Appendix 1: Additional details on modeling approaches and results
Team A
Team members:
J.A. Moore, johnallanmoore@gmail.com, Northwestern University, Evanston IL, USA
K. Elkhodary, khalile@aucegypt.edu, The American University in Cairo, New Cairo, Egypt
Approach
Simulations were performed using the finite element (FE) software ABAQUS 6.12. A dynamic analysis, with explicit time stepping was used. Failure surfaces were generated by the element removal method. An element is removed when a proposed damage variable \(\delta ^{*}\ge \delta ^{*Crit}\). Domain decomposition parallelization was also applied for computational expediency.
Material Law Isotropic elasticity was assumed. For inelastic behavior, the von Mises stress \(r({\varvec{\sigma }})\) is tested against an isotropically hardening yield surface, whose radius \(r(\sigma _{y})\) is defined via a proposed simple modification to the Johnson-Cook flow rule. The modification herein proposed aims to account for the evolving damage with a single variable and a corresponding parameter. As such,
$$\begin{aligned} r({\varvec{\sigma }})= & {} \sqrt{\frac{1}{2}\left[ (\sigma _{1}-\sigma _{2})^{2} +(\sigma _{1}-\sigma _{3})^{2}+(\sigma _{2}-\sigma _{3})^{2}\right] }\nonumber \\\le & {} r(\sigma _{y}), \end{aligned}$$
(1)
with a modified Johnson-Cook (JC) rule defined as,
$$\begin{aligned} r(\sigma _{y})= & {} \left( A+B\varepsilon _{p}^{n}\right) \left( 1+C\hbox {In }\dot{\varepsilon }^{*}\right) \nonumber \\&\left( 1+T^{*m}\right) \left( 1-D\delta ^{*2}\right) . \end{aligned}$$
(2)
Here, A, B, C, n and m are the ordinary JC material parameters, \(\varepsilon _{p}\) is the effective plastic strain, \(\dot{\varepsilon }^{*}\) is a normalized strain rate, and \(T^{*}\) is the homologous temperature. The first term on the right hand side of (2) accounts for hardening, the second accounts for rate effects, and the third for thermal softening. Temperature rise is computed herein from adiabatic heating due to plastic dissipation, which ignores heat transfer. To avoid over-heating at slow (quasi-static) rates, a small fraction for the rate of conversion from plastic dissipation to heat is taken (i.e. 0.3 instead of 0.9).
In addition, a fourth term in (2) is proposed, and accounts for damage nucleation and evolution at a material point. D is a single parameter introduced to pre-multiply the newly defined damage variable \(\delta ^{*}\) outlined below. Of importance, \(\delta ^{*}\) does not require any additional parameters in its definition. This model was implemented as a user material subroutine (VUMAT), using a radial return algorithm.
Damage Law In nonlinear solids, the necessary and sufficient condition for compatible deformation may be expressed by the first-order differential law (Kroner 1981), \(Curl(\mathbf{F}^{e}{} \mathbf{F}^{p})\mathbf{=0}\), having assumed a multiplicative decomposition of the deformation gradient F into a plastic distortion \(\mathbf{F}^{p}\) followed by an elastic distortion \(\mathbf{F}^{e}\). Application of Stokes’ theorem to the differential law yields an incompatibility vector \(\delta \) (Kroner 1981),
$$\begin{aligned} \delta \equiv \oint _{c}{} \mathbf{F}d\mathbf{X}=\oint _{c}{} \mathbf{F}^{e}{} \mathbf{F}^{p}d\mathbf{X} \end{aligned}$$
(3)
Any non-zero value of \(\delta \) indicates the initiation of a topological defect at a material point, which we herein interpret as crack nucleation in the continuum.
In finite elements, the deformation gradient F is computed from unique nodal displacements and the assumption of differentiable displacement fields, irrespective of the constitutive behavior that evolves at the integration points. We will designate such a deformation gradient by \(\mathbf{F}^{FE}\) (i.e. \(\nabla _{\mathrm{X}}{{\mathbf {x}}}\equiv \mathbf{F}^{FE}\)). \(\mathbf{F}^{FE}\) is compatible throughout an FE simulation by definition (i.e. contour integral (3) vanishes identically with \(\mathbf{F} = \mathbf{F}^{FE})\). It follows that (3) can be re-hashed as,
$$\begin{aligned} -\delta =\mathbf{0}-\delta =\oint _{c}(\mathbf{F}^{FE}-\mathbf{F}^{e}{} \mathbf{F}^{p})d\mathbf{X}. \end{aligned}$$
(4)
As such, the integrand in (4) becomes non-zero only when the product \(\mathbf{F}^{e}{} \mathbf{F}^{p}\) is not equal to \(\mathbf{F}^{FE}\), which means that the evolving physical mechanisms in the solid cannot accommodate the deformation imposed in the FE simulation at that given time. Thus Eq. (4) serves as a condition for crack nucleation. We herein further simplify (4), and define a scalar measure of incompatibility \(\delta _{s}\) by Elkhodary and Zikry (2011),
$$\begin{aligned} \delta _{s}\left| \left| \mathbf{F}^{FE}-\mathbf{R}^{e}\mathbf{U}^{P}\right| \right| \mathbf{L}_{0}^{e} \end{aligned}$$
(5)
where \(L_{0}^{\mathrm{e}}\) is the initial element characteristic length. In (5) we have ignored elastic stretching; hence we took \(\mathbf{F}^{e}\cong \mathbf{R}^{\mathrm{e}}\), where \(\mathbf{R}^{e}\) is the rotation from the polar decomposition of \(\mathbf{F}^{e}\). We also assumed plasticity is irrotational, so that \(\mathbf{F}^{p}=\mathbf{U}^{p}\), where \(\mathbf{U}^{p}\) is the right plastic stretch tensor. Finally, in (2), \(\delta ^{*}=\delta _{s}/L^{e}\), where \(L^{e}\) is the current element characteristic length.
Selection of parameters for material model and failure
This modified Johnson-Cook rate, temperature and damage dependent flow rule was calibrated against tensile data using a finite element model of a dog-bone sample. The linear-elastic, density, and heat transfer properties were taken from MMPDS (Rice 2003); whereas, failure and Johnson-Cook parameters were fit to the provided tensile stress strain data. The dog-bone mesh and resulting stress strain curves are compared to experimental tensile test data in Fig. 25. Note, shear test-data was not used for calibration.
Modeling details for the challenge specimen
The nominal dimensions provided were used for modeling the test specimen. The region where FE boundary conditions where applied to the pin holes is shown in Fig. 26. This region was fixed for the upper hole and loaded with a prescribed velocity for the lower hole. For the slow load rate (0.0254 mm/s), a constant velocity was applied to the bottom hole and a fixed mass-scaling factor of 1000 was used, having removed the rate dependence from the flow rule. For the fast load rate (25 mm/s), two velocity profiles were tested. A smooth “actuator” type profile was applied to the bottom hole for the first 0.05 s; a constant velocity of 25 mm/s was applied after this time. For the fast simulation, no mass scaling was used. Lateral velocity was set to zero over this region for both holes. Out of plane velocity was also set to zero over the entire surface of both holes.
Element removal, based on the above defined fracture criterion, was used to model failure surface generation. No initial crack or preferential crack direction was applied. To allow for unbiased crack growth a uniform finite element mesh density was used throughout the model. The element type was an 8-node reduced integration linear hexahedral element with coupled thermal analysis capabilities (C3D8RT in ABAQUS\(\backslash \)Explicit). No formal mesh convergence study was performed on the sample; however, the mesh density was based on values from the previous fracture challenge. A mesh density of 12 elements through the thickness was thus used. This resulted in 36 elements around the radii of the large notch tips, and a total of 1,280,112 elements. A total of 48 processors were used in the parallelized analysis.
Table 7 Force displacement results
Blind predictions
The failed specimen shape for both the 0.0254 and 25.4 mm/s analyses are shown in Fig. 27. From these results the predicted crack path was B–D–E–A for both loading rates. The 25.4 mm/s analysis was not run to complete failure; however, based on the plastic strain and failure criteria profile, we were confident that the crack would terminate in the upper notch.
The maximum load and CODs at first crack initiation for both load rates are given in Table 7. The reaction force and COD are shown in Fig. 28.
Sources of discrepancy
We have identified three main sources of discrepancy with experimental results caused by the boundary conditions, post-processing method and failure criterion calibration method. We found that boundary conditions were very influential, particularly in the dynamic case. We tried two other sets of boundary conditions. Prior to blind predictions we did not constrain lateral motion at grips, and after the blind predictions we used rigid cylinders (Fig. 29) to load the specimen. For both these boundary conditions we observed fracture initiation in the top hole; when we constrained the lateral direction (which is what we reported) we predicted the experimentally observed failure sequence.
The post processing method we used for the reported blind predictions extracted forces only at boundary nodes and lead to an erroneous over-estimate of the force curve, in the dynamic model in particular, due to highly localized deformation at the grips. Therefore, this node averaging does not represent the experimentally observed values. For our corrected curves (Fig. 30), the forces at the reference nodes that drive the motion of the rigid cylinders was extracted. These forces give a better representation of the experimental results (less sensitive to grip localized behavior).
Finally, we believe the COD at failure predictions would be more accurate if we used a more systematic approach for determining the \(\delta _{\mathrm{critical}}^{*}\) (for element removal) as a function of mesh and sample size. For the analysis presented here, the same \(\delta _{\mathrm{critical}}^{*}\) was used for calibration and specimen failure predictions.
Team B
Team members:
C.H.M. Simha, Hari.Simha@NRCan-RNCan.gc.ca, CanmetMATERIALS, Natural Resources Canada, Hamilton, Ontario, Canada
B.W. Williams, Bruce.Williams@NRCan-RNCan.gc.ca CanmetMATERIALS, Natural Resources Canada, Hamilton, Ontario, Canada
Table 8 Hardening law coefficients for determined for Ti–6Al–4V from tensile tests in the rolling and transverse orientations
Finite element models, meshed with 8-noded brick elements, were used to simulate loading of the tensile, shear, and fracture specimens. The approximate element size near localization and failure in each of the specimens ranged from about 0.2 to 0.4 mm transitioning to larger elements away from the failure zone. Simulations were performed using the explicit dynamic solvers in ABAQUS and DYNA3D. For both solvers, a user-defined subroutine was implemented to describe the material behavior. The subroutine implemented in ABAQUS was based on von Mises yielding, with a hardening rule of the form,
$$\begin{aligned} \bar{\sigma }-\sigma _{y}(1+K\bar{\varepsilon }^{p})^{n} \end{aligned}$$
(6)
with the Xue–Weirzbicki damage model used to describe the failure of the material (Simha et al. 2014; Xue 2007). The DYNA3D subroutine also utilized the hardening rule given by Eq. 6 and the Xue-Wierzbicki damage model, but the deformation was based on the Cazacu–Plunkett–Barlat 2006 (CPB06) asymmetric/anisotropic yield function (Cazacu et al. 2006). Both subroutines used the Bazant-Pijaudier-Cabot non-local approach to mitigate the mesh dependence of finite element simulations (Simha et al. 2014). Though the two subroutines were very similar, there were small differences in the implementations of the two models, such as the tolerances utilized for convergence, which led to two slightly different predictions. The Cowpers-Symonds form, \((\dot{\bar{\varepsilon }}/\dot{\bar{\varepsilon }}_{ref})^{m}\) , was used to capture the effect of strain-rate in the DYNA3D simulations, whereas two different sets of hardening law coefficients were used for the slow and fast cases in the ABAQUS simulations. The hardening law coefficients at the slow and fast rates are provided in Table 8 and the hardening law is compared to the experimental stress versus strain curves in Fig. 31. Three simulation results are presented; ‘Mean’, ‘LB’, and ‘UB’ which correspond to the curves used in the mean, lower, and upper bound predictions of the fracture specimen.
In the determination of the yield function coefficients required for the CPB06 model, it was deemed reasonable to assume that the material behavior in the RD and TD directions of the sheet was equal. Odenberger et al. (2013) showed that the balanced biaxial flow stress state at room temperature for Ti–6Al–4V sheet was about 1.1 times the flow stress in the rolling direction. Hammer (Hammer 2012) showed that the compressive stress is about 1.1 times the tensile stress for Ti–6Al–4V sheet. These data were used to calibrate the in-plane coefficients for the CPB06 model with \(L_{11}=1.0, L_{12}=0.0, L_{13}=0.0, L_{22}=1.0\), \(L_{23}=0.0, L_{33}=0.82\), and \(k=-0.12\). The in-plane yield surface obtained from the CPB06 yield function is compared to the isotropic von Mises yield function (Fig. 32). The data from the shear tests conducted in the current work were used to determine the shear coefficient, \(L_{44}(=L_{55}=L_{66})=1.15\). The predicted shear behavior is compared with the experimental data in Fig. 33. Differences in predictions between ABAQUS and DYNA3D simulations were attributed to different boundary conditions employed to model the shear tests.
In addition to the tensile and shears tests, additional fracture data for Ti–6Al–4V provided by Hammer (2012) and Giglio et al. (2012) were used to determine the coefficients for the damage model. To account for the influence of mesh sensitivity on failure, a non-local implementation of the damage model was used in the simulations (Simha et al. 2014), in which the effective plastic strain used in the failure model was based on an averaged value between elements within a specified radius. In the current work, the non-local radius was about 1.5 mm in the ABAQUS simulations and 0.25 mm in the DYNA3D simulations. As discussed by Xue (2007), the damage model parameters can be estimated based on calculating an upper and lower bound to experimental fracture data. The upper and lower bound estimated for Ti–6Al–4V for the slow rate case is compared to experimental data in Fig. 34 for the DYNA3D sets of parameters. Though the upper and lower bounds do not bound the experimental data, they provide a reasonable estimate to capture the response of the Ti–6Al–4V alloy. The coefficients determined for Ti–6Al–4V based on the tensile and shear tests are provided in Table 9.
The predicted force versus COD at Location 1 (COD1) are shown in Figs. 35 and 36 for the ABAQUS and DYNA3D simulations of the fracture specimen and compared with the experimental data. A contour plot of the effective stress predicted from the slow DYNA3D simulation, after initial crack propagation, is shown in Fig. 37.
Table 9 Damage model coefficients utilized in ABAQUS and DYNA3D simulations for Ti–6Al–4V
Sources of discrepancy
The results show that the ABAQUS simulations over predicted the force response and under predicted COD1 at failure in both the slow and fast cases. The DYNA3D simulations better captured the force response, but marginally under predicted COD1 at crack propagation in both the slow and fast cases. After crack propagation, there were oscillations predicted in the DYNA3D simulations, which were not predicted in the ABAQUS simulations. The cause of these oscillations was unknown, but could be due differences applied boundary conditions of the two finite element models. Both the ABAQUS and DYNA3D simulations were able to predict the crack propagation path correctly (B–D–E). Results from the DYNA3D were used as the lower bound, and results from ABAQUS served to provide mean and upper bounds. For a discussion of the sources of the discrepancies between predictions and experiments and a description of Team B’s attempt to reduce the discrepancies, this team’s separate paper in this special issue can be consulted.
The difference in the size of the local radius is attributed to different implementations in the two codes and the mesh density; higher mesh density in DYNA3D and lower in the ABAQUS model. In the ABAQUS implementation during the stress-return procedure the stress, strain, and damage are calculated during each increment of the stress return. In the DYNA3D implementation, only the stresses and strain are computed during the stress-return procedure. The damage is computed after the stress return algorithm has completed. The different implementation might account for some of the discrepancy. Also, the radius of 1.5 mm is similar to what has been reported in the literature (see, for instance, Belnoue et al. 2010) and it is smaller than the diameter of the hole so that non-interacting boundaries do not impact the non-local calculation. Furthermore, it is our experience with the current non-local model in abaqus, that the non-local radius is usually a fraction of the smallest feature in the sample.
