Final Report: Cryo-mechanics of unsaturated frozen soils during freeze-thaw cycles

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Final Report: Cryo-mechanics of unsaturated
frozen soils during freeze-thaw cycles
6/30/2018
Contract Number: W911NF1510442
Grantee Proposal Number:
Period of Performance (Reporting Period for this report)
Start: Aug, 15 2015 End: Aug, 14 2016
SUBMITTED BY
WaiChing Sun
Assistant Professor
Department of Civil Engineering and Engineering Mechanics
Columbia University in the City of New York
Phone: 212-851-4371
Email: wsun@columbia.edu

Principal Investigator
• WaiChing Sun, Assistant Professor, Civil Engineering and Engineering Mechanics, Fu
Foundation School of Engineering and Applied Science, Columbia University
Postdoc Research Scientist
• Jinhyun Choo, postdoc research scientist (now assistant professor at the University of Hong
Kong).
Research Assistant
1. Kun Wang, PhD Candidate Civil Engineering and Engineering Mechanics, Fu Foundation
School of Engineering and Applied Science, Columbia University (supported by ARO and
Columbia cost share)
2. SeonHong Na, PhD Candidate Civil Engineering and Engineering Mechanics, Fu
Foundation School of Engineering and Applied Science, Columbia University (supported
by Columbia cost share). Will join McMaster University as assistant professor in Spring
2019.
3. Albert Martini, MS Candidate, Civil Engineering and Engineering Mechanics, Fu
Foundation School of Engineering and Applied Science, Columbia University (supported
by Columbia cost share)
4. Luca Tassini, Visiting Scholar, University of Perugia, Italy (supported by University of
Perugia, Italy)
Major Activities:
The support has been used to support research activities of the Sun Research Group at Columbia
University. This DURIP grant provides the funding to purchase an equipment that is critical for the
success of the number of research ideas that eventually lead to research works published in peer-
reviewed journals and the career growth of the PI and the PhD students. Due to the nature of the
research grant is for equipment purchase, the impact of the support goes beyond the original
performance period and will last as long as the equipment is continuously being used, as in the case
of this project.
The purchase request was sent by the PI as soon as the award letter was issued on August 15, 2015.
After finishing various types of paperwork and fulfilling a variety of requirements by Columbia
University required to make the purchase, the PI was able to make the purchase order with the
support of the Carleton laboratory, on 4/29/2016. The triaxial apparatus was arrived on 5/23/2016.
Following this, an engineering from the manufacturer GDS instruments, Mr. Diogo Teles was sent
to Columbia to help setting up the apparatus and complete the training. The training sessions are
attended by the PI, graduate students SeonHong Na, Kun Wang and Zhijun Cai and visiting scholar
Luca Tassini on 6/2/ to 6/3/2016.
Following the training session, the research group has conducted triaxial compression tests under
different combination of temperature, strain rate and confining pressure under the undrained
condition ever since. We are particularly interested at the behaviors of the frozen soil near the phase

transition temperature and hence we designed a series of tests, each with the same undrained
loading path, but are conducted under fixed temperature. The majority of the tests are taken place
during the visit of Luca Tassini, while graduate student Alberto Martini has undertaken the rest of
the test before his MS graduation. The major discoveries and findings are outlined in the next two
sections.
In the meantime, we have developed a new poromechanics theory that takes account of the pre-
melting theory, the phenomenological effect of thawing and freezing on the shear strength of the
materials but also, equally important, the hydraulic and thermal conductivities that vary with the
unfrozen water thin film exists between the solid grains and ice crystals. This eventually leads to
the first finite strain theory for thermo-hydro-mechanical responses of frozen soil and was
published in Computer Methods in Applied Mechanics and Engineering, one of the most
prestigious journals, not only in the field of computational geomechanics, but also in computational
mechanics. This work mainly done by PhD student SeonHong Na, who has worked with the PI to
derive the theory, implement the computer code, and invent new computer algorithms, including a
new operator-split algorithm and a new pre-conditioner to make the solver feasible for the highly
complex multi-physical problems.
Specific Objectives:
In this DURIP project, our goal is to find answers for the following important scientific questions
for frozen soil that could have significant impacts on cold region geotechnical engineering and
geomechanics.
1. How does macroscopic thermal effect vary under different degrees of saturation? One key
important objective we would like to achieve is to distinguish how yield surface, plastic
potential, hardening laws and thermo-elasticity of the soil differ when subjected to (i) small
temperature change but no phase transition occurs and (ii) large temperature change that
induces the water-ice phase transition?
2. How does stress anisotropy induced by liquid bridge evolve during freeze-thaw cycle?
While liquid bridge is well known to induce stress anisotropy in the pendular regime, this
anisotropy is often considered insignificant and neglected in the classical Bishop’s effect
stress theory as liquid bridge is prone to rupture. However, if a volume fraction of water is
frozen, this frozen part of the liquid bridge will be solidified and thus potentially intensify
the anisotropic effect. On the other hand, the experimental study can also shed light on how
anisotropy of stress evolves during thaw-weakening process.
3. How does spatial heterogeneity of material properties driven by temperature and unfrozen
water diffusion affect the validity of four-phase mixture theory and strength
homogenization scheme between ice- and solid-phase? This is a critical issue that has yet
been successfully addressed. Since the heat transfer and flow of the unfrozen water are not
in the same temporal and spatial scales, a macroscopic constitutive law without internal
length scale might not be valid or sufficient. The data set generated from this research

