An Attempt to Model the Surface Pressures of Apples Using the Finite Element Method

: Apples are the most popular fruits grown in Polish orchards. In order to obtain the best quality fruit, it is necessary to improve plantation maintenance, fruit harvesting, and processing. Given that many fruits are exposed to external factors, including forces that adversely affect their structure—causing them to crack, bruise, or crush—it is necessary to provide conditions that do not adversely affect their quality. Therefore, the aim of this article was to develop a simpliﬁed model of an apple that could be tested under different loads using the ﬁnite element method. The parameters of the model were selected to reﬂect the actual apple as accurately as possible. To assess the apples under impact load, as well as the construction of the FEM model, concrete and wooden substrates were used, where apples were dropped from height of 10 mm and 30 mm. Due to this research, an apple model was obtained that reﬂects the actual object very well (high R 2 coefﬁcient). In addition, the layering and distribution of surface pressures of the real and model objects from the distribution are presented. This shows that the constructed model corresponds to the behaviour of the biological material, subjected to loads in real conditions.


Introduction
Poland is one the world's leading apple producers. The annual production volume reaches about 3 million tonnes [1]. When assessing the quality of these fruits, it turns out that only one-third are intended for dessert purposes, while the remaining quantity is only suitable for processing. The production of fruit of the highest quality, for direct consumption, the parameters of which are described in the Official Journal of the European Union, requires procedures to be followed to classify it in one of three quality classes. The 'extra' class includes apples of superior quality, which have, among other qualities, the whole stalk, and do not show any damage to the flesh, i.e., they must have no bruising. Apples in Class I may have no stalks, if the areas of detachment are clean and the skin in that area is not damaged. It is permissible for the flesh to be damaged (bruised) with an area not exceeding 1 cm 2 . Class II apples, i.e., the lowest class allowed on the consumer market, includes fruit, the flesh of which is damaged, with the damaged area not exceeding 2.5 cm 2 [2,3].
It appears that the "structure of destination" should change in favour of dessert apples; therefore, high quality fruits are required. As emphasised by Konopacka and Rutkowski [4], and Nadulski [5], high quality apples translates into higher levels of export and, above all, greater consumption by increasingly demanding consumers. In order to ensure high quality fruit, care and harvesting should be carried out in such a way to prevent the the need to perform experimental studies. It is a particularly important technique used in the design and construction of engineering structures [21]. Petru et al. [22] described a mechanical behaviour of seeds that were exposed to external linear loading. Thanks to the tests carried out, the researchers identified seed sites that were particularly vulnerable to changes in the structure of biological material [22,23]. Guner et al. [24] examined the behaviour of the structure of the biological material using the example of a nut that was squeezed between two parallel plates until the shell and grain began to crack [24]. The behaviour of a biological material in individual layers of beans was also studied using the example of beans and peas. The maximum deformation that occurred during the external load application was determined. The finite element method was used to model the deformation [25]. Similar studies were conducted by Akangbe et al. [26], who, under external loading, studied the behaviour of interacting rapeseeds in the stack [26].
Theoretical aspects of computer integration-in regard to the method-were solved in the 1960s and 1970s. Research and development from the European Union's research base in the 1990s resulted in extensive scientific output in various fields of engineering [27]. This method also proved useful in agricultural engineering research, in the context of deformation of agricultural products. Many tests were carried out to estimate the complex stress field of organic materials under different boundary conditions [28][29][30][31][32][33][34]. It should be stressed that most of these tests were carried out considering static or quasi-static load cases-small deformation and linear contacts with the linear assumptions of the elastic material model through implicit solvers. The consideration of non-linearity in partial contact and ductility in material models (especially in cases of impact loads) in such tests were absent or very limited [27].
Many experimental methods can measure the level of damage, stress distribution, surface pressure, and the reaction of the fruit to external forces. Works involving damage measurements, surface pressure using the finite element method are beginning to appear [21][22][23]. For example, the extent of deformation of the tomato fruit because of hard surface contact, between tomatoes, has been mapped [35]. In this study, a 3D model of the tomato was generated in CATIA, and then the compression tests were simulated using MES to determine the mechanical properties of the tomato. The comparison of experimental data with the conducted finite element simulation confirmed that the two sets of results were consistent [5]. The tests in which the Golden Delicious apple variety was tested also used a 3D scanner, a high-speed digital camera, and the FEM technique mentioned above; the aim of the tests was to determine the degree of deformation of these fruits arranged in a box. The experimentally validated FEM model in this study will be used to determine the appropriate conditions related to the transport, storage, and processing of apples. In this way, the authors confirmed the results of a simulation using SolidWorks during the fruit drop, where they also used a high-speed camera, and consequently noticed that the results of visual tests and simulations were very similar [35].
