting the real RVE into an equivalent RVE. However, the predicted values are on the high side and there are still large errors.

[0069] The method of the present invention is as follows:

[0070] A method for predicting the macroscopic mechanical properties of fiber-reinforced composites, as Figure 1 shown, includes:

[0071] Step 1: Take a representative volume element (RVE) from the fiber-reinforced composite, apply stress in the 2 direction, where the 2 direction is the y direction of the coordinate system, and calculate the fiber strain and matrix strain of the fiber and matrix in the 2 direction. Specifically:

[0072] Step 1.1: As Figure 2 shown, take an RVE from the fiber-reinforced composite. An RVE is composed of a matrix wrapping the fiber. The cross-section of the fiber is circular. It is a transversely isotropic material, and the matrix is an isotropic material. Define f to represent the fiber and m to represent the matrix. Use 1, 2, 3 to represent the x, y, z directions of the Cartesian coordinate system, where the 1 direction is the longitudinal direction of the fiber. Then the engineering elastic constants of the fiber are defined as the elastic modulus E f,11 of the fiber in the 1 direction, the elastic modulus E f,22 of the fiber in the 2 direction, the elastic modulus E f,33 of the fiber in the 3 direction, the Poisson's ratio ν f,12 of the fiber in the 1-2 direction, the Poisson's ratio ν f,13 of the fiber in the 1-3 direction, the Poisson's ratio ν f,23 of the fiber in the 2-3 direction, the shear modulus G f,12 of the fiber in the 1-2 direction, the shear modulus G f,13 of the fiber in the 1-3 direction, the shear modulus G f,23 of the fiber in the 2-3 direction. And due to the transverse isotropy of the fiber, E f,22 =E f,33 , ν f,12 =ν f,13 , G f,12 =G f,13 ; The engineering elastic constants of the matrix are defined as the elastic modulus E m of the matrix, the Poisson's ratio ν m of the matrix, and the shear modulus G m of the matrix;

[0073] Various mixing methods (such as the Chamis method, RoM method) have high accuracy in predicting the elastic modulus E in the macroscopic 1 direction 11 However, there are large errors in predicting the elastic moduli E 22 and E 33 in the 2 and 3 directions.

[0074] In the process of predicting the macroscopic mechanical properties of fiber composites by the RoM method, when calculating E 11 along the 1 direction with an applied stress σ 1 , assuming that the fiber strain ε f,11 and the matrix strain ε m are equal. If the fiber volume fraction is c f , and the matrix volume fraction is c m , and the macroscopic strain in the 1 direction is ε 11 , according to the equality of stresses, σ 1 is:

[0075] σ 1 = E 11 ε 11 = c f E f,11 ε f,11 + c m E m ε m

[0076] According to the above equation, since ε f,11 = ε m , then E 11 can be obtained as:

[0077] E 11 = c f E f,11 + c m E m

[0078] When calculating E 22 along the 2 direction with an applied stress σ 2 , assuming that the fiber stress σ f,22 and the matrix stress σ m are equal, and the macroscopic strain in the 2 direction is ε 22 , according to the equality of strains, ε 22 is:

[0079] ε 22 = c f ε f,22 + c m ε m

[0080] Since σ f,22 = σ m , then E 22is:

[0081]

[0082] However, the calculated E using the above formula 22 is less than the true value. If we assume ε f,22 = ε m , we get E 22 as:

[0083] E 22 = c f E f,22 + c m E m

[0084] But the calculated E using the above formula 22 is greater than the true value. Obviously, there are problems with the above two solution formulas. In reality, the stresses of the fiber matrix are not equal, and the strains are not equal either. This indicates that there is a problem of deformation incoordination in the RoM method when calculating E 22 . The reason for the deformation incoordination is the interaction between the fibers and the matrix. The fibers hinder the deformation of the matrix, and the deformation of the matrix in turn affects the deformation of the fibers, resulting in additional deformation.

