A research program is in progress to develop strain rate dependent deformation and failure models for the analysis of polymer matrix composites subject to high strain rate impact loads. Strain rate dependent inelastic constitutive equations have been developed to model the polymer matrix, and have been incorporated into a micromechanics approach to analyze polymer matrix composites. The Hashin failure criterion has been implemented within the micromechanics results to predict ply failure strengths. The deformation model has been implemented within LS-DYNA, a commercially available transient dynamic finite element code. The deformation response and ply failure stresses for the representative polymer matrix composite AS4/PEEK have been predicted for a variety of fiber orientations and strain rates. The predicted results compare favorably to experimentally obtained values.

Recently applications have exposed polymer matrix composite materials to very high strain rate loading conditions, requiring an ability to understand and predict the material behavior under these extreme conditions. In this second paper of a two part report, a three-dimensional composite micromechanical model is described which allows for the analysis of the rate dependent, nonlinear deformation response of a polymer matrix composite. Strain rate dependent inelastic constitutive equations utilized to model the deformation response of a polymer are implemented within the micromechanics method. The deformation response of two representative laminated carbon fiber reinforced composite materials with varying fiber orientation has been predicted using the described technique. The predicted results compare favorably to both experimental values and the response predicted by the Generalized Method of Cells, a well-established micromechanics analysis method.

AcknowledgmentsThe authors would like to acknowledge Fiberite, Inc./or providing the material used for theexperimental tests on the IM7/977-2 system along with additional information and data. The authors would alsolike to acknowledge Cincinnati Testing Laboratories, Inc. for conducting the tensile testson the IM7/977-2 material presented in this report.Trade names or manufacturers' names are used in this report foridentification only. This usage does n_)t constitute an officialendorsement, either expressed or im plied, by the NationalAeronautics and Space AdJ ninistration.NASA Center for Aerospace Information7121 Standard DriveHanover, MD 21076Price Code: A03Available fromNational Technical Information Service5285 Port Royal RoadSpringfield, VA 22100Price Code: A03

This paper describes the concept of state variable approach to model the mechanical response of nickel base superalloys. This approach relies on metallurgical principles to develop equations to model the defor mation of materials subjected to monotonic, creep and fatigue loads. State variables are used to model the effects of microstructural features such as precipitate particles & grain boundaries on the movement of dislocations. The aim of this work was to develop & apply a state variable constitutive model to predict the response of Rene 80, a cast nickel base superalloy. The accuracy of model predictions between 760C & 982C under combined thermal and mechanical loads is demonstrated.

Composite materials are heterogeneous materials on the microscopic level. For short-fiber reinforced composites, the fiber-matrix and fiber-fiber interactions play a significant role in the strength and global stiffness of the material. A unit cell model is developed to study the influence of fiber clustering patterns on the local stress distribution and global composite properties. Design variables are established to account for the variations in fiber clustering patterns, aspect ratio, volume fraction, and fiber packing. Design constraints are used to represent composite moduli, thermal coefficients of expansion, maximum interfacial stress and thermal residual stress. The model is analyzed by the p-version finite element method. The study is limited to elastic analysis. The p-version finite element method offers rapid convergence on localized stress fields using only coarse mesh and allows geometric shape variations without remeshing. Solution accuracy can be checked systematically using a series of solutions with increasing numerical complexity that is generated from a given mesh. A parametric study of the effect of fiber cluster patterns, fiber packing, and fiber aspect ratio is conducted. The study shows that the fiber cluster pattern and fiber packing has a significant impact on all of the composite properties studied.

This work describes the development of a unified nonisothermal constitutive model to predict the thermomechanical fatigue (TMF) response of a Nickel base superalloy, Rene 80. Nonisothermal deformation mechanisms are modeled using state variables. The flow equation of the Ramaswamy-Stouffer model was rewritten in the form of an Arrhenius equation with explicit temperature dependence. The isothermal predictions were correlated with the test data at four test temperatures between 538-degrees-C and 982-degrees-C. Material parameters were verified using nonisothermal tensile calculations. This verification showed that modeling the transition between planar slip and dislocation climb accurately is crucial for obtaining reliable TMF predictions. The revised constitutive model could successfully predict Rene 80 response from several TMF tests between 760-degrees-C and 982-degrees-C.

The unified constitutive equations for Rene´ 80 developed by Bhattachar and Stouffer (1992) are used to predict the thermomechanical fatigue (TMF) response of a Nickel base superalloy Rene´ 80 between 649°C and 1093°C. Predictions using these equations suggest that temperature history effects are significant during TMF, and that the TMF response of Rene´ 80 cannot be predicted completely using only isothermal parameters. It is postulated without metallurgical observations that the two deformation mechanisms in Rene´ 80, planar slip at low temperatures and dislocation climb at high temperatures, produce characteristic microstructures which interact under nonisothermal conditions to produce extra hardening that is not present during isothermal deformation. A state variable approach has been used to model this interaction. The nonisothermal model with temperature history effects could successfully predict the initial and saturated TMF response, and block isothermal response of Rene´ 80 from several tests between 649°C and 1093°C.

