CN111855458A - A method for solving the constitutive relation of porous materials based on nanoindentation theory - Google Patents

Abstract

The invention relates to the technical field of nanomechanical performance testing of electronic packaging, in particular to a method for solving the constitutive relationship of porous materials based on nanoindentation theory. The specific technical scheme is as follows: a method for solving the constitutive relationship of porous materials based on the nanoindentation theory, using a nanoindenter to perform multiple indentations on a porous material substrate to obtain multiple displacement-load curves, and remove the curves with large errors, The remaining curves are averaged to obtain the average curve, and the average elastic modulus of the average curve is taken as the experimental elastic modulus E; then the characteristic stress σ _r is determined, and the hardening exponent n is determined according to the dimensionless function; then the characteristic strain is determined ε _r and determine the yield stress σ _y ; finally the constitutive curve is obtained according to the hardening exponent n, the yield stress σ _y and the elastic modulus E. The invention solves the problems in the prior art that the material properties and the stress-strain curve are not in a one-to-one correspondence, and the number of iterations during simulation is relatively large and the time spent is relatively long.

Description

A method for solving the constitutive relation of porous materials based on nanoindentation theory

technical field

The invention relates to the technical field of nanomechanical performance testing of electronic packaging, in particular to a method for solving the constitutive relationship of porous materials based on nanoindentation theory.

Background technique

Nanoindentation technology is an effective method to evaluate the mechanical properties of coatings and thin films. As an advanced micro/nano scale mechanical testing technique, indentation load and displacement maps are widely used in the study of mechanical properties of materials. The nanoindentation response is essentially related to the stress-strain curves of elastoplastic isotropic materials through nanoindentation of different indentation types. The elastic modulus, hardness and plasticity of materials can be determined from the indentation curves, usually existing The analysis methods based on finite element simulation can be divided into two categories: forward analysis and inversion analysis. Forward analysis refers to predicting the P-h curve of a material based on a series of known mechanical parameters of the material without the aid of finite element simulation. Forward analysis is fairly straightforward because the properties of the material are available and the computational accuracy of the finite element model can often be well verified. In contrast, inversion analysis refers to the determination of the mechanical properties of a material based on known indentation P-h curves, often with more sophisticated methods. But obviously, inversion analysis is more useful, because in engineering practice, the mechanical properties of some materials are always unknown.

In addition, the inversion analysis methods can be classified into two categories according to whether the dimensionless analysis theory is adopted or not. One is to first compare the finite element simulation results with the experimental results, adjust the parameters until the fitting error is acceptable, and finally achieve the mechanical properties of the material. This inverse analysis-based approach was widely used in the early stages of nanoindentation research. However, its numerical error may not be well controlled, and the correctness of the predicted material parameters depends to a large extent on the correctness of the input material parameters, so the problem of uniqueness often occurs, and the relationship between materials and stress-strain curves is not one-to-one. , and the number of iterations during the simulation is large, which takes a long time. The second is to first carry out a dimensionless analysis, link the finite element results with the dimensionless functions, and form a series of nonlinear fitting equations. By calculating these dimensionless equations, the mechanical constitutive relationship of the material is finally determined.

SUMMARY OF THE INVENTION

In view of the deficiencies of the prior art, the present invention provides a method for solving the constitutive relation of porous materials based on the nanoindentation theory, which solves the problem of uniqueness that often occurs in the prior art, that is, the material properties and the stress-strain curve are not one-to-one Corresponding relationship, as well as the problem that the number of iterations in the simulation is relatively large and it takes a long time.

To achieve the above purpose, the present invention is achieved through the following technical solutions:

The invention discloses a method for solving the constitutive relation of porous materials based on nano-indentation theory, comprising the following steps:

(1) Use a nano-indenter to perform multiple indentations on the porous base material to obtain multiple displacement-load curves, remove the curves with large errors, and perform average curve fitting on the remaining curves to obtain the average curve. The average curve is used as For the experimental curve, take the average elastic modulus of the average curve as the experimental elastic modulus E;

(2) Determination of the characteristic stress σ _r : Assuming two extreme characteristic stresses, the finite element simulation is carried out continuously by the dichotomy method until the displacement-load curve obtained by the finite element simulation is completely consistent with the experimental curve obtained in step (1), so as to determine characteristic stress;

(3) Determine the hardening index n according to the dimensionless function;

(4) Determination of characteristic strain _εr : Assuming the range of characteristic strain, the finite element simulation is carried out continuously by the dichotomy method until the displacement-load curve of the finite element simulation is completely consistent with the experimental curve obtained in step (1), so as to determine the characteristic strain ;

(5) Determine the yield stress σ _y ;

(6) According to the hardening exponent n, the yield stress σ _y and the elastic modulus E calculated by the above steps, the constitutive curve is obtained.

