Computational Modeling of Tool-Rock Frictional Contact With Anisotropic Damage

Abstract

The mechanism of frictional contact is investigated between a blunt tool and quasi-brittle rocks with anisotropic damage. A recently developed anisotropic elasto-plastic-damage model is further validated using experimental results under monotonic and cyclic loadings for rocks and concrete. A finite element model of tool-rock frictional contact is validated by analytical results in the two asymptotic regimes of an elastoplastic rock. The mesh sensitivity is reduced using a fracture energy-based method for anisotropic damage. The tool-rock frictional contact is predominantly controlled by three dimensionless parameters: $\eta$, $\xi$, and $\zeta$, which characterize elastoplasticity, brittleness, and anisotropic damage, respectively. The newly introduced damage coefficient $\zeta$ controls the ratio of damage in different directions. As the elastoplastic parameter $\eta$ increases with more plastic deformation, the dimensionless average contact stress ${\widetilde{\Pi }}$ increases and then slightly varies before stabilizing. As the brittleness number $\xi$ increases in a more brittle mode, the contact stress ${\widetilde{\Pi }}$ generally decreases. When the damage coefficient $\zeta$ decreases from isotropic to anisotropic damage, the contact stress ${\widetilde{\Pi }}$ generally increases. The magnitudes of the average contact stress in numerical modeling are closer to experimental results when considering anisotropic damage.

FormalPara Highlights

• An elasto-plastic-damage model is further validated by experimental data under uniaxial loadings for rocks and concrete.

• Tool-rock frictional contact is primarily controlled by three dimensionless parameters characterizing elastoplasticity, brittleness, and anisotropic compressive damage.

• The magnitudes of average contact stress in numerical modeling are closer to typical experimental results by considering anisotropic damage.

• A mesh regularization method is extended from isotropic to anisotropic damage and then validated by numerical and experimental tests.

1 Introduction

Rock drilling is an essential process in many subsurface operations, such as hydrocarbon production, geothermal exploration and carbon sequestration. While many works are focused on the mechanism of the pure cutting component with a sharp cutter (Liu et al. 2022; Zhang et al. 2022; Zhou and Lin 2013), this study is emphasized on the mechanism of frictional contact on the wear flat of a blunt cutter. The mechanisms of tool-rock frictional contact have been studied using extensive scratch tests (Almenara and Detournay 1992; Rostamsowlat 2018; Rostamsowlat et al. 2018, 2019; Zhou et al. 2017). In the analytical and numerical perspectives, rocks are idealized as different materials, such as elastic (Adachi 1996), elastoplastic (Zhou and Detournay 2014), rigid-plastic (Zhou 2017), elasto-plastic-damage (Voyiadjis and Zhou 2019), and poroelastoplastic materials (Zhou and Voyiadjis 2021). For an elastoplastic rock, the mechanism is mainly governed by a dimensionless elastoplastic parameter $\eta$ defined as $\eta ={\overline{E}}'\tan \beta /\sigma _{c}$ (where ${\overline{E}}'$ is the plane strain elastic modulus, $\beta$ is the inclination angle of a blunt tool, and $\sigma _{c}$ is the uniaxial compressive strength). The term ${\overline{E}}'\tan \beta$ can be interpreted as a characteristic elastic contact stress (Johnson 1970). Three regimes exist: asymptotic elastic with small $\eta$ ($0<\eta \lesssim 1$), elastoplastic with intermediate $\eta$ ($1 \lesssim \eta \lesssim 10$), and asymptotic rigid-plastic with large $\eta$ ($\eta \gtrsim 10$) (Zhou and Detournay 2014). The bounds of $\eta$ are estimated from the relationship between the dimensionless average contact stress ${\widetilde{\Pi }}$ and $\eta$ (where ${\widetilde{\Pi }}$ is the average of dimensionless stress $\Pi$ with $\Pi =\sigma /\sigma _{c}$, and $\sigma$ is the normal contact stress).

The average stress ${\widetilde{\Pi }}$ increases with the elastoplastic parameter $\eta$ before leveling off (Zhou and Detournay 2014). In the asymptotic elastic regime, ${\widetilde{\Pi }}$ is proportional to $\eta$ (Adachi 1996). In the asymptotic rigid-plastic regime, ${\widetilde{\Pi }}$ is a constant that is influenced by both the internal and interface friction angles (Zhou 2017). The three regimes have been confirmed by extensive experiments, in which $\eta$ is varied through the inclination angle $\beta$ while fixing other parameters (Rostamsowlat 2018; Rostamsowlat et al. 2019). Contact stress increases with the inclination angle and slightly decreases before leveling off at a limit value, which increases with a decrease in the interface friction coefficient. For numerical modeling with an elastoplastic rock, contact stress is overestimated when compared with experimental results (Zhou and Detournay 2014). The gap may be attributed to the negligence of material length scales in the elastoplastic material model.

Material length scales are accounted for by elasto-plastic-damage models. Most current elasto-plastic-damage models consider isotropic damage using damage scalars (Ahmed et al. 2020; Chaboche 1988). For a quasi-brittle rock with isotropic damage, frictional contact is also controlled by the dimensionless brittleness number $\xi$ in addition to $\eta$. The brittleness number is defined as $\xi =d/\Lambda _{c}$, where d represents a geometrical length scale, such as the depth of sliding. The parameter $\Lambda _{c}$ is a material length scale defined as $\Lambda _{c}=(1-\nu ^{2})G_{fc}/\sigma _{c}$, where $\nu$ is the Poisson’s ratio, and $G_{fc}$ is the compressive fracture energy. The influence of $\eta$ on contact stress ${\widetilde{\Pi }}$ is similar to that for an elastoplastic rock. Whereas, ${\widetilde{\Pi }}$ is also affected by the brittleness number $\xi$ with a negative correlation (Voyiadjis and Zhou 2019). The numerical result of ${\widetilde{\Pi }}$ is slightly underestimated when compared to experimental results (Almenara and Detournay 1992).

The load-induced damage is generally anisotropic for initially isotropic materials, and the anisotropic damage behaviors are different depending on loading conditions. For example, under uniaxial tension, cracks are generally normal to the axial direction, resulting in larger axial damage compared to lateral damage (Chow and Wang 1987; Vilppo et al. 2021; Voyiadjis and Park 1996). Whereas under uniaxial compression, cracks and shear bands are more inclined to the axial direction, leading to smaller axial damage compared to lateral damage (Hu et al. 2010; Pellet et al. 2005; Vilppo et al. 2021). Damage leads to the reduction of stiffness, and the axial elastic modulus is larger than the lateral elastic modulus due to smaller axial damage, based on experiments on rocks and concrete under uniaxial and triaxial compression (Chiarelli et al. 2003; Hu et al. 2010; Lu et al. 2022; Shao 1998). Anisotropic damage can be modeled using a set of damage vectors (Kuhl et al. 2000; Lu et al. 2022), preferably with second-order damage tensors, within the framework of thermodynamics (Chaboche 1993; Cicekli et al. 2007; Voyiadjis et al. 2008; Xu and Arson 2014). An anisotropic elasto-plastic-damage model is recently developed using the hypothesis of strain energy equivalence, and the implemented user material (UMAT) model is validated by experimental data under simple loading conditions (Voyiadjis et al. 2022).

Limited works are available on modeling anisotropic damage for tool-rock interactions. Percussive drilling is simulated to analyze the influence of pre-existing cracks on the penetration stiffness and fracture pattern (Saadati et al. 2015). Rock indentation with a spherical indenter is modeled to investigate force-penetration curves by considering anisotropic damage (Shariati et al. 2019, 2022). Metal forming is analyzed using both isotropic and anisotropic damage models. The anisotropic damage model is advantageous in considering the directions of microcracks with varying loading paths (Badreddine and Saanouni 2017; Badreddine et al. 2010). However, there has been a lack of modeling tool-rock frictional contact by considering anisotropic damage.

This study aims to fill the gap by investigating tool-rock frictional contact with anisotropic damage. An anisotropic constitutive model is first described, focusing on additional adjustments. The slightly revised model is validated by experimental data under monotonic and cyclic loadings for rocks and concrete using single-element tests. A finite element model is constructed for the frictional contact between a blunt tool and a quasi-brittle rock. The model is validated by analytical results in asymptotic regimes without considering damage. The anisotropic damage and contact stress are analyzed in a typical case using the anisotropic damage model. The mechanism of frictional contact is investigated focusing on three governing parameters. Numerical and experimental results are compared, followed by a discussion of the significance and limitations.

