CN114397210B - Bayesian method for measuring anisotropic plasticity of material based on spherical indentation - Google Patents
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Abstract
The invention discloses a Bayesian method for measuring anisotropic plasticity of a material based on spherical indentation, which comprises the steps of carrying out a spherical indentation experiment on a metal material to be tested, and obtaining an indentation snapshot of the spherical indentation experiment; developing finite element simulation of spherical indentations according to Holloon hardening rule, and establishing average and difference indentation snapshot likelihood functions; based on Bayes theory, establishing a relation between an indentation contour snapshot in the spherical indentation and the constitutive parameters of the material, acquiring a posterior probability density function of the constitutive parameters of the measured material based on a transfer Markov chain method, and solving and giving out anisotropic plastic performance and credible domain distribution of the reversely-deduced material. The invention is based on a statistical Bayesian inference algorithm and a spherical indentation test, considers possible uncertainty in the numerical value optimization process, can give out a parameter identification result in the form of probability distribution, and has high reliability of parameter identification.
Description
Technical Field
The invention belongs to the technical field of material plastic mechanical property testing methods, and particularly relates to a method for reversely deducing tested metal anisotropic property parameters by adopting spherical indentation morphology information in a pressing experiment.
Background
The anisotropy has obvious influence on the structural strength and deformation behavior of the material, and the measurement of the anisotropic property has been a long-standing problem for many years. In the conventional method, the anisotropic properties are measured by performing a plurality of uniaxial stretching/compression tests on a test piece in orthogonal directions. However, this experimental method is destructive, time consuming, and not suitable for limited volume samples, such as coatings and articular cartilage. Furthermore, uniaxial experiments are generally not suitable for in situ measurement of changes in mechanical properties of materials at intermediate forming stages.
In recent years, instrument indentation experiments have received attention because of their flexible and convenient test procedures. The method provides a novel method for inducing local contact deformation of materials. The mechanical measuring process is nondestructive and is very suitable for on-site mechanical measuring areas, especially small samples. During the last decades, researchers have done a lot of work with indentations to measure various mechanical properties of materials, such as isotropic plasticity, residual stress, fracture toughness, etc. Because of the above advantages, research into measuring material anisotropy by indentation is of great importance.
The earliest ideas of studying material anisotropy using indentation can be traced back to the work of Vlassak and Nix (j.j. Vlassak, w.d.nix, measuring the elastic properties of anisotropic materials by means of indentation experiments, j.mech. Phys.solids 42 (1994) 1223-1245.). He found that the multiaxial stress state under the indenter may lead to an average anisotropic spring effect. However, vlassak and Nix did not further study the plastic anisotropic behavior of the material under indentation. It should be noted that non-uniform deformation under indentation is essentially different from deformation in uniaxial experiments, which makes it very difficult to extract mechanical information of the material directly from the indentation measurements. To obtain the anisotropic properties of the material from indentation experiments, researchers have used inverse methods of finite element modeling and numerical optimization.
Bocciarelli et al (M.Bocciarelli, G.Bolzon, G.Maier, parameter identification in anisotropic elastoplasticity by indentation and imprint mapping, mech. Mater.37 (2005) 855-868.) use parameters of an anisotropic yield model calibrated with conical indentations. The indentation load-displacement curve and the residual indentation map are weighted in a well-defined objective function, and then the material parameters are solved numerically by trust domain optimization. Nakamura and Gu (T.Nakamura, Y.Gu, identification of elastic-plastic anisotropic parameters using instrumented indentation and inverse analysis, mech. Mater.39 (2007) 340-356.) established a method of measuring the anisotropic properties of thermal spray coatings that considered the indentation load curves of spherical indenters and bosch indenters. Furthermore, they found that the load curves obtained from the two rams show opposite trends as the depth values increased. Yonezu et al (A.Yonezu, K.Yoneda, H.Hirakata, M.Sakihara, K.Minoshima, A simple method to evaluate anisotropic plastic properties based on dimensionless function of single spherical indentation-application to SiC whisker-reinforced aluminum alloy, mater. Sci. Eng., A527 (2010) 7646-7657.) propose a framework for enhancing the plastic anisotropy of aluminum alloys using spherical indentation estimation whiskers. They established a size function relating the material anisotropy parameters to indentation load and depth of contact values. The anisotropic parameters are then numerically determined by matching the indentation test response to a response predicted using a dimensionless model.