Preferably, the nano-indenter is a Berkovich indenter, and the angle between the edge and the center of the indenter is 65.3° and 77.05°.

Preferably, the formula of the hardening exponent n is as follows:

In the formula, A=0.010100×n ² +0.0017639×n-0.0040837,

B=0.14386×n ² +0.018153×n-0.088198,

C=0.59505×n ² +0.03407×n-0.65417,

D=0.58180×n ² -0.088460×n-0.67290;

hr is the residual depth of the displacement-load curve in step (2).

preferably,

where Ei is the Young's modulus of the nanoindenter;

V is the Poisson's ratio of the matrix material;

Vi is the Poisson's ratio of the nanoindenter.

Preferably, the formula of yield stress σ _y is as follows:

where R is the hardening coefficient.

Preferably, the formula for the hardening coefficient R is:

The present invention has the following beneficial effects:

1. Compared with forward analysis, the method for solving the constitutive relation of nanoporous materials based on the related theory of nanomechanics is more suitable for engineering, and is more suitable for the determination of unknown material properties, and the method is simpler. Moreover, the displacement-load curve obtained in the present invention completely coincides with the known experimental curve, so that the obtained unknown material parameters are exactly the same, and the accuracy is good.

2. The present invention is oriented to packaging materials, and there is no uniqueness problem (the uniqueness problem means that the material properties of multiple materials may correspond to the same stress-strain relationship); moreover, the number of iterations in the simulation is small, and the time used is relatively short. Compared with the existing method, the fitting situation is better, and there is a one-to-one correspondence between material properties and stress-strain curves. Moreover, when the elastic modulus of the material is known, the desired material properties can be obtained by one indentation.

Description of drawings

Fig. 1 is the structural representation of the present invention;

Fig. 2 is the indentation response curve of Sn-Bi alloy;

Fig. 3 is the elastic modulus curve of Sn-Bi alloy indentation as a function of depth;

Fig. 4 is the characteristic stress determination diagram in the embodiment;

Figure 5 is the comparison between the experimental curve Test 002 and the P-h curve with a characteristic stress of 80MPa output;

Figure 6 is a diagram for determining the hardening index;

Figure 7 is the comparison between the experimental curve Test 002 and the P-h curve output with a characteristic strain of 0.027;

Figure 8 is the determination of yield stress;

Fig. 9 Constitutive curves deduced under different parameters;

Figure 10 is the load-displacement curve obtained by finite element simulation;

Figure 11 shows the stress-strain curve of the material.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

Unless otherwise specified, the technical means used in the implementation examples are conventional means well known to those skilled in the art.

The invention discloses a method for solving the constitutive relation of porous materials based on nano-indentation theory. The constitutive relationship of the materials.

Specifically include the following steps:

(1) Referring to Figure 1, use a nano-indenter to perform multiple indentations on the porous material substrate to obtain multiple (or a series) displacement-load curves, remove the curves with large errors or obvious deviations, and put the rest The average curve fitting is performed on the curve of the original, specifically, the average curve fitting is performed in the origin software to obtain the average curve, the average curve is used as the experimental curve, and the average elastic modulus of the average curve is taken as the experimental elastic modulus E; The head is a Berkovich indenter, and the angles between the edge and the center of the indenter are 65.3° and 77.05°.

(2) Determination of the characteristic stress σ _r : Assuming that the matrix of the porous material is an elastic-plastic material and the initial yield stress is the characteristic stress, the magnitude of the characteristic strain can be ignored. Assuming two extreme characteristic stresses, the finite element simulation is carried out continuously by the dichotomy method until the displacement-load curve obtained by the finite element simulation is completely consistent with the experimental curve obtained in step (1), so as to determine the characteristic stress; it should be understood that: various The approximate range of the characteristic stress of the material is clear, such as 0-1000MPa, first assume that the characteristic stress is 500MPa, put it into the finite element simulation software, derive the displacement-load curve, and compare the curve with the known experimental curve, such as If there is a gap, select a number between 0-500MPa or 500MPa-1000MPa, repeat the dichotomy method, and finally determine the characteristic stress, and the same is true for the determination of the characteristic strain.

