A method for predicting defect concentration in amorphous alloys by dynamic relaxation analysis

Abstract

This invention belongs to the field of dynamic relaxation behavior modeling technology for amorphous alloys, and specifically relates to a method for predicting the defect concentration of amorphous alloys through dynamic relaxation analysis. A fractional-order viscoelastic model is employed to simulate the dynamic relaxation behavior of amorphous alloys. The model has relatively few parameters and can reflect the evolution of defect concentration during the dynamic relaxation process. The established fractional-order viscoelastic model can be verified through dynamic relaxation experiments, and the evolution of defect concentration can be obtained by fitting the experiments. This method comprehensively utilizes multiple key parameters such as high-frequency shear modulus, low-frequency shear modulus, and relaxation time to achieve quantitative analysis of the complex interaction between the material's microstructure and macroscopic properties. This verification method not only proves the reliability of the model but also allows for the quantitative calculation of the defect concentration evolution. The application of this technology in the alloy design and preparation stages provides new theoretical guidance and experimental methods for the prediction and evaluation of amorphous alloy properties.

Description

Technical Field

[0001] This invention belongs to the field of dynamic relaxation behavior modeling technology for amorphous alloys, and specifically relates to a method for predicting the defect concentration of amorphous alloys through dynamic relaxation analysis. Background Technology

[0002] Amorphous alloys are obtained by rapidly cooling supercooled liquids. They are characterized by long-range disorder and short-range order, combining properties of both metals and glasses, and exhibiting unique mechanical and physical properties such as high strength, high fracture toughness, large elastic strain limit, superplasticity in the supercooled liquid phase, good corrosion resistance, and excellent soft magnetic properties. These superior properties make them promising for wide applications in microelectronics, defense, aerospace, biomedicine, and sports and leisure industries.

[0003] Dynamic mechanical relaxation is a crucial indicator for understanding the mechanical / physical properties of viscoelastic amorphous solids. Due to the inhomogeneity of their microstructure, the relaxation behavior of amorphous solids typically deviates strongly from Debye relaxation. Because amorphous structures lack long-range order, traditional crystal models cannot be directly applied, making the prediction of their relaxation behavior under stress complex. Furthermore, current techniques lack models that correlate microscopic defects with macroscopic properties, resulting in predictions of defect concentration relying on experimental data and lacking accuracy. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a method for predicting the defect concentration of amorphous alloys through dynamic relaxation analysis. Specifically, for the dynamic relaxation behavior of amorphous alloys, a viscoelastic fractional-order model is constructed that is easy to apply, has clear physical meaning, and meets the accuracy requirements. The viscoelastic fractional-order model provided by this invention can well describe the dynamic relaxation master curve and accurately predict the evolution of defect concentration, verifying the rationality of the model.

[0005] To achieve the above objectives, the technical solution adopted by this invention is a method for predicting the defect concentration of amorphous alloys through dynamic relaxation analysis, comprising the following steps:

[0006] Step 1: Using dynamic relaxation analysis, constitutive equations describing the dynamic relaxation behavior of amorphous alloys are established to analyze the dynamic relaxation behavior of amorphous alloys and obtain the complex modulus and loss modulus.

[0007] Step 2: Under fixed temperature conditions, the complex modulus and loss modulus obtained in Step 1 are used to predict the defect concentration of the amorphous alloy.

[0008] Furthermore, the dynamic relaxation analysis method described in step one includes:

[0009] Step 1: The constitutive equation for the fractional viscoelastic element is defined using the Riemann-Liouville type fractional derivative operator, as shown in equation (1):

[0010]

[0011] In the formula, σ represents stress, c represents the viscoelastic coefficient of the material, α represents the order of differentiation, ε represents strain, and t represents time; when α = 0, σ = cε, and the fractional-order viscoelastic element corresponds to the Hooke spring element; when α = 1, ... Fractional-order viscoelastic elements correspond to Newtonian stickpot elements;

[0012] Step 2: Based on the constitutive equation of the fractional-order viscoelastic element obtained in Step 1, replace the spring in the Maxwell element with the fractional-order viscoelastic element, and connect a spring in series outside the entire model to describe the asymmetric loss modulus peak of the amorphous alloy.

