Probing Constitutive Models of Bohus Granite with In Situ Spherical Indentation and Digital Volume Correlation

Abstract

Spherical indentation of granite was investigated using Digital Volume Correlation (DVC) aiming at probing constitutive laws of the studied rock. In situ indentation was performed within an X-ray tomograph. Finite element simulations of the problem, using different constitutive models, were carried out and their trustworthiness was assessed thanks to DVC residuals. Three laws were investigated, namely, pure elasticity, then compressible elastoplasticity, and finally compressible elastoplasticity coupled with damage. Frictional contact effects were studied as well. The results show that compressible elastoplasticity should be accounted for to achieve high accuracy of results, and that frictional effects are of importance in terms of damage extent. If macrocrack initiation is also sought, then damage features should be included in the model.

Highlights

• Spherical indentation of granite was investigated using Digital Volume Correlation.

• Different constitutive laws were probed and their trustworthiness assessed thanks to registration residuals.

• Compressible elastoplasticity is to be accounted for to obtain faithful results.

• Frictional effects are of importance in terms of damage extent.

• Macrocrack initiation was described with a damage model.

Appendices

Appendix A: DVC Hardware Parameters

Tomograph	North Star Imaging X50+
X-ray source	XRayWorX XWT-240-CT
Target / Anode	W (reflection mode)
Filter	None
Voltage	140 kV
Current	220 µA
Focal spot size	5 µm
Tube to detector	938 mm
Tube to object	192 mm
Detector	Dexela 2923
Definition	$1536\times 1944$ pixels ($2\times 2$ binning)
Number of projections	1000
Angular amplitude	360°
Frame average	25 per projection
Frame rate	7 fps
Acquisition duration	1 h 24 min
Reconstruction algorithm	Filtered back-projection
Gray Levels amplitude	8 bits
Volume size	$811\times 808\times 707$ voxels (after crop)
Field of view	$25\times 25\times 22$ mm$^3$ (after crop)
Image scale	30.8 µm/voxel
Pattern	Natural (Fig. 2)

Appendix B: Constitutive Laws

In this appendix, the three investigated constitutive models are detailed. The last two models required user defined subroutines to be implement within the explicit version of Abaqus.

1.1 Elasticity (E)

Isotropic elasticity is assumed. The Cauchy stress tensor $\pmb {\sigma }$ is related to the elastic strain tensor $\pmb {\epsilon }^e$ by

$$\begin{aligned} \pmb {\epsilon }^e = \frac{1+\nu }{E} \pmb {\sigma } -\frac{\nu }{E} (\pmb {\sigma }:\mathbf{{1}})\mathbf{{1}} \end{aligned}$$

(9)

where E denotes the Young’s modulus, $\nu$ the Poisson’s ratio, and $\mathbf{{1}}$ the second order identity tensor. In elasticity, the elastic strain tensor $\pmb {\epsilon }^e$ is equal to the infinitesimal strain tensor $\pmb {\epsilon }$.

1.2 Elastoplasticity (EP)

Since the strain levels are assumed to remain small, the strain tensor is additively partitioned into elastic and inelastic parts

$$\begin{aligned} \pmb {\epsilon } = \pmb {\epsilon }^e + \pmb {\epsilon }^i \end{aligned}$$

(10)

where $\pmb {\epsilon }^i$ denotes the inelastic strain tensor. The Drucker-Prager (1956) yield function $F_y$ reads

$$\begin{aligned} F_y = q - p \tan (\beta ) - d \end{aligned}$$

(11)

where q is von Mises’ equivalent stress, p the hydrostatic pressure (i.e., $p=-1/3(\pmb {\sigma }:\mathbf{{1}})$), d the cohesion, and $\beta$ the friction angle.

The flow potential reads

$$\begin{aligned} G = q - p \tan (\psi ) \end{aligned}$$

(12)

where $\psi$ is the dilation angle (Shariati et al. 2019)

$$\begin{aligned} \psi = \tan ^{-1}\left( \frac{3}{3\frac{{\dot{\epsilon }}^i_a}{{\dot{\epsilon }}^i_v}-1}\right) \end{aligned}$$

(13)

${\dot{\epsilon }}^i_a$ the inelastic axial strain rate, and ${\dot{\epsilon }}^v_a$ the inelastic volumetric strain rate.