In previous work, numerical algorithms for measuring material properties were essentially based on minimizing experimental and simulation differences, using finite element simulation and parameter updates. However, these works treat model parameters as deterministic variables, giving only a single point estimate of material parameters. On the one hand, uncertainties, such as non-uniformity of the microstructure and inaccuracy of the numerical model, are unavoidable when measuring material properties using indentations. Therefore, these methods are particularly sensitive to experimental uncertainty and are prone to falling into local minima during numerical iterations. On the other hand, when the single-point estimation result of the deterministic optimization method involves the selection of parameters in mathematical modeling and material design, the model information of the inversion problem is very limited.
Disclosure of Invention
In order to solve the defects in the prior art, the invention aims to provide a method for acquiring the anisotropic parameters of the metal material by adopting the indentation profile morphology in an indentation experiment and combining the statistical Bayesian inverse calculation.
The invention is realized by the following technical scheme.
The invention provides a Bayesian inverse calculation method for measuring anisotropic plastic properties of a material based on a spherical indentation experiment, which comprises the following steps:
(1) Carrying out a spherical indentation experiment on a metal material to be tested, obtaining the profile morphology of the indentation remained on the surface of the test piece through a laser confocal microscopic test, and obtaining an indentation snapshot S of the spherical indentation experiment exp And vertical displacement values in the lateral and longitudinal directions are defined as imprint snapshots S, respectively T And S is L The method comprises the steps of carrying out a first treatment on the surface of the Creating an imprint snapshot S T And S is L Average magnitude and difference imprint snapshot of (a)
And Δs;
(2) According to Holloon hardening rule, developing spherical indentation finite element simulation, and generating indentation snapshot database O according to anisotropic parameters in design space s ；
(3) Assuming that the prior information of the model parameters is uniformly distributed, the prior information function is constant, and an average and difference indentation snapshot is established
And a likelihood function of Δs;
(4) Combining the mean and difference indentation snapshots given in the step (3) according to the Bayesian inference theory
And establishing an orthogonal subspace of the indentation snapshot based on an orthogonal decomposition algorithm by taking a likelihood function of delta S and a priori information function as constants; based on a cubic polynomial basis function, establishing a relation between an indentation snapshot subspace coordinate coefficient and a material anisotropy parameter c;
(5) Replacing the imprinting snapshot in the log-likelihood function with the subspace coordinate coefficient, and directly performing model evaluation according to the given material anisotropy parameter in the c; the log-likelihood function can replace the imprint snapshot with subspace coordinate coefficients;
(6) Sampling the posterior probability density function based on a transfer Markov chain algorithm to obtain posterior probability distribution of the anisotropic plastic performance parameter of the tested material.
Due to the adoption of the technical scheme, the invention has the following beneficial effects:
(1) According to the invention, the anisotropy of the metal material is obtained by adopting the indentation experiment, so that the preparation and processing processes of the test piece, which are complicated in the original anisotropy experiment, are avoided, and the defects of destructiveness, serious material waste and the like of the conventional anisotropy experiment are overcome. On the other hand, the invention can be applied to the detection of the local mechanical property of the metal material and the detection of the evolution of the mechanical property of the material in the processing process, and is also further applied to the measurement of the mechanical property of the in-service part.
(2) According to the invention, the profile morphology of the indentation left after unloading is used as effective material response information in parameter identification, so that accurate measurement of a load-displacement curve in the whole indentation loading and unloading process is avoided, and the deformation history of the material in indentation loading is not required to be known additionally. The unloading residual indentation profile can be obtained through confocal microscopic test, the experimental complexity is lower, and the experimental process is simpler, more convenient and more flexible.
(3) Based on a statistical Bayesian inversion theory, model parameters are processed according to random variables in modeling of a parameter identification method. The method has the advantages that uncertainty in experiments and numerical models can be considered, the solving result of the inverse problem can be given out in the form of probability distribution, model information of the parameter identification inverse problem is reflected to the greatest extent, the problem that the solution of the inverse problem falls into a local minimum value is avoided, influences of data disturbance, numerical oscillation and the like can be ignored, and the reliability of parameter identification is improved.
