Abstract
Microscopic analysis of materials is generally complex; hence engineers always look for alternative methods of analysis, by appropriate modelling. In specific, studying the damage of materials is important in almost every discipline of science/engineering/technology. Material damage includes the beginning, advancement and fracture of materials. The most important tool so far, used to analyze this damage, is called damage mechanics which relies heavily on continuum mechanics. It encompasses useful theories for the representation, modelling, and analysis of material damage. Broadly speaking, damage mechanics utilizes state variables to show the effects of damage. For instance, the variations in stiffness, and the lifetime of the material are two common factors affected by damage. Examples of state variables used include crack density and coefficient of thermal expansion. These variables pave the way for further damage due to the presence of conjugate thermodynamic forces. The major aim of damage mechanics is to be able to predict future damage in materials. Since the propagation of damage is not spontaneous after the damage initiation, modelling the damage evolution is important. A hardening function is used for this purpose in case of plasticity like formulations. However this is not very efficient; micromechanics modelling have proven to be more superior. Damage mechanics is required in almost every field. In particular, its utility in the field of geology has been progressing rapidly. Recent research studies in the damage mechanics of geological materials have given engineers a deeper insight into material damage prevention and control. Specifically, soil and rock mechanics, subsets of engineering geology, have made the maximum progress in utilizing damage mechanics. Predicting the mechanical behaviour of soils and rocks, and consequently their damage mechanics properties have been very fruitful. This paper analyzes in detail such research carried out in the recent past. It also illustrates the scope for future work in this area.
Key Words: damage mechanics, continuum mechanics, damage properties, geological material properties, rock formation
Introduction
Continuum Damage Mechanics
The Classical Uniaxial damage theory: Continuum damage mechanics is a continuum mechanics based framework that can be used at the macroscopic level to characterize, model, and represent the effects of defects (distributed) and their formation on material behavior. There are many key concepts involved in the understanding of continuum mechanics, the foundation of which is the uniaxial tension experiment as described below:
Classical uniaxial damage theory: The damage amplitude in a specified plane can be computed by finding out the area associated with the intersection of all defects in the plane. For instance, consider conditions specified by Figure 1. It can be seen that the uniaxial tension is effectively in the area represented by S-SD, where SD denotes the defects associated with the plane. Damage variable w, a positive scalar quantity, is then defined as: SD/S.
Figure 1: Example
For an undamaged material, SD is 0 which implies that w is also 0. In general, the damage variable w can take a value between 0 and 1, where 1 implies that the material is fully damaged. In other words, w is a representation of the extent of damage on a material, on a scale of 0 to 1.
Simple Isotopic Damage Theory: The above discussion is for 1D which can be extended to 3D by using the popular Lemaitre-Chaboche elastic model. According to this 3D model, the stiffness D associated with a damaged material is = (1 – w)D0, where w is again the damage variable, and D0 represents the elastic stiffness associated with the material when undamaged.
- Literature Survey
History of Continuum Damage Mechanics
Janusz Murzewski was the first one to develop Continuum Damage Theory (CDT) in 1957. Initially, it was aimed at analyzing a mechanical body by describing its probabilistic degradation of cohesion. By the end of 1960s, many improvements to the theory were proposed, and during the 1970s, CDT gained international popularity as an analytical tool. The essence of CDT is this: Consider a body with many distributed defects. The non-uniformity makes it difficult to analyze the damage properties of the bodies. This theory helps to remodel the situation as a homogenous problem having reduced effective properties.
The first research that utilized this theory well was done by Dr. Richard Schapery. He used it to explain the behavior of solid rocket fuels subject to tensile loading as follows: Solid rocket fuels are comparable to hot mix asphalt or HMA with regard to certain characteristics. For instance, they are both visco-elastic materials, are semi solids, and are sensitive to both loading temperature and time. This made Dr. Schapery come up with the work potential theory, which laid the foundation for modern continuum damage models for HMA. The applications of this theory have since been researched upon extensively by many engineers across the globe. Specifically, the joint research team from the Texas A&M University, and the North Carolina State University, led by Dr. Robert Lytton, Dr. Richard Kim, and Dr. Dallas Little has been involved in a 15-year long research that proves the validity of the theory. They have shown the behavior of HMA materials and pavements during fatigue loading [1].
