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DOE OSTI2022

Calibrating constitutive models with full‐field data via physics informed neural networks

Abstract The calibration of solid constitutive models with full‐field experimental data is a long‐standing challenge, especially in materials that undergo large deformations. In this paper, we propose a physics‐informed deep‐learning framework for the discovery of hyperelastic constitutive model parameterizations given full‐field surface displacement data and global force‐displacement data. Contrary to the majority of recent literature in this field, we work with the weak form of the governing equations rather than the strong form to impose physical constraints upon the neural network predictions. The approach presented in this paper is computationally efficient, suitable for irregular geometric domains, and readily ingests displacement data without the need for interpolation onto a computational grid. A selection of canonical hyperelastic material models suitable for different material classes is considered including the Neo–Hookean, Gent, and Blatz–Ko constitutive models as exemplars for general non‐linear elastic behaviour, elastomer behaviour with finite strain lock‐up, and compressible foam behaviour, respectively. We demonstrate that physics informed machine learning is an enabling technology and may shift the paradigm of how full‐field experimental data are utilized to calibrate constitutive models under finite deformations.

Hamel, Craig M.↗

NASA NTRS2017

Grain Size-Dependent Crystal Plasticity Constitutive Model for Polycrystal Materials

A new method to introduce grain size-dependence in classical crystal plasticity constitutive model is developed by considering the resistance to dislocation motion in the grain boundary influence region as equivalent to that of a work hardening. A general framework for the size-dependent constitutive model is derived by implementing this method on a core and mantle model. The work hardening, equivalent to grain boundary effect, is realized by introducing a fictitious, pre-existing, plastic strain grain boundary influence region (mantle) following the principles of classical crystal plasticity. This fictitious plastic strain, in effect, increases the yield strength and decreases the initial hardening coefficient of the grain. With the thickness of grain boundary influence region and the distribution of introduced plastic strain remaining the same, the grain boundary effect increased as the grain size becomes smaller. A simplified model that considers the grain boundary effect on a grain average sense is also developed under this general framework. Implementation of this general framework to the specific cases of crystal plasticity constitutive models is demonstrated by considering the case of power law flow rule and hyperbolic-secant hardening rule. Finally, the grain size-dependent constitutive model is validated by comparing the predicted stress-strain behavior of polycrystal samples with different average grain sizes under uniaxial loading with the experimental results.

Grain Size↗

DOE OSTI2020

Calibration and modification of the constitutive model for the mechanical response of P91

The objective of this first milestone was to calibrate and if necessary, modify a constitutive model predicting the primary and secondary creep response of P91. The overall path to be followed consists of first calibrating this constitutive model against experimental data available from the literature (this milestone). In parallel, data acquired by indentation tests will be used to calibrate the model (this involves transforming the indentation data into equivalent uniaxial tests). An overall comparison of model parameters will allow assessing to what extent these rapid tests can be used to calibrate models predicting the overall mechanical response of the bulk. Given that COVID naturally hindered the gathering of new experimental data, the choice was made to ensure that LANL would have sufficient funds in year two to perform the bulk of the new calibration work. The following report details (i) how and which of the bulk experimental data was selected, (ii) constitutive model formulation, (iii) predictions of the constitutive model: effects of imposed stress, temperature and precipitate content, (iv) future work.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

NASA NTRS1986

Life prediction and constitutive models for engine hot section anisotropic materials program

This report presents the results of the first year of a program designed to develop life prediction and constitutive models for two coated single crystal alloys used in gas turbine airfoils. The two alloys are PWA 1480 and Alloy 185. The two oxidation resistant coatings are PWA 273, an aluminide coating, and PWA 286, an overlay NiCoCrAlY coating. To obtain constitutive and/or fatigue data, tests were conducted on coated and uncoated PWA 1480 specimens tensilely loaded in the 100 , 110 , 111 , and 123 directions. A literature survey of constitutive models was completed for both single crystal alloys and metallic coating materials; candidate models were selected. One constitutive model under consideration for single crystal alloys applies Walker's micromechanical viscoplastic formulation to all slip systems participating in the single crystal deformation. The constitutive models for the overlay coating correlate the viscoplastic data well. For the aluminide coating, a unique test method is under development. LCF and TMF tests are underway. The two coatings caused a significant drop in fatigue life, and each produced a much different failure mechanism.

