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DOE OSTI2026

A finite viscoelastic constitutive model for low to high strain rate response of elastomers with application of strain rate-induced glass transition

Amorphous elastomers exhibit significant rate-stiffening and unique viscous flow characteristics across a wide range of strain rates, often undergoing glass transition above a strain rate threshold. We have developed a thermodynamically-consistent and micromechanically-inspired constitutive model for soft elastomeric materials to capture the rate-dependent stress-strain behavior and hysteresis when subjected to low to high strain rates. Here, our proposed constitutive model encapsulates the viscous flow of materials through molecular motion at low strain rates and local rearrangement and alignment of the molecules trying to overcome the intermolecular resistance at high strain rates, essentially covering the glass transition. We applied our constitutive model to uniaxial compression experiments performed at low and high strain rates for polyborosiloxane (PBS) to identify the material parameters, and subsequently, performed numerical simulations of single and multi-cycle compression, stress relaxation, and small amplitude oscillatory tension-compression. Our analyses indicate that the model predicts higher total energy dissipation with increasing strain rate; however, dissipation associated with molecular relaxation decreases (forming a cusp) because, beyond a crossover strain rate, intermolecular rearrangement and alignment become dominant, which is consistent with the onset of the glass transition. For cyclic loading-unloading, we observed that dissipation over a cycle remains constant at low strain rates but decreases non-monotonically at high strain rates before becoming constant, with the peak stress over the cycle becoming higher, which can be interpreted as more loading being carried elastically by the polymer network as the intermolecular rearrangement process occurs. Additionally, our model was able to predict the qualitative nature of the storage modulus and loss modulus in the limit of small strain over a wide range of frequency sweeps.

36 MATERIALS SCIENCE↗

NASA NTRS1984

Constitutive modeling for isotropic materials

A state-of-the-art review of applicable constitutive models with selection of two for detailed comparison with a wide range of experimental tests was conducted. The experimental matrix contained uniaxial and biaxial tensile, creep, stress relaxation, and cyclic fatigue tests at temperatures to 1093 C and strain rates from .0000001 to .001/sec. Some nonisothermal cycles will also be run. The constitutive models will be incorporated into the MARC finite element structural analysis program with a demonstration computation made for advanced turbine blade configuration. In the code development work, particular emphasis is being placed on developing efficient integration algorithms for the highly nonlinear and stiff constitutive equations. Another area of emphasis is the appropriate and efficient methodology for determing constitutive constants from a minimum extent of experimental data.

Lindholm, U. S.↗

NASA NTRS2001

The Constitutive Modeling of Thin Films with Randon Material Wrinkles

Material wrinkles drastically alter the structural constitutive properties of thin films. Normally linear elastic materials, when wrinkled, become highly nonlinear and initially inelastic. Stiffness' reduced by 99% and negative Poisson's ratios are typically observed. This paper presents an effective continuum constitutive model for the elastic effects of material wrinkles in thin films. The model considers general two-dimensional stress and strain states (simultaneous bi-axial and shear stress/strain) and neglects out of plane bending. The constitutive model is derived from a traditional mechanics analysis of an idealized physical model of random material wrinkles. Model parameters are the directly measurable wrinkle characteristics of amplitude and wavelength. For these reasons, the equations are mechanistic and deterministic. The model is compared with bi-axial tensile test data for wrinkled Kaptong(Registered Trademark) HN and is shown to deterministically predict strain as a function of stress with an average RMS error of 22%. On average, fitting the model to test data yields an RMS error of 1.2%

Murphey, Thomas W.↗

NASA NTRS1990

Application of symbolic computations to the constitutive modeling of structural materials

In applications involving elevated temperatures, the derivation of mathematical expressions (constitutive equations) describing the material behavior can be quite time consuming, involved and error-prone. Therefore intelligent application of symbolic systems to faciliate this tedious process can be of significant benefit. Presented here is a problem oriented, self contained symbolic expert system, named SDICE, which is capable of efficiently deriving potential based constitutive models in analytical form. This package, running under DOE MACSYMA, has the following features: (1) potential differentiation (chain rule), (2) tensor computations (utilizing index notation) including both algebraic and calculus; (3) efficient solution of sparse systems of equations; (4) automatic expression substitution and simplification; (5) back substitution of invariant and tensorial relations; (6) the ability to form the Jacobian and Hessian matrix; and (7) a relational data base. Limited aspects of invariant theory were also incorporated into SDICE due to the utilization of potentials as a starting point and the desire for these potentials to be frame invariant (objective). The uniqueness of SDICE resides in its ability to manipulate expressions in a general yet pre-defined order and simplify expressions so as to limit expression growth. Results are displayed, when applicable, utilizing index notation. SDICE was designed to aid and complement the human constitutive model developer. A number of examples are utilized to illustrate the various features contained within SDICE. It is expected that this symbolic package can and will provide a significant incentive to the development of new constitutive theories.

