An Experimental and Numerical Study on the Influence of Filling Materials on Double-Crack Propagation

Abstract

Filling materials such as clay or sand widely exist in natural rock joints and work as weak bonds between the joint surfaces. The fillings affect rock deformation and failure behavior, and show different influences in terms of single crack or multiple cracks. While most of the literature have focused on unfilled cracks in brittle materials, this study aims to investigate various filling materials on the crack behavior, e.g., initiation, secondary cracks and peak strength. In this paper, the crack propagation in rock-like specimens with double-filled and unfilled cracks are investigated experimentally and numerically. Uniaxial compression tests were conducted and the experimental observations indicate that the peak stress and first crack initiation stress of the specimens vary with different geometries and different filling materials, while the crack initiation location and the pattern of crack coalescence show similar behavior between filled and unfilled cracks. In parallel to the experimental tests, numerical simulations were carried out using a modified phase field model (PFM) to complement the experiments and provide a new perspective. The PFM is found to produce consistent stress–strain curve, strength, and crack patterns with those observed in the experimental tests for both unfilled and filled cracks.

Appendix: Finite Element Discretization

Introducing virtual functions, we convert the governing equations of strong form for the phase field modeling to the weak form as

$$\begin{aligned} \int _{\Omega }\left( -{\varvec{\sigma }}:\delta {\varvec{\varepsilon }}\right) \mathrm {d}\Omega +\int _{\Omega }{\varvec{b}} \cdot \delta {\varvec{u}} \mathrm {d}\Omega +\int _{\Omega _{t}}{\varvec{t}} \cdot \delta {\varvec{u}} \mathrm {d}S=0, \end{aligned}$$

(19)

and

$$\begin{aligned} \int _{\Omega }-2(1-k)H_\mathrm{u}(1-\phi )\delta \phi \mathrm {d}\Omega +\int _{\Omega } G_\mathrm{c}\left( l_0\nabla \phi \cdot \nabla \delta \phi +\frac{1}{l_0}\phi \delta \phi \right) \mathrm {d}\Omega =0. \end{aligned}$$

(20)

Meanwhile, the standard vector-matrix notation is applied in the finite element discretization; therefore, the displacement field and phase field are approximated as

$$\begin{aligned} {\varvec{u}} = {\varvec{N}}_u {\varvec{d}},\;\; \phi = {\varvec{N}}_{\phi } \hat{{\varvec{\phi }}}, \end{aligned}$$

(21)

where ${\varvec{d}}$ and $\hat{{\varvec{\phi }}}$ are the vectors composed of the node values of the displacement field and phase field. ${\varvec{N}}_\mathrm{u}$ and ${\varvec{N}}_{\phi }$ are shape function matrices and in 2D they are defined as

$$\begin{aligned} {\varvec{N}}_\mathrm{u} = \left[ \begin{array}{ccccc} N_{1}&{}0&{}\dots &{}N_{n}&{}0\\ 0&{}N_{1}&{}\dots &{}0&{}N_{n} \end{array}\right] , \;\; {\varvec{N}}_\phi = \left[ \begin{array}{cccc} N_{1}&N_{2}&\dots&N_{n}, \end{array}\right] \end{aligned}$$

(22)

where n denotes the node number in an element and $N_i$ is the shape function related to node i. The test functions are approximated as

$$\begin{aligned} \delta {\varvec{u}} = {\varvec{N}}_u \delta {\varvec{d}},\;\; \delta \phi = {\varvec{N}}_{\phi } \delta \hat{{\varvec{\phi }}}, \end{aligned}$$

(23)

where $\delta {\varvec{d}}$ and $\delta \hat{{\varvec{\phi }}}$ compose the node values of the test functions.

Afterwards, the gradients of the unknowns and test functions are defined as

$$\begin{aligned} {\varvec{\varepsilon }} = {\varvec{B}}_\mathrm{u} {\varvec{d}},\;\; \nabla \phi = {\varvec{B}}_\phi \hat{{\varvec{\phi }}}, \;\;{\varvec{\delta }} \varepsilon = {\varvec{B}}_\mathrm{u} \delta {\varvec{d}},\;\; \nabla \phi = {\varvec{B}}_\phi \delta \hat{{\varvec{\phi }}}, \end{aligned}$$

(24)

where ${\varvec{B}}_u$ and ${\varvec{B}}_\phi$ are defined based on the derivatives of the shape function:

$$\begin{aligned} {\varvec{B}}_\mathrm{u}=\left[ \begin{array}{ccccc} N_{1,x}&{}0&{}\dots &{}N_{n,x}&{}0\\ 0&{}N_{1,y}&{}\dots &{}0&{}N_{n,y}\\ N_{1,y}&{}N_{1,x}&{}\dots &{}N_{n,y}&{}N_{n,x} \end{array}\right] ,\; {\varvec{B}}_\phi =\left[ \begin{array}{cccc} N_{1,x}&{}N_{2,x}&{}\dots &{}N_{n,x}\\ N_{1,y}&{}N_{2,y}&{}\dots &{}N_{n,y}. \end{array}\right] \end{aligned}$$