program can provide evidence to test the validity of the four-phase (ice- solid-water-air)
mixture theory originally proposed in Zhou and Meschke [2013] and determine whether
classical homogenization scheme based on equivalent inclusion method is sufficient for
modeling frozen soil undergoing phase transitions.
4. How do temperature change, phase transition and flow of unfrozen water in pore space
affect the onset and propagation of micro-cracks and affect macroscopic behaviors at high
confining pressure?
Significant Results:
Experimental Studies on Frozen Soil
During the report period, the Columbia research team has leveraged the cost share to conduct tests
on frozen soils. To validate the correct functioning of the machine and the procedure used, the
results of one of the tests performed in the laboratory, at room temperature, have been compared
with the results provided by VELACS (Verification of Liquefaction Analysis by Centrifuge
Studies), also obtained on Nevada Sand samples, with failure point at room temperature. The
comparison showed a perfect qualitative correspondence between the results.
Once the machine has been tested, a VELACS test (carried out on a sample of Nevada Sand with
a relative density of 40%) has been used for the comparison with the tests carried out at Columbia
University. During these experiments, to highlight the basic characteristics of frozen soils (all the
tests have been conducted at the same confining pressure), the parameters that were changed are
the temperature and axial strain rate.
In addition to the expected changes in the constitutive responses due to the formation and thawing
of ice crystals in the pore space, we also found that the crystallization of ice in the pores has a
profound influence on the failure mode. Our experiments indicate that while shear band may still
form at a temperature low enough to trigger ice crystallization, a slight reduction in temperature
may significantly strengthen the ice in the pore space, leads to highly undrained shear strength but
also replace the highly localization failure mode with a diffuse one as shown in Figure 1. The
research group is currently working on a manuscript to summarize the experimental findings and
also to obtain more evidences from experiments conducted by PhD student Nicholas Vlassis.
Figure 1: specimen at the end of the undrained triaxial compression test under different temperature (LEFT), the

deviatoric stress vs. axial strain of tests at different temperature and loading rates.
Computational Thermo-hydro-mechanics of Frozen Soil at the Finite Deformation Range
In this paper we present for the first time a finite strain poromechanics theory that fully considers
the thermo-hydro-mechanical coupling effect of the mass-exchanging, phase-transiting porous
media. Previously, significant contributions have been made to derive thermal-sensitive or degree-
of-saturation-sensitive constitutive laws for the frozen soil. An implicit total Lagrangian finite
element framework is formulated, while thermal and cryo-suction effects are explicitly captured by
a generalized hardening rules that allow the yield surface to evolve based on the volume fraction
of ices in the pore space and the temperature. In addition, we also highlight a number of numerical
issues that are crucial for developing a practical and robust numerical implementation. Numerical
examples are provided for elucidating the mechanical behavior of frozen soil under thawing and
freezing conditions. The results indicate that a comprehensive model that explicitly captures the
multiple thermo-hydro-mechanical coupling mechanisms of frozen porous media (instead of
lumping them together through phenomenological laws) may yield far more accurate and reliable
results. This elegant approach also eliminates the needs to introducing excessive amount of ad hoc
parameters solely for curve-fitting, and is therefore easier to calibrate and more practical.
Figure 2: Constitutive laws for the thermo-hydro-mechanical coupling responses of frozen soil. Figure reproduced
from Na and Sun, CMAME 2017.
The details of the constitutive law is summarized in Figure 2 in which the degree of saturation of
ice is obtained from pre-melting theory such that the ice crystal pressure is a function of temperature,