Team C
Team members:
A.R. Cerrone, albert.cerrone@ge.com, GE Global Research Center, Niskayuna, NY, USA
A. Nonn, aida.nonn@oth-regensburg.de, Ostbayerische Technische Hochschule (OTH) Regensburg, Germany
J.D. Hochhalter, jacob.d.hochhalter@nasa.gov, NASA Langley Research Center, Hampton, VA, USA
G.F. Bomarito, geoffrey.f.bomarito@nasa.gov, NASA Langley Research Center, Hampton, VA, USA
J.E. Warner, james.e.warner@nasa.gov, NASA Langley Research Center, Hampton, VA, USA
B.J. Carter, bjc21@cornell.edu, Cornell University, Ithaca, NY, USA
D.H. Warner, dhw52@cornell.edu, Cornell University, Ithaca, NY, USA
A.R. Ingraffea, ari1@cornell.edu, Cornell University, Ithaca, NY, USA
Approach
An over-the-counter methodology was used to predict fracture initiation and propagation in the challenge specimen. Specifically, the finite element software Abaqus/Explicit was used to simulate deformation and damage in the challenge specimen geometry with nominal dimensions. The continuum (Ti–6Al–4V) was modeled as linear elastic isotropic with the von Mises yield criterion. Hardening was defined by a tabular function of plastic strain. Damage initiation, in turn, was modeled with a ductile damage initiation criterion wherein the equivalent plastic strain at failure was a function of stress triaxiality. Damage propagation, finally, was modeled with an energy-based law with exponential softening. To discriminate between the fast (dynamic) and slow (static) actuation rates, a different set of plasticity and damage parameters was calibrated for each individually.
Material models
Linear elastic isotropic material parameters
The linear elastic isotropic (LEI) properties of room temperature Ti–6Al–4V used in this study are given in Table 10.
Table 10 LEI properties of room temperature Ti–6Al–4V
Calibration of hardening laws
For both loading rates, yielding in shear started at approximately a 12 % lower von-Mises stress than in tension. Clearly, the material exhibited anisotropic hardening. Hardening curves from both the tensile and shear tests were fit for both the static and dynamic cases. These curves are plotted in Fig. 38.
The anisotropic Drucker–Prager yield criterion in Abaqus/Explicit could have been employed to address the issue of anisotropic hardening; however, due to lack of time for calibration, it was passed over in favor of a simpler approach. Sensitivity studies showed that when assigned to the entire discretization, both the tensile and shear flow curves consistently yielded the A–C–F failure path; however, based on engineering good judgment, the B–D–E–A path was expected. To establish upper bound and best (“expected”) predictions, the tensile flow curve was assigned to the entire discretization. To establish lower bound predictions, the model was split into two regions. The shear and tensile flow curves were assigned to the shear-stress controlled region (beyond the lower notch including the B–D–E–A path) and axial-stress controlled region (beyond the upper notch including the A–C–F path), respectively, Fig. 39a.
Calibration of damage initiation and propagation laws
Two damage initiation laws were considered, Fig. 40. The first is a failure locus calibrated against the slow tensile (stress triaxiality \(= 0.8\)) and shear (stress triaxiality \(= 0\)) test data. However, as evident from Fig. 39b, the relevant triaxiality levels for the challenge specimen are in the range between 0.2 and 0.6. Consequently, the remainder of the locus had to be estimated with good engineering judgment. Based on results from the shear test wherein failure strain for the fast actuation rate was approximately 20 % lower than for the slower one, this locus was decreased by 20 % to give its dynamic counterpart. Denoted “estimated” in forthcoming sections, it was employed to establish all blind predictions. The second is another failure locus given by Giglio et al. (2013) for room-temperature Ti–6Al–4V under quasi-static loading. This locus was also reduced by 20 % to give a locus appropriate for dynamic loading. Due to lack of available calibration data, damage was assumed to evolve based on a critical fracture energy (10 N/mm) criterion with exponential softening.
Modeling details
Geometry and boundary conditions
The nominal dimensions of the tensile, shear, and challenge specimens were considered in the calibration and prediction phases. With regards to boundary conditions, rigid bodies were used for both the shear and challenge specimens. For the shear specimen specifically, all contact nodes were tied to a rigid body. The challenge specimen, in turn, showed considerable sensitivity to how its pins were modeled—in general, frictionless rigid body pins favored the B–D–E–A failure path whereas rigid body pins with friction favored the A–C–F path. It is noteworthy that Pack et al. (2014) explored this issue during the First Sandia Fracture Challenge (Boyce et al. 2014) and noted that boundary condition selection had no influence on the crack path and minimal influence on the specimen’s response. Clearly, this was not the case for the challenge geometry. To resolve this ambiguity, kinematic coupling constraints were considered which yielded the A–C–F path. Additionally, based on good engineering judgment, pins with friction were deemed to be more representative of the actual loading conditions. Consequently, rigid body pins with friction (with coefficient 0.10) were adopted for this study.
Crack propagation
Elements began to accumulate damage once a critical failure strain was reached. Damage evolved according to an energy-based criterion. If a given element’s stiffness degraded beyond acceptable limits, it was removed from the discretization (no remeshing or element-state mapping was required). This removal introduced new free surface into the discretization, and with subsequent removals, a faceted “crack” began to form in the discretization.
Mesh refinement
A mesh refinement study on the challenge specimen was conducted. It was observed that doubling the number of elements from four to eight in the thickness direction beyond the notches resulted in steeper load drops. A mesh size of 0.2 mm beyond the notches was adopted based on the authors’ past experiences with high-strength materials and plasticity-based damage models (Cerrone et al. 2014). 98,460 eight-noded brick elements with reduced integration discretized the geometry.
Blind predictions
The predictions and experimental results are given in Fig. 41. The A–C–F path was predicted for both actuation rates when no anisotropy was considered (the “expected” predictions). In the case of the lower bound predictions, the B–D–E–A failure path was predicted. In experiments, all eight samples failed B–D–E–A in the fast test while ten of eleven failed B–D–E–A in the slow test. It is noteworthy that the lower bound predictions captured maximum load accurately; however, the predicted COD1 at failure was off by well over 50 %. Moreover, the predicted maximum load and COD1 at failure in the expected predictions were off by approximately 15 %. Additionally, the predictions were too stiff during loading. In hindsight, the simplifications made to address the issue of anisotropic hardening were inadequate. Furthermore, it seems that the estimated failure locus over-predicted damage initiation for low stress triaxialities.
These two issues were addressed shortly after submission of the blind predictions to Sandia. First, earlier onset of yielding due to shear stress was enforced by leveraging the tensile flow curve in conjunction with the “*POTENTIAL” option in Abaqus/Explicit. As noted earlier, yielding in shear starts at approximately a 12 % lower von-Mises stress than in tension in this titanium alloy. Consequently, the R12 parameter was set to 0.88 to obtain the reduction of the yield stress by 12 % under shear. Additionally, the failure locus given by Giglio et al. (2013) was used as it gave more realistic failure strains for low stress triaxialities. As was done for the blind prediction, the static failure locus was decreased by 20 % to give the dynamic locus. These two minor adjustments resulted in a marked improvement in the predictions, Fig. 41—not only was the B–D–E–A crack path now predicted for both actuation rates, but the maximum loads and COD1 values at failure were well within the scatter of the experimental data.
Sources of discrepancy
Failure to account for anisotropic hardening/yielding favored the A–C–F failure path while discriminating between tensile and shear-dominance favored the experimentally-consistent B–D–E–A path. Under the current scheme, this implies that accounting for anisotropy in the nonlinear regime is necessary to predict the correct failure path. The reader is directed to Cerrone et al. (2016) for a detailed discussion on this topic. Additionally, applying a failure locus curve that gave lower failure strains for lower stress triaxialities resulted in a more accurate prediction of damage initiation. While these modifications improved the quality of the predictions considerably, the discrepancy of stiffness during early loading was left unaddressed. The predictions did not account for some compliance in the challenge specimen. This is an indication that perhaps the proposed boundary conditions were incorrect; for example, the 0.10 friction coefficient assigned between the loading pin and specimen might have been too high. Additionally, differences between the as-tested geometries and specimen with nominal dimensions and variation in the elastic properties could have also contributed to the discrepancy.
Team D
Team members:
T. Zhang, tzhang@gem-innovation.com, Global Engineering and Materials Inc., Princeton, NJ, USA
X. Fang, xfang@gem-innovation.com, Global Engineering and Materials Inc., Princeton, NJ, USA
J. Lua, jlua@gem-innovation.com, Global Engineering and Materials Inc., Princeton, NJ, USA
Summary of XSHELL methodology
The XSHELL toolkit developed in house is employed to predict the fracture patterns and its associated load-deflection curves at two different loading rates for the 2014 Sandia Challenge problems. Plane strain core model in XSHELL is used to capture the stress triaxiality effect in the vicinity of the crack tip. Crack initiation and propagation is accomplished through an element-wise crack insertion with cohesive injection once its accumulative plastic strain reaches a critical value. Maximum plastic strain direction is employed as the crack growth direction law.
XSHELL is an extended finite element based toolkit for Abaqus, which is developed for dynamic failure prediction of thin walled shell structures (Zhang et al. 2014). The use of the extended finite element methodology (XFEM) allows a mesh topology independent of any arbitrary crack surface and is proven to have great potential in automating the process as the crack grows with time with fixed mesh. The XSHELL toolkit features the kinematic representation of a cracked shell via its phantom paired elements, crack initiation prediction using an accumulative plastic strain criterion, mesh independent crack insertion through a cracked shell along the direction determined by the crack growth law, cohesive injection for characterization of the energy dissipation during crack growth, and display of the fractured pattern and the load displacement curve using a customized Abaqus CAE interface.
Table 11 Summary of material properties for Sandia challenge problem
Calibration of material properties and failure parameters
Based on the uniaxial tensile experimental testing data from Sandia National Lab, the true stress strain curves at two different loading rates have been iteratively calibrated (Fig. 42). Abaqus 3D FEA model with data look rate dependent plasticity is employed to determine the material constitutive properties by matching the numerically predicted load displacement curve with the experimentally measured load displacement curves. An additional stress strain curve at strain rate 400/s is selected by referring to the paper by Lee and Lin (1998) with a small modification to ensure that it is always higher than the other two stress stain curves at lower strain rates (Fig. 42). This curve serves as the upper bound for the numerical analysis. The failure strains at two testing rates can be also calibrated using uniaxial tensile tests, which are the same as around 0.6.
With Sandia Challenge 3D Abaqus model and rate dependent data look model, two analyses have been performed: one with the displacement loading rate 0.0254 mm/s; and another with the displacement loading rate 25.4 mm/s. Average strain rates at critical area can be estimated for these two loading cases: 0.015/s for slow loading and 40/s for fast loading. Based on these two average strain rates, the stress strain curves for Sandia Challenge problem at low loading rate and high loading rate can be interpreted from the material stress strain curves at different strain rates as in Fig. 42. The resulting stress strain curves for these two loading cases and the elastic material properties are listed in Table 11.
Mesh sensitivity study has also been performed using Abaqus shell FEA model with S4R element to identify a rational mesh size. The load displacement curves obtained from a dense mesh and a coarse mesh agree with each other, which indicates that the coarse mesh is good enough for the numerical simulation of the Sandia Challenge problem.
Blind prediction for Sandia challenge problem with XSHELL
Sandia Challenge FEA model for XSHELL plane strain core application is shown in Fig. 43. Given the location of the notch and holes in the Sandia challenge problem, a user defined XSHELL zone denoted by the red box in Fig. 43 is used with plane strain core option. Due to the nature of this Sandia Challenge problem, the plane strain core model will be invoked only at the crack tip. The edges of the loading holes are linked with two Reference Points by using kinematic coupling. A constant velocity loading condition is applied at the upper Reference Point while the lower Reference Point is fixed. Abaqus explicit solver is used to perform the numerical analysis.
The Sandia Challenge Problem has been analyzed with the calibrated failure strain as 0.6 and the approximate plane strain core coefficient \(\alpha \) as 0.001 based on experience. As shown in Fig. 44c, the predicted crack path for 0.0254 mm/s loading rate is B–D–E, which agrees with experimental tested crack path shown in Fig. 44d. The predicted Load-CODs curve for 0.0254 mm/s loading rate is compared with the experimental testing curve in Fig. 44a. The solid red line in this figure represents GEM’s prediction, while the dotted blue line represents the experimental testing curve. The predicted COD1 value at the instant of the first crack initiation is 2.21 mm, and the peak load from the numerical simulation is 23,830.9 N which is higher than the experimental tested average peak force value 19,643.36 N.
The predicted crack path for 25.4 mm/s loading rate is shown in Fig. 45c with crack initiation at Hole D followed by its growth to Hole B and the second crack initiation at Hole D followed by its growth to Hole E. The predicted Load-CODs curves are compared with experimental result in Fig. 45a. The predicted COD1 value at first crack initiation is 2.4 mm, and the peak load from the XSHELL prediction is 23,815.8 N, which is higher than the experimental tested average peak value 20,435.88 N.
Sources of discrepancy
The initial stiffness of Load-COD1 curves predicted by the XSHELL toolkit has a discrepancy in comparison with the test data. This is mainly due to the use of kinematic coupling boundary condition with all degrees of freedom fixed for the loading hole. The high peak load prediction may be related to the use of a plasticity model without \(J_{3}\) dependence.
Team E
Team members:
V. Chiaruttini*, vincent.chiaruttini@onera.fr, Onera, Université Paris-Saclay, Châtillon, France.
M. Mazière, matthieu.maziere@mines-paristech.fr, MINES ParisTech, PSL Research University, Centre des Matériaux, CNRS UMR 7633, Evry, France.
S. Feld-Payet, sylvia.feld-payet@onera.fr, Onera, Université Paris-Saclay, Châtillon, France.
V.A. Yastrebov, vladislav.yastrebov@mines-paristech.fr, MINES ParisTech, PSL Research University, Centre des Matériaux, CNRS UMR 7633, Evry, France.
J. Besson, jacques.besson@mines-paristech.fr, MINES ParisTech, PSL Research University, Centre des Matériaux, CNRS UMR 7633, Evry, France.
J.-L. Chaboche, jean-louis.chaboche@onera.fr, Onera, Université Paris-Saclay, Châtillon, France.
*corresponding author: vincent.chiaruttini@onera.fr
For the blind round robin prediction of ductile fracture, we used finite element simulation with a visco-elasto-plastic material model in finite strain. This model has been calibrated for both tensile and shear tests. Afterwards some numerical analyses were performed to identify a suitable approach to obtain a satisfactory crack path and force-displacement curve for different loading rates.
Visco-elasto-plastic constitutive model
To identify the model, a finite element analysis has been carried out on uni-axial tests on samples oriented mainly in the rolling direction, which is the most representative loading condition for the Sandia Fracture Challenge. The strain rate has been estimated close to the gauge for both loading rates (0.66 10\(^{-3}\) s\(^{-1}\) and 0.66 s\(^{-1})\). At the higher rate, the hardening effects should be lower than expected due to a considerable thermal heat produced. The provided test curves were transformed to represent the effective strain and stress but they do not take into account the necking effects.