The susceptibility of pears to different impact conditions was determined experimentally and implemented using FEM [21]. In this test, the authors used three drop heights (0.2, 0.5, and 1 m), two impact surfaces (steel and wood), two fruit impact orientations (vertical and horizontal), and three ripeness levels (unripe, ripe, and overripe). The lowest and highest differences between the predicted and observed upholstery surfaces calculated by ANSYS software were 0.00% and 60.53%, respectively. Comparing experimental and model data, high compliance was observed for pears dropped from 0.2 and 0.5 m heights. These studies show the high efficiency of the applied FEM method as an accurate tool to study the deformation of biological material of agricultural origin.
The aim of the study was to develop a simplified model of an apple using the finite element method. The validation consisted in analyses and comparing the results obtained with the experimental results presented in the paper 'Influence of Contact Surface Type on the Mechanical Damages of Apples Under Impact Loads' [36], published in the journal Food Bioprocess Technology (2017) 10:1479-1494, doi:10.1007/s11947-017-1918-z. An inno-vative area in this study was the assessment of fruit damage based on the stress contour distribution based on the surface pressures.

Materials and Methods
The model of the apple ('Golden Delicious' variety) was presented as an ideal sphere with a diameter of 71.7 mm and a weight of 163 g, determined from an average measurement of 300 apples. Based on the geometric model, a discrete model (Figure 1) was made for finite element method (FEM) calculations.

Materials and Methods
The model of the apple ('Golden Delicious' variety) was presented as an ideal sphere with a diameter of 71.7 mm and a weight of 163 g, determined from an average measurement of 300 apples. Based on the geometric model, a discrete model (Figure 1) was made for finite element method (FEM) calculations. The apple was modelled with an eight-node hexahedral solid element, a volume elements with 3 degrees of freedom in each node. The average size of a finite element was 1 mm. In the areas most stressed during the test, the grid of the elements was additionally compacted. Since, during the test, a partial plasticisation of the material of which the apple model was made may have occurred, and a significant change in the model may have occurred because of large deflections, all of the elements were adjusted to non-linear calculations, in terms of material and geometry. Considering that the model would have been exposed to fast-changing phenomena, specially prepared explicit elements were used. The material was homogeneous throughout the entire volume of the model. The choice of material was so difficult that it had to be suitable for studying dynamic phenomena and match the viscoelastic nature of the biological material. An additional difficulty was the behaviour of the actual object during the impact tests. The material of the apples was liquefied upon impact, changing the state. Modelling of phase transitions would considerably complicate the adjustment of the numerical model to the results obtained from the experiment. However, due to the way the object was deformed, a non-linear constitutive description of the material as a function of significant deformation had to be applied. The model was developed in the Abaqus CAE system. Finally, the authors decided on a material called Crushable Foam, available in the software used, designed for low-density polymer foams. It was considered that the behaviour of the apple material during dynamic events most resembles the crispy foam. It is an isotropic material, partially strengthening as the model volume changes. This reinforcement is directly related to the deformation mechanism in the apple, which is why the volumetric type reinforcement was chosen in the properties of the material. The description of this material assumes that the dependence of the flow surface in the deviant plane on pressure is expressed by means of an ellipse and uses three components of plasticizing stress (uniaxial compression, hydrostatic-in compression and tension) to determine the parameters of this ellipse. When the material is reinforced, the range of the described ellipse increases (the hydrostatic tensile The apple was modelled with an eight-node hexahedral solid element, a volume elements with 3 degrees of freedom in each node. The average size of a finite element was 1 mm. In the areas most stressed during the test, the grid of the elements was additionally compacted. Since, during the test, a partial plasticisation of the material of which the apple model was made may have occurred, and a significant change in the model may have occurred because of large deflections, all of the elements were adjusted to non-linear calculations, in terms of material and geometry. Considering that the model would have been exposed to fast-changing phenomena, specially prepared explicit elements were used. The material was homogeneous throughout the entire volume of the model. The choice of material was so difficult that it had to be suitable