Swanson, G. A.↗

DOE OSTI2024

Constitutive Models via Automatic Differentiation v.1.0.0

SAND2024-00905O Constitutive Models via Automatic Differentiation (CMAD) provides a software framework for solving constitutive or material model calibration problems. It relies on JAX's automatic differentiation capabilities to compute the derivatives needed for both forward and adjoint sensitivity analyses. The software can be used to implement constitutive models and calibrate the parameters from experimental data. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Seidl, Daniel↗

DOE OSTI2020

Calibration of hyperelastic constitutive models: the role of boundary conditions, search algorithms, and experimental variability

The calibration of hyperelastic constitutive models of soft tissue and tissue surrogates is often treated as an exercise in curve-fitting to the average experimental response, and many of the complicating factors such as experimental boundary conditions and data variability are ignored. Here, we focus on three questions that arise in this area: the ramifications of ignoring the experimental boundary conditions, the use of local optimizers, and the role of data variability. Using data from a uniaxial extension experiment on a tissue surrogate, we study how these three factors affect the calibration of isotropic hyperelastic constitutive models. Our results show that even with the simplest of constitutive models, it is necessary to look beyond a "good fit" to the average.

60 APPLIED LIFE SCIENCES↗

NASA NTRS1987

Life prediction and constitutive models for engine hot section anisotropic materials

The results are presented of a program designed to develop life prediction and constitutive models for two coated single crystal alloys used in gas turbine airfoils. The two alloys are PWA 1480 and Alloy 185. The two oxidation resistant coatings are PWA 273, an aluminide coating, and PWA 286, an overlay NiCoCrAlY coating. To obtain constitutive and fatigue data, tests were conducted on uncoated and coated specimens loaded in the CH76 100 CH110 , CH76 110 CH110 , CH76 111 CH110 and CH76 123 CH110 crystallographic directions. Two constitutive models are being developed and evaluated for the single crystal materials: a micromechanic model based on crystallographic slip systems, and a macroscopic model which employs anisotropic tensors to model inelastic deformation anisotropy. Based on tests conducted on the overlay coating material, constitutive models for coatings also appear feasible and two initial models were selected. A life prediction approach was proposed for coated single crystal materials, including crack initiation either in the coating or in the substrate. The coating initiated failures dominated in the tests at load levels typical of gas turbine operation. Coating life was related to coating stress/strain history which was determined from specimen data using the constitutive models.

Swanson, G. A.↗

DOE OSTI2025

ASME Code change proposal to implement new universal high temperature constitutive models for Section III, Division 5

This report completes work on a universal high temperature constitutive model suitable for use with the ASME Boiler & Pressure Vessel Code Section III, Division 5 rules for the design by inelastic analysis of Class A nuclear reactor components. The goals of this work are to provide a simple model form that adequately captures the high temperature response of materials and can be applied to any future Code material. Additionally, the report describes an automated process for calibrating a model against test data. The idea is to simplify the effort required to generate a constitutive model for an arbitrary material, provided test data is available. This will accelerate the process of qualifying new Code materials in the future. In addition, the report provides calibrated models and detailed validation comparisons to test data for five currently-qualified or soon-to-be qualified materials: 316H, Grade 91, Alloy 800H, Alloy 617, and Alloy 709. The report surveys the available data for the remaining two ASME Class Materials --- 2.25Cr-1Mo and 304H --- concluding that there is enough data data to generate a model for 2.25Cr-1Mo steel provided some additional sources of non-public data can be included in the test database, but that a dedicated cyclic test campaign would be needed for 304H. Supplemental material includes the full text of an ASME Code change proposal to incorporate the models for the four currently-qualified Class A material, detailed validation comparisons to test data for the five material models, and input files for reference implementations of the constitutive models in the NEML and NEML2 modeling frameworks.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS2003

Development of a Pressure-Dependent Constitutive Model with Combined Multilinear Kinematic and Isotropic Hardening

The development of a pressure-dependent constitutive model with combined multilinear kinematic and isotropic hardening is presented. The constitutive model is developed using the ABAQUS user material subroutine (UMAT). First the pressure-dependent plasticity model is derived. Following this, the combined bilinear and combined multilinear hardening equations are developed for von Mises plasticity theory. The hardening rule equations are then modified to include pressure dependency. The method for implementing the new constitutive model into ABAQUS is given.

Allen Phillip A.↗

DOE OSTI2023

An assessment of Callahan's crushed salt constitutive model

This memorandum investigates the Callahan crushed salt constitutive model developed and used at Sandia National Laboratories for geomechanics applications. The formulation is reviewed, calibration against a novel long-term experiment with complex loading history is performed and validation against other experiments is attempted. Areas of improvement and deficiencies are identified that support the need for an alternative or updated constitutive model for crushed salt.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

NASA NTRS2026

Development, Characterization, and Validation of Elevated-Temperature Constitutive Models: Deformation and Damage