2 Anisotropic Elasto-Plastic-Damage Model

The anisotropic elasto-plastic-damage model for quasi-brittle materials is briefly described here (Voyiadjis et al. 2022). An emphasis is placed on additional revisions, including the damage evolution functions, the weight factors for damage scalars, and the regularization of mesh size sensitivity. The model is formulated within the thermodynamics framework using strain energy equivalence and considers anisotropic tensile and compressive damage behaviors. The stress tensors in the two configurations are related with (Cordebois and Sidoroff 1982):

$$\begin{aligned} {\overline{\sigma }}_{ij}=M_{ijkl}\sigma _{kl} \end{aligned}$$

(1)

where $M_{ijkl}$ is an anisotropic damage effect tensor related to a damage tensor $\varphi _{ij}$, ${\overline{\sigma }}_{ij}$ is the stress tensor with the bar denoting the fictitious undamaged configuration, and $\sigma _{kl}$ is the stress tensor in the damaged configuration.

Even though the adopted damage model is phenomenological, the damage tensor can be interpreted from the microstructure with microcracks and microvoids. For example, on the principal plane of damage tensor normal to the $x_{1}$ axis, the damage component $\varphi _{11}$ can be interpreted as (Chaboche 1987):

$$\begin{aligned} \varphi _{11}=\frac{A_{\varphi 1}}{A_{1}} \end{aligned}$$

(2)

where $A_{\varphi 1}$ is the damage area of microcracks and microvoids (Fig. 1), and $A_{1}$ is the total area on the corresponding plane in the representative volume.

Fig. 1

Sketch of damage areas on three principal planes (Chaboche 1987)

Two other damage components $\varphi _{22}$ and $\varphi _{33}$ can be interpreted similarly (Fig. 1). The three principal damage indexes are generally different, due to different damage areas ($A_{\varphi 1}$, $A_{\varphi 2}$, and $A_{\varphi 3}$) on the three principal planes. The anisotropic damage effect tensor recovers an isotropic one for isotropic damage.

The undamaged and damaged configurations are linked through the hypothesis of strain energy equivalence (Badreddine et al. 2010; Cordebois and Sidoroff 1982). For a special case under uniaxial loadings, the stress–strain relationship is:

$$\begin{aligned} \sigma _{11}=E_{11}\varepsilon _{11}^{e}\,\text {with} \,E_{11}=(1-\varphi _{11})^{2}{\overline{E}}_{11} \end{aligned}$$

(3)

where $\sigma _{11}$ is the axial stress, $\varepsilon _{11}^{e}$ is the axial elastic strain, $E_{11}$ is the damaged axial elastic modulus, ${\overline{E}}_{11}$ is the undamaged axial elastic modulus simplified as ${\overline{E}}$, and $\varphi _{11}$ is the axial damage.

The axial stress is also related to the lateral strain with (Chiarelli et al. 2003; Voyiadjis et al. 2022):

$$\begin{aligned} \sigma _{11}=-E_{22}\varepsilon _{22}^{e}\,\text {with} \,E_{22}=(1-\varphi _{11})(1-\varphi _{22}){\overline{E}}_{22} \end{aligned}$$

(4)

where $\varepsilon _{22}^{e}$ is the lateral elastic strain, $E_{22}$ is the damaged lateral elastic modulus, ${\overline{E}}_{22}$ is the undamaged lateral elastic modulus, with ${\overline{E}}_{22}={\overline{E}}/\nu$, and $\varphi _{22}$ is the lateral damage.

The hardening stresses are (Voyiadjis et al. 2022):

$$\begin{aligned} c^{\pm }=(1-\varphi _{a}^{\pm }){\overline{c}}^{\pm } \end{aligned}$$

(5)

where $c^{\pm }$ are the damaged hardening stresses, ${\overline{c}}^{\pm }$ are the undamaged hardening stresses, $\varphi _{a}^{\pm }$ are damage scalars, the superscript “$^{+}$” means tension, and “$^{-}$” denotes compression.

The damage scalars $\varphi _{a}^{\pm }$ are defined as the weighted averages of principal damage indexes as:

$$\begin{aligned} \varphi _{a}^{\pm }=\sum _{k=1}^{3}w_{k}^{\pm }{\hat{\varphi }}_{k}^{\pm } \end{aligned}$$

(6)

with

$$\begin{aligned} w_{k}^{\pm }=\frac{\left( {\hat{\varphi }}_{k}^{\pm }\right) ^{n^{\pm }}}{\sum _{k=1}^{3}\left( {\hat{\varphi }}_{k}^{\pm }\right) ^{n^{\pm }}} \end{aligned}$$

(7)

where $w_{k}^{\pm }$ are weighting factors of the $k\text {th}$ principal damage indexes ${\hat{\varphi }}_{k}^{\pm }$, and $n^{\pm }$ are newly introduced exponents to better adjust the weighting factors with $n^{\pm }\ge 0$. The weighting factors $w_{k}^{\pm }$ recover the previous expressions with $n^{\pm }=1$ (Voyiadjis et al. 2022), and the $w_{k}^{\pm }$ of larger principal damage indexes increase with $n^{\pm }$.

The hardening stresses increase with the plastic strains in the undamaged configuration due to strain hardening, and the hardening stresses are defined as (Voyiadjis et al. 2022):

$$\begin{aligned} {\overline{c}}^{\pm }=f_{0}^{\pm }+{\overline{h}}^{\pm }{\overline{\varepsilon }}_{eq}^{\pm } \end{aligned}$$

(8)

where $f_{0}^{\pm }$ are the initial yield strengths, ${\overline{h}}^{\pm }$ are the hardening moduli that mainly govern the strain hardening process, ${\overline{\varepsilon }}_{eq}^{\pm }$ are the equivalent plastic strains. Dimensionless hardening ratios $\alpha _{h}^{\pm }$ are defined as $\alpha _{h}^{\pm }={\overline{h}}^{\pm }/{\overline{E}}$. The hardening ratios $\alpha _{h}^{\pm }$ are generally positive, and negative values of $\alpha _{h}^{\pm }$ mean softening.

The damage criteria are defined as (Chow and Wang 1987):

$$\begin{aligned}{} & {} F^{\pm }(Y_{eq}^{\pm },K^{\pm })=Y_{eq}^{\pm }-K^{\pm }=0\,\text {with}\nonumber \\{} & {} \quad Y_{eq}^{\pm } =\sqrt{\frac{1}{2}Y_{ij}^{\pm }L_{ijkl}^{\pm }Y_{kl}^{\pm }} \end{aligned}$$

(9)

where $Y_{eq}^{\pm }$ are the equivalent damage driving forces, $K^{\pm }$ are the thresholds of damage driving forces, $Y_{ij}^{\pm }$ are the total damage driving forces, and $L_{ijkl}^{\pm }$ are symmetric fourth-order tensors that consider anisotropic damage.

The tensors $L_{ijkl}^{\pm }$ vary under different loading conditions. The tensile component $L_{ijkl}^{+}$ is defined as (Chow and Wang 1987):

$$\begin{aligned} L_{ijkl}^{+}=6\zeta ^{+}I_{ijkl}^{vol}+2(1-\zeta ^{+})I_{ijkl}^{s} \end{aligned}$$

(10)

where $\zeta ^{+}$ is the anisotropic tensile damage coefficient between 0 and 1 (Chow and Lu 1989), and $I_{ijkl}^{vol}$ and $I_{ijkl}^{s}$ are the volumetric and symmetric fourth-order unit tensors, respectively.

The compressive component $L_{ijkl}^{-}$ is defined as (Pellet et al. 2005):

$$\begin{aligned} L_{ijkl}^{-}=6I_{ijkl}^{vol}+2(\zeta ^{-}-1)I_{ijkl}^{s} \end{aligned}$$

(11)

where $\zeta ^{-}$ is the anisotropic compressive damage coefficient between 0 and 1.

The anisotropic damage coefficients $\zeta ^{\pm }$ are key parameters introduced for anisotropic damage, with $\zeta ^{\pm }=1$ for isotropic damage. The damage coefficients $\zeta ^{\pm }$ can be conveniently interpreted for special cases without lateral damage driving forces ($Y_{11}^{\pm }>Y_{22}^{\pm }=Y_{33}^{\pm }=0$). Under uniaxial tension with larger axial damage, $\zeta ^{+}$ is the ratio of lateral damage to axial damage. While under uniaxial compression with larger lateral damage, $\zeta ^{-}$ is the ratio of axial damage to lateral damage. The focus is on the anisotropic compressive damage coefficient $\zeta ^{-}$ that is simplified as $\zeta$ later.

