# Finite Element Model Updating Combined with Multi-Response Optimization for Hyper-Elastic Materials Characterization

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## Abstract

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_{i}that will define the behavior elastomers. Obtaining these experimental curves requires the use of expensive and complex experimental equipment. For years, a direct method called model updating, which is based on the combination of parameterized finite element (FE) models and experimental force-displacement curves, which are simpler and more economical than stress-strain curves, has been used to obtain the C

_{i}constants. Model updating has the disadvantage of requiring a high computational cost when it is used without the support of any known optimization method or when the number of standardized tests and required C

_{i}constants is high. This paper proposes a methodology that combines the model updating method, the mentioned standardized tests and the multi-response surface method (MRS) with desirability functions to automatically determine the most appropriate C

_{i}constants for modeling the behavior of a group of elastomers. For each standardized test, quadratic regression models were generated for modeling the error functions (ER), which represent the distance between the force-displacement curves that were obtained experimentally and those that were obtained by means of the parameterized FE models. The process of adjusting each C

_{i}constant was carried out with desirability functions, considering the same value of importance for all of the standardized tests. As a practical example, the proposed methodology was validated with the following elastomers: nitrile butadiene rubber (NBR), ethylene-vinyl acetate (EVA), styrene butadiene rubber (SBR) and polyurethane (PUR). Mooney–Rivlin, Ogden, Arruda–Boyce and Gent were considered as the hyper-elastic models for modeling the mechanical behavior of the mentioned elastomers. The validation results, after the C

_{i}parameters were adjusted, showed that the Mooney–Rivlin model was the hyper-elastic model that has the least error of all materials studied (MAEnorm = 0.054 for NBR, MAEnorm = 0.127 for NBR, MAEnorm = 0.116 for EVA and MAEnorm = 0.061 for NBR). The small error obtained in the adjustment of the C

_{i}constants, as well as the computational cost of new materials, suggests that the methodology that this paper proposes could be a simpler and more economical alternative to use to obtain the optimal C

_{i}constants of any type of elastomer than other more sophisticated methods.

## 1. Introduction

^{2}, which is something that makes clear the difference between elastomers and metals, crystals or glasses. Their macroscopic behavior is complex and depends on the time of load application, the temperature, the kind of vulcanizing, the historic loads and the state of deformation. Nevertheless, it is possible in general to achieve large deformations of the order of 100-700%, without entering the plastic region [2]. All those material characteristics make it very difficult to define a model that is able to define their complete behavior. Over the years, several theoretical models have been proposed, and more are being proposed currently. However, there are a few that are used most commonly. These include Mooney–Rivlin [3,4], Arruda–Boyce [5], Gent [6] and Ogden [7]. All of these explain the material behavior macroscopically. The theoretical models are different expressions of the deformation energy functions that are used in the large deformation theory that are determined by a series of invariant λ

_{i}values and constants of material C

_{i}[8]. That is, due to the complexity of the material, a simple explanation has not been found despite attempts by many authors. No matter what material is selected for a theoretical model, it is necessary to test the elastomer in as many tensional configurations as required. The experimental characterization of these C

_{i}constants requires the use of several standardized tests for: Tensile stress [9], compression [10], plane stress, volumetric compression and shear [11]. Currently, there are several methods for measuring the stress-strain field by these tests to obtain the constants of material C

_{i}. For example, there are those that are known as direct methods [12]: The Moiré method [13,14], electronic speckle pattern interferometry method [15], grid methods [16] and digital image correlation method (DIC) [17,18]. Normally, these methods require expensive and complex experimental equipment to correctly determine the relationship between stresses and strains. Obtaining these elastic constants C

_{i}directly from the stress-strain curves is usually achieved by a direct process that is known as curve-fitting. Another classical method that is known as "model updating" is based on inverse problems that are applied to numerical simulations of the finite element method (FEM) [19,20,21]. Model updating bases its operating principle on the hyper-elastic models and their corresponding constants C

_{i}that are implemented in the theory of FEM. The implementation of theoretical models with hyper-elastic behavior in the FEM makes it possible to conduct a very precise finite element (FE) analysis in which the same standardized tests that are mentioned above can be modeled. Model updating uses the experimental results (e.g., load-displacement data) for comparison with those load-displacement data that are obtained from the FE simulations. The force-displacement experimental curves that are obtained from standardized tests do not require equipment that is as expensive and complex as that required to obtain the stress-strain experimental curves of these same standardized tests. [16,22,23,24]. If the data that are obtained experimentally and those that have been obtained from the FEM differ significantly, it means that hyper-elastic model that was considered and its corresponding constants C

_{i}are inappropriate for simulation of the behavior of the elastomer that is being studied. The theoretical hyper-elastic model and its corresponding constants C

_{i}will be varied iteratively. The adjustment will be completed when the difference between the experimentally obtained data and the data obtained from the FEM do not differ significantly. Although this method of adjustment is well-known and used, it usually involves a high computational cost. This problem is amplified when the model of behavior that is being considered is more complex or when the number of standardized tests that are used to adjust the behavior of the studied elastomer is high. For this reason, model updating has never been used individually to adjust the constants C