This report provides a brief review of experimentally observed hereditary and nonhereditary material behavior along with background information on standard as well as advanced internal state variable constitutive modeling at elevated temperature. A description of exploratory, characterization, and validation testing is presented along with a detailed outline of what constitutes “sufficient” data content (i.e., quality and quantity) for developing or enhancing, characterizing, and validating a particular sophisticated nonlinear time- and history-dependent (hereditary) class of constitutive models known as GVIPS (generalized viscoplasticity with potential structure). The tests described are necessary to reveal a material’s behavior in the reversible (or viscoelastic) and irreversible (or viscoplastic) regimes, for the identification of both deformation and damage model parameters. Results presented are primarily for metallic materials. In all cases, both uniaxial and multiaxial tests are described, and the linkage between the specific tests and the parameters within the model that can be characterized from the results of these tests are also discussed. Discussion is also provided relative to the role information management must play relative to material data collection, analysis, maintenance, and dissemination. The need for such an information system is particularly important as both analyst and designer move toward utilizations of sophisticated, nonlinear time- and history dependent (hereditary) constitutive models. Lastly, the concept of state space and its utility in understanding and establishing constitutive models is addressed throughout. The intent behind this document is to help both the modeler and experimentalist understand each other’s specific points of view and provide guidance for both model development and characterization. Emphasis has been placed on providing the mechanician with information regarding how tests are performed and what issues to be aware of when interpreting results. It is hoped that experimentalists will take away a new perspective on the types of information that modelers and analysts are looking for from them.

Constitutive Modeling↗

NASA NTRS1986

Structural analysis of turbine blades using unified constitutive models

The utility of advanced constitutive models and structural analysis methods in predicting the cyclic life of an air-cooled turbine blade is assessed. Five structural analysis methods were exercised in calculating the cyclic stress-strain response at the airfoil critical location. The methods studied were a cyclic elastic finite-element analysis, nonlinear finite-element analyses based on classical inelastic models and the unified models of Bodner and Walker, and a simplified inelastic procedure. These analyses were compared in terms of computing times and of predicted crack initiation lives using the Strainrange Partitioning method.

Kaufman, A.↗

NASA NTRS1988

A constitutive model with damage for high temperature superalloys

A unified constitutive model is searched for that is applicable for high temperature superalloys used in modern gas turbines. Two unified inelastic state variable constitutive models were evaluated for use with the damage parameter proposed by Kachanov. The first is a model (Bodner, Partom) in which hardening is modeled through the use of a single state variable that is similar to drag stress. The other (Ramaswamy) employs both a drag stress and back stress. The extension was successful for predicting the tensile, creep, fatigue, torsional and nonproportional response of Rene' 80 at several temperatures. In both formulations, a cumulative damage parameter is introduced to model the changes in material properties due to the formation of microcracks and microvoids that ultimately produce a macroscopic crack. A back stress/drag stress/damage model was evaluated for Rene' 95 at 1200 F and is shown to predict the tensile, creep, and cyclic loading responses reasonably well.

Sherwood, J. A.↗

DOE OSTI2021

Calibration of elastoplastic constitutive model parameters from full-field data with automatic differentiation-based sensitivities

Here, we present a framework for calibration of parameters in elastoplastic constitutive models that is based on the use of automatic differentiation (AD). The model calibration problem is posed as a partial differential equation-constrained optimization problem where a finite element (FE) model of the coupled equilibrium equation and constitutive model evolution equations serves as the constraint. The objective function quantifies the mismatch between the displacement predicted by the FE model and full-field digital image correlation data, and the optimization problem is solved using gradient-based optimization algorithms. Forward and adjoint sensitivities are used to compute the gradient at considerably less cost than its calculation from finite difference approximations. Through the use of AD, we need only to write the constraints in terms of AD objects, where all of the derivatives required for the forward and inverse problems are obtained by appropriately seeding and evaluating these quantities. We present three numerical examples that verify the correctness of the gradient, demonstrate the AD approach's parallel computation capabilities via application to a large-scale FE model, and highlight the formulation's ease of extensibility to other classes of constitutive models.

42 ENGINEERING↗

NASA NTRS1985

Constitutive modeling for single crystal superalloys

A crystallographic approach to constitutive modeling of single crystal superalloys is discussed. The approach is based on identifying the active slip planes and slip directions. The shear stresses are computed on each of the slip planes from applied stress components. The slip rate is then computed on each slip system and the microscopic inelastic strain rates are the sum of the slip in the individual slip systems. The constitutive model was implemented in a finite element code using twenty noted isoparametric solid elements. Constants were determined for octahedral and cube slip systems. These constants were then used to predict tension-compression asymmetry and fatigue loops. Other data was used to model the tensile and creep response.

