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Journal articles on the topic 'Brittle damage'

Author: Grafiati

Published: 6 September 2023

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1

Lawn, B. R., K. S. Lee, H. Chai, A. Pajares, D. K. Kim, S. Wuttiphan, I. M. Peterson, and X. Hu. "Damage-Resistant Brittle Coatings." Advanced Engineering Materials 2, no. 11 (November 2000): 745–48. http://dx.doi.org/10.1002/1527-2648(200011)2:11<745::aid-adem745>3.0.co;2-e.

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2

Brannon, Rebecca M., Joseph M. Wells, and O. Erik Strack. "Validating Theories for Brittle Damage." Metallurgical and Materials Transactions A 38, no. 12 (September 28, 2007): 2861–68. http://dx.doi.org/10.1007/s11661-007-9310-7.

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3

Zheng Li, Yundong Shou, Deping Guo, and Filippo Berto. "A coupled elastoplastic damage model for brittle rocks: elastoplastic damage model for brittle rocks." Frattura ed Integrità Strutturale 14, no. 53 (June 11, 2020): 446–56. http://dx.doi.org/10.3221/igf-esis.53.35.

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4

Liu, Dong Xin, Lu Ming Shen, Itai Einav, and Francois Guillard. "Numerical Investigation on the Failure Behavior of Brittle Granular Chain under Impact." Applied Mechanics and Materials 846 (July 2016): 205–10. http://dx.doi.org/10.4028/www.scientific.net/amm.846.205.

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In brittle granular materials, the fragmentation waves have received far less attention due to their complexity despite of their significant role in mineral processes, earthquake hazards control, etc. In this research, the Material Point Method (MPM) is used to analyze how fragmentation waves propagate in a 3-dimensional 10 brittle beads chain with a rate-dependent elasto-damage model. The simulations show that generally, the second bead will become the most severely damaged one, followed by the third bead. Most failure points will appear near the contact surface between the brittle spheres and extend to interior conically. An interesting phenomenon is that with a lower damage threshold or fracture energy, despite of the increase of total damage in the whole chain, less damage is developed in some beads after a period of time. This is mainly because more damage in the beginning dissipates excessive stress wave energy to the extent such that the reflected wave will not be able to cause more damage in the local system.

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5

Kubik, Jan, and Zbigniew Perkowski. "Description of Brittle Damage in Concrete." Communications - Scientific letters of the University of Zilina 4, no. 3 (September 30, 2002): 9–12. http://dx.doi.org/10.26552/com.c.2002.3.9-12.

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6

Deng, H., and S. Nemat-Nasser. "Dynamic damage evolution in brittle solids." Mechanics of Materials 14, no. 2 (December 1992): 83–103. http://dx.doi.org/10.1016/0167-6636(92)90008-2.

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7

Welemane, Hélène, and Cristina Goidescu. "Isotropic brittle damage and unilateral effect." Comptes Rendus Mécanique 338, no. 5 (May 2010): 271–76. http://dx.doi.org/10.1016/j.crme.2010.04.005.

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8

Basista, M., and D. Gross. "A note on brittle damage description." Mechanics Research Communications 16, no. 3 (May 1989): 147–54. http://dx.doi.org/10.1016/0093-6413(89)90052-9.

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9

Mudunuru, Maruti Kumar, Nishant Panda, Satish Karra, Gowri Srinivasan, Viet T. Chau, Esteban Rougier, Abigail Hunter, and Hari S. Viswanathan. "Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials." Applied Sciences 9, no. 13 (July 3, 2019): 2706. http://dx.doi.org/10.3390/app9132706.