The invention adopts the indentation experiment to obtain the anisotropic plastic property of the metal material, effectively avoids the complicated cutting processing process of the test piece, and has the potential of being applied to the detection of the local property of the metal material and the performance measurement of the in-service part. On the other hand, the method is based on a statistical Bayesian inversion theory, considers the influence of potential uncertainty in indentation experiments and numerical modeling, treats model parameters as random variables, gives out the distribution characteristics of parameter identification results in the form of probability distribution, and has high reliability.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate and do not limit the invention, and together with the description serve to explain the principle of the invention:
FIG. 1 is a graph showing the determination of an anisotropic plastic parameter σ using an established numerical method YT N and R 22 Is a basic step diagram;
FIG. 2 is a press-in test model under a spherical indentation load, 1 is a spherical indenter, 2 is a test piece, and 3 is a material coordinate;
FIG. 3 is a snapshot of imprinting in two orthogonal directions, longitudinal and transverse, 1 being the pile-up phenomenon, 2 being the reference line;
FIG. 4 is an average and differential imprint snapshot of SiCw/A6061
And ΔS,1 is the differential imprint snapshot ΔS, 2 is the average imprint snapshot +.>
FIG. 5 is a quarter finite element model, local mesh and material coordinates, 1 being material coordinates, 2 being test piece, 3 being indenter;
FIG. 6 is a Markov chain identifying an obtained posterior probability distribution of an anisotropic parameter using an established numerical method: (a) Representation sigma YT The method comprises the steps of carrying out a first treatment on the surface of the (b) represents n; (c) R represents 22 ；
FIG. 7 is a material anisotropy parameter inversion set σ YT The posterior edge distribution of the test pattern;
FIG. 8 is a graph showing the identification of posterior edge distribution of the material anisotropy parameter inversion set n;
FIG. 9 is a material anisotropy parameter inversion set R 22 The posterior edge distribution of the test pattern;
fig. 10 shows a uniaxial stress-strain curve of SiCw/a6061 and a stress-strain curve obtained by a numerical method based on an indentation test and an establishment, 1 is a uniaxial experimental stress-strain curve in the L direction, 2 is a stress-strain curve obtained by a numerical method based on an indentation test and an establishment in the L direction, 3 is a uniaxial experimental stress-strain curve in the T direction, and 4 is a stress-strain curve obtained by a numerical method based on an indentation test and an establishment in the T direction.
Detailed Description
The present invention will now be described in detail with reference to the drawings and the specific embodiments thereof, wherein the exemplary embodiments and descriptions of the present invention are provided for illustration of the invention and are not intended to be limiting.
The invention discloses a Bayesian inverse calculation method for measuring anisotropic plastic properties of a material based on a spherical indentation experiment, which comprises the following steps:
Spherical indentation test indentation snapshot S exp ＝[S 1 ,S 2 ,…,S K ]Is expressed in terms of (a); wherein K represents a spherical indentation experimental indentation snapshot S exp Is the dimension of the vector containing the value S 1 ,S 2 ,…,S K Is the sequence amount of vertical displacement of the experimental indentation snapshot; impression snapshot S T And S is L Average magnitude and difference imprint snapshot of (a)
And ΔS>
And Δs= (S T -S L ) Form of/2.
The Hollomon hardening law is expressed as:
in sigma T Sum sigma L Stress in the transverse and longitudinal directions, respectively; e is the elastic modulus; n is a hardening exponent; sigma (sigma) YT Sum sigma YL Yield stress in the transverse and longitudinal directions; epsilon T And epsilon L Respectively along the transverse direction and the longitudinal directionYield stress in the direction.
Indentation snapshot database O s Expressed as:
wherein O is s An imprint database obtained by extensive finite element modeling according to different parameter combinations in a material anisotropy parameter design space is represented.
And DeltaO s Is an average and difference print snapshot matrix, e.g
ΔO s ＝[ΔS 1 ΔS 2 …ΔS M ]. M is the number of anisotropic parameter combinations for the indentation simulation material.
