Slip Gradient Research Articles

Unlocking the effects of friction on fault damage zones

Two-dimensional, numerical models of a linear fault embedded within a linear elastic medium show the generation of off-fault tensile failure that results from inelastic slip along the fault. We explore quasi-static models with slip-weakening friction to assess the effects of spatially and temporally variable friction on the damage patterns. Tensile fractures form where tangential normal stresses along the fault exceed the tensile strength of the rock. These stresses result from locally high slip gradients at the rupture tip. Because faults of different displacement history and rock type could have varying slip-weakening distances, we examine the effect of changing the slip-weakening distance on the damage pattern and find that this parameter is important in determining off-fault fracture intensity and continuity along strike. Faults with short slip-weakening distance produce greater off-fault damage and significantly greater seismic radiated energy than faults with longer slip-weakening distances. We also investigate the effect of pre-existing damage on the subsequent development of fractures in second generation slip episodes and find that damage localizes onto pre-existing patches. These results could guide field studies of small faults as to whether the fault failed in one slip event or multiple small events.

Evidence from coseismic slip gradients for dynamic control on rupture propagation and arrest through stepovers

Analysis of historic slip distributions from large‐magnitude continental strike‐slip earthquake ruptures reveals a pronounced correlation between the distance over which slip decreases as a rupture approaches a structural “step” in the fault and the ability of the rupture to propagate through the step. Our analysis of coseismic slip gradients near these stepovers indicates that in earthquakes in which slip decreases gradually toward a step, rupture will not continue on the next segment. Conversely, in earthquakes in which slip decreases abruptly, rupture commonly renucleates on the next segment. The fact that ruptures that stopped had low slip gradients near stepovers, relative to those that continued, indicates that rupture dynamics control the propagation of rupture through stepovers. These results corroborate dynamic rupture models that show that ruptures in which slip decreases abruptly at a step generate strong seismic waves that serve to renucleate rupture on the opposite side of the structural step. There are several potential causes for gradual decrease of displacement as rupture approaches a stepover, including transference of deformation onto subsidiary structures, rheological contrasts that could dictate favored rupture propagation directions, the existence of stress shadows from previous earthquakes within the system, or other material or frictional heterogeneities. All of these are potentially observable, and, if mapped systematically, could provide the basis for a strong constraint on the likely end points of future earthquakes. Inasmuch as earthquake magnitude is strongly dependent on the size of the rupture, such predictions would be of great utility as a basic component of scenario‐based earthquake rupture forecasts.

AN ANALYTIC SOLUTION FOR ONE-DIMENSIONAL DISSIPATIONAL STRAIN-GRADIENT PLASTICITY

AbstractAn analytic solution is developed for the one-dimensional dissipational slip gradient equation first described by Gurtin [“On the plasticity of single crystals: free energy, microforces, plastic strain-gradients”, J. Mech. Phys. Solids48 (2000) 989–1036] and then investigated numerically by Anand et al. [“A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results”, J. Mech. Phys. Solids53 (2005) 1798–1826]. However we find that the analytic solution is incompatible with the zero-sliprate boundary condition (“clamped boundary condition”) postulated by these authors, and is in fact excluded by the theory. As a consequence the analytic solution agrees with the numerical results except near the boundary. The equation also admits a series of higher mode solutions where the numerical result corresponds to (a particular case of) the fundamental mode. Anand et al. also established that the one-dimensional dissipational gradients strengthen the material, but this proposition only holds if zero-sliprate boundary conditions can be imposed, which we have shown cannot be done. Hence the possibility remains open that dissipational gradient weakening may also occur.

