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A fiber-bundle model for the continuum deformation of brittle material

Abstract

The deformation of brittle material is primarily accompanied by micro-cracking and faulting. However, it has often been found that continuum fluid models, usually based on a non-Newtonian viscosity, are applicable. To explain this rheology, we use a fiber-bundle model, which is a model of damage mechanics. In our analyses, yield stress was introduced. Above this stress, we hypothesize that the fibers begin to fail and a failed fiber is replaced by a new fiber. This replacement is analogous to a micro-crack or an earthquake and its iteration is analogous to stick–slip motion. Below the yield stress, we assume that no fiber failure occurs, and the material behaves elastically. We show that deformation above yield stress under a constant strain rate for a sufficient amount of time can be modeled as an equation similar to that used for non-Newtonian viscous flow. We expand our rheological model to treat viscoelasticity and consider a stress relaxation problem. The solution can be used to understand aftershock temporal decay following an earthquake. Our results provide justification for the use of a non-Newtonian viscous flow to model the continuum deformation of brittle materials.

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Acknowledgements

The author thanks the Editor K. Ravi-Chandar and two anonymous reviewers for constructive comments. A part of this study was conducted under the auspices of the MEXT Program for Promoting the Reform of National Universities.

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Correspondence to K. Z. Nanjo.

Appendix 1

Appendix 1

Derivation of the equation of the rate of energy release in Eq. 15.

Using the stored elastic energy \(e_0 =E_0 \varepsilon _0^2 /2\) and the recovered elastic energy \(e_1 =\bar{\sigma }\varepsilon _0 /2\), we obtain the energy lost in aftershocks \(e_{{\textit{AF}}} =\frac{E_0 \varepsilon _0^2 }{2}-\frac{\bar{\sigma }\varepsilon _0 }{2}\). Substitution of Eq. 13 into this equation gives

$$\begin{aligned} e_{{\textit{AF}}}= & {} \frac{E_0 \varepsilon _0 }{2}\left( {\varepsilon _0 -\varepsilon _y } \right) \nonumber \\&\times \left[ {\frac{1}{\left\{ {1+\left( {n-1} \right) \left( {\varepsilon _0 -\varepsilon _y } \right) ^{n-1}\left( {\frac{t}{\tau _c }} \right) } \right\} ^{1/\left( {n-1} \right) }}} \right] .\nonumber \\ \end{aligned}$$
(16)

Using the total energy of aftershocks \(e_{AFT} =E_0 \varepsilon _0 \left( {\varepsilon _0 -\varepsilon _y } \right) /2\), we rewrite Eq. 16 as

$$\begin{aligned}&e_{{\textit{AF}}} \nonumber \\&\quad =e_{{\textit{AFT}}} \left[ {\frac{1}{\left\{ {1+\left( {n-1} \right) \left( {\varepsilon _0 -\varepsilon _y } \right) ^{n-1}\left( {\frac{t}{\tau _c }} \right) } \right\} ^{1/\left( {n-1} \right) }}} \right] .\nonumber \\ \end{aligned}$$
(17)

Taking the time derivative of Eq. 17, we obtain the rate of energy release

$$\begin{aligned} \frac{1}{e_{{\textit{AFT}}} }\frac{{\textit{de}}_{{\textit{AF}}} }{{\textit{dt}}}=\frac{\frac{1}{\tau _c }\left( {\varepsilon _0 -\varepsilon _y } \right) ^{n-1}}{\left\{ {1+\left( {n-1} \right) \left( {\varepsilon _0 -\varepsilon _y } \right) ^{n-1}\left( {\frac{t}{\tau _c }} \right) } \right\} ^{\frac{n}{n-1}}}.\nonumber \\ \end{aligned}$$
(18)

Using \(c=\frac{\tau _c }{\left( {n-1} \right) \left( {\varepsilon _0 -\varepsilon _y } \right) ^{n-1}}\), we rewrite Eq. 18 as

$$\begin{aligned} \frac{1}{e_{{\textit{AFT}}} }\frac{{\textit{de}}_{{\textit{AF}}} }{{\textit{dt}}}=\frac{\left( {\frac{1}{n-1}} \right) c^{\frac{1}{n-1}}}{\left( {c+t} \right) ^{\frac{n}{n-1}}}. \end{aligned}$$
(19)

If we take \(n=\frac{p}{p-1}\), Eq. 19 is rewritten as

$$\begin{aligned} \frac{1}{e_{{\textit{AFT}}} }\frac{{\textit{de}}_{{\textit{AF}}}}{{\textit{dt}}}=\frac{\left( {p-1} \right) c^{p-1}}{\left( {c+t} \right) ^{p}}. \end{aligned}$$
(20)

This is the equation of the rate of energy release in aftershocks in Eq. 15.

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Nanjo, K.Z. A fiber-bundle model for the continuum deformation of brittle material. Int J Fract 204, 225–237 (2017). https://doi.org/10.1007/s10704-016-0175-x

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Keywords

  • Fracture
  • Brittle deformation
  • Rheology
  • Fiber-bundle
  • Yield stress
  • Viscoelasticity