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IN THIS ISSUE

Guest Editorial

J. Appl. Mech. 2012;79(3):030301-030301-2. doi:10.1115/1.4005965.
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This issue of the Journal of Applied Mechanics is dedicated, with our admiration and affection, to Professor Jim Rice of Harvard University. It serves as the proceedings of the 3 days symposium on Mechanics in Geophysical and Materials Sciences, which was held at the California Institute of Technology during Jan. 20–22, 2011 to celebrate Jim’s brilliant career on the occasion of his 70th birthday.

Commentary by Dr. Valentin Fuster

Research Papers

J. Appl. Mech. 2012;79(3):031001-031001-10. doi:10.1115/1.4005878.

The fixed-point iteration algorithm is turned into a quadratically convergent scheme for a system of nonlinear equations. Most of the usual methods for obtaining the roots of a system of nonlinear equations rely on expanding the equation system about the roots in a Taylor series, and neglecting the higher order terms. Rearrangement of the resulting truncated system then results in the usual Newton-Raphson and Halley type approximations. In this paper the introduction of unit root functions avoids the direct expansion of the nonlinear system about the root, and relies, instead, on approximations which enable the unit root functions to considerably widen the radius of convergence of the iteration method. Methods for obtaining higher order rates of convergence and larger radii of convergence are discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031002-031002-10. doi:10.1115/1.4005957.

Rice’s internal variables formalism [1975, “Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms,” in Constitutive Equations in Plasticity, edited by A. Argon, MIT Press, Cambridge, MA, pp. 23–75] is one of the basic tools in the micromechanics of materials. One of its implications is the possibility to relate the compliance/resistivity contributions of cracks—the key quantities in the problem of effective elastic/conductive properties—to the stress intensity factors (SIFs) and thus to utilize a large library of available solutions for SIFs. Examples include configurations that are common in materials science applications: branched and intersecting cracks, cracks with partial contact between crack faces, and cracks emanating from pores. The formalism also yields valuable physical insights of a qualitative character, such as the impossibility to correlate, in a quantitative way, the strength of microcracking materials and their stiffness reduction.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031003-031003-5. doi:10.1115/1.4005879.

Motivated by observations of the subglacial drainage of water, we consider a hydraulic fracture problem in which the crack grows parallel to a free surface, subject to fully turbulent fluid flow. Using a hybrid Chebyshev/series-minimization numerical approach, we solve for the pressure profile, crack opening displacement, and crack growth rate for a crack that begins relatively short but eventually becomes long compared with the distance to the free surface. We plot nondimensionalized results for a variety of different times, corresponding with different fracture lengths, and find substantial differences when free-surface effects are important.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031004-031004-10. doi:10.1115/1.4005880.

The stability of steady slip and homogeneous shear is studied for rate-hardening materials undergoing chemical reactions that produce weaker materials (reaction-weakening process), in drained conditions. In a spring- slider configuration, a linear perturbation analysis provides analytical expressions of the critical stiffness below which unstable slip occurs. In the framework of a frictional constitutive law, numerical tests are performed to study the effects of a nonlinear reaction kinetics on the evolution of the instability. Slip instabilities can be stopped at relatively small slip rates (only a few orders of magnitude higher than the forcing velocity) when the reactant is fully depleted. The stability analysis of homogeneous shear provides an independent estimate of the thickness of the shear localization zone due to the reaction weakening, which can be as low as 0.1 m in the case of lizardite dehydration. The potential effect of thermo-chemical pore fluid pressurization during dehydration is discussed, and shown to be negligible compared to the reaction-weakening effect. We finally argue that the slip instabilities originating from the reaction-weakening process could be a plausible candidate for intermediate depth earthquakes in subduction zones.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031005-031005-7. doi:10.1115/1.4005893.

