A micro‐mechanical damage model based on gradient plasticity: algorithms and applications

As soon as material failure dominates a deformation process, the material increasingly displays strain softening and the finite element computation is significantly affected by the element size. Without remedying this effect in the constitutive model one cannot hope for a reliable prediction of the ductile material failure process. In the present paper, a micro‐mechanical damage model coupled to gradient‐dependent plasticity theory is presented and its finite element algorithm is discussed. By incorporating the Laplacian of plastic strain into the damage constitutive relationship, the known mesh‐dependence is overcome and computational results are uniquely correlated with the given material parameters. The implicit C¹ shape function is used and can be transformed to arbitrary quadrilateral elements. The introduced intrinsic material length parameter is able to predict size effects in material failure. Copyright © 2002 John Wiley & Sons, Ltd.     

A gradient flow theory of plasticity for granular materials

Summary A flow theory of plasticity for pressure-sensitive, dilatant materials incorporating second order gradients into the flow-rule and yield condition is suggested. The appropriate extra boundary conditions are obtained with the aid of the principle of virtual work. The implications of the theory into shear-band analysis are examined. The determination of the shear-band thickness and the persistence of ellipticity in the governing equations are discussed.     

Variational projection methods for gradient crystal plasticity using Lie algebras

A computational method is developed for evaluating the plastic strain gradient hardening term within a crystal plasticity formulation. While such gradient terms reproduce the size effects exhibited in experiments, incorporating derivatives of the plastic strain yields a nonlocal constitutive model. Rather than applying mixed methods, we propose an alternative method whereby the plastic deformation gradient is variationally projected from the elemental integration points onto a smoothed nodal field. Crucially, the projection utilizes the mapping between Lie groups and algebras in order to preserve essential physical properties, such as orthogonality of the plastic rotation tensor. Following the projection, the plastic strain field is directly differentiated to yield the Nye tensor. Additionally, an augmentation scheme is introduced within the global Newton iteration loop such that the computed Nye tensor field is fed back into the stress update procedure. Effectively, this method results in a fully implicit evolution of the constitutive model within a traditional displacement‐based formulation. An elemental projection method with explicit time integration of the plastic rotation tensor is compared as a reference. A series of numerical tests are performed for several element types in order to assess the robustness of the method, with emphasis placed upon polycrystalline domains and multi‐axis loading. Copyright © 2016 John Wiley & Sons, Ltd.     

An implicit BEM formulation for gradient plasticity and localization phenomena

The aim of this paper is to discuss a boundary element formulation for non‐linear structural problems involving localization phenomena. In order to overcome the well‐known mesh dependency observed in local plasticity, a gradient plasticity model is used. An implicit boundary element formulation is proposed and the underlying consistent tangent operator defined. This formulation is based on the classical displacement and strain integral representations combined with an integral representation of the plastic multiplier. First numerical examples are presented to illustrate the application of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical approximation of incremental infinitesimal gradient plasticity

We investigate a representative model of continuum infinitesimal gradient plasticity. The formulation is an extension of classical rate‐independent infinitesimal plasticity based on the additive decomposition of the symmetric strain tensor into elastic and plastic parts. It is assumed that dislocation processes contribute to the storage of energy in the material whereby the curl of the plastic distortion appears in the thermodynamic potential and leads to an additional nonlocal backstress tensor. The formulation is cast into a numerical framework by a saddle point approximation of the corresponding minimization problem in each incremental loading step. This allows one to reformulate the (nonlocal) dissipation inequality to a point‐wise flow rule and yields a solution scheme, which is a direct extension of the standard approach in classical plasticity. Our numerical results show the regularizing effects of the additional physically motivated terms. Copyright © 2008 John Wiley & Sons, Ltd.     