_{i}on the basis of the standardized tests mentioned above. In recent years, researchers have used several techniques to adjust the constants or parameters that define the behavior of FE models on the basis of experimental data. For example, the bio-inspired soft computing techniques and multi-response surface method (MRS) are some of the most frequently used [25,26,27,28]. These techniques usually reduce considerably the number of simulations required for the constants of the FE models that must be adjusted. Other researchers have used the model updating method or a similar procedure to find the parameters that best define the mechanical behavior of a given elastomer. However, in their adjustment, they have used a relatively small number of standardized tests [29,30]. This paper proposes a method to determine automatically the theoretical hyper-elastic material behavior models (Mooney–Rivlin, Ogden, Arruda–Boyce and Gent) and the corresponding C

_{i}constants that are most appropriate for modeling the behavior of a group of elastomers. The proposed adjustment method is based on the experimental force-displacement data from the standardized tests, parameterized FE models based on the standardized tests and the multi-response surface method (MRS) with desirability functions. This combination of methods has never been used previously to determine automatically the theoretical hyper-elastic material behavior models and the corresponding C

_{i}constants that are most appropriate for modeling the behavior of a group of elastomers using all standardized tests mentioned above. The proposed method was developed as follows: Firstly, several standardized tests (tensile, plane stress, compression, volumetric compression, and shear) were developed in order to obtain the experimental force-displacement curves of each standardized test. Then, each standardized test was replicated with respective FE models so that the Mooney–Rivlin, Ogden, Arruda–Boyce and Gent hyper-elastic models and their corresponding C

_{i}constants were parameterized. From these parameterized FE models, the force-displacement curves were obtained. A Box–Behnken design of experiments (DoE) was developed in order to generate, for each of the theoretical hyper-elastic models, a design matrix with which to vary each corresponding C

_{i}of the parameterized FE models. Using the force-displacement curves that were obtained from the FE models, and the MRS, quadratic regression models were generated for each standardized test for modeling an error function (ER). The latter will be minimized by applying it to each of the standardized tests. The EF represented the distance between the force-displacement curves that were obtained experimentally and those that were obtained by means of the parameterized FE models. This enabled us to mathematically model the distance between both force-displacement curves according to the C

_{i}proposed in the design matrix. The best hyper-elastic materials models and their corresponding C

_{i}constants were found when the ER functions defined for each of the standardized tests were minimized. This adjustment process was undertaken using desirability functions. As a practical example, the proposed methodology is validated with nitrile butadiene rubber (NBR), ethylene-vinyl acetate (EVA), styrene butadiene rubber (SBR) and polyurethane (PUR) elastomers. Agreement between the force-displacement curves that were obtained by the most appropriate FE models and those that were obtained experimentally demonstrates that the proposed methodology may be valid for determining automatically the most appropriate hyper-elastic model and its corresponding constants C

_{i}to use to correctly define the hyper-elastic behavior of several elastomers. Also, the method that is proposed can be applied quickly for the optimal search of the C

_{i}constants for a variety of elastomers. That is, only the experimental phase to obtain the force-displacement curve and the quadratic regression that is used to obtain the ER and its minimization by means of desirability functions would be the new phases that are required to adjust the constants C

_{i}of the new elastomer. This means that the method that this paper proposes could be a faster, simpler and more economical way to obtain the C

_{i}constants of any type of elastomer than other more sophisticated methods.