[0013] The governing equations are obtained by considering the relationships between the units, as shown in equation (2):

[0014]

[0015] Where E1, E2, and η1 are the spring constants and viscosity of the sticky pot in the corresponding unit, respectively;

[0016] Step 3: Based on the governing equations obtained in Step 2, apply the combination rules of fractional calculus and Fourier integrals to obtain the complex modulus, as shown in equation (3):

[0017]

[0018] The parameters in the formula have the relationship shown in formula (4):

[0019]

[0020] Parameters G0 and G ∞ These correspond to the low-frequency modulus and the high-frequency modulus, respectively, τ MF This is the characteristic relaxation time of the model, κ is the numerical factor, and ω is the angular frequency;

[0021] Step 4: Based on the complex modulus relationship obtained in Step 3, the expression for the loss modulus is further obtained as shown in Equation (5):

[0022]

[0023] The parameter A in the formula has the relationship shown in equation (6).

[0024]

[0025] The above establishes constitutive equations for the viscoelastic mechanical behavior of amorphous alloys, describes the dynamic relaxation behavior of amorphous alloys, and uses constitutive equations to analyze the dynamic relaxation behavior of amorphous alloys in order to predict the defect concentration.

[0026] The predicted defect concentration of amorphous alloys includes,

[0027] (1) Based on the complex modulus and loss modulus obtained in the dynamic relaxation analysis, the expression for internal friction tanδ is obtained as shown in equation (7):

[0028]

[0029] (2) When ωτ MF When sufficiently large, i.e. under low temperature or high frequency measurement, ωτ MF >>1, with an approximate relationship as shown in equation (8):

[0030] tanδ~(ω τM F) -α (8)

[0031] (3) Based on the approximate relationship in equation (8), and taking into account... According to equation (9):

[0032]

[0033] In the formula, K is the Boltzmann constant, and N is a constant;

[0034] According to the formula (9) given above, at a fixed temperature, after performing dynamic relaxation analysis on the amorphous alloy, the slope x of ln(tanδ) and ln(ω) is used as the correlation factor of the defect concentration of the amorphous alloy, thus completing the prediction of the defect concentration of the amorphous alloy.

[0035] The beneficial effects of this invention are as follows: This invention provides a method for predicting the defect concentration of amorphous alloys through dynamic relaxation analysis. It employs a fractional-order viscoelastic model to simulate the dynamic relaxation behavior of amorphous alloys. The model has few parameters and can reflect the evolution of defect concentration during the dynamic relaxation process. The established fractional-order viscoelastic model can be verified through dynamic relaxation experiments, and the evolution of defect concentration can be obtained by fitting the experiments. This verification method not only proves the reliability of the model but also allows for the quantitative calculation of the defect concentration evolution. Attached Figure Description

[0036] Figure 1 This is a flowchart of the present invention;

[0037] Figure 2 This is a schematic diagram of a fractional-order viscoelastic model in an embodiment of the present invention;

[0038] Figure 3 Cu at different temperatures in the embodiments of the present invention 46 Zr 35 Hf 12 Experimental graph showing the evolution of normalized loss modulus and normalized storage modulus of Al7 amorphous alloy with frequency.

[0039] Figure 4 The figures shown are the master curve obtained by the time-temperature equivalence principle and the curve fitted by the MF model in the embodiments of the present invention.

[0040] Figure 5 This invention presents the double logarithmic variation of the loss factor tanδ with the driving frequency and the evolution of the defect concentration with temperature at different temperatures in this example. Detailed Implementation

[0041] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0042] like Figure 1 As shown, the method for predicting the defect concentration of amorphous alloys through dynamic relaxation analysis according to the present invention includes the following steps:

[0043] First, dynamic relaxation analysis was performed on the model alloy.