GRAIN-SIZE DEPENDENT YIELD BEHAVIOR UNDER LOADING, UNLOADING AND REVERSE LOADING

Two types of higher-order stresses work-conjugate to slip gradient in single crystals are investigated to analyze the grain-size dependent yield behavior of polycrystals. The first is the higher-order stress due to the self-energy of geometrically necessary dislocations (GNDs). The second higher-order stress examined is an extension based on the non-recoverable energy that is postulated to be proportional to the accumulated density of incrementally defined GNDs. It is shown that the second higher-order stress is co-directional with the in-plane gradient of slip-rate and consequently causes isotropic hardening, whereas the first higher-order stress is co-directional with the in-plane gradient of slip. These higher-order stresses are incorporated into a strain gradient plasticity theory of single crystals. Subsequently, using a finite element method, 2D model polycrystals are analyzed to demonstrate the influence of the two types of higher-order stresses on the grain-size dependent yield behavior under loading, unloading and reverse loading.

Rupture Termination and Jump on Parallel Offset Faults

Proper evaluation of seismic hazard depends on accurate estimates of potential earthquake size. In areas with complex, multisegment fault systems, such an estimate in turn depends on an ability to predict the circumstances under which rupture may jump between fault segments. Many observational and numerical studies have analyzed the phenomenon of jumping rupture, but none has focused on how the process of rupture termination on the primary (nucleating) fault segment affects the ability of rupture to jump to a secondary fault segment. In the current study, I model the dynamics of a simple 2D strike-slip fault system with two parallel segments arranged with either a compressional or extensional stepover. I vary the suddenness with which the initial shear stress tapers to zero on the primary section. If the initial shear stress goes to zero over a very small (100 m) distance, rupture readily jumps both compressional and extensional stepovers of 1 km. If the initial shear stress tapers to zero over 1 km, rupture can jump the compressional stepover, but not the extensional stepover. If the initial shear stress tapers to zero over 2.5 km, rupture cannot jump either the compressional or the extensional stepover. The results illustrate the importance of the slip gradient (and the resultant static stress field) and the acceleration of the rupture front (and the resultant generation of stopping phases) in determining the probability of jumping rupture.

断層破壊過程の複雑さが強震動予測結果に及ぼす影響とその支配的パラメータの抽出

The procedure for source modeling according to a characterized source model has been proposed and established in prevenient researches on the estimation of strong ground motions from scenario earthquakes along active faults. In recent researches, such procedure is also applied to the strong ground motion prediction for some great subduction earthquakes. In these cases, it seems that short-period motions are underestimated while long-period ground motions explain the observed records well. We examine the effects of complexity of fault rupture process on strong ground motions in period range form 5 to 10 s by handling source parameters of the 2003 Tokachi-Oki earthquake. For that purpose, we referred to the source model obtained by inversion analysis of near-field strong motion records in the antecedent research. First, we have made several simplified models from original source model. In the models, we gradually simplified the source parameters, such as slip amount, rupture time, rake angle, and the form of slip velocity time function of fault elements. Contribution of each source parameter to the waveform complexity has been evaluated by comparing spectral levels of synthetic waveforms from different source models in three frequency ranges and by comparing synthetic waveforms themselves. As a result, the complexity of rupture propagation is most important to correctly simulate ground motions in the period of several seconds for M 8 earthquake. We have also examined relation between the rupture propagation complexity and the slip distribution. The result shows that acceleration of rupture propagation is seen in the areas where either slip or spatial gradient of slip is large.

Trishear in 3D. Algorithms, implementation, and limitations

The algorithms and implementation of pseudo-3D and true-3D trishear models are explained, including a strategy to model lateral fault propagation. I show that the pseudo-3D algorithm is adequate and sufficient to model trishear in three-dimensions. Although ad-hoc, the pseudo-3D algorithm preserves volume in simulations without and with lateral fault propagation. A disadvantage of the pseudo-3D algorithm is that it produces very simple, and perhaps not realistic hanging wall geometries, specially in simulations in which the fault slip varies along strike. The true-3D algorithm has a more elaborate and richer kinematics that produces more realistic hanging wall geometries. However, the true-3D algorithm contains mathematical inconsistencies that result in considerable volume changes when the slip gradients along the tip line are high and the tip line is highly oblique to the slip vector and/or the fault strike. The volume changes occur to a large extent in the hanging wall, and to a minor extent in the forelimb and footwall areas.