for studying dynamic phenomena and match the viscoelastic nature of the biological material. An additional difficulty was the behaviour of the actual object during the impact tests. The material of the apples was liquefied upon impact, changing the state. Modelling of phase transitions would considerably complicate the adjustment of the numerical model to the results obtained from the experiment. However, due to the way the object was deformed, a non-linear constitutive description of the material as a function of significant deformation had to be applied. The model was developed in the Abaqus CAE system. Finally, the authors decided on a material called Crushable Foam, available in the software used, designed for low-density polymer foams. It was considered that the behaviour of the apple material during dynamic events most resembles the crispy foam. It is an isotropic material, partially strengthening as the model volume changes. This reinforcement is directly related to the deformation mechanism in the apple, which is why the volumetric type reinforcement was chosen in the properties of the material. The description of this material assumes that the dependence of the flow surface in the deviant plane on pressure is expressed by means of an ellipse and uses three components of plasticizing stress (uniaxial compression, hydrostatic-in compression and tension) to determine the parameters of this ellipse. When the material is reinforced, the range of the described ellipse increases (the hydrostatic tensile stress (pt) remains constant, while the compression values (pc) move. An example of stress distribution for such a material before and after reinforcement is shown in Figure 2.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 16 stress (pt) remains constant, while the compression values (pc) move. An example of stress distribution for such a material before and after reinforcement is shown in Figure 2. Appropriate parameters of the material were determined, based on an experimental compression test and dynamic analysis of the apples. The elastic values of tested material were as follows: Young's modulus (E) = 4.069 MPa, Poisson's ratio (ν) = 0.32, which were calculated in a paper prepared by Komarnicki et al. [36]. A density equal to 845 kg·m −3 was determined, based on the measurement of the whole series of objects. Table 1 presents the material parameters that were adopted. The numerical tests consisted of dropping the model from a range of height (10-150 mm) on to a horizontal concrete and wooden slab. Their properties are presented in Table  2. Due to the structure of the wood material, Young's modules for this material were specified in three directions of the grains: 1-longitudinal, 2-radial, and 3-tangential. Poisson's ratios ν12 ν13 ν 23 were also determined, based on the grain direction. Linear models of materials were adopted, as the stresses achieved in them were significantly below the yield point. The calculations were performed in the Abaqus/Explicit module for the analysis of dynamic phenomena [37]. The model was forced kinematically, by means of speed, applied to the model over its entire volume. The speed was determined by the height from Appropriate parameters of the material were determined, based on an experimental compression test and dynamic analysis of the apples. The elastic values of tested material were as follows: Young's modulus (E) = 4.069 MPa, Poisson's ratio (ν) = 0.32, which were calculated in a paper prepared by Komarnicki et al. [36]. A density equal to 845 kg·m −3 was determined, based on the measurement of the whole series of objects. Table 1 presents the material parameters that were adopted. The numerical tests consisted of dropping the model from a range of height (10-150 mm) on to a horizontal concrete and wooden slab. Their properties are presented in Table 2. Due to the structure of the wood material, Young's modules for this material were specified in three directions of the grains: 1-longitudinal, 2-radial, and 3-tangential. Poisson's ratios ν12 ν13 ν23 were also determined, based on the grain direction. Linear models of materials were adopted, as the stresses achieved in them were significantly below the yield point. The calculations were performed in the Abaqus/Explicit module for the analysis of dynamic phenomena [37]. The model was forced kinematically, by means of speed, applied to the model over its entire volume. The speed was determined by the height from which the object was dropped. The speed range during analysis ranged from 443 to 1715 mm·s −1 . Stopa et al., in their observations, assumed and confirmed the hypothesis that permanent deformations indicated an occurrence of the first signs of damage in the parenchyma tissue, and it followed an increase of the impact energy corresponding with the initial phase of maximum surface pressure stabilisation [20].
To provide this hypothesis, the following was conducted: detailed analyses of the contours and the distribution of average surface pressure during individual impacts for successive impact energies. The surfaces of the bruises were also measured as a measure of apple flesh damage.