pore water pressure of the unfrozen water and the cryo-suction pressure. Using the idea of
generalized critical state plasticity, the Cam-clay model is extended such that the hardening rules
of the yield surface depends on the degree of saturation of ice. The upshot of this approach is that
the model may seamlessly reduce into the classical soil model in room temperature but also
incorporate the effects of freezing and thawing.
Microstructural Analysis of Geomaterials via micro-CT images
In this study, we have adopted a hierarchical multi-scale approach [White et al., 2006; Sun et al.,
2011a; 2011b] to simulate hydraulic and electrical transport in three samples of Fontainebleau
sandstone that had been imaged using X-ray CT by Lindquist et al. [2000]. Each specimen was
partitioned into 8 unit cubes with linear dimensions of 1.08 mm for pore-scale LB calculation of
permeability and formation factor. Our analysis of the geometry and percolative structure indicate
that dimensions of the unit cubes are sufficiently large for treating them as REV for numerical
simulation of the transport properties. The image resolution also seems adequate, and there is not
a need to acquire CT images at a more refined resolution for such digital rock physics applications,
in agreement with the analysis of Thovert et al. [2001] that focused on the percolation and electrical
conductivity.
Unlike previous studies that used different numerical techniques for the hydraulic and electrical
transport, we here employed LB simulations for both, which provide a more consistent basis for
synthesizing permeability, formation factor and geometric attributes. The LB method has seldom
been used to simulate formation factor of porous rock. The methodology we adopted here proved
to be effective, but we should also note that by no means is it the only feasible approach for
simulating electrical transport. For example, Chai and Shi [2008] have proposed an alternative
formulation by adding an additional term on the right-hand side of the evolution equation such that
the Chapman-Enskog expansion in time and space would recover the exact Poisson equation when
diffusivity is non-zero.
Figure 3: Permeability of Fontainebleau sandstone as a function of porosity. Simulated values from this study (with
pressure gradient along the axial direction) are shown as solid squares. For comparison, laboratory data from the
following studies are also plotted: Bourbie and Zinszner, Doyen, Fredrich et al. and Gomez et al..(LEFT) and Formation
factor of Fontainebleau sandstone as a function of porosity. Values from this study from LB and FE simulations are
shown as blue and brown circles, respectively. For comparison, laboratory data from the following studies are also
plotted: Doyen, David and Darot, Fredrich et al., and Gomez et al..(RIGHT). Figure reproduced from Sun & Wang,

IJRMMS, 2018.
Our permeability calculation, shown in Fig. 2 is closer to the experimental measurements than
previous numerical studies (most of which also used the LB method), especially when the porosity
is below 10%. The improvement here may be attributed to several factors. First, advance in
computation capability has allowed us to consider a unit cube of larger dimension, that more closely
approximates a REV of the porous sandstone. Whereas we considered a sub-volume of 1903 voxels,
Arns et al.35 started off with a sub-volume of only 1203 voxels, and to circumvent numerical issues
related to percolation in the low-porosity regime, they had to “fine grain” the CT image by replacing
each voxel by 2 × 2 × 2 voxels with half the linear dimension. Second, difference in the LB
methodology may have contributed to the discrepancy, but in the absence of more detailed
description of their numerical procedure, it is difficult to assess this issue. Lastly, difference in
segmentation algorithms can also result in subtle differences in pore geometry of segmented images,
which would impact permeability values derived from LB simulations.
In our hybrid scheme, the pore-scale LB results were linked with FE simulation in a homogenized
approach to compute and upscale formation factor and permeability tensor at specimen scale. White
et al.15 have demonstrated the feasibility and effectiveness of such an approach for calculating the
permeability of Castlegate sandstone (with porosity of 20–24%). Our study has shown that this
hybrid scheme can be extended to calculating both effective permeability and formation factor of
a porous sandstone, with porosities over a relatively broad range that is associated with an
appreciable change in connectivity.
These results indicate that flow simulations at pore-scale could be used as a tool for predicting both
formation factor and permeability of geological materials. This is significant also for the frozen soil
problems where important material parameters such as relative permeability for the unfrozen water
and the thermal conductivities might not be so easy to extract from experiments. In those cases,
microCT images are not only useful to capture the growth of the ice lens and the freezing of the
water in the pore space, but also for predicting the evolution of the effective material parameters as
a function of porosity, temperature, confining pressure and microstructural attributes.
Figure 4: MicroCT images of ice crystral growth in chalk specimen. Figure obtained from collaborator Cino Viggiani
Currently, the PI is preparing a new set of theoretical work that incorporate an important set of
experimental work where the growth mechanism of the ice crystal is captured in great details in a
series of microCT images done by the Grenoble research group. The PI has obtained agreement
between the researchers there to share their microCT images for future analysis, should the
opportunity comes for future study on frozen soil.
Brittle-ductile transition
The test results from experiments also lead us to we to derive a framework that couples a phase-
field description of fracture with pressure-sensitive plasticity, for encapsulating a wide range of
failure modes of geomaterials from brittle fracture to ductile flow. Particular attention is paid to a

