CN109900554B - A Method of Calculating Fracture Toughness Using Indentation Method - Google Patents

Abstract

The invention belongs to the technical field of fracture toughness testing, and provides a method for calculating fracture toughness by using the indentation method, comprising the following steps: (1) obtaining the relationship between the depth and the load of the material indentation and unloading stages; Load-displacement curve during unloading, to obtain the state parameters of plastic residual depth and contact depth; (3) Use finite element software to predict the material under the action of the indenter, including the radial displacement correction factor and the non-axisymmetric correction factor of the Berkovich indenter; (4) The uniaxial tensile experiment of the material obtains the elastic modulus of the material and combines the finite element software with the GTN theory to obtain the critical porosity of the material; (5) correlates the indentation work with the fracture energy and is equivalent to the strain energy density, thereby obtaining the critical porosity Fracture toughness equation characterized by plastic residual depth. The fracture toughness test method can obtain the elastic modulus and fracture toughness values of the material quickly and with low loss through a small-scale indentation test of the material.

Description

A Method of Calculating Fracture Toughness Using Indentation Method

technical field

The invention relates to the technical field of fracture toughness testing, in particular to a method for calculating fracture toughness by using an indentation method.

Background technique

With the continuous development of aerospace engineering, reactor engineering, welding technology and petroleum engineering, the demand for testing the mechanical properties of metal materials is increasing. There are a large number of in-service equipment in thermal power, nuclear power, metallurgy, petrochemical and other industries. Traditional sampling tests can obtain more comprehensive performance parameters of in-service equipment materials, but sampling tests are generally destructive and are not suitable for in-service equipment. The indentation method does not require sampling, has non-destructive characteristics, and can accurately and reliably obtain the mechanical properties of in-service equipment materials. The volume of the indentation test device is relatively small, and the positioning of the tested area is easy to realize during the test, and it can be applied to the materials (such as weld seam, heat-affected zone, base metal) for measuring the gradient change of surface properties. In view of the above advantages, the fracture toughness test method of the indentation test can quickly and non-destructively obtain the fracture toughness value of the material.

The indentation initiation energy (IEF) model considers that the test energy of the indentation is equal to the energy required for crack initiation when the critical load is reached. When the continuous ball indentation test of IEF is used to test the fracture toughness, because of the geometric shape of the ball, it needs to be indented to a greater depth, and it cannot be tested on some gradient materials and bends; while the Berkovich indenter calculates the error of the reduced elastic modulus Larger, the existing indentation method is unreasonable, so it needs to be improved urgently.

Contents of the invention

In order to overcome the deficiencies of the prior art, the present invention provides a method for calculating the fracture toughness by using the indentation method. By continuously loading and unloading the measured material on the indentation device using the Berkovich indenter, the corresponding unloading of the measured material is obtained. Point effective elastic modulus and plastic residual depth, according to the established correlation formula between indentation work and fracture energy, and then obtain the fracture toughness of the material. The process of obtaining the fracture toughness value of the material is fast and low-loss, which can effectively save the cost of experimental materials.

The present invention adopts following technical scheme:

A kind of method utilizing indentation method to calculate fracture toughness, comprises the steps:

(1) Based on the indentation experiment, at a constant rate, obtain the full load-displacement curve under the condition of no less than 6 times of unloading;

(2) Perform a power function parameter fitting on the 30%-70% part of each unloading curve, obtain the parameters of the residual plastic deformation depth h _p and the contact depth h _c after unloading, and calculate the unloading stiffness at the unloading point of each curve;

Then perform a straight line fitting on each unloading curve to obtain the unloading stiffness of each curve;

The unloading stiffness of the first curve is the straight line fitting value, and the second line starts to compare the unloading stiffness calculated by the power function fitting with the straight line fitting value, and if the error between the two is within 10%, it is judged that the power function fitting parameter is valid ;

Use the following formula to fit valid data points P and h _p :

Among them, h _p is the residual plastic deformation depth after unloading, P is the press-in load, C ₁ , C ₂ , and C ₃ are fitting parameters and satisfy C ₂ *C ₃ <0, and then obtain the parameter relationship between h _p and P;