The chosen constitutive model is initially based solely on a modified Norton flow with an anisotropic Hill criterion. To insure an accurate description of hardening at both low and high rates, a softening evolution is added using a negative value for K:
$$\begin{aligned} \dot{p}=\frac{{\left( {\frac{3}{2}\hat{\sigma }:M_{{Hill}}:\hat{\sigma }} \right) -R}}{{K_{0}+K\left( 1-e^{{-bp}}\right) }}^{n} \end{aligned}$$
To allow a finer tuning of the hardening in the studied plastic strain domain up to four terms were tested in combination with nonlinear isotropic terms in the final model:
$$\begin{aligned} R=R_{0}+\sum \nolimits _{i=1}^{N} Q_{i}\left( {1-e^{-\beta _i p}}\right) \end{aligned}$$
This model was calibrated using implicit 3D finite element computation within the Z-set finite element software (Besson and Foerch 1997). Such an approach allows us to accurately fit uni-axial tests using a finite strain implicit quasi-static finite element solution process, the mesh size was chosen to ensure the correct deformation level for the appearance of the necking effect (Fig. 46).
To deal with material anisotropy, the provided shear test result was studied. The first stage was to correctly fit the initial stiffness of the numerical model with the testing data, dealing with modification on the prescribed boundary conditions. To satisfactory reproduce those experiments, an anisotropic Hill criterion was finally used, whose calibration was done using results from Gilles et al. (2012, 2011).
Failure modeling
Our driving idea was to identify the simplest model that could produce sufficiently accurate results on the Sandia challenge’s specimen. Due to the specific specimen geometry, only two different kind of failure can be observed: dominant shear failure under low triaxiality for crack path B–D–E–A, or high triaxiality failure along crack path A–C–F. Thus, it is not strictly necessary to apply a Gurson-type model, which can predict a failure path whose geometry would highly depend on triaxiality. A pragmatic approach could be adopted, involving a cohesive zone model to insure the softening process due to the material damaging. The chosen model (Alfano and Crisfield 2001) (allowing to decouple the material toughness for traction and shear failure modes), was calibrated using both tests (uni-axial and shear, see Fig. 47), with a special attention paid to the region of the negative slope observed in the shear experiments, the mesh size was chosen according to the necking effect.
Numerical simulation and discussion
To predict the crack path, complete 3D finite element computations were performed with only visco-elastic-plastic constitutive model up to the maximal force before the necking appears. Only after the peak force cohesive elements were introduced in the zones with high accumulated plastic strain, which were observed in certain ligaments when reaching the peak load.
Both fast and slow loading cases were simulated using a 3D mesh in finite strain, with a quasi-static implicit solution and quadratic elements producing about 90,000 degrees of freedom. Approximately 250 incremental loading steps were used to reach the E–A ligament failure (with a computational time limited to less than 2 h).
The obtained result confirmed that, as expected, the shear dominant failure occurs through the B–D–E–A crack path with the first B–D–E failure stage followed a second crack initiation process on the E–A final ligament. As the test COD gauges were not able to deliver global Force-COD plots after the first failure, a simple image analysis from the organizers movie (on the fast case) has been carried out to extend the curve after the unstable crack propagation (Fig. 48). This analysis shows quite a good agreement with our blind prediction and the experimental data during the stable failure phases (see Fig. 49).
Sources of discrepancy
The pragmatic approach that was applied for our predictions has some evident drawbacks and limitations: the quasi-static simulation, combined with excessively dissipative cohesive zone, failed to start the damage process sufficiently early (especially at high speed) and hence produced the softening slope which is not sufficiently steep. Furthermore, for a much more complex case (where triaxiality has a significant impact on the crack path), such a method cannot be successful. However, this rather simple approach (from computational point of view) offers the possibility to properly capture the peak force and to correctly fit the testing curve during the ultimate crack initiation process on this specific Sandia Fracture Challenge conditions.
Team F
Team members:
J. Lian, junhe.lian@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
Y. Di, yidu.di@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
B. Wu, bo.wu@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
D. Novokshanov, denis.novokshanov@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
N. Vajragupta, napat.vajragupta@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
P. Kucharczyk, pawel.kucharczyk@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
V. Brinnel, victoria.brinnel@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
B. Döbereiner, benedikt.doebereiner@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
S. Münstermann, sebastian.muenstermann@iehk.rwth-aachen.de, RWTH Aachen University, Aachen, Germany
Approach
The material model used in the team is based on a hybrid damage mechanics model (Lian et al. 2013) which was further extended to incorporate the effect of strain rate and temperature for Charpy test and machining under adiabatic condition (Buchkremer et al. 2014; Münstermann et al. 2012). The distinction of the model is that it differentiates the damage initiation and fracture and it is developed in a hybrid way by combining the uncoupled model to act as the damage threshold and coupled model to represent the microstructure degradation till final fracture. The model is implemented into the FE code Abaqus/Explicit by means of a user material subroutine (VUMAT) and all the simulations presented here are conducted in this environment.
The yield function of the model is given in Eq. 7. It is noted that the coupling effect of the damage into the yield function is only valid once the damage initiation criterion is fulfilled.
$$\begin{aligned} \varPhi =\bar{\sigma }-\left( 1-D\right) \sigma _{y}\left( \bar{\varepsilon }^{p},\eta ,\theta ,\dot{\bar{\varepsilon }}^{p},T\right) \le 0 \end{aligned}$$
(7)
Generally, the isotropic yielding and hardening are employed based on the negligible difference between the flow responses from tensile tests along rolling and transverse direction for the investigated material. However, a more general plasticity model (Bai and Wierzbicki 2008) to account for the stress state effect on yielding is employed, as defined in Eq. 8. In addition, the influence of strain rate and temperature on the yielding is also defined (Eq. 8).
$$\begin{aligned}&\sigma _{y}\left( {\bar{\varepsilon }^{p}, \dot{\varepsilon }^{p},T,\eta , \uptheta }\right) \nonumber \\&\quad = \left[ {\sigma _{y} \left( {\bar{\varepsilon }^{p}}\right) \cdot \left( c_{{\dot{\bar{{\varepsilon }}}^{{\mathrm{p}}}}}^{1} ln\dot{\bar{\varepsilon }}^{{\mathrm{p}}} + c_{{\dot{\bar{{\varepsilon }}^{{\mathrm{p}}}}} ^{2}} \right) + c_{{\dot{\bar{{\varepsilon }}}^{{\mathrm{p}}}}} ^{3}\cdot \dot{\bar{\varepsilon }}^{p}} \right] \nonumber \\&\quad \quad \times \left[ {c_{{\mathrm{T}}}^{1} exp\left( {c_{{\mathrm{T}}}^{2} T} \right) + c_{{\mathrm{T}}}^{3}} \right] \left[ {1 - c_{{{\upeta }}}\cdot \left( {\eta - \eta _{0}} \right) } \right] \nonumber \\&\qquad \times \left[ {c_{{{\uptheta }}}^{{\mathrm{s}}} + \left( {c_{{{\uptheta }}}^{{{\mathrm{ax}}}} - c_{{{\uptheta }}} ^{{\mathrm{s}}}} \right) \cdot \left( {\lambda - \frac{{\lambda ^{{m + 1}}}}{{m + 1}}} \right) } \right] \end{aligned}$$
(8)
In the equation, \(\sigma _y ({\bar{\varepsilon }}^{\mathrm{p}})\) stands for the flow curve under the reference condition, in the context, i.e. quasistatic tensile test at room temperature; \(c_{\dot{\bar{\varepsilon }}^{\mathrm{p}}}^1 -c_{\dot{\bar{\varepsilon }}^{\mathrm{p}}}^3 \) are the material parameters for the strain rate effect; \(c_\mathrm{T}^1 - c_\mathrm{T}^3 \) are the material parameters for temperature effect; \(c_{\upeta }\) and \(\eta _0\) are the material parameters for the effect of stress triaxiality; \(c_{\uptheta }^\mathrm{s} \), \(c_{\uptheta }^{\mathrm{ax}}\) and m are the material parameters for the effect of the Lode angle. For details, readers are referred to Bai and Wierzbicki (2008) and Münstermann et al. (2013).
It is also noted that under the adiabatic condition, the temperature evolution is defined according to Eq. 9.
$$\begin{aligned} \dot{T}= \frac{\delta \cdot \bar{\sigma }\cdot {\dot{\bar{{\varepsilon }}}}^{{\mathrm{p}}}}{\rho \cdot c_{\mathrm{p}}} \end{aligned}$$
(9)
where \(\delta , \rho \) and \(c_{\mathrm{p}}\) are the specific heat fraction, material density and heat capacity, respectively.
For damage modelling, a discontinuous damage evolution law is assumed, i.e. the initiation of damage is not associated with the plastic deformation but a characteristic strain-based criterion which depends on stress triaxiality and Lode angle. Afterwards, a simple linear increase is assumed with respect to the equivalent plastic strain, and the rate of damage evolution is governed by the energy dissipation between damage initiation and the complete fracture of the material point. The critical damage to fracture is assumed to be a function of Lode angle. Element deletion technique is used for the crack propagation.
$$\begin{aligned} D=\left\{ {{\begin{array}{ll} 0;&{}\quad \bar{\varepsilon }^{\mathrm{p}}\le \bar{\varepsilon }_{\mathrm{i}}^\mathrm{p} \\ \int _{\bar{\varepsilon }_\mathrm{i}^\mathrm{p}}^{\bar{\varepsilon }^{\mathrm{p}}} \frac{\sigma _{\mathrm{y}0}}{2G_\mathrm{f}} d\bar{\varepsilon }^{\mathrm{p}}; &{}\quad {\bar{\varepsilon }_\mathrm{i}^\mathrm{p} < \bar{\varepsilon }^{\mathrm{p}}<\bar{\varepsilon }_\mathrm{f}^\mathrm{p}} \\ {D_{\mathrm{cr}} ;}&{}\quad {\bar{\varepsilon }_\mathrm{f}^\mathrm{p} \le \bar{\varepsilon }^{\mathrm{p}}} \\ \end{array}}}\right. \end{aligned}$$
(10)
where \(G_\mathrm{f}\) is the material parameter and \(\sigma _{\mathrm{y}0} \) is the stress at damage initiation. The damage initiation strain, \(\bar{\varepsilon }_\mathrm{i}^\mathrm{p} \), and the critical damage accumulation for fracture are defined in Eqs. 11 and 12, respectively.
$$\begin{aligned} \bar{\varepsilon }_i^p= & {} \left[ {C_1 e^{-C_2 \eta } -C_3 e^{-C_4 \eta }} \right] \bar{\theta }^{2}+C_3 e^{-C_4 \eta } \end{aligned}$$
(11)
$$\begin{aligned} D_{\mathrm{cr}}= & {} c_{\mathrm{cr}}^1 \cdot \bar{\theta }^{2}+c_{\mathrm{cr}}^2 \end{aligned}$$
(12)
where \(C_1 -C_4 \) are material parameters for damage initiation locus and \(c_{\mathrm{cr}}^1 \) and \(c_{\mathrm{cr}}^2 \) are the material parameters for the fracture locus.
Table 12 Calibrated material parameters for the model
Material parameter calibration
The model involves a large number of material parameters for accurate characterization of material behavior. For a complete material parameter calibration procedure, several types of tests and specimens are required as described by Lian et al. (2013, 2015) and Buchkremer et al. (2014). In the SFC2, as only limited tests were provided, the calibration of the plasticity parameters, including the effect of stress state, strain rate and temperature, and damage initiation locus was assisted by material database of the Steel Institute. The reference flow curve was derived from the tensile tests along the rolling direction (specimen RD5) provided by Sandia National Laboratory. The Ludwik equation was used to extrapolate the flow curve to large strains. The most important parameters for fracture prediction, the damage evolution and critical damage accumulation parameters, were calibrated by an iterative fitting procedure such that force–displacement responses of tensile and shear tests from simulations can meet the experiments. The 8-node linear brick element with reduced integration (C3D8R) was used for both models and a mesh size of 0.1 mm was applied to the critical region to represent the high order of strain gradient. The comparison of the force–displacement response between the experiment and simulation with the final calibrated material parameters are shown in Figs. 50 and 51 for both tensile and shear tests, respectively. It is noted that only the tests under slow loading rate were used for the calibration and it is assumed that the damage parameters keep the same for fast loading rate for simplicity as the simulations of tensile and shear tests with the same damage parameters under fast loading rate also give reasonably good results compared to the experiments. All the calibrated material parameters are listed in Table 12.
Blind prediction
For the simulation of the blind prediction challenge, the numerical model was constructed based on the received ideal geometry and the dimension scattering as a result from the manufacturing process was neglected. The same element type (C3D8R) was applied to the model. To account for the high strain gradient in the critical deformation region, a finer mesh was implemented in this area whereas the coarser mesh was applied on the rest of the model and the model finally consisted of about 800,000 elements. As the calibration models, the mesh size of 0.1 mm was chosen for the fine mesh. The loading roll and the fixing roll were modeled as the rigid body while the contact interaction between the specimen and the rolls were assumed to be frictionless. Regarding the boundary condition assignment, the fixing roll was constrained in all degrees of freedom and only displacement along the vertical axis was applied to the loading roll. The overview of the model before deformation and the critical region after loading for both slow and fast loading rates are shown in Fig. 52. To balance the predictive quality and computational efficiency, time and mass scaling were applied to the simulations for slow and fast loading rates, respectively. A convergence of the force–displacement response was reached for the time scaling and a minor effect of the kinetic energy over the internal energy was also met for the mass scaling.
As illustrated in Fig. 52, the crack path prediction of the slow loading rate is A–C–(B–D–E)–F. In general, the crack path A–C–F is in a competition with the crack path B–D–E, and these two are roughly corresponding to the fracture patterns under tensile and shear loadings, respectively. Based on the calibrated parameters, the model gives very sensitive and similar responses of these two modes as these two crack paths are concurrent. This observation is also confirmed by the experiments. Although most of the fracture path for slow loading rate is B–D–E, A–C–F was also observed for one specimen. For the fast loading rate, the prediction agrees with the experiments. Despite the small crack from A to C during the deformation owning to the sensitivity of the model, the dominant crack path is B–D–E–A. The force versus COD1 for both loading rates are plotted in Fig. 53. For both rates, the maximum force is overestimated 10–15 %, but the predicted fracture displacements are in a reasonable agreement with the experiments consistently for both slow and fast loading rates.
Sources of discrepancy
Although overall accepted prediction was achieved with several assumptions and literature data, there is still certain degree of space to improve the blind prediction, such as an optimized strain hardening extrapolation, mesh regularization or non-local formulation to decrease the scaling factors. The overestimation of the force level for both slow and fast condition is clearly related to a higher description of the stress–strain response of the material. Both the strain hardening extrapolation and the damage evolution parameters are responsible for this, as the material owns a quite early damage initiation from microstructural point of view. Another critical point for the challenge simulation is the expensive computational cost. As the mesh size is pre-defined in the calibration test, to maintain consistent softening response of the material, a large number of elements resulted in the challenge simulation. To balance the computational time and predictive accuracy, high time and/or mass scaling factor is employed, which also results in the oscillation of the force response under fast loading condition. In this regard, the implementation of mesh regularization or non-local formulation is of interest. The further improvement together with the details for the modeling is reported in a full length paper in the same Special Issue as this article.