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In brittle fracture applications, failure paths, regions where the failure occurs and damage statistics, are some of the key quantities of interest (QoI). High-fidelity models for brittle failure that accurately predict these QoI exist but are highly computationally intensive, making them infeasible to incorporate in upscaling and uncertainty quantification frameworks. The goal of this paper is to provide a fast heuristic to reasonably estimate quantities such as failure path and damage in the process of brittle failure. Towards this goal, we first present a method to predict failure paths under tensile loading conditions and low-strain rates. The method uses a k-nearest neighbors algorithm built on fracture process zone theory, and identifies the set of all possible pre-existing cracks that are likely to join early to form a large crack. The method then identifies zone of failure and failure paths using weighted graphs algorithms. We compare these failure paths to those computed with a high-fidelity fracture mechanics model called the Hybrid Optimization Software Simulation Suite (HOSS). A probabilistic evolution model for average damage in a system is also developed that is trained using 150 HOSS simulations and tested on 40 simulations. A non-parametric approach based on confidence intervals is used to determine the damage evolution over time along the dominant failure path. For upscaling, damage is the key QoI needed as an input by the continuum models. This needs to be informed accurately by the surrogate models for calculating effective moduli at continuum-scale. We show that for the proposed average damage evolution model, the prediction accuracy on the test data is more than 90%. In terms of the computational time, the proposed models are ≈ O ( 10 6 ) times faster compared to high-fidelity fracture simulations by HOSS. These aspects make the proposed surrogate model attractive for upscaling damage from micro-scale models to continuum models. We would like to emphasize that the surrogate models are not a replacement of physical understanding of fracture propagation. The proposed method in this paper is limited to tensile loading conditions at low-strain rates. This loading condition corresponds to a dominant fracture perpendicular to tensile direction. The proposed method is not applicable for in-plane shear, out-of-plane shear, and higher strain rate loading conditions.

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10

Kim, Jong Ho, Young Gu Kim, Do Kyung Kim, Kee Sung Lee, and Soon Nam Chang. "Static and Dynamic Indentation Damage in SiC and Si₃N₄." Key Engineering Materials 287 (June 2005): 410–15. http://dx.doi.org/10.4028/www.scientific.net/kem.287.410.

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Hertzian and explosive indentations were used to determine the damage behavior of SiC and Si3N4 ceramics. Specimens were selected with different microstructures. In order to observe the subsurface damaged zone, the bonded interface technique was adopted. It was found that the damage response depends strongly on the microstructure of ceramics. Examination of subsurface damage reveals a competition between brittle and quasiplastic damage mode: brittle fracture mode is dominant in fine grain microstructure; quasiplastic deformation occurs in coarse grain. Dynamic indentation induces subsurface yield zone which contains extensive micro-cracks. The role of microstructure on static and dynamic damage behavior are discussed in terms of the weakness of grain boundary and grain size.

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11

Krajcinovic, Dusan, Michal Basista, and Dragoslav Sumarac. "Micromechanically Inspired Phenomenological Damage Model." Journal of Applied Mechanics 58, no. 2 (June 1, 1991): 305–10. http://dx.doi.org/10.1115/1.2897186.

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The paper suggests a phenomenological damage theory for perfectly brittle response of solids. The theory is based on the micromechanics of brittle deformation processes. The inelastic change of the compliance is identified as the flux and a properly averaged energy release rate as the affinity. The paper identifies conditions under which the damage potential exists. The proposed model is illustrated on the examples of plain concrete specimens subjected to uniaxial tension and compression.

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12

Li, Fei, Shuang You, Hongguang Ji, and Hao Wang. "Study of Damage Constitutive Model of Brittle Rocks considering Stress Dropping Characteristics." Advances in Civil Engineering 2020 (October 26, 2020): 1–9. http://dx.doi.org/10.1155/2020/8875029.

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Deep brittle rock exhibits characteristics of rapid stress dropping rate and large stress dropping degree after peak failure. To simulate the whole process of deformation and failure of the deep brittle rock under load, the Lemaitre strain equivalent theory is modified to make the damaged part of the rock has residual stress. Based on the damage constitutive model considering residual strength characteristics, a correction factor reflecting stress dropping rate is added, the Weibull distribution is used to describe the inhomogeneity of rock materials, and Drucker–Prager criterion is used to quantitatively describe the influence of stress on damage; a damage constitutive model of deep brittle rock considering stress dropping characteristics is established. According to the geometric features of the rock stress-strain curve, the theoretical expressions of model parameters are derived. To verify the rationality of the model, triaxial compression experiments of deep brittle rock under different confining pressures are conducted. And the influence of model parameters on rock mechanical behaviour is analysed. The results show that the model reflects the stress dropping characteristics of deep brittle rock and the theoretical curve is in good agreement with the experimental results, which indicates that the proposed constitutive model is scientific and feasible.

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13

Yakin, H. N., M. R. M. Rejab, Nur Hashim, and N. Nikabdullah. "A new quasi-brittle damage model implemented under quasi-static condition using bond-based peridynamics theory for progressive failure." Theoretical and Applied Mechanics, no. 00 (2023): 6. http://dx.doi.org/10.2298/tam230404006y.

