CN111855458B

CN111855458B - A method for solving the constitutive relation of porous materials based on nanoindentation theory

Info

Publication number: CN111855458B
Application number: CN202010715942.3A
Authority: CN
Inventors: 龙旭; 贾啟普; 李震
Original assignee: Northwestern Polytechnical University
Current assignee: Northwestern Polytechnical University
Priority date: 2020-07-23
Filing date: 2020-07-23
Publication date: 2022-03-01
Anticipated expiration: 2040-07-23
Also published as: CN111855458A

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 curves with large errors, The average curve fitting is performed on the remaining curves 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, which takes a long time.

Description

Porous material constitutive relation solving method based on nanoindentation theory

Technical Field

The invention relates to the technical field of electronic packaging nanometer mechanical property testing, in particular to a porous material constitutive relation solving method based on a nanometer indentation theory.

Background

The nano-indentation technology is an effective method for evaluating the mechanical properties of coating and film materials. The indentation load and displacement graph is used as an advanced micro/nano-scale mechanical testing technology and is widely applied to the research of mechanical properties of materials. The nanoindentation response is essentially related to the stress-strain curve of an elastoplastic isotropic material through nanoindentation of different indentation types, the elastic modulus, hardness and plasticity of the material can be determined through the nanoindentation curve, and generally, the existing analysis method based on finite element simulation can be divided into two categories, namely forward analysis and inversion analysis. Forward analysis refers to the prediction of the P-h curve of a material from a series of known mechanical parameters of the material without the aid of finite element simulations. Forward analysis is quite straightforward, since the properties of the material are available, the computational accuracy of finite element models can often be well verified. In contrast, inversion analysis refers to the determination of the mechanical properties of a material from a known indentation P-h curve, and is usually studied using more complex methods. It is clear that the role of the inversion analysis is greater, since in engineering practice the mechanical properties of some materials are not known at all.

In addition, the inversion analysis method can be classified into two types according to whether a dimensionless analysis theory is adopted or not. Firstly, comparing a finite element simulation result with an experimental result, adjusting parameters until a fitting error is acceptable, and finally achieving the mechanical property of the material. This reverse analysis based approach has been widely used in the early stages of nanoindentation studies. However, the numerical error may not be well controlled, and the correctness of the predicted material parameter depends on the correctness of the input material parameter to a great extent, so that the uniqueness problem often occurs, the material and the stress-strain curve are not in a one-to-one correspondence relationship, and the iteration times are more during simulation, and the time is longer. Secondly, firstly carrying out dimensionless analysis, connecting finite element results with dimensionless functions to form a series of nonlinear fitting equations, and finally determining the mechanical constitutive relation of the material by calculating the dimensionless equations.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides a porous material constitutive relation solving method based on a nanoindentation theory, and solves the problems that uniqueness often occurs in the prior art, namely, material properties and stress-strain curves are not in one-to-one correspondence, iteration times are more during simulation, and time is long.

In order to achieve the purpose, the invention is realized by the following technical scheme:

the invention discloses a porous material constitutive relation solving method based on a nanoindentation theory, which comprises the following steps of:

(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 dichotomy until a displacement-load curve obtained by finite element simulation and the step (1)) The obtained experimental curves are completely consistent, so that the characteristic stress is determined;

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

(4) characteristic strain epsilon_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.

Preferably, the nanoindenter is a Berkovich indenter with the angular lines between the edge and the center of the indenter being 65.3 ° and 77.05 °.

Preferably, the formula of 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；

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

Preferably, the first and second liquid crystal materials are,

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.

Preferably, the yield stress σ_yThe formula of (1) is as follows:

wherein R is a hardening coefficient.

Preferably, the formula of the hardening coefficient R is:

the invention has the following beneficial effects:

1. compared with forward analysis, the method for solving the constitutive relation of the nano porous material based on the relevant theory of nano mechanics is more suitable for engineering and measurement of unknown material properties, and is simpler. Moreover, the displacement-load curve obtained in the invention is completely superposed with the curve known by experiments, so that the obtained unknown material parameters are completely the same and the accuracy is good.

