Elastic-plastic Material Behavior Research Articles

Micromechanical investigation of DLC based coating systems by means of combined characterization and simulation methods

The assessment of the elastic-plastic material behavior is imperative for benchmarking the performance of thin coating systems. Linear-elastic approaches can lead to deviating results and experimental investigations, such as fatigue tests, frequently overlook the intricacies of the failure mechanism. Therefore, the elastic-plastic material behavior of DLC based coating systems with CrN support interlayer, deposited on 100Cr6 substrates and Si, was investigated. Using an inverse approach, a nanoindentation simulation with spherical indenter geometry was iteratively optimized to experimental curves by adjusting the material parameters Young's modulus E, yield strength Y, hardening coefficient K and hardening exponent n for DLC and CrN layers. Mechanical parameters were measured and the yield strength of DLC was analyzed with a scratch procedure to implement appropriate initial values. The results showed an overall good agreement between the measured and optimized force-displacement curves with reasonable determined material parameters. The control parameter Y exhibited nearly identical outcomes for scratch measurements and the calculated value from simulations although DLC was deposited on different substrates. The employment of a spherical indenter tip mitigates the influence of the microstructure in the CrN layer through homogenization effects and a larger affected area of the plastic zone. Furthermore, enhancing the performance of the coating systems can be achieved by increasing the DLC or CrN interlayer thickness.

A covariant formulation for finite strain modelling of orthotropic elasticity and orthotropic plasticity with plasticity-induced evolution of orthotropy: Application to natural fibres

We introduce a rate-independent model for orthotropic elastic and orthotropic plastic material behaviour in a hyper-elasto-plastic framework at finite strains. The model is based on the postulate of covariance and does not rely on a multiplicative decomposition of the deformation gradient. Furthermore, a plastic-deformation-induced evolution of orthotropy is considered, similar to the notion of plastic spin. We propose that the orthotropic strain energy function and the orthotropic yield criterion are guided by identical structural tensors that evolve with plasticity. The modelled material behaviour is significant for natural fibres such as flax, hemp, or pulp fibres. Our formulation has three findings. Firstly, the covariant formulation of plasticity provides rate equations for the plastic variables and the structural tensors suitable for reproducing stress–strain diagrams of natural fibres. Secondly, the introduction of plastic-deformation-induced evolution of orthotropy in the proposed covariant setting results in a non-associative plasticity algorithm. Thirdly, the covariant setting allows the incorporation of suitable constitutive equations for the structural tensors to evolve orthotropy. The latter successfully models the stiffness increase in stress–strain diagrams of cyclic tensile tests of natural fibres.

Linking the microstructure with strain-life curves for improved utilization of the lightweight potential of thick-walled nodular cast iron

The assessment of the fatigue life of critical, highly stressed component areas is an important aspect in the design of thick-walled components for wind turbines made of nodular cast iron. With the demand for resource efficiency through lightweight design combined with a certain capability to withstand extreme loads and special events, this issue is becoming increasingly important. However, there are still gaps in the description of local strain-based material behaviour. For the design and local strain-based fatigue assessment of large cast components, possibilities to determine the fatigue life as accurately as possible must be provided by standards and guidelines. The paper aims to provide a wide range of material and fatigue data from one common source for general design interests. During research projects at Fraunhofer LBF, material properties for the PRAM-based fatigue life assessment and the elastic-plastic material behaviour have been derived from 9 nodular cast iron grades with 44 microstructural variations. To describe the influence of the technological size effect on cyclic stress-strain and PRAM-N curves the quasi-static parameters (tensile strength and elongation at break) and the microstructure (nodularity and graphite particle density) are considered. In addition, suggestions for scatter bands in the strain-based design approach are given to the designers of thick-walled GJS components.

