Experimental Model Validation of Foam Encapsulated Components

Many systems utilize foam encapsulants to mitigate shock and vibration. To accurately model the dynamic response of components encapsulated in foam, one must have a detailed understanding of the encapsulant material and how this material should be modeled. A series of experiments with increased complexity have been conducted to develop parameters for constitutive material models and provide model validation data for component models. This paper presents a model validation approach for analytical models with foam encapsulated components. This approach accounts for the stochastic nature of the foam material and the experimental uncertainty. In addition, the validation approach defines calibration and validation data and when the data can be used with the analytical model. Experimental data are presented with corresponding analytical predictions and assessments of model accuracy are performed using validation tools and metrics. Introduction Developing a linear structural finite element model that is able to accurately predict structural dynamic component response to various inputs requires a detailed understanding of the system and its intended use. Among the various modeling details required to develop a predictive model are: 1) physics 2) model form 3) material parameters 4) boundary conditions and 5) input loads. Incorporating the appropriate physics into a model is important and may significantly influence predictions made by the model. For example, joint models have been shown to be important when attempting to accurately model component responses as the joint causes most of the energy dissipation [1]. Model form is important when developing a finite element model as inaccurate representation or over simplification of system geometries can adversely impact model predictions. Additionally, accurate representation of material physics (e.g. isotropic/orthotropic) and material parameters are essential to a good analytical model. Finally, one must accurately model boundary and loading conditions when attempting to correlate or validate an analytical model with experimental data. An analyst is required to address each of these issues, some in more detail than others, if an adequate or predictive model is desired. It is often difficult to decide when a model is adequate. Many times, experimental data are solely used to tune the parameters of a finite element model without performing validation to determine if the model is "adequate" for a specific use. This practice, although common, does not incorporate a formal validation process. Such a process involves defining metrics of comparison, calibration and validation against the predefined metrics. As part of an ongoing effort to develop adequate finite element models for foam encapsulated components, a formal validation process is used to characterize the predictive capability of these models. The effort to develop and validate models with foam encapsulated components is being accomplished through a series of experiments with increasing complexity. A diagram representing this series is shown in Figure 1. The validation series begins with baseline material/component model validation and is followed by benchmark validation and subsystem validation. The material and component validation effort is accomplished using constitutive (Phase 0) coupon samples and Phase I geometries. Constitutive testing consists of quasi-static tension and compression tests, glass-transition (Tg) tests and Dynamic Mechanical Analysis (DMA) tests. The Phase I or dumbbell geometries are simple foam coupon samples sandwiched between two end-blocks. The Phase I samples have attractive geometries for modal characterization as the foam properties are the only unknowns in this system and the modal responses are very sensitive to the material properties of interest. The Phase II geometry consists of two rigid masses encapsulated in foam material and surrounded by an aluminum case. Finally, the subsystem validation geometry consists of a single rigid mass encapsulated in foam. The subsystem geometry, however, includes three bolted lap joints. A parallel effort to develop the bolted lap joint model is being conducted and both efforts will be combined into the final subsystem mock validation of the Low Fidelity Unit (LFU) geometry. This paper will focus only on the material/component and benchmark validation efforts for the foam. Figure 1. Summary of three levels of validation with increased complexity In the following sections, experimental material characterization, model development, calibration and validation efforts are presented as an example of a formal process used to assess the predictive capability of an analytical foam model. Approach The objective of this effort is to improve the predictive modeling capability of systems using foam encapsulants. This effort seeks to validate a linear viscoelastic foam material for structural dynamics and quantify the variability of the model parameters. The effectiveness of this effort will be assessed by incorporating a detailed model validation process. This process is outlined in Figure 2 and clearly distinguishes between the use of calibration and validation data. Calibration efforts are those that seek "some degree of agreement with complex integrated tests in the required application domain of the code by explicit tuning" [2]. Therefore, experimental calibration data are collected for the purpose of tuning the analytical model. Validation data, on the other hand, are those data that are used specifically in "the process of comparing the results" with the analytical model [2]. Validation data are not used for calibration unless the assessment of the model indicates poor predictive accuracy and it is deemed appropriate to do so after failing to meet the validation criteria (in this case, new validation data would need to be collected to assess the newly calibrated model). Figure 2. Validation process used for development of foam encapsulated component models The development of foam encapsulated component models begins with characterizing the parameters of a linear viscoelastic material model using constitutive and Phase I data. These data are also used to develop a probabilistic model of the material parameters that will be needed in the finite element model. From the probabilistic model, several realizations of material parameters are generated to be used in the model predictions. After the Phase I calibration data are used with the analytical model, the newly calibrated model is exercised, with parameters from the probabilistic model, to calculate model predictions. Additionally, a new set of validation data is collected (preferably with a system that varies in some manner from that used for calibration). Finally, the predictive accuracy of the model is assessed using prescribed validation metrics. These metrics may vary in difficulty and should be chosen in a manner that best critiques the performance of the model for a specified use. If the model validates, the process is repeated at the benchmark level. This paper focuses on the validation process rather than selection of validation metrics. Constitutive Model Development and Calibration Data A linear viscoelastic model is used to represent the foam encapsulation material. To begin constitutive testing, compressive and tensile modulii are estimated from quasi-static tests. Five "replica" samples are used for each test type which required ten total samples. Because of the randomness in the foam coupon samples, the average densities vary from 16.9 to 19.9 lb/ft for a desired 20 lb/ft material. The resulting stress-strain relationships for the compressive and tensile tests are shown in Figures 3 and 4, respectively. The resulting average tensile modulus (38.2 ksi) is larger than the average compressive modulus (32.7 ksi) which is common among cellular solids and is consistent with the mechanics of foam discussion presented in [3]. Figure 3. Compressive modulus data (5 replicas) Figure 4. Tensile modulus data (5 replicas) In addition to the quasi-static compressive and tensile modulus testing, Tg and DMA tests are conducted to characterize the temperature and frequency dependence of the material. Five replicas are used to characterize the glass-transition temperature by imparting a torsional strain (1 Hz strain-rate) on a coupon sample and varying the temperature. The glass-transition temperature is identified by a knee in the stress-temperature curve and the plot in Figure 5 indicates a Tg of approximately 95 ̊C. In addition to the Tg shown in Figure 5, one can see the modulus decreases by approximately two orders of magnitude when the temperature is changes from 80 to 120 ̊C. Additionally, the plots shift up or down systematically according to the test sample density. In Figure 5, the sample from the top most curve has a density of 20.7 lb/ft and the sample from the bottom most curve has a density of 18.2 lb/ft. Figure 5. Stress-temperature curves showing glass-transition temperature The final set of constitutive experiments consist of DMA tests. Like the Tg tests, torsional strain is applied to the foam sample. This time, however, the strain rate is varied while the temperature is held constant and data are collected for a variety of temperature set points. Figure 6 shows a plot of DMA frequency sweeps at several temperature set-points. One begins to observe the frequency dependence of the foam material between 90 and 100 ̊C, which is expected given the Tg is identified in Figure 5 to be 95 ̊C. Figure 6. DMA data showing frequency sweeps at temperature set-points from -30 to 130 ̊C In addition to constitutive data, Phase I data are used to develop and calibrate the models for encapsulated foam components. As mentioned above, the Phase I geometry consists of a foam sample bonded between two endblocks. Both end-blocks for the Phase I calibration samples are made of steel and are shown in Figure 7. Moda