The study lists and compares experimental results [36] with model results for discharges on two substrates: concrete and wood. Comparison of substrates from corrugated cardboard was abandoned due to difficulties in creating its numerical model. All drop heights, i.e., from 10 to 150 mm, were compared with a change in height every 10 mm. The heights of 10 and 30 mm were analysed in detail for both substrates. Komarnicki el al. [36], in their tests, considered these heights crucial, representing the limit beyond which the permanent deformation that causes the damage occurs.  which the object was dropped. The speed range during analysis ranged from 443 to 1715 mm·s −1 . Stopa et al., in their observations, assumed and confirmed the hypothesis that permanent deformations indicated an occurrence of the first signs of damage in the parenchyma tissue, and it followed an increase of the impact energy corresponding with the initial phase of maximum surface pressure stabilisation [20].

Results and Discussion
To provide this hypothesis, the following was conducted: detailed analyses of the contours and the distribution of average surface pressure during individual impacts for successive impact energies. The surfaces of the bruises were also measured as a measure of apple flesh damage.
The study lists and compares experimental results [36] with model results for discharges on two substrates: concrete and wood. Comparison of substrates from corrugated cardboard was abandoned due to difficulties in creating its numerical model. All drop heights, i.e., from 10 to 150 mm, were compared with a change in height every 10 mm. The heights of 10 and 30 mm were analysed in detail for both substrates. Komarnicki el al. [36], in their tests, considered these heights crucial, representing the limit beyond which the permanent deformation that causes the damage occurs.       The results were presented on the cut-off axis, and on the ordinate axis-the va calculated with the proposed model. The graph shows that the relation presenting average maximum values of surface pressures can be considered correct because the erage error of matching the model to the variables does not exceed ±10%. It shoul noted that the material tested is of biological origin and is characterised by consider anisotropy and irregularity within a single object (apples), influenced by conditions as sunlight, humidity, temperature, and other conditions.  The results were presented on the cut-off axis, and on the ordinate axis-the values calculated with the proposed model. The graph shows that the relation presenting the average maximum values of surface pressures can be considered correct because the average error of matching the model to the variables does not exceed ±10%. It should be noted that the material tested is of biological origin and is characterised by considerable anisotropy and irregularity within a single object (apples), influenced by conditions such as sunlight, humidity, temperature, and other conditions. Figure 3 shows a comparison of mean values of surface pressures, p, obtained from experimental tests with model results for both substrates. The average model fitting error for each substrate was determined based on the average value of surface pressure +/−5% observations around the trend line (p < 0.95). The high density of the results indicates that the model is correctly matched to the real object, because the model values are slightly different from the experimental values. Based on the linear regression analysis and the coefficient of determination obtained from it, the model fit is good-for concrete substrate R 2 = 0.68-and high (for wood substrate R 2 = 0.84). The higher surface pressure p values obtained for both experimental and model values for the concrete type substrate are related to the fact that it was a non-deformable substrate with a high modulus of elasticity [20].

Results and Discussion
In Figure 4, a comparison of contact surface A obtained through experimental and model tests for both substrates is presented. An even distribution of results within an error limit of 10% can be observed. For both substrates, the coefficient of determination obtained, based on linear regression analysis, was very good. For instance, concrete R 2 = 0.90 and wood R 2 = 0.94. Stopa et al. [20], Komarnicki et al. [36], and Unuigbe and Onuoha [38] proved that the size of the contact surface depended on the type of the resistance surface and was capable of deformation, i.e., the flexural modulus.
In Figure 5, a comparison of force F values for experimental and model values is presented. The results obtained are within an error limit of 10%. For both substrates, the coefficient of determination obtained based on linear regression analysis is very good. For instance, concrete R 2 = 0.90 and wood R 2 = 0.92. The obtained values are influenced by the height of the drop, the type of retaining surface, and the radius of curvature of the tested object. The deviation of some measurements from the mean value has no significant effect because these are the values obtained for the drop height at which the damage to the flesh tissue occurs [20,36].
