Peridynamic simulations of rock indentation

This paper presents an ordinary state‐based peridynamic model to simulate failure in porous rocks and its application in the rock excavation processes. Rock indentation is a fundamental process for mechanical rock fracturing, which is frequently encountered in rock excavation. To model crushing occurring under high triaxial loads observed in the rock indentation experiments, peridynamic model has been extended to consider the pore‐collapse phenomenon. Rock heterogeneity is considered in the simulations by sampling the strength parameters from a probability distribution function. The model is qualitatively validated using the acoustic emission data from the indentation experiments. The developed model is able to form a crushed zone, which serves as the point of initiation for tensile cracks, which split the specimen. Quantitative validation relies on the comparison of the force‐penetration data as well as the indentation pressure‐penetration data obtained from the simulations and experiments performed on Bentheim Sandstone specimens.


PERIDYNAMIC CONTINUUM
Peridynamics continuum formulation [6] allows for the modeling of complex fragmentation processes.The reason is that the internal forces are computed, instead of a displacement gradient, from a displacement difference of material point , with a set of material points  ′ within a volume defined by a cutoff radius , known as the peridynamic horizon [11].This notion of direct connectivity between the material points is referred to as a bond.Fracture processes are modeled by irreversibly deleting the bonds between the material points once they reach a critical value.Once a bond fails, it does not further contribute to the internal force calculation.Balance of linear momentum for a material point  in a peridynamic body is given by where [, ] is the force state at  and [, ]⟨ ′ − ⟩ is the force, which a material point  exerts on  ′ .Angular brackets are used to denote quantities that  operates on.
The deformation state [, ] is defined as [, ]⟨ ′ − ⟩ =  ′ −  = ( ′ +  ′ ) − ( + ), where  ′ −  and  ′ −  are the deformed relative position vector and the relative displacement vector of the bond  ′ − , respectively.Analogous to the deformation gradient of the classical continuum theory, it provides information on the change of relative positions in the neighborhood of a point x.In the following, the relative position vector is denoted as  =  ′ −  and the relative displacement vector is denoted as  =  ′ − , which gives [, ]⟨  ⟩ = ( + ).
The force state [, ] for the ordinary state-based peridynamics is characterized by a magnitude, that is, a scalar state [, ] and a direction, provided by the unit vector state [, ]: where [, ] is a unit vector state, given by The force state [, ] depends on a scalar stretch-like quantity, denoted as the extension state [, ], which characterizes the kinematics of the model.Extension state  is defined as the difference of the length of the deformed and undeformed relative position vectors and can be further decomposed additively into an isotropic (  ) and a deviatoric (  ) extension state, it is given by where  ′ −  =  .The isotropic extension state can be represented in terms of a scalar-valued volume dilatation (, ) that is defined to match the volumetric strain of a classical continuum model under isotropic loading conditions.
where (, ) is defined as: where () in known as the weighted volume, and is defined as: TA B L E 1 Material properties for Bentheim sandstone.

Model extension to consider pore-collapse phenomenon
Ordinary state-based peridynamic model has been thoroughly investigated for the tension-dominated fracture problems, including failure strengths and crack dynamics [9,12].However, for cases where the dominant loading is in compression [13] and additional nonlinear dissipation phenomena, such as the pore-collapse, are present, the model underpredicts the failure strengths [14].To this end, the standard ordinary state-based peridynamic model [7,8] is extended to consider the pore-collapse phenomenon in porous materials.
The mechanical properties and the deformation characteristics of Bentheim sandstone were studied under triaxial loads by Klein et al. [4].The material parameters for Bentheim sandstone are presented in Table 1.It is a relatively homogeneous porous rock, and these pores collapse under compressive loads leading to stiffening of the material, this effect can be seen by comparing the volumetric strain predicted by a constant bulk modulus with the one obtained in the experiments under hydrostatic compression, as shown in Figure 1.This stiffening under compressive pressure, due to pore-collapse, contributes to a significant increase in the total dissipated energy.
The material stiffening due to pore-collapse is modeled by using the experimental relationship between the hydrostatic compression and volumetric strain (Figure 1) instead of just using a constant Bulk modulus.According to Klein et al. [4], this relationship has one inflection point leading up to a compressive volumetric strain of 3.0% and after that, this relationship becomes linear again.So, the range of the volumetric strains from 0 to −3.0% is fitted with a third-order polynomial and for compressive volumetric strain greater than 3.0%, a linear relationship is adapted.Additionally, these volumetric deformations are assumed to be irreversible, as pore-collapse is an irreversible process.
For the modification of the ordinary state-based linear elastic model, an additive split is defined for the dilatation (Equation 6) under compressive deformation as Here  e (, ) is the reversible elastic part of the volumetric strain and  pc (, ) is the irreversible volumetric strain representing the pore-collapse under compression.The scalar force state [, ] (Equation 2) is modified to consider only the elastic part of the dilatation as where (, ) is the pressure which is computed according to Here,  is the Bulk modulus,  1 ,  2 ,  3 , and  4 are calibrated for a third-order polynomial and  5 and  6 are calibrated for a first-order polynomial using the experimental data from Klein et al. [4], as shown in Figure 1.The applicable range of volumetric strain for the third-and first-order polynomial are shown in Equation (10).For further details regarding the model, the interested reader is referred to Butt [14].