## 2. Testing Process

## 3. Finite Element Models Proposed

_{i}for the elastomers that were studied. For steel parts, an isotropic model was considered with an elastic modulus and Poisson ratio (E = 210000 MPa and μ = 0.3) respectively. Although some of the proposed FE models could be simplified by flat elastic conditions in order to reduce the simulation time because of its large size (i.e., plane stress test), all parameterized FE models in this paper were configured as three-dimensional FE models. The parameterized FE model that was proposed for the tensile test was based on the type 2 test specimen according to ISO 37:2017 [9]. In this case, hexahedral elements with linear formulation were selected for modeling the specimen. Also, full integration with Herrmann formulation was undertaken. In order to guarantee the stability of the FE model and facilitate its convergence, the movement of the central nodes in the direction perpendicular to the axis of the testing machine was restricted (Z axle). This restriction of the displacement of the central nodes of the FE model represented a symmetry condition in the simulation. In the lower part of the FE model, a group of nodes had their entire displacement restricted in order to create the conditions of the pneumatic fixing system of the jaw. In the upper part of the FE model, a displacement ramp that will deform a specimen to a maximum value of 400 mm was applied to a group of nodes in order to cause the displacement of the mobile jaw. Also, due to the size of the required dimensions of the specimen together with the steel plates for the planar stress test, and to reduce the computational cost of the numerical simulations, a parameterized symmetric FE model was proposed for this test. This FE model considers the steel plates but does not consider the screws and nuts, since they are not relevant for the FE simulation. The entire set was built in a single block, thereby eliminating the need to impose contact conditions between the steel plates and the elastomers to be tested. The symmetry condition in the FE model was modeled by imposing null displacement in a direction that is perpendicular to the axis of the testing machine (Z axle) for the central nodes. This condition of symmetry (in the Y axle), while it reduced the computational cost of the FE model, also reduced the instabilities of the FE model, facilitating its convergence. Also, the nodes that correspond to the lower hole were fixed and a displacement ramp was applied on those nodes that correspond to the upper hole. This displaced the FE model to a maximum value of 50 mm. In practice, the cylindrical elastomer material was glued to a 50 × 50 mm steel sheets for the compression test. For the parameterized FE model, a square steel plate was considered to be a circular sheet with a diameter of 50 mm. This simplification was carried out on the parameterized FE model, mainly because the areas near the corners of the steel square did not influence the results that were obtained from the force-displacement curves. Also, a quarter of the FE model was created, and symmetry conditions for the XZ and YZ planes were imposed, whereas the nodes were clamped in the lower plane of the FE model. In the upper plane, movement was permitted in the direction of compression and a maximum displacement of -3.9 mm was undertaken. The volumetric compression test is similar to the compression test. In this case, the cylindrical elastomer was deformed inside a cylindrical chamber. Therefore, its displacement in the radial direction, as well as in its base (Z direction), was restricted. In order to reduce computational cost and improve the convergence of the proposed FE model, a quarter of the specimen with symmetry conditions in the XZ and YZ planes was proposed to model the volumetric compression test. Also, displacement in the radial direction was restricted for the periphery nodes, whereas displacement of the lower nodes was fixed with a null value and the upper nodes were displaced by a maximum value of -0.6 mm. Finally, the proposed parameterized FE model for the shear test is similar to the one that was proposed for planar stress due to the similarity in the arrangement of the elastomer parallelepipeds to the steel parts (Figure 1a). In this case, the four steel plates and the four elastomer parallelepipeds are modeled by symmetry (XZ) along the longitudinal direction of the assembly to be tested (Figure 1b). The boundary conditions, in this case, consisted of the imposition of zero displacement in the direction perpendicular to the plane of symmetry XZ. The nodes of the lower hole were fixed and a displacement ramp was added to those nodes that correspond to the upper hole that deformed the FE model by a maximum value of 10 mm (Figure 1c). The configuration of the remaining parameterized FE models that are proposed according to the other standardized tests can be found in the Additional Material that is attached to this work (See Section S2).

#### Mesh Sensitivity Analysis

_{i}constants with the Mooney–Rivlin model (C

_{10}= 0.030747, C

_{01}= 0.042667 and C

_{11}= 0.027110). The correct size of the elements, as well as the type of element and its numerical formulation, correspond to those parameterized FE models whose force-displacement curves obtained do not differ significantly from those curves that were obtained experimentally at the lowest computational cost. Figure 2 shows, for each of the standardized tests, the force-displacement curves that were obtained from the parameterized FE models with element sizes of 0.25, 0.5 and 1 mm and linear and quadratic formulation (8 and 20 nodes), respectively, and the curves that were obtained experimentally. As a general rule, these figures show that, as the size of the element decreases and its number of nodes increases (20 nodes and quadratic formulation), the difference between the force-displacement curves that were obtained from the FE models and those that were obtained experimentally is smaller. For example, Figure 2a shows the force-displacement curves from the FE model for the compression test and the one obtained experimentally when the displacement was the maximum (10 mm). In this case, the forces obtained from the FE models for this displacement of 10 mm was 167.05, 159.41, and 154.58 N. for linear formulation with element sizes of 0.25, 0.5 and 1 mm, respectively, and was 161.96, 157.10 and 149.25 N., respectively, for the quadratic formulation with the same element size. For the same displacement, the force that was obtained experimentally was 147.01 N. Figure 2b shows the force-displacement curves for the shear test. In this case, the force obtained from the FE models was -441.28, -184.33 and -184.24 N. for linear formulation with element sizes of 0.25, 0.5 and 1 mm respectively, whereas the force was -203.39, -197.28 and -177.36 N. for the quadratic formulation with the same element sizes. The force value that corresponds to a displacement of 10 mm, in this case, was -177.94 N. Finally, Figure 2c shows the force-displacement curves for the tensile test. In this case, the force that was obtained from the FE models was -9.92, -101.31 and -99.68 N. for linear formulation with element sizes of 0.25, 0.5 and 1 mm respectively, but was -98.27, -96.19and -94.84 N. for the quadratic formulation with the same element sizes. The force value that corresponds to a displacement of 10 mm, in this case, was -93.70 N.

_{EXP}are the forces that were obtained experimentally for a value of displacements k, and Y

_{FEM}are those forces that were obtained from the FE simulations for the corresponding values of displacement k and m is the number of force-displacement values that were used to make the adjustment. Table 1 shows that, as the size of the elements decreased and when the proposed FE models had a quadratic formulation (20 nodes), the computational cost increased for all standardized tests that were conducted. The table also shows that, as the size of the elements decreased, the MAE also decreased for both FE models with a linear and quadratic formulation. The smallest MAEs obtained were for mesh sizes of 0.25 mm and quadratic formulation.

## 4. Modeling with the RSM

_{n}factors and an error coefficient e. The functions that are used most frequently are linear or quadratic. Cross products among the variables being studied provide the basis of the model. The quadratic model appears below.

