Data-Driven Framework for Mechanical Behavior Characterization from Instrumented Indentation

Abstract

This study developed a novel data-driven indentation computation framework to characterize the indentation response and behavior by combining the reduced-order model (ROM), optimal technology, and indentation response curve. ROM was utilized to build a surrogated model to approximate the accuracy of the indentation response. The dataset was generated using the indentation test or physical model for ROM. Simplicial Homology Global Optimization (SHGO) was considered an optimal technology for searching for mechanical properties in inverse analysis. The developed framework was illustrated and verified using a numerical example. The results were compared with the actual value obtained by an indentation test. The results show that the developed framework characterizes the material mechanical property and indentation response well and agrees with the engineering practice. The proposed framework provides a feasible, scientific, helpful, and promising way to capture the material mechanical behavior and indentation response. Meanwhile, it also has essential reference significance for another engineering field.

1. Introduction

A mechanical property is the essential index for characterizing and evaluating the material’s performance and is directly related to material design, numerical analysis, and engineering application. Characterization and determination of mechanical properties have always been essential to material design, analysis, evaluation, and performance inspection [1]. In order to understand the mechanical properties, various test technologies and experimental facilities have been proposed to capture and determine the physical and mechanical properties of materials. However, characterization of the mechanical properties is still a challenging task and a hot topic due to the complexity and diversity of materials. In recent years, nanoindentation, a convenient and effective technology used to investigate material properties at nanoscale, has received extensive attention in the field of materials and has been applied to characterize and analyze the material mechanical properties [2,3,4]. Instrumented indentation provides valuable insights into material behavior and is regarded as one of the most versatile and practical methods for characterizing material mechanical properties [5].

In the 1970s, nanoindentation was developed to characterize the hardness of small volumes of materials. Now, nanoindentation has emerged as one of the most commonly used tools for investigating the mechanical properties of materials at very small scales. For several decades, assessing mechanical properties through instrumented indentation has been a key focus in materials science and industrial applications [6,7,8,9,10,11,12,13,14]. Comprehensive theory and methods have been developed to extract material properties from P versus h curves obtained through instrumented indentation. Doerner and Nix (1986) proposed determining Young’s modulus and hardness using the maximum load and the initial unloading slope from P versus h curves [15]. Giannakopoulos and Suresh (1999) also developed a procedure to characterize the elastic and plastic properties [9]. Self-similar concepts and solutions have been developed to capture the plastic indentation of a power law plastic material under spherical indentation [16]. Dimensional analysis is valuable for understanding the complex relationship between indentation data and mechanical properties in material sciences [8]. Giannakopoulos et al. (1994) derived an analytical framework to capture elastoplastic properties from the force–depth (p-h) relationship of the indentation [17]. A routine was developed to independently calibrate the effective tip radius and the machine stiffness using standard reference materials [18]. However, obtaining an analytical solution for predicting the indentation response and characterizing mechanical properties from indentation test data is challenging due to the complexity and nonlinearity of the behavior. Nanoindentation was utilized to explore the mesoscopic mechanical properties of the primary granite minerals and the associated interfaces and investigate the fracture characteristics of the failed Brazilian disc of granite using with micro X-ray computed tomography [19]. The numerical simulation could extract the uniaxial stress–strain parameters from the nanoindentation experimental data [20]. The microscopic mechanical properties of granite minerals were studied, and a calibration process was established using nanoindentation tests. The discrete element method was employed to simulate the evolution of microcracks and the crack characteristics of various minerals found in granite. This simulation was based on micro-X-ray computed tomography, scanning electron microscopy, and numerical results [21]. However, the numerical simulation of the nanoindentation test is cost-consuming due to its complexity.

With the development of machine learning, various machine learning technologies have been widely adopted to understand and characterize material mechanical properties from the laboratory test and nanoindentation data [22,23,24]. The neural network was employed to determine the hardening parameters and the constitutive properties by combining them with the numerical spherical indentation data [25,26,27,28]. Zhang et al. (2019) developed a Bayesian statistical model to identify plastic properties from conical indentation [29]. Lu et al. (2020) applied deep learning to characterize the elastoplastic properties of metals and alloys based on instrumented indentation [5]. Lü et al. (2024) developed a novel framework for characterizing the shear strength parameters of rock minerals at the nanoscale by combining an artificial neural network, numerical simulation, and nanoindentation [30]. Various neural network technology was developed to establish the relationship between their scratch hardness and material parameters based on empirical formulas and machine learning [31]. Indentation was combined with machine learning to evaluate the residual stress of metallic materials and analyze the effect of residual stress on the indentation responses [32]. However, these models did not characterize the physical mechanism of the material deformation and failure [33]. Recent developments in data science provide an excellent tool for discovering the mechanism behind data and dealing with uncertainty [34]. Data sciences have been successfully applied in various fields [35,36,37,38,39]. Recently, a reduced-order model (ROM) was developed using insights about the engineering structure under consideration. ROM has demonstrated outstanding performance in replacing the numerical simulation during parameter determination [40]. This study introduced a ROM-based data-driven model to effectively capture the indentation response of materials.

