Nanoindentation inversion method for shape memory alloy constitutive model based on neural network

Abstract

The present invention discloses a method for inverting a shape memory alloy constitutive model based on a nanoindentation based on a neural network, comprising: S1, establishing a nanoindentation experimental model and a uniaxial tensile unloading experimental model; S2, extracting the force-displacement curve of the indenter under different material parameters; S3, extracting the stress-strain curve of the tensile surface under different material parameters; S4, extracting the force parameter of each point of the force-displacement curve and the stress parameter of each point of the stress-strain curve, respectively as the input and output of a fully connected neural network; S5, training the neural network, respectively predicting the stress-strain response of the loading and unloading stages of the uniaxial tensile unloading test model under different material parameters. The present invention further reduces the scale requirements for the data set, can expand the predicted model from the traditional power law constitutive model to the complex hyperelastic constitutive model, and can completely bypass the constitutive model to perform nanoindentation experimental inversion to predict the material stress-strain response.

Description

Technical Field

[0001] The invention relates to the technical field of material mechanical property testing, and in particular to a method for inverting a shape memory alloy constitutive model by nanoindentation based on a neural network. Background Art

[0002] The characterization of mechanical properties can be described by establishing a mechanical model of the material, that is, the constitutive relationship. The superelasticity and shape memory effect of shape memory alloys make it often difficult to establish a constitutive model that can describe their mechanical behavior. Compared with traditional hardness testers, nanoindenters have the advantages of less damage and high resolution. At this stage, they have become an important instrument for measuring the mechanical parameters of materials. The existing methods for obtaining the mechanical parameters of materials by inverting the indentation curve mainly include the OP method and the energy method. The former has developed into a mature commercial method, but it has limitations in the case of indentation protrusions and shallow indentation depths, and is highly dependent on the accuracy of the indenter area function measurement; although the latter overcomes the problem of needing to measure the indenter area function, the accuracy of its test results depends on its indentation energy scaling relationship and contact stiffness.

[0003] With the comprehensive improvement of computer software and hardware capabilities, data-driven constitutive modeling has gradually emerged. The main idea is to embed experimental material data directly into mechanical simulations, thereby bypassing the material's constitutive equations for modeling. Existing methods of inverting material constitutive parameters through machine learning are often only applicable to simple elastic-plastic materials, and cannot predict the reverse phase transformation process of shape memory alloys well, and the number of data sets required is no less than 1,000 groups. Summary of the invention

[0004] In order to solve the problem that the existing method of inverting material constitutive parameters cannot predict the reverse phase transformation process of shape memory alloys and the required data set size is large, the present invention proposes a nanoindentation inversion shape memory alloy constitutive model method based on neural network to solve the above problems.

[0005] The present application discloses a method for inverting a shape memory alloy constitutive model by nanoindentation based on a neural network, comprising the following steps:

[0006] S1. A nanoindentation test model and a uniaxial tensile unloading test model were established based on finite element numerical simulation software. The constitutive model used the hyperelastic constitutive model built into the finite element numerical simulation software.

[0007] S2, applying displacement loading of the same depth to the indenter of the nanoindentation test model, and unloading after loading, and extracting the force-displacement curve of the indenter under different material parameters;

[0008] S3, applying displacement loading of the same distance to the tensile surface of the uniaxial tensile unloading test model, and unloading after loading, and extracting the stress-strain curve of the tensile surface under different material parameters;

[0009] S4, extracting the force parameter of each point of the force-displacement curve in S2, converting the force parameter into a tensor form and normalizing it, and using it as the input of the fully connected neural network; extracting the stress parameter of each point of the stress-strain curve in S3, converting the stress parameter into a tensor form and normalizing it, and using it as the output of the fully connected neural network;

[0010] S5. Split the input and output data sets of the fully connected neural network in S4 into loading and unloading stages, train the fully connected neural network separately, and predict the stress-strain response of the loading and unloading stages of the uniaxial tensile unloading test model under different material parameters.

[0011] Preferably, the constitutive model in S1 is an Auricchio hyperelastic model, and the Auricchio hyperelastic model includes a phase change process and an inverse phase change process;

[0012] The phase transformation process is the transformation from austenite phase to single martensite phase;

[0013] The reverse phase transformation process is the transformation of a single martensite phase into an austenite phase;

[0014] During the phase transformation process and the reverse phase transformation process, the volume of martensite will change, and the martensite volume change rate is as follows:

[0015]

[0016] in, is the volume change rate of martensite during phase transformation, is the volume change rate of martensite during the reverse phase transformation process.