[0085] To intuitively analyze the interaction between the fibers and the matrix, we can refer to the principle of the Chamis method. The fibers are equivalent to a hexahedron with a constant volume. Then the equivalent RVE converted from the true RVE is as Figure 3 shown.

[0086] From Figure 3 (a)(b), it can be found that when calculating E 1 by applying σ 11 in the 1 direction, there is only a tangential interaction between the fibers and the matrix at the interface, which is the shear stress and can be expressed by the cohesive force model. The error between the predicted E 11 by the RoM method and the Chamis, M-T, etc. methods is very small, indicating that the shear stress between the fiber matrix hardly affects the calculation results. From Figure 3 (b), it can be found that when calculating E 2 by applying σ 22 in the 2 direction, there is not only shear stress but also normal stress between the fibers and the matrix, which can also be expressed by the cohesive force model. Obviously, if the influence of the shear stress is not considered, it can be considered that the normal stress at the fiber matrix interface is the main reason for the large calculation error of E 22 .

[0087] Step 1.2: For simplicity in modeling, a true RVE plane is intercepted in the RVE along the 2-3 direction, as Figure 4 (a) shown. Define the fiber radius as r and the fiber volume fraction as c f, if the side length of the matrix is \(l\), then \(c\) f = \(\pi r\) 2 \(l / l\) 3 ;

[0088] Step 1.3: Convert the real RVE plane in Step 1.2 into an equivalent RVE plane with the same fiber volume, and calculate \(c\) f = \(a\) 2 \(l / l\) 3 , then

[0089] Step 1.4: Apply \(\sigma\) in the 2 - direction of the equivalent RVE plane in Step 1.3 2 When, according to the classical mixture method in composite mechanics, the stresses borne by the fiber and the matrix are both \(\sigma\) 2 When, that is, \(\sigma\) f,22 = \(\sigma\) m = \(\sigma\) 2 , \(\sigma\) f,22 is the fiber stress, and the matrix stress is \(\sigma\) m , the fiber strain \(\varepsilon\) f,22 and the matrix strain \(\varepsilon\) m can be obtained:

[0090]

[0091] Step 1.5: Compare the fiber strain \(\varepsilon\) f,22 and the matrix strain \(\varepsilon\) m in Step 1.4, and the ratio is while \(E\) f,22 > \(E\) m , so in the length \(l\), the deformation of the matrix is much greater than that of the fiber, resulting in deformation incoordination between the fiber and the matrix in the 2 - direction. Regarding the fiber as an inclusion phase, the deformation difference \(\Delta l\) of the fiber and the matrix in \(l\) is the difference between the actual deformation length \(l\) m of the matrix and the actual deformation length \(a\) f of the fiber:

[0092]

[0093] The strain difference \(\Delta\varepsilon\) 22 along the 2 - direction in \(l\) is:

[0094]

[0095] Step 2: Based on the theoretical fiber strain and matrix strain of the fiber and the matrix along the 2 - direction obtained in Step 1, since the stresses and strains borne by the matrix and the fiber are different during the loading process of the real RVE, therefore, according to the actual deformation of the fiber - matrix, introduce a deformation - coordination plane and a tensile deformation - coordination factor \(\alpha\) in the deformation region, calculate the real strain of the fiber - matrix when loaded along the 2 - direction, and then solve the real macroscopic elastic modulus in the 2 - direction;

[0096] Step 2.1: In fact, due to the interaction (such as cohesion) between the fiber and the matrix, such a large deformation difference and strain difference will not occur under non-destructive external loads. Instead, the deformation should be consistent, indicating that the normal stress at the fiber-matrix interface affects the stresses of the fiber and the matrix respectively.