Study on the applicability of a modified strain approach to predict the fatigue life of HFMI-treated transverse stiffeners under variable amplitude loading

AbstractThis paper analyses the applicability of a modified strain approach to predict the fatigue life of HFMI-treated transverse stiffeners under variable amplitude loading (VAL) with random load sequences of a p(1/3) and linear shaped spectrum. Local stresses are determined using linear-elastic finite element analyses. The measured weld geometry and component imperfections are considered. From the hardness of the HFMI-treated zone and the base material, the elastic–plastic material behaviour and Coffin-Manson parameters to describe the damage parameter Woehler curve are estimated. Based on a hysteresis counting method (HCM), the damage for each closed hysteresis is calculated. The applied notch strain approach includes the impact of residual stresses and the influence of surface roughness. Thus far, the application of similar approaches has only been validated for welded components with comparatively low residual stresses and HFMI-treated welds subjected to constant amplitude loading. To validate the accuracy of the approach for HFMI-treated welds under variable amplitude loading, the approximated fatigue life is compared to the number of cycles derived from experimental investigations. In this study, it is shown in conjunction with experimental results that it is essential to consider the strength of the base material near the weld when assessing the service life. This area can be more critical than the HFMI-treated weld toe.

Strain-life behavior of thick-walled nodular cast iron

Abstract For the design of thick-walled cast iron components used for wind energy turbines, large machines, presses, and engine parts made of nodular cast iron, the assessment of fatigue life of critical component areas is a crucial point to consider. Importance increases with the demand to safe resources by applying a lightweight design in combination with a certain resistance against extreme loads and misuse situations. Moreover, this gets more important when notches or deviations in the local microstructure are present. Nevertheless, due to accessibility, a lack in the description of the local strain-based material behavior still exists. Especially new cast iron grades such as ADI are not well investigated but offer an often-discussed potential for a lightweight design of the structure or to increase the highest admissible load on the structure. To determine the cyclic material behavior of nodular cast iron, different grades EN-GJS-400-18LT, EN-GJS-450-18, EN-GJS-500-14, EN-GJS-600-3, EN-GJS-700-2, and ADI in different wall thicknesses were investigated concerning their elastic–plastic material behavior under strain control. Moreover, new possibilities for the trilinear description of the strain-life and the stress–strain curve and the statistical and technological size effect are discussed in the following based on cyclic stress–strain and strain-life curves.

Non-equibiaxial residual stress evaluation methodology using simulated indentation behavior and machine learning

Measuring the residual stress in the components in nuclear power plants is crucial to their safety evaluation. The instrumented indentation technique is a minimally invasive approach that can be conveniently used to determine the residual stress in structural materials in service. Because the indentation behavior of a structure with residual stresses is closely related to the elastic-plastic behavior of the indented material, an accurate understanding of the elastic-plastic behavior of the material is essential for evaluation of the residual stresses in the structures. However, due to the analytical problems associated with solving the elastic-plastic behavior, empirical equations with limited applicability have been used. In the present study, the impact of the non-equibiaxial residual stress state on indentation behavior was investigated using finite element analysis. In addition, a new nonequibiaxial residual-stress prediction methodology is proposed using a convolutional neural network, and the performance was validated. A more accurate residual-stress measurement will be possible by applying the proposed residual-stress prediction methodology in the future.

Overload and variable amplitude load effects on the fatigue strength of welded joints

Different load spectra and individual load peaks might substantially relax high residual stresses as well as induce compressive residual stresses in welded components and, consequently, affect the fatigue performance of these joints. Consideration of peak loads and resulting relaxation of residual stresses in fatigue analyses can substantially enhance the accuracy of life prediction. The aim of the current study is to experimentally investigate the fatigue strength of welded joints subjected to different levels of overloads and variable amplitude (VA) loads and to develop local fatigue assessment method to account for the relaxation of residual stresses via a mean stress correction using the 4R method. The 4R method applies a local stress ratio for the mean stress correction considering material strength, residual stresses, applied stress ratio of external loading and local weld geometry in elastic–plastic material behaviour. Fatigue tests were carried out on fillet-welded longitudinal gusset joints made of S700 high-strength steels under applied stress ratio R = 0–0.1. A mild strength steel (S355) and ultra-high-strength steel (S1100) were selected as reference steel grades for the fatigue testing to study the material strength effects. Numerical analyses were conducted to evaluate the fatigue notch factors using the effective notch stress concept with the reference radius of rref = 1.0 mm and theory of critical distance (TCD) using the point method. The experimental results indicated that a substantial improvement in the fatigue strength capacity can be claimed in the joints subjected to tensile overloads, particularly in the studied S700 and S1100 steels. The higher-level overload (0.8fy), corresponding to the nominal cross-sectional area, improved the mean fatigue strength of the welded joints manufactured of high-strength S700 steel by approximately 60%, while the lower overload (0.6fy) improved the mean fatigue strength by 20%. In addition, a use of equivalent nominal stresses for the joints subjected to VA loads resulted in conservative assessments when employing S–N curves obtained for the CA loading. The 4R method, via the local mean stress correction for individual cycles, provided higher accuracy for the fatigue assessments.

