Research Papers

Identification of Nonlinear Viscoelastic Models of Flexible Polyurethane Foam From Uniaxial Compression Data

[+] Author and Article Information
Yousof Azizi

Ray W. Herrick Laboratories,
School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: yazizi@purdue.edu

Patricia Davies, Anil K. Bajaj

Ray W. Herrick Laboratories,
School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received September 3, 2013; final manuscript received October 26, 2015; published online February 5, 2016. Assoc. Editor: Harley Johnson.

J. Eng. Mater. Technol 138(2), 021008 (Feb 05, 2016) (13 pages) Paper No: MATS-13-1162; doi: 10.1115/1.4032169 History: Received September 03, 2013; Revised October 26, 2015

Flexible polyethylene foam is used in many engineering applications. It exhibits nonlinear and viscoelastic behavior which makes it difficult to model. To date, several models have been developed to characterize the complex behavior of foams. These attempts include the computationally intensive microstructural models to continuum models that capture the macroscale behavior of the foam materials. In this research, a nonlinear viscoelastic model, which is an extension to previously developed models, is proposed and its ability to capture foam response in uniaxial compression is investigated. It is hypothesized that total stress can be decomposed into the sum of a nonlinear elastic component, modeled by a higher-order polynomial, and a nonlinear hereditary type viscoelastic component. System identification procedures were developed to estimate the model parameters using uniaxial cyclic compression data from experiments conducted at six different rates. The estimated model parameters for individual tests were used to develop a model with parameters that are a function of strain rates. The parameter estimation technique was modified to also develop a comprehensive model which captures the uniaxial behavior of all six tests. The performance of this model was compared to that of other nonlinear viscoelastic models.

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Fig. 4

An illustration of linearizing of the initial response of the compression test data. (a) Red straight line is tangent to the linear section of stress–strain curve observed shortly after the contact is initiated; t0 is the intersection of the red line and zero stress. (b) Adjusted data.

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Fig. 3

Response of the foam block during the initial small strain part of the compression test

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Fig. 2

Results of six compression tests of different durations. (a) Stress versus time and (b) the same data plotted against strain. Light to dark blue indicates decreasing strain rate from 0.0088/s to 0.00053/s.

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Fig. 1

The experimental setup for compression tests on a cubic 76.2 mm × 76.2 mm × 76.2 mm foam sample and the displacement path profile of the top plate during the compression tests. The top plate looses contact at time t = T – Ψ with the top of the foam. The downward direction is taken to be the positive direction.

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Fig. 5

Simulated 150 s compression test (strain rate equals 88 × 10−4 s−1) using a model with N = 10 and M1 = M2 = 1. Blue: experiment; dashed red: simulation. Note that two plots are on top of each other.

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Fig. 6

Estimated parameters normalized by their true values for 100 noise realizations; x-axis represents the 14 parameters in the order a, α, b, β, k1,⋯, and k10: (a) simulated 150 s compression test (strain rate equals 88 × 10−4 s−1); (b) simulated 290 s compression test (strain rate equals 45 × 10−4 s−1); (c) simulated 631 s compression test (strain rate equals 21 × 10−4 s−1); (d) simulated 1233 s compression test (strain rate equals 11 × 10−4 s−1); (e) simulated 1233 s compression test (strain rate equals 80 × 10−5 s−1); and (f) simulated 1650 s compression test (strain rate equals 53 × 10−5 s−1). The cross represents mean value of the estimated parameters and the bars indicate the standard deviation of the estimates.

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Fig. 16

Results of simultaneously fitting model 4 to all six data sets for another foam sample (foam D). R2: 0.984, 0.991, 0.996, 0.988, 0.995, and 0.993, respectively. N = 10, M1 = 12, and M2 = 9. Solid blue and dashed red curves represent experimental data and estimation, respectively. (a) T1, (b) T2, (c) T3, (d) T4, (e) T5, and (f) T6 data.

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Fig. 7

Response prediction resulting from fitting model 4 to data from tests: (a) T1, (b) T2, (c) T3, (d) T4, (e) T5, and (f) T6. Each model is assumed to have N = 10 and M1 = M2 = 1. Solid blue: experimental data; dashed red: predicted response; black (square): elastic component σe; purple (triangle): first viscoelastic component σv1; and green (star): second viscoelastic component σv2. Solid blue and dashed red graphs are very close to each other.

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Fig. 8

Estimated parameters for a typical data set: ((a) and (b)) first viscoelastic parameters (a1 and α1) convergence during internal loop and ((c) and (d)) second viscoelastic parameters (b1 and β1) convergence during outer loop

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Fig. 12

Fitting simple functions to the estimated viscoelastic parameters as functions of input strain rates: (a) a(ε˙), (b) b(ε˙)−1, (c) α(ε˙), and (d) β(ε˙). Red crosses represent the estimated parameters. Black line is the fitted functions.

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Fig. 13

Estimated elastic stress σe for T1,⋯, T4 (N = 10, M1 = 1, and M2 = 1). Light to dark blue curves indicate increasing strain rate from 0.0011 s−1 to 0.0088 s−1.

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Fig. 14

Effect of variations of different parameters on the predicted responses. (a) Low-pass filter cutoff frequency varies between 2 Hz and 20 Hz while N = 10, M1 = 11, and M2 = 9. (b) M1 varies between 5 and 31 while M2 = 9, N = 10, and Fc = 5 Hz, (c) M2 varies between 5 and 23 while M1 = 11, N = 10, and Fc = 5 Hz. Square, circle, triangle, plus, cross, and star signs correspond to T1 to T6 tests, respectively. Light to dark blue also indicates decreasing strain rate from 0.0088/s to 0.00053/s (T1 to T6, respectively).

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Fig. 15

Results of simultaneously fitting model 4 to all six data sets (foam A). R2: 0.988, 0.988, 0.996, 0.996, 0.995, and 0.996, respectively. N = 10, M1 = 11, and M2 = 9. Solid blue and dashed red curves represent experimental data and estimation, respectively. (a) T1, (b) T2, (c) T3, (d) T4, (e) T5, and (f) T6 data.

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Fig. 10

Response predictions resulting from model 4 to data captured in T6 with N = 10, M1 = 1, and M2 = 5. (a) Overall response, (b) low strain region (start of test), and (c) high stress region. Solid blue: T6 test (slowest test), dashed red: predicted response.

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Fig. 9

Stress versus time results for (a) low strain region and (b) high stress region. Solid blue: T6 test (slowest test) and dashed red: predicted response for model 4 with M1 = M2 = 1 and N = 10.

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Fig. 11

Predictions of the response resulting from fitting model 1, model 2, model 3, and model 4 to T1 test data. Experimental data (blue) and model 1 (red/star), model 2 (black/triangle), model 3 (brown/square), and model 4 (green/circle) fits (see Table 1 for an explanation of model numbers). (a) complete response and models 2 and 3 estimations; (b) complete response and models 1 and 4 estimations; (c) the response at the start of the test; and (d) estimation errors with different models.




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