Abstract
The stress and strain relationship in the gauge region of six cruciform geometries is studied: the ISO standard geometry with slits in arms, two geometries with thinned gauge areas, two geometries with thinned gauge areas and slits in arms, and one modified ISO standard geometry with slits in arms and a thinned gauge area. For all the geometries, finite element simulations are performed under uniaxial loading to compare the plastic strain, the von Mises stress distribution and the inplane stress evolution. Results show that less plastic strain can be achieved in the gauge of the two ISO standard geometries. For the remaining cruciform geometries, a strong nonlinear coupling between applied forces in arms and gauge stresses is generated. The evolution of this nonlinear coupling depends on the geometry type, applied biaxial load ratio and the elasticplastic properties of the material. Geometry selection criteria are proposed to reduce this nonlinear coupling.
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Introduction
Sheet metals and alloys are subjected to biaxial stresses and changing strain paths during their forming processes and under service conditions. Under these conditions, the mechanical properties can differ from those obtained under uniaxial stress conditions. Several biaxial deformation devices have been developed to test materials, each of them having advantages and disadvantages [1, 2]. Amongst these, biaxial deformation rigs using cruciform shaped samples allow to deform materials under different inplane loading modes [2–5]. The cruciform shape offers the possibility for applying any arbitrary load ratio, thus providing access to a large portion of the 2dimensional stress space. It allows fullfield insitu strain measurements on either side of the gauge region, in contrast to Marciniak samples, punch test samples, and tubular samples, whose internal sides are inaccessible for techniques such as digital image correlation (DIC). Additionally, when the axes of the device can operate independently, it allows to perform nonproportional strain path changes [6–8]. It should be noted that the eigenstress coordinate system cannot be changed during nonproportional loadings, thus making it difficult to perform indepth analysis of nonproportional strain path change tests. Furthermore, when biaxial testing is performed insitu during neutron or Xray diffraction, footprints of the microstructural evolution such as texture, inter and intragranular stresses and dislocation density can be followed [9–11]. The cruciform geometry also has its drawbacks. One of the major challenges is to develop a cruciform geometry that can achieve significant amount of plastic deformation in the gauge section while allowing analytical computation of the gauge stresses.
To that end, significant efforts have been directed towards optimizing the cruciform geometry. Recently, an ISO standard [12] for biaxial tensile testing was established. According to this standard, the cruciform design should have a uniform thickness and slits in the arms, as shown in Fig. 1(a). The slit based design was originally proposed in the work of Hayhurst [13], and later adopted by Kelly [14], Makinde et al. [15, 16], and Kuwabara and coworkers [2, 17, 18]. The purpose of the slits is twofold: (i) to reduce the stress heterogeneity within the square gauge area so that the inplane normal stress components can be computed as the force divided by area along a given direction, and (ii) to prevent shear loadings in the cruciform arms or machine grips in case the machine is not perfectly aligned. This geometry has been used in several studies [19–21] to perform nonproportional strain path changes and determine yield surface evolution. One of the main drawbacks of this design is that large stress concentrations can develop at the slitends during uniaxial or biaxial loading. Consequently, the amount of plastic strain achieved in the gauge region is low. Another drawback was revealed in the FE simulations performed by Hanabusa et al. [22], who showed the presence of inplane compressive stresses normal to the uniaxial loading direction. Their influence on the computation of stresses as force divided by area was not addressed. Not accounting for this may affect the prediction of yield surface and its evolution for different material systems.
It is well known that the achievable amount of gauge plastic strain can be increased by locally thinning the gauge section [3, 23]. Two such thinned geometries are presented in the Figs. 1(b) and (c). The cruciform geometry in Fig. 1(b) was proposed in the work of Baptista et al. [24]. They used an optimization procedure to obtain (i) the shape of the intersection of cruciform arms (henceforth known as crossarms) and (ii) the depth of thinned region, for a given thickness of the sheet material. The gauge area was continuously thinned down using a spline curve. In contrast, the geometry in Fig. 1(c) has a constant thickness in the gauge region and a gradual thickness reduction from the arms to the gauge region. Thinning of the cruciform sample, however, does not facilitate fulfilling the other requirements, i.e. a homogeneous stress distribution and a correct analytical computation of the gauge stresses. Makinde and coworkers [15, 16, 25] were the first to argue that computing gauge stresses as force divided by area may be erroneous for some cruciform geometries because of the difficulty in accurately determining the crosssectional gauge areas. Based on this argument, Green et al. [26] suggested the use of finite element (FE) simulations to obtain the gauge stresses. They proposed an iterative procedure to match the simulation predicted forces in the arms and the gauge strains with experimental data. Simulation predicted gauge stress vs strain plots were obtained for different yield functions. The evolution of gauge stresses as a function of the forces in the arms was however not reported. Hoferlin et al. [27] demonstrated, for their cruciform geometry, that a uniaxial load in one of the arms results in a biaxial stress state in the gauge region. Furthermore, when loading with inplane force ratios between 0 and 0.36, the inplane stress component corresponding to the lower load is significantly under predicted when computed as force divided by area. For a geometry similar to the one in Fig. 1(c), Bonnand et al. [28] proposed to relate the applied forces (F _{1} and F _{2}) in the arms and gauge stresses (S _{11} and S _{22}) using a linear relationship: S _{11} = aF _{1} − bF _{2} and S _{22} = − bF _{1} + aF _{2}, with a and b as constants. A similar relationship was already proposed in [27]. Claudio et al. [29] estimated the values of a and b from FE simulations of elastic deformation under uniaxial tension and used them to compute the stresses for all load ratios. Such a linear coupling between forces and stresses is however only valid in the elastic regime [27]. For plastically deforming materials, this coupling will be inherently nonlinear.