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A novel quasi-brittle damage model implemented under quasistatic loading condition using bond-based peridynamics theory for progressive failure is proposed to better predict damage initiation and propagation in solid materials. Since peridynamics equation of motion was invented in dynamic configuration, this paper applies the adaptive dynamic relaxation equation to achieve steady-state in peridynamics formulation. To accurately characterise the progressive failure process in cohesive materials, we incorporate the dynamic equation with the novel damage model for quasi-brittle materials. Computational examples of 2D compressive and tensile problems using the proposed model are presented. This paper presents advancement by incorporating the adaptive dynamic equation approach into a new damage model for quasi-brittle materials. This amalgamation allows for a more accurate representation of the behavior of damaged materials, particularly in static or quasi-static loading situations, bringing the framework closer to reality. This research paves the way for the peridynamics formulation to be employed for a far broader class of loading condition behaviour than it is now able to.

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14

Alizadeh, Ali, and Behrouz Gatmiri. "Plasto-damage modelling for semi-brittle geomaterials." E3S Web of Conferences 9 (2016): 17005. http://dx.doi.org/10.1051/e3sconf/20160917005.

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15

Sun, Xin, Stephanie A. Wimmer, and Dale G. Karrt. "Shear Band Initiation of Brittle Damage Materials." International Journal of Damage Mechanics 5, no. 4 (October 1996): 403–21. http://dx.doi.org/10.1177/105678959600500404.

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16

Yazdani, S., and S. Karnawat. "Mode I Damage Modeling in Brittle Preloading." International Journal of Damage Mechanics 6, no. 2 (April 1997): 153–65. http://dx.doi.org/10.1177/105678959700600202.

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17

Gambarotta, Luigi. "Friction-damage coupled model for brittle materials." Engineering Fracture Mechanics 71, no. 4-6 (March 2004): 829–36. http://dx.doi.org/10.1016/s0013-7944(03)00020-1.

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18

Fish, Jacob, and Qing Yu. "Two-scale damage modeling of brittle composites." Composites Science and Technology 61, no. 15 (November 2001): 2215–22. http://dx.doi.org/10.1016/s0266-3538(01)00115-4.

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19

Fujii, Y., T. Kiyama, and Y. Ishijima. "‘Condition insensitive damage indicator’ for brittle rock." International Journal of Rock Mechanics and Mining Sciences 34, no. 3-4 (April 1997): 86.e1–86.e13. http://dx.doi.org/10.1016/s1365-1609(97)00110-x.

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20

Garroni, Adriana, and Christopher J. Larsen. "Threshold-based Quasi-static Brittle Damage Evolution." Archive for Rational Mechanics and Analysis 194, no. 2 (October 2, 2008): 585–609. http://dx.doi.org/10.1007/s00205-008-0174-9.

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21

Challamel, Noël, Christophe Lanos, and Charles Casandjian. "Creep damage modelling for quasi-brittle materials." European Journal of Mechanics - A/Solids 24, no. 4 (July 2005): 593–613. http://dx.doi.org/10.1016/j.euromechsol.2005.05.003.

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22

Valanis, K. C. "A theory of damage in brittle materials." Engineering Fracture Mechanics 36, no. 3 (January 1990): 403–16. http://dx.doi.org/10.1016/0013-7944(90)90288-r.

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23

Gambarotta, Luigi, and Sergio Lagomarsino. "A microcrack damage model for brittle materials." International Journal of Solids and Structures 30, no. 2 (1993): 177–98. http://dx.doi.org/10.1016/0020-7683(93)90059-g.

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24

Zimmermann, A. "Damage evolution during microcracking of brittle solids." Acta Materialia 49, no. 1 (January 2001): 127–37. http://dx.doi.org/10.1016/s1359-6454(00)00294-9.

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25

Chiu, Chin-Chen, and Yung Liou. "Low-velocity impact damage in brittle coatings." Journal of Materials Science 30, no. 4 (February 1995): 1018–24. http://dx.doi.org/10.1007/bf01178439.

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26

PEERLINGS, R. H. J., R. DE BORST, W. A. M. BREKELMANS, and J. H. P. DE VREE. "GRADIENT ENHANCED DAMAGE FOR QUASI-BRITTLE MATERIALS." International Journal for Numerical Methods in Engineering 39, no. 19 (October 15, 1996): 3391–403. http://dx.doi.org/10.1002/(sici)1097-0207(19961015)39:19<3391::aid-nme7>3.0.co;2-d.