[0017]

[0018] in, is the martensite volume change rate from austenite phase to single martensite phase, is the volume change rate of martensite during the transformation from single martensite phase to austenite phase.

[0019] Preferably, the loading function of the phase change process is:

[0020] F AS (τ) = ‖t‖ + 3ap;

[0021] Where τ is Kirchhoff stress, t is stress deviator, p is hydrostatic pressure, α is material parameter, and ‖·‖ is Euclidean norm;

[0022] The function at the beginning of the phase transition is:

[0023]

[0024] The function at the end of the phase transition is:

[0025]

[0026] In the formula,

[0027]

[0028] in, and is the internal variable, represents the internal stress state of the material when the phase change begins. Indicates the internal stress state of the material when the phase change is completed. and is the material parameter, represents the starting stress of martensitic transformation, represents the end stress of martensitic transformation;

[0029] The conditions for the transformation of austenite phase to single martensite phase are as follows:

[0030]

[0031] For the phase change process, there is the following equation:

[0032]

[0033] Among them, β AS is a material parameter, indicating the phase change speed, ξ S is the volume fraction of martensite, H AS is the activation function of the phase transition process, H AS Specifically, it takes the value of 1 when the conditions for the transformation from austenite to single martensite are met, and takes the value of 0 in other cases. To load the function F AS The rate of change.

[0034] Preferably, the loading function of the reverse phase change process is:

[0035] F SA (τ) = ‖t‖ + 3αp;

[0036] The function at the beginning of the reverse phase transition is:

[0037]

[0038] The function at the end of the reverse phase transition is:

[0039]

[0040] In the formula,

[0041]

[0042] in, and is the internal variable, represents the internal stress state of the material when the reverse phase transition begins. It indicates the internal stress state of the material when the reverse phase transformation is completed. is the material parameter, represents the starting stress of reverse martensitic transformation, represents the end stress of the reverse transformation of martensite;

[0043] The conditions for the transformation of a single martensite phase to austenite phase are as follows:

[0044]

[0045] For the reverse phase change process, there is the following equation:

[0046]

[0047] Among them, β SA is the material parameter, indicating the reverse phase transition speed, H SA is the activation function of the inverse phase transition process, H SA Specifically, it takes the value of 1 when the condition of transformation from a single martensite phase to austenite phase is met, and takes the value of 0 in other cases. To load the function F SA The rate of change.

[0048] Preferably, the Auricchio hyperelastic model also includes an elastic process, the elastic body is isotropic, and the free energy equation is an isotropic equation;

[0049] Perform multiplicative decomposition of the left Cauchy-Green deformation tensor:

[0050] b e =F e F eT ;

[0051] Among them, b e is the left Cauchy-Green deformation tensor, F e is the elastic deformation gradient;

[0052] Kirchhoff stresses can be written in terms of volumetric and deviatoric stress components:

[0053] τ=pI+t;

[0054] Where, p = Klog(Je ),

[0055] Where I is the second-order identity tensor, K is the bulk modulus, G is the shear modulus, and J is the e is the volume of the elastic body, n A is the feature vector, t A is the deviatoric principal stress, It is mainly elongated.

[0056] Preferably, the curves extracted by S2 and S3 are extracted from the result file of the finite element numerical simulation software using a python script, and the results are saved as a txt file, wherein the first column of the txt file obtained by the curve extracted by S2 is the displacement parameter, and the second column is the force parameter, and the first column of the txt file obtained by the curve extracted by S3 is the strain parameter, and the second column is the stress parameter.

[0057] Preferably, in the process of extracting parameters from the curve in S4, a python script is used to separate the force parameters and stress parameters in the second column of the txt file, save them as a new file for backup, and convert the parameters into tensor form for normalization.

[0058] Preferably, the data set described in S5 is split according to monotonicity to obtain loading segment data and unloading segment data.

[0059] Preferably, the fully connected neural network consists of an input layer, multiple hidden layers and an output layer, wherein the number of neurons in the input layer is the number of force parameters extracted from the force-displacement curve, and the number of neurons in the output layer is the number of stress parameters extracted from the stress-strain curve.