Cruciform geometries combining slits and thinner gauge areas have also been proposed. One such geometry was introduced in the work of Zidane et al. [30], similar to the one shown in Fig. 1(d). This geometry has four equallength slits per arm. It has a width that is twice the gauge thickness, and is more than an order of magnitude higher than the ISO standard geometry. Similar to the ISO standard geometry, there are no notches in the arms but the crossarm radius of curvature is much higher. A twostep thickness reduction is introduced on one side of the cruciform in the gauge region; first a square zone with an abrupt thickness reduction is introduced, then a second circular zone is added with a gradual thickness reduction. This specimen was used to obtain forming limit curves for aluminum alloy AA5086. Fracture always occurred at the center of the sample independent of the loading conditions. Leotoing et al. [8] used this geometry to developed a predictive numerical model for the forming limit curve using FE simulations. In contrast, Liu et al. [31] introduced slits with different lengths in their cruciform geometry, circular notches at crossarms with smooth ends at the arms and a steep transition to the gauge area (Fig. 1(e)). Deng et al. [32] proposed a modified slit design (Fig. 1(f)), that conforms to the ISO standard specifications, in order to increase the amount of plastic deformation prior to failure while keeping the gauge shape for the analytical computation of the stresses. Similar to the works of Hayhurst [13] and Kelly [14], the arm thickness is increased with respect to the gauge region. Equispaced slits are drawn over the entire length of the arms. A comparison of the plastic strain reached in the gauge area and at the location of stress concentrations is not reported. Furthermore, the role of cruciform geometry on the coupling between force in the arms and gauge stresses has not been discussed for any of the slitthinned geometries.
Understanding the nonlinear coupling between forces and stresses is important since they can significantly affect the inter (type II) and intra (type III) granular stress state and microstructural evolution. In a recent work involving the authors [10], insitu neutron diffraction studies during biaxial strain path changes were performed on 316 L stainless steel cruciform samples having the geometry shown in Fig. 1(c). During monotonic uniaxial tensile loading, the presence of significant inplane compressive stresses normal to the loading direction was confirmed using DIC strain measurements and FE simulations. Furthermore, in the plastic regime the evolution of diffraction peak positions i.e. microscopic strains associated with type I stresses (applied stresses), for different grain families as a function of the stress was significantly different from that obtained for dogbone (DB) samples. Similar cruciform geometric effects in DIC strains were also observed by [9] during insitu xray diffraction studies of biaxial deformation of their ferritic sheet steel cruciform samples (not shown here).
In light of the above, the main objective of the present work is to highlight the role of cruciform geometry on the gauge stress evolution during elastic and plastic deformation under monotonic uniaxial and biaxial tensile loading. FE simulations are performed on six cruciform geometries shown in Fig. 1, i.e. (i) the ISO standard slit geometry [2, 17, 18], (ii) the elliptical crossarm steeply thinned geometry without slits [24], (iii) the circular crossarm gradually thinned geometry without slits [10, 28, 29], (iv) the twostep gradually thinned geometry with slits [8], (v) the uneven slit, circular notched and sharply thinned geometry [31], and (vi) the modified ISO standard slit geometry [32]. These are classified according to the presence of slits and/or gauge thinning as SLITI, THINI, THINII, SLITTHINI, SLITTHINII, SLITTHINIII, respectively. The results are compared with those obtained for DB samples used in [10]. The remainder of this paper is divided as follows. Section 2 describes the experimental procedure and section 3 the FE simulation setup. Section 4 begins with the experimental validation of the simulation framework using the THINII geometry. Then the equivalent plastic strain and von Mises stress distributions in the six cruciform geometries are plotted to study the stress concentrations and maximum achievable plastic strains for each geometry. Next, the inplane gauge stress evolution for the six cruciform geometries is analyzed and compared with the results of DB samples. The THINII geometry is then studied for different biaxial load ratios and different materials. The results of these simulations are used to propose geometry selection criteria in section 5. The main conclusions of this study are presented in section 6.
Experimental Procedure
THINII and DB samples are deformed using a biaxial deformation rig developed in collaboration with Zwick/Roell (Ulm, Germany) [6, 10]. A detailed description of the machine is presented in [10]; for brevity, only the relevant details of the machine are recalled here. The biaxial testing rig is equipped with independent arm control and allows to deform up to 50 kN along direction 1 and 100kN along direction 2. A twocamera ARAMIS4M DIC system from GOM is installed on the biaxial rig to measure the surface strains on the gauge region of the samples. The error associated with the DIC measurement for the THINII and DB samples is given using the equation err(%) = x x strain(%) + y; where x and y are in the range [0.014, 0.024] and [0.05, 0.09], respectively.