The equivalent damage indexes $\varphi _{eq}^{\pm }$ are defined as (Monnamitheen Abdul Gafoor and Dinkler 2020):

$$\begin{aligned} \varphi _{eq}^{\pm }=G^{\pm }(r^{\pm })=1-\left( \frac{r_{0}^{\pm }}{r^{\pm }}\right) ^{A^{\pm }} \text {exp}\left[ B^{\pm }\left( 1-\frac{r^{\pm }}{r_{0}^{\pm }}\right) \right] \end{aligned}$$

(12)

where $r^{\pm }$ are suitable norms of damage thresholds $K^{\pm }$, $r_{0}^{\pm }$ are the initial values of $r^{\pm }$, and $A^{\pm }$ and $B^{\pm }$ are shape factors that affect the post-yield stress–strain curves.

Equation (12) recovers the damage evolution function for isotropic damage with $A^{\pm }=1$ (Oliver et al. 1990). The parameters $A^{\pm }$ and $B^{\pm }$ are constrained by $0\le \varphi _{eq}^{\pm }<1$ and the damage rates ${\dot{\varphi }}_{eq}^{\pm }\ge 0$, resulting in $B^{\pm }\ge 0$ and $A^{\pm }+B^{\pm }\ge 0$.

Mesh sensitivity is reduced by maintaining the same fracture energy for different mesh sizes (Oliver et al. 1990). The regularization method is extended from isotropic to anisotropic damage here. For anisotropic damage under a general loading condition, the energy dissipation rates ${\dot{\gamma }}^{\pm }$ per unit volume are approximated as:

$$\begin{aligned} {\dot{\gamma }}^{\pm }=Y_{eq}^{\pm }{\dot{\varphi }}_{eq}^{\pm } \end{aligned}$$

(13)

Following a similar procedure of the regularization for isotropic damage (Voyiadjis and Zhou 2019), the following equation is derived to solve for parameters $B^{\pm }$ given parameters $G_{f}^{\pm }$ and $A^{\pm }$ for an element size $\ell$:

$$\begin{aligned}{} & {} \mathop {\int }\limits _{1}^{{\infty }}\left[ A^{\pm }\left( \varOmega ^{\pm }\right) ^{1-A^{\pm }} +B^{{\pm }}\left( \varOmega ^{\pm }\right) ^{2-A^{\pm }}\right] \nonumber \\{} & {} \quad \text {exp}\left[ B^{\pm }\left( 1 -\varOmega ^{\pm }\right) \right] d\varOmega ^{\pm }=\frac{\ell _{ch}^{\pm }}{\ell } \end{aligned}$$

(14)

where $\varOmega ^{\pm }$ are dimensionless norms of damage driving forces, $\ell _{ch}^{\pm }$ are tensile and compressive characteristic lengths with $\ell _{ch}^{\pm }=G_{f}^{\pm }{\overline{E}}/$ $(f_{0}^{\pm })^{2}$, and $G_{f}^{\pm }$ are tensile and compressive fracture energy.

The materials properties under uniaxial loadings are used for simplicity (Oliver et al. 1990). For example, the initial yield strengths are $f_{0}^{+}=\sigma _{t}$ and $f_{0}^{-}=\sigma _{c}$, where $\sigma _{t}$ is the uniaxial tensile strength. Similarly, the fracture energy is $G_{f}^{+}=G_{ft}$ and $G_{f}^{-}=G_{fc}$, and the characteristic lengths are $\ell _{ch}^{+}=\ell _{ch}^{t}$ and $\ell _{ch}^{-}=\ell _{ch}^{c}$.

For the special case with $A^{\pm }=1$, the parameters $B^{\pm }$ can be simplified as:

$$\begin{aligned} B^{\pm }=\frac{2}{\ell _{ch}^{\pm }/\ell -1} \end{aligned}$$

(15)

3 Calibration and Validation of Material Model

The anisotropic damage model is validated by experimental data under monotonic and cyclic loadings for rocks and concrete using single-element tests.

3.1 Single-Element Test Under Monotonic Loading

The developed UMAT subroutine of the anisotropic damage model is tested under simple loadings, including uniaxial, biaxial, and three-point bending tests (Voyiadjis et al. 2022). The slightly revised model is further validated by experimental data under monotonic and cyclic loadings for rocks and concrete.

A typical experiment is first described for monotonic uniaxial compression of rock. The experiment is performed with a servo-controlled rock mechanics testing system MTS-815 (Walton et al. 2015, 2017). The MTS-815 test machine includes three systems for data collection, hydraulic control, and loading, as illustrated in a schematic diagram (Chen and Wang 2022). The rock samples are cylindrical with a diameter of 54 mm and a length of 100 mm. The axial loading can be applied with stress or strain-controlled mode, with the strain-controlled mode being adopted. In this mode, a constant downward velocity is applied at the top of the rock sample before and after yield strength. The strain rate is generally on the order of magnitude of $10^{-5}$ s$^{-1}$ under quasi-static condition (Qin and Wang 1999), even though the specific value is not available for this test. The axial force is measured through load cells, and the axial deformation is measured with linear variable differential transformers (LVDTs). The stress–strain curves are averaged based on 20 tests.

A single square element is used to calibrate and validate the elasto-plastic-damage material model with an element size of 100 mm. The four-node plane stress element (CPS4R) is adopted with reduced integration. The left and bottom sides have fixed normal displacements, while the right side remains free, as illustrated in a sketch depicting the boundary conditions (Cicekli et al. 2007). A constant downward velocity of $6.2\times 10^{-3}$ mm/s is applied at the top with a strain rate of $6.2\times 10^{-5}$ s$^{-1}$. It is unnecessary to use the same strain rate in numerical and experimental tests, as the adopted material model is rate-independent.

Essential modeling parameters include elastic parameters ${\overline{E}}=31$ GPa, $\nu =0.2$, plastic parameters $\sigma _{c}=66$ MPa, $\alpha _{h}^{-}$= 0.8, $\alpha =0.12$ ($\phi \simeq 10{^\circ }$), $\alpha _{p}=0$, damage parameters $G_{fc}=1.64\times 10^{4}$ N/m, $\ell _{ch}^{c}=117$ mm, $\zeta ^{-}=0.8$, $A^{-}=3$, $B^{-}=10$, $n^{-}=5$, and the geometrical parameter $\ell =100$ mm for rock, as summarized in Table 1 (Walton et al. 2017). The internal friction angle $\phi$ is around $10{^\circ }$ with a failure envelope slope of $\alpha =0.12$ based on $\alpha =2\sin \phi /(3-\sin \phi )$ (Roscoe and Burland 1968), and the dilation angle is $0{^\circ }$ with a dilation coefficient of $\alpha _{p}=0$. The material parameters can be generally divided into two categories, including physically based and phenomenologically based parameters, which are measured from and fit by experimental data, respectively. The phenomenologically-based empirical parameters include the hardening ratio $\alpha _{h}^{-}$, damage coefficient $\zeta ^{-}$, shape factors $A^{-}$ and $B^{-}$, and damage exponent $n^{-}$.

Table 1 Parameters of the elasto-plastic-damage model in monotonic uniaxial compression for rock (Walton et al. 2017) and uniaxial tension for concrete (Gopalaratnam and Shah 1985)

The calibration procedures of material properties are briefly described in uniaxial compression. Parameters such as ${\overline{E}}$ and $\sigma _{c}$ are estimated from the experimental stress–strain curve, and typical values are assumed for most other parameters. The post-yield behavior is predominantly governed by the two shape factors $A^{-}$ and $B^{-}$, which are adjusted through trial and error to approximate match experimental and numerical stress–strain curves. Empirical parameters are not unique, and one set of experimental data may be fit by different sets of empirical parameters. The compressive characteristic length $l_{ch}^{c}$ and the compressive fracture energy $G_{fc}$ are calculated from Eq. (14) based on the two shape factors and the element length.