[0060] Beneficial effects of the present invention:

[0061] (1) The fully connected neural network training method mentioned in the present invention converts the response curve into discrete points as input and output, further reducing the size requirement of the data set.

[0062] (2) The present invention divides the data of the loading and unloading stages and trains them separately, which can expand the prediction model from the traditional power-law constitutive model to a complex hyperelastic constitutive model.

[0063] (3) The data set used in this invention is sufficient in experimental data (about 10 3 When the data set is used as a group, it can replace the data set generated by the numerical simulation, thereby completely bypassing the constitutive model of the material and performing nanoindentation experimental inversion to predict the material stress-strain response. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1Schematic diagram of the process of the method for inverting constitutive model by nanoindentation based on fully connected neural network according to an embodiment of the present invention;

[0065] Figure 2 A schematic diagram of a fully connected neural network structure according to an embodiment of the present invention;

[0066] Figure 3 The coefficient of determination R is predicted by the embodiment of the present invention. 2 Minimal curves;

[0067] Figure 4 The coefficient of determination R is predicted by the embodiment of the present invention. 2 The biggest curve;

[0068] Figure 5 The determination coefficient R of the embodiment of the present invention is 2 Schematic diagram of stress regression analysis of the minimum prediction results;

[0069] Figure 6 is the determination coefficient R of the embodiment of the present invention 2 Schematic diagram of the maximum predicted results of stress regression analysis. DETAILED DESCRIPTION

[0070] In order to make the objectives, technical solutions and advantages of the present application more clearly understood, the present application is further described in detail below with reference to the accompanying drawings and examples.

[0071] The present application discloses a method for inverting the constitutive model of shape memory alloys by nanoindentation based on a neural network, such as Figure 1 As shown, the following steps are included:

[0072] S1. A nanoindentation test model and a uniaxial tensile unloading test model are established based on finite element numerical simulation software. The constitutive model uses the hyperelastic constitutive model that comes with the finite element numerical simulation software. The finite element numerical simulation software used in this embodiment is ABAQUS, and the constitutive model is the Auricchio hyperelastic model. The main control equations of the Auricchio hyperelastic model are as follows:

[0073] In the large deformation framework, the deformation gradient tensor is multiplicatively decomposed:

[0074] F=F e F tr ;

[0075] Among them, F e is the elastic deformation gradient, F tr is the phase transformation deformation gradient;

[0076] The Auricchio hyperelastic model includes a phase transformation process and a reverse phase transformation process. The phase transformation process is the transformation from austenite to single martensite (A→S); the reverse phase transformation process is the transformation from single martensite to austenite (S→A).

[0077] During the phase transformation and reverse phase transformation, the volume of martensite changes. The volume change rate of martensite is as follows:

[0078]

[0079] in, is the volume change rate of martensite during phase transformation, is the volume change rate of martensite during the reverse phase transformation process.

[0080] For the phase transformation process from austenite to single martensite, the thermomechanical coupling is not considered, the temperature term is ignored, and the Drucker-Prager loading function is introduced:

[0081] F AS (τ) = ‖t‖ + 3ap;

[0082] where τ is Kirchhoff stress, t is deviatoric stress, p is hydrostatic pressure, α is material parameter, and ‖·‖ is the Euclidean norm.

[0083]

[0084] Where I is the second-order identity tensor.

[0085] The functions at the beginning and end of the phase transition are:

[0086]

[0087] In the formula,

[0088]

[0089] in, and is the internal variable, represents the internal stress state of the material when the phase change begins. Indicates the internal stress state of the material when the phase change is completed. and is the material parameter, represents the starting stress of martensitic transformation, represents the end stress of martensitic transformation;

[0090] The conditions for the transformation of austenite phase to single martensite phase are as follows:

[0091]

[0092] For the phase change process, there is the following equation:

[0093]

[0094] Among them, β AS is a material parameter, indicating the phase change speed, ξ S is the volume fraction of martensite, H AS is the activation function of the phase transition process, H AS Specifically, it takes the value of 1 when the conditions for the transformation from austenite to single martensite are met, and takes the value of 0 in other cases. To load the function F AS The rate of change.