(3) Determine the hardening index n according to the dimensionless function, the formula is as follows:

In the formula, A=0.010100×n ² +0.0017639×n-0.0040837,

B=0.14386×n ² +0.018153×n-0.088198,

C=0.59505×n ² +0.03407×n-0.65417,

D=0.58180×n ² -0.088460×n-0.67290;

hr is the residual depth of the displacement-load curve derived in step (2). According to the known experimental curve, the residual depth hr can be measured. The residual depth is the displacement after complete unloading in the displacement-load curve, as shown in Figure 10, where We _e is the elastic work, W _p is the plastic work, and W _total is the total work in the indentation, and its value is equal to W The sum of _e and W _p . The displacement corresponding to the maximum load is the maximum displacement h _m , the displacement at the time of complete unloading is the residual displacement _hr , and the slope just before unloading (at the maximum load) is the stiffness S.

where Ei is the Young's modulus of the nanoindenter;

V is the Poisson's ratio of the matrix material;

Vi is the Poisson's ratio of the nanoindenter.

The above formula (1) applies to the analysis of the indenter by Berkovich, and the angles between the edge and the center of the indenter are 65.3° and 77.05°.

(4) Determination of characteristic strain ε _r : The determination process of characteristic strain is similar to the determination of characteristic stress. By providing the possible range of characteristic strain, the finite element simulation is carried out continuously by the bisection method, and the value of characteristic strain is adjusted until the finite element simulation is completed. The displacement-load curve is completely consistent with the experimental curve obtained in step (1), so as to determine the characteristic strain; it should be noted that, unlike the ideal elastic-plastic, the power-law function can be used to describe the plastic behavior of metals and their alloys, so this In-step constitutive properties are estimated based on a power-law function model.

(5) Determine the yield stress σ _y , the formula is as follows:

In the formula, R is the hardening coefficient, and the formula is:

It should be understood that the total strain includes two parts ε _p and ε _y , and ε _p represents the nonlinear part of the total strain, as shown in Fig. 11, and ε _y is very small and can be ignored relative to ε _p ; Fig. 11 Among them, when _σ≤σy belongs to the elastic stage, after σ> _{σy is the elastic-plastic stage, the strain corresponding to the yield stress σy is ε y , the} strain corresponding to the characteristic stress σ _r is the characteristic strain ε _r , and ε _p represents The nonlinear part of the total strain.

(6) According to the hardening exponent n, the yield stress σ _y and the elastic modulus E calculated by the above steps, the constitutive curve is obtained.

Example

Referring to Fig. 2 and Fig. 3, according to the response curve of Sn-Bi alloy indentation method test and the elastic modulus curve that changes with depth provided by Guilin University of Electronic Science and Technology, the present invention adopts Test 001 and Test 002 in the experimental curve of Fig. 2. An inversion calculation was performed, and the elastic modulus was determined by averaging in Figure 3.

On the above basis, the present invention takes the test 002 indentation result as an example to carry out detailed inversion analysis, and the specific steps are as follows:

(1) First determine the characteristic stress σ _r , as shown in Fig. 4, assuming that the porous material matrix is ideal elastic-plastic, given two extreme characteristic stresses, the finite element simulation is carried out continuously by the dichotomy method, until the displacement obtained by the finite element simulation is - The load curve is completely consistent with the experimental curve, so that the characteristic stress σ _r = 80MPa is determined, and the results are shown in Fig. 5 . In Fig. 4, ε _y is the corresponding strain when the stress goes to the yield stress. (σ=Rε ⁿ is the graph described in Figure 4, σ _r =Rε _r ⁿ is the relational expression of a point when the stress is σ _r (the strain corresponding to σ _r is ε _r ). Similarly, σ=Eε is In the first half of the linear stage, σ _y =Eε _y is a relational expression for a point when ε is ε _y ).

(2) The hardening exponent n is determined according to the dimensionless function, and the result is shown in Fig. 6 . According to the above formula (1), the hardening exponent n=0.305.