Consideration of material heterogeneity
Materials like rocks and concrete have inherent heterogeneity and random defects that must be accounted for in the model.To address rock heterogeneity in the simulations, strength parameters such as the fracture energy are sampled from a probability distribution.According to the literature [15,16], Weibull's distribution has been found to well represent the distribution of micro-defects in the rocks.Therefore, the simulations will consider the strength parameters sampled from a Weibull distribution [17].The probability density function for the Weibull's distribution is given by where,  is the sampled parameter,  0 is the mean value of the parameter to be sampled, and  is known as the shape parameter.The shape parameter  represents the level of homogeneity of the material [18], where a larger  represents a more homogeneous material and vice versa (Figure 2).Simulations use a value of  = 3 for the shape parameter according to Liu et al. [16].
The open-source software Peridigm [19,20] was extended to consider the pore-collapse model as well as the distributed strengths in the simulation domain.This extended version of Peridigm is used to carry out the simulations presented in the next section.

ROCK INDENTATION SIMULATIONS
Numerical and experimental investigations [10,21,22] have shown that the rock fragmentation under indentation involves several progressive deformation mechanisms including the elastic deformations at an initial loading stage, then the volumetric compaction, plastic deformation, and finally the macro fracture.Thus, it is necessary for a simulation model to be able to reproduce these qualitative features observed in the indentation experiments.Simulations performed in this study for the indentation tests cover a total of six specimen sizes, with a combination of three diameters 30, 50, and 84 mm and two heights 50 and 100 mm.For validating the current model, we use the experimental data from indentation tests on Bentheim Sandstone reported in Yang et al. [10].The temporal evolution of fractures occurring due to the indentation load is presented in Figure 3 for a specimen with 30-mm diameter and 100-mm height.The simulation captures the experimental observation of the successive formation of the crushed zone and the initiation of a central macroscopic tensile fracture splitting the sample in half.Once the tensile stresses at the perimeter of this zone exceed a critical value, a tensile crack is formed which splits the specimen.As this tensile crack propagates,  it exceeds a critical crack propagation velocity, at which the crack tip bifurcation takes place and the crack branches [12], as shown in the fourth column of Figure 3. Finally, the specimen splits into two main and several small fragments (fifth column of Figure 3).
The indentation pressure resulting from the indentation loads is calculated using Here,  is the indentation load,  is the indentation depth,  ind is the radius of the indenter tip,  0 denotes the half width of the indenter tip, and  is the angle between the indenter and the specimen surface (here 40 • ).For further details of the indenter geometry, the interested reader is referred to Yang et al. [10].The indentation force-penetration and the pressure-penetration relationship computed from the simulations are compared with the experimental data in Figure 4.The loading stiffness, the peak load, as well as the peak indentation pressure predicted by the simulation are in a good agreement with the experiments.

CONCLUSION
The standard constitutive framework of the ordinary state-based peridynamics was extended to include the pore-collapse phenomenon, which was modeled as permanent deformations.The extended model was validated through qualitative comparisons with the AE data and quantitative comparisons with the force-penetration and the pressure-penetration data from the indentation experiments performed on a Bentheim sandstone sample.These results suggest that the extended simulation model has a promising potential for modeling the indentation and excavation processes in porous rocks.

F
I G U R E 2 Weibull distribution for a scale factor 1.0 and various shape parameters  (left).The distribution of the strengths sampled for an indentation specimen from a Weibull distribution with a shape parameter  = 3 (right).F I G U R E 3Temporal evolution (left to right columns) of the fracture process during the indentation test (top row), the associated damaged and cracked regions are filtered out for visualization (bottom row).

F I G U R E 4
Comparison of the indentation force-penetration (left) and the indentation pressure-penetration (right) relationship measured in the experiments and computed from the simulations.