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27

Xu, Xiao, Chuanhua Xu, Jianhua Hu, Shaowei Ma, Yue Li, Lei Wen, and Guanping Wen. "Strength Estimation of Damaged Rock Considering Initial Damage Based on P-Wave Velocity Using Regression Analysis." Sustainability 14, no. 22 (November 9, 2022): 14768. http://dx.doi.org/10.3390/su142214768.

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High dispersion of rock mass strength causes significant difficulties in strength prediction. This study aims to investigate experimentally the strength prediction model for brittle damaged rock with multiscale initial damage based on P-wave velocity using regression analysis. Intact dolomitic limestone was collected from a deep metal mine in Southern China. Rock specimens with different initial damage degrees were prepared through the application of uniaxial compressive stress. Both intact rock and damaged rock specimens were tested for P-wave velocity and uniaxial compressive strength (UCS). The test results indicate that the method of prefabricating initial damage to the rock mass through uniaxial compressive stress is feasible. The UCS values of the damaged rock specimens were correlated with the square of the P-wave velocity (linearly positive) and the initial damage (linearly negative). The parameters of the new strength prediction model have a physical significance, and its results are within the upper and lower limits of the 95% confidence interval of the UCS. The strength prediction model considering multiscale initial damage based on P-wave velocity could reasonably predict the strengths of brittle rock masses.

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28

Lee, Kang-Hyun, In-Mo Lee, and Young-Jin Shin. "Brittle Rock Property and Damage Index Assessment for Predicting Brittle Failure in Excavations." Rock Mechanics and Rock Engineering 45, no. 2 (September 30, 2011): 251–57. http://dx.doi.org/10.1007/s00603-011-0189-7.

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29

Pijaudier-Cabot, Gilles, and Christian La Borderie. "Mechanical damage, chemical damage and permeability in quasi-brittle cementitious materials." Revue européenne de génie civil 13, no. 7-8 (August 14, 2009): 963–82. http://dx.doi.org/10.3166/ejece.13.963-982.

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30

Zuo, Q. H., L. E. Deganis, and G. Wang. "Elastic waves and damage quantification in brittle material with evolving damage." Journal of Physics D: Applied Physics 45, no. 14 (March 20, 2012): 145302. http://dx.doi.org/10.1088/0022-3727/45/14/145302.

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31

Pijaudier-Cabot, Gilles, and Christian La Borderie. "Mechanical damage, chemical damage and permeability in quasi-brittle cementitious materials." European Journal of Environmental and Civil Engineering 13, no. 7-8 (September 2009): 963–82. http://dx.doi.org/10.1080/19648189.2009.9693163.

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32

Arutyunyan, R. A. "Problem of long-term high temperature strength of metal materials." Izvestiya MGTU MAMI 8, no. 2-4 (July 20, 2014): 5–14. http://dx.doi.org/10.17816/2074-0530-67380.

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The concept of brittle and ductile-brittle fracture under high-temperature Kachanov-Rabotnov creep laid the foundations of continuum damage mechanics. The article specifies the parameter of damage and formulates a system of interconnected consistent kinetic equations of the creep rate and damage. According to the obtained solutions, theoretical curves of the change in density, creep and long-term strength are constructed.

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33

Lawn, Brian R., Yan Deng, Pedro Miranda, Antonia Pajares, Herzl Chai, and Do Kyung Kim. "Overview: Damage in brittle layer structures from concentrated loads." Journal of Materials Research 17, no. 12 (December 2002): 3019–36. http://dx.doi.org/10.1557/jmr.2002.0440.

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In this article, we review recent advances in the understanding and analysis of damage initiation and evolution in laminate structures with brittle outerlayers and compliant sublayers in concentrated loading. The relevance of such damage to lifetime-limiting failures of engineering and biomechanical layer systems is emphasized. We describe the results of contact studies on monolayer, bilayer, trilayer, and multilayer test specimens that enable simple elucidation of fundamental damage mechanics and yet simulate essential function in a wide range of practical structures. Damage processes are observed usingpost mortem(“bonded-interface”) sectioning and directin situviewing during loading. The observations reveal a competition between damage modes in the brittle outerlayers—cone cracks or quasiplasticity at the top (near-contact) surfaces and laterally extending radial cracks at the lower surfaces. In metal or polymeric support layers, yield or viscoelasticity can become limiting factors. Analytical relations for the critical loads to initiate each damage mode are presented in terms of key system variables: geometrical (layer thickness and indenter radius); material (elastic modulus, strength and toughness of brittle components, hardness of deformable components). Such relations provide a sound physical basis for the design of brittle layer systems with optimal damage thresholds. Other elements of the damage process—damage evolution to failure, crack kinetics (and fatigue), flaw statistics, and complex (tangential) loading—are also considered.