2. The invention faces to the packaging material, and the uniqueness problem (the uniqueness problem means that the material properties of various materials may correspond to the same stress-strain relation) does not occur; moreover, the iteration times in the simulation are less, and the used time is less; compared with the existing method, the fitting condition is better, and the material property and the stress-strain curve are in one-to-one correspondence. Also, with a known modulus of elasticity of the material, the desired material properties can be obtained by one indentation.

Drawings

FIG. 1 is a schematic structural view of the present invention;

FIG. 2 is an indentation response curve of a Sn-Bi alloy;

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

FIG. 4 is a characteristic stress determination diagram in an example;

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

FIG. 6 is a graph of hardening index determination;

FIG. 7 is a comparison of the experimental curve Test 002 with a P-h curve with a characteristic strain of 0.027 output;

FIG. 8 is a determination of yield stress;

FIG. 9 is a constitutive curve derived from inverse extrapolation under different parameters;

FIG. 10 is a load-displacement curve obtained by finite element simulation;

fig. 11 is a stress-strain curve of a material.

Detailed Description

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.

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

The invention discloses a method for solving the constitutive relation of a porous material based on a nanoindentation theory, which has the basic principle that according to abaqus software, characteristic stress and characteristic strain are calculated according to a dichotomy in sequence, then yield stress is calculated, and finally the constitutive relation of the material can be reversely deduced.

The method specifically comprises the following steps:

(1) referring to fig. 1, a nanometer indenter is used to perform multiple indents on a porous material substrate to obtain multiple (or a series of) displacement-load curves, remove curves with large errors or significant deviations, perform average curve fitting on the remaining curves, specifically perform average curve fitting in origin software to obtain an average curve, use the average curve as an experimental curve, and use the average elastic modulus of the average curve as an experimental elastic modulus E; wherein the nanometer pressure head is a Berkovich pressure head, and the angular lines between the edge and the center of the pressure head are 65.3 degrees and 77.05 degrees.

(2) Characteristic stress sigma_rDetermination of (1): assuming that the porous material matrix is an elastoplastic material and the initial yield stress is a characteristic stress, the magnitude of the characteristic strain can be ignored.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; it should be understood that: the approximate range of the characteristic stress of various materials is clear, such as 0-1000MPa, the characteristic stress is assumed to be 500MPa, the materials are put into finite element simulation software, a displacement-load curve is derived, the curve is compared with a known experimental curve, if a difference exists, a number is selected from 0-500MPa or 500MPa-1000MPa, the dichotomy is repeated, the characteristic stress is finally determined, and the characteristic strain is also determined.

(3) Determining a hardening index n according to a dimensionless function, the formula being 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；

h_rFor the residual depth of the displacement-load curve derived in step (2), the residual depth h can be determined from the known experimental curve_r. The residual depth is the displacement after complete unloading in the displacement-load curve, as shown in FIG. 10, where W is the value in FIG. 10_eIs elastic work, W_pIs plastic work, W_totalIs the total work in indentation, and has a value equal to W_eAnd W_pAnd (4) summing. The displacement corresponding to the maximum load is the maximum displacement h_mThe displacement at full unload is the residual displacement h_rFurther, the slope immediately upon entering unloading (at maximum load) is the stiffness S.

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.

Equation (1) above is suitable for analysis by the Berkovich indenter and the angular lines between the edge and the center of the indenter are 65.3 ° and 77.05 °.

(4) Characteristic strain epsilon_rDetermination of (1): determining the characteristic strain in a process similar to the determination of the characteristic stress, continuously performing finite element simulation by adopting a dichotomy by providing a possible range of the characteristic strain, and adjusting the value of the characteristic strain until a displacement-load curve simulated by the finite element is completely consistent with an experimental curve obtained in the step (1), so as to determine the characteristic strain; it should be noted that: unlike ideal elastoplasticity, the constitutive properties in this step are estimated based on a power law function model, since the power law function can be used to describe the plastic behavior of metals and their alloys.