realistic capture of the transition of these failure modes with changes in confining pressure and
strain rate conditions. In the process of our development, several new contributions are presented
to make the two-way coupling between phase-field and plasticity theoretically consistent and
numerically robust. They include a microforce-based derivation of a phase-field model that honors
the dissipative nature of the fracturing process, the dilative/compactive split and rate-sensitive
storage of plastic work, and the use of phase-field effective stress for plasticity. The results are
published in Choo and Sun, CMAME 2018 and summarized in Figure 5.
By coupling the phase field fracture model with a cap plasticity, we introduce a new approach
where the brittle fracture, quasi-brittle damage, ductile fracture, shear banding and the diffusive
failures in high confining pressure can all be simulated and predicted by one single model with the
same material parameter. This consistency is critical for extending the frozen soil model for real-
world application, as the formation of the ice will profoundly strength the materials such that the
soil may exhibit rock-like behaviors. This is a very interesting problem. As the ice and solid grains
may in that case behaves like a two-phase composite, the constitutive responses will become
inherently anisotropic and heterogeneous.
Figure 5: Modeling of brittle-ductile transition of geological materials. Figure reproduced from Choo & Sun, CMAME,
2018.
Key outcomes or other achievements:
From the scientific viewpoint, this work has led to 7 journal articles publication, 1 PhD thesis and
3 conference papers. This project is instrumental to the success of the PI and his students as
exemplified by the following major honors the PI received since the beginning of the project. In
particular, the PI is the only person who has ever won both the EMI Leonardo da Vinci Award from
American Scoeity of Civil Engineers and the Zienkiewicz Numerical Methods in Engineering Prize
from the United Kingdom Institution of Civil Engineers, both are the most prestigious honors for

