CN111855458B - Porous material constitutive relation solving method based on nanoindentation theory - Google Patents
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Abstract
The invention relates to electronic packaging nanomechanical propertyThe technical field of energy testing, in particular to a porous material constitutive relation solving method based on a nanoindentation theory. The specific technical scheme is as follows: a porous material constitutive relation solving method based on a nano indentation theory is characterized in that a nano pressure head is used for carrying out multiple indentation on a porous material substrate to obtain multiple displacement-load curves, curves with large errors are removed, the other curves are subjected to average curve fitting to obtain an average curve, and the average elastic modulus of the average curve is taken as an experimental elastic modulus E; then determining the characteristic stress sigmarDetermining a hardening index n according to a dimensionless function; re-determination of the characteristic strain epsilonrAnd determining the yield stress sigmay(ii) a Finally according to the hardening index n and the yield stress sigmayAnd the elastic modulus E to obtain the constitutive curve. The invention solves the problems that the material attribute and the stress-strain curve are not in one-to-one correspondence in the prior art, and the iteration times are more and the time is longer during simulation.
Description
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 sigmarDetermination 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 epsilonrDetermination 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 sigmay；
(6) The hardening index n and the yield stress sigma are calculated according to the stepsyAnd 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；
hrIs 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 σyThe 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 sigmarDetermination 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；
hrFor the residual depth of the displacement-load curve derived in step (2), the residual depth h can be determined from the known experimental curver. 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. 10eIs elastic work, WpIs plastic work, WtotalIs the total work in indentation, and has a value equal to WeAnd WpAnd (4) summing. The displacement corresponding to the maximum load is the maximum displacement hmThe displacement at full unload is the residual displacement hrFurther, 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 epsilonrDetermination 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 sigmayThe 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 epsilonpAnd εy，εpThe non-linear portion representing the total strain is shown with reference to FIG. 11, andyvery small relative to epsilonpIt can be ignored; in FIG. 11, at σ ≦ σyWhile belonging to the elastic phase, sigma > sigmayFollowed by an elastoplastic phase, yield stress sigmayCorresponding strain is εyCharacteristic stress σrCorresponding strain is characteristic strain epsilonr，εpRepresenting the nonlinear part of the total strain.
(6) The hardening index n and the yield stress are calculated according to the stepsσyAnd 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 sigmarAs 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 σrThe results are shown in fig. 5 at 80 MPa. In FIG. 4,. epsilonyIs the strain corresponding to the stress at which the stress is reduced to yield stress. (sigma ═ R epsilonnIs the graph depicted in fig. 4, σr＝Rεr nWhen the stress is σrTime (sigma)rCorresponding strain is εr) In the same way, σ ═ E ∈ is the first half of the linear phase, σ ═ E ∈ isy＝EεyIs epsilon asyA 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 εrMethod and determination of characteristic stress sigmarIn the same way, the characteristic strain εr0.027, the results are shown in fig. 7.
(4) Determination of the yield stress sigmayAfter 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 stepsyAnd 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 (3)
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; 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 sigmarDetermination of (1): assuming two extreme characteristic stresses, finite element simulation is continuously carried out by adopting a dichotomy,until the displacement-load curve obtained by finite element simulation is completely consistent with the experimental curve obtained in the step (1), so as to determine the characteristic stress;
(3) determining a hardening index n according to a dimensionless function, the formula of the hardening index n 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；
hr is the residual depth of the displacement-load curve in step (2);
hm is the maximum displacement corresponding to the maximum load;
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 pressure head;
(4) characteristic strain epsilonrDetermination 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 sigmay；
(6) The hardening index n and the yield stress sigma are calculated according to the stepsyAnd the elastic modulus E to obtain the constitutive curve.
2. The method for solving constitutive relation of porous material based on nanoindentation theory as claimed in claim 1, wherein the method comprisesIs characterized in that: yield stress sigmayThe formula of (1) is as follows:
wherein R is a hardening coefficient; the total strain comprises two parts epsilonpAnd εy，εpRepresenting the nonlinear part of the total strain.
3. The method for solving the constitutive relation of the porous material based on the nanoindentation theory as recited in claim 2, wherein: the hardening coefficient R is given by:
in the formula, epsilonyRepresenting the yield stress sigmayThe corresponding strain.
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