The dynamic tensile behavior of granite samples, when some preexisting cracks are introduced artificially, is investigated. Spalling tests using a Hopkinson bar are performed and strain rates of ~10^{2} 1/s are obtained in both specimen types (with and without initial cracks). This experimental technique is employed being of the same order as strain rates in rock materials during percussive drilling, the application of interest here. The dynamic tensile responses of both samplesets are compared using the velocity profile measured on the freeend of the sample. Furthermore, an anisotropic damage model based on the concept of obscuration probability describes the response without preexisting cracks. Here, a term of cohesive strength in obscuration zones is added to accurately handle the softening behavior of the material in tension. Results from the spalling tests are used to validate the model prediction of the dynamic tensile strength and also to calibrate the cohesive model parameters. Damaged elements are numerically introduced in the finite element calculations simulating the spalling experiments performed on predamaged samples. The results are compared with the experimental ones. Good agreement is obtained showing that a twoscale approach may constitute a suitable method to simulate numerically the tensile response of predamaged granite.
The strain rate dependency of the mechanical response in brittle materials has been widely investigated in the literature. Considerable rate dependency is reported especially in the case of tensile strength [
There is a wide range of applications pertinent to the dynamic tensile behavior of brittle materials from blasting in open quarries and concrete structures exposed to impact loading to screen rupture of cell phones due to free fall. Percussive drilling, which is the application of interest in this investigation, is just one of them. The main goal in this work is to develop a reliable numerical tool for simulating the rock fragmentation mechanism during percussive drilling. In modeling such problems, a constitutive model is needed to cover both the tensile behavior of the brittle materials at high strain rate, because of the rapid indentation, and also confined compression behavior that occurs underneath the indenter. The Krieg, Swenson, and Taylor [
Spalling tests are performed to investigate the dynamic tensile behavior of Bohus granite. As the DFH model predicts the dynamic tensile strength of the brittle materials at high strain rates, this work can be seen as a validation step for the model prediction of the granite tensile strength. The strain rate in the rock during the percussive drilling process is in order of 10^{2} 1/s based on previous numerical simulations [
The experimental results from the spalling tests on Bohus granite are presented. The dynamic tensile strength of the material (without preexisting cracks) is measured and compared with the quasistatic results. When there are cracks in the specimens that are introduced in addition to the material default cracks and defects, called structural cracks in this work, the material response changes considerably. The effect of the structural cracks on the mechanical response and the fracture pattern in EdgeOn Impact (EOI) tests, that is, impact of an aluminum projectile onto a rock slab, was previously studied [
Spalling test with Hopkinson bar is suggested as a suitable technique to measure the tensile strength of brittle materials at strain rates between 10^{1} and 10^{2} 1/s [
Accordingly, the spalling test with Hopkinson bar is an appropriate method to investigate the dynamic tensile behavior of the material in the interesting range of strain rates pertinent to percussive drilling. The experimental setup used in this work is shown in Figure
Spalling experiment setup with Hopkinson bar.
The results from the spalling tests are presented in this section. A typical result for both the rear face velocity (obtained from the laser) and the strain data (obtained from the strain gauges) is shown in Figure
Spalling test results for a specimen without preexisting cracks. Rear face velocity profile from the laser (a) and strain data from the three strain gauges (b).
Furthermore, the nominal stress level (
Nominal stress obtained from gauge G1 (
The effect of the preexisting cracks on the mechanical response and the fracture pattern in EdgeOn Impact (EOI) tests, that is, impact of an aluminum projectile onto a rock slab, was previously studied [
When preexisting structural cracks are present in the specimen, the dynamic response of the material during a spalling test changes considerably. A typical result of the spalling test with preexisting structural cracks is shown in Figures
Spalling test results for a specimen with preexisting structural cracks. Rear face velocity from the laser (a) and strain data from the three strain gauges (b).
Nominal stress obtained from gauge G1 (
It should be mentioned that the measurement from gauge G2 seems to be extremely high in the tensile part. The reason for this could be that this gauge was glued near one of the structural cracks in which the tensile axial strain gets locally large. The pullback velocity in this test is
The fragmentation process in brittle materials exposed to dynamic loading, with particular application to percussive drilling, is of most interest in this investigation. The stress state in the material beneath the drilling tool consists of both compressive and tensile stresses. It is well known that brittle materials such as rocks behave differently in compression and tension. Therefore the constitutive model for these materials, to account for such types of phenomena, should include this difference and should be able to distinguish between the two different stress sign dependent responses. For this reason and based on a previous investigation [
The KSTDFH material model is composed of two separate parts in order to deal with both compressive and tensile responses of the material. A plasticity model (KST) is employed to simulate the compressive behavior of geomaterials accounting for the effect of hydrostatic and deviatoric parts of the stress tensor. The fragmentation process, due to the opening of cracks, is defined by using a damage model (DFH), which is explained in detail in [
In the DFH model, defects with different sizes and orientations are assumed to be randomly distributed within the brittle material. Under static loading the weakest defect is triggered leading to a rapid failure of the sample. Consequently the failure stress is a random variable. Accordingly, a probabilistic approach may be employed to explain the material response to tensile loading at high strain rates. The weakest link theory and Weibull model are adopted as a framework for the damage model [
Obscuration phenomenon and multiple fragmentation process.
The interaction law between cracks already initiated and the critical defects of the material is given by the concept of probability of nonobscuration
More recently, Erzar and Forquin [
In the eigenstress frame, the compliance tensor is defined by
In the numerical analysis discussed below, the Bohus granite rock characterized in [
Material parameters used in the DFH material model.