The test material is a warm rolled 10 mm thick sheet of 316 L stainless steel with composition (in %wt) 17.25Cr, 12.81Ni, 2.73Mo, 0.86Mn, 0.53Si and 0.02C. An electron backscattering diffraction analysis of this steel reveals a mild texture, but the uniaxial mechanical response in rolling and transverse direction were the same as was reported in [10]. This implies that the macroscopic uniaxial mechanical response of the material can be assumed to be independent of the rolling direction. The isotropic elastic Young’s modulus (Y) is 190 GPa and the Poisson’s ratio (ν) is 0.31. Figure 2 shows the true stress v/s true strain curve (in black) for this steel obtained from uniaxial tests on DB samples (different from the one shown in Fig. 1(g)) performed by Dr. Niffenegger (see acknowledgements).
The THINII and DB samples are prepared from this steel. The outer geometry of the THINII and DB samples is water cut and the gauge section is mechanically ground in a symmetric manner to a thickness of 3 mm. Mechanical tests have not been performed for the remaining geometries.
Simulation Setup
ABAQUS/Standard software [33] is used to perform the FE simulations for the six cruciform and DB geometries. In order to improve the computational efficiency, only 1/8th of the entire geometry is simulated with symmetric boundary conditions on appropriate surfaces. Figure 3 shows the FE mesh for all the geometries tested in this work. A structured hexahedron mesh is employed with linear 8node mesh elements (C3D8 in ABAQUS) for the THINII and DB geometries as shown in Fig. 3(c), (g). A hexahedron mesh is also used for the SLITI and SLITTHINIII geometries, except near the slitends where a mixed hexahedron/tetragonal mesh is used as shown in the inset of Fig. 3(a), (f). The meshing procedure has been based on the simulations performed by [22] for the SLITI geometry and [32] for the SLITTHINIII geometry which show that the stress concentrations occur at the slitends (inset of Fig. 3(a), (b)). The number of elements for the DB, SLITI, THINI, THINII, SLITTHINI, SLITTHINII, SLITTHINIII, and DB geometries are 3333, 29,756, 23,966, 4626, 22,964, 17,408 and 11,336, respectively.
Isotropic elastic properties of 316L stainless steel were used [10]. The plastic response is modeled using the ABAQUS material model that is based on the VM yield criterion and the associated flow rule. The builtin rateindependent combined nonlinear isotropic and kinematic hardening law with 5 backstresses is used. The stress v/s strain curve from the monotonic tensile loading test on DB samples (black curve in Fig. 2) is provided as an input to ABAQUS/Standard. To account for microplasticity, the initial yield point is taken at 135 MPa. The ABAQUS/Standard algorithm uses this experimental curve to fit the backstress parameters; manual parameter fitting is not required. Figure 2 also shows the VM stress v/s strain curve fitted by ABAQUS FE simulation (red line). As can be seen, the fitted and experimental curves have a good match. It should be noted that since the present work does not deal with strain path changes, using an isotropic hardening model instead of a combined hardening model with 5 backstress parameters will not result in significant differences in the predicted stress v/s strain curves. Changing the number of backstress parameters, however, affects the fitting procedure and in this case the combined hardening model with 5 backstress terms provides the best fit.
The experiments on the THINII and DB samples are performed under load control, therefore load control is also used to simulate these geometries. The simulated forces are applied as surface tractions on the inside of the holes in the arms of these geometries using a linear ramp. This is illustrated in Figs. 3(c) and (g). The SLITI and SLITTHINIII geometries are designed to be deformed using clamps attached to the arms. These geometries were deformed under displacement control. Therefore, a linearly ramped displacement on the surface in contact with the clamps is applied in the FE simulations. The remaining geometries have also been deformed under displacement control, as illustrated in Figs. 3(a), (b), (d), (e), and (f).
To facilitate comparison between the different geometries, the strain values for all geometries reported in all the line plots will be averaged on the surface of a 3.8 mm × 3.8 mm area at the center of the gauge area (this becomes 1.9 mm × 1.9 mm area for the 1/8th geometry as shown in the inset of Fig. 3(c) for the THINII geometry) and the stresses will be averaged along the thickness of the sample beneath this area. The averaging procedure is motivated from the insitu neutron diffraction experiments in the work of [10] where surface strains are measured using DIC and the neutrons measurements are obtained from the 3.8 mm × 3.8 mm × 3 mm volume.
Results
Experimental Validation of the FE Procedure
The following set of simulations are carried out for the THINII and DB samples and compared with experimental data from [10]: uniaxial tensile loading of DB sample to ~16.4 kN, uniaxial tensile loading along axis 1 of the THINII to 50kN and equibiaxial tensile loading of THINII (axis 1 and 2) to 50kN on each axis. Figure 4 shows a good match between the two inplane strain components E _{11} and E _{22} from the three simulations and the experiments. Minor differences between them may be due to the tolerances in gauge thickness (range of 0.1 mm) associated with manufacturing the samples.