A finite element formulation to solve a non-local constitutive model with stresses and strains due to slip gradients

Solving constitutive models that incorporate the effects of plasticity and slip gradients is often complicated by the non-local nature of the models. This work presents a finite element solution to a crystal plasticity constitutive model that includes kinematic and stress effects due to slip gradients. The foundation of the model is a three term multiplicative decomposition of the deformation gradient that results in a second order differential equation in terms of the stress that drives slip. Converting the equation into a weak form results in an integral equation that includes first order derivatives of the stress as well as boundary conditions for the stress and gradients of slip rate for each slip system. Using this weak form, an incremental finite element method is developed to solve the constitutive model within a finite element solution to the equilibrium equation. Results for the compression of a two-dimensional plate show the effects of including slip gradient effects in the constitutive model and indicate the tendency for localization of the slip and dislocation density into narrow bands separating regions of nearly constant dislocation density and long range strain.

Bending of a single crystal thin foil of material with slip gradient effects

The simulation of the cyclic bending of a single crystal thin foil is presented. The simulations demonstrate a constitutive model that incorporates the effects of geometrically necessary dislocations that accompany slip gradients and includes a slip gradient proportional hardness and energetically conjugate long range stresses and strains proportional to the second gradient of slip. Results are presented for three cycles of bending and straightening for foil thicknesses of 25, 50 and 100 µm. Various combinations of the gradient hardness coefficient and the long range strain length parameter are examined. The results show an isotropic type hardening behavior due to the gradient hardness and a kinematic type hardening behavior due to the long range strains. Comparing the results from the three different thickness specimens shows an obvious decrease in the effects of the gradient hardness and the long range strain with an increase in the foil thickness.

A crystal plasticity model that incorporates stresses and strains due to slip gradients

This work is concerned with incorporating the kinematic and stress effects of excess dislocations in a constitutive model for the elastoplastic behavior of crystalline materials. The foundation of the model is a three term multiplicative decomposition of the deformation gradient in which the two classical terms of plastic and elastic deformation are included along with an additional term for long range strain due to the collective effects of excess dislocations. The long range strain is obtained from an assumed density of Volterra edge dislocations and is directly related to gradients in slip. A new material parameter emerges which is the size the region about a continuum point that contributes to long range strains. Using Hookean elasticity, the stress at a point is linearly related to the sum of the elastic plus the long range strain fields. However, the driving force for slip is postulated to be due only to the elastic stress so that the long range stress is a back stress in the constitutive relationship for plastic deformation. A consistent balance of the total deformation rate with the three proposed mechanisms of deformation leads to a set of differential equations that can be solved for the elastic stress, rotation and pressure which then implicitly defines the material state and equilibrium stress. Results from the simulation of a tapered tensile specimen demonstrate that the constitutive model exhibits isotropic and kinematic type hardening effects as well as changes in the pattern of plastic deformation and necking when compared to a material without slip gradient effects.

Higher-Order Stress due to Self Energy of Geometrically Necessary Dislocations

The self energy of geometrically necessary dislocations (GNDs) in single crystals is considered to inevitably introduce a higher-order stress as the work conjugate to slip gradient. It is pointed out that this higher-order stress changes stepwise in response to in-plane slip gradient and thus explicitly influences the initial yielding of polycrystals. The self energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin (2002). The resulting theory is applied to 2D and 3D model crystal grains of diameter D, leading to a D-1-dependent term with a coefficient determined by grain shape. This size effect term is verified using published experimental data of several polycrystalline metals. It is thus found that the D-1-dependent term is successful for predicting not only the grain size dependence of initial yield stress but also the dislocation cell size dependence of flow stress in the submicron to several micron range of D.

