A method for determining the allowable damage scale of concrete based on the fracture extremum theory

Abstract

The application discloses a method for determining the allowable damage scale of concrete based on a fracture extremum theory, which comprises the following steps: S1, carrying out a fracture test to obtain a peak load; S2, calculating a stress intensity factor generated by an external load and cohesive force according to a stress intensity factor manual; S3, establishing a crack tip stress intensity factor balance equation according to a crack propagation criterion, and then deducing an expression of the external load; S4, determining a critical effective crack length of the concrete according to the fracture extremum theory; and S5, calculating the allowable damage scale according to the critical effective crack length. The method for determining the allowable damage scale of concrete based on the fracture extremum theory is adopted, the fracture extremum characteristics of the concrete are fully considered, the peak load of the fracture test needs to be measured only when the method is applied to analyze the crack propagation of the concrete, the test operation is simple, the calculation result is high in accuracy, and the method is suitable for test pieces of different sizes and types.

Description

Technical Field

[0001] This invention relates to a fracture and damage analysis technique for concrete, and more particularly to a method for determining the allowable damage scale of concrete based on fracture extreme value theory. Background Technology

[0002] Concrete, as a multiphase composite material, has initial defects within it. Under changes in the external environment or the action of loads, these initial defects gradually evolve and develop into a damaged zone. From the load-crack opening displacement curves obtained from fracture tests, it can be observed that there is an extreme point at the critical instability moment. Combining damage and fracture mechanics theories, Li Qingbin et al. (Li Qingbin, Zhang Chuhan, Wang Guanglun. Dynamic and static damage fracture analysis of type I cracks in concrete [J]. Journal of Civil Engineering, 1993, 26(6):20-27.) proposed an allowable damage scale for judging the stability of concrete cracks, namely, the length of the damaged zone at the critical instability moment.

[0003] Currently, methods for determining the allowable damage scale include the Williams stress function method (Qing Longbang, Wang Tuo, Guan Junfeng, Liu Jinchun. Analytical study on allowable damage scale of finite-size concrete specimens [J]. Engineering Mechanics, 2017, 34(01): 213-218.) and the experimental measurement method (Qing Longbang, Cao Guorui, Guan Junfeng. Experimental study on allowable damage scale of concrete based on DIC method [J]. Engineering Mechanics, 2019, 36(10): 115-121.). Among them, the Williams stress function method obtains the analytical expression of the allowable damage scale of finite-size concrete specimens based on the mathematical expression of the stress field near the crack propagation tip of concrete, while considering the influence of stress relaxation factors. However, this method only studies the first three order solutions of the stress field near the crack tip; for different specimen types, the stress function is expanded to different orders, and currently only analytical solutions for two specimen types are given. The experimental measurement method, based on fracture tests, utilizes digital image correlation (DIC) technology to obtain the damage scale at any load moment during crack propagation. However, this method is complex to operate, and setting up the observation instruments requires high-quality experimental conditions. Both of these methods are mainly used for specimens with small aggregate and specimen sizes, and neither considers the extreme characteristics of concrete fracture.

[0004] Based on the above analysis, it is necessary to propose a method for determining the permissible damage scale applicable to specimens of different sizes and types. Summary of the Invention

[0005] The purpose of this invention is to provide a method for determining the allowable damage scale of concrete based on the fracture extreme value theory. This method fully considers the fracture extreme value characteristics of concrete. When applying this method to analyze the crack propagation of concrete, only the peak load of the fracture test needs to be measured. The test operation is simple, the calculation results are highly accurate, and it is applicable to specimens of different sizes and types.

[0006] To achieve the above objectives, this invention provides a method for determining the allowable damage scale of concrete based on fracture extreme value theory, comprising the following steps:

[0007] S1: Conduct fracture tests to obtain peak load;

[0008] S2: Calculate the stress intensity factor generated by external load and cohesion according to the stress intensity factor handbook;

[0009] S3: Based on the crack propagation criterion, establish the stress intensity factor equilibrium equation at the crack tip, and then derive the expression for the external load;

[0010] S4: Determine the critical effective crack length of concrete based on the fracture extreme value theory;

[0011] S5: Calculate the allowable damage scale based on the critical effective crack length.

[0012] Preferably, the peak load of the fracture test in step S1 is recorded.

[0013] Preferably, in step S2:

[0014] Stress intensity factor K generated by external load I P The expression for (a,P) is:

[0015]

[0016] Where P is the external load applied to the specimen; B, D, S, L, and W are the thickness, height, length, span, and self-weight of the specimen, respectively; the crack height ratio α = a / D, where a represents the effective crack length; the shape function k(α) is expressed as follows:

[0017]

[0018] Stress intensity factor K generated by cohesion I c The expression for (a,σ(ω)) is:

[0019]

[0020] Where σ is the cohesion within the damaged zone; ω is the crack opening displacement; the parameter g(a) related to the cohesion near the crack tip is expressed as follows:

[0021]

[0022] Where, A1 = σ(CTOD); A2 = [f t -σ(CTOD)] / (a–a0); a0 is the initial crack length; f t For tensile strength; s = 1 - a0 / a; M1, M2, and M3 are expressed by polynomials of seam height ratio;

[0023] The relationship between the cohesion σ and crack opening displacement ω within the allowable damaged zone of concrete is represented by a nonlinear softening curve:

[0024]

[0025] Where c1 and c2 are material parameters; w0 is the maximum crack opening displacement.