(3) Determining the characteristic strain ε _r , the method is the same as the method for determining the characteristic stress σ _r , and the determined characteristic strain ε _r = 0.027, and the result is shown in Fig. 7 .

(4) Determine the yield stress σ _y , after determining the value of the characteristic strain, the yield stress can be calculated to be 20.5 MPa according to formula (2), and the result is shown in Fig. 8 .

这个函数表达式，该表达式的曲线与80MPa的交点就是屈服应力的解。The right side of equation (2) in Figure 8 refers to

This functional expression, the intersection of the curve of this expression and 80MPa is the solution for the yield stress.

(5) According to the hardening exponent n, the yield stress σ _y and the known elastic modulus E calculated by the above steps, the inverse constitutive curve can be drawn, and then the yield stress and elastic modulus of the material can be obtained according to the constitutive curve. and the stress-strain relationship of the material.

Comparative ratio

Using Test 001 in the experimental curve of Fig. 2 as the control experimental curve, using the method of the above-mentioned embodiment to obtain n=0.254, the characteristic stress is 80MPa, the characteristic strain is 0.029, the yield stress is 26.8MPa, and the curve difference between Test001 and Test002 is mainly It is because the residual indentation depth and the maximum indentation depth are different, which leads to different values of the parameter n during the inversion calculation of the constitutive model. Referring to Figure 9, Figure 9 is the reverse inference when Test 002 and Test 001 are used as the control test curves. Constitutive Curve.

The above-mentioned embodiments are only to describe the preferred mode of the present invention, but not to limit the scope of the present invention. Without departing from the design spirit of the present invention, those of ordinary skill in the art can Variations and improvements should fall within the protection scope determined by the claims of the present invention.

Claims (6)

1. A porous material constitutive relation solving method based on a nanoindentation theory is characterized by comprising the following steps: the method comprises the following steps:

(1) carrying out multiple indentation on the porous matrix material by using a nanometer pressure head to obtain a plurality of displacement-load curves, removing curves with large errors, carrying out average curve fitting on the other curves to obtain an average curve, taking the average curve as an experimental curve, and taking the average elastic modulus of the average curve as an experimental elastic modulus E;

(2) characteristic stress sigma_rDetermination of (1): assuming two extreme characteristic stresses, continuously performing finite element simulation by adopting a bisection method until a displacement-load curve obtained by finite element simulation is completely consistent with an experimental curve obtained in the step (1), and determining the characteristic stresses;

(3) determining a hardening index n according to a dimensionless function;

(4) characteristic strain_rDetermination of (1): assuming the range of the characteristic strain, continuously performing finite element simulation by adopting a bisection method until a displacement-load curve simulated by the finite element is completely consistent with the experimental curve obtained in the step (1), and determining the characteristic strain;

(5) determination of the yield stress sigma_y；

(6) The hardening index n and the yield stress sigma are calculated according to the steps_yAnd the elastic modulus E to obtain the constitutive curve.

2. The method for solving the constitutive relation of the porous material based on the nanoindentation theory as recited in claim 1, wherein: the nano indenter was a Berkovich indenter with angular lines between the edge and the center of the indenter of 65.3 ° and 77.05 °.

3. The method for solving the constitutive relation of the porous material based on the nanoindentation theory as recited in claim 1, wherein: the formula for the hardening index n is as follows:

wherein A is 0.010100 Xn²+0.0017639×n-0.0040837，

B＝0.14386×n²+0.018153×n-0.088198，

C＝0.59505×n²+0.03407×n-0.65417，

D＝0.58180×n²-0.088460×n-0.67290；

hr is the residual depth of the displacement-load curve in step (2).

4. The method for solving the constitutive relation of the porous material based on the nanoindentation theory as recited in claim 3, wherein:

wherein Ei is the Young modulus of the nanometer indenter;

v is the Poisson's ratio of the base material;

vi is the Poisson's ratio of the nanometer indenter.

5. The method for solving the constitutive relation of the porous material based on the nanoindentation theory as recited in claim 1, wherein: yield stress sigma_yThe formula of (1) is as follows:

wherein R is a hardening coefficient.

6. The method for solving the constitutive relation of the porous material based on the nanoindentation theory as recited in claim 5, wherein: the hardening coefficient R is given by:

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