A D V A N C E D M A T E R I A L S & P R O C E S S E S | O C T O B E R 2 0 2 2 4 0 The ratio of compressive plateau stress (σCPS) to the tensile plateau stress (σUPS) in the sample result above is approximately 1.0. This is a reasonable number for a typical NiTi sheet material that has a strong < 1 1 0 > texture component. PROPAGATION OF MATERIAL PROPERTY UNCERTAINTY TO SIMULATIONS OF CYCLIC LOADING Here, an example of propagating the uncertainty in the material parameters determined using the BI method is demonstrated using a simulation of NiTi fatigue loading. The material parameter probability distributions shown in the sample results above are used in a simulation of cyclic loading of the diamond-shaped NiTi specimen introduced in Fig. 1a. The cyclic loading path is shown in Fig. 4a and the extrema of the cyclic loading path are annotated by labels D and E. The commonly used fatigue indicator parameters for NiTi—mean strain and strain amplitude—are calculated using the tensor method[6]. A scatter plot of strain amplitude vs. mean strain obtained from a simulation is shown in Fig. 4b. Fatigue safety factors are typically estimated using the mean strain and strain amplitude at the critical point in the model. In this example, the critical point is taken as the point where the largest strain amplitude occurs. The critical point is shown by a red square in Fig. 4b. A series of simulations with material parameters in the 95% credible intervals of the material parameter distributions shown in the sample result above are carried out and probability distributions are constructed from the mean strain and the strain amplitude at the critical point in each simulation. These probability distributions are shown in Figs. 4c and d. The strain amplitude shows a range of approximately 0.007, which may translate to a large uncertainty in the fatigue safety factor calculated using these strain amplitude data. CONCLUSIONS In this article, the authors have developed and described the implementation of a method for determining the material parameter inputs to the superelastic constitutive model for nickel-titanium with their accompanying uncertainty. The inputs to the method are surface full-field strain data and global load data obtained from one or more tensile tests on an appropriate specimen and a simulation library that provides a dataset with various combinations of material parameter inputs and corresponding strain and load outputs. This method uses Bayesian Inference to obtain a probability distribution of the input parameters. The numerical implementation of the method uses Markov Chain Monte Carlo sampling accelerated by a machine learning method that augments the results obtained from the simulation library. There are four main benefits of this model calibration method: i. The probability distribution of the material parameters determined using this method automatically furnishes a quantification of uncertainty in the parameters. The uncertainty can be communicated using measures such as credible intervals. ii. The uncertainty in the material parameters can be propagated to subsequent simulations such as simulation of fatigue loading of NiTi samples. iii. The diamond specimen geometry used in this method enables determination of tensile as well as compressive plateau stresses from a single test. iv. The machine learning element of this method allows use of a relatively small simulation library compared to performing calibration without such library augmentation. This is a versatile method in the sense that it can be applied to any constitutive model implemented in finite element solvers. The statistical methods such as Markov Chain Monte Carlo sampling used in this scheme are implemented in a wide variety of software tools such as Matlab and Python. While the overall accuracy of the simulation results primarily depends on the accuracy of the underlying constitutive model, methods such as these can help quantify, communicate, and propagate uncertainty in the simulation FEATURE Fig. 4 — Propagation of uncertainty in material parameters to a simulation of cyclic loading: (a) cyclic loading path; (b) strainmap obtained using a fatigue simulation; and (c, d) probability distribution of mean strain and strain amplitude respectively at the critical point obtained from a series of simulations. Adapted from Paranjape et al.[3]. (a) (b) (c) (d) 1 2
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