Plastic Deformation in the Cruciform Geometries
The predicted gauge stress and strain evolution for the six geometries are compared. All the geometries are subjected to uniaxial deformation along direction 1 whereas the arms along direction 2 are kept free i.e. at zero force. The THINII cruciform simulations are performed under load control with a linearly ramped force F _{1} = 100kN (2000 steps). All the remaining cruciform simulations are performed under displacement control. A linearly ramped displacement is applied at the end of the arm along direction 1. The simulations are stopped when the equivalent plastic strain reaches 40% at the location of the highest stress concentration in each geometry.
Figure 5 shows the contour plots for the distribution of equivalent plastic strain and VM stress in and near the gauge region of the six cruciform geometries. The highest stress concentration occurs at the end of the second outermost slit for the SLITI, center of the gauge area for the THINI, crossarms for the THINII, gauge area for the SLITTHINI, the innermost slit for the SLITTHINII, and the outermost slit for SLITTHINIII (shown in Figure 5). For the SLITI and SLITTHINIII geometries, the VM stress and the equivalent plastic strain are negligible in comparison with those at the location of the slitends. For the SLITI and SLITTHINIII geometries, the location of the slits and their small radius of curvature (0.1 mm) at the slitends are responsible for the large difference in the stresses at slitends and cruciform gauge area. Furthermore, for the SLITTHINIII geometry, the effect of gauge area thinning by a factor of 2.5 to increase the gauge strain is overshadowed by the stress concentrations at the slitend. For the SLITTHINII geometry, the combined effect of the larger radius of curvature at slitend, increased distance from the slitend and gauge area, larger armtogauge thickness ratio of 3.33 and the notch in the arms, results in higher gauge equivalent plastic strain in comparison to the SLITI and SLITTHINIII geometries.
The THINII geometry, has the same armtogauge thickness ratio of 3.33 as the SLITTHINII geometry. However, the VM stress and equivalent plastic strain near the center of the gauge region are comparable to those at the cross arms (Figures 5(e), (f)). This indicates that the gradual thickness reduction from 10 mm in the arms to 3 mm in the gauge region, larger gauge area and the absence of slits play a role in the improved gauge stress. Note that the crossarms have a variable thickness, which is greater than the gauge area but lesser than the arms. For the THINI and SLITTHINI geometries, the maximum equivalent plastic strain and von Mises stresses occur near the center of the specimen. These geometries have an armtogauge thickness ratio of 6.67 and 5.33, respectively. For the THINI geometry, the region with the highest stress concentration is smaller in comparison to the SLITTHINI geometry. This is a consequence of the sharp thickness change in the gauge region of the THINI geometry.
From these results, we can conclude that a large armtogauge thickness ratio is necessary to obtain the highest stress concentrations in the gauge area, irrespective of the presence of slits or notches at crossarms. Furthermore, for the slit geometries, the slit width should be comparable to the gauge thickness to avoid high stress concentrations at the slit ends. In addition, the crossarms should be thicker than the gauge area to reduce the stress concentrations.
Gauge Strain and Stress Evolution during Uniaxial Deformation in Cruciform and DB Samples
In this section, the gauge stress and strain evolutions for the six cruciform geometries and the DB are studied during uniaxial loading. The results are shown up to the point where a maximum 40% equivalent plastic strain is reached at the locations of the highest stress concentrations. Figure 6(a) shows the plot of the predicted total gauge strain component E _{22} as a function of E _{11} for all samples; the DB results are obtained from the simulations performed in section Experimental Validation of the FE Procedure. The ratio \( \raisebox{1ex}{${E}_{22}$}\!\left/ \!\raisebox{1ex}{${E}_{11}$}\right. \) for the DB and cruciform geometries in the elastic (limited to E _{11} = 0.1% as shown in inset of Figure 6(a)) and plastic regimes are shown in Table 1. In the elastic regime, for the SLITI and SLITTHINIII geometries, \( \raisebox{1ex}{${E}_{22}$}\!\left/ \!\raisebox{1ex}{${E}_{11}$}\right. \) is similar to the Poisson’s ratio, however, for the remaining cruciform geometries, it is significantly different. At the end of loading, the SLITI, SLITTHINI and SLITTHINIII geometries result in a ratio \( \raisebox{1ex}{${E}_{22}$}\!\left/ \!\raisebox{1ex}{${E}_{11}$}\right. \)close to −0.5. Whereas, this ratio is less than −0.5 for the THINI and THINII geometries and more than −0.5 for the SLITTHINII geometry. These results indicate that the SLITI, THINI, THINII and SLITTHINI geometries must have a compressive stress component in the direction 2 normal to the loading direction, whereas SLITTHINII must have a tensile stress component along direction 2. The SLITTHINIII geometry should have nearly the same stress state as the DB sample.