[0097] For the entire equivalent RVE, according to Equation (3), the sum of the additional strains of the fiber and the matrix along the 2 direction is obtained as:

[0098]

[0099] Step 2.2: Let the deformation-consistent plane of the fiber and the matrix be the deformation coordination plane N of the equivalent RVE plane. 1 -N 2 When loaded, the interaction between the fiber and the matrix is divided into normal stress and shear stress. The shear stress has no effect on the macroscopic mechanical properties, while the normal stress has a greater influence. Let the normal stress between the fiber and the matrix be σ s , and the additional strain of the fiber caused by σ s is The additional strain of the matrix is Then there is:

[0100]

[0101] Step 2.3: Because under the action of σ s then the expressions of are obtained, and the tensile deformation coordination factor and is defined. Then there is:

[0102]

[0103] Step 2.4: Let the nominal deformation of the fiber be Δa f , and the nominal deformation of the matrix be Δl m , then the nominal deformations of the fiber and the matrix on the equivalent RVE are:

[0104]

[0105] Step 2.5: Let the nominal strain of the fiber be The nominal strain of the matrix is Then there is:

[0106]

[0107] Step 2.6: On the basis of Step 2.5, due to the interaction between the fiber and the matrix, the true strain of the fiber is defined as The true strain of the matrix is By combining equations (6) and (8), the true macroscopic strain of the equivalent RVE in the 2-direction is obtained.

[0108]

[0109] Furthermore, the true macroscopic elastic modulus in the 2-direction is calculated. It is:

[0110]

[0111] Step 3: Introduce the shear deformation coordination factors β and γ to solve the true macroscopic engineering elastic constants of the fiber-reinforced composite. Specifically, it includes:

[0112] Step 3.1: According to the RoM method and the Chamis method, the expressions of G 12 、G 13 、G 23 and E 22 are similar. Therefore, when shear stresses τ 12 and τ 23 are applied to the equivalent RVE in the 1-2 direction and the 2-3 direction respectively, there is also a deformation coordination relationship between the fiber and the matrix. By analogy with the expression of the tensile deformation coordination factor α in equation (6), two shear-tensile deformation coordination factors β and γ are introduced, and their expressions are:

[0113]

[0114] Step 3.2: Referring to the process of solving the true macroscopic strain of the equivalent RVE in the 2-direction in Step 2.6 it can be known that the true shear strain of the equivalent RVE in the 1-2 direction is defined as and the true shear strain in the 2-3 direction is By analogy with equation (9), and

[0115]

[0116] Accordingly, the true macroscopic shear modulus in the 1-2 direction can be obtained and the true macroscopic shear modulus in the 2-3 direction

[0117]

[0118] Define the true macroscopic elastic modulus in the 1-direction and the true macroscopic elastic modulus in the 3-direction and the true macroscopic elastic modulus in the 1-3 direction and the true Poisson's ratio in the 1-2 direction True Poisson's ratio in the 1-3 direction True Poisson's ratio in the 2-3 direction Since the fiber is a transversely isotropic material and the matrix is an isotropic material, the fiber-reinforced composite material is also transversely isotropic Then the true macroscopic engineering elastic constants of the fiber-reinforced composite material are obtained as follows:

[0119]

[0120] Simulation experiment:

[0121] To verify the accuracy of the uRoM method of the present invention in predicting the macroscopic mechanical properties of fiber-reinforced composite materials, the engineering elastic constants of the fiber and the matrix in Table 2 are used to calculate the fiber volume fraction c f The effects on the macroscopic engineering elastic constants of IM7 / 8552, T700S / Epoxy, T800H / Epoxy and E-Glass / PMR-15, and comparison with the Chamis, RoM and M-T methods are carried out to obtain the prediction results of the macroscopic mechanical properties of fiber-reinforced composite materials under different methods, as Figure 4 ~shown in Figure 7

[0122] Table 2 Engineering elastic constants of fiber and matrix

[0123]