[0044] Step 1: The constitutive equation for the fractional viscoelastic element is defined using the Riemann-Liouville type fractional derivative operator, as shown in equation (1):

[0045]

[0046] In the formula, σ represents stress, c represents the viscoelastic coefficient of the material, α represents the order of differentiation, ε represents strain, and t represents time; when α = 0, σ = cε, and the fractional-order viscoelastic element corresponds to the Hooke spring element; when α = 1, ... Fractional-order viscoelastic elements correspond to Newtonian stickpot elements;

[0047] Step 2: Based on the constitutive equation of the fractional-order viscoelastic element obtained in Step 1, replace the spring in the Maxwell element with the fractional-order viscoelastic element, and connect a spring in series outside the entire model to describe the asymmetric loss modulus peak of the amorphous alloy.

[0048] The governing equations are obtained by considering the relationships between the units, as shown in equation (2):

[0049]

[0050] Where E1, E2, and η1 are the spring constant and viscosity of the sticky pot in the corresponding unit, respectively. Figure 2.

[0051] Step 3: Based on the governing equations obtained in Step 2, apply the combination rules of fractional calculus and Fourier integrals to obtain the complex modulus, as shown in equation (3):

[0052]

[0053] The parameters in the formula have the relationship shown in formula (4):

[0054]

[0055] Parameters G0 and G ∞ These correspond to the low-frequency modulus and the high-frequency modulus, respectively, τ MF This is the characteristic relaxation time of the model, κ is the numerical factor, and ω is the angular frequency;

[0056] Step 4: Based on the complex modulus relationship obtained in Step 3, the expression for the loss modulus is further obtained as shown in Equation (5):

[0057]

[0058] The parameter A in the formula has the relationship shown in equation (6).

[0059]

[0060] like Figure 2 As shown, in order to verify the feasibility and effectiveness of the present invention, Cu was selected. 46 Zr 35 Hf 12 Al7 amorphous alloy was used as the model alloy; Cu was tested using an inverted torsion pendulum internal friction apparatus. 46 Zr 35 Hf 12 Dynamic relaxation experiments were conducted on Al7 amorphous alloy to obtain the dynamic modulus evolution curves at different temperatures. The experimental results of the normalized storage modulus and loss modulus as a function of driving frequency are shown below. Figure 3 As shown.

[0061] Figure 3 (a) shows the curve of normalized energy storage modulus versus frequency, with a temperature interval of 5K. Figure 3 (b) shows the normalized loss modulus as a function of frequency, with a temperature interval of 5K. Based on the time-temperature equivalence principle, the master curve of the loss modulus at a reference temperature of 700K is obtained, as follows: Figure 3 As shown.

[0062] The principal curve is fitted using formula (5), as follows: Figure 4 As shown, the fractional-order viscoelastic model agrees well with the data points across the entire observation frequency range, and can describe Cu over a relatively wide frequency range. 46 Zr35 Hf 12 The viscoelastic behavior of Al7 amorphous alloys is now complete, and the dynamic relaxation analysis of the alloys in the model is finished.

[0063] The following is about Cu 46 Zr 35 Hf 12 Defect concentration prediction for Al7 amorphous alloy.

[0064] Step 5: Based on the complex modulus and loss modulus obtained in Step 4 and Step 3, the expression for internal friction tanδ can be obtained as shown in Equation (7):

[0065]

[0066] Step 6: When ωτ MF When sufficiently large, i.e. under low temperature or high frequency measurement, ωτ MF >>1, with an approximate relationship as shown in equation (8),

[0067] tanδ~(ωτ MF ) -α (8)

[0068] Step 7: Based on the approximate relationship in Step 6, and taking into account... According to equation (9),

[0069]

[0070] In the formula, K is the Boltzmann constant, and N is a constant.

[0071] This invention utilizes an inverted torsional pendulum internal loss meter to conduct a loss factor tanδ measurement experiment on a model alloy, obtaining the double logarithmic variation of the loss factor tanδ with the driving frequency at different temperatures. The experimental results are as follows: Figure 5 As shown in (a). Using formula (9) to... Figure 5 By fitting the experimental results in (a), the evolution of the defect concentration correlation factor χ with temperature was finally obtained, as follows: Figure 5 As shown in (b).