Fig. 2

Influence of shape factors a $A^{-}$ and b $B^{-}$ on stress–strain curves under uniaxial compression. Experimental data for rock are from Walton et al. (2017)

The numerical stress–strain curve is generally close to the experimental result for the baseline case with $A^{-}=3$ and $B^{-}=10$ for the uniaxial compression of rock in Fig. 2 (Walton et al. 2017). The axial stress $\sigma _{1}$ increases linearly to the yield strength and then decreases with an increase in the axial strain $\varepsilon _{1}$ in the post-peak region with damage. The axial stress component $\sigma _{11}$ is simplified as $\sigma _{1}$, and other parameters, including strains, damage, and displacements, are similarly simplified. The axial stress keeps increasing with the axial strain after yielding due to strain hardening in the undamaged configuration. The slope ${\overline{E}}^{- ep}$ after yielding is affected by the hardening modulus ${\overline{h}}^{-}$ in the undamaged configuration (Voyiadjis et al. 2022), with ${\overline{E}}^{- ep}=13.8$ GPa for ${\overline{E}}=31$ GPa and ${\overline{h}}^{-}=24.8$ GPa ($\alpha _{h}^{-}=0.8$). A sensitivity analysis is conducted for the shape factors $A^{-}$ and $B^{-}$. After yielding, the stress generally increases as $A^{-}$ decreases from 3 to $-10$ while keeping $B^{-}=10$ (Fig. 2a). The stress also increases as $B^{-}$ decreases from 10 to 0 while keeping $A^{-}=3$ (Fig. 2b). The shape factor $A^{-}$ has a larger influence on the stress immediately after yielding, and pre-peak hardening occurs with a small $A^{-}$ under the constraints of $B^{\pm }\ge 0$ and $A^{\pm }+B^{\pm }\ge 0$.

The fracture energy-based regularization method is then validated by experimental data for concrete under monotonic uniaxial tension. The tension test is also conducted in an MTS servo-controlled test machine, as shown in a schematic diagram (Gopalaratnam and Shah 1985). Rectangular prism specimens are used with the dimensions of 76 mm (width) by 19 mm (thickness) by 305 mm (length). The two ends of the concrete sample are held by wedge-type frictional grips to apply tensile load on the specimen in strain-controlled mode. The axial deformation is measured using both 13 mm strain-gage extensometers and 83 mm LVDTs, and the axial strain is calculated based on the 83 mm length.

Fig. 3

Stress-displacement curves in uniaxial tension using different element sizes. Experimental data for concrete are from Gopalaratnam and Shah (1985)

The boundary conditions and calibration procedures are generally similar in single-element tests for uniaxial tension and compression. A constant upward velocity of $7\times 10^{-4}$ mm/s is applied at the top, corresponding to a strain rate of $1.4\times 10^{-5}$ s$^{-1}$ for a typical element size of $\ell =50$ mm. The stress-displacement curve after yielding is first calibrated by using parameters in Table 1 (Fig. 3). The post-peak axial displacement $\delta _{1}^{post}$ is defined as $\delta _{1}^{post}=\delta _{1}-\delta _{1}^{pre}$, where $\delta _{1}$ and $\delta _{1}^{pre}$ are the total and pre-peak displacements, respectively. The two shape factors $A^{+}$ and $B^{+}$ are adjusted while fixing tensile fracture energy $G_{ft}$ for three cases with different element sizes. The corresponding values are $A^{+}=$ 0.4, 0.9, and 2.5, $B^{+}=$ 0.29, 0.3, and 0.013, respectively, with $\ell =$ 25 mm, 50 mm, and 100 mm, and with $\ell _{ch}^{t}=428$ mm based on Eq. (14). The post-peak stress-displacement curves are similar using different element sizes, and the numerical result is close to the experimental result in Fig. 3 (Gopalaratnam and Shah 1985).

3.2 Single-Element Test Under Cyclic Loading

The material model is then validated by experimental data under cyclic uniaxial compression. Table 2 summarizes the modeling parameters calibrated by experimental data for rock (Hudson and Harrison 2000) and concrete (van Mier 1984). Consider first the cyclic compression of rock. Detailed experimental information is not available except for the force-displacement curve (Hudson and Harrison 2000). The force-displacement curve is converted to the stress–strain curve by assuming a typical cubic size of 50 mm. A constant downward velocity of $2.5\times 10^{-3}$ mm/s is first applied at the top of a square element, and the strain rate is $5\times 10^{-5}$ s$^{-1}$ with $\ell =50$ mm in numerical modeling. Cyclic loading is applied by properly changing the directions of applied velocity.

Table 2 Parameters of the material model under cyclic uniaxial compression for rock (Hudson and Harrison 2000) and concrete (van Mier 1984)

Fig. 4

Evolution of a axial stress $\sigma _{1}$ in cyclic compression and b axial damage $\varphi _{1}^{-}$ and dimensionless axial stiffness $E_{1}/{\overline{E}}_{1}$. Experimental data for rock are from Hudson and Harrison (2000)

Fig. 5

Evolution of a axial stress $\sigma _{1}$ in cyclic compression, b axial damage $\varphi _{1}^{-}$ and lateral damage $\varphi _{2}^{-}$, and c dimensionless axial stiffness $E_{1}/{\overline{E}}_{1}$ and lateral stiffness $E_{2}/{\overline{E}}_{2}$. Experimental data for concrete are from van Mier (1984)

The numerical and experimental stress–strain curves are generally close (Fig. 4a). Only the first five unloading-loading cycles are modeled due to convergence issues with a larger strain. The energy dissipation in each unloading-loading cycle is not considered for the sake of simplicity, and the unloading-loading curves overlap in one cycle in numerical modeling. The axial damage $\varphi _{1}^{-}$ increases with the axial strain $\varepsilon _{1}$ after yielding, and the dimensionless axial stiffness $E_{1}/{\overline{E}}_{1}$ decreases in both numerical and experimental results (Fig. 4b). The nondecreasing damage remains constant in an unloading-loading cycle, and the monotonic relationship between axial damage $\varphi _{1}^{-}$ and axial strain $\varepsilon _{1}$ is obtained through corresponding monotonic compression in numerical modeling. The dimensionless axial stiffness is calculated based on Eq. (3)$_{2}$ with $E_{1}/{\overline{E}}_{1}=\left( 1-\varphi _{1}^{-}\right) ^{2}$ for the numerical result, and $E_{1}/{\overline{E}}_{1}$ is estimated from the unloading-loading stress–strain curves for the experimental result.

Consider now the cyclic compression of concrete. The experiments are conducted using 100 mm cubes with the strain rate of $5\times 10^{-6}$ s$^{-1}$ (van Mier 1984). The element size is 100 mm with the strain rate of $7\times 10^{-5}$ s$^{-1}$ in numerical modeling. Stress–strain curves are similar in numerical and experimental results (Fig. 5a), by using material parameters summarized in Table 2 (van Mier 1984). The post-peak stress decreases more gently, and the elastic modulus is smaller in the unloading-loading cycles of the numerical result. The axial damage $\varphi _{1}^{-}$ is smaller than the lateral damage $\varphi _{2}^{-}$ after yielding with anisotropic damage (Fig. 5b), resulting in a larger axial stiffness compared to the lateral stiffness (Fig. 5c). The dimensionless lateral stiffness is calculated based on Eq. (4)$_{2}$ with $E_{2}/{\overline{E}}_{2}=\left( 1-\varphi _{1}^{-}\right) \left( 1-\varphi _{2}^{-}\right)$ for the numerical result, and thus $E_{1}/{\overline{E}}_{1}>E_{2}/{\overline{E}}_{2}$ with $\varphi _{1}^{-}<\varphi _{2}^{-}$. The axial stiffness is underestimated, and the lateral stiffness is overestimated in numerical modeling by comparing it with the experimental result. Although there is room for improvement to better match the numerical and experimental results, the model can capture the anisotropic reduction of stiffness.

4 Finite Element Model

A finite element model is developed for tool-rock frictional contact under plane strain. Numerical data are validated by analytical results in two asymptotic regimes. Anisotropic damage in frictional contact is illustrated through a typical case, followed by the mesh size sensitivity analysis.

4.1 Finite Element Mesh and Validation

Two coupled processes exist in rock cutting with a blunt cutter, including a cutting process in front of the cutter and a frictional contact process at the bottom (Detournay and Defourny 1992). Only the decoupled frictional contact process is considered here under plane strain, and the wear flat of a blunt cutter is idealized as a slightly inclined slider. The slider moves on a rock with a horizontal velocity V, an inclination angle $\beta$, a depth d, and a contact length 2a (Fig. 6a). The shear stress $\Gamma$ at the interface is expressed by a frictional law with $\Gamma =\mu \sigma$, where $\mu$ is the interface friction coefficient.