In this paper the delamination of coating subjected to compressive stress on a cylindrical substrate is considered. This problem is particularly interesting in oxide coatings on wire elements exposed to extreme temperatures and in ceramic coatings on turbine engine blades or other components that operate at high temperatures. Using the results of Hutchinson (Hutchinson, 2001, “Delamination of Compressed Films on Curved Substrates,” J. Mech. Phys. Solids, 49 , pp. 1847–1864) the aforementioned problem is discussed from the aspect of application of the linear elastic fracture mechanics (LEFM) concept for an interfacial crack. The energy release rate and mode mixity for the case of the coating delamination in the axial and radial directions are determined. It is shown that the results also depend on whether the substrate is convex or concave. Delamination in the radial direction in the case of the concave substrate is harder, but it is more likely when the substrate is convex. Delamination in the axial direction is equally likely in both cases. The results presented in this paper justify the application of the concept of linear elastic fracture mechanics for an interfacial crack for explaining the influence of the elastic characteristics of the substrate on the buckling delamination of the coating.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031006-031006-10. doi:10.1115/1.4005881.

The paper reviews a recently developed finite chain model for the weakest-link statistics of strength, lifetime, and size effect of quasi-brittle structures, which are the structures in which the fracture process zone size is not negligible compared to the cross section size. The theory is based on the recognition that the failure probability is simple and clear only on the nanoscale since the probability and frequency of interatomic bond failures must be equal. The paper outlines how a small set of relatively plausible hypotheses about the failure probability tail at nanoscale and its transition from nano- to macroscale makes it possible to derive the distribution of structural strength, the static crack growth rate, and the lifetime distribution, including the size and geometry effects [while an extension to fatigue crack growth rate and lifetime, published elsewhere (Le and Bažant, 2011, “Unified Nano-Mechanics Based Probabilistic Theory of Quasibrittle and Brittle Structures: II. Fatigue Crack Growth, Lifetime and Scaling,” J. Mech. Phys. Solids, 1322–1337), is left aside]. A salient practical aspect of the theory is that for quasi-brittle structures the chain model underlying the weakest-link statistics must be considered to have a finite number of links, which implies a major deviation from the Weibull distribution. Several new extensions of the theory are presented: (1) A derivation of the dependence of static crack growth rate on the structure size and geometry, (2) an approximate closed-form solution of the structural strength distribution, and (3) an effective method to determine the cumulative distribution functions (cdf’s) of structural strength and lifetime based on the mean size effect curve. Finally, as an example, a probabilistic reassessment of the 1959 Malpasset Dam failure is demonstrated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031007-031007-10. doi:10.1115/1.4005883.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031008-031008-8. doi:10.1115/1.4005885.

A hydrostatically coupled dielectric elastomer (HCDE) actuator consists of two membranes of a dielectric elastomer, clamped with rigid circular rings. Confined between the membranes is a fixed volume of a fluid, which couples the movements of the two membranes when a voltage or a force is applied. This paper presents a computational model of the actuator, assuming that the membranes are neo-Hookean, capable of large and axisymmetric deformation. The voltage-induced deformation is described by the model of ideal dielectric elastomer. The force is applied by pressing a rigid flat punch onto one of the membranes over an area of contact. The computational predictions agree well with experimental data. The model can be used to explore nonlinear behavior of the HCDE actuators.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031009-031009-9. doi:10.1115/1.4005894.

Novel indentation studies combined with in situ transmission electron microscopy correlate large load drops with instabilities involving dislocation substructure. These instabilities are captured in finite element simulations of indentation that employ quantized crystal plasticity (QCP) in the vicinity of a nanoindenter tip. The indentation load-displacement traces, slip patterns, and creation of gaps are correlated with the scale, strength, and shear strain burst imparted by slip events within microstructural cells. Large load drops (ΔP/P ∼ 25%) are captured provided these cellular slip events produce shear strain bursts ∼ 8%, comparable to 8 dislocations propagating across a 25 nm microstructural cell. The results suggest that plasticity at the submicron, intragranular scale involves violent stress redistributions, triggering multi-cell instabilities that dramatically affect the early stages of a nanoindentation test.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031010-031010-10. doi:10.1115/1.4005958.