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34

Feng, Xi-Qiao, and Shou-Wen Yu. "Analyses of damage localization at crack tip in a brittle damaged material." Engineering Fracture Mechanics 53, no. 2 (January 1996): 169–77. http://dx.doi.org/10.1016/0013-7944(95)00110-7.

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35

Chenchiah, Isaac Vikram, and Christopher J. Larsen. "Quasi-Static Brittle Damage Evolution in Elastic Materials with Multiple Damaged States." Archive for Rational Mechanics and Analysis 215, no. 3 (September 27, 2014): 831–66. http://dx.doi.org/10.1007/s00205-014-0795-0.

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36

Kano, Takeshi, Eiki Sato, Tatsuya Ono, Hitoshi Aonuma, Yoshiya Matsuzaka, and Akio Ishiguro. "A brittle star-like robot capable of immediately adapting to unexpected physical damage." Royal Society Open Science 4, no. 12 (December 2017): 171200. http://dx.doi.org/10.1098/rsos.171200.

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A major challenge in robotic design is enabling robots to immediately adapt to unexpected physical damage. However, conventional robots require considerable time (more than several tens of seconds) for adaptation because the process entails high computational costs. To overcome this problem, we focus on a brittle star—a primitive creature with expendable body parts. Brittle stars, most of which have five flexible arms, occasionally lose some of them and promptly coordinate the remaining arms to escape from predators. We adopted a synthetic approach to elucidate the essential mechanism underlying this resilient locomotion. Specifically, based on behavioural experiments involving brittle stars whose arms were amputated in various ways, we inferred the decentralized control mechanism that self-coordinates the arm motions by constructing a simple mathematical model. We implemented this mechanism in a brittle star-like robot and demonstrated that it adapts to unexpected physical damage within a few seconds by automatically coordinating its undamaged arms similar to brittle stars. Through the above-mentioned process, we found that physical interaction between arms plays an essential role for the resilient inter-arm coordination of brittle stars. This finding will help develop resilient robots that can work in inhospitable environments. Further, it provides insights into the essential mechanism of resilient coordinated motions characteristic of animal locomotion.

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37

Du, Chun Zhi, and Wen Yan Qiu. "An Advanced Damage Propagation Model of Brittle Fracture." Advanced Materials Research 538-541 (June 2012): 1711–15. http://dx.doi.org/10.4028/www.scientific.net/amr.538-541.1711.

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To reduce the coal mining risk, drainage of methane with drills and hydraulic fracture has been put in practice. In the paper, an advanced damage propagation model of brittle fracture is studied based on mode I crack. The localization band of damage is inducted, and the modal of localization of damage around crack-tip is established using fracture and damage mechanics. Besides, the stress distribution in localization of damage is assumed as obeying a quadratic function, thus the stress field around the crack is obtained, and the calculation formula of crack length is presented, which can provide reference for mine production.

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38

Wang, Haijing, Bo Zhou, Shifeng Xue, Xuejing Deng, Peng Jia, and Xiuxing Zhu. "An Anisotropic Damage Model of Quasi-Brittle Materials and Its Application to the Fracture Process Simulation." Applied Sciences 12, no. 23 (November 25, 2022): 12073. http://dx.doi.org/10.3390/app122312073.

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Accurate predictions of the failure behaviors of quasi-brittle materials are of practical significance to underground engineering. In this work, a novel anisotropic damage model is proposed based on continuous damage mechanics. The anisotropic damage model includes a two-parameter parabolic-type failure criterion, a stiffness degradation model that considers anisotropic damage, and damage evolution equations for tension and shear, respectively. The advantage of this model is that the degradation of elastic stiffness only occurs in the direction parallel to the failure surface for shear damage, avoiding the interpenetration of crack surfaces. In addition, the shear damage evolution equation is established based on the equivalent shear strain on the failure face. A cyclic iterative method based on the proposed anisotropic damage model was developed to numerically simulate the fracture process of quasi-brittle materials. The developed model and method are important because the ready-made finite element software is difficult to simulate the anisotropic damage of quasi-brittle materials. The proposed anisotropic damage model was tested against a conventional damage model and validated against two benchmark experiments: uniaxial and biaxial compression tests and Brazilian splitting tests. The results demonstrate that the proposed anisotropic damage model simulates the mesoscale damage mode, macroscale fracture modes, and strength characteristics more effectively and accurately than conventional damage models.