Average and differential indentation snapshot
And the likelihood function of Δs is expressed as:
in the method, in the process of the invention,
and f (delta S) exp C, Φ) is average and differential indentation snapshot +.>
And a likelihood function of Δs; phi is a mapping model of the relation between the constitutive parameter c and the deformation behavior (such as a residual indentation snapshot) of the material under the indentation; />
And delta sigma 2 Is the variance; k represents a spherical indentation experimental indentation snapshot S exp Dimension of (2); />
And DeltaS exp Snapshot of the average and differential indentations observed for the experiment; c represents a material anisotropy parameter; />
And f (DeltaS|c, phi) is a snapshot of the experimental imprint according to the given anisotropy parameter in the material anisotropy parameter c and the assumed constitutive law phi>
And DeltaS exp The index j represents the j-th value in the associated imprint vector.
f(S exp Φ) is a normalization constant, independent of c, and can be expressed as:
f(c|S exp ,Φ)∝f(S exp |c,Φ)×f(c|Φ)
wherein f (c|S exp Phi) is a posterior probability density function, f (S) exp C, Φ) is a likelihood function, f (c|Φ) is a priori probability density function of model parameters.
The relationship between the indentation snapshot subspace coordinate coefficient and the material constitutive parameter c is expressed as:
in the method, in the process of the invention,
(or delta alpha) i ) Is a subspace coordinate matrix +.>
Column i (or Δα), matrix β is the transpose of subspace coordinate matrix α, denoted +.>
Δβ＝Δα T The method comprises the steps of carrying out a first treatment on the surface of the The subscript j value corresponds to c in design space j Regression coefficients of the j-th material parameter combination; m is the number of anisotropic parameter design space combinations of the indentation simulation materials; k is expressed as an approximation of a cubic polynomial basis function, which is defined as:
k＝[1,x,y,z,xy,xz,yz,x 2 ,y 2 ,z 2 ,x 2 y,x 2 z,xy 2 ,xz 2 ,y 2 z,yz 2 ,xz 2 ,yz 2 ,xyz,x 3 ,y 3 ,z 3 ] T
wherein the parameters x, y and z represent the anisotropic parameters sigma, n and R, respectively 22 ；
(or delta alpha) i ) Is a polynomial regression coefficient, can be obtained by +.>
And->
Obtained.
The linear combination of the indentation snapshots using basis functions is expressed as:
in the middle of
And DeltaS i Imprinting the snapshot for the ith average and difference, corresponding to the ith combination of material anisotropy parameters in the design space; />
And Deltau i Is a base vector; m is the number of combinations in the anisotropic parameter design space; />
And delta alpha i Corresponding coordinate coefficients in the subspace; />
And DeltaS ave Is an arithmetic mean snapshot of the print matrix, defined as
Step 5, replacing the imprinting snapshot in the log-likelihood function with the subspace coordinate coefficient, and directly performing model evaluation according to the given material anisotropy parameter in the step c; the log-likelihood function may replace the imprint snapshot with subspace coordinate coefficients.
The imprint snapshot subspace coordinate coefficient representation likelihood function may be expressed as:
in the method, in the process of the invention,
and f (delta S) exp C, Φ) is average and differential indentation snapshot +.>
And a likelihood function of Δs; u is O s A middle-imprinted snapshot orthogonal base; />
And DeltaS exp Respectively representing an average indentation snapshot and a differential indentation snapshot obtained by indentation experiments, and defining the corresponding subspace coordinate coefficients as +.>
And delta alpha exp They are Tongzhi>
And delta alpha exp ＝U T (ΔS exp -ΔS ave ) Obtaining a relation; u is O s An orthogonal base of the middle profile snapshot; c represents a material anisotropy parameter; phi is a mapping model of the relation between the constitutive parameter c and the residual indentation snapshot under the indentation; k represents a spherical indentation experimental indentation snapshot S exp Dimension of (2); />
And Δs is the average and difference imprint snapshot; />
And Δα indentation snapshots are corresponding coordinate coefficients in the subspace.
The log-likelihood function may replace the imprint snapshot with subspace coordinate coefficients may be expressed as:
where λ is the weighting factor.
And 6, sampling the posterior probability density function based on a transfer Markov chain algorithm to obtain posterior probability distribution of the anisotropic plastic performance parameters of the tested material.