An anisotropic elastoplastic bounding surface model with non-associative flow rule is developed for simulating the mechanical behavior of different types of clays. The non-associative flow rule allows for the simulation of not only strain-hardening but also strain-softening response. The theoretical framework of the model is given, followed by the verification of the model as applied to the experimental results of a strain-hardening Kaolin tested under different undrained stress paths. The undrained behavior of Boston Blue clay, which exhibits a strain-softening behavior, is also simulated. It is shown that the non-associative nature of the model gives more accurate results than those of the same model employing an associative flow rule, especially for normally consolidated Kaolin specimens. The results show that the model is also capable of simulating the strain-softening behavior of Boston blue clay with reasonable accuracy.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031011-031011-9. doi:10.1115/1.4005887.

Deformation of granular materials is often characterized by strain localization in the form of shear bands, which exhibit a characteristic width of about 10–20 particle diameters. Much of the relative motion of particles within a shear band is accompanied by rolling, as opposed to sliding, since the latter requires more dissipative work. However, in a densely packed assembly, rolling cannot be accomplished without some sliding. This dissipative constraint implies a characteristic rotation transmission distance, i.e., the distance to which the information about rotation of a particle propagates. Here, we use the discrete element method to investigate this length and its directional dependence as function of the force chain network. We found that the rotation transmission distance correlates with the shear band width observed in experiments and previous numerical simulations. It is strongly dependent on the particle size distribution and the coefficient of interparticle friction, and weakly dependent on pressure. Moreover, the transmission of rotations is strongly directionally dependent following the pattern of force chains. To describe this dependence, we define a nonlocal tensorial description of force chain directionality.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031012-031012-5. doi:10.1115/1.4005895.

A commonly used idealization when describing separation of a chemical bond between molecules is that of an energy well which prescribes the dependence of energy of interaction between the molecules in terms of a reaction coordinate. The energy difference between the peak to be overcome and the root of the well is the so-called activation energy, and the overall shape of the well dictates the kinetics of separation through a constitutive assumption concerning transport. An assumption tacit in this description is that the state of the bond evolves with only a single degree of freedom—the reaction coordinate—as the system explores its energy environment under random thermal excitation. In this discussion we will consider several bonds described by one and the same energy profile. The cases differ in that the energy profile varies along a line extending from the root of the well in the first case, along any radial line in a plane extending from the root of the well in a second case, and along any radial line in space extending from the root of the well in a third case. To focus the discussion we determine the statistical rate of escape of states from the well in each case, requiring that the profile of the well is the same in all three cases. It is found that the rates of escape each depend exponentially on the depth of the well but that the coefficients of the exponential vary with depth of the well differently in the three cases considered.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031013-031013-10. doi:10.1115/1.4005896.

Geophysical observations have shown that transient slow slip events, with average slip speeds v on the order of 10−8 to 10−7 m/s, occur in some subduction zones. These slip events occur on the same faults but at greater depth than large earthquakes (with slip speeds of order ∼ 1 m/s). We explore the hypothesis that whether slip is slow or fast depends on the competition between dilatancy, which decreases fault zone pore pressure p, and thermal pressurization, which increases p. Shear resistance to slip is assumed to follow an effective stress law $τ=f(σ-p)≡ fσ¯$. We present two-dimensional quasi-dynamic simulations that include rate-state friction, dilatancy, and heat and pore fluid flow normal to the fault. We find that at lower background effective normal stress ($σ¯$), slow slip events occur spontaneously, whereas at higher $σ¯$, slip is inertially limited. At intermediate $σ¯$, dynamic events are followed by quiescent periods, and then long durations of repeating slow slip events. In these cases, accelerating slow events ultimately nucleate dynamic rupture. Zero-width shear zone approximations are adequate for slow slip events but substantially overestimate the pore pressure and temperature changes during fast slip when dilatancy is included.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031014-031014-10. doi:10.1115/1.4005888.