Gradient single-crystal plasticity with free energy dependent on dislocation densities

This study develops a small-deformation theory of strain-gradient plasticity for single crystals. The theory is based on: (i) a kinematical notion of a continuous distribution of edge and screw dislocations; (ii) a system of microscopic stresses consistent with a system of microscopic force balances, one balance for each slip system; (iii) a mechanical version of the second law that includes, via the microscopic stresses, work performed during viscoplastic flow; and (iv) a constitutive theory that allows: • the free energy to depend on densities of edge and screw dislocations and hence on gradients of (plastic) slip; • the microscopic stresses to depend on slip-rate gradients. The microscopic force balances when augmented by constitutive relations for the microscopic stresses results in a system of nonlocal flow rules in the form of second-order partial differential equations for the slips. When the free energy depends on the dislocation densities the microscopic stresses are partially energetic, and this, in turn, leads to backstresses in the flow rules; on the other hand, a dependence of these stresses on slip-rate gradients leads to a strengthening. The flow rules, being nonlocal, require microscopic boundary conditions; as an aid to numerical solutions a weak (virtual power) formulation of the flow rule is derived.

Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations

The higher-order stress work-conjugate to slip gradient in single crystals at small strains is derived based on the self-energy of geometrically necessary dislocations (GNDs). It is shown that this higher-order stress changes stepwise as a function of in-plane slip gradient and therefore significantly influences the onset of initial yielding in polycrystals. The higher-order stress based on the self-energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32] and applied to single-slip-oriented 2D and 3D model crystal grains of size D. It is thus found that the self-energy of GNDs gives a D - 1 -dependent term for the averaged resolved shear stress in such a model grain under yielding. Using published experimental data for several polycrystalline metals, it is demonstrated that the D - 1 -dependent term successfully explains the grain size dependence of initial yield stress and the dislocation cell size dependence of flow stress in the submicron to several-micron range of grain and cell sizes.

A gradient theory for single-crystal plasticity

This study develops a continuum theory of small-deformation single crystal plasticity that includes ‘strain gradients’ and ‘strain rate gradients’ in the spirit of our recent papers (Gurtin and Anand 2005a, 2005b) for isotropic plasticity. Specifically, we develop a constitutive theory that allows forenergetic microscopic stresses that arise as a consequence of an assumed dependence of the free energy on edge and screw dislocations densities induced by slip gradients; anddissipative microscopic stresses that depend on slip-rate gradients.Our theory is based on a system of microscopic force balances—one balance for each slip system—with derivation based on the principle of virtual power. When augmented by constitutive relations for the microscopic stresses these force balances result in non-local flow rules in the form of second-order partial differential equations for the slips.

Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved

By exploiting the strain gradient crystal plasticity theory put forward by [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids 54, 128–160], we show that a modelling involving only energetic material length scales through the defect energy (i.e., a function of Nye’s dislocation density tensor added to the free energy; see, e.g., [Gurtin, M.E., 2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids 50, 5–32]) may not be enough in order to describe the size effect exhibited by metallic components. In fact, strain gradients that enter the constitutive modelling by taking Nye’s tensor as an independent kinematic variable allow the description of the increase in strain hardening accompanied with diminishing size, but they do not help in capturing the related strengthening; such a size effect can be instead qualitatively described by incorporating (in a standard, phenomenological way) the gradient of the plastic slip (rate), as a further independent kinematic variable, in the isotropic hardening function that provides the resistance to flow on each slip system (Bardella, 2006). In this way, (at least) one dissipative length scale L is introduced in the modelling, and its presence may even lead to a change in the “higher-order” (i.e., non-standard) boundary conditions to be imposed in the inherent boundary value problems. By making use of a simple example that, by taking a proper limit, also provides isotropic plasticity, we explicitly show how the nature of the relevant boundary value problems changes whether L is set to zero or not, and, by analysing the modelling capability, we give an insight on the influence of the crystallography and conclude that it is recommendable that at least one dissipative length scale be always incorporated in the modelling.