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39

Zhang, Ji. "Three-Parameter Damage Evolution for Quasi-Brittle Solids." Applied Mechanics and Materials 580-583 (July 2014): 3119–24. http://dx.doi.org/10.4028/www.scientific.net/amm.580-583.3119.

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This paper aims to investigate the performance of a new three-parameter damage mechanics model which describes three basic damage mechanisms of quasi-brittle solids: tension, shear, and hydrostatic compression. The stress is first decomposed into its positive part and negative part, and then the latter is further decomposed into its deviatoric part and hydrostatic part, whereby a three-parameter damage description is formulated. Through matrix representation of the tensor formulation, specific forms of the three-parameter damage theory are illustrated in various stress states. It is found that the proposed framework of three-parameter damage theory can inherit the existing two-parameter models and extend them to a broader scope of application.

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40

Hentati, Hamdi, Ilyes Ben Naceur, Wassila Bouzid, and Aref Maalej. "Numerical Analysis of Damage Thermo-Mechanical Models." Advances in Applied Mathematics and Mechanics 7, no. 5 (July 21, 2015): 625–43. http://dx.doi.org/10.4208/aamm.2014.m517.

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AbstractIn this paper, we present numerical computational methods for solving the fracture problem in brittle and ductile materials with no prior knowledge of the topology of crack path. Moreover, these methods are capable of modeling the crack initiation. We perform numerical simulations of pieces of brittle material based on global approach and taken into account the thermal effect in crack propagation. On the other hand, we propose also a numerical method for solving the fracture problem in a ductile material based on elements deletion method and also using thermo-mechanical behavior and damage laws. In order to achieve the last purpose, we simulate the orthogonal cutting process.

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41

Shibasaka, Toshiro. "Machining Damage of Sintered Brittle Materials (1st Report)." Journal of the Japan Society for Precision Engineering 56, no. 3 (1990): 551–56. http://dx.doi.org/10.2493/jjspe.56.551.

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42

Chia, J., and J. W. Hancock. "Finite Element Modelling Damage in Brittle Matrix Composites." Key Engineering Materials 227 (August 2002): 55–60. http://dx.doi.org/10.4028/www.scientific.net/kem.227.55.

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43

Yoh, E. G., Y. I. Kim, and Yong Sin Lee. "Modeling Brittle Damage Evolution in Forging/Extrusion Die." Key Engineering Materials 233-236 (January 2003): 119–26. http://dx.doi.org/10.4028/www.scientific.net/kem.233-236.119.

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44

WANG, Ningchang. "Review on Brittle Material Subsurface Damage Detection Technology." Journal of Mechanical Engineering 53, no. 9 (2017): 170. http://dx.doi.org/10.3901/jme.2017.09.170.

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45

Pensée, V., D. Kondo, and L. Dormieux. "Micromechanical Analysis of Anisotropic Damage in Brittle Materials." Journal of Engineering Mechanics 128, no. 8 (August 2002): 889–97. http://dx.doi.org/10.1061/(asce)0733-9399(2002)128:8(889).

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46

Murzewski, Janusz W. "Brittle and Ductile Damage of Stochastically Homogeneous Solids." International Journal of Damage Mechanics 1, no. 3 (July 1992): 276–89. http://dx.doi.org/10.1177/105678959200100302.

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47

Mohamad-Hussein, A., and J. F. Shao. "An elastoplastic damage model for semi-brittle rocks." Geomechanics and Geoengineering 2, no. 4 (November 29, 2007): 253–67. http://dx.doi.org/10.1080/17486020701618329.

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48

Perras, Matthew A., and Mark S. Diederichs. "Predicting excavation damage zone depths in brittle rocks." Journal of Rock Mechanics and Geotechnical Engineering 8, no. 1 (February 2016): 60–74. http://dx.doi.org/10.1016/j.jrmge.2015.11.004.

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49

Pires-Domingues, S. M., H. Costa-Mattos, and F. A. Rochinha. "Modelling of nonlinear damage on elastic brittle materials." Mechanics Research Communications 25, no. 2 (March 1998): 147–53. http://dx.doi.org/10.1016/s0093-6413(98)00018-4.

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50

Chandra, Dipankar, and Theodor Krauthammer. "Rate-Sensitive Micromechanical Damage Model for Brittle Solid." Journal of Engineering Mechanics 122, no. 5 (May 1996): 412–22. http://dx.doi.org/10.1061/(asce)0733-9399(1996)122:5(412).

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