the respectively society that has age limits.
1. EMI Leonardo da Vinci Award, the Engineering Mechanics Institute of American Society of
Civil Engineers, 2018. The purpose of the award is to recognize outstanding young investigators
early in their careers for promising ground-breaking developments in the field of Engineering
Mechanics and Mechanical Sciences as relevant to Civil Engineering, understood in the broadest
sense. The award is given annually to a young investigator, generally under 35 years of age or have
worked no more than 7 years since receiving their doctoral degree, and whose contributions have
the promise to define new directions in theory and application of Engineering Mechanics, in the
vein of Leonardo da Vinci (1452-1519), a man of unquenchable curiosity and feverishly inventive
imagination. The EMI of ASCE selected the PI "for his fundamental contributions to computational
multiscale poromechanics".
2. Zienkiewicz Numerical Methods in Engineering Prize, Institution of Civil Engineers (ICE) and
John Wiley & Sons, 2017. Instituted following a donation by John Wiley & Sons Ltd to
commemorate the work of Professor Olgierd Cecil Zienkiewicz CBE. DSc FRS FREng of the
Institute for Numerical Methods in Engineering, University of Wales , Swansea. The medal is
awarded biennially by the Institution of Civil Engineers (ICE) to a researcher under 40 for the paper
which contributes most to research in numerical methods in engineering, among 8 prime peer-
reviewed journals published by ICE or Wiley, i.e., Geothechnique, Geothechnique Letters,
International Journal for Numerical Methods in Engineering, International Journal for Numerical
Methods in Biomedical Engineering, International Journal for Numerical Methods in Fluids,
International Journal for Numerical and Analytical Methods in Geomechanics, International
Journal of Numerical Modelling: Electronic Networks, Devices and Fields, and ICE Proceedings.
3. AFOSR Young Investigator Program Award, Air Force Office of Scientific Research, US Air
Force, 2017. The Air Force’s Young Investigator Program (YIP) award is one of the most
prestigious honors bestowed by the US Air Force to outstanding scientists beginning their
independent careers. The program is designed to identify and support talented scientists and
engineers who show exceptional promise for doing creative research in order to encourage their
teaching and research careers.
More importantly, the research group has successfully secured three tenure-track positon for three
former members, PhD student SeonHong Na (McMaster University, Canada), Jinhyun Choo (the
University of Hong Kong, Hong Kong) and Yang Liu (Northeastern University, USA). These
results are the testimonial of the quality of the work of this research group.
The following peer-reviewed journal articles and conference papers are completed either fully or
partially supported by this grant:
1. K. Wang, W.C. Sun, Anisotropy of a tensorial Bishop's coefficient for wetted granular
materials, Journal of Engineering Mechanics, doi:10.1061/(ASCE)EM.1943-
7889.0001005, 2015. [PDF] [Bibtex]
2. Y. Liu, W.C. Sun, J. Fish, Determining material parameters for critical state plasticity
models based on multilevel extended digital database, Journal of Applied
Mechanics, 88(1), doi: 10.1115/1.4031619, 2015. [PDF][Bibtex]
3. Y. Liu, W.C. Sun, Z-F. Yuan, J. Fish, A nonlocal multiscale discrete-continuum model
for predicting mechanical behavior of granular materials, International Journal for
Numerical Methods in Engineering, 106(2):129-160, doi: 10.1002/nme.5139, 2016.

[PDF] [Bibtex] (PhD Student Yang Liu won 2015 best poster competition at USNCCM
San Diego).
4. N. Guo, J. Zhao, W.C. Sun, Multiscale analysis of shear failure of thick-walled hollow
cylinder in dry sand, Géotechnique Letters, 6(1), doi:10.1680/jgele.15.00149, 2016.
[PDF][Bibtex]
5. S. Na, W.C. Sun, Wave propagation and strain localization in a fully saturated softening
porous medium under the non-isothermal conditions, International Journal for Numerical
and Analytical Methods in Geomechanics, 40(10):1485-
1510, doi:10.1002/nag.2505, 2016. [PDF][Bibtex]
6. Z. Zheng*, W.C. Sun, J. Fish, Micropolar effect on the cataclastic flow and brittle-ductile
transition in high-porosity rocks, Journal of Geophysical Research,
doi:10.1002/2015JB012179, 2016. [Bibtex]
7. K. Wang, W.C. Sun, A semi-implicit discrete-continuum coupling method for porous
media based on the effective stress principle at finite strain, Computer Methods in
Applied Mechanics and Engineering, 304(1):546-583, doi:10.1016/j.cma.2016.02.020,
2016. [PDF][Bibtex]
8. K. Wang, W.C. Sun, S. Salager, S. Na, G. Khaddour, Identifying material parameters for
a micro-polar plasticity model via X-ray micro-CT images: lessons learned from the
curve-fitting exercises, International Journal of Multiscale Computational Engineering,
14(4):389-413, doi:10.1615/IntJMultCompEng.2016016841, 2016. [PDF][Bibtex]
9. A.G. Salinger, R.P. Pawlowski, Eric T. Phipps, R.A. Bartlett, G.A. Hansen, I.
Kalashnikova, J.T. Ostien, W.C. Sun, Q. Chen, A. Mota, R.A. Muller, E. Nielsen, X.
Gao. Albany: A Component-Based Partial Differential Equation Code Build on
Trilinos, International Journal of Multiscale Computational Engineering, 14(4):415-
438, doi:10.1615/IntJMultCompEng.2016017040, 2016. [PDF][Bibtex]
10. W.C. Sun, Foreword: computational poromechanics, International Journal of Multiscale
Computational Engineering, doi:10.1615/IntJMultCompEng.2016018596, 2016.
[PDF][Bibtex]
11. K. Wang, W.C. Sun, A unified variational eigen-erosion framework for interacting
fractures and compaction bands in brittle porous media, Computer Methods in Applied
Mechanics and Engineering, 318:1-32 doi:10.1016/j.cma.2017.01.017, 2017.
[PDF][Bibtex]
12. S. Na, W.C. Sun, Computational thermo-hydro-mechanics for multiphase freezing and
thawing porous media in the finite deformation range, Computer Methods in Applied
Mechanics and Engineering, 318:667-700, doi:10.1016/j.cma.2017.01.028, 2017. (PhD
Student SeonHong Na selected as runner-up for the 2017 best paper competition at EMI
Nashville) [PDF][Bibtex]
13. W.C. Sun, Z. Cai, J. Choo", Mixed Arlequin method for multiscale poromechanics
problems, International Journal for Numerical Methods in Engineering, 111:624-659,
doi:10.1002/nme.5476, 2017. [PDF][Bibtex]
14. I. Wollny, W.C. Sun, M. Kaliske, A hierarchical sequential ALE poromechanics model
for tire-water-road interaction on fluid-infiltrating roads, International Journal for
Numerical Methods in Engineering, 112(8):909-938, doi:10.1002/nme.5537, 2017.
[PDF][Bibtex]
15. S. Na, W.C. Sun, H. Yoon, M. Ingraham, Effects of elastic heterogeneity on the fracture
pattern and macroscopic effective toughness of Mancos Shale in Brazilian Tests, Journal
of Geophysical Research:Solid Earth, B013374, 122(8): 6202-
6230 , doi:10.1002/2016JB013374, 2017. [PDF][Bibtex]
16. H. Xin*, W.C. Sun, J. Fish, a surrogate modeling approach for additive-manufactured
materials, International Journal of Multiscale Computational
Engineering, doi:10.1615/IntJMultCompEng.2017024632, 2017.