(3) Using the finite element method to simulate the pressing process of the indenter, obtain the radial displacement correction coefficient and the non-axisymmetric correction factor of the Berkovich indenter, so as to obtain the reduced elastic modulus;

(4) Use finite element to simulate the uniaxial stretching process to obtain the critical porosity of the material;

(5) Calculate the fracture toughness based on the proposed correlation formula between indentation work and indentation fracture energy, combined with the critical porosity;

The correlation formula between described indentation work and indentation fracture energy is:

Among them, G _IEF is the indentation fracture energy, F(h _p ) is the indentation work at the depth of h _p , P(h _p ) is the load at the depth of h _p , and A _p (h _p ) is the projection surface of the indentation at the depth of h _p The area is calculated as:

Among them, γ is the radial displacement correction coefficient, h _p is the residual plastic deformation depth after unloading, is the critical residual plastic deformation depth;

Combined with the fitting formula of P and h _p obtained in step (2), the indentation fracture energy can be obtained with respect to The function is:

where A, B, and C are respectively

critical residual plastic deformation depth pass It can be obtained from the relationship with lnh _p , and its relationship is:

in is the effective elastic modulus, K and b are fitting parameters;

Effective modulus of elasticity The calculation formula is:

Among them, υ is the Poisson's ratio of the tested material, _υi is the Poisson's ratio of the indenter material, _Er is the reduced elastic modulus, and _Ei is the elastic modulus of the indenter;

exist At the time of critical fracture, the corresponding ordinate of E ^* is the critical residual plastic deformation depth

The indentation fracture energy is equivalent to the strain energy density, and the equivalent formula is:

Where S _cr is the critical strain energy density, υ is the Poisson's ratio of the tested material, and SED is the strain energy density;

and there are Then the final solution of fracture toughness is:

Where K _IC is the fracture toughness and E is the elastic modulus of the tested material.

Preferably, the indentation experiment in step (1) adopts a constant Berkovich indenter continuous loading and unloading method, and the indenter performs no less than 6 equal differential loadings at a loading rate of 50mN/s, and unloads after each loading , to obtain the full load-displacement curve.

Preferably, the power function parameter fitting formula in step (2) is:

P=B(hh _p ) ^m

Among them, B and m are fitting parameters, h _p is the residual plastic deformation depth after unloading, and h is the indentation depth of the indenter;

The linear fitting parameter relationship is:

P=a+S ₂ *h

Among them, a is the fitting parameter, and S2 is the unloading stiffness in the straight line fitting parameter _;

The unloading stiffness S1 in the power function parameter is obtained by the Oliver _- Pharr method, which is calculated by the following formula,

S ₁ =Bm(h _max -h _p ) ^m-1

Among them, h _max is the maximum indentation depth in the indentation experiment.

Preferably, the calculation formula of the reduced elastic modulus in the step (3) is:

Among them, E _r is the reduced elastic modulus, β is the non-axisymmetric correction factor of Berkovich indenter, γ is the radial displacement correction coefficient, A _c is the contact depth area, S is the unloading stiffness, and S is at the first unloading curve Take S ₂ , and then each unloading curve adopts the unloading stiffness S ₁ in the power function parameter.

The beneficial effects that the present invention has are:

1. The fracture toughness calculation method of this indentation method can obtain the effective elastic modulus and plastic residual depth of the corresponding unloading point of the measured material by continuously loading and unloading the measured material through the Berkovich indenter on the indentation equipment, according to the established The correlation formula between indentation work and fracture energy can be used to obtain the fracture toughness of the material, and the fracture toughness value of the material can be obtained more quickly and with low loss, effectively saving the cost of experimental materials.

2. The present invention optimizes the fitting range of the material unloading curve, and analyzes the error between the fitting value and the experimental value to ensure the validity of the fitting data.

3. The present invention correlates the critical plastic residual depth of the material through the critical porosity, and the critical porosity is obtained through the combination of uniaxial tension and finite element, avoiding the artificial factors obtained by the traditional statistical method, so that the fracture toughness of the material can be predicted more accurately value.