Arnold, Steven M.↗

NASA NTRS1986

Anisotropic constitutive model for nickel base single crystal alloys: Development and finite element implementation

A tool for the mechanical analysis of nickel base single crystal superalloys, specifically Rene N4, used in gas turbine engine components is developed. This is achieved by a rate dependent anisotropic constitutive model implemented in a nonlinear three dimensional finite element code. The constitutive model is developed from metallurigical concepts utilizing a crystallographic approach. A non Schmid's law formulation is used to model the tension/compression asymmetry and orientation dependence in octahedral slip. Schmid's law is a good approximation to the inelastic response of the material in cube slip. The constitutive equations model the tensile behavior, creep response, and strain rate sensitivity of these alloys. Methods for deriving the material constants from standard tests are presented. The finite element implementation utilizes an initial strain method and twenty noded isoparametric solid elements. The ability to model piecewise linear load histories is included in the finite element code. The constitutive equations are accurately and economically integrated using a second order Adams-Moulton predictor-corrector method with a dynamic time incrementing procedure. Computed results from the finite element code are compared with experimental data for tensile, creep and cyclic tests at 760 deg C. The strain rate sensitivity and stress relaxation capabilities of the model are evaluated.

Dame, L. T.↗

NASA NTRS2002

Constitutive Modeling of Nanotube-Reinforced Polymer Composites

In this study, a technique is presented for developing constitutive models for polymer composite systems reinforced with single-walled carbon nanotubes (SWNT). Because the polymer molecules are on the same size scale as the nanotubes, the interaction at the polymer/nanotube interface is highly dependent on the local molecular structure and bonding. At these small length scales, the lattice structures of the nanotube and polymer chains cannot be considered continuous, and the bulk mechanical properties can no longer be determined through traditional micromechanical approaches that are formulated by using continuum mechanics. It is proposed herein that the nanotube, the local polymer near the nanotube, and the nanotube/polymer interface can be modeled as an effective continuum fiber using an equivalent-continuum modeling method. The effective fiber serves as a means for incorporating micromechanical analyses for the prediction of bulk mechanical properties of SWNT/polymer composites with various nanotube lengths, concentrations, and orientations. As an example, the proposed approach is used for the constitutive modeling of two SWNT/polyimide composite systems.

Odegard, G. M.↗

NASA NTRS2002

Constitutive Modeling of Nanotube-Reinforced Polymer Composites

In this study, a technique is presented for developing constitutive models for polymer composite systems reinforced with single-walled carbon nanotubes (SWNT). Because the polymer molecules are on the same size scale as the nanotubes, the interaction at the polymer/nanotube interface is highly dependent on the local molecular structure and bonding. At these small length scales, the lattice structures of the nanotube and polymer chains cannot be considered continuous, and the bulk mechanical properties can no longer be determined through traditional micromechanical approaches that are formulated by using continuum mechanics. It is proposed herein that the nanotube, the local polymer near the nanotube, and the nanotube/polymer interface can be modeled as an effective continuum fiber using an equivalent-continuum modeling method. The effective fiber serves as a means for incorporating micromechanical analyses for the prediction of bulk mechanical properties of SWNT/polymer composites with various nanotube shapes, sizes, concentrations, and orientations. As an example, the proposed approach is used for the constitutive modeling of two SWNT/LaRC-SI (with a PmPV interface) composite systems, one with aligned SWNTs and the other with three-dimensionally randomly oriented SWNTs. The Young's modulus and shear modulus have been calculated for the two systems for various nanotube lengths and volume fractions.

Odegard, G. M.↗

NASA NTRS1987

Anisotropic constitutive modeling for nickel base single crystal superalloy Rene N4 at 982 C

A back stress/drag stress constitutive model based on a crystallographic approach to model single crystal anisotropy is presented. Experimental results demonstrated the need for the back stress variable in the inelastic flow equations. Experimental findings suggested that back stress is orientation dependent and controls both strain hardening and recovery characteristics. Due to the observed stable fatigue loops at 1800 F, drag stress is considered constant for this temperature. The constitutive model operated with constraints determined only from tensile data was extensively tested from simple tensile and fatigue to complicated strain hold tests. The model predicted very well under those conditions.