[0095] H AS The expression is as follows:

[0096]

[0097] Similarly, for the reverse phase transformation process from a single martensite phase to austenite phase, there is a loading function:

[0098] F SA (τ) = ‖t‖ + 3αp;

[0099] The functions at the beginning and end of the reverse phase transition are:

[0100]

[0101] In the formula,

[0102]

[0103] in, and is the internal variable, represents the internal stress state of the material when the reverse phase transition begins. It indicates the internal stress state of the material when the reverse phase transformation is completed. is the material parameter, represents the starting stress of reverse martensitic transformation, represents the end stress of the reverse transformation of martensite;

[0104] The conditions for the transformation of a single martensite phase to austenite phase are as follows:

[0105]

[0106] For the reverse phase change process, there is the following equation:

[0107]

[0108] Among them, β SAis the material parameter, indicating the reverse phase transition speed, H SA is the activation function of the inverse phase transition process, H SA Specifically, it takes the value of 1 when the condition of transformation from a single martensite phase to austenite phase is met, and takes the value of 0 in other cases. To load the function F SA The rate of change.

[0109] H SA The expression is as follows:

[0110]

[0111] For the elastic segment, the elastic body is considered to be isotropic, and the free energy equation is also an isotropic equation, so the left Cauchy-Green deformation tensor is multiplicatively decomposed:

[0112] b e =F e F eT ;

[0113] Among them, b e is the left Cauchy-Green deformation tensor, F e is the elastic deformation gradient;

[0114] Kirchhoff stresses can be written in terms of volumetric and deviatoric stress components:

[0115] τ=pI+t;

[0116] Where, p = Klog(J e ),

[0117] Where K is the bulk modulus, G is the shear modulus, and J is the e is the volume of the elastic body, n A is the feature vector, t A is the deviatoric principal stress, It is mainly elongated.

[0118] S2. Apply displacement loading of the same depth to the indenter of the nanoindentation test model, and unload after loading, and extract the force-displacement curve of the indenter under different material parameters. In this embodiment, the maximum displacement applied to the indenter is 500nm, and the loading rate is 5nm / s.

[0119] S3, applying displacement loading of the same distance to the tensile surface of the uniaxial tensile unloading test model, and unloading after loading, extracting the stress-strain curve of the tensile surface under different material parameters. In this embodiment, the maximum strain is 0.06 and the strain rate is 0.0012 / s.

[0120] The maximum displacement and maximum strain in S2 and S3 are only examples, as long as they are kept the same under different material parameters and meet the boundary conditions of actual working conditions. The number of discrete points taken in this implementation is 800 points. The material parameters used are shown in Table 1:

[0121] Table 1 Material parameters (unit: MPa)

[0122]

[0123] 729 sets of material parameters were obtained by permutation and combination according to the parameters in Table 1. 729 sets of force-displacement curves and stress-strain curves were calculated.

[0124] The curves extracted from S2 and S3 are too numerous, so they are extracted from the odb file (ABAQUS result file) using a python script through the interface provided by ABAQUS, and the results are saved as txt files. The first column is the displacement and strain parameters, and the second column is the force and stress variables. Each file has 800 rows of data.

[0125] S4, extract the force parameters of each point of the force-displacement curve in S2, convert the force parameters into tensor form and normalize them as the input of the fully connected neural network; extract the stress parameters of each point of the stress-strain curve in S3, convert the stress parameters into tensor form and normalize them as the output of the fully connected neural network.

[0126] In the process of extracting parameters from the curve, a python script is also used to separate the force and stress parameters in the second column of the txt files in S2 and S3, and save them as new files for backup. Then the numpy library is used to convert the above parameters into tensor form. Then the MinMaxScaler function in the sklearn library is used for normalization. The mathematical model of the function is as follows:

[0127]

[0128] X scaled (i) = X std (i)*(Max-Min)+Min

[0129] Among them, X min (i) is the minimum value of the tensor X in the i-th dimension, X max (i) is the maximum value of the tensor X in the i-th dimension, Min is the minimum value of the normalized target range, and Max is the maximum value of the normalized target range. In this embodiment, the normalized target range is [0, 1].

[0130] S5. Split the input and output data sets of the fully connected neural network in S4 into loading and unloading stages, train the fully connected neural network separately, and predict the stress-strain response of the loading and unloading stages of the uniaxial tensile unloading test model under different material parameters.