Figure 6(b) shows the inplane stress component S _{22} as a function of S _{11} for all the geometries. Simulations have revealed that the outofplane stress component (S _{33}) is negligible in comparison with the inplane stress components. In the elastic regime, the stress ratio \( (R)\ \raisebox{1ex}{${S}_{22}$}\!\left/ \!\raisebox{1ex}{${S}_{11}$}\right. \) is constant for all the geometries and shown in Table 1. Clearly, S _{22} is negligibly small for SLITI and SLITTHINIII but not for the remaining cruciform geometries. Amongst the latter, the SLITTHINII geometry has a tensile stress component along direction 2 while the remaining have a compressive component; the tensile component along direction 2 is counterintuitive because under uniaxial loading along direction 1, one would expect a compressive component along the transverse direction 2. Immediately following the onset of plasticity, R decreases for the SLITI and THINII geometries and increases for the remaining geometries. Between S _{11} = 200 and 400 MPa, the stress evolution becomes highly nonlinear for the THINI, THINII, SLITTHINI and SLITTHINII geometries. The ratio R at the end of loading is also shown in table 1. These results imply that the coupling between the forces in the arms is weak in the SLITI and SLITTHINIII geometries, intermediate in the SLITTHINI geometry and strong in the THINI, THINII and SLITTHINII geometries.
The SLITI and SLITTHINIII geometries are ideal for decoupling the gauge stresses and having negligible transverse stress components under uniaxial loading, however, due to the high stress concentrations at the slit ends these are not suitable to achieve more than a few percent plastic strain in the gauge area. On the other hand, the remaining geometries are suitable to achieve large gauge strains but suffer from a nonlinear coupled biaxial gauge stress state at the center of the sample. Furthermore, for the same type of material, the stress evolution in the elastic and plastic regime is significantly different.
Nonlinear Gauge Stresses during General Biaxial Loading
In this section, the nonlinearity in the S _{22} v/s S _{11} evolution is further analyzed for general biaxial loading for the THINII geometry. The THINII geometry is analyzed because the model predictions have been validated at two extremes of inplane tensile loading i.e. uniaxial and equibiaxial loading; thus, providing a higher confidence on the accuracy of the predicted results under other biaxial loads. Furthermore, the THINII geometry has been used during insitu neutron diffraction studies in [10, 11]. The nonlinear macroscopic stress evolution of the THINII geometry influences the inter (typeII) and intra (type III) granular stresses. This is evidenced from the nonlinear kink in the lattice strain evolution during uniaxial loading of the THINII sample [10, 11]. In the following, the macroscopic stress evolution in the THINII sample is studied for 10 different loading ratios F _{1}:F _{2} – 10:0, 10:1, 10:2, 10:3, 10:4, 10:5, 10:6, 10:7, 10:8, 10:9 and 10:10, while F _{1} is always increased from 0 to 50kN. The results for the inplane gauge stress components and the stressforce relationships are shown in Figure 7.
In the elastic regime, the linear stressforce relationship i.e. S _{11} = aF _{1} − bF _{2} and S _{22} = − bF _{1} + aF _{2} [27–29], is tested. For all loadings, the values for a and b are found to be constant and equal to 10,750 (m ^{−2}) and 2500 (m ^{−2}), respectively. This is in accordance with the work of Hoferlin et al. [27]. The elasticplastic transition is characterized by a kink in R and in the stressforce relationships. The kink evolves differently for different load ratios: it is very pronounced in all three plots for the force ratios 10:0 to 10:6 and less pronounced for the force ratios 10:7 to 10:10. Once full plasticity sets in, the nonlinear deviations between the inplane stresses and stressforce relationships become less pronounced.
Figure 7(b) shows the evolution of S _{11} as a function of F _{1}. Since for all the loading ratios F _{1} is ramped to 50kN, the evolution of S _{11} is similar for all the load ratios. A closer inspection shows that in S _{11} the point of inflection only appears for load ratios between 10:0 to 10:6. In addition, with increasing magnitude of F _{2} from a load ratio between 10:0 up to 10:6, the inflection point tends to occur at higher values of S _{11}. This is because the stress evolution follows the von Mises plastic behavior. Figure 7(c) shows that the evolution of S _{22} as a function of F _{1} is qualitatively similar to the evolution of S _{22} with respect to S _{11}.
In order to better understand the nonlinear behavior for different load ratios, Figure 8 shows the deviation from linear evolution of the elastic regime for each stress component i.e. \( M=\left[{S}_{11}{F_1}^{\ast }{\left(\frac{S_{11}}{F_1}\right)}_{elastic}\right] \) and \( N=\left[{S}_{22}{F_1}^{\ast }{\left(\frac{S_{22}}{F_1}\right)}_{elastic}\right] \) as a function of the applied force F _{1}. The nature of the nonlinearity varies significantly as function of the applied force. There are three interesting observations to be made. First, although the kink during the nonlinear behavior of S _{22} vs S _{11} in Figure 7(a) appears to be most pronounced for the load ratio of 10:5, the largest deviation from the linear elastic behavior in S _{11} occurs for uniaxial loading i.e. 10:0. The largest deviation from linearity in S _{22} occurs first for the load ratio 10:4 at F _{1} = 36kN and then for 10:2 at 50kN. Second, equibiaxial loading results in a nonlinear evolution of the stress components as a function of the applied force. Third, the load ratio of 10:6 results in a linear evolution of S _{11} with the same slope as in the elastic regime up to F _{1} = 30kN. This is at the beginning of the elasticplastic transition. After F _{1} = 30kN, S _{11} deviates away from the linear behavior. For the load ratio of 10:7, the evolution of S _{22} as a function of F _{1} is found to be nearly linear till the end of loading. The origin of these counterintuitive stress evolutions under different load ratios lies in the geometry of the sample. These should vary depending on the cruciform shape and size, and the material properties.