## 5. Combining FEM and MRS to Optimize Mechanical Problems

_{i}. The force-displacement curves of these parameterized FE models were obtained. A Box-Behnken Design of Experiments (DoE) was developed in order to generate, for each of the proposed hyper-elastic models, a design matrix with which to vary each of the corresponding C

_{i}of the parameterized FE models. The main objective of the DoE is to cover the entire range of possibilities in the most normal way possible and with the least possible data. The best hyper-elastic materials models and their corresponding constants C

_{i}were adjusted when the difference between the force-displacement curves that were obtained experimentally and those that were obtained by the FE simulations on the basis of standardized tests did not differ significantly. This adjustment process was performed using desirability functions, and previously required the definition of the error function (ER), which will be applied to each of the standardized tests to be minimized. The EF represented the distance between the force-displacement curves obtained experimentally from those force-displacement curves that were obtained by means of the parameterized FE models. It was defined as follows:

_{jk EXP}are the forces that were obtained experimentally for a value of displacements k according to the standardized j, and Y

_{FEM}are those forces that were obtained from the FE simulations for the corresponding values of displacement k according to the same standardized j, whereas m is the number of force-displacement values that were used to make the adjustment. Then, using MRS, quadratic regression models were generated for each of the standardized tests in order to model mathematically all EF

_{j}. After each of the EF

_{j}was calculated for each of the standardized tests, it was normalized to adjust the hyper-elastic models and their corresponding C

_{i}using desirability functions. The data is usually normalized in statistical processes to ensure that all variables are changed to the same scale (i.e., from 0 to 1). Normalization enables a comparison of mathematical models of outputs with different units. In this case, the normalization consists of subtracting the minimum value from each original value of EF

_{j}and then dividing the result by the range (See Equation (9)).

_{j norm}are the normalized errors functions, EF

_{j}are the errors functions that were obtained from the regression models that were developed with RSM and range (EF) is the range where the EF

_{j}are defined. The subscript j represents each of the standardized tests (tensile, plane stress, compression, volumetric compression, and shear). After the EF

_{j norm}was obtained, the adjustment process was undertaken using desirability functions.

## 6. Case Study

_{i}constants that are most appropriate for modeling the behavior of a group of elastomers. As a practical example, the proposed methodology is validated with nitrile butadiene rubber (NBR), ethylene-vinyl acetate (EVA), styrene butadiene rubber (SBR) and polyurethane (PUR) elastomers.

#### 6.1. Experimental Results

#### 6.2. Design of Experiments

_{i}were generated for each of the standardized tests according to the input constants C

_{i}and levels that appear in Table 2. After each of the FE models was simulated, a total of 21 results with their corresponding constants C

_{i}and their outputs (forces) were obtained for each of the standardized tests, as well as for each of the proposed hyper-elastic models. Thus, for example, the 5k Mooney–Rivlin model had a total of 125 × 21 = 2,625 results and their corresponding constants C

_{i}and outputs (forces) (See Table 3). The remaining tables that relate to the materials that were studied can be found in the Additional Material that is attached to this work (See Section S4).

_{norm}were calculated for each standardized test according to Equation (9).

#### 6.3. Modeling the EF for the Materials Studied According to the Standardized Tests

_{i}that defined the four proposed hyper-elastic material behaviors as input, a polynomial fitting regression process was developed using the RMS “R” package [38]. The objective of this process was to obtain, for each of the four hyper-elastic material behavior proposals and each of the four materials studied, a second-order polynomial model to model the EF according to each of the standardized tests. Equation (10) shows the second-order polynomial model that was obtained for the EF corresponding to the Mooney–Rivlin hyper-elastic model for the NBR material in the shear test. The remaining equations that correspond to the materials that were studied, as well as those that correspond to the standardized tests can be found in Section S5 of the Additional Material that is attached to this work.

_{(FEM)K}and EF

_{(R_MODELS)K}were, respectively, each of the error functions (ER) that were obtained from the parameterized FE models and from the second-order polynomial regression models when considering each of the constants C

_{i}(i.e., from 1 to m) that appear in Table 2. These MAE and RMSE that were calculated directly from the constants C

_{i}are shown in Table 3. They were used to generate the regression models and are known as training errors (train.MAE and train.RMSE). In addition, new samples or parameterized FE models that are based on each standardized test that is defined with different constants C

_{i}for the hyper-elastic material behavior and each of the four materials that were studied were used to test the regression models obtained and, thus, to obtain the testing errors of said regression models (test.MAE and test.RMSE). Ten parameterized FE models with their corresponding constants C

_{i}for Mooney–Rivlin, five for Arruda–Boyce, five for Gent and five for Ogden were generated respectively for each standardized test. The new constants C

_{i}for use in defining the parameterized FE models were generated randomly. They were not used previously to generate regression models and differ from those constants C

_{i}that appear in Table 2. After these new parameterized FE models were simulated, their force-displacement curves were obtained, and the values of the ER

_{(FEM)k}were recalculated according to Equation (8) for each standardized test. For each of the four materials studied, the new samples with their corresponding constants C

_{i}were used to calculate the ER

_{(R_MODELS)K}values with each of the second-order polynomial model Equations. Finally, with the values of ER