[0131] Since the curves in the loading and unloading stages are monotonic, the data set can be directly split according to the monotonicity to obtain two sets of data for the loading and unloading stages. Figure 2 As shown, in this implementation, the established fully connected neural network consists of an input layer, 6 hidden layers and an output layer, wherein the number of neurons in the input layer is the number of force parameters extracted from the force-displacement curve, and the number of neurons in the output layer is the number of stress parameters extracted from the stress-strain curve. In this embodiment, the number of neurons in the input layer and the number of neurons in the output layer are both 800, and the 6 hidden layers each contain 512 neurons. The activation function is the ReLU function, and its mathematical model is as follows:

[0132] ReLU = max(0,x);

[0133] Here, x is the input value of the function.

[0134] The loss function is MSELoss, which stands for mean square error, and its mathematical model is as follows:

[0135]

[0136] Among them, x i is the true value, y i is the predicted value.

[0137] The training dataset is divided into training set, validation set, and test set, accounting for 66%, 17%, and 17% respectively, and randomly sorted. The batch_size used in the training is 8, the learning rate is 0.001, and the number of iterations is 500.

[0138] like Figure 3 The figure shows the determination coefficient R in the stress-strain response of the uniaxial tensile unloading experiment predicted by the fully connected neural network. 2 Schematic diagram of the comparison between the minimum curve and the stress-strain response obtained by numerical simulation. Figure 3 It can be seen that the prediction deviations for the beginning and end stages of the positive phase transition and the overall stage of the reverse phase transition are relatively large, while the prediction effect for the elastic segment is relatively good.

[0139] like Figure 4 The figure shows the determination coefficient R in the stress-strain response of the uniaxial tensile unloading experiment predicted by the fully connected neural network. 2 Schematic diagram of the comparison between the maximum curve and the stress-strain response obtained by numerical simulation. Figure 4It can be seen that the prediction effects for the elastic section and the phase change section are relatively good.

[0140] like Figure 5 The figure shows the determination coefficient R in the stress-strain response of the uniaxial tensile unloading experiment predicted by the fully connected neural network. 2 Schematic diagram of stress regression analysis of the minimum curve, and its determination coefficient R 2 The value of is 0.992409359.

[0141] like Figure 6 The figure shows the determination coefficient R in the stress-strain response of the uniaxial tensile unloading experiment predicted by the fully connected neural network. 2 Schematic diagram of stress regression analysis of the largest curve, with its coefficient of determination R 2 The value of is 0.999976745.

[0142] The above shows and describes the basic principles, main features and advantages of the present invention. It should be understood by those skilled in the art that the present invention is not limited to the above embodiments, and the above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention to be protected. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A method for inverting the constitutive model of shape memory alloys by nanoindentation based on neural networks, characterized in that: The following steps are involved: S1. A nanoindentation test model and a uniaxial tensile unloading test model are established based on finite element numerical simulation software, and boundary conditions are set as needed. The constitutive model uses the hyperelastic constitutive model provided in the finite element numerical simulation software: the Auricchio hyperelastic model, which includes a phase change process and an inverse phase change process. The phase transformation process is the transformation from austenite phase to single martensite phase; The reverse phase transformation process is the transformation of a single martensite phase into an austenite phase; During the phase transformation and reverse phase transformation, the volume change rate of martensite is as follows: in, is the volume change rate of martensite during phase transformation, is the volume change rate of martensite during the reverse phase transformation process; S2, applying displacement loading of the same depth to the indenter of the nanoindentation test model, and unloading after loading, and extracting the force-displacement curve of the indenter under different material parameters; S3, applying displacement loading of the same distance to the tensile surface of the uniaxial tensile unloading test model, and unloading after loading, and extracting the stress-strain curve of the tensile surface under different material parameters; S4, extracting the force parameter of each point of the force-displacement curve in S2, converting the force parameter into a tensor form, and using it as the input of the fully connected neural network; extracting the stress parameter of each point of the stress-strain curve in S3, converting the stress parameter into a tensor form and normalizing it, and using it as the output of the fully connected neural network; S5. Split the input and output data sets of the fully connected neural network in S4 into loading and unloading stages, train the fully connected neural network separately, and predict the stress-strain response of the loading and unloading stages of the uniaxial tensile unloading test model under different material parameters.