A more general outcome of these simulations is that decoupling the cruciform gauge stresses and computing them as force divided by area is not possible for any stress ratio R. Furthermore, this coupling will be different for different geometries. This reinforces the need to use FE simulations to predict the gauge stress state.
Nonlinear Gauge Stress Evolution for Different Hardening Behavior
In this section, we study the gauge stress evolution for the THINII cruciform geometry during uniaxial loading for different materials. Figure 9(a) shows the true stress v/s true strain curve for six new materials in addition to the 316 L stainless steel: (i) Aluminum alloy AA5086, similar to the one from [31], with isotropic elastic properties Y = 73 GPa but a slightly modified ν = 0.31, yield stress at 135 MPa and maximum plastic strain of ~20%, (ii) a nearly perfectly plastic (NPP) pseudomaterial with the same elastic properties, yield stress and maximum strain as AA5086, but with a maximum stress of 206 MPa at a maximum plastic strain of ~20%, (iii) an artificial isotropic linear elastic material with Y=1GPa and ν = 0.31 (ISOLINI), (iv) an artificial isotropic linear elastic material with Y = 73 GPa and ν = 0.12 (ISOLINII), (v) an artificial isotropic linear elastic material with Y = 73 GPa and ν = 0.45 (ISOLINIII) and (vi) an artificial orthotropic linear elastic material (ORTHOLIN) with the engineering constants Y _{1} = 190GPa, Y _{2} = 73 GPa, Y _{3} = 190 GPa, ν _{12} = 0.31, ν _{13} = 0.31, ν _{23} = 0.31, G _{12} = G _{13} = G _{23} = 65 GPa. For the ABAQUS simulations of the ORTHOLIN material, a local coordinate system needs to be defined for each element in the FE mesh. For simplicity, the local coordinate system of all the elements is defined as the global coordinate system. Figure 9(b) shows S _{22} as a function of S _{11} for the THINII geometry for all the materials in this work. Note that the results for the ISOLINI material are only shown until S _{11} = ~ 60 MPa. Beyond this stress, there are significant elastic deflections in the arms at the location of applied force due to the very low stiffness of ISOLINI. This results in a nonlinear evolution of the S _{22} v/s S _{11} which is not representative of the elastic deformations of the THINII geometry made from other stiffer materials.
Focusing first on the stresses in the elastic regime, under uniaxial loading we have S _{11} = aF _{1} and S _{22} = bF _{1}. This gives R = S _{22}/S _{11} = b/a. Next, the Hooke’s law for an orthotropic elastic material with a negligible outofplane stress component gives E _{11} = S _{11}/Y _{1} − ν _{12} S _{22}/Y _{2} and E _{22} = S _{22}/Y _{2} − ν _{21} S _{11}/Y _{1}. Combining the last three equations gives,
Note here that the −E _{22}/E _{11} is the “effective Poisson’s ratio” for the cruciform geometry and it is not equal to ν _{12}. This implies that in the elastic regime the biaxial stress ratio R, and thus the ratio of the equivalent area measures b/a, are dependent on the material’s elastic properties. For an isotropic material with Y = Y _{1} = Y _{2} and ν = ν _{12} = ν _{21}, equation (1) reduces to \( R=\left(\frac{\raisebox{1ex}{${E}_{22}$}\!\left/ \!\raisebox{1ex}{${E}_{11}$}\right.+\nu}{1+\nu\ \raisebox{1ex}{${E}_{22}$}\!\left/ \!\raisebox{1ex}{${E}_{11}$}\right.}\right) \). Interestingly, for all the simulated isotropic elastic materials with the same Poisson’s ratio have the same R value equal to −0.232. This result suggests that for an isotropic material, R depends only on the Poisson’s ratio. This implies that the “effective Poisson’s ratio” i.e. \( \raisebox{1ex}{${E}_{22}$}\!\left/ \!\raisebox{1ex}{${E}_{11}$}\right. \) for a given cruciform geometry should also depend only on the Poisson’s ratio. Therefore, all the isotropic materials with the same Poisson’s ratio should give the same R, irrespective of the Young’s modulus. Next, R is computed for the ISOLINII and ISOLINIII materials having different Poisson’s ratio. These materials, whose Poisson’s ratios are 0.12 and 0.31, respectively, result in an R value that are ~5% less and ~5% more, respectively, than the R value for the isotropic linear elastic materials with ν = 0.31. The ORTHOLIN material results in R = − 0.129 which results in a 44.4% difference with respect to the isotropic linear elastic materials with ν = 0.31. Note that for the ORTHOLIN material, the difference in Young’s moduli Y _{1} and Y _{2} also results in a difference in the Poisson’s ratio ν _{12} and ν _{21}; from the relationship ν _{ ij }/Y _{ i } = ν _{ ji }/Y _{ j }, we have ν _{21} = 0.12. These results, along with equation (1), imply that a and b, which have dimensions m ^{−2}, are dependent on the material properties. This is fundamentally different from the gauge stressapplied force relationship in DB samples, where in the elastic regime this relationship is solely dependent on the DB gauge crosssection.