A unified constitutive model incorporating internal state variables based upon the deformation phenomena that are observed to occur at the microstructural level has been developed and applied to Rene´ 95. Material hardening is modeled using dragstress and back-stress state variables, while the reduction in the material’s load-carrying capability is described by using a damage-accumulation state variable. Application of the model to the tensile, cyclic, and creep loadings of Rene´ 95 at 650°C demonstrated that the model is capable of capturing cyclic hardening, damage accumulation, and tertiary creep by using one inelastic flow equation in concert with the state-variable-evolution equations.

This paper presents an extension of the Ramaswamy, Stouffer, and Laflen unified elastic–viscoplastic theory which uses internal state variables to represent a strain rate insensitive aluminium alloy namely 7050-T7451 alloy. The model constants are evaluated from the results of a uniaxial tensile test, with strain hold at saturation, and a fatigue loop. Strain holds in the saturated region of tensile monotonic curves resulted in significant amounts of stress relaxation. The material response is cyclically stable and reveals a strong Bauschinger effect. There is a significant reduction in the yield stress between the initial yield and the subsequent tensile yield stress observed after a fully reversed fatigue cycle. All of these material characteristics were predicted successfully.

A mathematical model was developed for an implantable force transducer to be inserted within the midsubstance of a ligament or tendon. The model was generated by performing both equilibrium and strain-displacement analyses on a metallic, curved beam structure placed within a parallel-fibered tissue. The analysis permitted the transverse pressure acting between the device and fibers to be calculated along with peak device strain and sensitivity (ratio of strain output to axial tissue force). Transducer pressure and transducer strain were expressed in terms of nondimensionalized design factors. A parametric analysis of the key design factors was then performed. The transverse pressure was shown to vary little for large changes in these factors whereas device strain changed markedly. The analysis was verified by a bench test on an example device. Such a model permits a proposed design to be evaluated without having to conduct costly experiments.

Local surface strains in bone-fascicle-bone subunits from human patellar tendon and anterior and posterior cruciate ligaments were measured between markers using low-speed photography during low rate subfailure testing. A simple stress-strain relationship of the power form was found to describe the bone-to-bone responses up to four percent strain for all three tissue types examined. The regional material behavior were best fit using an inverted strain-stress relationship, however. The power model, fitted to the experimental data, conformed to the expected stress-strain relationship better than either the quadratic or cubic models. With few exceptions, for a given stress, the strains near the proximal and distal bone ends were not significantly different from each other, but were significantly higher than the strains in the tissue midregions. Local strain patterns generally varied among subunits from the same tissue.

The objective of this research is to develop a constitutive equation for the uniaxial monotonic and cyclic response of Rene’80 between the temperatures of 538C and 982C. The constitutive equation is accompanied by experimental data for the evaluation of the material constants. Extensive verification is achieved through the successful correlation of tensile and creep response and prediction of mechanical cycling experiments including mean stress shifts. These results also serve as a starting point for reformulating the model for the prediction of the high temperature multiaxial response of Rene’80 that includes torsion, proportional, and nonproportional uniaxial and torsional loading histories.

An anisotropic constitutive model based on crystallographic slip theory was formulated for nickel-base single crystal superalloys. The current equations include both drag stress and back stress state variables to model the local inelastic flow. Specially designed experiments have been conducted to evaluate the existence of back stress in single crystals. The results showed that the back stress effect of reverse inelastic flow on the unloading stress is orientation dependent, and a back stress state variable in the inelastic flow equation is necessary for predicting anelastic behavior. Model correlations and predictions of experimental data are presented for the single crystal supperalloy Rene´ N4 at 982°C.

This paper contains an extension of the uniaxial state variable constitutive model of Ramaswamy et al. (1988) to the case of multiaxial loading. The correlation between uniaxial and multiaxial loading conditions is achieved through the assumptions of material isotropy and conservation of inelastic volume. The multiaxial extension is based only on the material parameters evaluated from uniaxial loading. The research is accompanied by a multiaxial experimental program to evaluate the response of Rene’ 80 at 871°C and 982° C. Experiments in the program include torsion, proportional axial and torsion, and nonproportional loading. It was shown experimentally that there is no extra hardening from the multiaxial loading than results from uniaxial loading. Further, it is shown that the multiaxial model is successful in predicting the experimental results using only the parameters determined from the uniaxial experiments.

An anisotropic constitutive model based on crystallographic slip theory was formulated for nickel-base single crystal superalloys. The current equations include both drag stress and back stress state variables in the inelastic flow equation for slip in each slip system. Experiments were conducted to evaluate the need of back stress in the model. The test results showed the effect of reverse inelastic flow on unloading is orientation dependent, and that a back stress state variable in the inelastic flow equation is necessary for predicting the anelastic behavior. Model correlations and predictions with experimental data are presented for the single crystal superalloy Rene’ N4 at 760C and 982C.

This paper contains a review of the experimental and analytical results achieved in the development of a unified state variable constitutive equation for Rene'80 in the temperature range from 538C to 982C. The constitutive equation is based on back stress and drag stress to model strain hardening and cyclic softening, respectively. The model parameters were determined from uniaxial experiments at 538C, 760C, 871C and 982C, and could correlate both the rate dependent and rate independent tensile response of Rene'80. The model could also predict strain rate dependence and mean stress shifts in fatigue. The same material constants were used to successfully predict the torsion, proportional and non-proportional tension/torsion fatigue response at 871C and 982C. The numerical procedure includes a method to predict any fatigue loop in the cyclic history without integrating through the entire fatigue history. This produced a significant speedup in the numerical calculations.