Scattering of elastic waves by structural inhomogeneities such as cylindrical cavities has been a subject of intensive study for decades. The time-harmonic elastodynamic analysis making use of the wave function expansions is one of the typical approaches for such problems, and since it gives semianalytical solutions that may show the effect of parameters of the problem rather explicitly, it is still repeatedly used in the study of dynamic response of elastic structures including inhomogeneities. Here, motivated by the observation of the unique underground structural failure patterns caused by the 1995 Hyogo-ken Nanbu (Kobe), Japan, earthquake, we analyze scattering of a plane harmonic body wave by a uniformly lined circular tunnel (cylinder), and from the structural failure patterns we evaluate possible mechanical characteristics of the associated incident seismic waves. In the two-dimensional, in-plane time-harmonic elastodynamic model employed, the lined circular tunnel may be located at a finite depth from an approximate flat free surface of a homogeneous isotropic linear elastic medium (half-space), and the plane wave impinges upon the tunnel at an arbitrary incident angle. We compare the effect of P and SV wave incidences by calculating the dynamic amplification of stresses and displacements around this simplified tunnel, and also show the influence of the wavelength and the incident angle of the plane wave, the overburden thickness, and the relative compliance of the linear elastic lining with respect to the surrounding medium. The results suggest that the observed underground structural failures, the exfoliation of the lining concrete and buckling of the reinforcing steel bars on the sidewall as well as the detachment of the subgrade from the invert, might have been induced by the incidence of P waves in a relatively high frequency range.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031015-031015-8. doi:10.1115/1.4005959.

Experimental observations have shown that the roughness of fracture surfaces exhibit certain characteristic scaling properties. Here, calculations are carried out to explore the extent to which a ductile damage/fracture constitutive relation can be used to model fracture surface roughness scaling. Ductile crack growth in a thin strip under mode I, overall plane strain, small scale yielding conditions is analyzed. Although overall plane strain loading conditions are prescribed, full 3D analyses are carried out to permit modeling of the three dimensional material microstructure and of the resulting three dimensional stress and deformation states that develop in the fracture process region. An elastic-viscoplastic constitutive relation for a progressively cavitating plastic solid is used to model the material. Two populations of second phase particles are represented: large inclusions with low strength, which result in large voids near the crack tip at an early stage, and small second phase particles, which require large strains before cavities nucleate. The larger inclusions are represented discretely and various three dimensional distributions of the larger particles are considered. The scaling properties of the predicted thickness average fracture surfaces are calculated and the results are discussed in light of experimental observations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031016-031016-12. doi:10.1115/1.4005897.

The micromechanical damage mechanics formulated by Ashby and Sammis, 1990, “The Damage Mechanics of Brittle Solids in Compression,” Pure Appl. Geophys., 133 (3), pp. 489–521, and generalized by Deshpande and Evans 2008, “Inelastic Deformation and Energy Dissipation in Ceramics: A Mechanism-Based Constitutive Model,” J. Mech. Phys. Solids, 56 (10), pp. 3077–3100. has been extended to allow for a more generalized stress state and to incorporate an experimentally motivated new crack growth (damage evolution) law that is valid over a wide range of loading rates. This law is sensitive to both the crack tip stress field and its time derivative. Incorporating this feature produces additional strain-rate sensitivity in the constitutive response. The model is also experimentally verified by predicting the failure strength of Dionysus-Pentelicon marble over strain rates ranging from ∼10− 6 to 103 s− 1 . Model parameters determined from quasi-static experiments were used to predict the failure strength at higher loading rates. Agreement with experimental results was excellent.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031017-031017-8. doi:10.1115/1.4005899.

The boundary integral equation method (BIEM) has been applied to the analysis of rupture propagation of nonplanar faults in an unbounded homogeneous elastic medium. Here, we propose an extended BIEM (XBIEM) that is applicable in an inhomogeneous bounded medium consisting of homogeneous sub-regions. In the formulation of the XBIEM, the interfaces of the sub-regions are regarded as extended boundaries upon which boundary integral equations are additionally derived. This has been originally known as a multiregion approach in the analysis of seismic wave propagation in the frequency domain and it is employed here for rupture dynamics interacting with medium interfaces in time domain. All of the boundary integral equations are fully coupled by imposing boundary conditions on the extended boundaries and then numerically solved after spatiotemporal discretization. This paper gives the explicit expressions of discretized stress kernels for anti-plane nonplanar problems and the numerical method for the implementation of the XBIEM, which are validated in two representative planar fault problems.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031018-031018-6. doi:10.1115/1.4005900.

