Peridynamic simulations of rock indentation

1 INTRODUCTION

Heterogeneous solids, such as rocks and concrete, contain a distribution of flaws. When subjected to tensile loads, cracks initiate from these flaws and propagate according to the linear elastic fracture mechanics (LEFM) criterion. This results in one or more cracks reaching a critical length, at which point the most critical crack propagates unstably, leading to failure. However, under compression, flaws in these materials undergo a complex sequence of failure processes. These include crack closure, pore-collapse, crack initiation in various modes, crack sliding, and the wing crack formation [1-4]. Due to the activation of these mechanisms, failure under compression results in diffused damage. This means that damage is observed at various locations, unlike in tensile failure where only a single crack localizes. The activation of these additional mechanisms under compression result in an order of magnitude higher dissipation as compared to the energy dissipated during tensile failure. This leads to significant variations in tensile and compressive strengths [5]. Hydrostatic pressure plays a crucial role in compressive failure by governing the activation of several of the aforementioned mechanisms. Therefore, it is a critical factor in the failure process.

In this paper, the standard ordinary state-based peridynamic simulation model [6-9] is extended to consider the additional dissipation due to the pore-collapse phenomenon and applied to the indentation test of a sandstone. Qualitative features of indentation are analyzed through acoustic emission (AE) data obtained from the experiments performed on Bentheim sandstone [10]. Quantitatively, the model is validated by comparison of force-penetration and indentation-penetration data obtained in the experiments by the simulation results.

2 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 x, with a set of material points within a volume defined by a cutoff radius δ, 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 x in a peridynamic body is given by

(1)

where is the force state at x and is the force, which a material point x 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 :

(2)

where is a unit vector state, given by

(3)

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

(4)

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.

(5)

where is defined as:

(6)

where in known as the weighted volume, and is defined as:

(7)

2.1 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.

TABLE 1. Material properties for Bentheim sandstone.

Material parameter	Value
Young's modulus, E (Pa)	9.5 × 109
Poisson's ratio, ν	0.18
Density, ρ (Kg/m3)	2011
Fracture energy, (J/m2)	10
Porosity, ϕ (%)	24

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 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

(8)

Here is the reversible elastic part of the volumetric strain and 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

(9)

where is the pressure which is computed according to

(10)

Here, K is the Bulk modulus, , and K₄ are calibrated for a third-order polynomial and K₅ and K₆ 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].

2.2 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

(11)

where, σ is the sampled parameter, σ₀ is the mean value of the parameter to be sampled, and k is known as the shape parameter. The shape parameter k represents the level of homogeneity of the material [18], where a larger k represents a more homogeneous material and vice versa (Figure 2). Simulations use a value of 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.

3 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

(12)

Here, F is the indentation load, d is the indentation depth, R_ind is the radius of the indenter tip, a₀ denotes the half width of the indenter tip, and α 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.

4 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.