_{(FEM)k}and ER

_{(R_MODELS)K}the tests’ errors were calculated (test.MAE and test.RMSE) using Equations (11) and (12). The correlation (Corr), the p-value and the MAE and the RMSE for both the training and testing phases were calculated for each polynomial model obtained. All of the polynomial models that were analyzed provided correlation values (Corr) close to “1” whereas most p-values were less than 0.01. This indicates that the inputs (or the constants C

_{i}that define the hyper-elastic behavior of the elastomers that were studied) that composed the second-order polynomial models are statistically significant. However, most values of MAE and RMSE that were obtained in the training phase (train.MAE and train.RMSE) and testing phase (test.MAE and test.RMSE) are less than 10%. This indicates that the second-order polynomial models have a good predictive capacity. In addition, these results indicate that the Mooney–Rivlin is the hyper-elastic material behavior of minor test errors, whereas the Arruda–Boyce is the material behavior of a major error for each standardized test and each of the four elastomers that were studied. These results of correlation (Corr), p-value, MAE and RMSE for each of the polynomial models can be found in Section S6 of the Additional Material that is attached to this work.

#### 6.4. Multi-Response Optimization

_{i}constants that are most appropriate for modeling the mechanical behavior of NBR, EVA, SBR and PUR materials according to hyper-elastic models (Mooney–Rivlin, Arruda–Boyce, Gent and Ogden) was developed with the use of the RMS “R” package [39] by desirability functions. In this case, the EF

_{norm}that was calculated by Equation (9), and was used, instead of EF because it was intended that all standardized tests be of equal importance in the adjustment of the C

_{i}constants. Section S7 of the Additional Material that is attached to this work (Tables S25–S28), provides the results of the adjustment of the constants C

_{i}of the hyper-elastic models of the studied material. For the NBR material (Table S25), the values of the overall desirability obtained for each of the hyper-elastic models were: 0.951 for the Mooney–Rivlin, 0.753 for the Arruda–Boyce, 0.777 for the Gent model, and 0.759 for the Ogden model. In addition, this table shows that some of the values are very close to the targets that were proposed. For example, the minimum target proposed for the EF

_{Norm,Shear}was 96.174 and the optimal value that was obtained was 96.07 with a desirability value of one for the Mooney–Rivlin model. In contrast, for the Arruda–Boyce model, the proposed target for the EF

_{Norm,Tens}was 115.47 and the optimal value that was obtained was 292.62 with a desirability value of 0.482. For the PUR material (Table S26), the values of the overall desirability that was obtained in this case for each hyper-elastic model were: 0.821 for the Mooney–Rivlin, 0.532 for the Arruda–Boyce, 0.645 for the Gent and 0.671 for the Ogden. Like the results in Table S25, the results in Table S26 show that some of the values that were obtained are also close to the proposed targets. For example, the minimum target proposed for the EF

_{Norm,Shear}was 112.854 and the optimal value that was obtained was 111.625 with a desirability value of one. This was also true for the Mooney–Rivlin model. In contrast, for the Gent, the proposed target for the EF

_{Norm,ComVol}was 249.261 and the optimal value that was obtained was 6276.983 with a desirability value of 0.246. For the EVA material (Table S27), the overall desirability values that were obtained in this case for the Hyper-elastic models were: 0.952 for the Mooney–Rivlin, 0.674 for the Arruda–Boyce, 0.644 for the Gent and 0.682 for the Ogden. This table also shows that the minimum target proposed for the EF

_{Norm,Comp}was 78.213 and that the optimal value that was obtained was 113.191 with a desirability value of 0.981. These values were the same for the Mooney–Rivlin model. In contrast, for the Gent, the proposed target for the EF

_{Norm,Comp}was 13.739 and the optimal value was 331.24 with a desirability value of 0.472. Finally, for the SBR material (Table S28), the values of the overall desirability, in this case, were: 0.949 for the Mooney–Rivlin, 0.747 for the Arruda–Boyce, 0.741 for the Gent and 0.765 for the Ogden. This table also shows that the minimum target proposed for the EF

_{Norm,Shear}was 91.287 and that the optimal value that was obtained was 162.006 with a desirability value of 0.959 also for the Mooney–Rivlin. In contrast, for the Arruda–Boyce, the proposed target for the EF

_{Norm,ComVol}was 63.944 and the optimal value that was obtained was 2677.111 with a desirability value of 0.501. As a result of the optimization process, it can be concluded that the Mooney–Rivlin is the hyper-elastic model that provides the best fit between the force-displacement curves obtained experimentally and by FEM for the four materials studied because its overall desirability is the highest. Gent and Arruda–Boyce are the hyper-elastic models that provided the worst adjustment between the force-displacement curves because they have the lowest overall desirability. In addition, the shear is the standardized test that achieves the best fit between the force-displacement curves for the NBR and PUR materials, because it has desirability of one. Also, for the EVA and SBR materials, the standard compression and tensile tests best adjust the force-displacement curves, as evidenced by desirability of 0.981 and 0.993, respectively. Table 4 indicates the most appropriate or optimal constants C

_{i}to correctly define the hyper-elastic behavior that were adjusted by MRS with desirability functions.