CDT and Asphalt – A deeper probe
When an asphalt pavement is made, there are many areas that are potential sources of cracks. These areas are generally present in the form of micro voids or air voids that are induced as a consequence of the granular nature of the asphalt, and are formed during the construction process itself. This is the base for the development of fatigue cracks in the concrete pavement. Loading is usually in the form of vertical load as given by a contact medium that is flexible, like a tire for instance. Since this loading is applied to the pavement surface and is mostly regular, consistent cycles start accumulating. Environmental influences that include UV radiation, thermal cycling, oxidation, moisture infiltration, and so on can also have an effect – they generally degrade the material from the load related factors. Further, they can even change the behavior of the material, or impose their own cyclic influences like done by the thermal effects. With the continuous existence of such disturbances, the asphalt concrete starts developing micro-cracks. Due to their very small size, these micro cracks are in most cases not noticed or detected. However, acoustic emission measurements can be used in the place of existing imaging techniques, to detect some obvious signs of these cracks. Also, the influence of these defects on macro scale measurements like displacement and force prove as an indirect method for their detection. With time, the micro cracks propagate, coalesce and finally become macro cracks that are visible to the naked eye. Hence, somewhere in between the initiation and formation of macro cracks, the detection needs to take place, which is done with the help of CDT. Beyond the point of formation of macro cracks, approximate cohesive zone methods, smeared CDT, and fracture mechanics take over as more accurate modeling tools to describe further damage [1].
S-VECD
The discussion above shows how asphalt concrete can be characterized using CDT. The actual CDT analysis behind this involves many protocols, using highly specialized equipments and complex mathematics involving higher levels of calculus. Since the late 1990s and the early 2000s, there have been extensive research studies in the evaluation of mechanical properties of paving materials. This was introduced by the asphalt concrete community as part of routine evaluation. This move led to the designing of the Asphalt Mixture Performer Tester and other tests required for the identification of permanent damage and linear visco-elastic characteristics of asphalt concrete. However, these tests could not identify an appropriate fatigue performance evaluation protocol. Dr. Underwood’s ensuing work that started in 2006, gave a simplified version of the popular visco-elastic continuum damage model or S-VECD. The its predecessor VECD, the S-VECD model provides a relation between material integrity, represented by variable C, and the amount of damage in the asphalt concrete body, represented by S. This relation is depicted in Figure 3, from which it can be observed that it is independent of most factors like temperature and testing that influence the fatigue performance. Hence the behavior of the material can be described using comparatively less testing. This is shown in Figure 4.
The overall functional relationships described by the S-VECD model are as shown in Figure 5. Though the equations look complex, their importance can be explained in simpler terms. Unlike functions in the area of simplified CDT methods, these functions are mathematically rigorous. Every step in the derivation needs a detailed understanding, and has broad implications. Essentially, these equations show that approximations and/or special allowances are not required in the modeling of the damage characteristic curves that are used in structural analytical methods like finite element models. In other words, though the S-VECD model has now been simplified in terms of characterization, the number and scope of applications remain the same. An outline of one possible application using this model is shown in Figure 6 [1].
- M-VCED
Most CDT theories applied in the asphalt concrete field focus on 1D tension behavior model for the material. A more practical model will be one that extends and incorporates the CDT theories in 3D. In order to form a unified, multiaxial description of the material in 3D, more complex and sophisticated techniques that can account for 3D realities are required. The test samples need to be subjected to a uniform confining pressure during the process of loading. The analysis of such tests is obviously more complex as the equations governing the model consider more parameters for a more complete relationship analysis between stress and strain. This can be understood better from Figures 7 and 8.
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And though more complex, the above investigations give more sophisticated results that aid a more fundamental understanding of the material at hand, leading to more sustainable and efficient pavement technologies. Figure 9 shows an example prediction from M-VECD model for a condition of simple loading [2].
Damage Mechanics and Earthquakes
Transient, steady-state and accelerating are three common states while observing creep under static loading constraints. The work by Dr. Ian suggests that the apparent tri-modal behavior between strain and event rate can be modeled as a simple damage mechanics problem. This is done by bringing in the mechanisms of local feedback – both positive and negative. The feedback mechanisms are applied to the corresponding rules that govern subcritical, time dependent crack growth. The individual rule governing the measured strain ε is of the form ε(t) = ε0 [1 + t/mτ]m, in which τ is the ratio between initial crack length and rupture velocity. The negative feedback results in a local hardening mechanism, in which, transient creep dominates and 0 < m < 1. The crack growth is observed to be stable and decelerating at this stage. The positive feedback results in a local softening mechanism leading to m<0. In this case the crack growth is observed to be unstable and accelerating. This represents a quasi-static instability criterion, i.e. ε → ∞. This is defined as a finite failure time which ultimately results in the localization of damage, leading to the development of a through going fracture.