Likelihood ratio gradient estimation for dynamic reliability applications

This paper investigates the issue of performing a first-order sensitivity analysis in the setting of dynamic reliability. The likelihood ratio (LR) derivative/gradient estimation method is chosen to fulfill the mission. Its formulation and implementation in the system-based Monte Carlo approach that is commonly used in dynamic reliability applications is first given. To speed up the simulation, we then apply the LR method within the framework of Z-VISA, a biasing (or importance sampling) method we have developed recently. A widely discussed dynamic reliability example (a holdup tank) is studied to test the effectiveness and behaviors of the LR method when applied to dynamic reliability problems and also the effectiveness of the Z-VISA biasing technique for reducing the variance of LR derivative estimators.     

Characteristics of acoustic plate modes on rotated Y-cuts of quartz utilized for biosensing applications

Acoustic plate modes (APM) on various quartz substrates have been investigated in order to determine their usefulness for liquid-sensing applications. The modes have been characterized in terms of their mass sensitivity, mode separation, temperature sensitivity, and reproducibility of the experimental results. Promising characteristics are found for rotated Y-cuts of quartz with the direction of acoustic mode propagation being perpendicular to the X-axis of the quartz crystal. Experiments on the detection of immunochemical reactions are performed using different quartz APM sensors, and the results are compared to similar experiments utilizing APM devices on ZX-LiNbO3.     

Assortment of technological terms for silicon-germanium thin films in gradient configuration

Phase diagrams have been studied to describe the RF PECVD process for intrinsic-hydrogenated silicon Si:H and silicon-low germanium alloy a-Si_1−xGe_x:H thin films using textured Al substrates that have been overdeposited with n-type amorphous Si:H (n⁺ a-Si:H). UV, vis, IR, atomic force microscopy (AFM), Raman spectroscopy, small angle X-ray and cross-section transmission electron microscopy (TEM) are used to establish the phase diagram. The a-Si:H, a-Si_1−xGe_x and μc-Si:H processes are applied for optimization of triple-junction thin silicon-based n-i-p solar cells.     

An arbitrary Lagrangian–Eulerian finite element method for finite strain plasticity

This paper presents a new arbitrary Lagrangian–Eulerian (ALE) finite element formulation for finite strain plasticity in non‐linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part ( F = F ^e F ^p), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi‐static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Processing of grain-size functionally gradient bioceramics for implant applications

This paper reports work on the processing of functionally gradient alumina bioceramics with a continuously decreasing grain size across the thickness, with the view of ultimately utilizing high-quality nano/ultrafine powders only at the surface of an implant to provide superior wear and mechanical properties. A model of disc geometry is used to examine the feasibility of producing this brand of materials. Wet processing/ball milling and sequential slip casting procedures were used to de-agglomerate alumina powders and deposit green layers of varying particle sizes from 50 to 250 nm. Both pressure-less sintering and hot pressing were evaluated as high temperature sintering/consolidation processes. The results indicate that pressure-less sintering may not be suitable. Hot pressing, however, achieved very promising results producing near fully dense product with a grain size that gradually changes across its thickness.     

Degenerate scale problem in antiplane elasticity or Laplace equation for quadrilaterals with arbitrary configuration

This paper provides numerical solutions of the degenerate scale for shapes of quadrilaterals with arbitrary configuration in an exterior boundary value problem of antiplane elasticity or Laplace equation. The first step is to find the parameters in the Schwarz–Christoffel mapping. The first prevertex on the unit circle can be placed in a particular position, or at −1. From the single-valued condition of the mapping function, only one prevertex is independent. The real preverteces can be found from the condition that the computed ratio of two edges is equal to a ratio of two real edges assumed beforehand. An iteration is suggested to obtain the preverteces numerically. After those parameters are obtained, the degenerate sizes of four edges can be evaluated by a numerical integration. Several numerical examples and the computed results were provided.     

Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity

The paper presents the theory and the numerics of a thermodynamically consistent formulation of gradient plasticity at small strains. Starting from the classical local continuum formulation, which fails to produce physically meaningful and numerically converging results within localization computations, a thermodynamically motivated gradient plasticity formulation is envisioned. The model is based on an assumption for the Helmholtz free energy incorporating the gradient of the internal history variable, a yield condition and the postulate of maximum dissipation resulting in an associated structure. As a result the driving force conjugated to the hardening evolution is identified as the quasi‐non‐local drag stress which incorporates besides the strictly local drag stress essentially the divergence of a vectorial hardening flux. At the numerical side, besides the balance of linear momentum, the algorithmic consistency condition has to be solved in weak form. Thereby, the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non‐linear programming. Moreover, different discretization techniques are proposed in order to compare the FE‐performance in local plasticity with the advocated gradient formulation both for hardening and softening. Copyright © 2001 John Wiley & Sons, Ltd.     

On an intermediate electronic configuration of cobalt (+III)

In a D_4h site with a strong elongation the ³A_2g term of cobalt (+III) can be stabilized with an electronic configuration intermediate between low spin (¹A_1g) and high spin (⁵E_g): d²_xzd²_yzd¹_xyd¹_z2d⁰_x2_?y2. The La₂Li_0.5Co_0.5O₄ phase, whose layer structure favours strongly such a distortion of the CoO₆ octahedra, illustrates very well this possibility for cobalt (+III). ¹A_1g → ³A_2g and ¹A_1g → ⁵E_g transitions appear as the temperature increases.     

Analysis of scattering from arbitrary configuration of elliptical obstacles using T-matrix representation

A new method of analysis of an arbitrary configuration of elliptical cylinders is presented. The treatment of such a problem is based on a combination of the cylindrical and elliptical coordinates systems. This approach is applied to the formulation of the scattered field at the surface of equivalent cylinder surrounding dielectric or metal elliptical object. To analyse the arbitrary set of elliptical cylinders, the iterative scattering procedure is used. A good agreement of the proposed method with the commercial FD simulator is achieved.     

High gradient magnetic separation theory and applications

This paper discusses key technical and economical achievements which have extended the range of application of magnetic separation methods into the commercial processing of micron size feebly magnetic materials. Commercial application of magnetic methods in the cleaning of kaolin clay is reviewed and a discussion of magnetic separation principles is given with emphasis on identification and utilization of important process parameters. Possible future developments in magnetic processing of municipal and industrial wastewaters and of applications of magnetic methods to the preparation of clean fuels from coal are discussed.     

An explicit finite element approach with patch projection technique for strain gradient plasticity formulations

Implicit and explicit finite element approaches are frequently applied in real problems. Explicit finite element approaches exhibit several advantages over implicit method for problems which include dynamic effects and instability. Such problems also arise for materials and structures at small length scales and here length scales at the micro and sub-micron scales are considered. At these length scales size effects can be present which are often treated with strain gradient plasticity formulations. Numerical treatments for strain gradient plasticity applying the explicit finite element approach appear however to be absent in the scientific literature. Here such a numerical approach is suggested which is based on patch recovery techniques which have their origin in error indication procedures and adaptive finite element approaches. Along with the proposed explicit finite element procedure for a strain gradient plasticity formulation some numerical examples are discussed to assess the suggested approach.     

A mixed finite element and mesh-free method using linear complementarity theory for gradient plasticity

A mixed finite element (FE) and mesh-free (MF) method for gradient-dependent plasticity using linear complementarity theory is presented. The assumed displacement field is interpolated in terms of its discrete values defined at the nodal points of the FE mesh with the FE shape functions, whereas the assumed plastic multiplier field required to express its Laplacian is interpolated in terms of its discrete values defined at the integration points of the FE mesh with the MF interpolation functions. A standard form of linear complementarity problem is constructed by combining the weak form of momentum conservation equation and pointwise enforcements of both non-local constitutive equation and non-local yield criterion. The discrete values of the plastic multiplier are taken as the only primary unknowns to be determined. The numerical results demonstrate the validity of the proposed method in the simulation of the strain localization phenomenon due to strain softening.