Stouffer, D. C.↗

NASA NTRS1991

Elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites

A constitutive model to describe the elastic/viscoplastic behavior of fiber-reinforced thermoplastic composites under plane stress conditions is presented. Formulations are given for quasi-static plasticity and time-dependent viscoplasticity. Experimental procedures required to generate the necessary material constants are explained, and the experimental data is compared to the predicted behavior.

Gates, T. S.↗

DOE OSTI2024

Initial Alloy 709 constitutive models for use with the ASME design by inelastic analysis and EPP+SMT design methods

This report details a preliminary inelastic constitutive model describing the behavior of Alloy 709. This model will serve two purposes: (1) integration into Nonmandatory Appendix HBB-Z of the ASME Boiler & Pressure Vessel Code Section III, Division 5 and (2) extrapolating cyclic test data to difficult to measure conditions for formulating improved creep-fatigue design methods. For both applications, the model must accurately capture the material behavior across a wide range of temperatures and a variety of test conditions, both monotonic and cyclic. For this purpose we adopt a universal model form under consideration to standardize the description of high temperature constitutive models in the ASME Code. This report briefly restates that model form and how we calibrate the model against the test data, summarizes the test database, and validates the final, trained model by comparison to the experimental tests.

36 MATERIALS SCIENCE↗

DOE OSTI2023

High temperature inelastic constitutive models for the ASME Section III, Division 5 Class A materials

This report describes the exploration of a universal high temperature inelastic constitutive model for use with the ASME Boiler & Pressure Vessel Code Section III, Division 5, Class A design rules. The idea, developed based on feedback from reactor vendors, is to simplify the current bespoke material models for Grade 91, 316H, and Alloy 617 and the new model for Alloy 800H into a simple, single model form. The report describes a new parallel time integration technique implemented in the pyoptmat package which allowed us to explore a wide variety of model forms, searching for a suitable common model. The report then describes preliminary work on models for monontonic deformation and then the development of a set of models suitable for capturing high temperature cyclic deformation. These models are suitable for use with the ASME design rules, except potentially for a narrow, material-specific band of temperatures at the edge of the creep regime. An incremental improvement the current models could overcome this discrepancy and provide a new, simpler set of constitutive model for all four materials.

36 MATERIALS SCIENCE↗

NASA NTRS2002

Constitutive Modeling of Nanotube/Polymer Composites with Various Nanotube Orientations

In this study, a technique has been proposed for developing constitutive models for polymer composite systems reinforced with single-walled carbon nanotubes (SWNT) with various orientations with respect to the bulk material coordinates. A nanotube, the local polymer adjacent to the nanotube, and the nanotube/polymer interface have been modeled as an equivalent-continuum fiber by using an equivalent-continuum modeling method. The equivalent-continuum fiber accounts for the local molecular structure and bonding information and serves as a means for incorporating micromechanical analyses for the prediction of bulk mechanical properties of SWNT/polymer composite. As an example, the proposed approach is used for the constitutive modeling of a SWNT/LaRC-SI (with a PmPV interface) composite system, with aligned nanotubes, three-dimensionally randomly oriented nanotubes, and nanotubes oriented with varying degrees of axisymmetry. It is shown that the Young s modulus is highly dependent on the SWNT orientation distribution.

Odegard, Gregory M.↗

NASA NTRS2001

Constitutive Modeling of Nanotube-Reinforced Polymer Composite Systems

In this study, a technique has been proposed for developing constitutive models for polymer composite systems reinforced with single-walled carbon nanotubes (SWNT). Since the polymer molecules are on the same size scale as the nanotubes, the interaction at the polymer/nanotube interface is highly dependent on the local molecular structure and bonding. At these small length scales, the lattice structures of the nanotube and polymer chains cannot be considered continuous, and the bulk mechanical properties of the SWNT/polymer composites can no longer be determined through traditional micromechanical approaches that are formulated using continuum mechanics. It is proposed herein that the nanotube, the local polymer near the nanotube, and the nanotube/polymer interface can be modeled as an effective continuum fiber using an equivalent-continuum modeling method. The effective fiber retains the local molecular structure and bonding information and serves as a means for incorporating micromechanical analyses for the prediction of bulk mechanical properties of SWNT/polymer composites with various nanotube sizes and orientations. As an example, the proposed approach is used for the constitutive modeling of two SWNT/polyethylene composite systems, one with continuous and aligned SWNT and the other with discontinuous and randomly aligned nanotubes.