In the presented hybrid model, the early stages of damage are dominated by transient creep and the later stages are dominated by accelerating creep. The superposition of both the stages is observed at intermediate times, during which spontaneous production of an apparent steady state can be seen. This phase seems to have a relatively constant strain rate and a power law rheology. The so predicted values of acoustic emission event rates for the transient and accelerating phases are shown to be identical to the improved Omori laws that govern aftershocks and foreshocks respectively. Further, they provide a physical interpretation to the empirical constants measured. In between the two stages, the event rate is seen to approach a more or less constant value. For a finite event rate during the main shock, the instability criterion needs to be satisfied. In other words, finite crack velocity must be met at the time of dynamic failure - dx/dt → VR , where VR denotes dynamic rupture velocity. This hybrid model can also be used for analyzing dynamic loading conditions, with minor modification. The scaling components that result show that dynamic loading systematically leads to more non-linear behavior [3].
Continuum Damage Mechanics to Analyze strength of Rocks
Rock materials’ strength can be characterized as a random distribution. The variation in strength is analyzed using Mohr’s criteria and damage mechanics. The comparison of results with set of data obtained by experimenting on Carrara marble shows constitute properties of rock materials can be described suitably using damage mechanics. The average square error of strength gives a measure of rock mechanics. This work deals with a new model that gives the relationship between the sample size and strength of rocks. Further, the relation between rock strength and earthquake activity has also been studied and reported.
Statistical and damage theory show the elasto-plastic deformation properties and damageproperties of rocks. Using these, an elasto-plastic damage constitutive statistical model has been developed. This model uses the damage evolution function to analyze the effective stresses. Further, the property of residual strength of rocks is also considered [4].
Damage Mechanics to Analyze Underground water and Softening of Argillite
A study on the damage properties and relation between water and rock and their interaction has been carried out. This is specifically for special geological conditions as follows: 1) 18 layers of argillite 2) Longtan Hydro power Station - highwall slope. A series of tests involving uniaxial compression for the 18 layers have been carried out using Rock mechanics Rigidity Servo Testing System such that there is different water content in each layer. The test results have shown that softening of compressive strength is dependent strongly on the water content in argillite. It also is seen to depend on the soaking time in water. Damage mechanics has been utilized to study the influence of ground water on rock strength. Rock damage as a function of water content has been established, which is also shown to be related to rock stress rate [5].
Conclusion
Various applications of continuum damage mechanics have been analyzed in the literature survey presented above. It is clear that there is a vast scope for future research in the field of damage mechanical properties of geological materials. Such data and results can help in practical applications like soil utilization and management of aquifers for instance. The theory related to continuum damage mechanics is continuously evolving and looks to have the potential to open new avenues for analysis and research.
References
[1] Zeiada, W.A. , K.E. Kaloush, B.S. Underwood, M. Mamlouk (2013). “Improved Method to Consider Air Void and Asphalt Content Changes on Long-Term Performance of Asphalt Concrete Pavements,” International Journal of Pavement Engineering, In Press.
[2] Underwood, B.S. and Y.R. Kim (2013). “A Nonlinear Viscoelastic Constitutive Model for Asphalt Cement and Asphalt Mastic,” International Journal of Pavement Engineering, In Press.
[3] Main, Ian G. "A damage mechanics model for power‐law creep and earthquake aftershock and foreshock sequences." Geophysical Journal International 142.1 (2000): 151-161.
[4] Youqing, Yang. "CONTINUUM DAMAGE MECHANICS ANALYSIS ON STRENGTH OF ROCK [J]." Chinese Journal of Rock Mechanics and Engineering 1 (1999).
[5] Zhende, Z. H. U., et al. "Analysis of strength softening of argillite underground water by damage mechanics." Chinese Journal of Rock Mechanics and Engineering 23.4 (2004): 739-4.
