A regularized continuum model for fault zones with rate and state friction

<p>Accurate representation of fault zones is important in many applications in Earth sciences, including natural and induced seismicity. The framework developed here can efficiently model fault zone localization, evolution, and spontaneous fully dynamic earthquake sequences in a continuum plasticity framework. The geometrical features of the faults are incorporated into a regularized continuum framework, while the response of the fault zone is governed by a rate and state-dependent friction. Although a continuum plasticity model is advantageous to discrete approaches in representing evolving, unknown, or arbitrarily positioned faults, it is known that either non-associated plasticity or strain-softening can lead to mesh sensitivity of the numerical results in absence of an internal length scale. A common way to regularize the numerical model and introduce an internal length scale is by the adoption of a Kelvin-type visco-plasticity element. The visco-plastic rheological behavior for the bulk material is implemented along with a return-mapping algorithm for accurate stress and strain evolution. High slip rates (in the order of 1 m/s) are captured through numerical examples of a predefined strike-slip fault zone, where a detailed comparison with a reference discrete fault model is presented. Additionally, the regularization effect of the Kelvin viscosity parameter is studied on the fault slip velocity for a growing fault zone due to an initial material imperfection.  The model is consistently linearized leading to quadratic convergence of the Newton solver. Although the proposed framework is a step towards the modeling of earthquake sequences for induced seismicity applications, the numerical model is general and can be applied to all tectonic settings including subduction zones.</p><div> <div> <div> </div> <div> <div> <div> </div> <div> <p> </p> <p> </p> </div> </div> </div> </div> </div>

Comparison of evolving gradient damage formulations with different activity functions

In the paper, two existing upgrades of the gradient damage model for the simulations of cracking in concrete are compared. The damage theory is made nonlocal via a gradient enhancement to overcome the mesh dependence of simulation results. The implicit gradient model with an averaging equation, where the internal length parameter is assumed as constant during the strain softening analysis, gives unrealistically broadened damage zones. The gradient enhancement of the scalar damage model can be improved via a function of an internal length scale, so an evolution of the gradient activity is postulated during the localization process. Two different modifications of the averaging equation and respective evolving gradient damage formulations are presented. Different activity functions are tested to see whether the formation of a too wide damage zone still occurs. Activating or localizing character of the gradient influence can be introduced and the impact of both approaches on the numerical results is shown in the paper. The aforementioned variants are implemented and examined using the benchmarks of tension in a bar and bending of a cantilever beam.

The Micromorphic Approach to Generalized Heat Equations

In this paper, the micromorphic approach, previously developed in the mechanical context is applied to heat transfer and shown to deliver new generalized heat equations as well as the nonlocal effects. The latter are compared to existing formulations: the classical Fourier heat conduction, the hyperbolic type with relaxation time, the gradient of temperature or entropy theories, the double temperature model, the micro-temperature model or micro-entropy models. A new pair of thermodynamically-consistent micromorphic heat equations are derived from appropriate Helmholtz-free energy potentials depending on an additional micromorphic temperature and its first gradient. The additional micromorphic temperature associated with the classical local temperature is introduced as an independent degree of freedom, based on the generalized principle of virtual power. This leads to a new thermal balance equation taking into account the nonlocal thermal effects and involving an internal length scale which represents the characteristic size of the system. Several existing extended generalized heat equations could be retrieved from constrained micromorphic heat equations with suitable selections of the Helmholtz-free energy and heat flux expressions. As an example the propagation of plane thermal waves is investigated according to the various generalized heat equations. Possible applications to fast surface processes, nanostructured media and nanosystems are also discussed.

Computations of anomalous phase change

Purpose – The purpose of this paper is to demonstrate how anomalous diffusion behaviors can be manifest in physically realizable phase change systems. Design/methodology/approach – In the presence of heterogeneity the exponent in the diffusion time scale can become anomalous, exhibiting values that differ from the expected value of 1/2. Here the author investigates, through directed numerical simulation, the two-dimensional melting of a phase change material (PCM) contained in a pattern of cavities separated by a non-PCM matrix. Under normal circumstances we would expect that the progress of melting F(t) would exhibit the normal diffusion time exponent, i.e., F∼t1/2. The author’s intention is to investigate what features of the PCM cavity pattern might induce anomalous phase change, where the progress of melting has a time exponent different from n=1/2. Findings – When the PCM cavity pattern has an internal length scale, i.e., when there is a sub-domain pattern which, when reproduced, gives us the full domain pattern, the direct simulation recovers the normal ∼t1/2 phase change behavior. When, however, there is no internal length scale, e.g., the pattern is a truncated fractal, an anomalous super diffusive behavior results with melting going as t n; n > 1/2. By studying a range of related fractal patterns, the author is able to relate the observed sub-diffusive exponent to the cavity pattern’s fractal dimension. The author also shows, how the observed behavior can be modeled with a non-local fractional diffusion treatment and how sub-diffusion phase change behavior (F∼t n; n < 1/2) results when the phase change nature of the materials in the cavity and matrix are inverted. Research limitations/implications – Although the results clearly demonstrate under what circumstances anomalous phase change behavior can be practically produced, the question of an exact theoretical relationship between the cavity pattern geometry and the observed anomalous time exponent is not known. Practical implications – The clear role of the influence of heterogeneity on heat flow behavior is illustrated. Suggesting that modeling heat and fluid flow in heterogeneous systems requires careful consideration. Originality/value – The novel direct simulation of melting in a two-dimensional PCM cavity pattern provides a clear illustration of anomalous behavior in a classic heat and fluid flow system and by extension provides motivation to continue the numerical investigation of anomalous and non-local behaviors and fractional calculus tools.

Gradient plasticity used for modeling extrinsic and intrinsic size effects in the torsion of Au microwires

The gradient plasticity theory proposed by Aifantis and coworkers has been successfully used to model size effect phenomena at the microscale and nanoscale, by introducing into the formulation an internal length scale associated with the phenomenological coefficients of the gradient plasticity model. In this paper, Aifantis’ gradient plasticity theory is applied to model the sample size-dependent torsion of thin wires, with a strain-dependent internal length scale as well as grain size dependence based on the Hall-Petch relationship. This study reveals that internal length scale is related with sample size and grain size, with such a connection determined by the ductility of the material.

A Finite Points Method Approach for Strain Localization Using the Gradient Plasticity Formulation

The softening elastoplastic models present an unsuitable behavior after reaching the yield strength: unbounded strain localization. Because of the material instability, which is reflected in the loss of ellipticity of the governing partial differential equations, the solution depends on the discretization. The present work proposes to solve this dependency using the meshless Finite Points Method. This meshfree spatial discretization technique allows enriching the governing equations using gradient’s plasticity and introducing an internal length scale parameter at the material model in order to objectify the solution.

Stress–elastic strain relation for a two-phase isotropic strain gradient plastic composite

The stress–elastic strain relationship is studied for a composite under a plastic deformation. The constitutive law of each component is described by a deformation theory of strain gradient plasticity which introduces an internal length scale. The conventional deformation plastic theory is obtained when the internal length scale tends to 0. The Hashin–Shtrikman upper bound for a two-phase composite governed by a power law is derived. It is predicted, by differentiating the bounds, that in most cases, the stress and the elastic strain follow a non-linear relation immediately after the elastic range. However, for some particular values of the ratio of the internal length scale and the micro-scale of the composite, this relation is linear. The prediction is illustrated by various numerical examples.