[0072] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for predicting the defect concentration of amorphous alloys using dynamic relaxation analysis, characterized in that, Includes the following steps: Step 1: The constitutive equation of the fractional viscoelastic element is defined using the Riemann-Liouville type fractional derivative operator. Then, based on the constitutive equation of the fractional-order viscoelastic element, the spring in the Maxwell element is replaced with a fractional-order viscoelastic element, and a spring is connected in series outside the entire model to describe the asymmetric loss modulus peak of the amorphous alloy. Then, the governing equation is obtained through the relationship between the elements. Finally, based on the control equations, the combination rules of fractional calculus and Fourier integrals are applied to obtain the complex modulus and loss modulus. Step 2: Under fixed temperature conditions, obtain the internal friction by using the complex modulus and the loss modulus. Expression, and based on internal friction By establishing an approximate relationship between the angular frequency and the complex modulus under low-temperature or high-frequency measurements, a constitutive equation for describing the dynamic relaxation behavior of amorphous alloys is established. This allows for the analysis of the dynamic relaxation behavior of amorphous alloys and the prediction of the defect concentration.

2. The method for predicting the defect concentration of amorphous alloys by dynamic relaxation analysis as described in claim 1, characterized in that, The first step is as follows: Step 1: The constitutive equation for the fractional viscoelastic element is defined using the Riemann-Liouville type fractional derivative operator, as shown in equation (1): (1)， In the formula, For stress, The viscoelastic coefficient of the material. To find the order of the derivative, In response, For time; when hour, Fractional-order viscoelastic elements correspond to Hooke's spring elements; when hour, Fractional-order viscoelastic elements correspond to Newtonian stick pot elements; Step 2: Based on the constitutive equation of the fractional-order viscoelastic element obtained in Step 1, replace the spring in the Maxwell element with the fractional-order viscoelastic element, and connect a spring in series outside the entire model to describe the asymmetric loss modulus peak of the amorphous alloy. The governing equations are obtained by considering the relationships between the units, as shown in equation (2): （2）， in , , These are the spring constant and the viscosity of the sticky pot, respectively, in the corresponding unit; Step 3: Based on the control equations obtained in Step 2, apply the combination rules of fractional calculus and Fourier integrals to obtain the complex modulus, as shown in equation (3): （3）， The parameters in the formula have the relationship shown in formula (4): （4）， parameter and These correspond to low-frequency modulus and high-frequency modulus, respectively. This is the characteristic relaxation time of the model. It is a numerical factor, and ω is the angular frequency; Step 4: Based on the complex modulus relationship obtained in Step 3, the expression for the loss modulus is further obtained as shown in Equation (5): （5）， The parameter A in the formula has the relationship shown in equation (6): （6）， The above establishes constitutive equations for the viscoelastic mechanical behavior of amorphous alloys, describes the dynamic relaxation behavior of amorphous alloys, and uses constitutive equations to analyze the dynamic relaxation behavior of amorphous alloys in order to predict the defect concentration. The second step is specifically as follows: Step 1: Based on the complex modulus and loss modulus obtained from the dynamic relaxation analysis, obtain the internal friction. The expression is as shown in equation (7): （7）， Step 2: When When large enough, i.e. under low temperature or high frequency measurement, The approximate relationship is as shown in equation (8): （8）， Step 3: Based on the approximate relationship in equation (8), the constitutive equation describing the dynamic relaxation behavior of amorphous alloys is obtained as follows: （9）， In the formula, K is the Boltzmann constant, and N is a constant; , ω is the characteristic relaxation time of the complex modulus, and ω is the angular frequency; At a fixed temperature, dynamic relaxation analysis of the amorphous alloy was performed, and... and slope As a correlation factor for the defect concentration of amorphous alloys, the prediction of the defect concentration of amorphous alloys is now complete.

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