Vol.:(0123456789) The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 https://doi.org/10.1007/s00170-025-15653-1 ORIGINAL ARTICLE Simulation of the flexural behavior of prestressed fiber‑reinforced polymer concrete Michelle Engert1  · Konstantin Frankenbach1 · Kim Torben Werkle1 · Hans‑Christian Möhring1 Received: 30 January 2025 / Accepted: 26 April 2025 / Published online: 15 May 2025 © The Author(s) 2025 Abstract Recently, prestressed fiber-reinforced polymer concrete (PFRPC) has shown potential for use in highly stressed machine components. In particular, the low density and thermal and dynamic properties of the material should be mentioned. How- ever, at present, there are still no possibilities for dimensioning the hybrid material, which consists of the granular material polymer concrete and prestressed carbon fibers. In particular, the numerical representation of the residual stress field poses a challenge. This paper is divided into 4 sections. First, the bending properties of pure polymer concrete, fiber-reinforced polymer concrete and PFRPC are determined experimentally. In the subsequent section, the numerical modeling of pure polymer concrete is carried out, first comparing various numerical and analytical models for determining the modulus of elasticity of granular materials. The model is then extended to include the integration of carbon fiber rovings, followed by an investigation into various methods for mapping the residual stress field. The Caquot model (15% accuracy) was found to be particularly suitable for mapping the polymer concrete, and a subdivision of the material into tensile and compressive areas proved to be essential. By determining the mechanical properties of the impregnated carbon fiber rovings using representa- tive volume elements, the fiber-reinforced polymer concrete could be mapped with an accuracy of 2.2%. The integration of the selected models, in conjunction with the introduction of a homogeneous residual stress field derived from experimental values, enabled the successful representation of the bending test with an accuracy of 12.6%. Keywords Simulation · Composite · Residual stress · Bending 1 Introduction Prestressed fiber-reinforced polymer concrete represents an advancement in machine tool construction and, according to [1], has the potential to improve vibration damping of machine tool structures while simultaneously reducing mass. The basis of the hybrid material is the vibration damping material polymer concrete [2]. Polymer concrete is an inho- mogeneous, granular material consisting of a reaction resin matrix and mineral fillers. Since the material became known with the development of cold-curing resins in the 1970 s, it has been used in the machine tool industry in the form of machine beds [3]. Further positive properties of polymer concrete can be found in its low density, high thermal sta- bility [3], and low production-related CO2 emissions [4]. In order to increase the strength of polymer concrete, fiber materials have already been integrated in numerous studies [5–12]. Suitability for highly stressed machine structures, however, only results from the integration of prestressed car- bon fiber rovings [13]. A machine component made of this material was presented for the first time in [1]. In addition to the enormous lightweight construction potential, the study showed above all that the dynamic and thermal properties can be improved compared to a conventional steel compo- nent. However, the choice of material impairs the static stiff- ness of the component. The authors assume that this effect could be counteracted by increasing the fiber volume frac- tion [1]. Due to the high material cost of the epoxy resin [14] used as the matrix material in the study [1], an experimental test of this hypothesis would involve a high financial outlay. Alternatively, a simulative test could be considered. How- ever, there is currently no validated method for the numeri- cal representation of prestressed fiber-reinforced polymer concrete. In this study, simulation methods known from the literature are used and extended in order to develop a * Michelle Engert michelle.engert@ifw.uni-stuttgart.de 1 Institute for Machine Tools (IfW), University of Stuttgart, Holzgartenstraße 17, 70174 Stuttgart, Germany http://crossmark.crossref.org/dialog/?doi=10.1007/s00170-025-15653-1&domain=pdf http://orcid.org/0000-0002-8099-963X 2592 The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 suitable method for the design of future components made of prestressed fiber-reinforced polymer concrete. In addition to methods known from literature for the numerical representa- tion of polymer concrete, known methods for the simulation of cement concrete are also used. Furthermore, the find- ings are supplemented by a proposal for the simulation of impregnated roving bundles, as well as several approaches for the representation of residual stress resulting from the prestressing. Uniaxial bending tests are performed to vali- date the simulation results. 