Multi-Layer Fabric Composites Combined with Non-Newtonian Shear Thickening in Ballistic Protection-Hybrid Numerical Methods and Ballistic Tests.

Multi-layer fabrics are commonly used in ballistics shields with a lower bulletproof class to protect against pistol and revolver bullets. In order to additionally limit the dynamic deflection of the samples, layers reinforced with additional materials, including non-Newtonian fluids compacted by shear, are additionally used. Performing a wide range of tests in each case can be very problematic; therefore, there are many calculation methods that allow, with better or worse results, mapping of the behavior of the material in the case of impact loads. The search for simplified methods is very important in order to simplify the complexity of numerical fabric models while maintaining the accuracy of the results obtained. In this article, multi-layer composites were tested. Two samples were included in the elements subjected to shelling. In the first sample, the outer layers consisted of aramid fabrics in a laminate with a thermoplastic polymer matrix. The middle layer contained a non-Newtonian shear-thickening fluid enclosed in hexagonal (honeycomb) cells. The fluid was produced using polypropylene glycol and colloidal silica powder with a diameter of 14 µm in the proportions of 60/40. The backing plate was made using a 12-layer composite made of Twaron® para-aramid fabrics with a DCPD matrix-not yet used in a wide range of ballistics. Then, numerical simulations were carried out in the Abaqus/Explicit dynamic analysis. The Johnson-Cook constitutive strength model was used to describe the behavior of elastic-plastic materials constituting the elements of the projectiles. For the non-Newtonian fluid, a Up-Us EOS was used. The inner layers of the fabric were treated as an orthotropic material. Complete homogenization of the sample layers was carried out, thanks to which each layer was treated as a homogeneous continuum. As a parameter of fracture mechanics for shield components, the strain criterion was used with the smooth particles hydrodynamics method (SPH). Then, the results of simulations were compared with the results of the ballistic test for both samples placed next to each other, which resulted in the formation of a multi-layer composite in one ballistic test subjected to impact loads during firing with a 9 × 19 mm Parabellum FMJ projectile with an initial velocity of 370 ± 10 m/s. The results of numerical tests are very similar to the ballistic tests, which indicates the correct mapping of the process and the correct conduct of layer homogenization. The applied proportions of the components in the non-Newtonian fluid allowed a reduction in the deflection compared to previous studies. Additionally, the proposal to use a DCPD matrix allowed to obtain a much lower deflection value compared to other materials, which is a novelty in the field of production of ballistic shields.

Mechanics and modeling of cold rolling of polymeric films at large strains — A rate-independent approach

Recently a new phenomenon of bonding polymeric films in solid-state, via symmetric rolling, at ambient temperatures (≈20 °C) well below the glass transition temperature (Tg≈78°C) of the polymer has been reported. In this new type of bonding, polymer films are subject to plane strain active bulk plastic compression between the rollers during deformation. Here, we analyze these plane strain cold rolling processes, at large strains but slow strain rates, by finite element modeling. We find that at low temperatures, slow strain rates, and moderate thickness reductions during rolling (at which the Bauschinger effect can be neglected for the particular class of polymeric films studied here), the task of material modeling is greatly simplified and enables us to deploy a computationally efficient, yet accurate, finite deformation rate-independent elastic–plastic material behavior (with the inclusion of isotropic-hardening). The finite deformation elastic–plastic material behavior based on (i) the additive decomposition of stretching tensor ( D = De+Dp, i.e., a hypoelastic formulation) with incrementally objective time integration and, (ii) multiplicative decomposition of deformation gradient (F=FeFp) into elastic and plastic parts, are programmed and carried out for cold rolling within Abaqus Explicit. Frictional interactions are modeled using a consistent rate-independent Coulombic law. Predictions from both formulations, i.e., hypoelastic and multiplicative decomposition, closely match the experimentally observed rolling loads. No specialized hyperelastic/visco-plastic material model is required to describe the behavior of the particular blend of polymeric films under the conditions described here, thereby significantly speeding up the computation for steady-state rolling simulations. It is revealed that under the deformation conditions when principal axes show negligible rotation, hypoelastic formulation can be valid at large elastic stretches. Moreover, the use of classical rigid-plastic modeling (often applicable to metals) is found to greatly underestimate the rolling loads for polymers due to large elastic stretches in the polymer films at large strains. Deformation aspects of solid-state polymeric sheets presented here are expected to facilitate the development of new processes involving (or related to) cold roll-bonding of polymers.

