A Microstructure-Based Model to Characterize Micromechanical Parameters Controlling Compressive and Tensile Failure in Crystallized Rock

Abstract

A discrete element model is proposed to examine rock strength and failure. The model is implemented by UDEC which is developed for this purpose. The material is represented as a collection of irregular-sized deformable particles interacting at their cohesive boundaries. The interface between two adjacent particles is viewed as a flexible contact whose stress–displacement law is assumed to control the material fracture and fragmentation process. To reproduce rock anisotropy, an innovative orthotropic cohesive law is developed for contact which allows the interfacial shear and tensile behaviours to be different from each other. The model is applied to a crystallized igneous rock and the individual and interactional effects of the microstructural parameters on the material compressive and tensile failure response are examined. A new methodical calibration process is also established. It is shown that the model successfully reproduces the rock mechanical behaviour quantitatively and qualitatively. Ultimately, the model is used to understand how and under what circumstances micro-tensile and micro-shear cracking mechanisms control the material failure at different loading paths.

Appendices

Appendix 1

1.1 An estimation for thickness of fracture process zone

A material cracks when sufficient stress and energy are applied to break the inter-molecular bonds. These bonds hold the molecules together and their strength is supplied by the attractive forces between the molecules. Many equations have been proposed to formulate this force and its potential energy. The Lennard-Jones’ potential (Griebel et al. 2007) is a simple and extensively used function:

$$\Uppi \left( z \right) = a\Upomega \left[ {\left( {\frac{\zeta }{z}} \right)^{n} - \left( {\frac{\zeta }{z}} \right)^{m} } \right] $$

(20)

where z denotes the separation distance between two adjacent molecules, and

$$a = \frac{1}{n - m}\left( {\frac{{n^{n} }}{{m^{m} }}} \right)^{{\frac{1}{n - m}}} $$

(21)

The depth of the potential, Ω, describes the energy needed to break the bond and thereby the strength of the molecular force (Fig. 11). It is called bond energy. The value ζ parameterizes the zero crossing of the potential; the integers m and n depend on the material molecular nature and are more commonly among 6–16.

Fig. 11

Lennard-Jones’ potential for real and homogenized material (left and middle), and homogenized inter-molecular force (right)

On close inspection, all real materials show a multitude of heterogeneities even if they macroscopically appear to be homogeneous. These deviations from homogeneity may exist in the form of cracks, voids, particles or regions of a foreign material, layers or fibres in a laminate, grain boundaries or irregularities in a crystal lattice. Heterogeneities of any kind can locally act as stress concentrators and thereby lead to the formation and coalescence of micro-cracks or voids as a source of progressive material damage. To take these microstructural defects into account, a homogenization approach is adopted by assuming the process zone as the representative volume element across which the fine-scale heterogeneous microstructure is “smeared out” and the material is described as homogeneous with spatially constant effective properties. The latter then accounts for the microstructure in an averaged sense. They, for our purpose, include the bound energy and zero crossing of the potential. As illustrated in Fig. 11, the effective potential of the homogenized process zone is thus formulized by

$$\Uppi^{ * } \left( z \right) = a\Upomega^{ * } \left[ {\left( {\frac{{\zeta^{ * } }}{z}} \right)^{n} - \left( {\frac{{\zeta^{ * } }}{z}} \right)^{m} } \right] $$

(22)

where the parameters superscripted by * denote the effective ones.

As the potential derivative with respect to z, the homogenized inter-molecular force $P^{ * } \left( z \right)$ is written as

$$P^{ * } \left( z \right) = \frac{{\partial \Uppi^{ * } }}{\partial z} = \frac{{a\Upomega^{ * } }}{z}\left[ { - n\left( {\frac{{\zeta^{ * } }}{z}} \right)^{n} + m\left( {\frac{{\zeta^{ * } }}{z}} \right)^{m} } \right] $$

(23)

The peak value of the homogenized inter-molecular force, which is called effective cohesive force, $P_{c}^{ * }$, takes place at $z_{m}^{ * }$ as shown in Fig. 11. Note that $P_{\text{c}}^{ * }$ is significantly smaller than the actual peak molecular force in the physical material as it includes an average effect of the entire material micro-defects. Solving the derivative of $P^{ * } \left( z \right)$ for z,

$$z_{m}^{ * } = \zeta^{ * } \left[ {\frac{{n\left( {n + 1} \right)}}{{m\left( {m + 1} \right)}}} \right]^{{\frac{1}{n - m}}} $$