17. H. Xin*, W.C. Sun, J. Fish, Thermo-mechanical discrete element simulations on powder-
bed sintering-based additive manufacturing, International Journal of Mechanical
Sciences, doi:10.1016/j.ijmecsci.2017.11.028, 2017.
18. O.I. Ulven*, W.C. Sun, Capturing the two-way hydro-mechanical coupling effect on
fluid-driven fracture in a dual-graph lattice beam model, International Journal for
Numerical and Analytical Methods in Geomechanics, 42(5):736-
767, doi:10.1002/nag.2763, 2018. [PDF]
19. J. Choo", W.C. Sun, Coupled phase-field and plasticity modeling of geological materials:
From brittle fracture to ductile flow, Computer Methods in Applied Mechanics and
Engineering, 330:1-32, doi:10.1016/j.cma.2017.10.009, 2018. [PDF][Bibtex]
20. J. Choo", W.C. Sun, Cracking and damage from crystallization in pores: Coupled chemo-
poro-mechanics and phase-field modeling, Computer Methods in Applied Mechanics and
Engineering, 335:347-379, doi:10.1016/j.cma.2018.01.044, 2018.[PDF][Bibtex]
21. K. Wang, W.C. Sun, A multiscale multi-permeability poroplasticity model linked by
recursive homogenizations and deep learning, Computer Methods in Applied Mechanics
and Engineering, 334(1):337-380, doi:10.1016/j.cma.2018.01.036, 2018. [PDF][Bibtex]
22. S. Na, W.C. Sun, Computational thermomechanics of crystalline rock. Part I: a combined
multi-phase-field/crystal plasticity approach for single crystal simulations, Computer
Methods in Applied Mechanics and Engineering, 338:657-
691, doi:10.1016/j.cma.2017.12.022, 2018. [PDF][Bibtex]
23. W.C. Sun, T-F. Wong, Prediction of hydraulic and electrical transport properties of
sandstone with multiscale lattice Boltzmann/finite element simulation on
microtomographic images, International Journal of Rock Mechanics and Mining
Sciences, 106, 269-277, doi:10.1016/j.ijrmms.2018.04.020, 2018. [PDF][bibtex]
24. G. Liu*, W.C. Sun, S.M. Lowinger**, Z. Zhang, M. Huang, J. Peng, Flow network and
discrete element modeling of injection-induced crack propagation and coalescence in
brittle rock, Acta Geotechnica, doi:10.1007/s11440-018-0682-1, 2018.