4. The accuracy of the elastic modulus obtained in the calculation process of the present invention at the initial press-in stage is significantly improved, and has engineering application value.

Description of drawings

Fig. 1 is the flow chart of calculating fracture toughness method by indentation method;

Fig. 2 is a schematic diagram of the Berkovich indenter pressing into the material;

Figure 3 is a model diagram of the radial displacement of the indenter;

Fig. 4 is the indentation experiment curve of Al6061 alloy;

Figure 5 is the indentation test curve of SS302 alloy.

Detailed ways

The present invention is specifically described below in conjunction with accompanying drawing:

As shown in Figure 1-2, a method for calculating fracture toughness by indentation method is characterized in that it includes the following steps:

Step (1), based on the indentation experiment, at a constant rate, obtain the full load-displacement curve under the unloading condition of no less than 6 times;

The indentation test adopts the method of continuous loading and unloading of the constant Berkovich indenter. The indenter is loaded at least 6 times at a loading rate of 50mN/s, and unloaded after each loading to obtain the full load-displacement curve.

In step (2), a power function parameter fitting is performed on each unloading curve to obtain the state parameters of the plastic residual depth and contact depth, and the unloading stiffness at the unloading point of each curve is calculated, and then a straight line fitting is performed on each unloading curve, Obtain the unloading stiffness of each curve, and obtain the relationship between h _p and P parameters;

During the fitting process, the 30%-70% part of the unloading curve is selected for fitting, and the power function parameter fitting formula is:

P=B(hh _p ) ^m (1)

Among them, B and m are fitting parameters, h _p is the depth of residual plastic deformation after unloading;

The linear fitting parameter relationship is:

P=a+S ₂ *h (2)

Among them, a is the fitting parameter, _S2 is the unloading stiffness in the straight line fitting parameter,

The unloading stiffness S ₁ in the power function parameter is obtained by the Oliver-Pharr method and calculated by the following formula,

S ₁ =Bm(h _max -h _p ) ^m-1 (3)

Among them, h _max is the maximum indentation depth in the indentation experiment; the unloading stiffness of the first curve is the linear fitting value, and the second curve starts to compare the unloading stiffness calculated by power function fitting with the linear fitting value, if The error of the two is within 10% to judge that the fitting parameters of the power function are valid;

Use the following formula to fit valid data points P and h _p :

Among them, C ₁ , C ₂ , and C ₃ are fitting parameters and satisfy C ₂ *C ₃ <0.

Step (3), using the finite element to simulate the indenter pressing process, obtain the radial displacement correction coefficient and the non-axisymmetric correction factor of the Berkovich indenter, so as to obtain the reduced elastic modulus;

The radial displacement model of the indenter is shown in Figure 3

Among them, r* the distance from the initial mark point A* to the centerline of the indenter, and r is the distance from the centerline of the indenter to A due to the change of the initial mark point from A* after being pressed in.

For the solution of the non-axisymmetric correction factor, the finite element models of the three-dimensional conical indenter and the three-dimensional Berkovich indenter are used to simulate the nanoindentation process of different materials at the same loading depth, and the contact stiffness and elastic modulus of the unloading curve are calculated. Among them, The ratio of the contact stiffness obtained by the three-dimensional Berkovich indenter to that obtained by the three-dimensional conical indenter is the correction factor β.

Reduced elastic modulus, the calculation formula is:

Among them, E _r is the reduced elastic modulus, β is the non-axisymmetric correction factor of Berkovich indenter, γ is the radial displacement correction coefficient, S is the unloading stiffness (that is, the unloading stiffness S ₁ in judging the effective power function parameters), and the unloading The stiffness S is obtained by the Oliver-Pharr method.

Step (4), using finite element to simulate the uniaxial tension process to obtain the critical porosity of the material;

In step (5), the fracture toughness is calculated from the proposed correlation formula between indentation work and indentation fracture energy combined with critical porosity.

