Spherocylindrical microplane constitutive model for shale and other anisotropic rocks

Abstract

Constitutive equations for inelastic behavior of anisotropic materials have been a challenge for decades. Presented is a new spherocylindrical microplane constitutive model that meets this challenge for the inelastic fracturing behavior of orthotropic materials, and particularly the shale, which is transversely isotropic and is important for hydraulic fracturing (aka fracking) as well as many geotechnical structures. The basic idea is to couple a cylindrical microplane system to the classical spherical microplane system. Each system is subjected to the same strain tensor while their stress tensors are superposed. The spherical phase is similar to the previous microplane models for concrete and isotropic rock. The integration of stresses over spherical microplanes of all spatial orientations relies on the previously developed optimal Gaussian integration over a spherical surface. The cylindrical phase, which is what creates the transverse isotropy, involves only microplanes that are normal to plane of isotropy, or the bedding layers, and enhance the stiffness and strength in that plane. Unlike all the microplane models except the spectral one, the present one can reproduce all the five independent elastic constants of transversely isotropic shales. Vice versa, from these constants, one can easily calculate all the microplane elastic moduli, which are all positive if the elastic in-to-out-of plane moduli ratio is not too big (usually less than 3.75, which applies to all shales). Oriented micro-crack openings, frictional micro-slips and bedding plane behavior can be modeled more intuitively than with the spectral approach. Data fitting shows that the microplane resistance depends on the angle with the bedding layers non-monotonically, and compressive resistance reaches a minimum at 60°. A robust algorithm for explicit step-by-step structural analysis is formulated. Like all microplane models, there are many material parameters, but they can be identified sequentially. Finally, comparisons with extensive test data for shale validate the model.Peer ReviewedPostprint (published version

Branching of hydraulic cracks enabling permeability of gas or oil shale with closed natural fractures

“…where C ijkl = transversely isotropic elastic compliance tensor of shale (for unloading); σij, ij are the stress and strain tensors in the rock, calculated from a constitutive model for smeared cracking damage (with a localization limiter (25)), for which the spherocylindrical microplane constitutive model (28) has been used. The coordinates are Cartesian, xi, i = 1, 2, 3 (x1 ≡ x, x2 ≡ y, x3 ≡ z).…”

Anisotropic elastic, strength, and fracture properties of Marcellus shale

“…These conclusions ought to be taken into account in various situations relevant to geological engineering design, construction, and operation where a large traction-free crack can grow prior to failure, and when extrapolation from small-scale laboratory tests to field-size structures is needed. In particular, the effect of size becomes extremely important in hydraulic fractures [77,78,79].…”

Conversion of explicit microplane model with boundaries to a constitutive subroutine for implicit finite element programs

“…Now we can proceed similarly to Equation (5) (analogous to eqs. (11)(12)(13)(14) of Bažant and Prat 4 ), which is based on the variational principle of virtual work and gives the corresponding incremental tensor of inelastic strain:…”

“…Although the M7 version of microplane model is selected in this study, thanks to its success in representing all kinds of material test data for concrete, similar explicit–implicit conversions can be made in other microplane models with boundary inequalitites. A few examples can be listed, such as the anisotropic spherocylindrical microplane model 11 for shale, the triad‐microplane model 12,47 for textile twill composites, or the fatigue microplane model for concrete 48 …”

“…Beginning from 1996, 10 a series of progressively improved explicit microplane damage models have been developed for plain and fiber-reinforced concrete, textile composites, shale and other rocks. 11,12 They have been extensively used as user's material models in the form of explicit Fortran subroutines in, for example, VUMAT of ABAQUS. Based on dynamic relaxation, the explicit FE programs are effective computationally, but sometimes the errors due to artificial inertia and damping may be troublesome.…”