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The Combined Effects of Moisture and Temperature on the Mechanical Response of Paper

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Abstract

To model advanced 3-D forming strategies for paper materials, the effects of environmental conditions on the mechanical behavior must be quantitatively and qualitatively understood. A tensile test method has been created, verified, and implemented to test paper at various moisture content and temperature levels. Testing results for one type of paper for moisture contents from 6.9 to 13.8 percent and temperatures from 23 to 168 degrees Celsius are presented and discussed. Coupled moisture and temperature effects have been discovered for maximum stress. Uncoupled effects have been discovered for elastic modulus, tangent modulus, hardening modulus, strain at break, tensile energy absorption (TEA), and approximate plastic strain. A hyperbolic tangent function is also utilized which captures the entire one-dimensional stress-strain response of paper. The effects of moisture and temperature on the three coefficients in the hyperbolic tangent function may be assumed to be uncoupled, which may simplify the development of moisture- and temperature-dependent constitutive models. All parameters were affected by both moisture and temperature with the exception of TEA, which was found to only be significantly dependent on temperature.

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Acknowledgements

The authors would like to acknowledge and thank BiMaC Innovation and its industrial partners for their financial support of this research. Additionally, the authors would like to thank Per Larsson and the Department of Fibre and Polymer Technology at KTH Royal Institute of Technology for their assistance with preparing the handsheets and Magnus Gimker and Innventia for their assistance with pulp beating.

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Authors and Affiliations

  1. Department of Solid Mechanics, KTH Royal Institute of Technology, SE - 100 44, Stockholm, Sweden

    E. Linvill & S. Östlund

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  1. E. Linvill
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  2. S. Östlund
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Correspondence to E. Linvill.

Appendices

Appendix A: Moisture Loss Analysis

Utilizing the results presented in Fig. 3, the amount of moisture loss at each tested environmental condition can be extrapolated as shown in Table 6. Each value in Table 6 represents the maximum amount of moisture loss at that respective test condition. The values for 23 degrees Celsius at all moisture contents and values for 168 degrees Celsius at 6.9 and 13.9 % moisture content represent maximum moisture losses which are known either from Fig. 3 or from the fact that tests without temperature application should retain constant moisture, while the values were linearly extrapolated from the known results.

Table 6 Maximum changes of moisture content at specimen failure
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The moisture loss during the tests should be greatest for the failure properties, so the effect of the moisture loss on the maximum stress and strain at break results are investigated. The maximum stress results with the adjustments presented in Table 6 are shown in Fig. 15. The results displayed in Fig. 15 show that the coupled effects of moisture and temperature are still present. The trends have not changed drastically compared to Fig. 11.

Fig. 15
figure 15

Dependency of maximum stress on moisture and temperature including moisture loss

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The results of the strain at break considering the adjustments in Table 6 are displayed in Fig. 16. The results in Fig. 16 appear to indicate the same trends as the results without correction for moisture loss in Fig. 12. While the moisture loss has some impact on the results, the trends and appearance of coupling do not appear to be considerably affected by the loss of moisture.

Fig. 16
figure 16

Dependency of strain at break on moisture and temperature including moisture loss

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Determining the actual amount of moisture loss during each test would be difficult and require many assumptions. Therefore, not adjusting for moisture loss is determined to be the least biased method to present the experimental results. The results which are not corrected for moisture loss are assumed to capture the trends and coupling of the effects of moisture and temperature on each parameter.

Appendix B: Measurement Error Analysis

Sources and magnitudes of possible measurement error are displayed in Table 7. Utilizing the first five possible measurement errors (displacement, force, width, thickness, and test length) in Table 7, estimates of the error induced into the numerical results caused by the test measurement system can be analyzed for each load case. The results of these analyses are displayed in Table 8. The maximum effects of the possible measurement error on each parameter as well as the measurements which dominate the errors are presented in Table 8. The errors on the three curve-fitting parameters for the hyperbolic tangent function (C 1, C 2 and C 3) are not presented, because determining the maximum measurement error on those parameters is not trivial. Additionally, the maximum error for those parameters is expected to be approximately the same magnitude as the errors presented in Table 8.

Table 7 Sources and magnitudes of possible measurement error
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Table 8 Maximum possible measurement error on the parameters
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The thickness of paper sheets varies considerably, especially in hand sheets like the ones utilized in this study. This thickness variation causes the greatest measurement error throughout the results. The other two factors which dominate the error magnitude are the specimen length and the strain range utilized in calculating the tangent modulus. The specimen length was limited by the temperature application method, while the strain range was limited by the length of the hardening regions (i.e. the region during which both reversible and irreversible straining occurs) for specimens with a small plastic straining region. Reducing these sources of measurement error would be difficult given the requirements of the temperature application method and given the testing condition at high temperature and low moisture content.

The errors in Table 8 are expected only in worst-case scenarios, so actual error values much less than these are expected. Therefore, the results should not be significantly affected by the possible measurement errors.

Table 8 does not take into consideration the potential moisture loss and the temperature error. The moisture content and temperature changes according to Table 7 would cause shifts along the horizontal axis for the figures presented in the results section of this paper, while the errors in Table 8 would cause shifts along the vertical axis for those figures.

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Linvill, E., Östlund, S. The Combined Effects of Moisture and Temperature on the Mechanical Response of Paper. Exp Mech 54, 1329–1341 (2014). https://doi.org/10.1007/s11340-014-9898-7

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