A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity

Strain gradient crystal plasticity attempts to predict material size effects by taking into account geometrically necessary dislocations that are required to accommodate gradients of crystallographic slip. Since these dislocations have a non-zero net Burgers vector within the material, dislocation induced long range stresses result in a back stress that influences the effective driving force for crystallographic slip. A dislocation induced back stress formulation is proposed in which the full tensorial nature of the dislocation stress state is included in the continuum description. The significance of this proposed back stress formulation is that it intrinsically includes latent kinematic hardening from dislocations lying on all slip systems. Using simple shearing of a semi-infinite cube oriented single crystal with either double-planar or octahedral slip system configurations, the proposed back stress formulation is examined in detail.

217 すべりの2次勾配を考慮した結晶塑性理論の数理構造について(結晶塑性解析・均質化法)

217 すべりの2次勾配を考慮した結晶塑性理論の数理構造について(結晶塑性解析・均質化法)

Evidence for self‐similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics

We characterize average slip distributions on earthquakes beyond their individual heterogeneity. For that, we analyze a large number of seismic slip distributions both measured at the surface after earthquakes (44 profiles) and derived from slip inversion models (76 models). Investigating the overall shape of these slip profiles, we find that they are roughly triangular both along strike and dip, and most of them (70–80%) are asymmetric. Long linear slopes and high slip gradients therefore are the key ingredients to describe earthquake slip profiles. The scaling relations between maximum displacement and length (or width) suggest furthermore that the triangular slip profiles are self‐similar. Such slip patterns make earthquakes dominated by one major zone of maximum slip hence one major “asperity.” Analyzing the position of hypocenters with respect to these “asperities,” we find that earthquakes nucleate at a distance from them that averages 20–30% of their total length. Compiling observations on 56 earthquakes, we show that this distance (i.e., the asperity size) is structurally defined. We then compare the earthquake slip profiles to cumulative slip profiles measured on long‐term faults of various ages and sizes and find that all profiles have a similar shape, triangular and asymmetric. Hence combining data for a large number of earthquakes leads to point out average, generic characteristics of the coseismic slip that are similar to those that emerge from the accumulation of events with time on a single fault. This suggests that these characteristics result from robust physical properties.

非局所結晶塑性を考慮した均質化理論の構築と有限要素離散化

In this study, we develop a homogenization theory and its finite element discretization based on the nonlocal crystal plasticity theory of Gurtin (J. Mech. Phys. Solids 48 (2000) ; J. Mech. Phys. Solids 50 (2002)), which includes slip gradients and additional slip boundary conditions with respect to microforce balances for each slip system. Although it is difficult to determine slip boundary conditions, the development of the homogenization theory for the nonlocal crystal plasticity makes it possible to obtain the unified treatment of slip boundary conditions on the periodic boundary in periodic materials. Furthermore, by applying a tangent modulus method to the finite element discretization of rate-dependent small deformation, a strong coupling boundary value problem is derived from homogenization equations which consist of the stress balance, the microforce balances and the macroscopic relation. Finally, the effectiveness of the proposed theory and method is confirmed through finite element analyses of a double slip single crystal model with periodic obstacles.

515 非局所結晶塑性を考慮した均質化理論による多結晶金属の有限要素解析(S03-1 結晶塑性解析,S03 有限要素法の最近の動向/メッシュフリー法)

In this study, we explore the role of grain boundaries in the size effect of polycrystal plasticity. For this purpose, we develop a homogenization theory and its finite element discretization based on the nonlocal single-crystal plasticity theory of Gurtin, which includes slip gradients and additional slip boundary conditions. For finite element polycrystal plasticity analysis, the additional slip boundary conditions are applied on grain boundaries. Two typical slip boundary conditions called micro-free and micro-clamped conditions are considered.