Fig. 6

Tool-rock frictional contact with a a sketch (Adachi 1996) and b the finite element mesh

Finite element models have been constructed for tool-rock frictional contact on different rocks (Voyiadjis and Zhou 2019; Zhou and Voyiadjis 2021). This study further adopts tensorial damage to account for the anisotropic damage behavior of rocks in frictional contact. The rock has a width of 100 mm, a height of 50 mm, and a thickness of 1 mm, under plane strain boundary conditions (Fig. 6b). The mesh is finer at the top 5 mm with an element size of 0.5 mm, gradually increasing towards the bottom with an element size of approximately 5 mm. There are 5192 elements with the type of eight-node brick element with reduced integration (C3D8R). The rock is initially isotropic and is characterized by the anisotropic damage model.

The tool can be characterized by a rigid analytical surface that is computationally efficient (Voyiadjis and Zhou 2019), whereas it is modeled by a deformable elastic solid here to improve computational convergence. The elastic modulus is set as a large value of 180 GPa, and the Poisson’s ratio is 0.2. Stress concentration occurs at the tip of the slider, and a fillet with a radius of 0.5 mm is adopted to alleviate stress concentration. The slider includes 110 C3D8R elements, and the element size is generally 0.5 mm except at the tip with a smaller element size of around 0.1 mm. The tool moves quasi-statically with $V=1$ mm/s, $\beta =0.5{^\circ }$, and $d=0.15$ mm. The interface friction coefficient $\mu$ varies in a wide range of around $0.02-1.0$ for different rocks with small inclination angles of $0{^\circ } \lesssim \beta \lesssim 2{^\circ }$ (Rostamsowlat et al. 2019). The interface friction angle $\psi$ and the internal friction angle $\phi$ are close in several experimental tests (Rostamsowlat et al. 2022b). A typical interface friction coefficient of $\mu =0.17$ is adopted with $\psi =\phi =10{^\circ }$. The surface-to-surface contact is adopted for the discretization of contact pair surfaces, as it is advantageous due to its smoothing effect compared to the node-to-surface contact (Hibbitt et al. 2001). There are two contact tracking methods: finite-sliding and small-sliding contact. Arbitrary relative motions are allowed for the contact surfaces in the former, and little sliding is assumed between contact surfaces in the latter. The small-sliding contact is generally used due to better computational convergence. The difference is small for the average stress in the steady state using the two algorithms.

Numerical results are validated by available analytical results of dimensionless average contact stress ${\widetilde{\Pi }}$. For an elastic rock with ${\overline{E}}=1$ GPa and $\nu =0.2$, ${\widetilde{\Pi }}$ is 0.24 in numerical modeling and 0.25 in the analytical solution (Adachi 1996). For a rigid-plastic rock with ${\overline{E}}=50$ GPa, $\nu =0.2$, and $\sigma _{c}=30$ MPa, the numerical and analytical values of ${\widetilde{\Pi }}$ are 2.6 and 2.7, respectively (Zhou 2017). The influence of $\eta$ on ${\widetilde{\Pi }}$ is analyzed in a special case, without considering hardening and damage in the anisotropic material model. For a series of cases with $\eta$ between 0.3 and 30.3, ${\widetilde{\Pi }}$ increases with $\eta$ before stabilizing at around 3.4.

4.2 Typical Case

A typical case of frictional contact is investigated using the anisotropic damage model (Voyiadjis et al. 2022). Essential material parameters are: ${\overline{E}}=5$ GPa, $\nu =0.2$, $\sigma _{t}=3$ MPa, $\sigma _{c}=30$ MPa, $\alpha _{h}^{\pm }=$ 0.83, $\alpha =0.12$ ($\phi \simeq 10{^\circ }$), $\alpha _{p}=0$, $G_{ft}=35$ N/m, $G_{fc}=1.75\times 10^{3}$ N/m, $\ell _{ch}^{t}=19$ mm, $\ell _{ch}^{c}=10$ mm, $\zeta ^{\pm }=0.8$, $A^{+}=0.8$, $B^{+}=0.083$, $A^{-}=0.8$, $B^{-}=0.15$, and $n^{\pm }=1$ (Table 3). The interface friction coefficient $\mu$ is 0.17. The compressive fracture energy $G_{fc}$ is 50 times the tensile fracture energy $G_{ft}$ (Zhou and Lin 2013). The three dimensionless numbers, $\eta$, $\xi$, and $\zeta$, are 1.5, 2.7, and 0.8, respectively.

Table 3 Modeling parameters used in a typical case of frictional contact

Anisotropic damage is depicted in damage contours, showing smaller vertical compressive damage $\varphi _{2}^{-}$ compared to horizontal compressive damage $\varphi _{1}^{-}$ (Fig. 7). Tensile damage is negligible, and the plastic yield zone is similar to the damage zone. Compressive damage is generally limited to one layer of elements beneath the slider. A boundary layer of grains and pulverized rocks has also been observed beneath a blunt cutter in experiments (Adachi 1996; Rostamsowlat et al. 2022b).

Fig. 7

Contours of anisotropic compressive damage with horizontal component $\varphi _{1}^{-}$ and vertical component $\varphi _{2}^{-}$

The contact stress ${\widetilde{\Pi }}$ is 1.0, according to the normal contact force of 277 N and the contact length of 9.5 mm. The distribution is shown, including horizontal compressive damage $\varphi _{1}^{-}$, vertical compressive damage $\varphi _{2}^{-}$, and contact stress $\Pi$ (Fig. 8). The dimensionless horizontal coordinate $\chi$ is defined at the tool-rock interface, ranging from $-1$ to 1, with $\chi =0$ representing the midpoint of the interface. The interface includes an elastic zone without damage and a damage process zone with anisotropic compressive damage. Damage increases in the damage process zone towards the tip with $\varphi _{2}^{-}<\varphi _{1}^{-}$. In the elastic region, $\Pi$ increases with an increase in elastic deformation. In the damage region, $\Pi$ first increases towards the tip due to strain hardening and then decreases due to an increase in damage. The peak value of $\Pi$ is around 1.9, when vertical damage $\varphi _{2}^{-}$ is approximately 0.3. Strain hardening behavior has also been observed in other contact problems, such as friction stir processing and metal forming (Ralls et al. 2021; Saanouni 2012).

Fig. 8

Distribution of horizontal damage $\varphi _{1}^{-}$, vertical damage $\varphi _{2}^{-}$, and dimensionless contact stress $\Pi$

Fig. 9

Influence of mesh size $\ell$ on the distribution of vertical damage $\varphi _{2}^{-}$ and dimensionless contact stress $\Pi$

4.3 Mesh Size Sensitivity

Besides the typical case, two other cases are analyzed with finer and coarser meshes to investigate the mesh size sensitivity. The three mesh sets primarily differ within the fine zone, with $\ell =$ 0.3 mm, 0.5 mm, and 0.7 mm, respectively. The studied mesh sizes are generally similar to the median grain sizes of typical limestones and sandstones (Rostamsowlat et al. 2019). The shape factor $B^{-}$ is 0.10, 0.15, and 0.21, respectively, with $\ell _{ch}^{c}=10$ mm and $A^{-}=0.8$ based on Eq. (14). The contact stress is close with ${\widetilde{\Pi }}=$ 1.01, 0.97, and 1.01, respectively, for the three cases. The normal contact force is 283 N, 277 N, and 275 N, and the contact length is 9.3 mm, 9.5 mm, and 9.1 mm, respectively. The distribution is similar with different mesh sizes for vertical damage $\varphi _{2}^{-}$ and contact stress $\Pi$ (Fig. 9).

5 Influence of Governing Parameters

The anisotropic model is adopted to investigate the effect of three dimensionless parameters, including the parameter $\eta$ characterizing elastoplasticity, the number $\xi$ depicting brittleness, and the coefficient $\zeta$ for anisotropic compressive damage. The effect of $\eta$ is investigated while keeping the two other parameters $\xi$ and $\zeta$ constant, and the influence of $\xi$ is analyzed by fixing the two other parameters $\eta$ and $\zeta$. The influence of $\zeta$ is analyzed by comparing results with anisotropic and isotropic damage. The numerical results are then compared with typical experimental results.