_{i}were obtained, they were validated by comparing the force-displacement curves that were obtained experimentally to the curves that were obtained by the parameterized FE models when constants C

_{i}are considered. Figure 4 shows the force-displacement curves for the NBR material and for each of the standardized tests when the optimal constants C

_{i}considered are those of Table 4. This figure shows that the distance between the force-displacement curve obtained experimentally and those obtained from the parameterized FE models considering the optimal constants C

_{i}is greater for the volumetric compression and planar stress tests (see Figure 4b,d), in which, its desirability values obtained for the hyper-elastic models are for Mooney–Rivlin 0.932 and 0.9, respectively, for Arruda–Boyce 0.513 and 0.482, for Gent 0.573 and 0.497, and finally for Ogden 0.643 and 0.485. For other materials, the adjustment between the force-displacement curves that were obtained experimentally and those that were obtained from the parameterized models is very similar to the NBR material, which can be found in Section S8 of the Additional Material that is attached to this work. Thus for example, for the PUR material (See Figure S14), EVA material (See Figure S15) and SBR material (See Figure S16), the distances between the force-displacement curves obtained experimentally and those obtained from the parameterized FE models is also greater for the volumetric compression and plane stress tests. Table 5 shows the average MAE

_{norm}that have been obtained for each of the materials that were studied and for each one of the hyper-elastic models, which were calculated from the force-displacement curves that were obtained experimentally and those that were obtained from the parameterized FE models by use of the optimal constants C

_{i}that appear in Table 4. Each of the MAE

_{norm}in this table has been calculated as the sum of the MAE

_{norm}obtained for each of the standardized tests according to Equations (34) and (51). For all the materials studied, it was observed that the hyper-elastic model that has the least error in adjusting the force-displacement curves is the Mooney–Rivlin model (MAE

_{norm}= 0.054 for NBR, MAE

_{norm}= 0.127 for NBR, MAE

_{norm}= 0.116 for EVA, and MAE

_{norm}= 0.061 for NBR). However, this table also shows the total computational cost to obtain the optimal C

_{i}constants for each of the materials studied (i.e., the FE simulations, the quadratic regression necessary to obtain the ER functions and its minimization by means of desirability functions). As mentioned previously, only the experimental phase to obtain the force-displacement curve, and the quadratic regression necessary to obtain the ER and its minimization by means of desirability functions, would be the new phases required to adjust the constants C

_{i}of any elastomer that differs from those discussed in this paper. This means that the method that is proposed in this paper could be a faster, simpler and more economical way to obtain the C

_{i}constants of any type of elastomer than other more sophisticated methods.

## 7. Conclusions

_{i}constants for correct modeling of the hyper-elastic behavior of elastomers requires the use of several standardized tests (tensile, plane stress, compression, volumetric compression, and shear). These tests require expensive and complex experimental equipment to correctly determine the relationship between stresses and strains. For years, a direct method called model updating, which is based on the combination of parameterized finite element (FE) models and experimental force-displacement curves, which are simpler and more economical than stress-strain curves, has been used to obtain the C

_{i}constants. This method has the disadvantage of a high computational cost when it is used without the support of a known optimization method or when the number of standardized tests and required C

_{i}constants is high. This paper proposes a methodology that combines the model updating method, the aforementioned standardized tests and the multi-response surface method (MRS) with desirability functions to determine automatically the most appropriate C

_{i}constants for modeling the behavior of a group of elastomers. The optimization ascribed the same value of importance to all the standardized tests. An error function (EF) is defined to determine the distance between the force-displacement curves obtained experimentally, and their homolog force-displacement curves obtained from the parameterized FE models that are based on the standardized tests mentioned. Second-order polynomial regression models are generated from these EFs and used to search for the C

_{i}constants. As a practical example, the proposed methodology was validated with the following elastomers: nitrile butadiene rubber (NBR), ethylene-vinyl acetate (EVA), styrene butadiene rubber (SBR), and polyurethane (PUR). Mooney–Rivlin, Ogden, Arruda–Boyce and Gent were considered to be the hyper-elastic models for modeling the mechanical behavior of the elastomers. After the optimal C

_{i}constants were found by use of the polynomial regression models by MRS with desirability functions, the validation results determined that the Mooney–Rivlin model was the hyper-elastic model with the least error for all materials that were studied (MAE

_{norm}= 0.054 for NBR, MAE

_{norm}= 0.127 for NBR, MAE

_{norm}= 0.116 for EVA and MAE

_{norm}= 0.061 for NBR). The optimal C

_{i}constants obtained with the Mooney–Rivlin for each of the studied material were: C

_{10}= 0.367 MPa, C

_{01}= -0.069 MPa, C

_{11}= 0.005 MPa for NBR material, C

_{10}= 0.982 MPa, C

_{01}-0.056 MPa, C

_{11}= 0.005MPa for PUR material, C

_{10}= 0.572 MPa, C

_{01}= -0.292MPa, C

_{11}= 0.002 MPa for EVA material, and C

_{10}= 0.112 MPa, C

_{01}= 0.152 MPa, C

_{11}= 0.005 MPa for SBR material. From these results, it is observed that the optimal values obtained for C

