Multiscale mechanical properties of shales: grid nanoindentation and statistical analytics

Abstract

The mechanical properties of shales, a type of heterogeneous and multiphase composite, are of multiscale characteristics in nature. A large number of indentation measurements were performed under the continuous stiffness measurement (CSM) mode on the Longmaxi shale, followed by data segmentation, Gaussian mixture modeling (GMM) deconvolution of segmented datasets, and results re-integration. Accompanying compositional analyses by X-ray powder diffraction and surface imaging were also conducted to assist data interpretation. Results showed that the studied shale consisted of a fine-grained, relatively homogeneous clay matrix with randomly embedded, coarse-grained solid inclusions of up to ~ 100 μm in size. The CSM mode enabled the analysis of phase angle lag, which is an effective indicator for surface roughness. The Young’s moduli of both microscopic constituent phases (e.g., clay matrix, carbonates, quartz, pyrite) and macroscopic bulk shale were precisely determined and, through the probability analysis of the indent locations, the characteristic lengths of the shale’s constituent phases were estimated. The clay matrix and carbonates have relatively large characteristic lengths than quartz and pyrite. The data analytics proposed in this study may provide a feasible framework to assess the multiscale mechanical properties as well as the characteristic lengths of the shale’s constituent phases via a single type of measurement technique on the same piece of sample.

Appendix

1.1 Analyses of the phase angles of the harmonic load and displacement

The continuous stiffness measurement (CSM) or dynamic modulus analysis (DMA) mode of nanoindentation testing has been increasingly used for the characterization of non-homogeneous or highly viscous materials, because the stiffness can be determined from the loading section of the indentation measurement. An additional advantage provided by the CSM mode is the determination of the phase angles of the harmonic load and displacement, which cannot be determined by the conventional quasi-static indentation loading or have simply been ignored even with the dynamic loading. Under the CSM mode, a small harmonic oscillation of either force or displacement with a given frequency (e.g., 45 or 75 Hz) and amplitude (e.g., 1 or 2 nm) is superimposed on the primary monotonic quasi-static loading signal. Based on the concept and theory of differential equations and vibrations, the total harmonic force acting on the sample can be described by a second-order ordinary differential equation:

$${\text{m}}\ddot{z} + {\text{D}}\dot{z} + {\text{Kz}} = {\text{F}}\left( {\text{t}} \right)$$

(4)

where m is the mass of the indenter column; z is the displacement; D is the combined viscosity coefficient in both the indentation head and sample; K is the equivalent stiffness, which includes the stiffness of the indenter-sample contact, S, the load frame stiffness, K_f, and the stiffness of the support springs, K_s; that is:

$$ {\text{ K = }}\left( {{\text{S}}^{{ - 1}} {\text{ + K}}_{{\text{f}}}^{{ - 1}} } \right){\text{ + K}}_{{\text{s}}}$$

(5)

${\text{F(t)}}$ and z(t) are the harmonic force and the resulting displacement, respectively, which can be described as:

$$ {\text{F(t) = F}}_{{\text{o}}} {\text{e}}^{{{\text{i}}\omega {\text{t}}}}$$

(6)

$${\text{z(t) = z}}_{{\text{o}}} {\text{e}}^{{{\text{i(}}\omega {\text{t - }}\phi {)}}}$$

(7)

As such, the harmonic displacement oscillates at a frequency same as that of the force, but with a phase angle lag $\phi$. Substituting Eq. (7) into (4) and equating the magnitudes yields:

$$ {\text{tan}}\phi = \omega {\text{D}} / ({\text{K}} - m\omega ^{2})$$

(8)

Based on Eq. (8), the phase angle lag $\phi$ is affected by both the equivalent stiffness K and damping ratio D. The former depends on the quality of contact between the indenter tip and sample surface or the stiffness of the contact. If the indenter is not in contact with the sample surface at all, the equivalent stiffness K is close to zero and, based on Eq. (8), ${\text{tan}}\phi$ is a negative value being equal to –${\text{D / (m}}\omega {)}$, resulting in a constant obtuse phase angle lag, as shown in Fig. 8c and d. As the indenter tip approaches the sample surface, the phase angle lag decreases rapidly when the indenter makes a stiff contact with the asperities of the surface (i.e., surface roughness), and finally becomes a small value constant upon the establishment of a stiff, conforming contact with increasing penetration depth into the sample surface. Thus, the relationship between the phase angle lag and indentation depth makes the phase angle lag an effective indicator for the surface roughness of the indented sample and the selection of the depth ranges in the data exclusion process.

However, the phase angle lag does not affect the accuracy of experiments or cannot be used to characterize the properties of materials except viscosity. On the other hand, the surface roughness can affect the measurement accuracy, especially at shallow depths. Therefore, through the phase angle analysis, the influence of surface roughness on the load and depth measurements can be reduced significantly, and thus the accuracy of data interpretation and analysis can be improved.