This study applied ROM to characterize the material mechanical behavior and approximate the indentation response during the nanoindentation test. The forward and inverse analysis of the indentation were developed by incorporating optimal technology, inverse analysis theory, and data science. A data-driven computational model was developed for instrumented indentation. The remainder of this study is organized as follows: Section 2 introduces the concept and theory of instrumented indentation. Section 3 introduces the idea, algorithm, and procedure of data-driven indentation computation. Section 4 uses a numerical example to illustrate and verify the developed method. Finally, Section 5 draws some conclusions.

2. Instrumented Indentation and Computation

Instrumented indentation is a widely used technique for assessing the mechanical properties of materials, making it one of the most versatile and practical tools available. The loading force (P) applied by the indenter tip and the corresponding penetration depth (h) into the material is continuously recorded during the test [41]. Figure 1 illustrates the typical p-h response curve for an elastoplastic material subjected to sharp indentation. The response generally adheres to Kick’s Law during the loading process, as defined in the following equation.

$$P=C{h}^2$$

(1)

where C represents the loading curvature, the average contact pressure $p_{ave}={\displaystyle \frac{P_m}{A_m}}$ (which corresponds to the true projected contact area A_m measured at the maximum load P_m) can be linked to the hardness of the indented material. The maximum indentation depth h_m occurs at P_m, and the initial unloading slope is defined as ${\displaystyle \frac{d{P}_u}{dh}}{|}_{h_m}$, where P_u is the unloading force. The term W_t refers to the total work performed by the load P during loading, while W_e is the elastic work released during unloading. The stored plastic work is represented as W_p = W_t − W_e. The residual indentation depth after complete unloading is denoted as h_r. As discussed by Giannakopoulos and Suresh [9], C, ${\displaystyle \frac{d{P}_u}{dh}}{|}_{h_m}$ and ${\displaystyle \frac{h_r}{h_m}}$ are three independent quantities that can be directly obtained from a single load–displacement p-h curve.

A power law can effectively describe the plastic behavior of many pure and alloyed metals, as illustrated schematically in Figure 1. The true stress–strain behavior of simple elastic–plastic materials can be represented as follows.

$$\sigma ={\sigma }_y(1+{\displaystyle \frac{E}{{\sigma }_y}}{\epsilon }_p{)}^n$$

(2)

where E represents Young’s modulus, n denotes the strain-hardening exponent, σ_y indicates the initial yield stress, and ε_y corresponds to the corresponding yield strain, such that ε_p depicts the nonlinear part of the total effective strain accumulated beyond ε_y.

To fully describe the mechanical behavior of materials, the effective stress theory of von Mises plasticity is applied along with Poisson’s ratio ν. E, n, σ_y, and ν could be used to characterize the mechanical and deformative mechanism.

Dimension analysis was employed to characterize the relationship between indentation data and elastoplastic properties based on the power plastic law. For a sharp indenter, the load P can be represented as follows.

$$P=P(h,E^{*},{\sigma }_y,n)$$

(3)

$$E^{*}=\left({\displaystyle \frac{1-{\upsilon }^2}{E}}+{\displaystyle \frac{1-{\upsilon }_i^2}{E_i}}\right)$$

(4)

where E_i and υ_i denote the Young’s modulus and Poisson’s ratio of the indenter, respectively. The Π theorem of dimensional analysis was applied to the above relationship. Equation (3) can be defined as follows.

$$P={\sigma }_y{h}^2{\Pi }_1\left({\displaystyle \frac{E^{*}}{{\sigma }_y}},n\right)$$

(5)

where Π₁ indicates a dimensionless function, so loading curvature C could be obtained based on the following equation.

$$C={\displaystyle \frac{P}{h^2}}={\sigma }_y{\Pi }_1^A({\displaystyle \frac{E^{*}}{{\sigma }_y}},{\displaystyle \frac{{\sigma }_r}{{\sigma }_y}})$$

(6)

where Π₁^A is a dimensionless function.