The correlation formula between indentation work and indentation fracture energy is as follows:

Among them, G _IEF is the indentation fracture energy, F(h _p ) is the indentation work at the depth of h _p , P(h _p ) is the load at the depth of h _p , and A _p (h _p ) is the projection surface of the indentation at the depth of h _p The area is calculated as:

Among them, γ is the radial displacement correction coefficient, h _p is the depth of residual plastic deformation after unloading, is the critical residual plastic deformation depth. Combined with the P(h _p ) fitting formula obtained in step (2), the indentation fracture energy can be obtained with respect to The function is:

where A, B, and C are respectively

critical residual plastic deformation depth pass It can be obtained from the relationship with lnh _p , and its relationship is:

in is the effective elastic modulus, K and b are fitting parameters,

Effective modulus of elasticity The calculation formula is:

Among them, υ is the Poisson's ratio of the tested material, υ _i is the Poisson's ratio of the indenter material, E _i is the elastic modulus of the indenter

exist At the time of critical fracture, the corresponding ordinate of E ^* is the critical residual plastic deformation depth

According to the metal toughness damage hole theory has the following formula:

Among them, D is the damage ratio, is the effective elastic modulus, and E is the elastic modulus of the measured material.

Among them, f is the porosity of metal ductile damage,

In the state of critical fracture of the metal, there are:

f ^* = f _F (14)

Among them, f _F is the critical porosity optimized by experiment and finite element in step (4)

The indentation fracture energy is equivalent to the strain energy density, and the equivalent formula is:

where S _cr is the critical strain energy density, υ is the Poisson's ratio of the material, and SED is the strain energy density.

and there are

Then the final solution of fracture toughness is:

where K _IC is the fracture toughness and E is the elastic modulus of the material.

Calculate the fracture toughness of Al6061 alloy and SS302 alloy. Fig. 4 is the indentation test curve of Al6061 alloy, and Fig. 5 is the indentation experiment curve of SS302 alloy. The parameters in the model proposed in the present invention are obtained by fitting test data. During the fitting process, the fitting parameters of Al6061, γ=1.1, β=1.049, f ^* =0.045, E ^* =58.3583GPa, G _IEF ＝5.495mJ/m ² , can be calculated obtained from conventional experiments The deviation of 3% is less than 15% of the current method; during the fitting process, the fitting parameters of SS302, γ=1.02, β=1.034, f ^* =0.195, E ^* =109.195GPa, G _IEF ＝5.750mJ/m ² , can be calculated obtained from conventional experiments The deviation of the method is 3%, which is less than 15% of the current method, and the fracture toughness data of the material can be obtained more quickly and non-destructively than the conventional method.

Table 1 The effective elastic modulus and elastic modulus error analysis of Al6061 alloy calculated by different methods

Table 2 The effective elastic modulus and elastic modulus error analysis of SS302 alloy calculated by different methods

During the initial loading process, the material does not start to be damaged, and the calculated effective elastic modulus should be close to the value obtained by the actual conventional mechanical test. As the experiment progresses due to damage, the effective elastic modulus should be further decreased and obtained by conventional experiments. Value, as can be seen from Table 1 and Table 2, the initial state records effective elastic modulus precision in the present invention and is obviously higher than the method of Sina Amiri, and the inventive method can well predict elastic modulus change development trend, so the present invention proposes The method can reasonably determine the critical fracture state of the material, and obtain the non-damage elastic modulus and fracture toughness value of the material.

Table 3 Error analysis of fracture toughness calculated by different methods

The results of table 3 reflect that the fracture toughness calculation model proposed by the present invention can reasonably obtain the fracture toughness value under different materials, therefore, the fracture toughness test method proposed by the present invention can accurately calculate the elastic modulus and fracture toughness of the material value.

Of course, the above descriptions are not intended to limit the present invention, and the present invention is not limited to the above examples. Changes, modifications, additions or replacements made by those skilled in the art within the scope of the present invention shall also belong to the present invention. protection scope of the invention.

