LAROMANCE Stress Update with Automatic Differentiation

Theory

The creep behavior of metals is governed by physical deformation mechanisms, such as dislocation glide, climb, and Coble creep Asaro (1983). If these physical mechanisms are not explicitly captured with the creep model formulation, transient creep behaviors cannot be readily captured Wen et al. (2017) Wen et al. (2019). Typical constitutive modeling frameworks are too computationally costly to be used effectively in engineering applications, so use of a simplified regression model in place of physics-based simulations, i.e. a reduced order model (ROM) is necessary.

The ROM method employed in this model depends on a database of physics-based simulation results. The formulation of that database must include the list of inputs which are needed to distinguish the desired output(s). In the case of the stainless steel alloy 316H, the degrees of freedom present in the full-fidelity model are too numerous to be included individually in a ROM database. They are reduced through instead relying on isotropic effective measures. For example, thermal creep in randomly textured polycrystalline 316H can be approximated with a J2 material response, i.e., a von Mises stress criterion Hutchinson (1976). This reduced list of inputs includes temperature, von Mises stress, accumulated effective von Mises strain (as a history tracking parameter), average dislocation density in the cell, and average dislocation density in the cell wall.

The physics-based simulations are expressed in a visco-plastic self-consistent (VPSC) framework Lebensohn and Tomé (1993), Lebensohn et al. (2007). VPSC model describes the polycrystal as a collection of orientations (grains) with associated volume fractions chosen to represent the initial texture of the aggregate. Each grain is regarded as a visco-plastic inclusion embedded in, and interacting with, a "homogeneous effective medium" (HEM), which has the average properties of the aggregate. The macroscopic response of the polycrystal results from the contribution of each grain. The visco-plastic compliance of the HEM is given by a self-consistent condition applied on the grain averages. The constitutive laws relating strain-rate and stress for the aggregate are written in a linearized form as, where and are the macroscopic visco-plastic compliance tensor and back-extrapolated terms for the aggregate, respectively. The inclusion formalism couples stress and strain-rate in the grain with the average stress and strain-rate in the effective medium through the interaction equation, where is the visco-plastic Eshelby tensor, is the macroscopic visco-plastic compliance tensor and the parameter "tunes" the stiffness of the inclusion-matrix interaction.

The constitutive laws describe the relative contributions of dislocation glide climb and Coble creep to inform deformation at the single crystal level. The total creep rate can be expressed as, Here, , and refer to the plastic deformation accumulated through dislocation glide, climb and Coble creep, respectively. The strain rates due to dislocation motion can be written as the sum of the mean shear/climb rates over all active systems in the grain: Here and denote the mean shear and climb rates, respectively, and the symmetric part of the Schmid tensor. is the climb tensor for edge dislocations Lebensohn et al. (2010).

This model relies on the original work of Wang et al. (2017) and Wang et al. (2016) in which the response of each material point (grain) is described in a statistical fashion (i.e. via the internal stress distribution) to allow the quantification of type III stresses associated with dislocations. With this approach, one must calculate the average strain rate in one grain accounting for the mechanical response in all sub-material points. This treatment provides a connection between the localized stress distribution within sub-material point and the average response of the material point. It is also required by the effective medium models, such as the VPSC framework, that assume the strain rate and stress within each grain are homogenous. As per Wang et al. (2017) and Wang et al. (2016), the mean shear rate of the slip system in the grain domain is expressed as: where is the shear rate of one sub-material point. is the local resolved shear stress. denotes the mean resolved shear stress in one grain, where is the deviatoric stress tensor. Notice that this law applies to the climb rate as well: with and , being the local and global resolved climb stress, respectively.

As a first approximation, each component of the stress field can be described by a Gaussian function. Therefore, the probability distribution function , representing the volume fraction of sub-material points with , can be written as: where is the distribution variance. Wang et al. (2016) suggest that the dispersion of intragranular stress depends on the total dislocation density, and express as, Where MPa/m is an effective scaling coefficient, introduced here for the sake of simplicity. In practice, though, is expected to be a function depending on the "Contrast factor" for each dislocation population and on dislocation arrangements. The material parameters in use for the generation of the ROM database can be found elsewhere Tallman et al. (2019). For the meaning and implementation of the parameters, readers are referred to the original work Wen et al. (2019).

The polynomial regression model used as the mathematical form of the ROM is formulated in terms of orthogonal polynomial terms, defined using Legendre polynomials. These terms are expressed as , polynomial of degree , evaluated for input , where the range of has been normalized to the interval. The model form includes polynomial terms of each input parameter and all interaction terms thereof, i.e., where is the maximum degree of polynomial to be used in the model, is the regression coefficient for the term formed from the product of -th degree polynomial of input , etc.

Three output values are required from the ROM: , , and . While the strain rate is the desired output for predicting creep behavior, The defect density rates are also needed, as they affect the evolution of the creep rate over time through their own evolution. An entirely independent regression model is employed to predict each of the three outputs required. For example, the regression model for is shown, i.e., where each is an index corresponding to the degree of the Legendre polynomial of the input parameter in the subscript. The coefficients are denoted as specific to the output with a superscript. Additionally, mappings are used to convert from engineering input units to terms less susceptible to error propagation for informing the ROM. All inputs and outputs are handled internally such that the use of the ROM is unchanged by the inclusion of these mappings.

The coefficients, , are evaluated by fitting the ROM to a database of results from the VPSC simulations using the physics-based constitutive laws. The maximum degree of the polynomial terms is selected to maximize fidelity to the data while avoiding an overfitting of the data, where overfitting is indicated by a stark difference in the regression fit to training and testing data. To validate the obtained regression coefficient values, initial conditions are given to the ROM and VPSC, and the resulting simulations are compared.

ROM Tiling

In order to widen the region of applicability without sacrificing ROM accuracy, a ROM tiling method can be used to cover a larger the input parameter space with several, separate ROMS. This requires smoothing from one ROM to another across regions of shared input space, which is performed internally via sigmoidal smoothing functions: (1) Here, is the strain rate for tile using inputs , and is a sigmoid function that smoothly varies from 0 to 1 for a given input value , (2) where and are the limits for tiles 1 and 2 over which the strain is smoothed, and .

ROM Input Windows

Due to the nature of formulating the ROM, the input values are limited to a window of applicability, outside of which, the ROM can no longer be guaranteed to be valid. LAROMANCEStressUpdateBase handles these limits for some of the coupled state variables internally via input parameters for each input that allow for error handling or extrapolation. If the input values are to be extrapolated, a sigmoidal function is utilized to extrapolate from the lower limit of the out-of-bound input to zero strain. Extrapolation can only be performed for temperature, stress, and the environmental factor state variables. The remaining ROM inputs (cell dislocations, cell wall dislocations, and previous strain) are calculated internally, and thus must be within the window of applicability at the start of the simulation.