During the unloading of the indentation test, the unloading slope of force P_u could be obtained as follows.

$${\displaystyle \frac{d{P}_u}{dh}}={\displaystyle \frac{d{P}_u}{dh}}(h,h_m,E^{*},{\sigma }_r,n)$$

(7)

Applying dimension analysis to the above equation, the unloading slope of force P_u and its derivation could be obtained.

$${\displaystyle \frac{d{P}_u}{dh}}=E^{*}h{\Pi }_2^0({\displaystyle \frac{h_m}{h}},{\displaystyle \frac{{\sigma }_r}{E^{*}}},n)$$

(8)

$${\displaystyle \frac{d{P}_u}{dh}}{|}_{h=h_m}=E^{*}{h}_m{\Pi }_2({\displaystyle \frac{E^{*}}{{\sigma }_r}},n)$$

(9)

After the specimen is completely unloaded, P_u = 0 and h = h_r. The value of h_r could be calculated as follows.

$${\displaystyle \frac{h_r}{h_m}}={\Pi }_3({\displaystyle \frac{{\sigma }_r}{E^{*}}},n)$$

(10)

3. Data-Driven Indentation Computation

The reduced-order model (ROM) offers an alternative approach to the high-fidelity models to approximate solutions with reasonable accuracy. This study adopted ROM to characterize the indentation test and approximate the complex and nonlinear relationship between the mechanical properties of materials and the corresponding p-h response curve of the indentation test. Based on the ROM indentation computational model, the forward analysis method was developed by combining the analytical model, material constitutive model, and mechanical properties. Meanwhile, an inverse analysis method was also proposed based on the ROM and optimal technology.

3.1. Reduced-Order Model

The ROM was employed to construct a low-order surrogate model for the indentation simulation based on the proper orthogonal decomposition [42]. Using the proper orthogonal decomposition procedure, we can derive the following equation for any x_i, i = 1, 2, …, I, and θ_j, j = 1, 2, …, J.

$${\tilde{u}}^h\left(x_i,{\theta }_j\right)=\sum _{k=1}^K{\beta }_k\left({\theta }_j\right){\phi }^k\left(x_i\right)+\tilde{g}(x_i,{\theta }_j)$$

(11)

The equations mentioned above can be reformulated in the following form.

$${\tilde{u}}^h=\phi \beta +\tilde{g}$$

(12)

where ${\tilde{u}}^h$ is the solution of deformation variables for the indentation test, x_i and θ_j denote the design and parameter variables of an indentation model, φ and β needs to be determined later, and $\tilde{g}(x,\theta )$ depicts an extension of the boundary conditions in the whole domain.

$$\tilde{g}\left(x,\theta \right)=\left\{\begin{array}{cc}g\left(x,\theta \right)\hfill & on \mathsf{\partial }\Omega \hfill \\ 0\hfill & elsewhere\hfill \end{array}\right.$$

(13)

Latin hypercube sampling (LHS) was employed to generate the set of design variables ${\theta }_j,j=\mathrm{1,2},\dots ,J$ to determine the unknown coefficient φ. Subsequently, utilizing the analytical solution, a series of discrete solutions (snapshots) $w_j=u^h\left({\theta }_j\right)-\tilde{g}\left({\theta }_j\right), j=\mathrm{1,2},\dots ,J,$ for the indentation simulation were acquired. We can refer to the spatial Gram matrix as M_x.

$$M_{ij}^x=\left(w_i·w_j\right), i,j=\mathrm{1,2},\dots ,J$$

(14)

where $\left(w_i·w_j\right)$ indicts the scalar product between w_i and w_j.

The positive eigenvalues of M_x are listed in descending order.

$${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_J\ge 0$$

(15)

The first K eigenfunctions ${\phi }^k\left(x\right), k=\mathrm{1,2},..K$ corresponded with the first K eigenvalues indicate the orthogonal principal directions of snapshots. If $r^k={\left(r_j^k\right)}_{j=i,i,\dots J}$ is the kth eigenvector of M_x, then the associated kth eigenfunction ${\phi }^k\left(x\right)$ is obtained by

$${\phi }^k\left(x\right)=\sum _{j=1}^K{r}_j^k{w}_j\left(x\right)$$

(16)

where K denotes the dimension of the proper orthogonal decomposition basis and can be determined using the following inequation.

$${\displaystyle \frac{{\sum }_{i=1}^K{\lambda }_i}{{\sum }_{i=1}^J{\lambda }_i}}>k$$

(17)

where k indicates the user-specified tolerance. In this study, the value of k is set to 0.9999.