5.1 Influence of $\eta$ and $\zeta$

Table 4 Modeling parameters for the influence of the elastoplastic parameter $\eta$

The effect of elastoplastic parameter $\eta$ is investigated for both anisotropic and isotropic elasto-plastic-damage rocks. Consider first anisotropic damage with a typical damage coefficient $\zeta =0.8$. The parameter $\eta$ is varied while fixing the two other parameters $\xi$ and $\zeta$. The elastoplastic parameter $\eta$ increases from 0.03 to 9.1 in ten cases, with ${\overline{E}}=$ 1 GPa, 2 GPa, 3 GPa, 4 GPa, 5 GPa, 7 GPa, 10 GPa, 15 GPa, 20 GPa, and 25 GPa (Table 4). Two brittleness numbers are considered with $\xi =$ 0.2 and 3.6, corresponding to low and high brittleness. For high elastic moduli and high brittleness, several adjustments are made in addition to a smaller time step to facilitate computational convergence. The initial depth d is 0.15 mm for lower elastic moduli (${\overline{E}}=1-7$ GPa) and 0.1 mm for higher elastic moduli (${\overline{E}}=10-25$ GPa). The corresponding compressive fracture energy ratio $\alpha _{g}$ is 15 and 10, respectively, to ensure the same brittleness number $\xi =$ 0.2 at different depths. The parameter $\alpha _{g}$ is defined as $\alpha _{g}=G_{fc}/G_{fc}^{r}$, where $G_{fc}^{r}$ is the compressive fracture energy with a typical reference value of $G_{fc}^{r}=1.75\times 10^{3}$ N/m. The fracture energy ratio $\alpha _{g}$ is 0.75 and 0.5, respectively, with lower and higher elastic moduli for brittleness number $\xi =$ 3.6. The shape factor $A^{-}$ is 0.8 for low brittleness and $0.5-0.8$ for high brittleness. A smaller $A^{-}$ leads to a gentler slope of strain softening in uniaxial compression.

The relationship between ${\widetilde{\Pi }}$ and $\eta$ is shown with both anisotropic and isotropic damage (Fig. 10). Three regimes can be approximately defined: asymptotic elastic for small $\eta$ ($\eta \lesssim$ 1), elastoplastic for intermediate $\eta$ (1 $\lesssim \eta \lesssim$ 4), and plasto-damage for large $\eta$ ($\eta \gtrsim$ 4). Plastic deformation increases with an increase in $\eta$. As $\eta$ increases in an anisotropic case with $\xi =$ 0.2 and $\zeta =0.8$, ${\widetilde{\Pi }}$ generally increases before stabilizing at around 1.6. The numerical and analytical results show good agreement when $\eta \lesssim$ 1 (Adachi 1996). The stress ${\widetilde{\Pi }}$ remains unaffected by $\xi$ for $\eta \lesssim$ 1, whereas ${\widetilde{\Pi }}$ decreases with an increase in $\xi$. As $\xi$ increases from 0.2 to 3.6 for the anisotropic case with $\zeta =0.8$, the limit value of ${\widetilde{\Pi }}$ decreases from 1.6 to 1.1.

Fig. 10

Influence of elastoplastic parameter $\eta$ on dimensionless average contact stress ${\widetilde{\Pi }}$ for both anisotropic and isotropic damage

Fig. 11

Influence of elastoplastic parameter $\eta$ and damage coefficient $\zeta$ on the distribution of a vertical damage $\varphi _{2}^{-}$ and b dimensionless contact stress $\Pi$

The relationship between ${\widetilde{\Pi }}$ and $\eta$ is investigated for both isotropic and anisotropic damage. As $\eta$ increases for an isotropic case with $\xi =$ 0.2 and $\zeta =1.0$, ${\widetilde{\Pi }}$ initially increases before slightly decreasing and stabilizing around 0.8 (Fig. 10). The stress ${\widetilde{\Pi }}$ is unaffected by the damage coefficient $\zeta$ in the asymptotic elastic regime. When $\zeta$ decreases from 1.0 with isotropic damage to 0.8 with anisotropic damage, the limit of ${\widetilde{\Pi }}$ increases from 0.8 to 1.6 for low brittleness with $\xi =0.2$, and from 0.5 to 1.1 for high brittleness with $\xi =3.6$.

The distribution is analyzed for vertical compressive damage and contact stress. The effect of $\eta$ and $\zeta$ is investigated through four cases, with $\eta =$ 1.2 and 4.5 corresponding to low and high elastic moduli, and with $\zeta =$ 0.8 and 1.0 corresponding to anisotropic and isotropic damage, respectively. The brittleness number $\xi$ is 0.2 corresponding to low brittleness. An emphasis is placed on vertical compressive damage $\varphi _{2}^{-}$ that is more related to the contact stress (Fig. 11a). When $\eta$ increases, the damage zone is expanded with an increase in $\varphi _{2}^{-}$. As $\zeta$ decreases from 1.0 with isotropic damage to 0.8 with anisotropic damage, the damage zone decreases with a decrease in $\varphi _{2}^{-}$. The contact stress $\Pi$ increases and then decreases towards the tip (Fig. 11b). The stress $\Pi$ is slightly affected by damage coefficient $\zeta$ for $\eta =$ 1.2 with a dominant elastic zone. Whereas the stress $\Pi$ is significantly affected by damage coefficient $\zeta$ for $\eta =$ 4.5 with a dominant damage zone. Specifically, as damage coefficient $\zeta$ decreases, vertical damage $\varphi _{2}^{-}$ decreases, resulting in an increase in contact stress $\Pi$. The peak value of $\Pi$ increases from 1.2 to 1.8, when damage coefficient $\zeta$ decreases from 1.0 to 0.8.

5.2 Influence of $\xi$ and $\zeta$

The influence of brittleness number $\xi$ is then analyzed for both anisotropic and isotropic elasto-plastic-damage rocks. Typical cases are analyzed with $\eta =$ 1.5 and 6.1, corresponding to a low elastic modulus ${\overline{E}}=$ 5 GPa and a high elastic modulus ${\overline{E}}=$ 20 GPa, respectively. Consider first the case with a low elastic modulus. The brittleness number $\xi$ decreases from 13.4 to 0.07, when the fracture energy ratio $\alpha _{g}$ increases from 0.2 to 40, with $\alpha _{g}=$ 0.2, 0.3, 0.5, 1, 2, 3, 5, 10, 20, and 40 in ten cases (Table 5). The damage coefficient $\zeta$ is 0.8 and 1.0, corresponding to anisotropic and isotropic damage, respectively. Some other parameters in the typical case are retained, such as the initial depth $d=$ 0.15 mm and the shape factor $A^{-}=$ 0.8.

Table 5 Modeling parameters for the influence of the brittleness number $\xi$

Fig. 12

Influence of brittleness number $\xi$ on the dimensionless average contact stress ${\widetilde{\Pi }}$ with a low elastic modulus for both anisotropic and isotropic damage

Fig. 13

Influence of brittleness number $\xi$ and damage coefficient $\zeta$ on the distribution of a vertical damage $\varphi _{2}^{-}$ and b dimensionless contact stress $\Pi$ with a low elastic modulus

Fig. 14

Influence of brittleness number $\xi$ on dimensionless average contact stress ${\widetilde{\Pi }}$ with a high elastic modulus for both anisotropic and isotropic damage

The relationship between ${\widetilde{\Pi }}$ and $\xi$ is summarized for both anisotropic and isotropic damage (Fig. 12). When brittleness number $\xi$ is small with $\xi \lesssim$ 0.5, ${\widetilde{\Pi }}$ is nearly unaffected by $\xi$ and $\zeta$ with ${\widetilde{\Pi }}\backsimeq$ 1.0. When $\xi$ increases from 0.07 to 13.4 for a typical case of anisotropic damage with $\zeta =$ 0.8, ${\widetilde{\Pi }}$ decreases from around 1.0 to 0.5 with a more brittle mode. As $\zeta$ decreases from 1.0 with isotropic damage to 0.8 with anisotropic damage while keeping $\xi =$ 13.4, ${\widetilde{\Pi }}$ increases from 0.4 to 0.5.

The distribution is then analyzed in four cases, with $\xi =$ 0.3 and 8.9 corresponding to low and high brittleness, and $\zeta =$ 0.8 and 1.0 corresponding to anisotropic and isotropic damage, respectively. When $\xi$ increases, the damage zone is expanded with an increase in $\varphi _{2}^{-}$ (Fig. 13a). As $\zeta$ decreases from isotropic to anisotropic damage, the damage zone slightly decreases with a decrease in $\varphi _{2}^{-}$. The stress $\Pi$ increases before decreasing, with the peak stress slightly shifting to the left for anisotropic damage, attributed to a slightly smaller damage process zone (Fig. 13b). Additionally, as $\xi$ increases, the damage zone is expanded with an increase in $\varphi _{2}^{-}$, resulting in a more significant stress drop. The stress $\Pi$ generally increases with a decrease in $\zeta$ within the damage zone.