(24)

Substituting $z_{m}^{ * }$ into Eq. 23 leads to

$$P_{\text{c}}^{ * } = a\frac{{\Upomega^{ * } }}{{\zeta^{ * } }}\left[ { - n\left( {\frac{{n\left( {n + 1} \right)}}{{m\left( {m + 1} \right)}}} \right)^{{\frac{ - n - 1}{n - m}}}\,+\,m\left( {\frac{{n\left( {n + 1} \right)}}{{m\left( {m + 1} \right)}}} \right)^{{\frac{ - m - 1}{n - m}}} } \right] $$

(25)

The homogenized equilibrium spacing between two molecules ($z_{0}^{ * }$) occurs when the potential energy is at a minimum or the force is zero. Solving Eq. 23 for z provides

$$z_{0}^{ * } = \zeta^{ * } \left[ \frac{n}{m} \right]^{{\frac{1}{n - m}}} $$

(26)

A tensile force is required to increase the separation distance from the homogenized equilibrium value. If this force exceeds the effective cohesive force, the bond is completely severed. The homogenized material then cracks and stress in a width equal to $z_{0}^{ * }$ is released. This means that $z_{0}^{ * }$ (which is significantly larger than real molecular equilibrium spacing) represents the homogenized process zone thickness (w), i.e. $w = z_{0}^{ * }$.

When a bond breaks, a quantity of energy equal to Ω* is dissipated. The accumulation of these energies over the process zone surface supplies the energy dissipation through fracturing. Therefore, the Griffith’s fracture energy, G _f, defined as the rate of energy release per unit cracked area, is expressed as

$$G_{\text{f}} = \frac{{\Upomega^{*} }}{{z_{0}^{*2} }} $$

(27)

Substituting Ω* obtained from Eq. 25 into the above relation yields

$$G_{f} = \frac{{\zeta^{ * } P_{c}^{ * } }}{{az_{0}^{ * 2} }}\left[ { - n\left( {\frac{{n\left( {n + 1} \right)}}{{m\left( {m + 1} \right)}}} \right)^{{\frac{ - n - 1}{n - m}}} + m\left( {\frac{{n\left( {n + 1} \right)}}{{m\left( {m + 1} \right)}}} \right)^{{\frac{ - m - 1}{n - m}}} } \right]^{ - 1} $$

(28)

On the other hand, the effective tensile strength of the homogenized material which represents the actual tensile strength of the material is estimated by

$$\sigma_{\text{t}} = \frac{{P_{\text{c}}^{ * } }}{{z_{0}^{ * 2} }} $$

(29)

Substituting $P_{\text{c}}^{ * }$ from Eq. 28 and $z_{0}^{ * }$ from Eq. 26 into Eq. 29 and solving it for $z_{0}^{ * }$, which actually represents the homogenized fracture process zone thickness, w, yields

$$w = \frac{1}{\beta }\frac{{G_{\text{f}} }}{{\sigma_{\text{t}} }} $$

(30)

where

$$\beta = \frac{{\frac{1}{m} - \frac{1}{n}}}{{\left( {\frac{m + 1}{n + 1}} \right)^{{\frac{m + 1}{n - m}}} - \left( {\frac{m + 1}{n + 1}} \right)^{{\frac{n + 1}{n - m}}} }} $$

(31)

depends on the integers m and n. Table 9 shows that β is relatively constant at 0.25 for common values of m ∈ [8,12] and n ∈ [13, 18].

Table 9 Values of β for common values of m and n

In mixed-mode fracturing,

$$G_{\text{f}} = \frac{{K_{\text{IC}}^{2} }}{{\tilde{E}}} + \frac{{K_{\text{IIC}}^{2} }}{{\tilde{E}}} $$

(32)

where $\tilde{E} = E$ for plane-stress, and $\tilde{E} = {E \mathord{\left/ {\vphantom {E {(1 - \nu^{2} )}}} \right. \kern-0pt} {(1 - \nu^{2} )}}$ for plane-strain.