The unknown coefficient β can be determined by solving the following penalized minimization problem.

$$\underset{{\beta }_j\in R^K}{\min }{‖u^{h,j}-\phi {\beta }_j-{\tilde{g}}_j‖}^2+\mu {‖{\beta }_j‖}^2$$

(18)

β_j can be obtained by solving the following normal equation

$$\left({\phi }^T\phi +\mu I_K\right){\beta }_j={\phi }^T\left(u^{h,j}-{\tilde{g}}_j\right),\hspace{1em}j=\mathrm{1,2},\dots ,J$$

(19)

where μ denotes a small regularization parameter.

3.2. SHGO

The Simplicial Homology Global Optimization (SHGO) method is a versatile algorithm designed for global optimization [43]. It utilizes the principles of simplicial integral homology and combinatorial topology. The SHGO method approximates the homology groups of a complex constructed on a hypersurface that is homomorphic to the complex of the objective function. One of its significant advantages is that it only requires function evaluations, eliminating the need for derivatives of the objective functions. This makes it particularly well-suited for solving black-box global optimization problems. Given these characteristics, the SHGO method may also be beneficial in characterizing mechanical properties based on indentation tests. SHGO has demonstrated outstanding performance in terms of material composition design [44,45]. This study applied SHGO to characterize the material mechanical behavior based on ROM and instrumented indentation.

3.3. ROM Bsed Data-Driven Indentation Model

Instrumented indentation has been widely used to characterize and predict the mechanical properties of materials. However, it is time-consuming and labor-intensive. It hinders the application and development of nanoindentation in the practical engineering field. Data science provides a promising approach to characterizing material properties and mechanisms. This study developed a novel data-driven framework to understand the procedure and capture mechanism of nanoindentation by combining the physical model of nanoindentation, ROM, and optimal technology. A physical model of nanoindentation was used to generate the snapshots for ROM based on the Latin hypercube sampling. ROM was employed to approximate and map the complex and nonlinear relationship between the mechanical properties and nanoindentation response. SHGO is considered an optimal technology for inverse analysis, which identifies the mechanical properties of a material based on the nanoindentation response.

3.3.1. Data Collection

Data is the core and driving force of data-driven technology and directly determines the quality and performance of data-driven models. This study synthesized the test procedure and generated the snapshots for ROM using a physical nanoindentation model. In order to improve the generalization of data, Latin hypercube sampling was utilized to build the combination of the mechanical properties, and the corresponding response of indentation was characterized using the physical model. The snapshots consisted of a combination of material mechanical properties and the nanoindentation response (p-h curve).

3.3.2. Forward Computation

Once snapshots were generated, the ROM algorithm was implemented to build the data-driven model. This study adopted POD and SVD to compute the base vector and the corresponding coefficient. The ROM characterizes the relationship between the material mechanical properties and the nanoindentation response well (Figure 2a).

3.3.3. Inverse Analysis

Once the ROM is completed, a forward analysis of the indentation can be performed. To determine the mechanical properties, the indentation test was commonly utilized to characterize the relationship between the indentation response and the material mechanical properties. This study employed the ROM to characterize the indentation behavior based on the indentation data. The indentation response was determined based on ROM and the material mechanical properties. The discrepancy between the indentation response characterized by ROM and the results obtained from the test was evaluated. SHGO was chosen as an optimal method for searching the material mechanical properties based on the above discrepancy (Figure 2b).

3.3.4. Objective Function

To determine the material’s mechanical properties through inverse analysis, optimal technology was employed to minimize the difference between the indentation response predicted by ROM and the response measured during testing. The objective function plays a crucial role in the inverse analysis based on optimal technology. In this study, the objective function was defined in the following equation to evaluate the consistency of the indentation response.

$$\mathrm{Obj}=\sqrt{{\displaystyle \frac{{(P_{rom}-P_{test})}^2}{n}}}+\sqrt{{\displaystyle \frac{{(h_{rom}-h_{test})}^2}{n}}}$$

(20)

where P_rom is the predicted indentation force by ROM, P_test is the indentation force measured by test, h_rom and h_test are the corresponding indentation depths using the indentation, respectively. n denotes the number of segments after the indentation curve is discretized.