(5) Determination of the yield stress sigma_yThe formula is as follows:

wherein R is a hardening coefficient, and the formula is as follows:

it is to be understood that: the total strain comprises two parts epsilon_pAnd ε_y，ε_pThe non-linear portion representing the total strain is shown with reference to FIG. 11, and_yvery small relative to epsilon_pIt can be ignored; in FIG. 11, at σ ≦ σ_yWhile belonging to the elastic phase, sigma > sigma_yFollowed by an elastoplastic phase, yield stress sigma_yCorresponding strain is ε_yCharacteristic stress σ_rCorresponding strain is characteristic strain epsilon_r，ε_pRepresenting the nonlinear part of the total strain.

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

Examples

Referring to fig. 2 and 3, according to the response curve of the Sn — Bi alloy indentation method Test and the elastic modulus curve with depth provided by the Guilin electronics science and technology university, the present invention uses Test001 and Test 002 in the experimental curve of fig. 2 to perform an inversion calculation, and the elastic modulus is determined by averaging in fig. 3.

On the basis, the detailed inversion analysis is carried out by taking the Test 002 indentation result as an example, and the specific steps are as follows:

(1) first determining the characteristic stress sigma_rAs shown in fig. 4, assuming that the porous material matrix is ideal elastic-plastic, two extreme characteristic stresses are given, and finite element simulation is continuously performed by adopting a bisection method until a displacement-load curve obtained by finite element simulation completely coincides with an experimental curve, so as to determine the characteristic stress σ_rThe results are shown in fig. 5 at 80 MPa. In FIG. 4,. epsilon_yIs the strain corresponding to the stress at which the stress is reduced to yield stress. (sigma ═ R epsilonⁿIs the graph depicted in fig. 4, σ_r＝Rε_r ⁿWhen the stress is σ_rTime (sigma)_rCorresponding strain is ε_r) In the same way, σ ═ E ∈ is the first half of the linear phase, σ ═ E ∈ is_y＝Eε_yIs epsilon as_yA point of time relationship).

(2) The hardening index n was determined from the dimensionless function, and the result was solved according to the above equation (1) as shown in fig. 6, and the hardening index n was 0.305.

(3) Determination of the characteristic Strain ε_rMethod and determination of characteristic stress sigma_rIn the same way, the characteristic strain ε_r0.027, the results are shown in fig. 7.

(4) Determination of the yield stress sigma_yAfter determining the value of the characteristic strain, the yield stress can be estimated to be 20.5MPa according to equation (2), and the result is shown in fig. 8.

The right hand side of formula (2) in FIG. 8 refers to

The intersection point of the curve of the functional expression and 80MPa is the solution of yield stress.

(5) The hardening index n and the yield stress sigma calculated according to the steps_yAnd the known elastic modulus E, drawing a reverse constitutive curve, and obtaining the relationship between the yield stress and the elastic modulus of the material and the stress-strain relationship of the material according to the constitutive curve.

Comparative example

By using Test001 in the experimental curve of fig. 2 as a comparison experimental curve, n is 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 Test 002 is mainly the difference between the residual indentation depth and the maximum indentation depth, so that the value of parameter n is different when the constitutive model is inverted and calculated, referring to fig. 9, fig. 9 is an constitutive curve reversely deduced by using Test 002 and Test001 as the comparison experimental curve.

The above-described embodiments are merely illustrative of the preferred embodiments of the present invention, and do not limit the scope of the present invention, and various modifications and improvements of the technical solutions of the present invention can be made by those skilled in the art without departing from the spirit of the present invention, and the technical solutions of the present invention are within the scope of the present invention defined by the claims.

Claims

1. a method for solving the constitutive relation of porous materials based on nanoindentation theory, is characterized in that: comprise 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, the average elastic modulus of the average curve is taken as the experimental elastic modulus E; 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°;

(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 exponent n according to the dimensionless function, and 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);

hm is the maximum displacement corresponding to the maximum load;

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;

(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.

2. a kind of method for solving the constitutive relation of porous materials based on nanoindentation theory according to claim 1, is characterized in that: the formula of yield stress _σy is as follows:

where R is the hardening coefficient; the total strain includes two parts ε _p and ε _y , and ε _p represents the nonlinear part of the total strain.

3. a kind of method for solving the constitutive relation of porous materials based on nanoindentation theory according to claim 2, is characterized in that: the formula of hardening coefficient R is:

In the formula, ε _y represents the strain corresponding to the yield stress σ _y .