Stouffer, Donald C.↗

NASA NTRSDate not supplied

A Nonlinear Viscoelastic-Viscoplastic Constitutive Model for Epoxy Polymers

The objective of this paper is to develop a nonlinear viscoelastic–viscoplastic constitutive model for epoxy polymers. The classic nonlinear viscoelasticity model is reformulated to yield a closed-form incremental constitutive relation, which relates the stress increments to the viscoelastic strain increments. A viscoplasticity model, which consists of the Drucker–Prager yield function, nonlinear isotropic and kinematic hardening laws, and the Perzyna viscosity function, is subsequently developed. The nonlinear viscoelasticity and the vis-coplasticity models are then implemented in a radial return algorithm. A closed-form algorithmic tangent operator is derived to facilitate model implementation. The nonlinear viscoelasticity model is validated by reproducing the test data for polymethyl methacrylate(PMMA). The present constitutive model’s capabilities are demonstrated through modeling viscoelastic–viscoplastic PMMA loaded at different strain rates. The mechanics of structure genome-based micromechanics approach, embedded with the present constitutive model, is used to homogenize a unidirectional fiber-reinforced composite with a PMMA matrix, subjected to uniaxial and shear loading at different strain rates. The viscoelastic characterization of PMT-F7 epoxy is evaluated by comparing the predicted creep–recovery responses by the present constitutive model with the experimental ones.

Liang Zhang↗

DOE OSTI2020

SNL will be able to provide well-calibrated models to weapons systems analysts by integrating an engine for constitutive model calibration with the GRANTA materials database

Researchers at Sandia National Laboratories have integrated the GRANTA materials database with the MatCal calibration engine to calibrate material models from characterization data. GRANTA is gaining acceptance across the NNSA Tri-lab complex and is being populated with weapons-specific test data by Sandia experimentalists. To use that data to create material models for use by weapons systems analysts, MatCal has been enabled import calibration data and test conditions from GRANTA to quickly and reproducibly produce a calibrated set of parameters for a given constitutive model. The team is currently working to store the parameters characterizing material behavior in GRANTA to make them accessible by all weapons analysts.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Calibrating the SPECTACULAR constitutive model using legacy Sandia data for two filled epoxy systems: 828/CTBN/DEA/GMB and 828/DEA/GMB

The SPECTACULAR model is a development extension of the Simplified Potential Energy Clock (SPEC) model. Both models are nonlinear viscoelastic constitutive models used to predict a wide range of time-dependent behaviors in epoxies and other glass-forming materials. This report documents the procedures used to generate SPECTACULAR calibrations for two particulate-filled epoxy systems, 828/CTBN/DEA/GMB and 828/DEA/GMB. No previous SPECTACULAR or SPEC calibration exists for 828/CTBN/DEA/GMB, while a legacy SPEC calibration exists for 828/DEA/GMB. To generate the SPECTACULAR calibrations, a step-by-step procedure was executed to determine parameters in groups with minimal coupling between parameter groups. This procedure has often been deployed to calibrate SPEC, therefore the resulting SPECTACULAR calibration is backwards compatible with SPEC (i.e. none of the extensions specific to SPECTACULAR are used). The calibration procedure used legacy Sandia experimental data stored on the Polymer Properties Database website. The experiments used for calibration included shear master curves, isofrequency temperature sweeps under oscillatory shear, the bulk modulus at room temperature, the thermal strain during a temperature sweep, and compression through yield at multiple temperatures below the glass transition temperature. Overall, the calibrated models fit the experimental data remarkably well. However, the glassy shear modulus varies depending on the experiment used to calibrate it. For instance, the shear master curve, isofrequency temperature sweep under oscillatory shear, and the Young's modulus in glassy compression yield values for the glassy shear modulus at the reference temperature that vary by as much as 15 %. Also, for 828/CTBN/DEA/GMB, the temperature dependence of the glassy shear modulus when fit to the Young's modulus at different temperatures is approximately four times larger than when it is determined from the isofrequency temperature sweep under oscillatory shear. For 828/DEA/GMB, the temperature dependence of the shear modulus determined from the isofrequency temperature sweep under oscillatory shear accurately predicts the Young's modulus at different temperatures. When choosing values for the shear modulus, fitting the glassy compression data was prioritized. The new and legacy calibrations for 828/DEA/GMB are similar and appear to have been calibrated from the same data. However, the new calibration improves the fit to the thermal strain data. In addition to the standard calibrations, development calibrations were produced that take advantage of development features of SPECTACULAR , including an updated equilibrium Helmholtz free energy that eliminates undesirable behavior found in previous work. In addition to the previously mentioned experimental data, the development calibrations require data for the heat capacity during a stress-free temperature sweep to calibrate thermal terms.

36 MATERIALS SCIENCE↗