_{11}are in a very small range (0.002 to 0.005). The proposed method can be applied quickly to the optimal search of the C

_{i}constants for a variety of elastomers. That is, the experimental phase to obtain the force-displacement curve, as well as the quadratic regression models that are necessary to obtain the ER and its minimization by desirability functions, would be the new phases that are required to adjust the constants C

_{i}of the new elastomer. The small error obtained in the adjustment of the C

_{i}constants and the computational cost for new elastomers suggest that the proposed methodology in this paper could be a simpler and more economical way to obtain the optimal C

_{i}constants of any type of elastomer than more sophisticated methods.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## List of Symbols/Nomenclature

Symbol | Nomenclature |

λi | Invariant |

ANOVA | Analysis of the variance |

C_{10}, C_{01}, C_{11} | Mooney–Rivlin material constants |

Chain | Arruda–Boyce material constant max length of a chain |

C_{i} | Constants of material |

DIC | Digital image correlation method |

DoE | Design of experiments |

E | Young’s Modulus; Gent material constant |

ER | Error function |

EVA | Ethylene-vinyl acetate |

FE | Finite element |

FEM | Finite element method |

Inv | Gent material constant, invariant |

K1, K2 | Ogden material constants |

MAE | Mean absolute error |

MRS | Multi-response surface method |

Nkt | Arruda–Boyce material constant, Number of chains in a material network |

NBR | Nitrile butadiene rubber |

PUR | Polyurethane |

RAM | Random access memory |

RMSE | Root mean square error |

SBR | Styrene butadiene rubber |

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**Figure 1.**(

**a**) Specimen mounted on the steel sheets designed for the shear test. (

**b**) Proposed parameterized symmetric finite element (FE) model. (

**c**) Details of the proposed FE model in which the nodes of the upper hole in which displacement was imposed are visible.

**Figure 2.**Comparison of the force-displacement curves that were obtained from the FE models and experimentally at different element sizes and formulations (linear/quadratic) for: (

**a**) the compression test, (

**b**) the shear test, and (

**c**) the tensile test.

**Figure 3.**(

**a**) Shear test for the nitrile butadiene rubber (NBR) material, (

**b**) shear test for the polyurethane (PUR) material, (

**c**) tensile test for the NBR material, and (

**d**) tensile test for the PUR material.

**Figure 4.**Force-displacement curve obtained from the FE simulations when the optimal constants C

_{i}are compared to the force-displacement obtained experimentally for the hyper-elastic Mooney–Rivlin model for NBR material in standardized tests (

**a**) compression, (

**b**) volumetric compression, (

**c**) shear, (

**d**) plane stress, and (

**e**) tensile.

**Table 1.**Element size, number of elements, number of nodes, computational cost and absolute mean error (MAE) corresponding to each of the standardized tests for FE models of a quadratic and linear formulation.

Test | Size [mm] | Nº Elements | N° Nodes | Time [min] | MAE |
---|---|---|---|---|---|

(Linear/ Quadratic) | (Linear/ Quadratic) | (Linear/ Quadratic) | |||

Tensile | 1 | 1568 | 1950/5850 | 1/5 | 65.05/3.54 |

0.5 | 11440 | 7190/21570 | 6/180 | 5.94/1.96 | |

0.25 | 91520 | 28762/86290 | 76/1728 | 4.71/0.90 | |

Planar Stress | 1 | 10150 | 10872/32612 | 16/106 | 12.66/8.57 |

0.5 | 81200 | 43482/130442 | 143/1232 | 8.62/2.08 | |

0.25 | 649600 | 173922/521762 | 1980/12000 | 3.35/1.81 | |

Compression | 1 | 4866 | 2223/6668 | 13/493 | 11.95/9.61 |

0.5 | 35904 | 8480/25438 | 201/1952 | 8.45/6.49 | |

0.25 | 281424 | 33544/100628 | 1598/9000 | 6.04/1.44 | |

Volumetric Compression | 1 | 2958 | 1371/3746 | 6/28 | 1968.48/35.14 |

0.5 | 22156 | 5361/15241 | 52/474 | 3.36/8.97 | |

0.25 | 165416 | 19441/58320 | 740/14000 | 2.98/2.55 | |

Shear test | 1 | 20560 | 8264/24792 | 36/280 | 174.20/20.33 |

0.5 | 163928 | 59496/179328 | 400/4833 | 5.17/15.47 | |

0.25 | 1308160 | 329702/992708 | 10163/ | 5.01/0.38 |

**Table 2.**Independent variables and experimental design levels used with the 3k and 5k full-factorial design for the proposed hyper-elastic models.