The effect of the method of the invention will be further illustrated by means of specific examples.
The first step: referring to fig. 1,2 and 3, a spherical indentation experiment is carried out on whisker reinforced aluminum alloy SiCw/A6061, indentation morphology information is obtained, and an indentation snapshot S of the spherical indentation experiment is obtained exp ＝[S 1 ,S 2 ,…,S K ]The method comprises the steps of carrying out a first treatment on the surface of the And vertical displacement values in the transverse direction and the longitudinal direction are defined as imprint snapshots S, respectively T And S is L The method comprises the steps of carrying out a first treatment on the surface of the Creating an imprint snapshot S T And S is L Average magnitude and difference imprint snapshot of (a)
And Δs. Table 1 shows the mechanical properties of SiCw/A6061 obtained by uniaxial experiments.
TABLE 1 mechanical Properties of SiCw/A6061 obtained by uniaxial experiments
And a second step of: referring to fig. 4, a finite element simulation of spherical indentations of metallic material was performed. The calculated material elastic modulus is fixed at 113GPa, usually assumed to be a known a priori, and the anisotropy parameters are within the previously defined range: 130MPa is less than or equal to sigma YT 310MPa or more, 0.082 n or less than 0.202 or less and 1.05 or less R or less 22 Less than or equal to 1.45, interval delta sigma YT =30 MPa, Δn=0.015 and Δr 22 =0.05. Developing a series of spherical indentation simulation in the given material parameter range, and using the indentation snapshot obtained by simulation to build an indentation snapshot database O s 。
And a third step of: referring to fig. 5, based on cubic polynomialsThe formula basis function, establish the direct relation between indentation snapshot subspace coordinate coefficient and material anisotropic parameter c, express as:
i=1, 2, …, M. Wherein (1)>
(or delta alpha) i ) Is a subspace coordinate matrix +.>
Column i (or Δα), matrix β is the transpose of subspace coordinate matrix α, denoted β=α T 。
k is a cubic polynomial basis defined as:
k＝[1,x,y,z,xy,xz,yz,x 2 ,y 2 ,z 2 ,x 2 y,x 2 z,xy 2 ,xz 2 ,y 2 z,yz 2 ,xz 2 ,yz 2 ,xyz,x 3 ,y 3 ,z 3 ] T
wherein the parameters x, y and z represent the anisotropic parameters sigma, n and R, respectively 22 ；
(or delta alpha) i ) Is a polynomial regression coefficient, can be obtained by +.>
And->
Obtained.
Fourth step: an indentation likelihood function is constructed.
The likelihood function may be expressed in the form of an imprint snapshot subspace coordinate coefficient:
the log-likelihood function may replace the imprint snapshot with subspace coordinate coefficients:
sampling the posterior probability density function by adopting a transfer Markov chain algorithm to obtain posterior probability distribution of the anisotropic plastic performance parameters of the tested material. Referring to FIG. 6, for identifying the anisotropy parameter σ using an established numerical method YT N and R 22 Iterative markov chains. Referring to fig. 7, 8 and 9, the anisotropy parameter σ is shown YT N and R 22 Is distributed at the rear edge of the frame. As can be seen from the figure, a better convergence result is obtained. The iterative result is unique in that the edge distribution provides only one peak for each anisotropic parameter. Table 2 lists the comparison of material parameters between indentation test and uniaxial test data, good agreement was found. The maximum error of both the average and the mapping result is less than 10%. The maximum error is the recognition rate of the parameter n (mapping estimation result) of-9.86%.
Table 2 comparing inverted anisotropic plasticity parameters with uniaxial test data
The average value obtained from the normal distribution fitting result essentially represents a statistic, and the fitting parameters are regarded as effective recognition results. The stress-strain curve reproduced by the anisotropic parameters in the average results was compared with the uniaxial experimental curve of SiCw/a6061, as shown in fig. 9. The results indicate that the numerical methods established herein are very efficient.