Odegard, Gregory M.↗

NASA NTRS2004

Constitutive Modeling of Nanotube-Reinforced Polymer Composite Systems

Odegard, Gregory M.↗

DOE OSTI2024

Simplified inelastic constitutive models for ASME Section III, Division 5 design by inelastic analysis

This report describes the development of simplified, universal constitutive model that captures the high temperature monotonic and cyclic behavior of a range of commonly-used high temperature materials. The goal of the work is to provide a simple, universal constitutive model to replace the current bespoke models for Grade 91, 316H, and Alloy 617 included in Nonmandatory Appendix HBB-Z of the ASME Boiler & Pressure Vessel Code, and to extend this model to cover Alloy 800H. We initiated this work in response to feedback from reactor vendors and other Code users requesting simplified models, compared to the current models, that are easier to implement and use in commercial finite element analysis software. This report describes the completion of this effort by developing a model to correct the defects in standard model forms presently used for high temperature material modeling, described in past work, developing and implementing new numerical methods to train this model against test data, and then actually training the model for the four materials. The report provides a complete mathematical description of the model along with the tabulated material coefficients for the four materials. The final step will be to formulate an ASME Code change to introduce the new models into the Code.

36 MATERIALS SCIENCE↗

NASA NTRS1988

Unified constitutive model for single crystal deformation behavior with applications

Single crystal materials are being used in gas turbine airfoils and are candidates for other hot section components because of their increased temperature capabilities and resistance to thermal fatigue. Development of a constitutive model which assesses the inelastic behavior of these materials has been studied in 2 NASA programs: Life Prediction and Constitutive Models for Engine Hot Section Anisotropic Materials and Biaxial Constitutive Equation Development for Single Crystals. The model has been fit to a large body of constitutive data for single crystal PWA 1480 material. The model uses a unified approach for computing total inelastic strains (creep plus plasticity) on crystallographic slip systems reproducing observed directional and strain rate effects as a natural consequence of the summed slip system quantities. The model includes several of the effects that have been reported to influence deformation in single crystal materials, such as shear stress, latent hardening, and cross slip. The model is operational in a commercial Finite Element code and is being installed in a Boundary Element Method code.

Walker, K. P.↗

NASA NTRS1988

Unified constitutive models for high-temperature structural applications

Unified constitutive models are characterized by the use of a single inelastic strain rate term for treating all aspects of inelastic deformation, including plasticity, creep, and stress relaxation under monotonic or cyclic loading. The structure of this class of constitutive theory pertinent for high temperature structural applications is first outlined and discussed. The effectiveness of the unified approach for representing high temperature deformation of Ni-base alloys is then evaluated by extensive comparison of experimental data and predictions of the Bodner-Partom and the Walker models. The use of the unified approach for hot section structural component analyses is demonstrated by applying the Walker model in finite element analyses of a benchmark notch problem and a turbine blade problem.

Lindholm, U. S.↗

DOE OSTI2024

Constitutive model development of aluminum alloy 1100 for elevated temperature forming process

Commercially pure aluminum alloy, AA1100, presents good electrical and thermal conductivity, high formability, and low cost. Those favorable characteristics have the potential to enable bipolar plates with improved economics and enhanced performance compared to current stainless steel bipolar plates for proton exchange membrane fuel cells. An accurate constitutive model is essential to develop and optimize processing parameters and effectively control the forming process. Here, the objective of this work is to develop a constitutive model of AA1100 that is able to simulate stress-strain relation, formed geometry, and predict the onset of fracture strain to avoid forming failure. Initially, a set of tensile tests at temperature between 300 and 500°C and strain rate between 0.005 and 1.0/s were conducted to examine the deformation behavior. Then, a set of damage-based unified visco-plastic constitutive equations is proposed and calibrated based on the results of stress-strain data. A genetic algorithm optimization method is applied to search for best fitting material constants in constitutive equations. The proposed model shows good predictability of both the stress-strain relation and fracture strain at low strain rate and high temperature conditions. The accuracy of proposed model is also evaluated statistically. A comparison of the proposed model with three popular models (Arrhenius-type mode, Johnson-Cook model and Zerilli-Armstrong model) was made. The proposed model shows the best experimental agreement with correlation coefficient of 0.96 in contrast to 0.25, 0.38 and 0.75 for the popular models, respectively. The proposed model can help to optimize the elevated temperature forming process and guide die design to enable optimal geometric features in the formed components.

08 HYDROGEN↗