In fact, a high coefficient of determination does not always guarantee a perfect fit of model to the experimental data. In this paper, discrepancies between the experimental and model data may result, among others, from properties of the material, conditions of fruit ripening, and conditions under which the tests are performed. These circumstances influence the differences in contact surface areas and surface pressures in the analysed Figures 3-5. Figure 6 presents the effect of the drop height on experimental and model values of surface pressures, p. It can be observed that model values start from a lower value, by about 0.04 MPa from the experimental values and their increase with height increase occurs faster than in the experimental results, and reaches most of them. Komarnicki et al. [36], in his studies, pointed out that, for a wood-type substrate, with a 10 mm drop height, damage occurs, so higher values obtained at higher drop heights do not have any effect, because, even at lower values, damage occurs. As previously stated, these results are within the error limit (10%) and should be considered as very good due to the value of the coefficient of determination R 2 = 0.99 obtained from the power model. The results are presented in Figure 6.
height of the drop, the type of retaining surface, and the radius of curvature of the tested object. The deviation of some measurements from the mean value has no significant effect because these are the values obtained for the drop height at which the damage to the flesh tissue occurs [20,36].
In fact, a high coefficient of determination does not always guarantee a perfect fit of model to the experimental data. In this paper, discrepancies between the experimental and model data may result, among others, from properties of the material, conditions of fruit ripening, and conditions under which the tests are performed. These circumstances influence the differences in contact surface areas and surface pressures in the analysed Figures 3-5. Figure 6 presents the effect of the drop height on experimental and model values of surface pressures, p. It can be observed that model values start from a lower value, by about 0.04 MPa from the experimental values and their increase with height increase occurs faster than in the experimental results, and reaches most of them. Komarnicki et al. [36], in his studies, pointed out that, for a wood-type substrate, with a 10 mm drop height, damage occurs, so higher values obtained at higher drop heights do not have any effect, because, even at lower values, damage occurs. As previously stated, these results are within the error limit (10%) and should be considered as very good due to the value of the coefficient of determination R 2 = 0.99 obtained from the power model. The results are presented in Figure 6     In Figure 8, a comparison of the obtained force F values for experimental and model values is presented. It can be observed that the strength values are close to each other and are characterised by an equal fast build-up rate as the height increases; the trend lines for both runs are characterised by a high coefficient of determination close to unity. Figure 7. Influence of the drop height on the experimental and model contact surface, for wood type substrates. Am-means model contact surface, Ac-means experimental contact surface. Error bars represent mean ± SD (standard deviation).
In Figure 8, a comparison of the obtained force F values for experimental and model values is presented. It can be observed that the strength values are close to each other and are characterised by an equal fast build-up rate as the height increases; the trend lines for both runs are characterised by a high coefficient of determination close to unity. In Figure 9, the effect of drop height on experimental and model values of surface pressures is presented. It can be observed that model values start from a lower value, around 0.1 MPa from the experimental values, and their increase with the drop height increase occurs faster than in the experimental results, but reaches a similar maximum value. However, as previously stated, these results were within the error ±10% compared to the base value calculated. In Figure 9, the effect of drop height on experimental and model values of surface pressures is presented. It can be observed that model values start from a lower value, around 0.1 MPa from the experimental values, and their increase with the drop height increase occurs faster than in the experimental results, but reaches a similar maximum value. However, as previously stated, these results were within the error ±10% compared to the base value calculated.   In Figure 10 concrete type substrates. pm-means model surface pressure, pe-means experimental surface pressure. Error bars represent mean ± SD (standard deviation).