[0026] The expressions for crack tip opening displacement CTOD and crack mouth opening displacement CMOD are as follows:

[0027]

[0028]

[0029] Preferably, in step S3, when the stress intensity factor at the crack tip reaches its critical value, the crack propagates, and then the expression for the external load is derived from the stress intensity factor equilibrium equation at the crack tip:

[0030]

[0031] Among them, K IC The stress intensity factor at the crack tip at the critical moment.

[0032] Preferably, in step S4, based on the fundamental assumptions of fracture extreme value theory, the partial derivative of the external load with respect to the effective crack length is zero at the peak time, thus obtaining the expression for the relationship between the external load and the effective crack length at the instability time:

[0033]

[0034] in,

[0035]

[0036]

[0037] Substituting the peak load obtained from step S1 into formula (9), the critical effective crack length a can be obtained. c .

[0038] Preferably, in step S5:

[0039] Based on the concept of permissible damage scale, the expression for permissible damage scale is determined as follows:

[0040] R = a c -a0 (12)

[0041] The critical effective crack length a c Substituting into formula (12), the allowable damage scale can be calculated.

[0042] Therefore, the present invention employing the above method has the following beneficial effects:

[0043] 1) The influence of the extreme characteristics of concrete fracture on crack propagation is considered, which simplifies the calculation and is applicable to different specimen types.

[0044] 2) Only the peak load of the fracture test needs to be measured. The operation is simple and the requirements for test conditions are reduced. It can be completed under ordinary laboratory conditions and can be promoted and popularized in the fields of scientific research and engineering.

[0045] 3) There are no strict requirements on specimen size and aggregate particle size.

[0046] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0047] Figure 1 This is a flowchart of the present invention. Detailed Implementation

[0048] The present invention will be further described below with reference to the accompanying drawings. It should be noted that this embodiment is based on the present technical solution and provides detailed implementation methods and specific operation processes, but the protection scope of the present invention is not limited to this embodiment.

[0049] Figure 1 The flowchart of this invention is as follows: Figure 1 As shown, the present invention includes the following steps:

[0050] S1: Conduct fracture tests to obtain peak load;

[0051] Preferably, the peak load of the fracture test in step S1 is recorded.

[0052] S2: The calculation process is illustrated using a three-point bending beam specimen as an example. The stress intensity factor generated by the external load and cohesion is calculated according to the stress intensity factor handbook.

[0053] Preferably, in step S2:

[0054] Stress intensity factor K generated by external load IP The expression for (a,P) is:

[0055]

[0056] Where P is the external load applied to the specimen; B, D, S, L, and W are the thickness, height, length, span, and self-weight of the specimen, respectively; the crack height ratio α = a / D, where a represents the effective crack length; the shape function k(α) is expressed as follows:

[0057]

[0058] Stress intensity factor K generated by cohesion I c The expression for (a,σ(ω)) is:

[0059]

[0060] Where σ is the cohesion within the damaged zone; ω is the crack opening displacement; the parameter g(a) related to the cohesion near the crack tip is expressed as follows:

[0061]

[0062] Where, A1 = σ(CTOD); A2 = [f t -σ(CTOD)] / (a–a0); a0 is the initial crack length; f t For tensile strength; s = 1 - a0 / a; M1, M2, and M3 are expressed by polynomials of seam height ratio;

[0063] The relationship between the cohesion σ and crack opening displacement ω within the allowable damaged zone of concrete is represented by a nonlinear softening curve:

[0064]

[0065] Where c1 and c2 are material parameters; w0 is the maximum crack opening displacement.

[0066] The expressions for crack tip opening displacement CTOD and crack mouth opening displacement CMOD are as follows:

[0067]

[0068]

[0069] S3: Based on the crack propagation criterion, establish the stress intensity factor equilibrium equation at the crack tip, and then derive the expression for the external load;

[0070] Preferably, in step S3, when the stress intensity factor at the crack tip reaches its critical value, the crack propagates, and then the expression for the external load is derived from the stress intensity factor equilibrium equation at the crack tip:

[0071]

[0072] Among them, K IC The stress intensity factor at the crack tip at the critical moment.