The following conclusion is drawn by analyzing the uniaxial anisotropy parameters of the whisker reinforced aluminum alloy SiCw/A6061 in FIG. 10 and identifying the resulting anisotropy parameters according to the method of the present invention: (1) Good consistency can be found by comparing the material parameters between the indentation test and the uniaxial test data; (2) The anisotropic performance parameter of the whisker reinforced aluminum alloy SiCw/A6061 is very small in error with uniaxial experiments, and the obtained whisker reinforced aluminum alloy is very good in accordance with expected effects; (3) The invention can give out the solving result of the inverse problem in the form of probability distribution by considering the potential experiment and numerical model uncertainty in parameter identification, and has high identification reliability.
The invention is not limited to the above embodiments, and based on the technical solution disclosed in the invention, a person skilled in the art may make some substitutions and modifications to some technical features without creative effort according to the technical content disclosed, and these substitutions and modifications are all within the protection scope of the invention.
Claims (10)
1. A bayesian method for measuring anisotropic plasticity of a material based on spherical indentation, comprising the following steps:
carrying out a spherical indentation experiment on a metal material to be tested, obtaining the profile morphology of the indentation remained on the surface of the test piece through a laser confocal microscopic test, and obtaining an indentation snapshot S of the spherical indentation experiment exp And vertical displacement values in the lateral and longitudinal directions are defined as imprint snapshots S, respectively T And S is L The method comprises the steps of carrying out a first treatment on the surface of the Creating an imprint snapshot S T And S is L Average magnitude and difference imprint snapshot of (a)
And Δs;
according to Holloon hardening rule, developing spherical indentation finite element simulation, and generating indentation snapshot database O according to anisotropic parameters in design space s ；
Creating mean and differential indentation snapshots
And a likelihood function of Δs;
according to Bayes inference theory, the given mean and difference indentation snapshots are combined
And establishing an orthogonal subspace of the indentation snapshot based on an orthogonal decomposition algorithm by taking a likelihood function of delta S and a priori information function as constants; based on a cubic polynomial basis function, establishing a relation between an indentation snapshot subspace coordinate coefficient and a material anisotropy parameter c;
replacing the imprinting snapshot in the log-likelihood function with the subspace coordinate coefficient, and directly performing model evaluation according to given material anisotropy parameters; replacing the imprint snapshot by the subspace coordinate coefficients by the log-likelihood function;
sampling the posterior probability density function based on a transfer Markov chain algorithm to obtain posterior probability distribution of the anisotropic plastic performance parameter of the tested material.
2. The bayesian method for measuring anisotropic plasticity of a material based on spherical indentations according to claim 1, wherein in the step (1),
spherical indentation test indentation snapshot S exp ＝[S 1 ,S 2 ,…,S K ]Is expressed in terms of (a); wherein K represents a spherical indentation experimental indentation snapshot S exp Is the dimension of the vector containing the value S 1 ,S 2 ,…,S K Is the sequence amount of vertical displacement of the experimental indentation snapshot;
impression snapshot S T And S is L Average magnitude and difference imprint snapshot of (a)
And ΔS>
And Δs= (S T -S L ) Form of/2.
3. The bayesian method for measuring anisotropic plasticity of a material based on spherical indentations according to claim 1, wherein the hollloon hardening law is expressed as:
wherein sigma T Sum sigma L Stress in the transverse and longitudinal directions, respectively; e is the elastic modulus; n is a hardening exponent; sigma (sigma) YT Sum sigma YL Yield stress in the transverse and longitudinal directions; epsilon T And epsilon L Strain in the transverse and longitudinal directions, respectively.
4. The bayesian method for measuring anisotropic plasticity of a material based on spherical indentations according to claim 2, wherein the indentation snapshot database O s Expressed as:
wherein O is s Representing an imprint database obtained by extensive finite element modeling in a material anisotropy parameter design space based on different combinations of parameters,
and DeltaO s Is the mean and difference print snapshot matrix, +.>
ΔO s ＝[ΔS 1 ΔS 2 …ΔS M ]M is the number of anisotropic parameter combinations.
