^{1}, Yoko Nitta

^{2}and Katsuyoshi Nishinari

^{3}

### Abstract

Gellan gel, a typical polysaccharide gel, is ruptured with different deformation behaviors from gelatin gel or rubber. It exhibits both strain hardening and softening; hardening is observed for moderate strain and softening occurs for larger strain. From the analyses of stress–strain curves of gellan gels, we propose forms of strain energy function. The fit with the proposed equation was excellent, while the existing models fail because they consider only one of hardening or softening effect. Furthermore, these equations are shown to be capable of extracting the hardening and softening effects separately from the observed stress–strain curves. By using these fitting equations, the concentration dependences of hardening and softening are investigated. It is shown that the degrees of hardening and softening both increase with increasing gellan concentration.

I. INTRODUCTION

II. EXPERIMENT

III. FITTING EQUATIONS

A. Affine/Phantom Network Model

B. Mooney equation

C. Langevin chain

D. BST equations

E. Nonaffine/double-tube models

F. Slip-tube model

G. Enthalpic elasticity

H. and equations

IV. RESULTS

V. DISCUSSION

VI. SUMMARY AND CONCLUSION

### Key Topics

- Gels
- 55.0
- Elasticity
- 17.0
- Langevin equation
- 17.0
- Fracture mechanics
- 13.0
- Polymers
- 10.0

## Figures

(Color online) Stress–strain curve of a gellan gel with gellan concentration . Diamond symbols show the experimental data. The results of fitting are shown by bold solid (Mooney equation), dashed (Langevin model), dotted (BST-1), and thin solid (BST-2) lines. The results of the Mooney and the BST-2 equations are almost the same.

(Color online) Stress–strain curve of a gellan gel with gellan concentration . Diamond symbols show the experimental data. The results of fitting are shown by bold solid (Mooney equation), dashed (Langevin model), dotted (BST-1), and thin solid (BST-2) lines. The results of the Mooney and the BST-2 equations are almost the same.

(Color online) Stress–strain curve of a gellan gel with gellan concentration . Diamond symbols show the experimental data (the same as in Fig. 1). The results of fitting are shown by bold solid (nonaffine tube model), thin solid (slip-tube model), dotted (double-tube model), and dashed (enthalpic elasticity) lines. The results of the nonaffine tube model and the slip-tube model are almost the same.

(Color online) Stress–strain curve of a gellan gel with gellan concentration . Diamond symbols show the experimental data (the same as in Fig. 1). The results of fitting are shown by bold solid (nonaffine tube model), thin solid (slip-tube model), dotted (double-tube model), and dashed (enthalpic elasticity) lines. The results of the nonaffine tube model and the slip-tube model are almost the same.

(Color online) Result of fitting by the equation. Diamond symbols show the same experimental data as in Figs. 1 and 2. The fitted curve shown by the solid line agrees remarkably well with the experiment.

(Color online) Result of fitting by the equation. Diamond symbols show the same experimental data as in Figs. 1 and 2. The fitted curve shown by the solid line agrees remarkably well with the experiment.

(Color online) Result of fitting by the BC-2 equation. Diamond symbols show the same experimental data as in Figs. 1–3. The fitted curve shown by the solid line agrees remarkably well with the experiment.

(Color online) Result of fitting by the BC-2 equation. Diamond symbols show the same experimental data as in Figs. 1–3. The fitted curve shown by the solid line agrees remarkably well with the experiment.

(Color online) Stress–strain curve of a sucrose-containing gellan gel. The concentrations of gellan and sucrose are and , respectively. The results of fitting are shown by solid (BC-3) and dashed (BC-2) lines. The BC-3 equation reproduces the experiment satisfactorily, whereas the BC-2 equation deviates from the experiment.

(Color online) Stress–strain curve of a sucrose-containing gellan gel. The concentrations of gellan and sucrose are and , respectively. The results of fitting are shown by solid (BC-3) and dashed (BC-2) lines. The BC-3 equation reproduces the experiment satisfactorily, whereas the BC-2 equation deviates from the experiment.

(Color online) Comparison of the contributions of the three terms in the equation. Diamond symbols show the same experimental data as in Figs. 1–4. The solid curve shows the fitted curve by the equation. The dashed curve is calculated with only the first term. The dotted curve is calculated with only the first two terms.

(Color online) Comparison of the contributions of the three terms in the equation. Diamond symbols show the same experimental data as in Figs. 1–4. The solid curve shows the fitted curve by the equation. The dashed curve is calculated with only the first term. The dotted curve is calculated with only the first two terms.

(Color online) Comparison of the contributions of the three terms in the BC-2 equation. Diamond symbols show the same experimental data as in Figs. 1–4. The solid curve shows the fitted curve by the BC-2 equation. The dashed curve is calculated with only the first term. The dotted curve is calculated with only the first two terms.

(Color online) Comparison of the contributions of the three terms in the BC-2 equation. Diamond symbols show the same experimental data as in Figs. 1–4. The solid curve shows the fitted curve by the BC-2 equation. The dashed curve is calculated with only the first term. The dotted curve is calculated with only the first two terms.

(Color online) Concentration dependence of the linear modulus obtained by fitting to the BC-2 equation. The solid straight line shows the result of fitting by the power law.

(Color online) Concentration dependence of the linear modulus obtained by fitting to the BC-2 equation. The solid straight line shows the result of fitting by the power law.

(Color online) Concentration dependence of the coefficient divided by . This quantity describes the degree of strain hardening.

(Color online) Concentration dependence of the coefficient divided by . This quantity describes the degree of strain hardening.

(Color online) Concentration dependence of the coefficient divided by . This quantity describes the degree of strain softening.

(Color online) Concentration dependence of the coefficient divided by . This quantity describes the degree of strain softening.

## Tables

Goodness of the fits for gellan gels. The smaller this value, the better the fit reproduced in the observation. The concentrations of gellan and sucrose in each sample are shown in the first and the second columns, respectively.

Goodness of the fits for gellan gels. The smaller this value, the better the fit reproduced in the observation. The concentrations of gellan and sucrose in each sample are shown in the first and the second columns, respectively.

Fracture stress and strain for each concentration. The concentrations of gellan and sucrose in each sample are shown in the first and the second columns, respectively.

Fracture stress and strain for each concentration. The concentrations of gellan and sucrose in each sample are shown in the first and the second columns, respectively.

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