Some examples of rank one convex functions in dimension two

We study the rank one convexity of some functions f(ξ) where ξ is a 2 × 2 matrix. Examples such as |ξ|2α + h(detξ) and | ξ |2α (| ξ |2 − γdet ξ) are investigated. Numerical computations are done on the example of Dacorogna and Marcellini, indicating that this function is quasiconve

A 3D Granular Model Of Equiaxed-Granular Solidification

The solidification of an aluminum-copper alloy has been simulated in 3D using a granular model. Compared to previous similar 2D approaches for where only one phase is continuous, the extension to 3D allows for concurrent continuity of the solid and liquid phases. This concurrent continuity is a key factor in the formation of the solidification defect known as hot tearing. In this 3D model, grains are modeled as polyhedrons based on a Voronoi tessellation of a pseudo-random set of nucleation centers. Solidification within each polyhedron is calculated using a back-diffusion model. By performing a series of simulations over a range of grain sizes and cooling rates, the percolation of the solid grains is determined. The results, which indicate that the grain size and cooling rates play an important role in hot tear formation, constitute a basis on which feeding and deformation calculations will be carried out further

Morphology and Growth Kinetic Advantage of Quenched Twinned Dendrites in Al-Zn Alloys

Twinned dendrites appearing in an Al-26 wt pct Zn alloy have been quenched during growth using a specifically designed setup that is positioned on top of a directional solidification experiment. X-ray tomography performed at the Swiss Light Source (SLS-beamline TOMCAT) allowed us to reconstruct the 3D morphology of these structures and to confirm previous observations performed on single 2D sections (Henry et al., Metall Mater Trans A 35A:2495-2501, 2004; Salgado-Ordorica and Rappaz, Acta Mater 56:5708-5718, 2008). Further characterization of these quenched specimens led to a better description of the mechanisms involved in the in-plane and lateral growth propagation of twinned dendrites. These were then put into relation with the competition mechanisms taking place during simultaneous solidification of twinned and regular dendrites. DOI: 10.1007/s11661-012-1539-0 (C) The Minerals, Metals & Materials Society and ASM International 201

Relationship between solidification microstructure and hot cracking susceptibility for continuous casting of low-carbon and high-strength low-alloyed steels: A phase-field study

© The Minerals, Metals & Materials Society and ASM International 2013Hot cracking is one of the major defects in continuous casting of steels, frequently limiting the productivity. To understand the factors leading to this defect, microstructure formation is simulated for a low-carbon and two high-strength low-alloyed steels. 2D simulation of the initial stage of solidification is performed in a moving slice of the slab using proprietary multiphase-field software and taking into account all elements which are expected to have a relevant effect on the mechanical properties and structure formation during solidification. To account for the correct thermodynamic and kinetic properties of the multicomponent alloy grades, the simulation software is online coupled to commercial thermodynamic and mobility databases. A moving-frame boundary condition allows traveling through the entire solidification history starting from the slab surface, and tracking the morphology changes during growth of the shell. From the simulation results, significant microstructure differences between the steel grades are quantitatively evaluated and correlated with their hot cracking behavior according to the Rappaz-Drezet-Gremaud (RDG) hot cracking criterion. The possible role of the microalloying elements in hot cracking, in particular of traces of Ti, is analyzed. With the assumption that TiN precipitates trigger coalescence of the primary dendrites, quantitative evaluation of the critical strain rates leads to a full agreement with the observed hot cracking behavior. © 2013 The Minerals, Metals & Materials Society and ASM International

High-throughput, nonperturbing quantification of lipid droplets with digital holographic microscopy.

In vitro differentiating adipocytes are sensitive to liquid manipulations and have the tendency to float. Assessing adipocyte differentiation using current microscopy techniques involves cell staining and washing, while using flow cytometry involves cell retrieval in suspension. These methods induce biases, are difficult to reproduce, and involve tedious optimizations. In this study, we present digital holographic microscopy (DHM) as a label-free, nonperturbing means to quantify lipid droplets in differentiating adipocytes in a robust medium- to high-throughput manner. Taking advantage of the high refractive index of lipid droplets, DHM can assess the production of intracellular lipid droplets by differences in phase shift in a quantitative manner. Adipocytic differentiation, combined with other morphological features including cell confluence and cell death, was tracked over 6 days in live OP9 mesenchymal stromal cells. We compared DHM with other currently available methods of lipid droplet quantification and demonstrated its robustness with modulators of adipocytic differentiation in a dose-responsive manner. This study suggests DHM as a novel marker-free nonperturbing method to study lipid droplet accumulation and may be envisioned for drug screens and mechanistic studies on adipocytic differentiation

Prediction of Hot Tear Formation in Vertical DC Casting of Aluminum Billets Using a Granular Approach