Hyper-Elastic Model | DoE | Input Constant | Magnitude | Levels | ||||
---|---|---|---|---|---|---|---|---|

−1 | −0.5 | 0 | 0.5 | 1 | ||||

Mooney–Rivlin | 5k | C_{10} | MPa | −0.25 | 0.13 | 0.5 | 0.88 | 1.25 |

C_{01} | MPa | −0.3 | 0.03 | 0.35 | 0.68 | 1 | ||

C_{11} | MPa | 0 | 0.13 | 0.25 | 0.38 | 0.5 | ||

Arruda–Boyce | 3k | Nkt | - | 0.26 | -- | 0.58 | -- | 0.9 |

Chain | - | 2 | -- | 13.5 | -- | 25 | ||

Gent | 5k | E | MPa | 0.6 | 1.375 | 2.15 | 2.925 | 3.7 |

inv | 63 | 67.125 | 71.25 | 75.375 | 79.5 | |||

Ogden | 5k | K1 | - | 0 | 0.125 | 0.25 | 0.375 | 0.5 |

K2 | - | −0.5 | −0.3125 | −0.125 | 0.062 | 0.25 |

**Table 3.**Design matrix and experiments obtained with a 5k DoE for the hyper-elastic Mooney–Rivlin model of the NBR material used in the shear test.

Inputs | Output | ||||
---|---|---|---|---|---|

Sample | C_{10} | C_{01} | C_{11} | Displacement | Force |

(MPa) | (MPa) | (MPa) | (mm) | (N) | |

1 | −0.25 | 0.35 | 0.125 | 0.00 | 0.000 |

2 | −0.25 | 0.35 | 0.125 | 0.50 | 12.117 |

3 | −0.25 | 0.35 | 0.125 | 1.00 | 24.695 |

4 | −0.25 | 0.35 | 0.125 | 1.50 | 38.228 |

5 | −0.25 | 0.35 | 0.125 | 2.00 | 53.264 |

6 | −0.25 | 0.35 | 0.125 | 2.50 | 70.419 |

7 | −0.25 | 0.35 | 0.125 | 3.00 | 90.374 |

8 | −0.25 | 0.35 | 0.125 | 3.50 | 113.829 |

9 | −0.25 | 0.35 | 0.125 | 4.00 | 141.470 |

10 | −0.25 | 0.35 | 0.125 | 4.50 | 173.949 |

11 | −0.25 | 0.35 | 0.125 | 5.00 | 211.880 |

… | … | … | … | … | … |

2623 | 1.25 | 1.00 | 0.500 | 9.00 | 7288.559 |

2624 | 1.25 | 1.00 | 0.500 | 9.50 | 7975.014 |

2625 | 1.25 | 1.00 | 0.500 | 10.00 | 8703.273 |

Hyper-Elastic Models | C_{i} | NBR | PUR | EVA | SBR |
---|---|---|---|---|---|

Mooney–Rivlin | C_{10} (MPa) | 0.367 | 0.982 | 0.572 | 0.112 |

C_{01} (MPa) | −0.069 | −0.056 | −0.292 | 0.152 | |

C_{11} (MPa) | 0.005 | 0.005 | 0.002 | 0.005 | |

Arruda–Boyce | Nkt | 0.578 | 0.643 | 0.567 | 0.579 |

Chain | 24.644 | 3.75 | 15.054 | 17.354 | |

Gent | E (MPa) | 2.144 | 2.982 | 2.237 | 1.899 |

inv1 | 76.465 | 70.645 | 63.1 | 79.46 | |

Ogden | k1 | 0.254 | 0.329 | 0.361 | 0.124 |

k2 | −0.261 | −0.499 | −0.119 | −0.426 |

**Table 5.**Average MAE

_{norm}obtained for each of the materials and each one of the hyper-elastic models studied.

Hyper-Elastic Models | NBR | PUR | EVA | SBR | ||||
---|---|---|---|---|---|---|---|---|

MAE_{norm} | Time (min.) | MAE_{norm} | Time (min.) | MAE_{norm} | Time (min.) | MAE_{norm} | Time (min.) | |

Mooney–Rivlin | 0.054 | 802 | 0.127 | 840 | 0.116 | 720 | 0.061 | 870 |

Arruda–Boyce | 0.194 | 668 | 0.536 | 715 | 0.246 | 621 | 0.225 | 742 |

Gent | 0.282 | 725 | 0.916 | 767 | 0.426 | 708 | 0.361 | 737 |

Ogden | 0.054 | 905 | 0.736 | 963 | 0.381 | 893 | 0.287 | 926 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Íñiguez-Macedo, S.; Lostado-Lorza, R.; Escribano-García, R.; Martínez-Calvo, M.Á.
Finite Element Model Updating Combined with Multi-Response Optimization for Hyper-Elastic Materials Characterization. *Materials* **2019**, *12*, 1019.
https://doi.org/10.3390/ma12071019

**AMA Style**

Íñiguez-Macedo S, Lostado-Lorza R, Escribano-García R, Martínez-Calvo MÁ.
Finite Element Model Updating Combined with Multi-Response Optimization for Hyper-Elastic Materials Characterization. *Materials*. 2019; 12(7):1019.
https://doi.org/10.3390/ma12071019

**Chicago/Turabian Style**

Íñiguez-Macedo, Saúl, Rubén Lostado-Lorza, Rubén Escribano-García, and María Ángeles Martínez-Calvo.
2019. "Finite Element Model Updating Combined with Multi-Response Optimization for Hyper-Elastic Materials Characterization" *Materials* 12, no. 7: 1019.
https://doi.org/10.3390/ma12071019