Fig. 15

Influence of damage coefficient $\zeta$ on the distribution of a vertical damage $\varphi _{2}^{-}$ and b dimensionless contact stress $\Pi$ with a high elastic modulus

Consider now the case with a high elastic modulus. Similar adjustments are made for computational convergence, such as a shallower depth $d=$ 0.1 mm and a smaller shape factor $A^{-}=0.5-0.8.$ A narrower range of the brittleness number is adopted with $\xi$ decreasing from 3.6 to 0.04, as the fracture energy ratio $\alpha _{g}$ increases from 0.5 to 40 with $\alpha _{g}=$ 0.5, 0.7, 1, 1.5, 2, 3, 5, 10, 20, and 40 in ten cases (Table 5). Three values of damage coefficient $\zeta$ are considered with $\zeta =$ 0.8, 0.9, and 1.0. The relationship between ${\widetilde{\Pi }}$ and $\xi$ is shown for both anisotropic and isotropic damage (Fig. 14). As $\xi$ increases from 0.04 to 3.6 while keeping $\zeta =$ 0.8, ${\widetilde{\Pi }}$ remains nearly unaffected by $\xi$ when $\xi \lesssim$ 0.2, and then decreases from approximately 1.6 to 1.1. When $\zeta$ decreases from 1.0 to 0.9 and 0.8 with $\xi =$ 0.9, ${\widetilde{\Pi }}$ increases from 0.7 to 1.0 and 1.3, respectively. The influence of $\zeta$ on ${\widetilde{\Pi }}$ is more significant with a larger $\eta$ due to a dominant damage region.

The distribution is analyzed for three cases with $\zeta =$ 0.8, 0.9, and 1.0. The influence of damage coefficient $\zeta$ on average stress ${\widetilde{\Pi }}$ is similar with different values of $\xi$ (Fig. 14), so a typical brittleness number of $\xi =$ 0.9 is chosen in the distribution. The damage process zone is dominant at the interface, and damage index $\varphi _{2}^{-}$ decreases as damage coefficient $\zeta$ decreases from isotropic to anisotropic damage (Fig. 15a). As damage index $\varphi _{2}^{-}$ decreases, the stress $\Pi$ generally increases with a decrease in damage coefficient $\zeta$ (Fig. 15b). The peak value of $\Pi$ increases from 1.0 to 1.2 and 1.4, respectively, when $\zeta$ decreases from 1.0 to 0.9 and 0.8 with $\eta =$ 6.1 and $\xi =$ 0.9.

5.3 Comparison With Experimental Data

The current numerical results are compared with experimental results using the anisotropic elasto-plastic-damage model by extending a previous study with isotropic damage (Voyiadjis and Zhou 2019). The essential modeling parameters are summarized in Table 6, with $\eta =16$, $\xi =0.8$, $\zeta =0.8-1.0$, ${\overline{E}}=26.4$ GPa, $\nu =0.2$, $\sigma _{c}=30$ MPa, $G_{fc}=3.9\times 10^{3}$ N/m $(\alpha _{g}=2.2$), $\ell _{ch}^{c}=114$ mm, $\Lambda _{c}=0.12$ mm, $A^{-}=0.6$, $B^{-}=0.03$, $\beta =1.0{^\circ }$, and $d=0.1$ mm. The damage coefficient $\zeta$ varies between 0.8 and 1.0 with intervals of 0.05. Other material parameters are similar to those in the typical case. The interface and internal friction angles are $10{^\circ }$. The inclination angle $\beta$ is doubled from $0.5{^\circ }$ to $1.0{^\circ }$, so the elastic modulus ${\overline{E}}$ is decreased by half to facilitate computational convergence and to ensure the same elastoplastic parameter $\eta$. A larger inclination angle leads to fewer elements in contact and thus adversely affects the stress distribution. Attempts have been made to increase the number of elements with a larger depth and a smaller element size, but additional convergence issues occur with anisotropic damage.

Table 6 Modeling parameters with a host of the damage coefficient $\zeta$

For isotropic damage with $\zeta =1.0$, the average stress ${\widetilde{\Pi }}$ is approximately 0.8, with the vertical force of 121 N and the contact length of 5 mm. The average stress ${\widetilde{\Pi }}$ increases with a decrease in damage coefficient $\zeta$ from isotropic to anisotropic damage. Specifically, ${\widetilde{\Pi }}$ increases to 1.1 and 1.6, when $\zeta$ decreases to 0.9 and 0.8, respectively (Fig. 16). The numerical results of ${\widetilde{\Pi }}$ between 0.8 and 1.4 show better agreement with experimental results (Almenara and Detournay 1992), by considering anisotropic damage with $\zeta$ decreasing from 1.0 to 0.85.

Fig. 16

Influence of damage coefficient $\zeta$ on dimensionless average contact stress ${\widetilde{\Pi }}$ with $\eta =$ 16 and $\xi =$ 0.8. The shaded area shows the range of typical experimental data with a similar set of $\eta$ and $\xi$ (Almenara and Detournay 1992)

The contact stress ${\widetilde{\Pi }}$ is influenced by interface friction angle $\psi$, which is affected by different factors including the roughness of the tool (Rostamsowlat 2018; Rostamsowlat et al. 2022b). When $\eta$ is small in the asymptotic elastic regime, the influence of $\psi$ on ${\widetilde{\Pi }}$ is minor (Adachi 1996). Whereas for a large $\eta$ in the plasto-damage regime, the influence is generally more significant (Rostamsowlat 2018). The influence of $\psi$ is analyzed for a baseline case in the plasto-damage regime with $\eta =$ 16, $\xi =$ 0.8, and $\zeta =0.9$. The interface friction angle $\psi$ decreases from $10{^\circ }$ to $0{^\circ }$ with decreasing roughness. Other parameters are similar to that in Table 4. The contact stress ${\widetilde{\Pi }}$ generally increases as $\psi$ decreases, with ${\widetilde{\Pi }}=$ 1.1, 1.1, 1.2, and 1.3 for $\psi =$ $10{^\circ }$, $8{^\circ }$, $6{^\circ }$, and $4{^\circ }$, respectively. The accuracy may be affected for smaller interface friction angles with convergence issues. The negative correlation between $\psi$ and ${\widetilde{\Pi }}$ is consistent with previous works, including experiments on quasi-brittle rocks (Rostamsowlat et al. 2019), analytical study on rigid-perfectly plastic rocks (Zhou 2017), and numerical modeling on poroelastoplastic rocks (Zhou and Voyiadjis 2021).

The influence of internal friction angle $\phi$ is also analyzed for the same baseline case in the plasto-damage regime. The internal friction angle $\phi$ is adjusted through the failure envelope slope $\alpha$ based on $\alpha =2\sin \phi /(3-\sin \phi )$ (Roscoe and Burland 1968), where $\phi$ varies between $5{^\circ }$ and $20{^\circ }$ at intervals of $5{^\circ }$. The contact stress ${\widetilde{\Pi }}$ increases with $\phi$, with ${\widetilde{\Pi }}=$ 1.0, 1.1, 1.4, and 1.7 for $\phi =$ $5{^\circ }$, $10{^\circ }$, $15{^\circ }$, and $20{^\circ }$, respectively. The positive correlation between $\phi$ and ${\widetilde{\Pi }}$ agrees with the analytical solution for rigid-perfectly plastic rocks (Zhou 2017).

6 Discussion

Calibration and validation of numerical models are discussed, including both the anisotropic elasto-plastic-damage material model and the finite element model of frictional contact. The three dimensionless governing parameters are summarized for frictional contact with an emphasis on anisotropic compressive damage coefficient $\zeta$. Limitations are then explained, including assumptions, adopted material model, and the influence of other empirical parameters.

6.1 Calibration and Validation

The main purpose is to investigate the mechanism of frictional contact with anisotropic damage. Proper material models should be chosen to analyze specific mechanisms of frictional contact. For example, the Mohr-Coulomb material model is applied to investigate three regimes, including the asymptotic elastic, elastoplastic, and asymptotic rigid-plastic (Zhou and Detournay 2014). A poroelastoplastic material model is used to analyze three pore pressure regimes, including low speed, transient, and high speed (Zhou and Voyiadjis 2021). These material models are available in commercial software, making it convenient to choose proper material parameters. However, a proper material model is not readily available in commercial software to analyze the mechanism of frictional contact with anisotropic damage. As a result, an anisotropic elasto-plastic-damage model is developed and implemented in ABAQUS as a UMAT (Voyiadjis et al. 2022). It is essential to calibrate and validate the UMAT material model under simple loadings, such as compression and tension. The calibration results from these simple tests cannot be simply generalized to more complicated problems, such as the tool-rock frictional contact. Instead, the calibration and validation provide credence to the material model before applying it to frictional contact. The calibration also provides typical values for both physically based and phenomenologically based material parameters of the rock.