0.15 




Weibull parameters  

23 

18.7 

1.0 

195 
Obscuration volume parameters  

3.74 

0.38 
At high strain rates, the ultimate strength is deterministic and is obtained from the DFH model as a function of the Weibull parameters as
The material strain rate sensitivity can be described by the DFH model using a multiscale description that is probabilistic at low strain rates and deterministic at high strain rates [
Ultimate strength in granite as a function of the strain rate in logarithmic scale based on the DFH model using the multiscale description and three values of random stress
The equation of motion is discretized using the FE method and the explicit time integration scheme is employed. The numerical simulation of the spalling tests is carried through with the DFH material model implemented as a VUMAT subroutine in the ABAQUS/Explicit software [
FE mesh used in the simulations of spalling tests with 38,000 8noded elements.
First the original DFH model with no cohesion is employed and the results are compared with the experiments. Later on, a cohesive strength is added to the original model to more realistically deal with the softening behavior of the material at dynamic loading. A parameter study is performed to obtain the cohesive model parameters that forms a best fit to the experimental results. It can be seen that adding the cohesive model makes the results more realistic and closer to the experimental results (see Figure
Material parameters used in the cohesion model.
Cohesion model parameters  

1.5 

12 

0.01 

1 
Finite element and experimental results from the spalling test for rear face velocity.
Finite element and experimental results from the spalling test for axial strain.
The numerical modeling of the spalling tests with preexisting structural cracks is performed. As the state of the initial damage in each specimen is not completely clear, a set of numerical analyses is needed to define this state for each test. This calibration stage mainly includes changing the amount of the preexisting cracks to obtain the similar stiffness reduction as the specimen that reflects itself mainly in the postpeak part in the rear face velocity profile.
Figure
Preexisting structural cracks in the analyzed specimen. The finite element mesh is also shown.
Finite element and experimental spalling test results. Preexisting structural cracks are present in the simulations.
Furthermore, a numerical (FEM) quasistatic tensile test is performed on the specimen with initial damage state according to Figure
The rate dependency of tensile strength in granite with and without preexisting cracks is investigated by means of spalling experiments. Considerable strain rate dependency of the tensile strength is obtained at strain rates of about 10^{2} 1/s, this loading rate being pertinent to the situation of rock materials at percussive drilling, which is the application of interest in this investigation. For instance, a dynamic tensile strength of 18.9 MPa is obtained at a strain rate of 70 1/s in a sample without preexisting cracks. This is more than twice the tensile strength of the specimen (with the same size) at quasistatic conditions, which is 8 MPa.
The DFH anisotropic damage model is used to explain the material response at dynamic loading. The DFH model allows predicting the dynamic tensile strength at strain rate of 70 1/s of 19.5 MPa which is fairly close to the experimental results.
Some specimens are exposed to coarser mechanical loading during the cutting process and new cracks, called structural cracks in this work, are introduced in addition to the default material cracks and defects. It is shown that the mechanical response of the material changes dramatically during spalling test due to such preexisting cracks. The lower effective stiffness of these specimens, in tension, reflects itself in the asymmetric postpeak part in the rear face velocity profile. Also the rebounding phenomenon is not seen or is negligible in these specimens and the rear face velocity curve is more plateaulike in the postpeak section.
Numerical modeling (finite element modeling) of the spalling tests is performed. First the original DFH model with no cohesion is employed and the results are compared with the experiments. Later on, a cohesive strength is added to the original model to more realistically deal with the softening behavior of the material at dynamic loading. It is shown that adding the cohesive strength makes the results more realistic and closer to the experimental results.
Furthermore, twoscale numerical modeling (FEM) of the spalling tests accounting for preexisting structural cracks is performed. As the state of the initial damage in each specimen is not completely clear, a first set of numerical analyses is conducted to define this state for each test. The initial cracks are introduced in the numerical model by selecting sets of elements and allocating them negligible tensile strength that leads to their immediate failure when loaded in tension. They can still carry compressive loads when crack closure occurs due to compressive stresses. It is shown that adding such cracks leads to results more similar to the experimental ones.
The authors declare that there are no competing interests regarding the publication of this paper.