Team G
Team members:
M.K. Neilsen, mkneils@sandia.gov, Sandia National Laboratories, Albuquerque, NM, USA
K. Dion, kdion@sandia.gov, Sandia National Laboratories, Livermore, CA, USA
Approach
Predictions for the SFC2 challenge were generated using a quasi-static finite element code in SIERRA solid mechanics (2011). A unified creep plasticity model was used to capture temperature and strain rate effects. For this model, the inelastic (creep + plastic) strain rate is given by
$$\begin{aligned} {{\varvec{\dot{\varepsilon }}}}^{in}=\frac{1}{3}\dot{\gamma }\mathbf{n}=\frac{3}{2}e^{f} \hbox { sinh}^{p}\left[ \frac{\tau }{\alpha D(1-cw^{d})}\right] \mathbf{n} \end{aligned}$$
(13)
where \(\dot{\gamma }\) is the equivalent plastic strain rate, \(\mathbf{n}\) is the associated flow direction, \(\tau \) is the vonMises effective stress, D is a user-prescribed function of equivalent plastic strain to define isotropic strain hardening, f, p, and \(\alpha \), are temperature-dependent material parameters, and w is a scalar measure of damage. Material parameters c and d define the reduction in isotropic strength due to damage. Damage evolution is given by Wilkins et al. (1980).
$$\begin{aligned} w=\int \left( \frac{1}{1+\frac{p}{\hat{p}}} \right) ^{\hat{a}}\left( 2-A\right) ^{\hat{\beta }}\hbox {d}\gamma \end{aligned}$$
(14)
where
$$\begin{aligned} s_1 \ge s_2 \ge s_3\qquad A=Max\left( \frac{s_2 }{s_1 },\frac{s_2 }{s_3}\right) \quad p=\frac{-1}{3}{\varvec{\sigma }}:{\varvec{i}}\nonumber \\ \end{aligned}$$
(15)
\(s_{i}\) are the eigenvalues of the stress deviator, p is pressure. Damage evolution depends on both pressure (first invariant of total stress) and third invariant of deviatoric stress. The pressure-dependent, first term in Wilkins et al. damage evolution equation is similar to the damage evolution equation proposed by Wellman (2012) and used by us in the initial Sandia Fracture Challenge (Boyce et al. 2014; Neilsen et al. 2014). Note that the stress dependence can be removed by setting the material parameters \({\hat{\alpha }}\) and \({\hat{\beta }}\) equal to zero in Eq. 14, then damage is simply accumulated equivalent plastic strain. When damage has reached a critical level, the element is not instantaneously removed nor is the stress instantaneously reduced to zero; instead the constitutive response is changed in five solutions steps to be that of a very flexible elastic material with moduli equal to 0.0001 times the original elastic moduli. This approach is used to make the acquisition of post failure equilibrium solutions possible for most problems.
The effects of heating due to plastic work were captured with fully coupled thermal stress simulations in which the volumetric heating rate, \(\dot{Q}\), was given by
$$\begin{aligned} \dot{Q}=\eta \dot{W}^{p}=\eta {\varvec{\sigma }}:{\varvec{\varepsilon }}^{in} \end{aligned}$$
(16)
where \(\eta \), the Taylor-Quinney coefficient prescribes the fraction of plastic work that is converted to heat, \(\dot{W}^{p}\) is the plastic work rate, \({\varvec{\sigma }}\) is the Cauchy stress, and \({\dot{\varvec{\varepsilon }}}^{in}\) the inelastic strain rate. A survey of literature yielded a wide range of values, 0.1 to 0.9, for the fraction of plastic work converted to heat. A value of 0.5 was used for \(\eta \) in these simulations.
Material parameters for the model were obtained from simulations of the uniaxial tension tests at different rates. Values for \(\alpha \) in Eq. 13 which define the effects of temperature on the isotropic strength were based on data from (Rice 2003). Strain hardening was based on a fit to the slow rate experiment (black curve in Fig. 54a). The high rate test was then simulated (blue curve in Fig. 54a). Finally, a fully coupled thermal stress simulation predicted that the apparent ductility of the material would be dramatically reduced when heating due to plastic work was included (red curve in Fig. 54a). This occurs because heating causes the apparent hardening of the material to decrease which leads to the initiation of necking and subsequent cracking earlier in the simulation. The predicted deformed shape at failure is in reasonably good agreement with experiments (Fig. 54b). In this figure, LIFE is simply the current damage divided by the critical damage, so when LIFE obtains a value greater than one the element is cracked and turns white. The model predictions were insensitive to changes in the damage parameters; however, these parameters were highly correlated with the failure strain.
Next, the shear test was simulated with y-displacements applied directly to all surface nodes on the specimen where the loading blocks contacted the specimen (Fig. 55a). These simulations predicted a crack similar to the experiment (Fig. 55b) but load-displacement curves that had a much higher initial slope and peak value than experiments (Fig. 55d). To try and understand this discrepancy, loading blocks were added to the shear test simulation (Fig. 55c). The loading blocks were preloaded by preventing normal displacement of the back surface of the back blocks and clamping the sample by displacing nodes on the front surface of the front blocks to generate a total clamping force of 170.8 kN which is equal to the expected clamping force of the eight bolts torqued to 5.65 N-m. This simulation matched the experimental load-displacement curve better (green curve in Fig. 55d) but did not fail because the sample was just rotating between the loading blocks. In the tests, the horizontal grip inserts (Fig. 7) which were hand-tightened would eventually prevent rotation of the sample and cause the observed shear failure. Due to this shear test discrepancy, we decided to go ahead and generate SFC2 challenge geometry predictions with parameters we had obtained from the uniaxial tension tests.
The SFC2 challenge geometry was then simulated using a model with 451,536 elements and a typical element edge length of 0.254 mm. Contact between the loading pins and sample was not included and instead half of each loading pin was modeled. The nodes at the centerline of the bottom pin were given zero displacement in all three directions while the nodes at the centerline of the top pin were given vertical displacement at the prescribed rate. Sliding was not allowed between the pin and the sample but the center of the pin could freely rotate about the pin axis for ‘free rotation’ simulations. Pin rotation was not allowed in ‘no rotation’ simulations. The free rotation and no rotation simulations were performed to bound expected behavior. Free rotation simulations were expected to be closer to the experiments. Free rotation simulations predicted a crack path of A–C–F (Fig. 56a) and no pin rotation simulations a crack path of B–D–E–A (Fig. 56b). The models bound the experimental displacements to failure but predict too high of a failure load.
Sources of discrepancy
The experimentally measured peak load for the challenge geometry was 86 to 91 percent of the predicted peak load. This discrepancy indicates that the material is likely weaker in shear than this model predicted with a von Mises yield. Simulations of the shear test with loading blocks indicated that the sample may be rotating more than expected which would contribute to the displacement discrepancy in the experiment. However, even with this discrepancy the model should have still predicted close to the correct loads. The discrepancy in load at yield and peak load in the shear test is similar in magnitude to load discrepancy with the challenge geometry again indicating that this material is likely weaker in shear than our model predicted. The most significant weakness in these simulations was that they did not account for this reduced strength in shear.
Team H
Team members:
K.N. Karlson, knkarls@sandia.gov, Sandia National Laboratories, Livermore, CA, USA
J.W. Foulk III, jwfoulk@sandia.gov, Sandia National Laboratories, Livermore, CA, USA
A.A. Brown, aabrown@sandia.gov, Sandia National Laboratories, Livermore, CA, USA
M.G. Veilleux, mgveill@sandia.gov, Sandia National Laboratories, Livermore, CA, USA
Approach
The material, time scale, and mode of loading dictated our path forward. Provided experimental data and literature advocate models that incorporate rate dependence, temperature dependence, and anisotropy in both the yield stress and the hardening. Void evolution must include multi-axial nucleation, growth, and coalescence. The low thermal conductivity of titanium and the time scales for characterization and testing requires thermomechanical coupling and implicit time integration. Local material softening requires regularized methods for solution.
Team Sandia California (Team H) used SIERRA Solid Mechanics (SIERRA SM) to capture the required physics and numerics for solution. SIERRA SM is a Lagrangian, three-dimensional, implicit code for the analysis of solids and structures. It contains a versatile library of continuum and structural elements, and an extensive library of material models. For all SFC2 related simulations, our team used Q1P0, 8 node hexahedral elements with element side lengths on the order 0.175 mm in failure regions. To model crack initiation and failure, element death removed elements from the simulation according to a continuum damage model. Exploratory studies were also conducted with regularized methodologies (nonlocality, surface elements). Unstable modes of fracture were resolved with implicit dynamics [HHT time integration with numerical damping (Hilber et al. 1977)].
Thermo-visco-poro-plasticity. We chose SIERRA SM’s isotropic Elasto Viscoplastic (EV) material model for our simulations because it contains the most relevant physics to accurately predict the SFC2 challenge problem such as the flexibility to include temperature and rate dependence for a material. However, since the EV model does not support anisotropic plastic behavior, the anisotropy evident in the provided data was included through other means described in detail in the following section.
The EV plasticity model is an internal state variable model for describing the finite deformation behavior of metals. The model incorporates strain rate and temperature sensitivity, as well as damage, and tracks history dependence through the use of internal state variables. In its full form, the model has considerable complexity, but most of the material parameters and resulting behavior are optional. The form of the material model specific to our use for SFC2 will now be outlined for the simplified case of uniaxial tension. For this simplified case, the stress evolves according to
$$\begin{aligned} \dot{\sigma }=E(\dot{\epsilon }-\dot{\epsilon }_{p}) \end{aligned}$$
(17)
where \(\epsilon \) is the total strain and \(\epsilon _{p}\) is the plastic strain. The flow rule is defined by
$$\begin{aligned} \dot{\epsilon }_{p}=f \hbox {sinh}^{n} \left( \frac{\sigma _{y}-k}{Y}-1\right) \end{aligned}$$
(18)
where \(\sigma _{y}\) is the equivalent stress; Y is a material parameter representing the rate independent, initial yield stress; f and n are material parameters that govern the material rate dependence; and \(\kappa \) is the isotropic hardening variable for the material, which evolves according to a hardening minus dynamic recovery model originally proposed by Kocks and Mecking (1980):
$$\begin{aligned} \dot{\kappa }=\kappa \frac{\dot{\mu }}{\mu }+(H-R_{d}\kappa ) \dot{\epsilon }_{p}. \end{aligned}$$
(19)
The temperature dependence for all material parameters (\(Y, f, n, H, R_d\)) can be specified explicitly with user specified scaling functions or using functional forms built into the model. Heat generation due to plastic work is calculated with
$$\begin{aligned} \dot{q}=\beta \sigma \dot{\epsilon }_p \end{aligned}$$
(20)
where the material parameter \(\beta \) is the fraction of plastic work dissipated as heat.
The EV model contains a void growth model and a void nucleation model to account for isotropic material damage. For void growth, damage evolves according to the model proposed by Cocks and Ashby (1980):
$$\begin{aligned} \dot{\phi }=\sqrt{\frac{2}{3}}\dot{\epsilon }_p\frac{1-(1-\phi )^{m+1}}{(1-\phi )^{m}}\hbox {sinh}\left[ \frac{1(2m-1)}{2m+1}\frac{p}{\sigma _{vm}}\right] \nonumber \\ \end{aligned}$$
(21)
where \(\sigma _{vm} \) is the von Mises stress, p is the hydrostatic stress, \(\phi \) is the void volume fraction of the material and the damage exponent m is a material parameter. With this void growth model, damage will only increase when \(p/\sigma _{vm} >0\). To account for damage resulting from other stress states, a void nucleation model based on \(J_{3}\) (Horstemeyer and Gokhale 1999; Nahshon and Hutchinson 2008) was also included in the material model:
$$\begin{aligned} \dot{\eta }=\eta \dot{\epsilon }_{p} N_1\left[ \frac{4}{27}-\frac{J_3^2}{J_3^2}\right] \end{aligned}$$
(22)
where \(N_1 \) is a material parameter, \(\eta \) is the number of nucleated voids, and \(J_i \) are the deviatoric stress invariants. These two damage models can be used independently or concurrently to model void nucleation and growth. Including damage evolution through these models reduces the material’s elastic modulus and shear modulus by a factor of \(1-\phi \), and the flow rule becomes
$$\begin{aligned} \dot{\epsilon }_{p}=f\hbox {sinh}^n\left[ \frac{\sigma _y-\kappa (1-\phi )}{Y(1-\phi )}-1\right] . \end{aligned}$$
(23)
The damage models require the definition of the initial void volume fraction \(\phi _0 \), the initial size of nucleated voids \(\phi _0^\eta \), and the initial void count per volume \(\eta _0 \). Void coalescence is modeled through \(\phi _{coal} \). The material point is unloaded for \(\phi >\phi _{coal} \). In contrast to surface elements, we remove elements (element death) when any integration points satisfy the coalescence criteria. Stabilization of fully integrated formulations is problematic with loaded and unloaded integration points.
Material parameter calibration
We populated the EV material parameters for Ti–6Al–4V sheet using a combination of the data provided in the challenge announcements and data from literature. Initially, the yield (Y, f, and n) and hardening (H and \(R_d\)) parameters were calibrated to the provided tensile data using a non-linear, least squares algorithm where the objective function consisted of the error between the provided data and model data. Since the rate dependence for the initial yield stress is not uniquely constrained by two data points, we used rate dependence data from Follansbee and Gray (1989) to supplement the data at two rates provided for the challenge. Temperature dependence was added to the initial yield stress Y and the elastic material properties according to data available in MMPDS-08 (Rice 2003). Various literature sources were employed to inform our choice of \(\beta \). Accurately modeling the temperature rise in the calibration specimens and the resulting softening required a coupled thermo-mechanical simulation with thermal expansion, specific heat, thermal conductivity and emissivity determined from MMPDS-08. Void growth damage parameters were chosen based on prior experience with the material model and a sensitivity study of the model to the damage exponent m. Figure 57 contains initial tension simulation results and parameter values.
After calibrating the model to the tension data, the shear data was incorporated into the model. Using material parameters calibrated to the tension data, a model of the shear test did not accurately predict the yield behavior of the specimen thus indicating that the material exhibits an anisotropic yield surface. By reducing the initial yield parameter Y by \(\sim 83\,\%\), the shear simulation results improved and compared well to the test data.
Since the triaxiality driven void growth model cannot evolve damage in pure shear, sensitivity studies for void nucleation lead to the selection of the appropriate \(N_1, \phi _0^\eta \) and \(\eta _0\) parameters to capture shear failure. Figure 58 contains the calibrated shear simulation results and the corresponding parameters.
Challenge specimen modeling details
Model development for the challenge specimen included specifying the appropriate boundary conditions and incorporating anisotropy. The solid mechanics boundary conditions consisted of a symmetry boundary condition along the half-thickness plane of the specimen and approximations of the pin boundary conditions in the test. A half-pin contiguously meshed into the specimen with the center node line having prescribed displacements approximated frictionless pins. The top pin’s centerline was fixed and the bottom pin’s centerline was displaced downward with a rate corresponding to the test rates. As stated previously, accurately modeling the calibration specimens required a coupled thermo-mechanical simulation. The thermal boundary conditions included radiation from the specimen surface to the room temperature surroundings and a symmetry boundary condition along the half-thickness plane of the specimen. Since the EV model cannot accommodate an anisotropic yield surface, the model of the specimen was split into two element blocks: Block 1 with a yield corresponding to the tension initial yield \(Y_{RT}^t \) and Block 2 with a lower yield \(Y_{RT}^{s*} =441\) MPa since that region is initially predominantly in shear. Figure 59 depicts Block 2 outlined in red with the remaining elements belonging to Block 1. Since the stress state in Block 2 does not directly correspond to that of the failure region in the shear model, a simulation of the challenge specimen at the slow rate using a rate and temperature independent Hill plasticity model influenced the selection of \(Y_{RT}^{s*} =441\) MPa. All simulations consisted of models constructed at the nominal dimensions according to the specimen drawings.