Effect of Surface Finishing State on Fatigue Strength of Cast Aluminium and Steel Alloys

The endurance limit of structural mechanical components is affected by the residual stress state, which depends strongly on the manufacturing process. In general, compressive residual stresses tend to result in an increased fatigue strength. Post-manufacturing processes such as shot peening or vibratory finishing may achieve such a compressive residual stress state. But within complex components, manufacturing-process-based imperfections severely limit the fatigue strength. Thus, the interactions of imperfections, residual stress state and material strength are key aspects in fatigue design. In this work, cast steel and aluminium alloys are investigated, each of them in vibratory finished and polished surface condition. A layer-based fatigue assessment concept is extended towards stable effective mean stress state considering the elastic–plastic material behaviour. Murakami’s concept was applied to incorporate the effect of hardness change and residual stress state. Residual stress relaxation is determined by elastic–plastic simulations invoking a combined hardening model. If the effective stress ratio within the local layer-based fatigue strength is evaluated as critical distance value, a sound calculation of fatigue strength can be achieved. Summing up, the layer-based fatigue strength design is extended and features an enhanced understanding of the effective stabilized mean stress state during cyclic loading.

Micromechanical deformation modeling and failure prediction of thermoplastic composites

Based on the multi-scale progressive homogenization theory, a micromechanical model is proposed that can predict the plastic deformation of thermoplastic fiber-reinforced composites. By developing microscopic representative volume elements (RVEs) that include three materials phases: fiber, matrix and fiber–matrix interface, the effect of elastic–plastic material behaviors and interface debonding on the mechanical properties of thermoplastic composites can be considered. Employing this model, the deformation and failure process of carbon fiber reinforced PEEK composite is studied, considering the effects of random size and distribution of carbon fiber, as well as the ratio of interface debonding. It can be found that the yielding strength is most sensitive to the proportion of interface debonding compared to Young’s modulus and ultimate strength. Interface debonding has the most significant impact on the mechanical performances of thermoplastic composites by introducing a higher degree of plastic deformation into the polymer matrix and changing the evolution path of material damage.

A Numerically Efficient Method to Assess the Elastic–Plastic Strain Energy Density of Notched and Imperfective Cast Steel Components

The fatigue strength of cast steel components is severely affected by manufacturing process-based bulk and surface imperfections. As these defect structures possess an arbitrary spatial shape, the utilization of local assessment methods is encouraged to design for service strength. This work applies the elastic–plastic strain energy density concept to study the fatigue strength properties of a high-strength cast steel alloy G12MnMo7-4+QT. A fatigue design limit curve is derived based on non-linear finite element analyses which merges experimental high-cycle fatigue results of unnotched and notched small-scale specimens tested at three different stress ratios into a unique narrow scatter band characterized by a scatter index of 1:TΔW¯(t)=2.43. A comparison to the linear–elastic assessment conducted in a preceding study reveals a significant improvement in prediction accuracy which is assigned to the consideration of the elastic–plastic material behaviour. In order to reduce computational effort, a novel approximation is presented which facilitates the calculation of the elastic–plastic strain energy density based on linear–elastic finite element results and Neuber’s concept. Validation of the assessment framework reveals a satisfying agreement to non-linear simulation results, showing an average root mean square deviation of only approximately eight percent in terms of total strain energy density. In order to study the effect of bulk and surface imperfections on the fatigue strength of cast steel components, defect-afflicted large-scale specimens are assessed by the presented elastic–plastic framework, yielding fatigue strength results which merge into the scatter band of the derived design limit curve. As the conducted fatigue assessment is based solely on linear–elastic two-dimensional simulations, the computational effort is substantially decreased. Within the present study, a reduction of approximately 400 times in computation time is observed. Hence, the established assessment framework presents an engineering-feasible method to evaluate the fatigue life of imperfective cast steel components based on rapid total strain energy density calculations.