Mechanical stresses and failure are believed to be a major cause for the limited cycle life of lithium-ion batteries employing high capacity Si electrodes. Recent experiments have shown that patterned Si thin film electrodes on substrate exhibit improved cycling stability and substantial sliding at the film/substrate interface. To facilitate experimental studies of stress evolution in such systems, we have developed a modified Stoney equation which accounts for the effect of interfacial sliding on the relationship between curvature and stress in patterned thin films on substrate.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031019-031019-9. doi:10.1115/1.4005960.

Wrinkling modes are determined for a two-layer system comprised of a neo-Hookean film bonded to an infinitely deep neo-Hookean substrate with the entire bilayer undergoing compression. The full range of the film/substrate modulus ratio is considered from the limit of a traction-free homogeneous substrate to very stiff films on compliant substrates. The role of substrate prestretch is considered wherein an unstretched film is bonded to a prestretched substrate with wrinkling arising as the stretch in the substrate is relaxed. An exact bifurcation analysis reveals the critical strain in the film at the onset of wrinkling. Numerical simulations carried out within a finite element framework uncover advanced post-bifurcation modes including period-doubling, folding and a newly identified mountain ridge mode.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031020-031020-7. doi:10.1115/1.4005961.

We characterize wave propagation along an infinitely long crack or conduit in an elastic solid containing a compressible, viscous fluid. Fluid flow is described by quasi-one-dimensional mass and momentum balance equations with a barotropic equation of state, and the wall shear stress is written as a general function of width-averaged velocity, density, and conduit width. Our analysis focuses on small perturbations about steady flow, through a constant width conduit, at an unperturbed velocity determined by balancing the pressure gradient with drag from the walls. Short wavelength disturbances propagate relative to the fluid as sound waves with negligible changes in conduit width. The elastic walls become more compliant at longer wavelengths since strains induced by opening or closing the conduit are smaller, and the fluid compressibility becomes negligible. As wavelength increases, the sound waves transition to crack waves propagating relative to the fluid at a slower phase velocity that is inversely proportional to the square-root of wavelength. Associated with the waves are density, velocity, pressure, and width perturbations that alter drag. At sufficiently fast flow rates, crack waves propagating in the flow direction are destabilized when drag reduction from opening the conduit exceeds the increase in drag from increased fluid velocity. This instability may explain the occurrence of self-excited oscillations in fluid-filled cracks.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031021-031021-16. doi:10.1115/1.4005962.

A linearized model is developed for lithium ion batteries, relying on simplified characterizations of lithium transport in the electrolyte and through the interface between the electrolyte and the storage particles of the electrodes. The model is valid as a good approximation to the behavior of the battery when it operates near equilibrium, and can be used for both discharge and charging of the battery. The rate of extraction of lithium from and to the electrode storage particles can be estimated from the results of the model, information that can be used in turn to estimate the shrinkage and swelling stresses that develop in the particles. Given specified rates of extraction for spherical particles, maps of the resulting shrinkage and swelling stresses can be developed connecting their values to battery parameters such as particles size, diffusion coefficient, lithium partial molar volume, and particle elastic properties. Since a constant rate of extraction can only be achieved for a limited period of time until the concentration of lithium at the particle perimeter constrains the lithium mass transport, plots of the average state of charge in the particle versus time are also produced.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031022-031022-6. doi:10.1115/1.4005963.

Epidermal electronic system (EES) is a class of integrated electronic systems that are ultrathin, soft, and lightweight, such that it could be mounted to the epidermis based on van der Waals interactions alone, yet provides robust, intimate contact to the skin. Recent advances on this technology will enable many medical applications such as to monitor brain or heart activities, to monitor premature babies, to enhance the control of prosthetics, or to realize human-machine interface. In particular, the contact between EES and the skin is key to high-performance functioning of the above applications and is studied in this paper. The mechanics concepts that lead to successful designs of EES are also discussed. The results, validated by finite element analysis and experimental observations, provide simple, analytical guidelines for design and optimization of EES with various possible functionalities.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031023-031023-11. doi:10.1115/1.4006024.