Blind predictions
Using the material model parameters and boundary conditions specified in the previous sections, the challenge specimen model predicted failure through crack path B–D–E–A for both rates. For both rates, the crack propagated unstably through B–D–E, as shown in Fig. 59, while the remaining ligament carried load until tensile failure occurred much further into the simulation (\({\sim }375\) s for the slow rate and \(\sim .36\) seconds for the fast rate). Table 13 lists the maximum loads and CODs at crack initiation for each rate and Fig. 60 displays the predicted load versus COD1 plot for both rates.
Table 13 Results predicted using the challenge specimen model
Sources of discrepancy
Several sources of error were present in the challenge specimen model. For example, an isotropic material model was used to simulate the anisotropic material through the use of separate element blocks and material parameters. Ideally, an anisotropic material model with rate and temperature dependence similar to EV would have been used. Additionally, material parameter uncertainties were large (e.g. \(\upbeta \)) and sensitivity studies show these uncertain parameters had significant effects on the simulation results. Numerical modeling issues also introduced error. SIERRA SM’s implicit contact algorithm would not converge. Consequently, we employed a contiguously meshed and rotating half-pin. Our inability to quickly resolve the evolution of local damage did not permit resources for regularized solutions. We were able to nicely resolve both crack initiation and propagation with surface elements (without thermomechanical coupling). Nonlocal studies will be the subject of future work.
Team I
Team members:
J.L. Bignell, jbignel@sandia.gov, Sandia National Laboratories, Albuquerque, NM, USA
S.E. Sanborn, sesanbo@sandia.gov, Sandia National Laboratories, Albuquerque, NM, USA
C.A. Jones, cajone@sandia.gov, Sandia National Laboratories, Albuquerque, NM, USA
P.D. Mattie, pdmatti@sandia.gov, Sandia National Laboratories, Albuquerque, NM, USA
Team I approached the problem using the commercially available general purpose finite element software Abaqus (Abaqus Standard Versions 6.13 and 6.14). Team I chose to use a “typical” finite element approach to this problem because in many real world applications (given time and budget constraints) this is the only viable approach. An implicit solver was used with numerical stabilization to overcome global instabilities during the fracture event. To reduce the run time, reduced integration 8-node hexahedral elements were used rather than fully integrated elements. Abaqus allows tabular input of yield, potential functions, and strain-to-failure curves. This feature was used to assign multiple yield functions and multiple strain-to-failure curves corresponding to different strain rates.
The Hill plasticity material model was used with rate dependent yield curves and isotropic hardening. In the Hill model, the strain rate is decomposed into the sum of an elastic strain rate and an inelastic strain rate. In the Abaqus implementation of the model, the user inputs six stress ratios \(R_{ij} \) to define the Hill yield surface. If they are all unity, the von Mises yield surface is recovered. The parameters of the Hill yield surface were determined for different strain rates during the calibration procedure.
To capture material degradation and failure, strain-to-failure curves, that are dependent on both the stress state (in terms of stress triaxiality \(\eta \)) and the strain rate, were employed:
$$\begin{aligned} \bar{\varepsilon }_D^{pl}\left( {\eta , \dot{\bar{\varepsilon }}_D^{pl}} \right) , \end{aligned}$$
where \(\eta =-\frac{p}{q}, p\) is the pressure stress, q is the von Mises equivalent stress, and \(\dot{\bar{\varepsilon }}_D^{pl}\) is the equivalent plastic strain rate. Because the stress and strain rate are changing throughout the simulation, degradation initiates when the state variable \(\omega _D \), given by the following integral:
$$\begin{aligned} \omega _{D}=\int \frac{\hbox {d}{\bar{\varepsilon }}_{D}^{pl}}{{\bar{\varepsilon }}_{D} ^{pl}\left( {\eta , {\bar{\varepsilon }}_{D}^{pl}}\right) } =1, \int \frac{\hbox {d}{\bar{\varepsilon }}_{D}^{pl}}{{\bar{\varepsilon }}_{D}^{pl} \left( {\eta , {\bar{\varepsilon }}_{D}^{pl}}\right) }=1 \end{aligned}$$
reaches unity. For the time discretized problem, the integral above is calculated by incrementing \(\omega _D\) at each time step, for each integration point, in the following way (note \(\omega _D\) can never decrease):
$$\begin{aligned} \varDelta \omega _{D}=\frac{\mathbf{\Delta }{\bar{\varepsilon }}_{D}^{pl}}{{\bar{\varepsilon }}_{D}^{pl}\left( \eta ,{{\bar{\varepsilon }}_{D}^{pl}} \right) }\ge 0 \end{aligned}$$
Once \(\omega _{D}\) reaches unity, the material stress is degraded in the following way,:
$$\begin{aligned} {\varvec{\sigma }} =\left( {1-D} \right) {\bar{{\varvec{\sigma }}}}, \end{aligned}$$
using a continuum damage variable D whose value ranges from zero to unity.
As the plastic strain increases the damage variable evolves as follows:
$$\begin{aligned} \dot{D}=\frac{L{\dot{\bar{\varepsilon }}}^{{pl}}}{\dot{\bar{u}}_{f}^{{pl}}}= \frac{{\dot{\bar{u}}}^{pl}}{{\dot{\bar{u}}}_{f}^{{pl}}},\quad \hbox {where }\quad {{\bar{u}}}_{f}^{pl}=\frac{2G_f}{\sigma _{y0}} \end{aligned}$$
Here \(G_f\) is the material fracture energy, L is the characteristic element length, and \(\sigma _{y0}\) is the value of the yield stress at the time of failure initiation. This method attempts to ensure that the energy dissipated during the damage evolution process equals the fracture energy for the material. While this method attempts to remove the dependence of failure on the size of the elements used, the same element size used in the calibration process was also used for the prediction. Degraded elements are removed from the model when D attains a value near unity (full degradation).
Finite element models representing the uniaxial tension and shear tests were constructed (see the mesh representations in Fig. 61). Two mesh sizes were initially investigated for this calibration, 0.5 and 0.25 mm, with 0.25 mm being adopted for the challenge predictions. Symmetry was utilized when possible. Using the two models, along with the given uniaxial tension and shear test data, the following steps were employed in the calibration process:
-
Use the uniaxial tension test data and model to determine the material hardening curves.
-
Use the shear test data and model to define the Hill plasticity potential ratios
-
Use the uniaxial tension test data and model to determine the failure initiation parameters in the tension regime (\(\eta \ge 0.33\))
-
Use the shear test data and model to determine the failure initiation parameters in the pure shear regime (\(\eta =0.0\))
-
Rerun the uniaxial tension and shear test models to verify all the inputs
-
Using the calibrated model inputs, make predictions for the two double-notch tension tests
For the calibration of the material hardening parameters the following strain rate relationship was assumed for the yield stress loaded at an arbitrary strain rate:
$$\begin{aligned} \sigma _{y}\left( {\dot{\bar{\varepsilon }}}^{pl}\right)= & {} \left( 1+C\left( {{\bar{\varepsilon }}}^{pl}\right) ln \left( \frac{{\dot{\bar{\varepsilon }}}^{pl}}{{\dot{\bar{\varepsilon }}}^{pl}_{ref}}\right) \right) \sigma _{y0.001}, \hbox {where},\\ C\left( {{\bar{\varepsilon }}}^{pl}\right)= & {} \frac{\left( \left( {\sigma _{y1.0}\left( {\bar{\varepsilon }}^{pl} \right) /\sigma _{y0.001} \left( {\bar{\varepsilon }}^{pl}\right) } \right) -1\right) }{ln\left( 1.0/0.001\right) }, \end{aligned}$$
along with the following assumed hardening relationships:
$$\begin{aligned} \sigma _{y0.001}\left( {\bar{\varepsilon }}^{pl} \right)= & {} A_{0.001}+B_{0.001}{\bar{\varepsilon }}^{{pl}^{n_{0.001}}}\; \hbox {and }\\&\sigma _{y1.0}\left( {\bar{\varepsilon }^{pl}}\right) =A_{1.0} +B_{1.0}{\bar{\varepsilon }}^{{pl}^{n_{1.0}}}. \end{aligned}$$
The parameters \(A_{0.001} \), \(B_{0.001} \), \(n_{0.001} \), \(A_{1.0} \), \(B_{1.0} \), \(n_{1.0} \) were determined by fitting the model response to the available tension test data. Figure 62a shows the yield stress vs. plastic strain curves at the 0.001 and 1.0 strain rates determined from the calibration process, along with the curves at other rates determined using the assumed relationships above. Since Abaqus linearly interpolates between the hardening curves that are defined in the input, it is necessary to enter a sufficient number of curves to maintain the assumed log-linear relationship above.
Using the rate dependent yield stress curves determined, the shear model was run with Hill stress ratios equal to unity. Results showed that the model over predicted the onset of yielding when compared with the available shear test data. Hill stress ratios were adjusted to \(R_{12} =R_{23} =R_{13} =0.88\) to bring the model’s shear response in line with the test data.
The strain to failure curves were determined starting with data found in the literature for Ti–6Al–4V (Giglio et al. 2012). This is referred to as the reference curve \(\bar{\varepsilon }_{D-ref}{}^{pl}\left( \eta \right) \) in the following discussion. The following rate dependent relationship was assumed:
$$\begin{aligned} \bar{\varepsilon }_{D}^{pl}\left( \eta ,{\bar{\varepsilon }}_{D}{}^{pl} \right)= & {} \left( 1+E\left( \eta \right) ln\left( \frac{\dot{\bar{\varepsilon }}^{pl}}{\dot{\bar{\varepsilon }}^{pl}{}_{ref}}\right) \right) \\&\bar{\varepsilon }_{D-ref}{}^{pl}\left( \eta \right) Q\left( \eta \right) . \end{aligned}$$
The scaling factors \(Q\left( \eta \right) \) and rate multiplier constants \(E\left( \eta \right) \) were determined through a fitting process using the shear and tensile test models and test data. Figure 62b shows the final fitted plastic strain to failure vs. stress triaxiality curves for the strain rates input into the model.
Figure 63 illustrates the response of the calibrated tension test model and Fig. 64 illustrates the response of the calibrated shear model. As mentioned above, only two mesh sizes were studied during the calibration and prediction efforts. Ultimately, the mesh size of 0.25 mm was used for the final calibration and prediction. Due to time constraints and Abaqus license restrictions, other mesh refinements were not considered. As discussed above, the material degradation and failure model used removes the dependence on the element size, no mesh sensitivity studies were performed to assess the degree to which this holds true.
Having fitted the parameters for the material model, the double notch coupon (challenge problem) model was run (Fig. 65 illustrates the mesh representation utilized). A half symmetry model was created with an element size of 0.25 mm in the potential failure regions. The loading pins were not explicitly modeled. Instead, velocities were prescribed to the top and bottom halves of the pin holes. It was recognized that this choice of boundary conditions artificially restricts rotation of the two ends of the coupon while the pins utilized in the testing likely allow for rotation. It was demonstrated later by additional analyses, that this choice of boundary conditions introduces only a small error in the predictions up to the first fracture in the specimens, whereas the behavior following the first fracture event (along ligament B–D–E) is significantly affected. Figure 67 illustrates the differences. The nominal dimensions provided were used to construct the model. Implicit dynamic simulations were performed with numerical dissipation added to stabilize the solution during the fracture event. Blind predictions were made for both the slow and fast loading rates. Table 14 summarizes the results of these analyses. Figure 66 shows the predicted location 1 load vs. COD1 curves, along with illustrations of the predicted fractures in the specimens for the two loading rates. The results show that the model over predicts both the peak force and COD1 at failure for both loading rates, with the discrepancy being more pronounced for the fast loading case. In addition, the model fails to capture the softening in the response corresponding to localization of the strain immediately preceding the first fracture event for both loading rates, again with the discrepancy being more pronounced for the fast loading case. It is thought that these discrepancies are directly attributable to the model’s neglect of the plastic strain induced heating and thermal softening of the material.
Sources of discrepancy
There are a number of potential sources of discrepancy between the blind prediction and the actual challenge problem results. The most significant source being the lack of inclusion of thermal effects in the material response and the accounting of temperature changes induced by plastic straining of the material (coupled thermal-mechanical response modeling). Results of the challenge problem tests indicated a significant temperature rise in the material in the failure regions, particularly for the fast loading cases. Another potential significant source of discrepancy is the boundary conditions used. After the results of the challenge were released, Team I reran the challenge predictions with “pin-like” boundary conditions achieved using multi-point constraints. It was found that there was not much difference in the response up to and including the peak load; however, beyond the peak load the response of the specimen was significantly different with a much larger COD achieved before fracture of the final ligament (Fig. 67). This is expected as with the updated boundary conditions, rotation of the specimen ends is allowed as fracture progresses.
Team J
Team members:
K. Pack, kpack@mit.edu, Massachusetts Institute of Technology, Cambridge, MA, USA
T. Wierzbicki, wierz@mit.edu, Massachusetts Institute of Technology, Cambridge, MA, USA
Plasticity modeling
The MIT team modeled the plasticity of the Ti–6Al–4V sheet using conventional description of continuum mechanics for metallic materials, namely a yield function, a hardening law, and a flow rule. The observation of negligible difference in the engineering stress–strain curve before necking between the rolling direction (RD) and the transverse direction (TD) at each loading speed led to the assumption of the identical flow stress under uniaxial tension along three perpendicular axes of RD, TD, and the thickness direction. However, additional information from a shear test required to make use of the Hill’48 yield function (Hill 1948) given in Eq. (24), which has the parameter N controlling the plastic flow under in-plane pure shear stress.
$$\begin{aligned}&\bar{\sigma }_{Hill}\left[ F(\sigma _{22}-\sigma {33})^{2} +G(\sigma _{33}-\sigma {11})^{2}\right. \nonumber \\&\quad \left. +H(\sigma _{11}-\sigma {22})^{2}+2L\sigma _{23}^{2}+2M_{13}^{2}+2N\sigma _{12}^{2}\right] ^{\frac{1}{2}}\nonumber \\ \end{aligned}$$
(24)
Parameters other than N are reduced to those of the von-Mises yield function, and N was calibrated so as for numerical simulation to predict a correct level of force in the V-notched rail shear test as demonstrated in Fig. 68a. The value of N larger than 1.5 implies the relative weakness in deformation resistance under pure shear loading.