Claims (4)

1. a kind of method for calculating fracture toughness using indentation method, which comprises the steps of:

(1) it is based on indentation test, under constant rate of speed, obtains and is no less than whole load-displacement curves under 6 unloading conditions；

(2) fitting of first power function parameter, residual plastic after being unloaded are carried out to the part each unloading curve 30%-70% Deforming depth h_p, contact depth h_cParameter calculates the unloading rigidity at every curve unloading；

It carries out a straight line fitting again to each unloading curve again, obtains the unloading rigidity of each curve；

The unloading rigidity cut-off line match value of first curve, Article 2 start by power function fitting calculate unloading rigidity value with The value of straight line fitting compares, if the two error judges that power function fitting parameter is effective in 10%；

To effective data point P and h_pIt is fitted using following equation:

Wherein, h_pFor afterflow depth after unloading, P is loading of pressing in, C₁、C₂、C₃For fitting parameter and meet C₂*C₃< 0, and then obtain h_pWith the parameters relationship of P；

(3) pressure head process of press in is simulated using finite element, obtains radial displacement correction factor and the non-axis of Berkovich pressure head The symmetry correction factor, to obtain decrement elasticity modulus；

(4) using finite element to process simulation is uniaxially stretched, the critical hole ratio of material is obtained；

(5) by the indentation function and indentation energy to failure incidence formula that propose, fracture toughness is calculated in conjunction with critical hole ratio；

The indentation function and indentation energy to failure incidence formula are as follows:

Wherein G_IEFTo be pressed into energy to failure, F (h_p) it is h_pDepth is pushed down into function, P (h_p) it is h_pLoad under depth, A_p(h_p) it is h_pDepth Lower impression perspective plane area, calculating formula are as follows:

Wherein, γ is radial displacement correction factor, h_pFor unloading after afterflow depth,For critical residual plastic deformation Depth；

In conjunction with the P and h obtained in step (2)_pFitting formula, must be pressed into energy to failure aboutFunction are as follows:

Wherein A, B, C are respectively

Critical residual is plastically deformed depthPass throughWith ln h_pRelationship acquires, relationship are as follows:

WhereinFor effective modulus of elasticity, K, b are fitting parameter；

Effective modulus of elasticityCalculation formula are as follows:

Wherein, υ is measured material Poisson's ratio, υ_iFor pressure head material Poisson's ratio, E_rTo reduce elasticity modulus, E_iFor pressure head springform Amount；

InIn E when Critical fracture^*Taking corresponding ordinate is critical residual plastic deformation depth

It is pressed into energy to failure and strain energy density is of equal value, equivalence formula are as follows:

Wherein S_crFor critical strain energy density, υ is measured material Poisson's ratio, and SED is strain energy density；

And haveThen fracture toughness finally solves formula are as follows:

Wherein K_ICFor fracture toughness, E is measured material elasticity modulus.

2. a kind of method for calculating fracture toughness using indentation method according to claim 1, which is characterized in that step (1) Middle indentation test continuously adds the method for unloading using constant Berkovich pressure head, and pressure head is carried out with the loading speed of 50mN/s No less than 6 equal difference loads, and the unloading after having loaded each time, obtain whole load-displacement curves.

3. a kind of method for calculating fracture toughness using indentation method according to claim 1, which is characterized in that step (2) Middle power function parameter fitting formula are as follows:

P=B (h-h_p)^m

Wherein, B, m are fitting parameter, h_pFor afterflow depth after unloading, h is pressure head compression distance；

Straight Line Fitting Parameters relationship are as follows:

P=a+S₂*h

Wherein, a is fitting parameter, S₂For the unloading rigidity in Straight Line Fitting Parameters；

Unloading rigidity S in power function parameter₁It is obtained by Oliver-Pharr method, is calculated and obtained by following formula,

S₁=Bm (h_max-h_p)^m-1

Wherein, h_maxFor compression distance maximum in indentation test.

4. a kind of method for calculating fracture toughness using indentation method according to claim 3, which is characterized in that step (3) The calculation formula of middle decrement elasticity modulus are as follows:

Wherein, E_rTo reduce elasticity modulus, β is Berkovich pressure head non-axis symmetry modifying factor, and γ is radial displacement amendment system Number, A_cTo contact depth area, S is unloading rigidity, and S takes S at first unloading curve₂, in each unloading curve later Using the unloading rigidity S in power function parameter₁。