[0124] By comparing the macroscopic mechanical properties of four materials, namely IM7 / 8552, T700S / Epoxy, T800H / Epoxy and E-Glass / PMR-15, in Figures 5 to 8, it can be found that when calculating , the results obtained by the Chamis, RoM, M-T and uRoM methods coincide, while when predicting the moduli in other directions (such as ), the results do not coincide, indicating that when loading in other directions, the normal stress between the fiber and the matrix affects both the strain of the fiber and the matrix, thus affecting the macroscopic mechanical properties

[0125] It can be found from Figures 5(b), 6(b), 7(b) and 8(b) that compared with the prediction results of the M-T method with higher accuracy as the benchmark, the calculated by the Chamis method is larger, the calculation result of the RoM method is smaller, and the error of the uRoM method is the smallest, which can prove the correctness of the proposed method (uRoM method). By comparing the calculation results of Figures 5(b)(c), 6(b)(c), 7(b)(c) and 8(b)(c), it can be seen that the and It has higher accuracy than the Chamis method and the RoM method, which indicates that the shear deformation coordination factors β and γ proposed by analogy with the stretching deformation coordination factor α are effective.

[0126] As can be seen from the above analysis, the uRoM method for predicting the macroscopic mechanical properties of fiber-reinforced composites proposed by considering deformation coordination has a small error compared with the M-T method, and can accurately predict the macroscopic elastic modulus and shear modulus of fiber-reinforced composites, which is convenient for engineering application calculations.

[0127] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. The above embodiments and the descriptions in the specification only illustrate the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials, characterized in that: comprising: Step 1: Take a representative volume element, i.e., RVE, from the fiber-reinforced composite material, apply stress in the 2 direction, where the 2 direction is the y direction of the coordinate system, and calculate the theoretical fiber strain and matrix strain of the fiber and the matrix in the 2 direction; Step 2: Based on the theoretical fiber strain and matrix strain of the fiber and the matrix in the 2 direction obtained in Step 1, according to the actual deformation of the fiber matrix, introduce a deformation coordination plane and a tensile deformation coordination factor α in the deformation region, calculate the true strain of the fiber and the matrix when loaded in the 2 direction, and then solve the true macroscopic elastic modulus in the 2 direction; in, σ s is the normal stress between fiber and matrix, Because σ s The additional strain on the fiber caused is, is the additional strain of the matrix; f is defined as the fiber, m is defined as the matrix, 1, 2, 3 are used to represent the x, y, z directions of the Cartesian coordinate system, where 1 is the longitudinal direction of the fiber; Step 3: Referring to the formula process for solving the true macroscopic elastic modulus in Step 2, introduce shear deformation coordination factors β and γ, and solve the true macroscopic engineering elastic constants of the fiber-reinforced composite material in the 1-2 direction, 1-3 direction, and 2-3 direction; The expressions of β and γ are: Among them, G m is the shear modulus of the matrix, and G f,12 is the shear modulus of the fibers in the 1-2 direction, and G f,23 is the shear modulus of the fibers in the 2-3 direction.