Table 14 Plasticity parameters of the Ti–6Al–4V sheet for the lower bound case
Accurate prediction of crack initiation in ductile metals depends highly on the hardening curve in the post-necking regime. Moreover, dynamic loading considered in the 2nd Sandia Fracture Challenge is inextricably linked with the thermal softening as well as the strain-rate effect. The MIT team employed a modified Johnson–Cook law, proposed by Roth and Mohr (2014), which uses a weight-average of the Swift and the Voce law for the strain hardening in the form of Eq. (25).
$$\begin{aligned}&K[\bar{\varepsilon }_{p}, \dot{\bar{\varepsilon }}_{p}, T] \nonumber \\&\qquad =\left[ \alpha \cdot A(\varepsilon _{0}+\bar{\varepsilon }_{p})^{n}+(1{-}\alpha ) \cdot (k_{0}+Q(1-e^{{\beta \bar{\varepsilon }}_{p}}))\right] \nonumber \\&\qquad \left[ 1+C\hbox {In}\left( \frac{\dot{\bar{\varepsilon }}_{p}}{\dot{\varepsilon }_{0}}\right) \right] \left[ 1-\left( \frac{T-T_{r}}{T_{m}-T_{r}}\right) ^{m}\right] \end{aligned}$$
(25)
In order to avoid high computational costs and ambiguity of boundary conditions, fully coupled thermo-mechanical analysis was simplified by purely mechanical analysis in which a fraction of incremental plastic work was converted into heat causing temperature rise based on Eq. (26).
$$\begin{aligned} dT=w\left[ \dot{\bar{\varepsilon }}_{p}\right] \frac{\eta _{k}}{\rho C_{p}}\bar{\sigma } d\bar{\varepsilon }_{p} \end{aligned}$$
(26)
\(\eta _{k}\) is the Taylor–Quinney coefficient assumed to be constant. The weighting factor changes smoothly from zero to unity by Eq. (27), which implies transition from isothermal to adiabatic condition.
$$\begin{aligned} w\left[ \dot{\bar{\varepsilon }}_{p}\right] = \left\{ {\begin{array}{ll} 0&{}\; \hbox {for }\,\dot{\bar{\varepsilon }}_{p}<\dot{\varepsilon }_{it}\\ \frac{(\dot{\bar{\varepsilon }}_{p}-\dot{\varepsilon }_{it})^{2}(3\dot{\varepsilon }_{a}-2\dot{\bar{\varepsilon }}_{p}\dot{\varepsilon }_{it})}{\dot{\varepsilon }_{a}-\dot{\varepsilon }_{it}^{3}} &{}\; \hbox {for }\, \dot{\varepsilon }_{it}\le \dot{\bar{\varepsilon }}_{p}\le \dot{\varepsilon }_{a}\\ 1&{}\;\hbox {for }\,\dot{\bar{\varepsilon }}_{a}<\dot{\bar{\varepsilon }}_{p}\\ \end{array}}\right. \nonumber \\ \end{aligned}$$
(27)
The optimized parameters for the lower bound case are provided in Table 14 together with six parameters of the yield function. The recommended calibration procedure through inverse analysis is explained in Roth and Mohr (2014). The only distinction was made in that a dog-bone specimen was used to find hardening parameters instead of a notched tensile specimen with circular cutouts, which was found to be more appropriate for the hardening curve optimization (see Pack et al. 2014). Careful attention was paid to the velocity profile applied to the boundary of the specimen for fast loading because it is not constant due to the compliance of the cross-head of a testing machine. It was confirmed that the engineering stress–strain curve predicted by finite element simulation showed a good agreement all the way to fracture with test results that showed the lowest stress after necking (regarded as lower bound cases) as depicted in Fig. 68b. Very fine mesh of 0.1 mm was used in the necked region, following the recommendation by Dunand and Mohr (2010).
Fracture modeling
Crack is assumed to initiate when the indicator D, calculated by a linear damage accumulation rule in Eq. (28) reaches unity. The corresponding finite element is then eliminated from a whole model.
$$\begin{aligned} D=\int _{0}^{\bar{\varepsilon }_{f}}\frac{d\bar{\varepsilon }_{p}}{\bar{\varepsilon }_{f}^{pr}(\eta , \theta )} \end{aligned}$$
(28)
The function \(\bar{\varepsilon }_{f}^{pr}\) defines the strain to fracture under proportional loading as a function of two stress-state dependent variables: the stress triaxiality \(\eta \) and the Lode angle \(\bar{\theta }\), whose combination specifies a loading path. Following Roth and Mohr (2014), the rate-dependent Hosford–Coulomb fracture model was taken for \(\bar{\varepsilon }_{f}^{pr}\) .
$$\begin{aligned}&\bar{\varepsilon }_{f}^{pr}(\eta , \bar{\theta }) =b_{0}\left( 1+\gamma \hbox {In} \left( \frac{\dot{\bar{\varepsilon }}_{p}}{\dot{\varepsilon }_{0}}\right) \right) \nonumber \\&\quad (1+c)^{\frac{1}{n}}\left[ \left\{ \frac{1}{2}((f_{1}-f_{2})^{a}((f_{2}-f_{3})^{a}\right. \right. \nonumber \\&\quad \left. \left. +\,(f_{3}-f_{1})^{a})^{\frac{1}{a}}\right\} +c(s\eta +f_{1}+f_{3})\right] ^{\frac{1}{n}} \end{aligned}$$
(29)
$$\begin{aligned}&f_{1}(\bar{\theta })=\frac{2}{3}\hbox {cos}\left[ \frac{\pi }{6}(1-\bar{\theta })\right] \end{aligned}$$
(30)
$$\begin{aligned}&f_{2}(\bar{\theta })=\frac{2}{3}\hbox {cos}\left[ \frac{\pi }{6}(1-\bar{\theta })\right] \end{aligned}$$
(31)
$$\begin{aligned}&f_{3}(\bar{\theta })=\frac{2}{3}\hbox {cos}\left[ \frac{\pi }{6}(1-\bar{\theta })\right] \end{aligned}$$
(32)
Mohr and Marcadet (2015) formulated the original rate-independent Hosford–Coulomb fracture model, inspired by the onset of microscopic shear localization. Even though shear tests were carried out by Sandia, and the test data were provided, the fact that slip occurred during the tests and the axial displacement was measured with an LVDT attached to fixtures whose compliance was not negligible is attributed to not including shear tests for the present fracture calibration. As a consequence, the identification of five fracture parameters had to rely solely on two uniaxial tensile tests. The MIT team was in possession of the fracture parameters for a similar alloy, so a, c, and n were taken from our database. This basically assumes that the current alloy of interest has the same dependence on \(\eta \) and \(\bar{\theta }\) as the similar alloy in our database. The remaining parameters of \(b_{0}\) and \(\gamma \) control the height of a fracture envelope and its strain-rate sensitivity, respectively. These two values were found such that the engineering strain (equivalently displacement) to fracture of a dog-bone specimen in both slow and fast loading condition is accurately captured as noted in Fig. 68b.
Blind prediction
In light of the symmetry through thickness, only a half of the specimen geometry with print dimensions was discretized by approximately 750,000 reduced-integration eight-node three-dimensional tri-linear solid elements (C3D8R of the Abaqus element library) with the smallest ones of \(0.1\times 0.1\times 0.1\,\hbox {mm}^{3}\) around two notches and three holes. This size of elements was chosen to be the same as the one used for the dog-bone and the shear specimen to minimize a possible mesh size effect. Abaqus/Explicit was used in simulation with the material models implemented through the user material subroutine (VUMAT). The upper and the lower pin were modeled as analytical rigid bodies, and the penalty contact with no friction was defined between the pins and the specimen. The lower pin was pulled down at 0.0254 mm/s for slow and 25.4 mm/s for fast loading with the first one tenth of the total simulation time reserved for acceleration. Uniform mass scaling was applied to reduce computational time in so far as it guarantees the negligible ratio of kinetic energy to internal energy before the first crack initiation. Crack propagation was considered to be nothing more than the problem of consecutive crack re-initiation, thus being modeled by continuing element deletion.
Figure 69 visualizes the sequence of damage accumulation and crack path for fast loading. Slow loading exhibited a similar non-uniform distribution of the damage indicator and ended up with the same crack path of B–D–E–A. In the early stage, plasticity comes into play in the ligament between A and C by tension and in two ligaments between B and D and D and E by combined shear and tension. Because of the relative weakness in shear resistance characterized by \(N=1.95\), the specimen prefers shear localization to necking, which results in rapid crack propagation. The two ligaments between B and D and D and E fracture almost at the same time, by which a huge amount of elastic energy is released all of a sudden. This causes the vibration of the whole system. Finally, the ligament between A and E deforms mostly due to bending and reaches complete failure.
Sources of discrepancy
The comparison of the force-COD1 curve between experiments and simulations is made in Fig. 70. Elastic region as well as the maximum force was very accurately predicted for both slow and fast loading with 3.6 % overestimation of the maximum force for slow loading. This error might have been caused partially by dimensional discrepancy between actual specimens and the finite element model. The COD1 at the first crack (abrupt load drop) for fast loading was predicted with great accuracy while it was 37 % overshot for slow loading. This is mainly due to the over-adjusted hardening curve after necking as can be deduced from the delayed COD1 at the maximum load. As noted by Pack et al. (2014), the flow stress after necking optimized by a dog-bone specimen deteriorates numerical prediction in other specimen geometries, and a notched tensile specimen with circular cutouts serves an improvement in the prediction.
Concluding remarks
Plasticity and fracture modeling on the basis of uniaxial tension and shear tests subjected to two different loading speeds made a satisfying prediction both in the slow and the fast challenge problem. The present plasticity model accounts for relatively low shear resistance of the Ti–6Al–4V sheet, which was crucial to capture the type of instability that led to a correct prediction of the crack path. More detailed explanation about the calibration procedure and the improvement in numerical prediction based on an exhaustive testing program performed on the leftover material can be found in Pack and Roth (2016).
Team K
Team members:
S.-W. Chi, swchi@uic.edu, University of Illinois at Chicago, USA
S.-P. Lin, slin46@ford.com, University of Illinois at Chicago, USA
A. Mahdavi, amahda2@uic.edu, University of Illinois at Chicago, USA
Approach
The simulation methodology in this work is based on an enriched Reproducing Kernel Particle Method (RKPM) (Chen et al. 1996; Dolbow and Belytschko 1999; Krysl and Belytschko 1997; Liu et al. 1995). The crack surface/tip discontinuity is embedded in the displacement approximation as follows.
$$\begin{aligned} \mathbf{u}^{h}= & {} \sum _{{I \in N - N_{{cut}} - N_{{tip}} }} {\Psi _{I} } \left( {\mathbf{x}} \right) {{\mathbf{d}}}_{I}\nonumber \\&+\sum _{{J \in N_{{cut}} }} {\sum _{{i = 1,2}} {S_{i}}} \left( {\mathbf{x}} \right) \Psi _{J} \left( {\text {x}} \right) a_{j}\nonumber \\&+ \sum _{{K \in N_{{tip}} }} {\sum _{{j = 1,2}} {f_{i}}} \left( {\mathbf{x}} \right) {\mathbf{b}}_{K} \equiv \sum _{I} \bar{\Psi }_{I}(\mathbf{x})\bar{d}_{I} \end{aligned}$$
(33)
Here \(\Psi _{I}\) is the reproducing kernel (RK) shape function with the Cubic B-spline (\(C^{2}\) continuous) as the kernel function centered at node \(\mathbf{x}_{I}\); \(N_{cut}\) and \(N_{tip}\) are node sets, in which the support of node contains the crack surface and crack tip, respectively; N is the total node set; \(\mathbf{d}_{I}\), \(\mathbf{a}_{J}\), and \(\mathbf{b}_{K}\) are nodal coefficients. \(S_{i}\) is introduced to represent the continuity across the crack surface and expressed as:
$$\begin{aligned} S_{I}(\mathbf{x})=\left\{ \begin{array}{ll} 1&{} \zeta ^{+}>0\\ 0&{} \zeta ^{-}<0\\ \end{array}\right. \hbox {and } S_{2}=1-S_{1}. \end{aligned}$$
(34)
where \(\zeta ^{+}\) and \(\zeta ^{+}\) denote the above and below crack regions, respectively. The crack tip enrichment function, \(f_{j}\), is formulated based on the visibility criterion (Krysl and Belytschko 1997; Lin 2013) and has the following form:
$$\begin{aligned} \begin{array}{ll} \begin{array}{l} \hbox {For } \theta _{0}>0, f_{1}(\mathbf{x})=\left\{ \begin{array}{ll} 1&{}\theta _{0}-\pi /2 \le \theta \le \pi \\ \sin (\theta -\theta _{0}+\pi )&{} -(\pi -\theta _{0}) \le \theta <\theta _{0}-\pi /2\\ 0&{} -\pi \le \theta <-(\pi -\theta _{0})\\ \end{array}\right. \\ \hbox {For } \theta _{0}\le 0, f_{1}(\mathbf{x})=\left\{ \begin{array}{ll} 0 &{}\pi +\theta _{0}\le \theta <\pi \\ \sin (\theta _{0}-\theta +\pi )&{} -\theta _{0}+\pi /2<\theta \le \pi +\theta _{0}\\ 1&{} -\pi \le \theta \le \theta _{0}+\pi /2\\ \end{array} \right. \end{array} &{}\hbox { and } f_2=1-f_1\\ \end{array} \end{aligned}$$
(35)
where \(\theta _{0}\) is the angle from the crack direction to the enriched node and \(\theta \) is the angle from the crack direction to the evaluation point. Note that the enrichment shape functions in (33), \(\bar{\Psi }_{I}\), satisfy the partition of unity condition, \(\sum \bar{\Psi }_{I}=1\). Therefore, the mass lumping in the explicit time integration in the Galerkin formulation is straightforward (Lin 2013).
Material model and calibration of material parameters
The material in simulations is modeled by the Johnson–Cook rate-dependent model with a bi-linear hardening \(J_{2}\) elastoplasticity:
$$\begin{aligned}&{H}(\bar{e}^p)=1 \end{aligned}$$
(36)
$$\begin{aligned}&K(\bar{e}^p, \dot{\bar{e}}^p) \left\{ \begin{array}{ll} K_s(\bar{e}^p)[1+C \ln (\dot{\bar{e}}^p)/\dot{\bar{e}}_0) ], &{}\quad \hbox { if } \dot{\bar{e}}^p<\dot{\bar{e}}^p_{\mathrm{crit}}\\ K_s(\bar{e}^p)[1+C \ln (\dot{\bar{e}}^p_{\mathrm{crit}})/\dot{\bar{e}}_0) ], &{}\quad \mathrm{esle} \end{array} \right. \nonumber \\ \end{aligned}$$
(37)
$$\begin{aligned}&K_s(\bar{e}^p)\nonumber \\&\quad =\left\{ \begin{array}{ll} [Y_0+\alpha _0 \bar{e}^p], &{}\quad \hbox {if } \bar{e}^p<\bar{e}^p_{\mathrm{crit}}\\ Y_1+\alpha _1(\bar{e}^p-\bar{e}^p_\mathrm{crit}), &{}\quad \hbox {if } \bar{e}^p<\bar{e}^p_{\mathrm{crit}}, Y_1=Y_0+\alpha _0 \bar{e}^p_\mathrm{crit}\end{array}\right. \nonumber \\ \end{aligned}$$
(38)
where H and K are the kinematic and isotropic hardening parameters, respectively; \(\bar{e}^{p}\) is the effective plastic strain. The initial yield stress, \(y_{0}=1.0474\) GPa, the hardening parameters \(\alpha _{0}=1.157\) GPa, and \(\alpha _{1}=0.45\) GPa, and the critical equivalent plastic strain \(\bar{e}_{\mathrm{crit}}^{p}=0.12\) were calibrated from slow-loading-rate tensile test data (Fig. 71). For rate effect, the reference plastic strain rate \(\dot{\bar{e}}_{0}\) was chosen to be the unity; the constant \(C= 0.015\) and the critical plastic strain rate \(\bar{e}_{\mathrm{crit}}^{p}=28\) were obtained from the numerical tensile tests as shown in Fig. 71. Considering the maximum principle tensile strain as the crack initiation criterion and assuming that the tensile failure is the predominant failure mode, the crack opening strain in the initiation criterion can be obtained according to the rupture point in the tensile test. The principal strain at the rupture point, 0.946, was assumed as the crack initiation strain for both slow- and fast- loading-rate cases.
Numerical simulations
An explicit updated Lagrangian reproducing kernel formulation with the enriched displacement approximation in (33) was employed to model the challenge problem. The crack propagation speed in the dynamic simulations was assumed to be 0.02 % of Rayleigh wave speed, based on analytical and empirical studies for non-branching crack propagation (Freund 1972, 1979). A RKPM discretization containing a total of 55,520 nodes, with 2 nodes in the thickness direction, was used for simulations. The mean in-plane nodal distance near the notched area in the discretization is 0.44 mm, which is consistent with the nodal density used for material parameter calibration.