In Figure 10   . However, based on course observations, it can be concluded that this difference is insignificant, because a higher value was obtained in the experimental studies and no damage was found due to the shape of the surface pressure distribution, whose maximum values were in the central point of contact. This indicates that we are dealing with an elastic deformation. This also confirms the absence of visible damage to the flesh tissue of the tested apple at the point of contact with the resistance element in the stroke test [36]. Based on the shape of the contact, it can be concluded that this difference may be due to the geometry of the objects. The experimental object could have a smaller radius of curvature (longitudinal shape of the contact) at this point [36], resulting in a higher surface pressure value, whereas the numerical model is the ideal ball. . However, based on course observations, it can be concluded that this difference is insignificant, because a higher value was obtained in the experimental studies and no damage was found due to the shape of the surface pressure distribution, whose maximum values were in the central point of contact. This indicates that we are dealing with an elastic deformation. This also confirms the absence of visible damage to the flesh tissue of the tested apple at the point of contact with the resistance element in the stroke test [36]. Based on the shape of the contact, it can be concluded that this difference may be due to the geometry of the objects. The experimental object could have a smaller radius of curvature (longitudinal shape of the contact) at this point [36], resulting in a higher surface pressure value, whereas the numerical model is the ideal ball. 22%). However, based on course observations, it can be concluded that this difference is insignificant, because a higher value was obtained in the experimental studies and no damage was found due to the shape of the surface pressure distribution, whose maximum values were in the central point of contact. This indicates that we are dealing with an elastic deformation. This also confirms the absence of visible damage to the flesh tissue of the tested apple at the point of contact with the resistance element in the stroke test [36]. Based on the shape of the contact, it can be concluded that this difference may be due to the geometry of the objects. The experimental object could have a smaller radius of curvature (longitudinal shape of the contact) at this point [36], resulting in a higher surface pressure value, whereas the numerical model is the ideal ball.   [20] and Komarnicki et al. [36]. Furthermore, they discovered that, in extreme cases, the surface pressure values in the central contact zone might decrease to zero.  [20] and Komarnicki et al. [36]. Furthermore, they discovered that, in extreme cases, the surface pressure values in the central contact zone might decrease to zero. Figures 16 and 17a,b presented the courses of values of surface pressures as functions of distance for the drop, from the height of 10 mm on the wooden substrate, as well as drawings of contour lines and distributions of surface pressures of the real and model objects, in the cross-section passing through the central point of contact. It can be observed that the difference in the maximum values of surface pressures, as opposed to the concrete substrates, are smaller and amount to about 0.03 MPa. Based on observations of the course, it can be concluded that this difference is not significant, because a higher value was obtained in the experimental studies, and no damage was found due to the shape of the distribution of surface pressures, whose maximum values were at the central point of contact. It can also be concluded that we are dealing with an elastic deformation. This confirms the absence of visible damage to the flesh tissue of the tested apple at the point of contact with the resistance element in the stroke test [20,36]. Based on the shape of the contact, it can be concluded that this difference may be due to the geometry of objects. The experimental object could have a smaller radius of curvature (longitudinal shape of the contact) [20,36], resulting in a higher surface pressure, while the numerical model is the ideal ball.
central point of contact. Figures 14 and 15a,b present the courses of values of surface pressures as functions of distance for the drop, from a height of 30 mm on the concrete substrate, as well as drawings of contour lines and distributions of surface pressures of the real and model objects, in the cross-section passing through the central point of contact. For drops from a height of 30 mm, Komarnicki et al. [36] found the damage, as evidenced by the shift of maximum surface pressures from the central zone to the peripheral zone of the contact area. In the analysed case, the surface pressure values, similar to the height of 10 mm, was 0.05 MPa lower for the numerical model than the experimental values. During the course of the maximum values of surface pressures, in the cross-section passing through the central point of contact, a decrease of these values could be observed in the central part of the contact, indicating the formation of plastic deformation, as confirmed by Stopa et al. [20] and Komarnicki et al. [36]. Furthermore, they discovered that, in extreme cases, the surface pressure values in the central contact zone might decrease to zero.    [36] found the damage, as evidenced by the shift of maximum surface pressures from the central zone to the peripheral zone of the contact area. In the analysed case, the surface pressure values, similar to the height of 10 mm, was 0.05 MPa lower for the numerical model than the experimental values. During the course of the maximum values of surface pressures, in the cross-section passing through the central point of contact, a decrease of these values could be observed in the central part of the contact, indicating the formation of plastic deformation, as confirmed by Stopa et al. [20] and Komarnicki et al. [36]. Furthermore, they discovered that, in extreme cases, the surface pressure values in the central contact zone might decrease to zero.   Based on observations of the course, it can be concluded that this difference is not significant, because a higher value was obtained in the experimental studies, and no damage was found due to the shape of the distribution of surface pressures, whose maximum values were at the central point of contact. It can also be concluded that we are dealing with an elastic deformation. This confirms the absence of visible damage to the flesh tissue of the tested apple at the point of contact with the resistance element in the stroke test [20,36]. Based on the shape of the contact, it can be concluded that this difference may be due to the geometry of objects. The experimental object could have a smaller radius of curvature (longitudinal shape of the contact) [20,36], resulting in a higher surface pressure, while the numerical model is the ideal ball.   [36] found the damage, as evidenced by the shift of maximum surface pressures from the central zone to the peripheral zone of the contact area. In the analysed case, the surface pressure values, as opposed to a drop from a height of 10 mm, reached similar values. In the course of the maximum values of surface pressures in the cross-section passing through the central point of contact, a decrease of these values was observed in the central part of the contact, which indicates the formation of plastic deformation, as confirmed by Stopa et al. [20] and Komarnicki et al. [36]. Furthermore, they found that, in extreme cases, the surface pressure values in the central contact zone might decrease to zero. contact) [20,36], resulting in a higher surface pressure, while the numerical model is the ideal ball.   values was observed in the central part of the contact, which indicates the formation of plastic deformation, as confirmed by Stopa et al. [20] and Komarnicki et al. [36]. Furthermore, they found that, in extreme cases, the surface pressure values in the central contact zone might decrease to zero.