2 Materials The subject of this investigation is the self-compacting poly- mer concrete EPUMENT 140/5 manufactured by RAMPF Machine Systems GmbH & Co. KG. The maximum grain size is 5 mm. As the precise composition of the grain mix- ture is uncertain, the aggregates are assumed to form a homogeneous mixture consisting of quartzitic particles. A Bisphenol-A based epoxy resin with a mass fraction of 9.2% is employed as binder. In addition to pure polymer concrete, the investigation comprises fiber-reinforced and prestressed fiber-reinforced polymer concrete. As fiber-reinforcement 24k carbon fiber rovings of type GRAFIL 34—700 (Mit- subishi Chemical Carbon Fiber and Composites) are applied, the fiber volume fraction varies between 0.2% (fiber-rein- forced samples) and 1.1% (prestressed samples). Table 1 provides an overview of the material characteristics of the used materials. Furthermore, the material characteristics of the polymer concrete itself are presented. All test samples are executed in identical dimensions of 50 × 50 × 500 mm3. The fiber-reinforced samples are rein- forced at five locations with one roving each (see Fig. 1). In contrast, the prestressed samples are reinforced at identical positions with six rovings each. During the manufacturing process of the prestressed samples, a total prestressing force of 3000 N is applied. The manufacturing process adheres to the recommendations set out in [1]. 3 Experimental investigation of the flexural properties To enable subsequent validation of the simulation results, it is necessary to supplement the parameters derived from literature with specific test results. In particular, data on the flexural properties of fiber-reinforced and prestressed fiber-reinforced polymer concrete must be determined. For this purpose, the samples are clamped between two steel plates using 4 screws over a clamping length of 60 mm, with a torque of 80 Nm each. At a distance of 405 mm from the clamping, the force is applied by means of a hydraulic cylinder. A load cell (KM26-10kN, ME-Meßsys- teme) situated between the cylinder and the samples trans- mits and measures the applied force. The deflection of the samples is measured using a dial gauge (Hahn & Kolb) placed vertically above the load cell. The test setup is illus- trated in Fig. 2. Table 1 Material characteristics known from literature  Density Tensile modulus Compression modulus Poisson’s ratio Volume ratio ρ E+ E- ν v [g/cm³] [GPa] [GPa] [-] [-] Epoxy resin 1.2 9.2 3.9 0.375 17.1% [15] [16] [17] [18] Aggregates 25.5 63.8 0.250 82.9% [19] [17] Carbon fibers 1.8 234 0.310 0.2 … 1.1% [20] [20] [20] Polymer concrete 2.3 32.5 0.222 [21] [21] [22] Fig. 1 Schematic illustration of the test samples with integrated car- bon fiber rovings 2593The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 In evaluating the test results, the primary focus is on determining the breaking force and the maximum deflec- tion. At least five samples for each material were tested. For further evaluation of the flexural behavior prestressed samples, the first failure is also determined. According to [13], this is defined as a drop in force of 10%. Observations revealed the formation of micro cracks close to the clamping at this point of the flexural tests. When further progressing the flexural tests, more micro cracks appear. This behavior is comparable to the behavior of conventional prestressed concrete in flexural tests [23]. The Young’s modulus of the samples can be calculated using Formula 1, considering the force at first failure F10%, the distance of the force applica- tion point to the restraint l, the deflection at first failure f10%, and the height h of the samples. The averaged test results are listed in Table 2. 4 Simulation of pure polymer concrete In order to map the mechanical properties of prestressed fiber-reinforced polymer concrete, it is first necessary to map the mechanical properties of pure polymer concrete with (1)E = F10% ⋅ l3 3 ⋅ f10% ⋅ I = 4 ⋅ F10% ⋅ l3 f10% ⋅ h4 sufficient accuracy. The initial step is to compare the various methods for determining the Young’s modulus of polymer concrete. In the subsequent step, the flexural test described in Section 3 is mapped in the simulation environment. All results are then validated using the real test results. 4.1 Determination of the Young’s modulus of polymer concrete To numerically assess the flexural behavior of a material, some material properties as the Young’s modulus are indis- pensable. As in some cases, experimental investigations might not be feasible, potentially in fact of high costs or the high number of material parameters in case of composite materials as polymer concrete, in the past, both analytical and numerical methods for the determination of the Young’s modulus of composite materials were presented within lit- erature. In the case of PFRPC, there are a number of mate- rial parameters that need to be taken into account. These include, in particular, the resin content and the grading curve. The resulting extensive range of parameters and the high curing time of the polymer concrete of 24 h lead to a significantly high time expenditure for experimental inves- tigations. Within this section, examples for both numerical and analytical methods will be presented and compared to the experimentally determined Young’s modulus. Fig. 2 Test setup for the experimental assessment of the flexural properties Table 2 Test results Pure Fiber-reinforced Prestressed Breaking force Fmax (N) 1324.07 1421.82 4325.8 Maximum deflection fmax (mm) 2.87 3.94 40.86 Failure force F10% (N) 1324.07 1421.82 1891.9 Failure deflection f10% (mm) 2.87 3.94 4.49 Young’s modulus E (GPa) 19.72 16.15 21.56 2594 The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 The most commonly referenced models in the literature for determining the Young’s modulus of granular materials, such as polymer concrete, are the Reuss series model, the Hashin model, and the Caquot model [24]. The series model conceptualizes granular materials as consisting of two phases superimposed in the load direc- tion, both subjected to the same stress. This model neglects the influence of the bonding zone between the phases [25]. The Young’s modulus EPC is calculated using Formula 2 [25], which incorporates the Young’s modulus of the matrix EM and the aggregates EA and the volume fraction of the matrix υM and of the aggregates υA [25]. Utilizing the values presented in Table 1, the tensile modulus is determined to be 19.74 GPa, while the compression modulus is calculated to be 18.01 GPa. This means a deviation of − 44.58% from the tensile modulus attained in the bending tests and respectively a deviation of + 0.1% in comparison to the compression modulus given on the data sheet [21]. Reasons for the high deviations might be the neglection of the bonding zone and the simplification of the load case. Additionally, the different grain sizes are not respected in this model. The Hashin model divides granular heterogeneous mate- rials in composite elements. Each element is constituted by a spherical aggregate that is concentrically surrounded by a shell out of the matrix material. The volume ratio is taken into account at the level of each composite element. As the loads are initiated by this surrounding surface, the compres- sion modulus KPC is calculated using Formulae 3, 4, and 5 [24, 26]. This is based on the compression modulus of the matrix KM and the aggregates KA, their Young’s modulus E, and their Poison’s ratio ν. The tensile modulus is 26.4 GPa (+ 58.6% compared to test results); the compression modu- lus is calculated to be 13.47 GPa (+ 33.9% compared to test results). The Hashin model shows even higher deviations than the series model even though a surrounding shell is respected. This indicates that the load case is not adequately represented by the model. In particular, the model cannot take into account a random arrangement of the mineral fillers, nor does it take into account the different grain sizes. A more in-depth assess- ment of the limitations of the model is not possible due to the limited scope of this paper. (2)EPC = EMEA EM�M + EA�A (3)KPC = KM + g ⋅ (KA − KM) 1 + (1−g)⋅KA−KM KA− 4 3 ⋅�M (4)K = E (1 − 2 ⋅ v) ⋅ 3 In 1935, Caquot proposed an empirical law for the cal- culation of the volume concentration of the matrix. Prereq- uisite is the assumption that all aggregates are enveloped by the matrix (see Formula 6) [24]. The minimum and maximum particle sizes, d and D, respectively, are taken into account [24]. The combination of the law put forth by Caquot with the findings of Hansen et al. leads to an tensile modulus of 15.99 GPa (− 11.1% compared to test results) under the assumption that the Poisson’s ratio of both the matrix and the aggregates equals 0.2 (see Formula 7) [24]. The compression modulus is determined to be 28.88 GPa (− 18.9% compared to test results). The calculation includes the Young’s modulus of the matrix EM and the aggregates EA. The Caquot model shows the lowest deviations out of the analytical methods. This might be due to the considera- tion of the minimum and the maximum grain size. However, the model is still limited as it cannot take into account the random arrangement of the mineral fillers. Next to the presented analytical methods, there are vari- ous numerical methods for the calculation of the Young’s modulus. Within this study, the method of the representative volume elements (RVE) will be presented as it was already successfully used for polymer concrete [27]. Next to the by Ciupan used three-dimensional representative volume ele- ments, also two dimensional volume elements will be taken into account. To reduce the computational effort required for the three-dimensional solid element, it is subdivided into three distinct layers. In the nomenclature used in the following sections, the smallest element is referred to as layer 1. This smaller element acts as the matrix for the next larger element, ensuring that even the smallest filler particles are accounted for in the calculations. Consequently, the RVE for layer 1 has an edge length of 0.2 mm, with an average filler particle diameter of 0.05 mm. Based on the grading curve analysis, a filler volume fraction of 45% is assumed. In accordance with the findings of Zhou et al., the geo- metric configuration of the aggregates is observed to bear resemblance to that of polyhedral [28]. In order to facili- tate the meshing process, only icosahedra were employed in this study (see Fig. 3a). The minimum distance between the randomly arranged filler particles is 0.001 mm. This also ensures that each filler grain is surrounded by matrix, (5)� = E 2 ⋅ (1 + �) (6)g = 1 − 0,47 ⋅ ( d D )0,2 = 1 − g∗ (7)EPC = (2 − g∗) ⋅ EM + g∗ ⋅ EA g∗ ⋅ EA + (2 − g∗) ⋅ EA ⋅ EA 2595The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 as would be expected in a realistically mixed system. The elementary cell is meshed with tetrahedral elements, as illustrated in Fig. 3b. The periodic boundary conditions are defined in accordance with the procedure outlined in [29]. Additionally, individual degrees of freedom are con- strained at the corners of the hexahedron-shaped unit cell. One of the corner points is also displaced along one of the coordinate axes to apply the load. Accurate prediction of material behavior requires the use of suitable material models. Since both the epoxy resin and the aggregates exhibit brittle behavior, the Drucker-Prager material model is employed to simulate their mechanical response [30]. The specific material parameters used for this model are detailed in Table 3. After simulating the elementary cell consisting of N finite elements, the global stress �RVE and strain �RVE pre- sent in the solid element are calculated by homogenization according to Formulae 8 and 9 [29]. VRVE represents the volume of the entire representative volume element. The individual kth finite element is characterized by its volume Vk, its stress σk, and its strain εk. The elementary relations of Hooke’s law are then used to determine the Young’s modulus of the volume element. The results of the calcula- tion are given in Table 4. Fig. 3 a Filler distribution for the determination of the Young’s modulus on layer 1 and layer 2 and (b) cross-linked filler geometry Table 3 Material properties for the Drucker-Prager model Tensile modulus Compression modulus Poisson’s ratio Friction angle Drucker-Prager Parameter Dilation angle Tensile strength E+ E- ν b K ψ σz [GPa] [GPa] [-] [°] [-] [-] [MPa] Epoxy resin 9.2 3.9 0.375 24.4 1 0 145.0 [16] [17] [18] [31] [32] [32] [16] Aggre-gates 25.5 63.8 0.250 24.4 1 0 12.04 [19] [17] [31] [32] [32] [31] Table 4 Material characteristics calculated by homogenization in the three-dimensional volume elements Tensile modulus Compression modulus Poisson‘s ration (ten- sion) Poisson’s ratio (compres- sion) E+ E- ν+ ν- [GPa] [GPa] [-] [-] Layer 1 16.2 13.2 0.234 0.237 Layer 2 21.2 26.8 0.149 0.160 Layer 3 24.2 40.2 0.120 0.134 2596 The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 The representative volume element on layer 2 has the same filler distribution as that on layer 1, but is scaled to an edge length of 0.5 mm. The average diameter of the fillers is 0.1 mm. The results for the material parameters for layer 2 are listed in Table 4. A separate geometry is generated for the representative volume element on layer 3 based on image analysis of an existing fracture cross section. For this purpose, the filler particles visible in this cross section are classified into six size classes according to their surface area. The size classes are characterized by their proportion of the total area and by the ratio of their average major to average minor ellipse axis. The respective base particles are represented as 20-sided polyhedra, which are compressed in their size ratios to account for the major and minor elliptical axes. Again, a random filler arrangement is generated (see Fig. 4). The filler content is 38%. The edge length of the volume element is 7.5 mm. As with layer 1 and layer 2, the stresses and strains in the volume element are calculated by homogenization. The results for the Young’s modulus calculation based on this are given in Table 4. On layer 3, the deviation from the test data is calculated by + 23.7% (tensile modulus) and respectively (8)�RVE = 1 VRVE ∫ V �dV = 1 VRVE N∑ k=1 �kVk (9)�RVE = 1 VRVE ∫ V �dV = 1 VRVE N∑ k=1 �kVk + 22.7% for the compression modulus. Reasons for the devi- ation from the test results could be found in the arrangement of the mineral fillers. As can be seen in Fig. 5, there is a very irregular filler distribution. Furthermore, determining the filler distribution on the basis of an image analysis could falsify it. Inaccurate assumptions in layers 1 and 2 or the neglection of pores could also have an influence. As part of this study, the suitability of representative volume elements for determining the Young’s modulus of polymer concrete was tested not only on a three-dimensional representative volume element, but also on a two-dimen- sional volume element based on an existing fracture cross section of one of the bending beams considered in Section 3. Similar to the investigations presented in [33], the basis for Fig. 4 Filler distribution for the determination of the Young’s modu- lus on layer 3 Fig. 5 Two-dimensional representative elementary volume 2597The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 the generation of the two-dimensional representative volume element is provided by the digital recording of the cross sec- tion of a beam broken during the flexural test. The image of the geometry is used to generate a two-dimensional CAD model of the polymer concrete. The element is scaled using the known external dimensions of the sample. In particular, the pores within the element are considered during the crea- tion of the geometry. The mesh size is determined based on the smallest visible grains, ensuring that each grain is repre- sented by at least two linear triangular elements (see Fig. 5). As with the three-dimensional representative volume ele- ment (RVE), periodic boundary conditions and constraints on external degrees of freedom are applied to prevent rigid body motion. The load is applied at node N2. The image analysis of the fracture edge revealed that the visible fillers constituted a volume fraction of 38%. How- ever, as demonstrated in Table 1, the volume fraction calcu- lated on the basis of the density is 83%. It is assumed that the filler content that cannot be imaged with the selected means is microfillers, such as fly ash. Given the significant influ- ence of microfillers on the mechanical properties of polymer concrete [34], it is assumed that the visible matrix consists largely of microfillers. The inhomogeneity of the material volume represented as a matrix precludes any direct assump- tion of material parameters for the simulation. However, to verify the method, the material parameters determined for the three-dimensional representative volume element of layer 2 are employed. The calculation is performed in two variants: initially, without consideration of the pores visible in the image section, and subsequently, with consideration of these. The material parameters obtained by homogenization are presented in Table 5. The deviation from the test data is calculated by + 12.9% (tensile modulus) and respectively + 16.6% (compressive modulus) for the 2D-RVE without pores. If the pores are respected in the calculation, the devia- tion decreases to + 10.5% (tensile modulus) and respectively + 15.1% (compressive modulus). The model is preliminary limited by the quality of the image analysis. However, as the 2D-RVE shows lower deviation than the 3D-RVE, the image analysis seems to cause less failure than the creation of a random volume element. The deviations of all calculated tensile and compressive moduli from the known test results and literature values are summarized in Fig. 6. The results indicate that the Caquot model provides the highest average accuracy among the analytical models, largely due to its inclusion of both minimum and maximum grain sizes, which is not accounted for in other analytical models. Additionally, the Caquot model tends to slightly underestimate the material properties, which can be advan- tageous in the design of structural components by incor- porating a conservative safety margin. The series model, however, demonstrates superior accuracy only when pre- dicting the compression modulus. It remains to be seen whether this exceptional accuracy, exceeding 99%, is spe- cific to the current material composition or can be general- ized to other compositions. Among the numerical methods, the two-dimensional RVE with pore consideration shows the highest degree of accuracy. This can be attributed to the use of actual geometries, which allow for more realistic replication of aggregate shapes. The higher Young’s modu- lus observed in the three-dimensional model aligns with Table 5 Material parameters obtained by homogenization in the two-dimensional representative volume elements Tensile modulus Compression modulus Poisson‘s ratio (tnesion) Poisson’s ratio (compres- sion) E+ E- ν+ ν- [GPa] [GPa] [-] [-] 2D-RVE (without pores) 23.0 36.7 0.136 0.138 2D-RVE (with pores) 22.7 35.9 0.186 0.197 +10.5% +15.1% -100% -50% 0% 50% 2D-RVE with pores 2D-RVE without pores 3D-RVE Caquot model Hashin model series model Tensile modulus Compression modulus -44.6% +0.1% -58.6% +33.9% -11.1% -18.9% +23.7% +22.7% +12.9% +16.6% Fig. 6 Accuracy of the calculation models for determining the Young’s modulus of pure polymer concrete 2598 The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 findings by Huang et al., who studied the applicability of two- and three-dimensional RVEs for cement concrete. The authors suggest that the more complex crack formation in three-dimensional systems, requiring higher energy input, could explain this behavior [35]. Overall, the precision of the numerical techniques is comparable to that of the Caquot model. However, experimental data should always be pre- ferred when determining the Young’s modulus of inhomoge- neous polymer concrete. To fully evaluate the applicability of these methods for polymer concrete, validation with other material compositions is required, which will be the focus of future studies. 4.2 Simulation of the bending tests This section presents an investigation into the efficacy of various methodologies for forecasting damage in flexural tests. In particular, the Concrete Damaged Plasticity (CDP) model is considered in all methods, in accordance with a recommendation by Józefiak et al. who successfully applied the failure model originally developed for cement concrete to polymer concrete [36]. This requires the definition of fur- ther material parameters, which are adopted from cement concrete as in [36]. The values are listed in Table 6. For the three-dimensional simulation of the bending test, the experimental setup depicted in Fig. 2 is first rep- licated in the simulation environment (see Fig. 7). The clamping is modeled using two steel blocks, with fixed boundary conditions applied to the surfaces opposite the clamping point. The clamping conditions are simulated with an overlap of 42 µm between the clamping jaws and the polymer concrete sample, which corresponds to the theoretical compression of the sample under a clamping force of 80 Nm per screw. A frictional contact with a static friction coefficient of 0.5 is applied between the sample and the clamping jaws, as per the methodology described by Kang et al. [37]. As in the test, a linearly increasing force up to a final value of 2000 N is defined at a distance of 405 mm from the fixture. To account for the very dif- ferent behavior of polymer concrete under compression and tension, the beam is divided at the neutral fiber into a compression side and a tension side. The compression side is assumed to have a Young’s modulus of 32.5 GPa, known from the data sheet, while the tension side is assumed to have a Young’s modulus of 19.72 GPa, calculated from the results of the flexural tests. The simulation results in failure of the beam at a load of 1298 N (− 1.96% compared to test results) and a deflection of 2.75 mm (− 4.18% com- pared to test results). Since it is not always possible to separate the tensile and compressive areas of complex geometries with com- plex load collectives, such as machine tool parts, a further three-dimensional simulation was carried out, this time without separation. The Young’s modulus of 32.5 GPa [21] given in the data sheet applies to both sides of the beam. All other parameters remain the same. The failure of the simulated sample occurs at a breaking force of 1159 N (− 12.46% compared to test results) and a deflection of 3.8 mm (+ 32.4% compared to test results). The result clearly illustrates the various properties of mineral casting and indicates that a load case-dependent assumption of the material properties may be necessary when mapping complex geometries. Further consideration of this circum- stance will be part of future work. As a comparison to the common three-dimensional meth- ods, a top-down calculation is built on the previously per- formed two-dimensional RVE calculation. As in [33], the previously generated model (see Fig. 5) is inserted into the area of maximum stress near the restraint (see Fig. 8). Due to the higher accuracy of the model with pores shown above, this model is reused for the simulation of the bending behav- ior. As with the three-dimensional model, the load is applied by means of a linearly increasing force up to a value of 2000 N. A fixed clamping is assumed on the 60 mm wide surfaces that are in contact with the clamping jaws in the test. Table 6 Material parameters for the definition of the CDP-model Dilation angle Eccentricity Initial equibiaxial compressive yield stress/ initial uniaxial compressive yield stress Second stress invariant on the tensile meridian/ second stress invariant on the compressive meridian Parameter of viscos- ity ψ e fb0/fc0 Kc μ (°) (-) (-) (-) (s) 36 0.267 1.05 0.667 0.005 [36] [32] [36] [36] [36] Fig. 7 Setup for the three-dimensional simulation of the bending test 2599The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 Various materials are defined within the model. In par- ticular, the visible aggregates within the inserted fracture surface are defined anew with the material parameters known from Table 1. As before, the values from the three- dimensional representative volume element (layer 2) are used for the matrix visible in this area. Outside the fracture zone, the material parameters obtained from the experiments with pure polymer concrete are assumed as in the three- dimensional calculation. For all materials, a distinction is made between the tensile and the compressive side. The simulation results in a failure of the sample at a force of 1042 N (− 22.64% compared to test results). The deflection at this point is 2.25 mm (− 21.6% compared to test results). In contrast to the determination of the Young’s modulus in Section 4.1, the usage of the real geometry does not seem advantageous in this case. This might be related to the com- plexity of the load case. The crack formulation might be of insufficient accuracy with the loss of the third dimension. Figure 9 compares the accuracy of the simulations carried out in terms of breaking force and maximum deflection. It is clear that the three-dimensional simulation with the divided beam gives the highest accuracy with a deviation of less than 5%. This can be attributed to the weaknesses of the 2D model in addition to the consideration of all the relevant properties of the polymer concrete. In particular, the com- plexity of generating a sufficiently high-resolution image should be mentioned. This is due to the uneven surface of the broken polymer concrete samples. 5 Simulation of fiber‑reinforced polymer concrete The second intermediate step prior to modelling the pre- stressed fiber-reinforced polymer concrete is the sufficiently accurate numerical modelling of fiber-reinforced polymer concrete. This requires a detailed model of the behavior of carbon fiber rovings impregnated with epoxy resin. To achieve this, another RVE is employed. It is known from measurements on real samples that the impregnated roving has a diameter of 4 mm. Based on the known number of individual filaments and their diameter of 7 μm, it can be deduced that the carbon fibers account for 7.45% of the sur- face area. Although the exact spatial distribution of fibers within the matrix remains undetermined, it can be reason- ably assumed, based on findings in [1], that the impregna- tion process ensures full encapsulation of each individual filament within the epoxy matrix. An edge length of 0.07 mm is defined for the representative volume element. Con- sidering the fiber volume fraction, this configuration results in exactly 10 individual filaments per RVE. The filaments are modelled in simplified form as bars. Fiber waviness, as discussed in [1], is disregarded for simplification purposes. The fibers are randomly arranged in the three-dimensional element. The Young’s modulus calculated by homogeniza- tion is 27.8 GPa. The calculated tensile strength is 514 MPa and the Poisson’s ratio is 0.302. In the subsequent step, the results obtained from the RVE are validated through the simulation of a flexural test using fiber-reinforced polymer concrete. A three-dimensional finite element model of the flexural test is developed based on the findings outlined in Section 4.2 (refer to Fig. 10). To simplify the model and reduce computational complexity, the clamping jaws are not explicitly modeled; instead, fixed boundary conditions are applied to the relevant surfaces to simulate the clamping effect. As in previous models, the beam is partitioned into two regions to differentiate between the tensile and compressive behavior of the polymer concrete under bending loads. The rovings are modeled as simple bars with a diameter of 4 mm. To further reduce the computation time, the beam is shortened to a length of 140 mm. The material properties of the polymer concrete are speci- fied as outlined in Section 4.2. For the rovings, the values obtained by homogenizing the RVE are assumed. The failure is again modeled according to the CDP model. The simula- tion results in a failure of the sample at a force F of 1474 N, corresponding to a deviation of 3.67%. The deflection f Fig. 8 Setup for the two-dimensional simulation of the bending test Fig. 9 Accuracy of the simulations of the bending test with pure pol- ymer concrete 2600 The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 is calculated by 3.94 mm using Formula 10, where l rep- resents the distance from the clamping, E is the Young’s modulus and h is the height of the sample. The deviation for the deflection can be calculated by 0.76%. Even though the accuracy of the model is quite high, the consideration of the in [13] shown waviness of the rovings could further improve the model quality. 6 Simulation of prestressed fiber‑reinforced polymer concrete To achieve a numerical representation of prestressed fiber- reinforced polymer concrete, it is essential to recalculate the mechanical properties of the impregnated carbon fiber rov- ings, as the number of rovings is increased compared to the non-prestressed fiber-reinforced samples. Due to the applied prestressing, the diameter of the impregnated roving bun- dles remains constant at 4 mm. This leads to a fiber volume fraction of 44.1%. The mechanical properties of the rovings are determined using the same homogenization procedure described in Section 5. The recalculated material properties include a Young’s modulus of 96.3 GPa, a tensile strength of 1.88 GPa, and a Poisson’s ratio of 0.317. Once the carbon fiber rovings are unloaded, residual stresses develop, as illustrated in [1]. In this study, three methods are considered for calculating these residual stresses: (1) analytical calculation based on the approach proposed by Mostafa et al. [38], (2) experimental determina- tion following the procedure outlined in [1], and (3) numeri- cal calculation by applying a compressive force equivalent to the prestressing force on the impregnated roving bundles. The aforementioned methods are validated by simulating the (10)f = 4F ⋅ l3 E ⋅ h4 flexural behavior using the model from Section 5 for the sim- ulation of fiber-reinforced polymer concrete (see Fig. 10). According to Mostafa et al., the mean value of the resid- ual stresses in the reinforcement component σF,res (see For- mula 11) [38] and the matrix σM,res (see Formula 12) [38] can be calculated from the stress σV present in the reinforce- ment component at the time of prestressing, the Young’s modulus of the reinforcement component EF and matrix EM, and the volume fraction of the reinforcement component vF [38]. According to Formula 13 [38], the stress resulting from the prestress in the reinforcement component σV is calculated from the prestressing force FV and the cross-sectional area of the reinforcement component AF. In order to minimize the computational effort, the impregnated roving bundles are assumed as the reinforcement component, as in Sect. 5. The resulting matrix stress σM,res of − 5.72 MPa and the resulting stress in the impregnated roving bundles σF,res of 221.79 MPa are given as initial values for the entire polymer concrete matrix or the entire roving bundles at the begin- ning of the simulation. These stresses are assumed to be uni- formly distributed, although this assumption is not entirely realistic based on observations from cement concrete stud- ies [39]. As in the previous experiments, a distinction is made between the first failure (formation of a first crack) and the fracture of the sample. The first failure occurs at a force of 1736 N (− 8.25% compared to test results), while the fracture of the sample occurs at a force of 4360 N (+ 0.8% compared to test results) according to the simulation. The deflection calculated by Formula 10 at the time of first failure is 3.42 mm (− 23.38% compared to test results), and at the time of fracture, it is 8.6 mm (− 78.95% compared to test results). Even though the first failure can be represented with a high accuracy, the model is clearly limited to uniaxial prestressed rovings. A further consideration of multiaxial prestressed rovings will be part of future work. To experimentally determine the residual stresses, a strain gauge is placed on one of the carbon fibers as described in [13]. The strain gauge measurements are taken 72 h after demolding. Following an appropriate curing period, the samples undergo the bending test outlined in Section 1 to (11)�F,res = ⎛ ⎜⎜⎝ 1 − 1 1 + EM EF ⋅ 1−vF vF ⎞ ⎟⎟⎠ ⋅ �V (12)�M,res = ⎛ ⎜⎜⎝ − 1 EF EM + 1−vF VF ⎞ ⎟⎟⎠ ⋅ �V (13)�V = FV AF Fig. 10 Setup for the simulation of the flexural test of fiber-reinforced polymer concrete 2601The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 assess their Young’s modulus. Based on Hooke’s law, an average residual stress of − 11.56 MPa is calculated from three samples. In the next step, the residual stresses are applied to the matrix in the same way as the analytically determined residual stresses before starting the simulation of the bending test. Since it is not possible to determine the residual stresses in the fibers using the method described, the analytically determined values are again assumed for these. In the simulation, the first matrix damage occurs at a force of 1806 N (− 4.53% compared to test results) and a deflec- tion of 3.56 mm (− 20.71% compared to test results), and the fracture of the sample occurs at 4767 N (+ 3.49% com- pared to test results) and 9.4 mm (− 76.99% compared to test results). As the procedure of the mapping of the residual stresses is the same as shown for the analytically determined residual stresses, the aforementioned limitations apply. For the simulative determination of the residual stresses, the prestressing force of 3000 N is applied as a compressive force to the impregnated roving bundles, assuming that they recover completely elastically. As illustrated in Fig. 11, the stress distribution within the matrix is inhomogeneous in both the longitudinal and transverse directions. The inho- mogeneity in the transverse direction can be attributed to the large spacing between the roving bundles. Given that polymer concrete is expected to