2. The method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials according to claim 1, characterized in that: the said Step 1 includes: Step 1.1: Take a representative volume element, i.e., RVE, from the fiber-reinforced composite material. An RVE consists of fibers wrapped by a matrix. The cross-section of the fiber is circular. It is a transversely isotropic material, and the matrix is an isotropic material. The engineering elastic constants of the fiber are defined as follows: the elastic modulus E of the fiber in the 1 direction, f,11 the elastic modulus E of the fiber in the 2 direction, f,22 the elastic modulus E of the fiber in the 3 direction, f,33 the Poisson's ratio ν of the fiber in the 1-2 direction, f,12 the Poisson's ratio ν of the fiber in the 1-3 direction, f,13 the Poisson's ratio ν of the fiber in the 2-3 direction, f,23 the shear modulus G of the fiber in the 1-2 direction, f,12 the shear modulus G of the fiber in the 1-3 direction, f,13 the shear modulus G of the fiber in the 2-3 direction, f,23 , and due to the transverse isotropy of the fiber, E f,22 = E f,33 , ν f,12 = ν f,13 , G f,12 = G f,13 ; The engineering elastic constants of the matrix are defined as follows: the elastic modulus E of the matrix, m the Poisson's ratio ν of the matrix, m the shear modulus G of the matrix, m ; Step 1.2: For simplicity in modeling, a real RVE plane is intercepted along the 2-3 direction in the RVE, the fiber radius is defined as r, and the fiber volume fraction is c f , the side length of the matrix is l, then c f = πr 2 l / l 3 ; Step 1.3: Convert the true RVE plane in Step 1.2 into an equivalent RVE plane with the same fiber volume fraction, and calculate c f = a 2 l / l 3 , then Step 1.4: Apply a stress σ in the 2 - direction of the equivalent RVE plane in Step 1.3 2 When this is the case, according to the classical mixture method in composite mechanics, the stresses borne by both the fiber and the matrix are σ 2 When this is the case, that is, σ f,22 = σ m = σ 2 , where σ f,22 is the fiber stress and the matrix stress is σ m , the fiber strain ε f,22 and the matrix strain ε m can be obtained as follows: Step 1.5: Compare the fiber strain ε f,22 and the matrix strain ε m as follows where E f,22 > E m . Therefore, the deformation of the matrix in the length l is much greater than that of the fiber, resulting in the deformation incoordination between the fiber and the matrix in the 2 direction. Considering the fiber as an inclusion, the deformation difference Δl between the fiber and the matrix in the length l is the difference between the actual deformation length l m of the matrix and the actual deformation length a f of the fiber: The strain difference Δη along the 2 direction on l 22 is as follows:

3. The method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials according to claim 2, characterized in that: the said Step 2 is specifically: Step 2.1: According to the actual deformation of the fiber matrix, for the entire equivalent RVE, the sum of the additional strains of the fiber and the matrix along the 2 direction is obtained according to Equation (3). It is as follows: Step 2.2: Assume that the deformation consistent plane of the fiber and the matrix is the deformation coordination plane N1-N2 of the equivalent RVE plane. When loaded, the interaction between the fiber and the matrix is divided into normal stress and shear stress. The shear stress has no influence on the macroscopic mechanical properties, while the normal stress has a greater influence. Assume that the normal stress between the fiber and the matrix is σ s , and the additional strain of the fiber caused by σ s is and the additional strain of the matrix is Then we have: Step 2.3: Since under the action of σ s acting we have: Step 2.4: Assume the nominal fiber deformation is Δa f , and the nominal matrix deformation is Δl m . Then the nominal deformations of the fiber and matrix on the RVE are as follows: Step 2.5: Assume that the nominal strain of the fiber is and the nominal strain of the matrix is Then we have: Step 2.6: On the basis of Step 2.5, the fibers interact with each other, and the true strain of the fiber is defined as The true strain of the matrix is By combining Equations (6) and (8), the true macroscopic strain of the equivalent RVE in the 2 direction is obtained as Furthermore, the true macroscopic elastic modulus in the 2 direction is calculated as follows:

4. The method for predicting the macroscopic mechanical properties of fiber-reinforced composite materials according to claim 3, characterized in that: the method for solving the true macroscopic engineering elastic constants of the fiber-reinforced composite material in the 1-2 direction, 1-3 direction, and 2-3 direction in the said Step 3 is: Solving for the True Macroscopic Elastic Modulus Based on Step 2 Expression defining the true macroscopic elastic modulus in the 1 direction True macroscopic elastic modulus in the 3 direction True macroscopic shear modulus in the 1-2 direction True macroscopic elastic modulus in the 1-3 direction True macroscopic elastic modulus in the 2-3 direction True Poisson's ratio in the 1-2 direction True Poisson's ratio in the 1-3 direction True Poisson's ratio in the 2-3 direction Then the true macroscopic engineering elastic constants of the fiber-reinforced composite are obtained as follows:

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