It is also necessary to validate numerical models using available theoretical and experimental results in tool-rock interactions. In previous works of rock cutting with a sharp cutter, strength theory is adopted in the dominant failure mode of crushing (Detournay and Defourny 1992), and linear elastic fracture mechanics is applied in the dominant failure mode of fracture (Guo et al. 1992). Different failure modes observed in experiments can be approximately captured in finite element modeling of rock cutting using an isotropic elasto-plastic-damage material model (Zhou and Lin 2013). In the current work, the model validation is conducted in two asymptotic regimes with analytical solutions (Adachi 1996; Zhou 2017). For a general case without analytical solution, the finite element model is then validated by experimental results (Almenara and Detournay 1992).

6.2 Three Governing Parameters

The mechanism of frictional contact is mainly controlled by three parameters: $\eta$, $\xi$, and $\zeta$, which characterize elastoplasticity, brittleness, and anisotropic damage, respectively. This study extends previous works in frictional contact by considering anisotropic damage through an anisotropic damage model (Voyiadjis et al. 2022). The anisotropic damage model includes anisotropic tensile and compressive damage coefficients to consider different behaviors under tension and compression. An emphasis is placed on the anisotropic compressive damage coefficient due to negligible tensile damage. The compressive damage coefficient $\zeta$ can be interpreted as the ratio of the axial damage to the lateral damage for a special case with only the axial component of damage driving force in uniaxial compression. The damage coefficient $\zeta$ is between 0 and 1, with isotropic damage being recovered when $\zeta =1$. For an elastoplastic rock, the elastoplastic parameter $\eta$ predominantly controls the mechanism of frictional contact (Zhou and Detournay 2014). For a rock with isotropic damage, the brittleness number $\xi$ also plays a governing role (Voyiadjis and Zhou 2019). For a rock with anisotropic damage, frictional contact is also controlled by the compressive damage coefficient $\zeta$. The compressive damage coefficient $\zeta$ governs the ratio of damage in different directions, and vertical damage is smaller than horizontal damage in frictional contact.

The main results are briefly summarized regarding the influence of three dimensionless governing parameters. As $\eta$ increases, the average stress ${\widetilde{\Pi }}$ increases before slightly varying and stabilizing. The influence of $\eta$ in numerical modeling is generally consistent with extensive experiments, in which $\eta$ is varied by changing inclination angle $\beta$ while fixing other parameters (Rostamsowlat 2018; Rostamsowlat et al. 2019, 2022a). As $\xi$ and $\zeta$ increase, the average stress ${\widetilde{\Pi }}$ generally decreases, due to the expansion of damage zone and an increase in damage index $\varphi _{2}^{-}$. The depth d also affects the average stress ${\widetilde{\Pi }}$, as $\xi$ scales with d. However, the depth of sliding in frictional contact is different from the depth of cut in scratch tests, which includes two components of cutting and frictional contact (Rostamsowlat 2018).

The magnitudes of average contact stress ${\widetilde{\Pi }}$ in numerical modeling are closer to experimental results by introducing an additional anisotropic compressive damage coefficient $\zeta$. The average stress ${\widetilde{\Pi }}$ is between 0.7 and 1.3 in typical experiments (Almenara and Detournay 1992). In a previous numerical study with isotropic damage, the numerical value of ${\widetilde{\Pi }}\backsimeq 0.8$ is slightly underestimated (Voyiadjis and Zhou 2019). In this study, ${\widetilde{\Pi }}$ increases from 0.8 to 1.4 as $\zeta$ decreases from 1 to 0.85, while keeping $\eta$ and $\xi$ constant. Additionally, this numerical study reveals a negative correlation between interface friction angle $\psi$ and contact stress ${\widetilde{\Pi }}$ observed in experiments (Rostamsowlat et al. 2019).

6.3 Limitation

The frictional contact problem is idealized with several assumptions in numerical modeling. For example, frictional contact on a wear flat is assumed to be decoupled with pure cutting on a cutting surface (Detournay and Defourny 1992). Coupled behaviors are observed in experiments, where part of the crushed particles in the cutting process flows underneath the wear flat (Rostamsowlat 2018; Rostamsowlat et al. 2019). In addition, perfect contact is assumed between the slider and rock. At a shallow depth of cut, the tool-rock interface may not be in full contact due to small cavities at the interface. The contact area becomes saturated at a threshold depth when these cavities are filled by crushed particles. The threshold depth of the cut depends on the depth of a wear flat surface. The contact stress increases with the depth before leveling off at values that depend on the inclination angle (Rostamsowlat 2018).

The anisotropic elasto-plastic-damage model can capture the general trend of decreasing moduli in different directions in uniaxial compression, whereas a gap still exists between numerical and experimental results (van Mier 1984). In addition, the mesh size sensitivity is reduced by maintaining the same fracture energy. The stress distribution is insignificantly affected by the element size after regularization. The thickness of the damage zone still depends on the mesh size, as damage is approximately limited to one layer of elements at the surface. A non-local damage model is preferable to further reduce mesh size sensitivity. Furthermore, computational convergence is an issue for anisotropic damage especially with high elastic moduli and high brittleness. The implementation algorithm is more complex in the anisotropic damage model with strain energy equivalence than in the isotropic damage model with strain equivalence. For example, a damage effect tensor is adopted for the stress mapping between undamaged and damaged configurations for anisotropic damage, while a damage scalar is sufficient for isotropic damage.

Besides the three dimensionless governing parameters, the essential responses of frictional contact are also affected by other empirical parameters, such as the hardening ratios that control strain hardening and the shape factors that control damage evolution. Strain hardening due to plastic evolution is coupled with strain softening due to damage evolution, and strain hardening/softening affect both stress-displacement curves and contact stress. Pre-peak strain hardening is generally neglected for simplicity, as the initial yield strength and peak strength are the input and output of the model, respectively. The input initial yield strengths are simply uniaxial strengths without considering pre-peak strain hardening. Otherwise, it is tedious to ensure the same uniaxial compressive strength while changing other material parameters.

Isotropic damage is considered using the anisotropic damage model for a special case with $\zeta =1$, and the results are slightly different from the previous study with isotropic damage. The maximum stress in stress distribution is generally larger than that in the previous study. The larger peak stress leads to a larger average stress at intermediate $\eta$. The larger peak stress may be related to the adopted initial damage thresholds that better consider strength enhancement under biaxial compression (Voyiadjis et al. 2022).

7 Conclusions

The recently developed anisotropic elasto-plastic-damage model is further calibrated by experiments under monotonic and cyclic loadings for rocks and concrete (Gopalaratnam and Shah 1985; Hudson and Harrison 2000; van Mier 1984; Walton et al. 2017). A fracture energy based method is extended from isotropic damage to anisotropic damage to regularize mesh size sensitivity, and the regularization method is validated by experimental data under uniaxial tension (Gopalaratnam and Shah 1985). The anisotropic constitutive model considers different tensile and compressive damage behaviors through anisotropic tensile and compressive damage coefficients, respectively. In uniaxial compression with anisotropic damage, axial damage is smaller, resulting in a larger axial elastic modulus compared to the lateral elastic modulus (van Mier 1984).

This work originally analyzes the mechanism of tool-rock frictional contact by accounting for anisotropic damage through finite element modeling. The mechanism is mainly governed by three dimensionless parameters: an elastoplastic parameter $\eta$, a brittleness number $\xi$, and an anisotropic compressive damage coefficient $\zeta$. The newly introduced parameter $\zeta$ controls the ratio of damage in different directions with $\zeta =1$ representing isotropic damage. As $\eta$ increases with more plastic deformation, the contact stress ${\widetilde{\Pi }}$ increases and then slightly varies before stabilizing. As $\xi$ increases with a more brittle behavior, the contact stress ${\widetilde{\Pi }}$ generally decreases. When $\zeta$ decreases from isotropic to anisotropic damage, the contact stress ${\widetilde{\Pi }}$ generally increases. The magnitudes of the average contact stress in numerical modeling are closer to experimental results when considering anisotropic damage (Almenara and Detournay 1992).