Figure 72 shows the predicted deformations and von Mises stress contours for both slow- and fast-loading-rate cases. The predicted crack paths for both cases follow a similar pattern, “B–D–E–A”, which agrees with experimental results. Figure 73 compares numerical load-COD curves to experimental ones for both loading cases. The numerical simulations over-predict the slopes of load-COD before yielding and peak loading values. This is likely mainly due to inappropriate boundary conditions. Unlike loaded at both bolt-holes through low-friction pins in the experiments, the specimen in the simulations was loaded through prescribed leftward displacement with fixed vertical direction on the left hole (Fig. 72). The extra constraint limits the COD development and therefore leads to an increase in force response. Furthermore, for the case with fast loading rate, the numerical model predicts much larger rupture COD than the experimental data. The main cause of the discrepancy may be attributed to an inaccurate crack initiation threshold and an inaccurate constant crack propagation speed for fast loading rate.
Sources of discrepancy
The crack paths obtained from simulations for both slow- and fast-loading-rate cases are in good agreement with experimental data; however, the force responses and the crack initiation COD are over-predicted. The over-prediction is mainly attributed to inappropriate boundary conditions, and inappropriate material parameters, including, but not limited to, rate-dependent hardening parameters and crack initiation thresholds. In the dynamic simulation, the crack tip speed also plays an important role and needs to be properly estimated based on a fracture energy release rate algorithm.
Team L
Team members:
J. Predan, jozef.predan@um.si, University of Maribor, Slovenia
J. Zadravec, zadravec.jozef@gmail.com, University of Maribor, Slovenia
Approach
The 2D plane strain simulation was performed with Abaqus CAE 6.14 software using an elasto-plastic model (continuum) and a damage model (damage initiation and damage evolution). The Abaqus Standard/Static solver and isotropic elastic plastic material behaviour including isotropic deformation hardening was employed. The elastic, plastic, damage initiation and evolution parameters were fit to experimental data. The parameter fitting utilized both tensile and shear data at both loading rates. The Extended Finite Element Method (XFEM) was used to simulate damage in the specimens. The finite elements mesh was refined in the regions where predicted crack propagation, as shown in Fig. 74. The damage initiation criterion was based on the quadratic traction-interaction law and the damage evolution criterion was energy dissipation with linear softening. Damage stabilization was used to reach convergence in the static solver.
Procedure for selection of parameters for material model and failure
For parameter calibration, CAE models of the tensile and shear specimens were constructed. The initial calibration utilized the elastoplastic tensile data. The cross section of a broken test specimen was used to calculate true stress and calibrate the plastic parameters. Further refinement of the parameters was achieved by first explicitly simulating the tensile test and subsequently simulating the shear test to achieve results that matched the experimentally reported outcomes.
Cohesive zone elements were used and the failure criteria chosen was a nominal stress based on a quadratic combination of all three ratios for crack initiation and maximal deformation energy for crack propagation. Results of both loading rates simulations were fit to the reported experimental calibration data.
Modeling details
A 2D assembly model was used for crack growth path prediction. This model consisted of the challenge specimen with exact nominal dimensions that were defined in challenge documentation and two pins. Between the pins and specimen the following contact properties were defined: for normal behaviour hard contact and for tangential behaviour the friction coefficient of 0.05. The center of the bottom pin was a fixed constraint in directions X and Y with free rotations about the Z axis. At the centre of upper pin, the following motions were prescribed: displacement of 4 mm in direction Y, and fixed in direction X, free rotation around axis Z. The model was quasi-static. A surface region was defined to allow possible cracking. The crack was formed from notch geometry according based on the cohesive zone damage criteria. For crack growth, XFEM technology was employed. The XFEM region was prescribed around all holes and notches to allow for possible crack initiation and propagation.
The Quads criterion was used for cohesive zone element placement, which is based on a quadratic combination of nominal stress ratios between a given stress value and the peak nominal stress value in each of three directions.
Mesh refinement
In the model, 61,315 linear quadrilateral plane strain elements were used of type CPE4. The maximum global size of any element was 1 mm or 0.25 mm in the XFEM crack prediction region. Refinement edges are shown in purple in Fig. 75. Because of the small, unstructured elements there was no influence of element orientation.
Blind predictions
Based on results of simulation shown on Figs. 76 and 77, Team L predicted, that the crack will follow path B–D–E–A. The predicted Force-COD1 curve for both the slow and fast loading rates are shown in Fig. 78.
Sources of discrepancies
The primary source of discrepancy likely was related to inadequate shear calibration due to convergence problems. Further, Team L also had made a post-processing mistake by exporting the reaction force for only 1 fixing point, whereas the pin had been constrained at 8 points and for this reason, our initial blind predictions reported forces that were too low.
Team M
Team members:
A.J. Gross, andrew.gross@mail.utexas.edu, University of Texas at Austin, Austin, TX, USA
K. Ravi-Chandar, ravi@utexas.edu, University of Texas at Austin, Austin, TX, USA
In some recent work we have identified that, for a class of ductile materials, plastic deformation proceeds without intervening damage until very large strain levels; this is confirmed through observations and measurements of deformation at multiple scales, from the macroscopic to the level of the grains (Ghahremaninezhad and Ravi-Chandar 2012, 2013; Haltom et al. 2013). The upshot of these investigations is twofold: first, it is essential that the plastic response of the material be calibrated to much larger strain levels than is usually achieved in a standard tensile test. Second, the mechanisms of final failure—void nucleation, growth and coalescence—occur within a highly localized zone in the plastically deformed material, and only at the very end of the material’s ability to withstand deformation. Thus, a model for the final failure of the material may be implemented numerically by a simple damage criterion such as element deletion. However, it is necessary to perform a careful evaluation of the plastic strain levels at which damage may initiate under multiaxial loading. We have adopted this approach in formulating the simulation of the challenge problem.
The plastic constitutive properties of Ti–6Al–4V are modeled by the flow theory of plasticity with isotropic hardening. Due to both the limited time to implement an appropriate yield criterion for HCP metals in ABAQUS (the FEM package that was used for this work) and the sparsity of stress paths used the calibration experiments, Hill’s 1948 anisotropic yield criterion (Hill 1948) was selected as the governing model for plasticity. The two parameters affecting the yield stress for out of plane shear conditions are assumed to be equal to their isotropic values since no data are available to calibrate them; this is considered to be appropriate because the corresponding stresses will be negligible in both the calibration experiments and challenge geometry. The remaining four parameters are subject to calibration. It is evident that uniaxial tensile test results cannot be used to determine the stress–strain behavior beyond a logarithmic strain of \({\sim }4\,\%\) because of the inhomogeneous deformation that occurs beyond the Considère strain. After this point, some model for the stress–strain relation must be considered. In this work, the behavior is represented by a monotonically increasing spline with seven segments beyond the Considère strain (Gross and Ravi-Chandar 2015). This form is chosen because it provides much more flexibility in the shape of the stress–strain curve than typical power law behavior, yet does not have an undue number of parameters. Temperature and rate effects were included by a multiplicative modification to the stress–strain curve defined by three parameters, following the form first used by Johnson and Cook (1983).
The parameters for the spline and the yield criterion are then found by an inverse procedure, where iterative finite element simulations of the slow rate calibration tests are performed with different trial parameters for the constitutive model. A nonlinear optimization scheme is used to minimize the sum of the errors for the rolling direction (RD) tensile, transverse direction (TD) tensile, and shear (VA; shear against grain) simulations. Error for each simulation is measured as the sum of the relative error between the net load in the experiment and simulation at 100 levels of global deformation. The resulting stress–strain curve and simulated load-elongation behavior for the calibration experiments are shown in Fig. 79. The VP shear data (shear parallel to grains) was not used, as the chosen constitutive model is not influenced by the orientation difference between VP and VA shear. Thus, only one shear test, or an average of the two could be used for calibration. VA shear was used exclusively for model calibration as it corresponds to the dominant orientation of shear loading in the challenge geometry. After calibration of the anisotropy and stress–strain curve from the slow rate tests was completed, the parameters for temperature and rate sensitivity were found using the high rate RD tensile and VA shear tests with the same inverse procedure. TD tension was not used because temperature and rate sensitivity are expected to be isotropic. The two tests used for calibration were chosen because they are dominated by different strain rates and temperature ranges.
Material failure is modeled by a strain to failure model where the failure strain, \(\upvarepsilon ^{\mathrm{f}}\), is dependent exclusively on stress triaxiality. When an element in the FEM simulations accumulates a damage parameter equal to unity according to the rule, \(\int \frac{d\varepsilon ^{p}}{\varepsilon ^{f}}\), its stiffness is set to zero, where \(d\varepsilon ^{p}\) is the plastic strain increment. The failure strain was calibrated by using the optimized RD tensile and VA shear simulations. For tension, the central element in the neck has both the highest triaxiality and strain. Since rupture of the specimen occurs rapidly, it corresponds to failure of this central element. By matching the experimental elongation at rupture in the simulation, the central element in the neck provides a strain to failure estimate under moderate levels of triaxiality. Strain to failure estimation in the shear specimen is based on past experimental experience indicating that the peak load in the test corresponds to the formation of a crack at one of the notch tips. Then the grip displacement at peak load in the experiment corresponds to global deformation state where the element at the current notch tip in the simulation must fail. This provides an estimate on the strain to failure under negative triaxiality conditions. After crack initiation, stable growth occurs in the experiment and could be used to perform a more detailed failure calibration. Due to time constraints this data was not used. It was found that the strains to failure over the large range of triaxiality spanned by these two tests were nearly identical (0.79 from tension and 0.82 from shear), so the strain to failure between them was simply interpolated linearly. For triaxialities in excess of those in the tensile test, a conservative strain to failure curve that is motivated by the exponential behavior first suggested by McClintock (1968) was adopted.
The challenge geometry is simulated with an ABAQUS/Explicit FEM model for both the slow and fast rate scenarios. Mass scaling is used to increase the stable time step to for both the slow and fast rate simulations. A highly refined mesh is used in the ligaments between the holes and notches, where strain localization is most likely to occur. Typical elements in this region were about 40 x 45 micron with 22 elements through the thickness. A total of about 700,000 eight-noded linear elements with reduced integration and hourglass control were used. 1.26 million and 0.93 million time steps were used in the slow and fast rate tests respectively, each simulation requiring several hundred hours of CPU time. Loading is applied on rigid, frictionless pins. The bottom pin was held fixed and a quadratic displacement rate was applied at the top pin for the slow loading scenario. For the fast loading a linear displacement rate at the top pin was imposed, consistent with the loading rate in the experiment. The results are summarized in Figs. 80 and 81 where the strain field at selected levels of deformation from the slow rate prediction and the load-COD1 variations for the slow and high rate predictions are shown.
For the slow rate test, early strain accumulation is the largest in ligament A–C, and remains so until a COD1 value of about 1.75 mm. After this point, both ligaments B–D and D–E have comparable levels of strain that exceed those in ligament A–C. The prediction indicates an increase in load carrying capability of the specimen until a COD1 of about 4.5 mm, where, on the cusp of localization, failure initiates, triggering fast fracture in ligament D–E, almost immediately followed by the fast fracture of ligament B–D. At initiation of failure, the load is predicted to drop abruptly to nearly zero, thus ending the simulation. It is projected that ligament EA will fail under continued loading, but this was not examined. The main details of the high rate test are similar to the slow rate.
Sources of discrepancy
The largest difference between predictions and experiments is that failure was predicted to occur at a COD1 \({\sim }4.1\) mm while it was observed at a COD1 \({\sim }3.03\) mm in the slow-rate tests, with a similar difference in the high-rate tests. All other aspects of the response were predicted within the desired tolerance. After a careful examination of the predictions in comparison to additional experiments that were instrumented to measure local deformation fields (see Gross and Ravi-Chandar 2015, 2016), it was determined that the elastic–plastic constitutive model did not satisfactorily represent the yield surface along the stress paths followed in the critical ligaments of the challenge geometry beyond a strain of \({\sim }0.2\); this deficiency in identifying and calibrating a proper constitutive model was the main source of the discrepancy between the predictions and the experiments. However, this deficiency did not manifest itself in the calibration exercise; this points to the need for a better approach to designing, performing and implementing experiments that are used in the calibration procedure.
Team N
Team Members:
L. Xue, xue@alum.mit.edu, Thinkviewer LLC, Sugar Land, TX, USA
Approach
The second Sandia Fracture Challenge gives an excellent opportunity to evaluate how different loading rate conditions can influence fracture behavior of a given configuration of ductile metals. In general, the effects of loading rate on mechanical behavior of metals can come in from several aspects, for instance (1) many metals display higher resistances under high strain rates; (2) the post-yield plastic hardening exhibits lower hardening capability at high strain rates; (3) the plastic work in the deformation zone converts to thermal energy at high strain rates, which heat up the material locally and hence reduces the material resistance; (4) high strain rates can activate different plastic flow mechanism and thus results in higher or lower fracture strain depending on the micro structures of the material. In the present study, the effects (1) and (2) are modeled by calibrating the material yielding and strain hardening behaviors separately for each loading rates, while the effect (3) and (4) are not considered explicitly, but rather they are inherently embedded in the calibration procedure because they are not singled out in the modeling.
Two pulling rates at the grips are used for the coupon tests of the material properties and the S-shaped specimen for fracture prediction. Several fracture initiation sites can be foreseen for the S-shaped flat plate with three holes in the vicinity of the roots of rounded slots. At first glance, this S-shaped structure is in several ways like the first Sandia Fracture Challenge, except that a second slot was added to the configuration. Previous studies have shown that, in this type of ductile fracture simulations, the constitutive relationship of the material plays a central role in the accuracy of the prediction results (Boyce et al. 2014). There are many choices of constitutive models that are capable of discerning between a mode I dominant and a mode II dominant ductile fracture (Xue 2007, 2008, 2009). The Xue (2009) damage plasticity model requires only one parameter to be calibrated for the simplest case. Yet, this model is enabled by a full 3D non-linear damage coupled yield function and has been shown to be able to capture different fracture modes through a series of numerical simulations. In the first Sandia Fracture Challenge, the Xue (2009) model was used and showed extraordinary capability in predicting fracture behavior of the given structure using limited simple material testing data. In this second Sandia Fracture Challenge, the same procedure was used to calibrate necessary material parameters and to obtain finite element predictions.
From the material testing results at the two loading rates, it is clear that the selected material Ti–6Al–4V exhibits some rate sensitivity. The nominal loading rates at the pins of the slow and fast condition are 0.0254 and 25.4 mm/s respectively. The strain rate in the deformation region of the S-shaped structure is further higher than that of the high rate in the tensile material testing, because the deformation zone in the material testing is greater than the deformation zone in the S-shaped structure while the end separation velocity are the same. This difference in strain rate should be about an order or so given the size of the holed zone in the S-shaped structure, which converts to about 3 % increase in the magnitude of yield stresses for the S-shaped structure considering a logarithmic relationship of the rate hardening coefficient. Note, this difference should be about the same in the slow rate scenario. However, it is considered relatively a small error and is not factored in the present study, i.e. the calibrated matrix stress–strain relationship from curve fitting of the tensile coupon in the rolling direction was used directly for the S-shaped specimen at the same nominal loading rate.
Table 15 Material parameters for Ti–6Al–4V for damage plasticity model
A parallel finite element simulation was used to calibrate the material parameters. In both calibration and prediction simulations, a one-point reduced explicit time integration scheme was adopted. In this type of ductile fracture analyses, it is well-known that the simulation results are subjected to mesh size of the finite element model. In order to minimize mesh size dependence, the element sizes were carefully chosen such that the elements in the material tests and the central region of the S-shaped structure used for prediction were about the same. A total of 16 elements through the thickness were used in order to capture the through thickness fracture pattern and this through thickness element length was used in meshing the in-plane elements. Thus, the element size was about 0.2 mm in all directions. Same mesh was used for both slow and fast loading rate.