Following the onset of plasticity, the evolution of S _{22} v/s S _{11} varies according to the hardening behavior of the material. An evolution trend is however not obvious. Simulations were also performed for the THINI, SLITTHINI and SLITTHINII geometries (results not shown). Similar to the THINII geometry, an evolution trend in S _{22} v/s S _{11} depending on the hardening behavior was not evident.
Engineering metals and alloys, that are macroscopically nearly elastic isotropic, often have a Poisson’s ratio that falls in the range 0.25–0.35. Geometrically same cruciform samples constructed from these materials will have nearly the same stress ratio R in the elastic regime. For these materials, R will strongly depend on the cruciform geometry and not the elastic properties of the material. A simple uniaxial DB test, along with the strain ratio E _{22}/E _{11} from cruciform samples, is sufficient to obtain the material parameters required for analytically approximating the stress ratio R. For anisotropic materials, such as cold rolled Mgalloys, in the elastic regime R will depend on the cruciform geometry shape and the degree of anisotropy; the full macroscopic anisotropic elastic stiffness tensor would be necessary to analytically obtain the stress ratio R. In the plastic regime, R varies according to both the cruciform geometry and the hardening behavior of the material with no obvious trend, thus making it difficult to analytically obtain its value. This further underlines the need to perform FE simulations to predict their stress evolution.
Discussion
In the following, we propose cruciform selection criteria to reduce the nonlinear gauge stress coupling and to improve the gauge plastic deformation based on the cruciform geometries simulated in the previous section.

(1)
Symmetric/asymmetric geometry: In the works of [8, 30], the SLITTHINI geometry was originally designed to be a onesided geometry due to the complexity associated with the design. However, onesided geometries could suffer from shear and bending stresses in the central region. All the cruciform geometries simulated in this work are symmetric twosided geometries. The results show negligible shear stresses and outofplane normal stress components in comparison to the inplane normal stresses. The cruciform should be designed symmetric along all three directions.

(2)
Number of slits and slit width: As seen from the simulations of SLITI and SLITTHINIII geometries in section 4.2, a relatively large number of slits results in homogenizing the stress field within the gauge area. These results also show that a slit width smaller than the cruciform thickness causes very high stress concentrations at the slitends. In contrast, for the SLITTHINI geometry, which has a slit width of 2.2 mm and slitends ~3 mm away from the gauge area, the highest stress concentrations occur in the gauge area. However, the biaxial stress ratio is appreciably large and the stress distribution is relatively less homogeneous. Based on these results, the slitwidths should be kept nearly the same size as the gauge thickness and should be as close as possible to the gauge area to obtain a homogeneous stress distribution.

(3)
Gauge thickness: The gauge thickness should be sufficiently smaller than the arm thickness in order to ensure that the stress concentrations occur in the center of the sample. Amongst the cruciform geometries studied in this work, those with an armtogauge thickness ratio greater than 5 have the highest stress concentration occurring in the gauge area. Note that when the region with reduced thickness encompasses the crossarms, the ratio of the crossarmtogauge thickness also becomes important.

(4)
Thickness gradient and gauge area: Cruciform geometries with gradual thickness reduction and larger gauge surface area have shown the highest stress concentrations and more homogeneous stress distribution near the center in comparison to geometries with steep thickness reduction and smaller surface area. The cruciform design should account for this aspect.

(5)
Crossarm thickness and shape: Amongst the simulated cruciform geometries, those with crossarms having the same thickness as the arms and thicker than the gauge area, do not result in stress concentrations in the gauge area.
Based on these geometry selection criteria, and the cruciform geometries simulated in this work, we propose a cruciform geometry design (nonoptimized) for the 316L steel that minimizes the nonlinear coupling of gauge stresses while attaining large plastic strains in the center of the gauge region. This is achieved by combining the geometric features from SLITTHINI and SLITTHINIII geometries. Figure 10(a) shows the new geometry design, henceforth known as SLITTHINNEW. This geometry has the same arm dimensions, gauge size and area, and crossarm shape and size as the SLITTHINI geometry. The only difference is the number of slits, slitwidth and slitends. A total of 9 slits of thickness 1 mm each are added per arm. On one end, all the slits terminate at 1 mm from the region of the first thickness reduction. In the opposite direction, similar to the SLITTHINIII geometry, the slits are drawn throughout the length of the arm. This geometry is simulated under uniaxial loading along direction 1. The ABAQUS FE mesh has 20,530, C3D8 mixed hexahedron and tetrahedron elements. The FE mesh is shown in Fig. 10(b). The simulation is stopped when a 40% equivalent plastic strain is first reached anywhere in the geometry. This occurs at the gauge area of the SLITTHINNEW geometry. Figures 10(c) and (d) show a snapshot of the equivalent plastic strain and von Mises stress distribution in the SLITTHINNEW geometry at the end of the simulation. The simulated S _{22} v/s S _{11} curve is shown in Fig. 10(e).
