Coupled Gradient Enhanced Damage - Viscoplasticity Model Using Phase Field Method

The aim of this dissertation is developing a framework to describe damage evolution as a phase transformation in solid materials and couple it to the well-known Perzyna type viscoplastic model to account for inelastic behavior of ductile materials. To accomplish this task, the following steps have been performed. First, a new nonlocal, gradient based damage model is proposed for isotropic elastic damage using the phase field method in order to show the evolution of damage in brittle materials. The general framework of the phase field model (PFM) is discussed and the order parameter is related to the damage variable in continuum damage mechanics (CDM). The time dependent Ginzburg-Landau equation which is also termed the Allen-Cahn equation is used to describe the damage evolution process. Specific length scale which addresses the transition region in which the process of changing the undamaged solid to the fully damaged material (microcracks) occurs is defined in order to capture the effect of the damaged localization zone. A new implicit damage variable is proposed through the phase field theory. Finite Difference Method is used and details of the different aspects and regularization capabilities are illustrated by means of numerical examples and the validity and usefulness of the phase field modeling approach is demonstrated. Subsequently, the theory is developed to address the anisotropic damage evolution and simulation in materials. The anisotropic damage is discussed and appropriate nonconserved order parameters in three mutually perpendicular directions are defined to find the growth of the components of a second order diagonal damage tensor corresponding to the principle directions of a general second order damage tensor. In contrast to the previous models, two new tensors are proposed to act as interpolation and potential functions along with three coupled Allen - Cahn equations in order to obtain the evolution of the order parameters, which is the basis of the definition of the damage rate. The tensor formulation of the growth of the components of the damage tensor using the phase field theory is proposed for the first time. It is shown that by introducing a set of material parameters, there is a robust and simplified way to model the nonlocal behavior of damage and predict the corresponding material behavior along the principal axes of the second order damage tensor. Finally, the framework of coupled nonlocal damage model through phase field method and viscoplasticity in continuum scale is developed. It is shown that the recently proposed non local gradient type damage model through the phase field method can be coupled to a viscoplastic model to capture the inelastic behavior of the rate dependent material. Free energy functional of the system containing two main parts including damage propagation as a phase transformation and viscoplastic deformation is proposed. Analogous to conventional viscoplastic models, two terms are incorporated in the viscoplastic free energy functional to appropriately address dissipation and the von Mises type viscoplastic surface. In this framework it is assumed that the damage variable covers summation of evolution of microcracks density in elastic and plastic region and the total strain represents the summation of the elastic and viscoplastic counterparts. Since all the examples of chapters two and three are solved using the Finite Element Method, appropriate algorithms and derivations are also summarized in the last chapter

Fundamentals and applications of spatial dissipative solitons in photonic devices : [Chapter 6]

We review the properties of optical spatial dissipative solitons (SDS). These are stable, self‐localized optical excitations sitting on a uniform, or quasi‐uniform, background in a dissipative environment like a nonlinear optical cavity. Indeed, in optics they are often termed “cavity solitons.” We discuss their dynamics and interactions in both ideal and imperfect systems, making comparison with experiments. SDS in lasers offer important advantages for applications. We review candidate schemes and the tremendous recent progress in semiconductor‐based cavity soliton lasers. We examine SDS in periodic structures, and we show how SDS can be quantitatively related to the locking of fronts. We conclude with an assessment of potential applications of SDS in photonics, arguing that best use of their particular features is made by exploiting their mobility, for example in all‐optical delay lines

Patchiness and Demographic Noise in Three Ecological Examples

Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only -at most- local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals. Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still difficult to numerically integrate due to the presence of the "square-root" intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three classic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.Comment: 20 pages, 5 figures. To appear in J. Stat. Phy

Origin of stabilization of macrotwin boundaries in martensites

The origin of stabilization of complex microstructures along macrotwin boundaries in martensites is explained by comparing two models based on Ginzburg-Landau theory. The first model incorporates a geometrically nonlinear strain tensor to ensure that the Landau energy is invariant under rigid body rotations, while the second model uses a linearized strain tensor under the assumption that deformations and rotations are small. We show that the approximation in the second model does not always hold for martensites and that the experimental observations along macrotwin boundaries can only be reproduced by the geometrically nonlinear (exact) theory

On the critical nature of plastic flow: one and two dimensional models

Steady state plastic flows have been compared to developed turbulence because the two phenomena share the inherent complexity of particle trajectories, the scale free spatial patterns and the power law statistics of fluctuations. The origin of the apparently chaotic and at the same time highly correlated microscopic response in plasticity remains hidden behind conventional engineering models which are based on smooth fitting functions. To regain access to fluctuations, we study in this paper a minimal mesoscopic model whose goal is to elucidate the origin of scale free behavior in plasticity. We limit our description to fcc type crystals and leave out both temperature and rate effects. We provide simple illustrations of the fact that complexity in rate independent athermal plastic flows is due to marginal stability of the underlying elastic system. Our conclusions are based on a reduction of an over-damped visco-elasticity problem for a system with a rugged elastic energy landscape to an integer valued automaton. We start with an overdamped one dimensional model and show that it reproduces the main macroscopic phenomenology of rate independent plastic behavior but falls short of generating self similar structure of fluctuations. We then provide evidence that a two dimensional model is already adequate for describing power law statistics of avalanches and fractal character of dislocation patterning. In addition to capturing experimentally measured critical exponents, the proposed minimal model shows finite size scaling collapse and generates realistic shape functions in the scaling laws.Comment: 72 pages, 40 Figures, International Journal of Engineering Science for the special issue in honor of Victor Berdichevsky, 201