3.3.5. Procedure of the Data-Driven Indentation Computation

This study developed a novel data-driven framework for indentation computation. ROM was employed to address the complex and nonlinear relationship between the material’s mechanical properties and the indentation response. The forward analysis determined the indentation response and properties based on the ROM and material mechanical properties. In the inverse analysis, SHGO was chosen as an optimal technology to search the material mechanical properties based on the objective function, represented by the indentation response curve discrepancy between the predicted by the ROM and monitored by the test. The flowchart of the developed data-driven indentation computation is shown in Figure 3. The detailed procedure can be described as follows.

Step 1: Determine the instrumented indentation test parameters and material information, such as the material’s constitution, strength model, mechanical properties range, maximum indentation force, etc.

Step 2: In a practical application, build the combination of the material mechanical properties based on the experimental design, e.g., Latin hypercube sampling.

Step 3: Compute the response curve of the indentation for each combination of material mechanical properties and obtain the snapshots for ROM.

Step 4: Based on the above snapshots, split the snapshots into training samples and testing samples for ROM, build ROM based on the POD and SVD, and then develop a data-driven indentation computational framework.

Step 5: For the forward analysis, compute the indentation response curve based on the ROM and material mechanical properties.

Step 6: For the inverse analysis, select the optimal technology, such as SHGO, to search for the material mechanical properties based on ROM and the monitored indentation response curve.

Figure 3. The data-driven computational model for indentation.

Figure 3. The data-driven computational model for indentation.

4. Application

To illustrate and verify the developed data-driven indentation computational framework, pure and alloyed engineering metal materials were adopted to implement the indentation test. The combination of the mechanical properties, which were utilized to generate the snapshots for ROM, was obtained using the Latin hypercube sampling method. The power law constitutive model was adopted to simulate the mechanical and deformation behavior of the indentation procedure. The indentation test with the Vickers indenter was utilized to simulate the indentation response.

Table 1. The range of mechanical properties for the material.

Young’s Modulus E (GPa)	Yield Strength σy (MPa)	Strain-Hardening Exponent n	Poisson’s Ratio υ
[10, 210]	[30, 3000]	[0, 0.5]	[0, 0.5]

lists the range of the mechanical properties. Five hundred combinations of the mechanical properties were generated based on Latin hypercube sampling. The corresponding indentation response curves were computed based on the physical model. The snapshots included a combination of the mechanical properties and corresponding responses.

According to the developed data-driven indentation computational framework, ROM was employed to address the complex relationship between the mechanical properties and the indentation response. This study used the above 500 snapshots to construct the ROM. The developed framework takes 17.26 s to obtain the ROM. The performance of the ROM is shown in Figure 4. Figure 4 shows that the predicted indentation response of ROM aligns excellently with the measured indentation curve during loading and unloading. The coefficient of determination is about 0.9970 and almost close to 1 for snapshots. It also shows that the ROM can capture the nonlinear relationship between the mechanical properties and the indentation response well. Therefore, it is feasible to approximate the physical model, which is costly and time-consuming, using ROM during the indentation test. In order to illustrate the generalization performance of the ROM, the indentation response curve of the non-snapshots was utilized to predict the difference for the snapshots. The computation time is about 3.17 s for the single non-snapshot. The generalization performance of the ROM is shown in Figure 5. Obviously, the indentation response predicted by ROM is very close to the response measured by the test. The ROM characterizes the indentation behavior well when loading and unloading. It has also proved that the ROM offers a scientific, reliable, and promising approach to characterize the indentation process more efficiently and accurately.

Nanoindentation is often used to determine mechanical properties. However, the complexity and time-consuming nature of the indentation test make it challenging to obtain accurate mechanical properties. In this study, the physical model obtained a set of synthetic indentation response curves. Then, the developed data-driven indentation computational framework was employed to determine the mechanical properties from the indentation response curve. The ROM was used to replace the indentation to improve efficiency and accuracy. The input mechanical properties for the synthetic indentation model are listed in

Table 2. The obtained mechanical properties and comparison with actual value.