Conclusions
The relationship between load and deformation may become non-linear; a non-linear analysis should be carried out at this point to obtain realistic results that reflect the actual behaviour. In this context, a clear approach to the solution was identified as valuable in solving recharging cases, such as the impact/collision and drop test event. The explicit values was observed in the central part of the contact, which indicates the formation of plastic deformation, as confirmed by Stopa et al. [20] and Komarnicki et al. [36]. Furthermore, they found that, in extreme cases, the surface pressure values in the central contact zone might decrease to zero.

Conclusions
The relationship between load and deformation may become non-linear; a non-linear analysis should be carried out at this point to obtain realistic results that reflect the actual behaviour. In this context, a clear approach to the solution was identified as valuable in solving recharging cases, such as the impact/collision and drop test event. The explicit

Conclusions
The relationship between load and deformation may become non-linear; a non-linear analysis should be carried out at this point to obtain realistic results that reflect the actual behaviour. In this context, a clear approach to the solution was identified as valuable in solving recharging cases, such as the impact/collision and drop test event. The explicit dynamics system was designed to simulate non-linear structural mechanic applications. In complex applications, open methods are more appropriate and an open approach provides an alternative process to solving problems [39][40][41][42][43]. Today's technology allows us to efficiently perform non-linear and time-dependent impact load cases using simulations based on numerical methods. However, these types of non-linearities (geometry, contact, and/or non-linearity of material), including dynamic simulations, have not yet become major practices in studies related to the deformation of agricultural products, because of their interactions [44].
In the literature, there are examples of modelling the behaviour of fruits (e.g., pears, apples, or tomatoes) under stress in quasi-static conditions. The proposed models accurately reproduced real objects, in terms of geometry, using 3D scanning; however, the three-layer structures and differences in physical and mechanical properties of the tissues of individual layers (skin, flesh tissue, and seminal nest) were often omitted [29]. The presented research, which included modelling of surface pressures, could replace long-lasting and time-consuming experimental studies. Moreover, a FEM model can be used to analyse and design new devices, technologies, or systems, e.g., during transport or processing, in order to reduce the risk of damage, as well as real deformation of the tissues of plant origin, at the stage of harvesting, transport, storage, and processing.
For this reason, we decided to create a three-layered model of the 'Golden Delicious' apple variety, taking into account the interactions between the individual layers and using experimental studies to determine the properties (Poisson's number, modulus of elasticity) of individual layers.
The next step of our study will obtain a creation of geometric model that will reflect with high precision a real object. Future plans include research on other apple varieties and other types of agricultural products, to define the damage of flesh tissue based on surface pressure. The created models will be used to analyse harvesting, transport, and storage processes, in order to reduce the occurrence of damage and design new technological lines.
Author Contributions: D.S.: main author, worked on all parts of the paper. Ł.K.: worked on the introduction, results, discussion, and the conclusion. P.D.: worked on the material and methods, mainly the FEM model. R.S. advised when the main author was in doubt. All authors have read and agreed to the published version of the manuscript.
Funding: Financial support from the Wrocław University of Environmental and Life Sciences is gratefully acknowledged. The study was financed by Wrocław University of Environmental and Life Sciences from research project D220/0004/17, under the program "Innovative Doctorate".

Conflicts of Interest:
The authors declare no conflict of interest.