A coupled hydromechanical granular model aimed at predicting hot tear formation and stress-strain behavior in metallic alloys during solidification is applied to the semicontinuous direct chill casting of aluminum alloy round billets. This granular model consists of four separate three-dimensional (3D) modules: (I) a solidification module that is used for generating the solid-liquid geometry at a given solid fraction, (II) a fluid flow module that is used to calculate the solidification shrinkage and deformation-induced pressure drop within the intergranular liquid, (III) a semisolid deformation module that is based on a combined finite element/discrete element method and simulates the rheological behavior of the granular structure, and (IV) a failure module that simulates crack initiation and propagation. To investigate hot tearing, the granular model has been applied to a representative volume within the direct chill cast billet that is located at the bottom of the liquid sump, and it reveals that semisolid deformations imposed on the mushy zone open the liquid channels due to localization of the deformation at grains boundaries. At a low casting speed, only individual pores are able to form in the widest channels because liquid feeding remains efficient. However, as the casting speed increases, the flow of liquid required to compensate for solidification shrinkage also increases and as a result the pores propagate and coalesce to form a centerline crack

Three-dimensional granular model of semi-solid metallic alloys undergoing solidification: Fluid flow and localization of feeding

A three-dimensional (3-D) granular model which simulates fluid flow within solidifying alloys with a globular microstructure, such as that found in grain refined Al alloys, is presented. The model geometry within a representative volume element (RVE) consists of a set of prismatic triangular elements representing the intergranular liquid channels. The pressure field within the liquid channels is calculated using a finite elements (FEs) method assuming a Poiseuille flow within each channel and flow conservation at triple lines. The fluid flow is induced by solidification shrinkage and openings at grain boundaries due to deformation of the coherent solid. The granular model predictions are validated against bulk data calculated with averaging techniques. The results show that a fluid flow simulation of globular semi-solid materials is able to reproduce both a map of the 3-D intergranular pressure and the localization of feeding within the mushy zone. A new hot cracking sensitivity coefficient is then proposed. Based on a mass balance performed over a solidifying isothermal volume element, this coefficient accounts for tensile deformation of the semi-solid domain and for the induced intergranular liquid feeding. The fluid flow model is then used to calculate the pressure drop in the mushy zone during the direct chill casting of aluminum alloy billets. The predicted pressure demonstrates that deep in the mushy zone where the permeability is low the local pressure can be significantly lower than the pressure predicted by averaging techniques. (C) 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved

A 3-D coupled hydromechanical granular model for simulating the constitutive behavior of metallic alloys during solidification

A three-dimensional (3-D) coupled hydromechanical granular model has been developed and validated to directly predict, for the first time, hot tear formation and stress strain behavior in metallic alloys during solidification. This granular model consists of four separate 3-D modules: (i) the solidification module is used to generate the solid liquid geometry at a given solid fraction; (ii) the fluid flow module (FFM) is used to calculate the solidification shrinkage and deformation-induced pressure drop within the intergranular liquid; (iii) the semi-solid deformation module (SDM) simulates the rheological behavior of the granular structure; and (iv) the failure module (FM) simulates crack initiation and propagation. Since solid deformation, intergranular flow and crack initiation are deeply linked together, the FFM, SDM and FM are coupled processes. This has been achieved through the development of a new three-phase interactive technique that couples the interaction between intergranular liquid, solid grains and growing voids. The results show that the pressure drop, and consequently hot tear formation, depends also on the compressibility of the mushy zone skeleton, in addition to the well-known contributors (lack of liquid feeding and semi-solid deformation). (C) 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved

A 3D coupled hydro-mechanical granular model for the prediction of hot tearing formation

A new 3D coupled hydro-mechanical granular model that simulates hot tearing formation in metallic alloys is presented. The hydro-mechanical model consists of four separate 3D modules. (I) The Solidification Module (SM) is used for generating the initial solid-liquid geometry. Based on a Voronoi tessellation of randomly distributed nucleation centers, this module computes solidification within each polyhedron using a finite element based solute diffusion calculation for each element within the tessellation. (II) The Fluid Flow Module (FFM) calculates the solidification shrinkage and deformation-induced pressure drop within the intergranular liquid. (III) The Semi-solid Deformation Module (SDM) is used to simulate deformation of the granular structure via a combined finite element/discrete element method. In this module, deformation of the solid grains is modeled using an elasto-viscoplastic constitutive law. (IV) The Failure Module (FM) is used to simulate crack initiation and propagation with the fracture criterion estimated from the overpressure required to overcome the capillary forces at the liquid-gas interface. The FFM, SDM, and FM are coupled processes since solid deformation, intergranular flow, and crack initiation are deeply linked together. The granular model predictions have been validated against bulk data measured experimentally and calculated with averaging techniques

3-D granular modeling and in situ X-ray tomographic imaging: A comparative study of hot tearing formation and semi-solid deformation in Al-Cu alloys

The mechanical behavior of partially solidified Al-Cu alloys is investigated to assess the influence of mushy zone deformation on hot tearing. For this purpose, the results of a semi-solid tensile test conducted in situ using X-ray microtomography are compared with the predictions of a coupled hydromechanical granular model in order to both validate the predictions of the model and explain the experimental observations. It is shown that hot tears initiate in the widest liquid channels connected to the free (oxidized) surfaces as long as there is contact between the intergranular liquid and the ambient air. The necking behavior is associated with the deformation-induced liquid pressure drop. Overall, the stresses predicted by the granular model under tensile and shear deformations agree well with the experimental data. Thus, the granular model achieves an important step in predicting hot tearing formation. (c) 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved