Investigation of Epoxy Resin under Uniaxial, Biaxial, and Triaxial Quasi-Static Loads

The epoxy investigated was a bisphenol-A/F-epichlorhydrin resin (Resin L) Supplier R&G Faserverbundwerkstoffe GmbH, Waldenbuch, Germany. The epoxy was processed with an isophorondiamin hardener (GL 1). Resin and hardener were carefully mixed for 5 min and vacuum processed for 2 min. The tensile specimens were cast in silicone molds. The compression and torsion specimens were first cast in a rod shape and then mechanically postprocessed. For the poker chip test, flat disc-shaped specimens were glued between steel rods. The manufacturing process for the poker chip specimens is described in detail in the “Poker Chip Test” section. The geometry of the test samples is shown in Fig. 1. To ensure a complete curing, different curing treatments were checked using tensile tests and differential scanning calorimetry (DSC) analysis:

For all tests, a strain rate of $\dot{\epsilon }{=10}^{-4}1/\mathrm{s}$ was applied. The selected strain rate results as a compromise between the objective of quasi-static loading and moderate testing time. Furthermore, comparability with, for example, Fiedler et al. (2001a), Morelle et al. (2017), Gerlach et al. (2008), and Littell et al. (2008) was ensured. The tensile tests were performed using a universal test machine Zwick/Roell Z010 (ZwickRoell GmbH, Ulm, Germany) with a 10 kN load cell. The compression and poker-chip tests were done using a universal test machine Instron 4505 (Instron GmbH, Darmstadt, Germany) with a 100 kN load cell. The torsion tests were performed using a universal test machine Zwick/Roell Z250 (ZwickRoell GmbH) with a 250 kN load cell for axial loading and a 1,000 Nm torque sensor. For tension, compression, and poker chip test, at least 20 specimens were used. Due to the long testing time, five specimens were used for each of the torsion tests. Since all tests were performed under quasi-static conditions, elastoplastic material behavior was assumed. However, dynamic mechanical analysis (DMA) was carried out to estimate the proportion of viscoelastic material behavior (see section “DMA”).

To ensure that the epoxy resin was completely cured, DSC analyses were performed to determine the degree of crosslinking. A Mettler Toledo type DSC 25 (Mettler Toledo, Columbus) heat flow differential calorimeter with a TC15 TA controller was used for the DSC analysis. Since the crosslinking process of epoxy resin is an exothermic reaction, the measured enthalpy of an uncured sample can be compared with that of a cured sample to draw conclusions on the degree of crosslinking of the epoxy resin. First, a freshly mixed sample was examined as a reference. The material was heated from room temperature to 300°C at a heating rate of $10\mathrm{K}/\min$. Fig. 2(a) shows the exotherm peak of the crosslink reaction. To evaluate the curing cycles, three epoxy resin samples were also heated from room temperature at $10\mathrm{K}/\min$ to 300°C for each curing cycle. Fig. 2(b) shows the curve of the DSC measurement for a specimen produced with curing cycle C. The glass transition temperature range of the resin was between 70°C and 90°C. Since the curves for the remaining samples appeared very similar, they were not plotted. No further exothermic peak could be measured; therefore, complete crosslinking of the resin can be assumed.

Two single cantilever DMA measurements were performed in the temperature range of 0°C to 120°C for 10 logarithmic distributed frequencies in the range of 1–25 Hz and with an amplitude of $20\mathrm{\mu m}$. The specimens with dimensions $20\times 10\times 2\mathrm{mm}$ were separated from a molded plate. A Cole-Cole plot was used to identify the range of evaluable measurement data (see Fig. 3). Therefore, the measured values in the temperature range from 0°C to 100°C were used to create a master curve at 25°C for the storage modulus as a function of the angular frequency using the time-temperature superposition principle [see Fig. 4(a)]. As shown in Zeltmann et al. (2016), DMA could be used to investigate the strain rate dependence of Young’s modulus. For this, a generalized Maxwell model was fitted to the master curve using the python package scipy version 1.10.0. The fit required 12 resp. 14 Prony terms. The normalized standard deviation of the parameters found did not exceed 9%. Using the generalized Maxwell model, the time-dependent relaxation modulus was calculated according to Eq. (9) as follows:with $E_0$, $e_i$, ${\tau }_i$ being the instantaneous Young’s Modulus and the parameters of the Prony term $i$. For the instantaneous Young’s Modulus, a value of $E_0=\mathrm{3,500}\mathrm{MPa}$ was chosen. This value corresponded to Young’s Modulus out of the technical datasheet (Magner and Kautz 2013).

For a constant strain rate at the beginning at $t=0$, the time-dependent stresses are given as (Zeltmann et al. 2017)

For a fixed strain (here, 0.005), stresses were calculated using Eq. (10) for a range of strain rates; thus, the secant modulus for the fixed strain was calculated at different strain rates. The secant modulus over strain rate is shown in Fig. 4(b). Since only a limited frequency range can be covered by the master curve, only a section of the time domain can be represented. An orientation for the limits of the time domain can be derived from the minimum (0.2 ms) and maximum ($>24\mathrm{years}$) relaxation times calculated for the fit of the generalized Maxwell model. The results of the DMA show that the selected strain rate of $\dot{\epsilon }={10}^{-4}1/\mathrm{s}$ was in a range where the stiffness of the material showed little dependence on the strain rate, and the effects of the viscoelastic properties of the epoxy were relatively small. Therefore, it was assumed that the material behavior can be approximately described using elastoplastic models. Accordingly, the material behavior was assumed to be elastoplastic in the following investigations unless otherwise indicated.

Uniaxial tensile tests were carried out to investigate the parameters of the epoxy resin until the specimen failed. The specimen shape is shown in Fig. 1(b). The longitudinal and transverse strain was measured using a video extensometer. Before investigating the mechanical properties in detail, the influence of the curing cycle was determined. The three different curing cycles were checked, each with 20 specimens. Representative curves for the comparison of the different curing cycles are shown in Fig. 5. No significant property changes resulted from the different curing cycles. Both curing cycles with higher curing temperature showed more scattering of the measured data than did the short curing cycle. This can be attributed to thermally induced residual stresses that lead to small deformations in the specimens. Following this, the shortest curing cycle (cycle C; see section “Material and General Conditions”) was used for every additional test.

The results of the uniaxial tensile test are shown in Fig. 6(a). The epoxy showed a linear-viscoelastic increase in stress followed by a nonlinear part. Young’s modulus of $E_T=3100\mathrm{MPa}$ was observed. Tensile strength of ${\sigma }_T^s=84\mathrm{MPa}$ and tensile strain at a break of ${\epsilon }_T=0.048$ were measured. To determine the onset of yielding, force-controlled quasistatic loading-unloading tests were performed [see Fig. 6(b)]. The tests showed that the nonlinear stress-strain relation consisted of two components: a viscoelastic part observed from the loading-unloading hysteresis and a plastic part observed from irreversible remaining strains. The irreversible strain remained even for small stresses. To check for strain recovery at zero force, for one specimen, force-controlled holding times of 600 s (approximately twice the test time of the regular tensile test) were held after each unloading cycle; see Fig. 6(c). The results show that irreversible strains remained even at small forces, and no significant recovery of the strains was expected beyond the holding time. Therefore, the yield stress was determined using the linear elastic limit ${\sigma }_{0.1}$ at which 0.1% plastic strain remained after unloading. This procedure was in general agreement with other investigations (Fiedler et al. 2001b; Morelle et al. 2017). A yield stress under tensile load of ${\sigma }_T^y=39\mathrm{MPa}$ was determined. In Fig. 6(d), the measured relationship between transverse and longitudinal strain under a tensile load is shown. An overall value of $\nu =0.41$ for the elastic Poisson’s ratio in the range of ${\epsilon }_1=0.3\%\dots 0.4\%$ was measured. The range for the evaluation of the elastic Poisson’s ratio was defined such that the measurement inaccuracy of the extensometer was $\approx 1\%$. After onset of yielding, an increase in Poisson’s ratio to values of ${\nu }^{pl}=0.5$ could be observed. A summary of the measured properties out of the tensile tests is given in Table 1.

For the compression test, a PTFE foil was used to minimize the friction between the sample [see Fig. 1(d)] and the pressure plates. The results of the compression tests are shown in Fig. 7. In agreement with the tensile tests, at the beginning, the stress increased in a linear-elastic manner before reaching a peak value. The stress-peak was followed by a softening part, leading to a plateau at which the stress stayed nearly constant while the strain increased. After this plateau, the stress increased after fracture of the specimen. From the linear part, Young’s modulus of $E_C=2800\mathrm{MPa}$ was observed. The yield stress was observed at ${\sigma }_C^y=81\mathrm{MPa}$. A video extensometer showed that the softening of the material was caused by the development of one macroscopic shear-band after the stress reached the first peak value. Whether this macroscopic shear band described the actual material behavior under compression or was caused by the specimen geometry and the friction between specimen and compression plate must be discussed. The latter represents a general problem in the realization of compression tests. After a strain at a break of ${\epsilon }_C^f=0.43$, a similar fracture pattern as that obtained by Fiedler et al. (2001b)—with cracks in the loading direction—was detected. As discussed in Roetsch and Horst (2022), material separation under uniaxial compressive stress was caused by shear stresses with simultaneous compressive pressure. The experimental compression strengths determined here represented the stress at which cracking was observed visually and acoustically. The compressive strength reached ${\sigma }_C^f=210\mathrm{MPa}$. Here, it is important to distinguish between true and engineering stresses. Despite the fact that the shear band formation could be recorded by the video extensometer, a measurement of the transverse strain was not possible; therefore, only the initial sample size could be used for stress calculations. Fig. 7 shows engineering stresses. An estimate of the true fracture stress was approximated by measuring a cracked specimen, leading to ${\sigma }_C^f\approx 135\mathrm{MPa}$. A summary of the measured properties out of the compression tests is given in Table 2.

Torsion tests with different varying axial tensile and compressive loads were performed. Clamping was realized using two precision vices with roughened clamping surfaces. A total of six different tests were carried out, each with five specimens [see Fig. 1(c)]. The tension/compression torsion tests were done with an axial load of $\pm 20\mathrm{MPa}$ and $\pm 40\mathrm{MPa}$. Furthermore, torsion tests were carried out in which the axial load was controlled to be zero. The deformation of the specimen was measured using the torsion of the clamping. For the evaluation of the torsion tests, assumptions of a uniform deformation of the specimen over the measuring length were made. The applied torque, twist angle, and axial loads and deformations were measured. The relationship between the shear stresses and the applied torque followed Seyda et al. (2020)

Assuming a linear dependence of the shear stress $\tau$ on the radius $r$, the shear stress is as follows:

However, the influence of the axial deformation of the specimen due to torsion (Poynting effect) was not taken into account. Assuming that the outer and inner radius $r_o$, $r_i$ of the specimens changed by the same difference $\mathrm{\Delta }r$, the radii of the specimens as a function of the axial deformations led toandwhere $t_0$ = initial wall thickness; and ${\epsilon }_{11}^h=\ln (1+{\epsilon }_{11})$ is the Hencky strain in the axial direction. Thus, using Eqs. (13) and (14), Eq. (12) gives the true shear stress. The distortion follows withwhere $\phi$ = twist angle. This approach takes into account the axial deformation of the specimen due to torsion (Poynting effect). A similar approach by Wu et al. (1997) was used, for example, for an analysis of aluminum and steel specimens under torsion (Lu et al. 2021).

The results of the torsion tests are shown in Fig. 8. For each test, a shear modulus of $G\approx 1250\mathrm{MPa}$ was measured. Yielding was defined using a linear elastic limit of ${\tau }_{0.2}$, leading to a yield stress of ${\tau }^y=26\mathrm{MPa}$ for an axial load of zero. No major changes were observed in the yield stresses at different axial loads. The combined compression-torsion tests with an axial load of $-40\mathrm{MPa}$ was not evaluated due to buckling of the specimens. The shear strength at an axial load of zero reached ${\tau }^s=51\mathrm{MPa}$. Given increasing axial tension, a decrease in the shear strength was observed. Moreover, a slightly decrease in shear strength was measured for increased axial compression load. After reaching the shear strength, a softening part and a constant stress plateau with large shear strain was observed. This large plastic deformation under torsion load seemed typical for epoxy (Fiedler et al. 2001b; Littell et al. 2008). With axial load, especially with tension load, a significant reduction in the strain at a break was observed. A summary of the measured properties out of the torsion tests is given in Table 3.

Various methods have evolved to generate triaxial tensile stress states in polymers. Lindsey (1966) used the so-called poker chip test to obtain hydrostatic tensile stress conditions in polyurethane rubber. Here, a disc-shaped specimen is glued between two steel cylinders and ripped apart [see Fig. 1(a)]. Asp et al. (1995) and Kim et al. (2013) adopted this test setup for the mechanical characterization of epoxy resin under triaxial tensile loads. A different method was used by Gross et al. (2017). Here, an epoxy resin was cooled from the curing temperature inside a quartz tube. As a result of the cooling, the epoxy resin contracts, which was hindered by its adhesion to the walls of the quartz tube. The results show similar reduced strengths to those observed in the poker chip test (Gross et al. 2017).

Therefore, the poker chip test appears to be an adequate way to investigate the mechanical behavior of the epoxy resin under triaxial tensile loading. The experimental setup of the poker chip test is shown in Fig. 1. The steel cylinders were sandblasted and decreased with acetone. A silicone O-ring was then placed on the lower cylinder and filled with epoxy resin. The upper steel cylinder was then placed on the still liquid epoxy resin and the O-ring. In contrast to Asp et al. (1995), this procedure does not require subsequent bonding of the specimen. The inner diameter (30 mm) and thickness (4 mm) of the O-ring specify the specimen’s geometry. The resin was cured in the same way as the tensile, compression, and torsion specimens.

The axial deformation was measured using the machine cross head (the deformation of the machine was corrected). Lindsey (1966) analyzed in detail the stress and deformation state of the poker chip sample in the elastic regime. The specimen with normalized radius $a$ was deformed in the $z$ direction by $2\epsilon$. The radius was normalized by half the specimen thickness $t/2$ such that the aspect ratio of the specimen corresponds to the normalized radius (Lindsey 1966; Asp et al. 1995). The stresses within the specimen are given in simplified form by Lindsey (1966)where $E$, $\nu$ = Young’s modulus and the Poisson’s ratio; ${\sigma }_z$, ${\sigma }_r$, ${\sigma }_{\theta }$, ${\tau }_{rz}$ = normal and shear stresses in the direction of the polar coordinates $z$, $r$, $\theta$; and $a$, $\epsilon$ = normalized specimen radius and nominal strain. $I_0$, $I_1$ are zero- and first-order modified Bessel functions. The stress distribution in the center of the sample is shown in Fig. 9.

All specimens tested showed linear elastic material behavior to failure (see Fig. 10). A total of 20 specimens were tested, and three different failure modes were observed (see Fig. 12). Half of the samples showed a cohesive fracture in the middle of the sample. The other half showed both complete adhesive fracture (failure of the interface) and a mixture of cohesive and adhesive fractures. Only the samples with cohesive failure were used for further investigation. An apparent Young’s modulus of $E^A=5500\mathrm{MPa}$ and an apparent strength of ${\sigma }_P^A=34\mathrm{MPa}$ with an elongation at a break of ${\epsilon }_P^f=0.006$ were measured. Eqs. (16)–(18) follow an actual stress distribution in the center of the specimens of ${\sigma }_z=44.66\mathrm{MPa}$, ${\sigma }_r={\sigma }_{\theta }=30.70\mathrm{MPa}$, which corresponds to a stress ratio $r:\theta :z$ of approximately $2:2:3$. From Eq. (1), a hydrostatic tensile strength of ${\sigma }_H=33.5\mathrm{MPa}$ was obtained for this stress distribution. Eq. (8) predicts a value of ${\sigma }_H=28.13$, which is a good estimate in the case of missing experimental data for $H$. A summary of the results for the poker chip test is given in Table 4.

Fig. 11 shows the tensile, compression, and torsion specimens used before and after the tests (only compression specimens after the tests). The tensile and compression tests show fracture surfaces perpendicular and parallel to the loading direction, respectively.

Fig. 12 shows the three failure modes for the poker chip test performed. The single air inclusions in the specimens are a result of the manufacturing process described. However, no correlation was observed between the location of the crack initiation and the presence of air inclusions. Moreover, no correlation with the presence of air inclusions could be established for the measured strengths such that the influence of these imperfections was classified as low. Fig. 12 shows a crack initiation in the center of the specimen near or at the interface. It is noteworthy that for half of the specimens, the crack area propagated to the center (thickness direction) of the specimen instead of progressing at the interface with the steel cylinders. This indicates a very critical stress state within the specimen. Scanning electron microscope (SEM) micrographs of the fracture surface were performed using a Zeiss EVO MA15 microscope (Carl Zeiss AG, Oberkochen, Germany). The SEM pictures of the fracture surface from the poker chip test are shown in Fig. 12 and for the uniaxial tensile tests in Fig. 13. Uniaxial tension typical fracture surfaces, such as those observed by Cantwell et al. (1988), Bandyopadhyay (1990), Fiedler et al. (2001b), and Morelle et al. (2017), can be investigated. Accordingly, a mirror-like surface is present around the crack initiation region due to continuously (slow) crack propagation. As the crack propagation progressed, the crack velocity increased, and secondary cracks developed because of the high local stresses that results in a rougher fracture surface.

The complete fracture surface of the poker chip specimen is mirror-like and nearly featureless, similar to the fracture surface for uniaxial tension near crack initiation. The smooth fracture surface corresponds to a continuously accelerating single crack (Cantwell et al. 1988; Bandyopadhyay 1990). As crack propagation progressed, radially aligned step-like (river-like) secondary cracks formed that partially coalesced toward the edge of the specimen. This is in general agreement with the observations from Asp et al. (1995) for a Diglycidyl Ether of Bisphenol A (DGEBA) epoxy system with aliphatic curing agents. Since the crack initiation is near the interface, and the crack progresses to the middle of the specimen, the single secondary cracks could be in different planes. The coalesce of those step-like cracks may be the result of the height adjustment of single cracks.

In the torsion test, nearly all torsion specimens failed at a 45° angle to the longitudinal axis of the specimen. This indicates a failure due to the largest principal tensile stress. Some fracture surfaces also showed smaller fracture angles, which indicates an additional contribution of the largest shear stress. The typical fracture surface of the torsion specimens is shown in Fig. 14. All fracture surfaces exhibited three characteristic details that were pronounced to different degrees. These characteristics included a mirror-like event-free surface, a finely textured surface with parallel river-like scratches, and areas with large steps. In Morelle et al. (2017), the smooth fracture surface (mirror-like) was associated with fracture initiation and slow fracture progression. The area with finely distributed scratches (river-like) follows this smooth fracture surface. This surface was also observed in the poker chip tests. With crack propagation, these fine scratches developed into larger steps, which could only be observed in this way in the torsion specimens and agree with observations by Fiedler et al. (2001b) (transition from river-like to step-like). Fiedler et al. (2001b) considered the cause for the structure of this fracture surface to be a superposition of normal and shear stress driven cracks, where the shear stresses lead to plastic deformation.

Fig. 15 shows the comparison of the determined yield stresses and strengths with the predictions of the paraboloid criterion and the extended paraboloid criterion. Since the extended paraboloid criterion is designed as a strength criterion, the comparison with the yield stresses is restricted to the predictions of the classical paraboloid criterion of Stassi-D’Alia (1967). The scaling of the criteria was done using the uniaxial tensile strength and the shear strength. The triaxial tensile strength determined in the poker chip test was also included in the yield criterion.

A physical interpretation of the paraboloid yield criterion when applied to epoxy resin could be the rotation of chain segments caused by deviatoric parts of the stress tensor and the initiation of nanovoids as a result of the volumetric part of the stress tensor, which could explain the pressure-dependent yield behavior. In addition, the final failure, especially under tensile load, could be explained by the formation of voids, which can be described by the volumetric dilatation energy criterion (Asp et al. 1996). The extended paraboloid criterion thus represents a combination of these mechanisms.

From the perspective of material modeling, the yield criterion transforms into a strength criterion through hardening processes. Assuming that no yielding occurs under hydrostatic tension (Roetsch and Horst 2022), the yield and strength criteria must match in this point. The paraboloid criterion shows good agreement with the measured yield stresses and tended to be a more conservative estimate.

A comparison of the two criteria with the measured strengths reveals the significant differences in the range of triaxial tension. While the criteria in the range of uniaxial tension, torsion, and uniaxial compression make almost identical predictions, the classical paraboloid criterion overestimates the measured triaxial tensile strength by approximately three times. The extended paraboloid criterion can provide exact agreement with the results from the poker chip test. For the uniaxial compression test, the estimated value of ${\sigma }_C^s\approx 135\mathrm{MPa}$ is included (see section “Compression Test”).

For the elastoplastic material behavior of the epoxy resin, the material model by Melro et al. (2013), Melro (2011), and van der Meer and van der Meer (2016) was used. The yield criterion corresponds to the paraboloid criterion of Stassi-D’Alia (1967) and has been reformulated into a dependence on the tensile yield stress ${\sigma }_T^y$ and the yield stress under shear ${\sigma }_S^y$

The plastic strain increment is obtained via a nonassociated yield rule towhere

Isotropic hardening is defined by the dependence of the yield stresses on the equivalent plastic strain ${\epsilon }_{eq}^p$ via two section-wise defined functionsandwhere the increment of the equivalent plastic strain is calculated as follows:

In the original model, the plastic multiplier $\mathrm{\Delta }\gamma$ is numerically calculated using a Newton-Raphson scheme (Melro et al. 2013). This algorithm shows high sensitivity on the initial guess value and, especially under compression load, no solution is found. To circumvent this problem, Brent’s method was implemented as an alternative solution method for the plastic multiplier. The implementation was accomplished according to Press (2007).

In continuum damage mechanics (CDM), the damage of a material is described using a damage parameter $D$ that degrades its stiffness ${\mathbf{C}}_{\mathbf{0}}$ (Simo and Ju 1987)

The implementation of Eq. (1) in the context of CDM follows the approach of Melro et al. (2013), where a damage function $F$ is introduced

Here, $r$ = local internal variable; and $\tilde{\varphi }$ = extended paraboloid criterion introduced in Eq. (1) formulated using effective stresses $\tilde{\sigma }={\mathbf{C}}_{\mathbf{0}}:\epsilon$. The degradation of the stiffness leads to a softening of the material. Implementing CDM models in FE analyses leads to a dependence of the solution on the size of the finite elements. To regularize the localization effect, an implicit gradient model introduced by Peerlings et al. (1996) is usedwhere $\overline{r}$ = nonlocal counterpart to the internal variable $r$. Additionally, the following Neumann boundary conditions are used

Damage onset is defined if $\tilde{\varphi }\ge 1$; therefore, in the damage regime, the condition $r\in (1,\infty )$ must hold. The evolution of the damage variable $D$ is defined using the following damage evolution law

The history variable $\kappa$ is related to the nonlocal variable $\overline{r}$ via Kuhn-Tucker relations (Peerlings et al. 1996)

The irreversibility of the damage process $\dot{D}\ge 0$ and the definition for $\kappa$ follows from the Kuhn-Tucker relations (Simo and Ju 1987)

To ensure that damage initiation remains a local event, this definition for $\kappa$ is extended (Seupel et al. 2018)

Therefore, even if the value of the nonlocal variable exceeds one, the history variable $\kappa$ is initially updated only if the local variable also exceeds one. This model was implemented as a user-subroutine UMAT in Abaqus version 2021, the commercial finite element analysis (FEM) software. The implementation of the implicit gradient model within UMAT is done using the heat-damage analogy introduced by Seupel et al. (2018). For this, the nondiagonal coupling terms of the displacement-phase field stiffness matrix ${\mathbf{K}}_{\mathbf{u}\overline{r}}={\mathbf{K}}_{\overline{r}\mathbf{u}}$ were set to zero. This leads to a slower convergence rate but makes the implementation easier since the derivative $\partial \overline{r}/\partial \epsilon$ is not needed (Navidtehrani et al. 2021, 2022). For the implementation using a UMAT subroutine, the following derivatives need to be defined:where ${\mathbf{C}}^T$ = consistent tangent operator out of the plasticity model. The principal deduction for ${\mathbf{C}}^T$ is shown in van der Meer and van der Meer (2016).

The input parameters used for the material model are listed in Table 5. The input variables required for the hardening law were taken directly from the measured curves of the tension and torsion tests and are provided in Table 6.

The comparison of the model predictions for a one element test case with the experimentally determined curves is shown in Figs. 16 and 17. Since the model was fitted via tensile and torsional curves, the material model provides very good agreement here. Given compression, the yield point is predicted well; however, the simulated curve deviates significantly from the experimental curve, for various possible reasons. On the one hand, technical stresses and strains are shown here for the compression case. Furthermore, a softening after an initial stress peak is observed in the measurements for both torsion and compression. Taking this softening into account via the hardening parameters under torsion leads to very good agreement for the values under compression. However, for finite element (FE) models with several elements, this softening needs to be regulated similarly to the case of damage to ensure a mesh-independent solution. This was not considered within the scope of this study. A possible approach to this was presented by, for example, Nguyen et al. (2016). However, the model shows good agreement with the measured fracture strains. Furthermore, both the very ductile material behavior under torsion and the ideally brittle material behavior under hydrostatic tension can in principle be represented by the model.

Fig. 17 shows the applicability of the extended paraboloid criterion to accounting for triaxial tensile stresses. The poker chip test is simulated using a rotational-symmetric FE model. Three different characteristic element sizes $l_e$ were used. The parameter $L$ was set to 0.5 mm. The strain at damage onset corresponds well with the strain at the breakout of the experiment. The model leads to a slight overprediction of the maximum stress and the strain at a break. The latter could be due to the damage propagating too slowly. No relevant plastic deformation was observed in the simulation of the poker chip test.