Cytotoxic, Elastic-Plastic and Viscoelastic Behavior of Aged, Modern Resin-Based Dental Composites

The development of resin-based composites (RBCs) is a delicate balance of antagonistic properties with direct clinical implications. The clear trend toward reducing filler size in modern RBCs solves esthetic deficiencies but reduces mechanical properties due to lower filler content and increases susceptibility to degradation due to larger filler-matrix interface. We evaluated a range of nano- and nano-hybrid RBCs, along with materials attempting to address shrinkage stress issues by implementing an Ormocer matrix or pre-polymerized fillers, and materials aiming to provide caries-protective benefit by incorporating bioactive fillers. The cytotoxic response of human gingival fibroblast (HGF) cells after exposure to the RBC eluates, which were collected for up to six months, was analyzed using a WST-1 assay. The microstructural features were characterized using a scanning electron microscopy and were related to the macroscopic and microscopic mechanical behaviors. The elastic-plastic and viscoelastic material behaviors were evaluated at the macroscopic and microscopic levels. The data were supplemented with fractography, Weibull analysis, and aging behavioral analysis. The results indicate that all RBCs are non-cytotoxic at adequate exposure. The amount of inorganic filler affects the elastic modulus, while only to a limited extent the flexural strength, and is well below the theoretical estimates. The nanoparticles and the agglomeration of nanoparticles in the RBCs help generate good mechanical properties and excellent reliability, but they are more prone to deterioration with aging. The pre-polymerized fillers lower the initial mechanical properties but are less sensitive to aging. Only the Ormocer retains its damping ability after aging. The strength and modulus of elasticity on the one hand and the damping capacity on the other are mutually exclusive and indicate the direction in which the RBCs should be further developed.

Dynamic plastic response and impulse saturation of rectangular plates subjected to pressure pulse loading

A combination of the Membrane Factor Method (MFM) and Saturation Analysis (SA) has proven to be a simple yet effective theoretical tool to predict the dynamic inelastic response of various structures subjected to long-duration pressure pulse loading. In this study, this theoretical tool is extended to the analysis of rectangular plates. Boundary conditions are treated in a unified manner in the analysis, so that the influence of the boundary conditions on the saturation phenomenon is clarified based on MFM. By taking the transient response phase and the exact yield conditions into consideration, complete solutions of saturation deflection and impulse are obtained, and the effect of the aspect ratio of rectangular plates is examined. Results from the present methods are also compared with finite element simulation utilizing elastic-plastic material behavior and modal solutions. The effects of the yield condition adopted and the transient response phase on the saturation phenomenon of rectangular plates under pressure pulse loading are respectively discussed.

Modelling residual stress and residual work hardening induced by surface mechanical attrition treatment

Shot peening induced residual stress is strongly related to residual work hardening which relies on both the process and the target material. In this work, residual stress and coupled residual work hardening are studied in a surface mechanical attrition treatment (SMAT) on a cylindrical structure by combining discrete element method (DEM) and finite element method (FEM). Shot, ultrasonic parameters, chamber structure, sonotrode surface, and cylindrical structure of the target are finely modeled in DEM simulation. On the basis of the DEM results, FEM simulation of the SMAT process is performed on a representative area considering the factual impact parameters and elastic-plastic behavior of the target material, and mechanism of residual stress generation is analyzed. It shows that isotropic hardening and kinematic hardening of the target material play a different role in residual stress generation. The obtained maximum compressive residual stress is mainly determined by residual yield stress, while the residual kinematic hardening is nearly unchanged when the coverage is increased. Further analyses indicate that the mechanisms of residual stress generation should be very different for the Johnson-Cook model and the model that includes kinematic hardening, even though these two kinds of models are reported to be able to well predict the shot peening induced residual stress. The gradient of the residual work hardening can be used for further evaluating mechanical properties of the SMATed cylindrical structure.