2. The method for inverting the shape memory alloy constitutive model by nanoindentation based on a neural network according to claim 1, characterized in that: The loading function of the phase change process is: F AS (τ)=‖t‖+3αp; Where τ is Kirchhoff stress, t is stress deviator, p is hydrostatic pressure, α is material parameter, and ‖·‖ is Euclidean norm; The function at the beginning of the phase transition is: The function at the end of the phase transition is: In the formula, in, and is the internal variable, represents the internal stress state of the material when the phase change begins. Indicates the internal stress state of the material when the phase change is completed. and is the material parameter, represents the starting stress of martensitic transformation, represents the end stress of martensitic transformation; The conditions for the transformation of austenite phase to single martensite phase are as follows: For the phase change process, there is the following equation: Among them, β AS is a material parameter, indicating the phase change speed, ξ S is the volume fraction of martensite, H AS is the activation function of the phase transition process, H AS Specifically, it takes the value of 1 when the conditions for the transformation from austenite to single martensite are met, and takes the value of 0 in other cases. To load the function F AS The rate of change.

3. The method for inverting the shape memory alloy constitutive model by nanoindentation based on a neural network according to claim 2, characterized in that: The loading function of the reverse phase change process is: F SA (τ)=‖t‖+3αp; The function at the beginning of the reverse phase transition is: The function at the end of the reverse phase transition is: In the formula, in, and is the internal variable, represents the internal stress state of the material when the reverse phase transition begins. It indicates the internal stress state of the material when the reverse phase transformation is completed. is the material parameter, represents the starting stress of reverse martensitic transformation, represents the end stress of the reverse transformation of martensite; The conditions for the transformation of a single martensite phase to austenite phase are as follows: For the reverse phase change process, there is the following equation: Among them, β DA is the material parameter, indicating the reverse phase transition speed, H SA is the activation function of the inverse phase transition process, H SA Specifically, it takes the value of 1 when the condition of transformation from a single martensite phase to austenite phase is met, and takes the value of 0 in other cases. To load the function F SA The rate of change.

4. The method for inverting the shape memory alloy constitutive model by nanoindentation based on a neural network according to claim 3, characterized in that: The Auricchio hyperelastic model also includes an elastic process, the elastic body is isotropic, and the free energy equation is an isotropic equation; Perform multiplicative decomposition of the left Cauchy-Green deformation tensor: b e =F e F eT ; Among them, b e is the left Cauchy-Green deformation tensor, F e is the elastic deformation gradient; Kirchhoff stresses can be written in terms of volumetric and deviatoric stress components: τ=pI+t; Where, p = Klog(J e ), Where I is the second-order identity tensor, K is the bulk modulus, G is the shear modulus, and J is the e is the volume of the elastic body, n A is the feature vector, t A is the deviatoric principal stress, It is mainly elongated.

5. The method for inverting the shape memory alloy constitutive model by nanoindentation based on a neural network according to claim 4, characterized in that: The curves extracted by S2 and S3 are extracted from the result file of the finite element numerical simulation software using a python script, and the results are saved as a txt file, wherein the first column of the txt file obtained by the curve extracted by S2 is the displacement parameter, and the second column is the force parameter, and the first column of the txt file obtained by the curve extracted by S3 is the strain parameter, and the second column is the stress parameter.

6. The method for inverting the shape memory alloy constitutive model by nanoindentation based on a neural network according to claim 5, characterized in that: In the process of extracting parameters from the curve in S4, a python script is used to separate the force parameters and stress parameters in the second column of the txt file, save them as a new file for backup, and convert the parameters into tensor form for normalization.

7. The method for inverting the shape memory alloy constitutive model by nanoindentation based on a neural network according to claim 6, characterized in that: The data set described in S5 is split according to monotonicity to obtain loading segment data and unloading segment data.

8. The method for inverting the shape memory alloy constitutive model by nanoindentation based on a neural network according to claim 7, characterized in that: The fully connected neural network consists of an input layer, multiple hidden layers and an output layer, wherein the number of neurons in the input layer is the number of force parameters extracted from the force-displacement curve, and the number of neurons in the output layer is the number of stress parameters extracted from the stress-strain curve.

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