Given β = 0.25, for fracturing under pure tension,

$$w = \frac{{4K_{\text{IC}}^{ 2} }}{{\tilde{E}\sigma_{\text{t}} }} $$

(33)

and under pure sliding

$$w = \frac{{4K_{\text{IIC}}^{ 2} }}{{\tilde{E}\sigma_{\text{t}} }} $$

(34)

Appendix 2

2.1 Relation between micro- and macro-tensile strength

Figure 12 presents a cutout of a representative collection of isotropic linear elastic triangles where a finite number of contacts are already broken to form a cracked surface. Assuming that the boundary is sufficiently far from the crack, LEFM suggests that for a through-thickness crack of half-width a the induced stress, σ _n, acting on the crack plane near the crack tip, i.e. r << a, is (Anderson 1995)

$$\sigma_{\text{n}} = \sigma_{\text{f}} \sqrt {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a { 2r}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${ 2r}$}}} $$

(35)

where σ _f denotes the remote tensile stress acting normal to the crack and r is the distance from the crack tip.

Fig. 12

Crack representation within a particle assemblage

The concentrated force F _c acting over the first contact right adjacent the crack tip is given by

$$F_{\text{c}} = \int_{0}^{{\frac{{d_{p} }}{2}}} {\sigma_{n} {\text{d}}r} \, = \,\,\sigma_{\text{f}} \sqrt {{\text{ad}}_{\text{p}} } $$

(36)

As soon as F _c exceeds the assumed strength of the contact, it breaks and the crack expands. Since the mode-I stress intensity factor for the system is defined as

$$K_{\text{I}} = \sigma_{\text{f}} \sqrt {\pi \,a} $$

(37)

Equation 36 can be re-written as

$$F_{\text{c}} = K_{\text{I}} \sqrt {{\raise0.7ex\hbox{${d_{\text{p}} }$} \!\mathord{\left/ {\vphantom {{d_{\text{p}} } \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} $$

(38)

Therefore, the tensile stress created at the first contact point is evaluated by

$$\sigma_{\text{c}} = \frac{{F_{\text{c}} }}{{{{d_{\text{p}} } \mathord{\left/ {\vphantom {{d_{\text{p}} } 2}} \right. \kern-0pt} 2}}} = \frac{{2K_{\text{I}} }}{{d_{\text{p}} }}\sqrt {{\raise0.7ex\hbox{${d_{\text{p}} }$} \!\mathord{\left/ {\vphantom {{d_{\text{p}} } \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} $$

(39)

For a regularly packed assemblage loaded along packing direction, at the incipient failure or crack extension, σ _c = t _c and K _I = K _IC. Substituting them into Eq. 39 yields

$$t_{\text{c}} = \frac{{2K_{\text{IC}} }}{{d_{\text{p}} }}\sqrt {{\raise0.7ex\hbox{${d_{\text{p}} }$} \!\mathord{\left/ {\vphantom {{d_{\text{p}} } \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} $$

(40)

Equation 40 indicates that material fracture toughness is in fact the macroscopic representation of the contact tensile strength and the adopted particle size. This result is anticipated as the concept of fracture toughness implies an internal length scale whereby the ratio of fracture toughness to material strength has the dimension of square root of length. The particle size supplies this internal length scale in the modelling.

The same approach can be followed to express K _IIC in terms of contact cohesion, c _c. Ultimately,

$$K_{\text{IC}} \propto t_{\text{c}} \sqrt {d_{\text{p}} } $$

(41)

$$K_{\text{IIC}} \propto c_{\text{c}} \sqrt {d_{\text{p}} } $$

(42)

It is generally accepted that K _IC and σ _t, measured by the Brazilian testing, are related together in a vast range igneous, metamorphic and sedimentary rocks (e.g. Zhang 2002; Gunsallus and Kulhawy 1984; Harison 1994). Zhang (2002) suggested the following empirical relation which provides good estimations with a coefficient of determination r ² = 0.94.

$$\sigma_{\text{t}} = 6.88K_{\text{IC}} $$

(43)

where the parameters involved are in the SI units. Substituting Eq. 43 in 41 leads to

$$\sigma _{{\text{t}}}\propto t_{{\text{c}}} \sqrt {d_{{\text{p}}} }.$$

(44)