(25)

The equations of weak form (19) and (20) are then converted to

$$\begin{aligned}&-(\delta {\varvec{d}})^\mathrm {T} \left[ \int _{\Omega } {\varvec{B}}_\mathrm{u}^\mathrm {T} {\varvec{D}}_\mathrm{e} {\varvec{B}}_\mathrm{u} \mathrm {d}\Omega {\varvec{d}} \right] + (\delta {\varvec{d}})^\mathrm {T} \left[ \int _{\Omega }{\varvec{N}}_u^\mathrm {T}{\varvec{b}} \mathrm {d}\Omega +\int _{\Omega _{h}} {\varvec{N}}_u^\mathrm {T} {\varvec{t}}\mathrm {d}S \right] =0, \end{aligned}$$

(26)

$$\begin{aligned}&-(\delta \hat{{\varvec{\phi }}})^{\mathrm {T}} \int _{\Omega }\left\{ {\varvec{B}}_\phi ^{\mathrm {T}} G_\mathrm{c} l_0 {\varvec{B}}_\phi +{\varvec{N}}_\phi ^{\mathrm {T}} \left[ \frac{G_\mathrm{c}}{l_0} + 2(1-k)H_u \right] {\varvec{N}}_\phi \right\} \mathrm {d}\Omega \hat{{\varvec{\phi }}}+ (\delta \hat{{\varvec{\phi }}})^\mathrm {T} \int _{\Omega }2(1-k)H_u{\varvec{N}}_\phi ^{\mathrm {T}} \mathrm {d}\Omega = 0. \end{aligned}$$

(27)

For arbitrary admissible test functions, one always has Eqs. (26) and (27), which leads to the discrete equations of weak form as follows:

$$\begin{aligned}&-\underbrace{\int _{\Omega } {\varvec{B}}_\mathrm{u}^\mathrm {T} {\varvec{D}}_\mathrm{e} {\varvec{B}}_\mathrm{u} \mathrm {d}\Omega {\varvec{d}}}_{{\varvec{F}}_\mathrm{u}^{int}={\varvec{K}}_\mathrm{u} {\varvec{d}}} + \underbrace{\int _{\Omega }{\varvec{N}}_\mathrm{u}^\mathrm {T}{\varvec{b}} \mathrm {d}\Omega +\int _{\Omega _\mathrm{t}} {\varvec{N}}_u^\mathrm {T} {\varvec{t}}\mathrm {d}S}_{{\varvec{F}}_\mathrm{u}^{ext}}=0, \end{aligned}$$

(28)

$$\begin{aligned}&-\underbrace{ \int _{\Omega }\left\{ {\varvec{B}}_\phi ^{\mathrm {T}} G_\mathrm{c} l_0 {\varvec{B}}_\phi +{\varvec{N}}_\phi ^{\mathrm {T}} \left[ \frac{G_\mathrm{c}}{l_0} + 2(1-k)H_\mathrm{u} \right] {\varvec{N}}_\phi \right\} \mathrm {d}\Omega \hat{{\varvec{\phi }}}}_{{\varvec{F}}_\phi ^{int}={\varvec{K}}_\phi \hat{{\varvec{\phi }}}}+ \underbrace{\int _{\Omega }2(1-k)H_\mathrm{u}{\varvec{N}}_\phi ^{\mathrm {T}} \mathrm {d}\Omega }_{{\varvec{F}}_\phi ^{ext}} = 0, \end{aligned}$$

(29)

where ${\varvec{F}}_\mathrm{u}^\mathrm{int}$ and ${\varvec{F}}_\mathrm{u}^{ext}$ are the internal and external forces for the displacement field; ${\varvec{F}}_\phi ^\mathrm{int}$ and ${\varvec{F}}_\phi ^\mathrm{ext}$ are the internal and external force terms for the phase field (Zhou et al. 2018b). In addition, the stiffness matrices for the displacement and phase fields are obtained through

$$\begin{aligned} \left\{ \begin{aligned}{\varvec{K}}_\mathrm{u}&= \int _{\Omega } {\varvec{B}}_\mathrm{u}^\mathrm {T} {\varvec{D}}_\mathrm{e} {\varvec{B}}_\mathrm{u} \mathrm {d}\Omega \\ {\varvec{K}}_\phi&= \int _{\Omega }\left\{ {\varvec{B}}_\phi ^{\mathrm {T}} G_\mathrm{c} l_0 {\varvec{B}}_\phi +{\varvec{N}}_\phi ^{\mathrm {T}} \left[ \frac{G_\mathrm{c}}{l_0} + 2(1-k)H_u \right] {\varvec{N}}_\phi \right\} \mathrm {d}\Omega , \end{aligned}\right. \end{aligned}$$

(30)

where ${\varvec{D}}_\mathrm{e}$ is the degraded elasticity matrix according to the degradation function.