Material parameters calibration
The material tension test data in the rolling directions for the two rates were used to calibrate the material stress–strain curve using Swift relationship, see Eq. (39) \(\upsigma _{\mathrm{M}}=\upsigma _{\mathrm{y0}}(1+\frac{\upvarepsilon _{\mathrm{p}}}{\upvarepsilon _{0}})^{\mathrm{n}}\) where \(\upsigma _{\mathrm{M}}\) is the material matrix resistance at given plastic strain \(\upvarepsilon _{\mathrm{p}}\), \(\upsigma _{{\mathrm{y}}0}\) is the initial yield stress, \(\varepsilon _{0}\) is a reference strain and n is the hardening exponent. A set of initial fitting parameters for low loading rate were used to run a detailed finite element analysis of the tensile tests and are then adjusted by matching the simulation load-displacement curve with the experimental one [Xue EFM 2009]. Due to the coupled nature of the stress–strain relationship with damage associated weakening in the damage plasticity model, an iterative process is needed to calibrate all the material parameters in Eqs. (39–42), where \(\upsigma _{\mathrm{f}0}\) is a reference fracture stress, \(\hbox {k}_{\mathrm{p}}\) is a pressure sensitivity parameter, and m and \(\beta \) are damage and weakening exponents. In the present study, both m and \(\beta \) were set to 2.0 according to previous studies on various metals. A reference fracture strain \(\upvarepsilon _{\mathrm{f}0}\) is substituted for the reference fracture stress \(\upsigma _{\mathrm{f}0}\), which is related to \(\upvarepsilon _{\mathrm{f}0}\) by \(\upvarepsilon _{\mathrm{f}0}=\upvarepsilon _{0}\left( {\frac{\upsigma _{\mathrm{f}0}}{\upsigma _{\mathrm{y}0}}-1} \right) ^{1/\mathrm{n}}\). After several iterations, the final fitted material parameters that give good match of the load-displacement curves for simple tension coupon tests are listed in Table 15 for the two loading rates. Then, the above process was repeated for the high loading rate.
$$\begin{aligned}&{\upsigma }_{\mathrm{M}}={\upsigma }_{\mathrm{y}0}\left( {1+\frac{\upvarepsilon _{\mathrm{p}}}{\upvarepsilon _{0}}} \right) ^{{\mathrm{n}}} \end{aligned}$$
(39)
$$\begin{aligned}&{\upvarepsilon }_{\mathrm{f}}={\upvarepsilon }_{0} \left\{ {\left( {\frac{{\upsigma }_{\mathrm{f}0}}{{\upsigma }_{\mathrm{y}0}}}\right) ^{1/{\mathrm{n}}}\left[ {\left( {1+\hbox {k}_{\mathrm{p}} \hbox {p}}\right) \frac{\sqrt{3}}{2\cos {\uptheta }_{\mathrm{L}}}}\right] ^{1/{\mathrm{n}}}-1}\right\} \nonumber \\ \end{aligned}$$
(40)
$$\begin{aligned}&\dot{\hbox {D}}=\hbox {m}\left( \frac{\upvarepsilon _{\mathrm{p}}}{\upvarepsilon }_{\mathrm{f}}\right) ^{\mathrm{m-1}}\frac{\dot{\upvarepsilon }_{\mathrm{p}}}{\upvarepsilon _{\mathrm{f}}} \end{aligned}$$
(41)
$$\begin{aligned}&{\upsigma }_{\mathrm{eq}}=\left( {1-\hbox {D}^{{\upbeta }}}\right) {\upsigma }_{\mathrm{M}} \end{aligned}$$
(42)
From the given tensile coupon tests, the material stress–strain curves show that the fracture strains under simple tension at the two loading rates are not too different from each other; however, the hardening exponent appears to be lowered significantly when the loading rate is high. It is also noticed that the material yield stress displays some strain rate hardening. At the high loading rate, the yield stress is about 10% higher when the strain rate is 1000 times higher.
Experiments in literature show that at higher loading rates, the fracture pattern often favors shear mode, e.g. it can change from a mode I to a mixed model I/III for a flat panel when a compact tension specimen loaded dynamically (Rivalin et al. 2001; Xue and Wierzbicki 2009). In the present study, path A–C–F represents a mode I fracture and path B–D–E–A represents an in-plane shear mode. These two modes should be the dominant fracture modes of the present structure for most metallic materials.
In addition to the tensile tests, a double V-notched specimen was also used for material shear testing. The fracture plane of the double V-notched specimen under simple shear is a little skewed and is not exactly along the straight line connecting the roots of the V-notches. The shear test shows the material yield stress under nominal simple shear is lower than that under simple tension. It was estimated that the initial yield stress was about 13 % lower under nominal simple shear condition. There are two possibilities to model this difference: (1) adopt a Tresca yield condition or similar J3-dependent yield condition or (2) assuming the material is anisotropic (likely due to rolling process, but we were not sure because of limited experimental data) and choose an anisotropic yield function such as Hill1948. In either way, it should be more accurate in describing the yield condition, at least in the initial yielding phase of the material. However, this J3-dependence yielding is ignored in the present prediction using the simple damage plasticity model (Xue 2009).
Results and discussions
Damage plasticity model for the material was used throughout the numerical simulation. Using the above described method and the calibrated material parameters, the pulling of the S-shaped structures at the two loading rates were simulated. The nominal geometry of the specimen was used. No variation in the actual size was considered for the geometrical tolerance. Two sets of hole-pin contacts used to pull the specimen apart were modeled explicitly, such that the rotation at the pin-hole contact was allowed. Fracture is simulated by element removal, i.e. when damage index D is accumulated to unity for an element, that element loses all its load carrying capacity. This can be seen from Eq. (42).
In the low loading rate case, a tensile fracture path (A–C–F) was predicted; in the high loading rate case, an in-plane shear fracture path (B–D–E–A) was predicted. The after fracture paths are shown in Fig. 82.
The predicted COD1-loading curves are shown in Fig. 83. There are some minor oscillations in the COD1-loading curve due to the exaggerated pulling rate used in the explicit numerical simulation. The load oscillation in the fast loading rate scenario after fracture occurred in the B–D–E segments is due to the sudden break of the structure.
The post-mortem experiment examination reveals that all S-shaped specimens fractured in the in-plane shear mode (B–D–E–A) under remote extension, except one that was believed to be not entirely flat that failed in A–C–F path. This differs from the numerical predictions in that the slow rate experiments result in a B–D–E–A fracture path as well as under the fast loading rate. The simulation of the fast loading rate case matches the experimental results well. In both cases, the predicted peak loads is about 10 % higher than that obtained experimentally.
Sources of discrepancy
In retrospect, neglecting the J3-dependence of the yield condition of the material appears to be the single most significant source of discrepancy in these predictions. For instance, it was shown by the MIT team that the Hill 1948 yield condition are capable of prediction the in-plane shear mode (path B–D–E–A) for both slow and fast loading rate. When the Hill 1948 yield condition degenerates to von Mises yield condition, the fracture path changes to A–C–F. Furthermore, the higher predicted peak loads were also resulted from the tensile stress–strain curve used in simulations. In both slow and fast loading the regions along the shear path were subjected to the most severe deformation, where the resistance was over-predicted. Meanwhile, the predicted COD1 values at fracture of both slow and fast loadings have a good match with the experimental data. This suggests the calibration procedure are fairly accurate under the assumption of the material obeys a von Mises yield criterion. It also suggests the adoption of a J3-dependent yield condition or an anisotropic yield function such as Hill 1948 will fix these observed errors in the current prediction. In either case, a re-calibration of the material parameters for damage accumulation and fracture initiation may be necessary.
Appendix 2: Further experimental details
Dimensional measurements
A calibrated coordinate measurement machine was utilized for in-plane dimensional measurements, and a calibrated micrometer was utilized for thickness measurements. All tested specimens had dimensions within the drawing tolerances. In addition to the in-plane dimensional measurements, thickness measurements were performed at Sandia with a calibrated Mitutoyo IP65 micrometer with 0.001-mm resolution in eight locations (TL, TC, TR, Pt. O, 5, BL, BC, and BR) shown in Fig. 84a; Tables 16 and 17 in “Appendix 2” provide the details of these measurements. Despite being within the thickness tolerance of the drawing, the challenge specimens had out-of-plane distortions that were caused by unbalanced milling of the plate thickness. The relative surface height measurements were taken using a Brown & Sharpe BestTest dial indicator with a 0.0127-mm precision attached to a height gage resting on a flat precision ground granite surface plate. The height was measured in 12 locations relative to the reference height “PT. O” as shown in Fig. 84a. Tables 18 and 19 provide the relative surface height measurements for each test sample. Sample 30 had the largest relative curvature out of all the samples, which may have contributed to this one sample failing by a different crack path than all other samples.
Table 16 Thickness measurements for all samples tested at the 0.0254 mm/s displacement rate
Table 17 Thickness measurements for all samples tested at the 25.4 mm/s displacement rate
Table 18 Surface height measurements for all samples tested at the 0.0254 mm/s displacement rate
Table 19 Surface height measurements for all samples tested at the 25.4 mm/s displacement rate
Fractographic observations
Images of the fracture surfaces and crack paths for Samples 11 (‘slow’ loading rate) and 27 (‘fast’ loading rate) are shown in Figs. 85 and 86 respectively. Both loading rates had remarkably similar fracture surfaces, so they will be described together. The ligament B-D was predominated by small ductile dimples, roughly 1 \(\upmu \hbox {m}\) in diameter. In addition, there were relatively flat patches \(\sim \)50–100 \(\upmu \hbox {m}\) in diameter on the fracture surface. In these flat patches, there were very few dimples, and the material had a smeared appearance somewhat reminiscent of a wear surface presumably associated with shear failure. This smeared zone has an appearance somewhat reminiscent of “slickenlines” associated with the shear fracture of geologic structures. Ligament D-E had similar features, although there was also a single rather large smeared flat patch 300–500 \(\upmu \hbox {m}\) in diameter with a cluster of very large 10–20 \(\upmu \hbox {m}\) diameter dimples on one side of the smear. These large dimples are thought to be associated with stable microvoid coalescence and the finer dimples are thought to be associated with fast fracture. While the large dimples and associated large smear patch was most evident in ligament D–E, it was also possible to find similar, albeit less pronounced, features in ligament B–D. The final rupture ligament, E-A, also had very pronounced cluster of large dimples as well as an even larger smear patch. In all cases, the dimples were not perfectly equiaxed as would be expected in a tensile failure, but had some degree of directionality consistent with a shear failure.
Sample 30 was the only exception that failed by path A–C–F. The fracture surface of the A–C ligament that failed apparently prematurely (at a lower COD1 than any other sample) is shown in Fig. 87. This ligament, A–C, should have nominally been loaded in pure tension. Therefore, it is reasonable to compare this ligament to the tensile fractures observed in the tensile bars provided for material calibration, specifically Fig. 3e. There are important differences that suggest that Sample 30 could have exhibited anomalously low ductility. Firstly, unlike the tensile bar the outer shape of the ligament A–C shows very little curvature suggestive of limited necking. Secondly, the center of the fracture surface shows marked differences. This central zone is typically called the ‘fibrous zone’ in tensile cup-and-cone failure, and it is surrounded by ‘shear lips’. The fibrous zone where fracture originates is typically flat, with only microscale perturbations from the mode-I crack path. In the tensile test fracture, Fig. 3e, this central fibrous zone is indeed reasonably flat with only minor ridges running horizontally in the image indicative of the lamellar microstructure developed during rolling. However, in Sample 30’s A–C ligament fracture surface, these ridges are markedly more pronounced and faceted indicating a much coarser and perhaps less uniform underlying crystallographic texture. In fact in Sample 30 there is no nominal mode-I crack plane and the entire central fibrous zone appears more like alternating shear lips. Inspection of the secondary crack ligament, C–F also revealed a pronounced ridge that deviated from the expected mode-I crack plane. This unusual fracture morphology raises suspicion that Sample 30 was not a nominal failure. This apparently anomalous behavior is discussed further in Sect. 3.3
Post-challenge experimental observations from the Ravi-Chandar Lab at University of Texas at Austin
The University of Texas volunteered to perform additional tests after the predictions had been made in order to produce a complimentary set of data to those already compiled by the two Sandia laboratories. Experiments were performed on three samples; 2, 5, and 31. These samples were obtained from the same manufacturing lot as the samples tested at Sandia National Laboratories. Due to the limited number of samples, only experiments with the slow loading rate were performed.
The experimental setup used in the University of Texas laboratory can be seen in Fig. 88. The experiments utilized a 100-kN Instron electromechanical load frame, with a 100-kN load cell (\(\pm \)0.25 % uncertainty of the measured value) at ambient temperature. The level of noise in the load signal was measured to be 2 N. The crosshead rate was maintained at 0.0254 mm/s, as prescribed in the challenge. Two universal joints were placed, one each at the upper and lower grips in order to minimize the effect of loading misalignments. In addition, the same clevises used by the Sandia laboratories were used. Instead of using COD gages to measure the displacements at the notch mouths, a digital single-lens reflex (DSLR) camera was used to view the entire specimen to allow the COD measurement to be made using DIC. Two additional cameras focused on the region between the notches were used to perform 3D-DIC and to obtain the kinematic fields in the regions of highest deformation and eventual failure. A high-speed video camera, with high frame rate capability, was positioned to view the ligaments B-D and D-E and resolve the sequence of failure; the camera was post-triggered with the falling signal from the load drop associated with specimen failure. Further details of the experimental methods, sensitivity, resolution, and results are described by Gross and Ravi-Chandar (2016).
Confirmation of the load-COD1 results produced at both of the Sandia laboratories is shown in Fig. 89. All three samples indicated the same crack path (B–D–E–A); failure occurred in ligaments B–D and D–E, although the loading on sample 31 was halted just after localization occurred in these ligaments, but before they failed. Follow up microscopy on this sample is presented in Gross and Ravi-Chandar (2016). For the two samples loaded until failure, fast fracture occurred in ligaments B-D and D-E nearly simultaneously. To resolve which ligament failed first, sample 2 was imaged at 20,000 fps and sample 5 was imaged at 40,000 fps. An image sequence showing three subsequent frames captured at 50-\(\upmu \hbox {s}\) intervals by the high speed camera at the time of failure for sample 2 is shown in Fig. 90. The overlaid color contours are of the vertical displacement field (relative to the trigger point) calculated with DIC from the high speed images. The first image shows the state of the sample just before cracking of any ligament, the second image shows ligament BD intact with ligament DE completely severed by a crack and the final image shows both ligaments fully cracked. Identification of cracking in the second image is made by observing a displacement field consistent with the elastic recovery expected after external loading is released from the ligament by the presence of a crack. A high speed video with more frames for this sample and without DIC processing is included as Supplementary Information for this article. Despite increasing the framing rate for sample 5, only three subsequent frames capture the same behavior observed for sample 2. Sufficient temporal resolution to determine the location of crack initiation in each ligament was not pursued. Thus, the greatest specificity that can be given for the cracking sequence of these two specimens is that fracture initiates in the ligament D–E, the associated unloading and the subsequent fracture of the ligament B–D occur within the next 100 \(\upmu \hbox {s}\), suggesting a very dynamic process of fracture. After continued loading ligament E–A is expected to fail as observed in the experiments performed by the Sandia laboratories.