1.3 Elastoplasticity Coupled with Damage (EPD)

The previous model essentially describes the behavior in compressive (i.e., confined) states. In the present case, it is activated in the immediate vicinity of the indentation zone (Shariati et al. 2019, 2020). Farther away from this compacted zone, cracks may initiate due to tensile stresses (Shariati et al. 2019). To describe the early stages of such mechanism, the so-called DFH model is considered (Denoual and Hild 2002; Forquin and Hild 2010) and coupled with the previous one. It was already applied in the study of dynamic fragmentation of Bohus granite (Saadati et al. 2014, 2015; Shariati et al. 2020).

An anisotropic damage model is considered. In the principal frame, the compliance tensor of damaged elements becomes

$$\begin{aligned}{}[\mathbf{{S}}^D] = \frac{1}{E} \left[ \begin{array}{ccc} \frac{1}{1-D_1} &{} -\nu &{} -\nu \\ -\nu &{} \frac{1}{1-D_2} &{} -\nu \\ -\nu &{} -\nu &{} \frac{1}{1-D_3} \\ \end{array}\right] \end{aligned}$$

(14)

and the growth law of each damage variable $D_i$ associated with principal stress $\sigma _i$ reads

$$\begin{aligned}&\frac{d^2}{dt^2}\left( \frac{1}{1-D_i} \frac{dD_i}{dt} \right) = 6S(kC_0)^3 {\widehat{\lambda }}_t({\widetilde{\sigma }}_i(t)) \quad \text{ if } \quad \sigma _i>0 \nonumber \\&\quad \text{ and } \quad \frac{d\sigma _i}{dt} >0 \end{aligned}$$

(15)

with

$$\begin{aligned} V_{FE} {\widehat{\lambda }}_t({\widetilde{\sigma }}_i(t)) = \left\{ \begin{array}{cl} 0 &{} \text{ if } \quad \sigma _i(t) < \sigma _k \\ \max \left( 1,V_{\mathrm {FE}}\lambda _0\left( \frac{{\widetilde{\sigma }}_i(t)}{\sigma _0}\right) ^m\right) &{} \text{ otherwise }\\ \end{array} \right. \end{aligned}$$

(16)

where ${\widetilde{\sigma }}_i$ is the i-th effective principal stress (i.e., ${\widetilde{\sigma }}_i = \sigma _i/(1-D_i)$, with no index summation), $V_{\mathrm {FE}}$ the volume of the considered finite element, m and $\sigma _0^m/\lambda _0$ the Weibull parameters (Weibull 1939), $C_0$ the longitudinal wave speed so that the crack propagation velicity is $kC_0$ (with $k=0.38$), and $S=3.74$ the dimensionless shape parameter of the relaxation zones (Saadati et al. 2015). The stress $\sigma _k$ is randomly selected for each finite element as the initiation level for the first crack according to the Weibull model

$$\begin{aligned} \sigma _k = \sigma _0 \left( -\frac{\log (1-P_k)}{\lambda _0V_{\mathrm {FE}}}\right) ^{1/m} \end{aligned}$$

(17)

where $P_k$ denotes the initiation probability, which is randomly selected in a uniform distribution ranging from 0 to 1. If the weakest link hypothesis applies, the initiation probability is equal to the cumulative failure probability of the considered element

$$\begin{aligned} P_F = 1-\exp \left( -\lambda _0V_{\mathrm {FE}} \left( \frac{\sigma _k}{\sigma _0}\right) ^m\right) \end{aligned}$$

(18)

and the corresponding mean failure stress becomes

$$\begin{aligned} {\overline{\sigma }}_F = \frac{\sigma _0}{\left( \lambda _0V_{\mathrm {FE}}\right) ^{1/m}} \Gamma \left( 1+\frac{1}{m}\right) \end{aligned}$$

(19)

where $\Gamma$ is the Euler function of the second kind (Abramowitz and Stegun 1965). If the failure probability of the whole structure is to be evaluated, the volume $V_{\mathrm {FE}}$ of any element has to be replaced by the effective volume $V_{\mathrm {eff}}$ (Davies 1973).