[0073] S4: Determine the critical effective crack length of concrete based on the fracture extreme value theory;

[0074] Preferably, in step S4, based on the fundamental assumptions of the fracture extreme value theory (which assumes that the external load P is continuously differentiable with respect to the effective crack length a during crack propagation, based on the load-effective crack length curve measured by quasi-brittle material fracture tests), the partial derivative of the external load with respect to the effective crack length is zero at the peak time, thus yielding the expression for the relationship between the external load and the effective crack length at the instability time:

[0075]

[0076] in,

[0077]

[0078]

[0079] Substituting the peak load obtained from step S1 into formula (9), the critical effective crack length a can be obtained. c .

[0080] S5: Calculate the allowable damage scale based on the critical effective crack length.

[0081] Preferably, in step S5:

[0082] Based on the concept of permissible damage scale, the expression for permissible damage scale is determined as follows:

[0083] R = a c -a0 (12)

[0084] The critical effective crack length a c Substituting into formula (12), the allowable damage scale can be calculated.

[0085] Example

[0086] Peak load was measured during a three-point bending fracture test on a concrete beam to calculate the allowable damage scale of the concrete. Three three-point bending beam specimens were fabricated for this test, with specimen dimensions of B×D×S = 100×100×400 mm. 3The tensile strength of C25 concrete is 1.87 MPa, and the elastic modulus is 28 GPa. A 300 kN universal testing machine was used, and displacement loading was employed. During the test, a static data acquisition device was used to collect data on the peak load P. max .

[0087] Table 1 shows the peak load of the experiment and the calculation results.

[0088]

[0089] Table 1 shows that the permissible damage scale of C25 concrete was determined using the method of the present invention. Because the present invention considers both the extreme characteristics and damage features of concrete fracture, the experimental operation is simple, requiring only the recording of the most easily measurable and highly accurate peak load in the fracture test, thus effectively improving the accuracy of the calculation.

[0090] Therefore, the present invention adopts the above-mentioned method for determining the allowable damage scale of concrete based on the fracture extreme value theory, which fully considers the fracture extreme value characteristics of concrete. When applying it to analyze the crack propagation of concrete, only the peak load of the fracture test needs to be measured. Moreover, the peak load is the peak load determined by a single fracture test, which is the only test data that needs to be measured in the fracture test. This simplifies the calculation, makes the test operation simple, and provides high accuracy of the calculation results. It is also applicable to specimens of different sizes and types.

[0091] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for determining the allowable damage scale of concrete based on fracture extreme value theory, characterized in that: Includes the following steps: S1: Conduct fracture tests to obtain peak load; S2: Calculate the stress intensity factor generated by external load and cohesion according to the stress intensity factor handbook; S3: Based on the crack propagation criterion, establish the stress intensity factor equilibrium equation at the crack tip, and then derive the expression for the external load; S4: Determine the critical effective crack length of concrete based on the fracture extreme value theory; S5: Calculate the allowable damage scale based on the critical effective crack length; In step S2: Stress intensity factor caused by external load K PI( a , P The expression is: （1） in, P External loads applied to the specimen; B , D , S , L and W These are the specimen's thickness, height, length, span, and self-weight; seam height ratio. α = a / D , a Indicates the effective crack length; shape function k ( α The expression is as follows: （2） Stress intensity factor generated by cohesion K c I( a , σ ( ω The expression is: （3） in, σ This refers to the cohesive force within the damaged area; ω The crack opening displacement; a parameter related to the cohesion near the crack tip. g ( a The expression is as follows: （4） in, A 1 = σ ( CTOD ); A 2 = [ f t - σ ( CTOD )] / ( a – a 0); a 0 represents the initial crack length; f t Tensile strength; s = 1 - a 0 / a ; M 1, M 2, M 3 is expressed as a polynomial of the seam height ratio; Cohesion within the allowable damage zone of concrete σ and crack opening displacement ω The relationship between them is represented by a nonlinear softening curve: （5） in, c 1. c 2 represents material parameters; w 0 represents the maximum crack opening displacement; Crack tip opening displacement CTOD and the displacement of the crack mouth opening CMOD The expression is as follows: （6） （7）； In step S3, when the stress intensity factor at the crack tip reaches its critical value, the crack propagates. Then, based on the stress intensity factor equilibrium equation at the crack tip, the expression for the external load is derived as follows: （8） in, K IC The stress intensity factor at the crack tip at the critical moment; In step S4, based on the fundamental assumptions of fracture extreme value theory, the partial derivative of the external load with respect to the effective crack length is zero at the peak time, thus yielding the expression for the relationship between the external load and the effective crack length at the instability time: （9） in, （10） （11） Substituting the peak load obtained from step S1 into formula (9), the critical effective crack length can be obtained. a c ; In step S5: Based on the concept of permissible damage scale, the expression for permissible damage scale is determined as follows: （12） Critical effective crack length a c Substituting into formula (12), the allowable damage scale can be calculated.

2. The method for determining the allowable damage scale of concrete based on fracture extreme value theory according to claim 1, characterized in that: Step S1 only requires recording the peak load of the fracture test. The fracture test is one of the three-point bending beam fracture test, wedge splitting fracture test, dynamic and static cube and cylinder splitting fracture test and arch fracture test.

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