Results show that the SLITTHINNEW geometry has the highest equivalent plastic strain in the gauge area of the sample throughout the simulation. The stress concentration is initially highest in the gauge area, however, after S _{11} ≈ 700 MPa, it shifts to the third slitend from the right in the arm along direction 2. In the elastic regime, the SLITTHINIII demonstrates a reduced stress ratio R in comparison to the SLITTHINI geometry; although not as much as the SLITTHINIII geometry. Following the onset of plasticity, the absolute value of ratio R approaches 0. The value of R fluctuates around 0 up to S _{11} ≈ S _{ mises } = 700 MPa. Following this, the highest stress concentration shifts from the gauge area to the slitend. In addition, R rapidly increases until the end of the simulation. At S _{11} ≈ S _{ mises } = 700 MPa, the obtained equivalent plastic strain at the center of the gauge area is ~20%.
Amongst all the cruciform geometries simulated in this work, the SLITTHINNEW geometry is the better geometry to obtain high gauge equivalent plastic strain with the lowest nonlinear coupling between the arms but only up to 700 MPa. The SLITTHINNEW geometry can be further optimized by tuning the slits, gauge area and crossarms. However, the results strongly indicate that designing a single cruciform geometry for all kinds of materials is not feasible when the additional requirements include decoupling the nonlinearly evolving inplane gauge stresses, homogeneous stress distribution, and achieving the highest stresses and large plastic strains in the gauge area.
Conclusion
In this work, an FE simulation study was undertaken to highlight the role of cruciform shaped sample geometry on the stress evolution within the gauge area. Six cruciform shaped sample geometries are studied: (i) the ISO standard slit geometry – SLITI, (ii) the elliptical crossarm steeply thinned geometry with no slits – THINI, (iii) the circular crossarm gradually thinned geometry with no slits THINII, (iv) the twostep gradually thinned geometry with slits – SLITTHINI, (v) the uneven slit, circular notched and sharply thinned geometry – SLITTHINII, and (vi) the modified ISO standard slit geometry – SLITTHINIII. The gauge stress and strain evolution for these geometries were compared with DB (dogbone) samples under uniaxial loading. Then the THINII geometry was tested under different biaxial tensile loads for different materials. Based on these studies, cruciform geometry selection criteria were suggested. Using these criteria, a new cruciform geometry (SLITTHINIII) was proposed that allowed to achieve medium plastic strain in the gauge area while significantly reducing the nonlinear coupling of gauge stresses during uniaxial loading. The main conclusions of this study are as follows:

1)
In the elastic regime, the SLITI and SLITTHINIII geometries result in uniaxial stress state at the center of the gauge region. In contrast, THINI, THINII, SLITTHINI and SLITTHINII geometries show a linear coupling between the forces in the arms and inplane gauge stresses in the elastic regime such that S _{11} = aF _{1} + bF _{2} and S _{22} = aF _{2} + bF _{1}. A uniaxial tensile load along one of the arms results in a stress component along the inplane direction normal to the loading direction. This transverse component is tensile for the SLITTHINI geometry and compressive for the THINI, THINII and SLITTHINII geometries. The transverse stress component can have a significant contribution in defining the gauge stress state. This makes it difficult to analytically compute the gauge stresses as a function of the applied force. The magnitude of the transverse stress component significantly varies according to the cruciform geometry and the macroscopic elastic anisotropy of the material.

2)
Following the onset of plasticity, the THINI, THINII, SLITTHINI and SLITTHINII geometries result in a nonlinear coupling between gauge stresses and applied forces in the arms. Its nature strongly depends on the cruciform geometry, applied biaxial load ratio and elasticplastic properties of the material. A material, geometry, and applied load ratio dependent trend on the nonlinear stress evolution couldn’t be established.

3)
In SLITI and SLITTHINIII, important stress concentrations are developing at the slit ends limiting the plastic strains that can be reached in the gauge region. These geometries are therefore not suitable to study the mechanical behavior of materials with low work hardening rates. In the THINII and SLITTHINII geometries, the stress concentrations occur at the crossarms and slitends, respectively. However, these are comparable to the gauge stresses. Consequently, relatively large plastic deformation can be achieved in the gauge area. The THINI and SLITTHINI geometries result in the largest von Mises stress and equivalent plastic strains in the gauge area.

4)
A new geometry is designed based on the knowledge gained from simulating the six geometries and novel cruciform geometry selection criteria. The new geometry helps reduce the nonlinear gauge stress coupling while attaining moderate gauge plastic strains.
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Acknowledgements
HVS and MVY thank the European Research Council for the financial support within the advanced grant MULTIAX (339245). The authors thank Markus Niffenegger at the Paul Scherrer Institute for providing the mechanical data for 316L stainless steel based on his experimental results.
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Upadhyay, M.V., Panzner, T., Van Petegem, S. et al. Stresses and Strains in Cruciform Samples Deformed in Tension. Exp Mech 57, 905–920 (2017). https://doi.org/10.1007/s1134001702706
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DOI: https://doi.org/10.1007/s1134001702706