Edge Pressures Obtained Using FEM and Half-Space: A Study of Truncated Contact Ellipses

In rolling or gear contacts, truncation of the contact ellipse can occur, for example, when an undercut extends into the contact area. For an elastic calculation approach, the edge constitutes a mathematical singularity, which is revealed by a theoretically infinitely high pressure peak. However, when elastic–plastic material behavior is taken into account, the pressure peak is limited by local hardening and yielding of the material, leading to plastic deformations. As a result, those calculations are rather challenging and the results partly unexpected due to the discontinuity contained in the geometry. Nevertheless, to the authors’ knowledge, hardly any published studies exist on elastic–plastic simulations of truncated contact ellipses. Therefore, a numerical study concerning the contact of a rigid ball with an elastic–plastic plane is presented. Due to an undercut in the plane, a quarter of the theoretical Hertzian contact ellipse is cut off. The aim of the study is to investigate the influence of the undercut angle on the pressure distribution and the elastic and plastic deformation at the edge. The use of FEM shows that the undercut angle has a significant effect on the characteristics of the contact. The results obtained using FEM are then used as a reference for comparison with a semi-analytical method (SAM). It is shown that the SAM, based on the half-space, provides comparable results only for very small undercut angles.

Computational model for bending fatigue prediction of surface hardened spur gears based on the multilayer method

A computational model for predicting both the location and the required number of cycles for bending fatigue failure in surface hardened spur gears is proposed. Linear elastic stresses and strains in a single tooth bending fatigue test are corrected for elastic–plastic material behavior. The tooth root region of spur gear is divided into layers. Fatigue properties are assigned to each layer via the hardness method. Based on the multiaxial fatigue criteria, fatigue failure location and corresponding fatigue lives are estimated. Predicted fatigue lives, failure locations, and transition from surface to subsurface fatigue failure show good agreement with the experimental results.

Analysis of Stresses and Strains in Stainless Steel 316 L Tubes Subjected to Die Expansion

Abstract SS316 L finned tubes are becoming very popular in high-pressure heat exchangers and particularly in CO2 cooler applications. Due to the high pressures present during operation, these tubes require an accurate residual stress evaluation generated by the die expansion process. Die expansion of air cooler fin tubes creates not only high stresses that can surpass the ultimate tensile strength when combined with operation stresses but microcracks during expansion when the process is not well controlled. This research work aims to study the elastic-plastic behavior and estimate the residual stress state of fin tubes subjected to the die expansion process. The stresses and deformations of the expanded SS316 L tube are analyzed numerically using the finite element method. The expansion and contraction process are modeled considering the elastic–plastic material behavior for different die sizes. The maximum longitudinal, tangential, and contact stresses are evaluated to verify the critical stress state of the joint during the expansion process. The importance of the material behavior in evaluating the residual stresses using kinematic and isotropic hardening is addressed. Finally, an experiment was conducted to assess the tangential and longitudinal strains of a 3/8 in. stainless steel tube subjected to expansion with an oval-shaped die.

Extension of the Freiberg Layer Model by Means of Solidification for Roll Casting

Finite element method (FEM) simulations offer deep and detailed modeling of complex processes, but with a high effort of computation power and time. For the special purpose of flat rolling, a layer model is developed by Schmidtchen earlier based on the classic elementary theory of plasticity with the purpose to model inhomogeneous behavior with a significant lower effort at computation. In a previous work, this model was extended with elastic–plastic material behavior. Herein, the model is extended by a viscose zone at the entry for the modeling of roll casting. It is now possible to model the inhomogeneous solidification of a melt in the roll gap and proceed with elastic–plastic inhomogeneous flat rolling in the solid region. The results are compared with FEM simulations from the literature and found to fit well together. But, the current model offers more plausible stress curves in the solid region, because an elastic–plastic material model is applied, where common FEM simulations are only able to work with viscose material laws in the whole roll gap.

Determination of Local Stresses and Strains within the Notch Strain Approach: The Efficient and Accurate Calculation of Notch Root Strains Using Finite Element Analysis

For the service life estimation of metallic components under cyclic loading according to strain-based approaches, a simulation of the elastic-plastic stress–strain path at the point of interest is necessary. An efficient method for determining this stress–strain path is the use of the load–notch-strain curve, as this is also implemented within the FKM guideline nonlinear. The load–notch-strain curve describes the relationship between the load on the component and the local elastic-plastic strain. On the one hand, this can be estimated from loads or theoretical elastic stresses by using notch root approximations. On the other hand, this can be determined in a finite element analysis based on the elastic-plastic material behaviour. This contribution describes how this latter option is carried out in general and how it can be optimised in such a way that the FEA requires significantly less calculation time. To show the benefit of this optimisation, a comparative calculation on an exemplary geometry is carried out.