In filled elastomers, the mechanical behavior of the material surrounding the fillers -termed interphasial material-can be significantly different (softer or stiffer) from the bulk behavior of the elastomeric matrix. In this paper, motivated by recent experiments, we study the effect that such interphases can have on the mechanical response and stability of fiber-reinforced elastomers at large deformations. We work out in particular analytical solutions for the overall response and onset of microscopic and macroscopic instabilities in axially stretched 2D fiber-reinforced nonlinear elastic solids. These solutions generalize the classical results of Rosen (1965, “Mechanics of Composite Strengthening,” Fiber Composite Materials, American Society for Metals, Materials Park, OH, pp. 37–75), and Triantafyllidis and Maker (1985, “On the Comparison between Microscopic and Macroscopic Instability Mechanisms in a Class of Fiber-Reinforced Composites,” J. Appl. Mech., 52 , pp. 794–800), for materials without interphases. It is found that while the presence of interphases does not significantly affect the overall axial response of fiber-reinforced materials, it can have a drastic effect on their stability.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031024-031024-9. doi:10.1115/1.4005898.

The overall elasto-plastic behavior of single crystals is governed by individual slips on crystallographic planes, which occur when the resolved shear stress on a critical slip system reaches a certain maximum value. The challenge lies in identifying the activated slip systems for a given load increment since the process involves selection from a pool of linearly dependent slip systems. In this paper, we use an “ultimate algorithm” for the numerical integration of the elasto-plastic constitutive equation for single crystals. The term ultimate indicates exact integration of the elasto-plastic constitutive equation and explicit tracking of the sequence of slip system activation. We implement the algorithm into a finite element code and report the performance for polycrystals subjected to complicated loading paths including non-proportional and reverse/cyclic loading at different crystal orientations. It is shown that the ultimate algorithm is comparable to the widely used radial return algorithm for J2 plasticity in terms of global numerical stability.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031025-031025-12. doi:10.1115/1.4006154.

We study volumetric deformation structures in stepover regions using numerical simulations and field observations, with a focus on small-scale features near the ends of rupture segments that have opposite-polarity from the larger-scale structures that characterize the overall stepover region. The reversed-polarity small-scale structures are interpreted to be generated by arrest phases that start at the barriers and propagate some distance back into the rupture segment. Dynamic rupture propagating as a symmetric bilateral crack produces similar (anti-symmetric) structures at both rupture ends. In contrast, rupture in the form of a predominantly unidirectional pulse produces pronounced reversed-polarity structures only at the fault end in the dominant propagation direction. Several observational examples at different scales from strike-slip faults of the San Andreas system in southern California illustrate the existence of reversed-polarity secondary deformation structures. In the examples shown, relatively-small pressure-ridges are seen only on one side of relatively-large extensional stepovers. This suggests frequent predominantly unidirectional ruptures in at least some of those cases, although multisignal observations are needed to distinguish between different possible mechanisms. The results contribute to the ability of inferring from field observations on persistent behavior of earthquake ruptures associated with individual fault sections.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2012;79(3):031026-031026-12. doi:10.1115/1.4005964.

Earthquakes occur as dynamic shear cracks and convert part of the elastic strain energy into radiated and dissipated energy. Local evolution of shear strength that governs this process, which is variable in space and time, can be studied from laboratory experiments and rupture models. At the same time, increasingly accurate measurements of radiated energy and other quantities characterize earthquakes in a rupture-averaged way. Here, we present and study two approaches to averaging frictional dissipation during dynamic rupture. The first one is based on the actual progression of dissipation, but the associated averaged shear stress does not reflect the local friction behavior. The second one is constructed to preserve prevailing features of local stress-slip response and performs well in the examples studied. The developed approach should be useful for visualizing energy partitioning in dynamic models and linking them to observations using diagrams that reflect dominant features of local stress evolution.

Commentary by Dr. Valentin Fuster