5. The bayesian method for measuring anisotropic plasticity of a material based on spherical indentation as claimed in claim 1, which is specificCharacterized in that the mean and difference indentation snapshots
And the likelihood function of Δs is expressed as:
in the method, in the process of the invention,
and f (delta S) exp C, Φ) is average and differential indentation snapshot +.>
And a likelihood function of Δs; phi is a mapping model of the relation between the constitutive parameter c and the deformation behavior (such as a residual indentation snapshot) of the material under the indentation; />
And delta sigma 2 Is the variance; k represents a spherical indentation experimental indentation snapshot S exp Dimension of (2); />
And DeltaS exp Stamping snapshots for experimentally observed averages and differences; />
And f (DeltaS|c, phi) is a snapshot of the experimental imprint according to the given anisotropy parameter in the material anisotropy parameter c and the assumed constitutive law phi>
And DeltaS exp The index j represents the j-th value in the associated imprint vector.
6. The bayesian method for measuring anisotropic plasticity of a material based on spherical indentations according to claim 5, wherein f (S exp Φ) is a normalization constant, independent of c, and can be expressed as:
f(c|S exp ,Φ)∝f(S exp |c,Φ)×f(c|Φ)
wherein f (c|S exp Phi) is a posterior probability density function, f (S) exp C, Φ) is a likelihood function, f (c|Φ) is a priori probability density function of model parameters.
7. The bayesian method for measuring anisotropic plasticity of a material based on spherical indentations according to claim 1, wherein the relation between the indentation snapshot subspace coordinate coefficient and the material anisotropy parameter c is expressed as:
wherein, the liquid crystal display device comprises a liquid crystal display device,
and delta alpha i Is a subspace coordinate matrix +.>
And the ith column of Δα, matrix β is the transpose of subspace coordinate matrix α, denoted +.>
Δβ＝Δα T The method comprises the steps of carrying out a first treatment on the surface of the The subscript j value corresponds to c in design space j Is the jth material parameter of (2)Regression coefficients of the number combinations; m is the number of anisotropic parameter combinations of the indentation simulation materials; k is expressed as an approximation of a cubic polynomial basis function, which is defined as:
k＝[1,x,y,z,xy,xz,yz,x 2 ,y 2 ,z 2 ,x 2 y,x 2 z,xy 2 ,xz 2 ,y 2 z,yz 2 ,xz 2 ,yz 2 ,xyz,x 3 ,y 3 ,z 3 ] T
wherein the parameters x, y and z represent the anisotropic parameters sigma, n and R, respectively 22 ；
Or Deltaalpha i Is a polynomial regression coefficient, can be obtained by +.>
And->
Obtained.
8. The bayesian method for measuring anisotropic plasticity of a material based on spherical indentations according to claim 1, wherein the linear combination of the indentation snapshots using basis functions is expressed as:
wherein the method comprises the steps of
And DeltaS i For the ith average and difference imprint snapshot, corresponding to the ith combination of material anisotropy parameters in design space, +.>
And Deltau i Is a basis vector +.>
And delta alpha i For the corresponding coordinate coefficients in the subspace, +.>
And DeltaS ave An arithmetic mean snapshot of the embossed snapshot matrix is defined as
9. The bayesian method for measuring anisotropic plasticity of a material based on spherical indentations according to claim 1, wherein the likelihood function is expressed in the form of indentation snapshot subspace coordinate coefficients, which can be expressed as:
in the method, in the process of the invention,
and f (delta S) exp C, Φ) are likelihood functions of average and differential indentation snapshots S and Δs, respectively; u is O s A middle-imprinted snapshot orthogonal base; />
And DeltaS exp Average indentation obtained for indentation experimentsSnapshot and differential indentation snapshot, their corresponding subspace coordinate coefficients are defined as +.>
And delta alpha exp They are Tongzhi>
And delta alpha exp ＝U T (ΔS exp -ΔS ave ) Obtaining a relation; u is the orthonormal of the contour snapshot; c is a material anisotropy parameter; phi is a mapping model of the relationship between the constitutive parameter c and the material residual indentation snapshot under indentation; k represents a spherical indentation experimental indentation snapshot S exp Dimension of (2);
and Δs is the average and difference imprint snapshot; />
And delta alpha is the indentation snapshot as the corresponding coordinate coefficient in the subspace.
10. The bayesian method of measuring anisotropic plasticity of a material based on spherical indentations according to claim 9, wherein the log likelihood function can be expressed as replacing the embossed snapshot with subspace coordinate coefficients:
where λ is the weighting factor.
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