exhibit behavior similar to cement concrete, a comparable stress distribution is antici- pated in reality. However, this hypothesis requires further investigation using precise measurement techniques. To further exploit the residual stresses field a complex integra- tion of various measurement systems like fiber Bragg grat- ings and strain gauges in different positions is planned. The inhomogeneity in the longitudinal direction can be explained by the Hoyer effect known from cement concrete [40]. The average residual stress is − 5.70 MPa. This means that the simulated and analytically determined residual stresses are in 99.7% agreement. Accordingly, the two simulations dif- fer primarily in the inhomogeneity of the residual stress field that is now present. In the next step, the determined residual stress distribution is applied to the bending beam. In this case, the first failure occurs at a force of 1714 N (− 9.41% compared to test results) and a deflection of 3.38 mm (− 24.72% compared to test results); the sample breaks at 3930 N (− 9.15% compared to test results) and 7.75 mm (− 81.03% compared to test results), respectively. Even though the model shows higher deviations than the afore- mentioned models, it might be feasible to represent multi- axial prestressed rovings and should be taken into account for future investigations. Figure 12 compares the accuracy of the simulations per- formed in terms of failure force, failure deflection, fracture force, and maximum deflection. On average, the use of the experimentally determined residual stresses provides the most accurate representation of the bending test. Only when the breaking force is considered, the analytically determined residual stresses provide a more accurate result. The high deviation of the maximum deflection is remarkable in all Fig. 11 Distribution of residual stresses in the bending beam -100% -50% 0% 50% Numerical Experimental Analytical -8.25% -23.38% +0.8% -78.95% -4.53% -20.71% +3.49% -76.99% -9.41% -24.72% -9.15% -81.03% Failure force Failure deflection Breaking force Maximum deflection Fig. 12 Accuracy of the simulations of the bending test with pre- stressed fiber-reinforced polymer concrete with different methods of determination of the residual stresses 2602 The International Journal of Advanced Manufacturing Technology (2025) 138:2591–2603 cases. This is presumably due to the crack gap opening, which is not considered in Formula 10. A more realistic evaluation would require a time-consuming calculation of the entire beam. However, since in most cases, a failure pre- diction is not required for the subsequent design of com- plex components, but instead a design in the range of elas- tic deformations is aimed at, the computationally intensive consideration is omitted at this point. 7 Conclusion Previous studies marked the first development of machine components made from the novel material, prestressed fiber- reinforced polymer concrete. These investigations empha- sized the necessity of a suitable design methodology for the material, which is inherently influenced by residual stresses and was not regarded in literature in the past. By employing a variety of analytical and numerical methods, the mechani- cal properties of pure polymer concrete were reproduced with sufficient accuracy. In the current study, the Caquot method for analytically calculating the Young’s modulus has proven particularly effective (15% accuracy), offering com- parable accuracy to numerical methods but with significantly lower computational effort. In order to be able to make a final statement on the suitability of the individual methods, validation using alternative material compositions is neces- sary. Furthermore, it was shown that the representation of the deviating tensile and compressive properties of polymer concrete plays a major role in the representation of the flex- ural properties of the material. In a second step, representa- tive volume elements were used to model the properties of the impregnated roving bundles found in fiber-reinforced polymer concrete. A validation of the properties determined in this way is pending, but the values obtained could be used to predict the bending behavior of fiber-reinforced polymer concrete with over 95% accuracy. In a final comparison of different methods for modeling residual stresses, it was found that a homogeneous distribution of residual stresses is a reasonable assumption for this material (12.6% accu- racy regarding the so defined first failure). If experimental determination of residual stresses is not feasible, analytical calculations can provide sufficiently accurate results (15.8% accuracy regarding the so defined first failure). Acknowledgements The authors would like to thank the German Research Foundation (DFG) for funding this work as part of the project “Prestressed fiber-reinforced mineral cast.” The authors would also like to thank RAMPF Machine Systems GmbH & Co. KG for providing the material data. Author contribution Michelle Engert: conceptualization, investigation, methodology, supervision, validation, visualization, writing—original draft. Konstantin Frankenbach: investigation, methodology, software, validation, visualization. Kim Torben Werkle: conceptualization, inves- tigation, project administration, supervision, writing—review and edit- ing. Hans-Christian Möhring: conceptualization, funding acquisition, project administration, supervision, writing—review and editing. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Competing interests The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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