Method	Young’s Modulus E (GPa)	Yield Strength σy (MPa)	Strain-Hardening Exponent n	Poisson’s Ratio υ
Actual	134	1.104146677	0.272536859	0.282475525
This study	191.810382	0.963285098	0.189510922	0.347130587

. The corresponding response curve is obtained from the indentation model (Figure 6). SHGO is then implemented to search the mechanical properties according to the developed framework. Finally, the mechanical properties solutions were obtained and listed in Table 2. It shows some differences between the actual and the obtained by this study, especially regarding Young’s modulus. It further proved that it is difficult to characterize the mechanical properties from instrument indentation due to the complexity and uniqueness problem during the inverse analysis. Figure 6 also illustrates the indentation response curve during the loading and unloading phases of test. The results show that the indentation response curve is in good agreement with the actual curve during the loading and unloading. The curve predicted by ROM is very close to the curve measured by the physical model. It shows that the ROM characterizes the indentation behavior and deformation mechanism well. Replacing the indentation test using the data-driven indentation computation is scientific and feasible. Meanwhile, the developed data-driven framework also provides an effective inverse analysis method to determine mechanical properties from the indentation curve. The developed data-driven indentation framework is a promising way to overcome the complex and time-consuming test.

The response of instrument indentation depends on the material mechanical parameters. This study utilized ROM to characterize the relationship between the material mechanical parameters and the indentation response, described in the p-h curve. The p-h curve of the instrument indentation was characterized by loading curvature C, true projected contact area at maximum load A_m, maximum indentation depth h_m, derivation ${\left.{\displaystyle \frac{d{P}_u}{dh}}\right|}_{h=h_m}$ of unloading slope of force P_u, the ratio ${\displaystyle \frac{h_r}{h_m}}$ between maximum indentation depth h_r and maximum indentation depth h_m, etc. This study investigated the sensitivity of material mechanical parameters to the feature of the p-h curve (Figure 7). It showed the complexity of the relationship between material mechanical parameters and indentation response. It also proved that it is challenging to determine the material mechanical properties of instrument indentation.

Figure 8 shows the material stress–strain relation based on the actual and obtained mechanical properties to further investigate the performance of the developed data-driven indentation computation framework. It shows that the obtained mechanical properties are significantly closer and almost in agreement with the actual value. It further proved that the developed data-driven framework characterized the mechanical properties and pattern behind the indentation response curve well. The developed framework offers a promising and feasible way to assess the mechanical properties and characterize the indentation behavior.

The converge process was evaluated to investigate the performance of the developed data-driven framework. Figure 9 shows the variation in the objective function with iteration. It also shows that the developed inverse analysis could determine the material mechanical properties with excellent converging performance. It further proved that the developed inverse analysis offers a practical and valuable method for characterizing material mechanical properties based on indentation response data.

5. Conclusions

This study addressed a novel data-driven indentation computation framework to characterize mechanical properties and indentation behavior based on the ROM, SHGO, and indentation tests. It has been successfully applied to a numerical example for determining a material’s mechanical properties, predicted the indentation response, and illustrated its performance. In the developed framework, the ROM is used to capture the nonlinear relationship between the material mechanical property and the indentation response. It is employed to replace the indentation test, thereby improving the efficiency of inverse analysis. The SHGO is an optimal technology for searching for material mechanical properties. The developed framework is implemented in Python 3.11. The material mechanical properties determined by the developed framework were compared to the results obtained by the nanoindentation test to verify its feasibility.

(1)

The ROM was used to analyze the complex relationship between the material’s mechanical properties and the corresponding indentation response during loading and unloading. The coefficient of determination is about 0.9970 and almost close to 1 for snapshots. The resulting mechanical properties were in excellent agreement with the actual values. The ROM improved the optimization efficiency and could be used to replace the indentation test in inverse analysis. It provided a reasonable and scientific surrogated model for determining mechanical properties based on an indentation test.

(2)

Inverse analysis, a crucial aspect of characterizing material mechanical properties, relies on optimal technology. This study adopted the SHGO, a practical and effective optimization method, to handle the optimization and seek the material mechanical property. The developed model is more in agreement with engineering practice, providing a sense of reassurance about its real-world applicability.

(3)

During the indentation computation, it is essential and is not easy to characterize material mechanical properties and corresponding indentation response. In this study, a ROM-based surrogate model was developed for an indentation test. This surrogate model is a valuable and promising tool for determining the mechanical properties during the indentation process.

(4)

This study adopted pure and alloyed engineering metal materials to validate and illustrate the developed framework through numerical examples and synthesized data. The developed framework is independent of the material. It can be extended and applied to various materials and models. The developed framework provides a helpful tool to characterize the material properties from the instrumental indentation.