Paul Lipinski

Elastoplasticity of micro-inhomogeneous metals at large strains

Abstract The aim of this paper is to develop a general approach to the problem of determination of elastoplastic properties of metallic polycrystals at finite transformations. First, the various physical parameters describing the internal structure of the polycrystal are discussed. Next, transition relations between local and overall levels are reviewed. Here, the framework developed by Hill is mainly used. Following, a new integral equation, linking the local and overall velocity gradients is presented. An analogous integral equation was proposed by Dederichs and Zeller for inhomogeneous elastic media characterized by symmetric tensors of elastic moduli and during small strain transformations. The Green technique has been applied to obtain the integral equation. The fundamental properties of the Green tensor are discussed from the point of view of its application at large transformations. Different methods of resolution of such an equation are presented here. Special attention is given to granular media for which the elastoplastic tangent moduli may be considered piecewise constant. A new “quasi” self-consistent model is given, the development and applications of which will be treated elsewhere.

Quantitative estimation of incompatibility stresses and elastic energy stored in ferritic steel

Plastic incompatibility second-order stresses were determined for different orientations of a polycrystalline grain, using X-ray diffraction data and results of the self-consistent elasto-plastic model. The stresses in cold rolled ferritic steel were determined both in as-received and under tensile loaded conditions. It has been shown that the Reuss model and the self-consistent model applied to near surface volume provide the best approaches to determine diffraction elastic constants. For the first time, the elastic energy in an anisotropic material (arising from plastic incompatibilities between grains having various lattice orientations) has been determined. The second-order incompatibility stresses and stored elastic energy are presented in Euler space.

Micromechanical modeling of ductile damage of polycrystalline materials with heterogeneous particles

Abstract A two-level homogenization approach is applied in this micro-mechanical modeling of the ductile damage of polycrystals containing intracrystalline non-shearable particles. Voids nucleate around these second phase particles by interface debonding and thereafter, grow due to the plastic straining of the crystalline matrix. The equivalent behavior of the single crystal containing voids is derived from the description of the single crystal with particles. Moreover, this behavior is deduced from the classical formulation of the single crystal plasticity based on crystallographic gliding and Schmids law. At microscopic scale, void nucleation is possible when the elastic strain energy stored in the particle and released during debonding exceeds the energy of creation of new surfaces at the crystal-particle interface. The material pre-damage behavior is described by the hardening by intra-crystalline particles model proposed in Bonfoh et al. (2003).

Modeling of intra-crystalline hardening of materials with particles

Abstract A two-level homogenization approach is developed for the micromechanical modeling of the elastoplastic behavior of polycrystals containing intracrystalline non-shearable particles. First, a micro-meso transition is employed to establish a constitutive relation for a single crystal containing particles. The behavior of an equivalent single crystal with particles is derived from the classical formulation of plasticity of the single crystal based on the Schmids law and crystallographic gliding. Then, the transition to the macroscopic scale is performed with a self-consistent scheme to determine the elastoplastic behavior of the macro homogeneous material. The obtained global behavior is characterized by a mixed anisotropic and kinematic hardening related to an evolution of inter- and intra-granular material microstructure. Results have been analyzed in light of second and third order internal stresses developed during the plastic flow. Especially, yield surfaces have been determined for various preloadings and particle volume fractions.

New Type of Diffraction Elastic Constants for Stress Determination

A new method for calculation of the diffraction elastic constants, based on the selfconsistent model, is proposed and tested. This method is especially useful in the interpretation of the results of X-ray measurements since the ellipsoidal inclusion near the sample surface is considered. In X-ray diffraction the information volume of the sample is defined by absorption, causing unequal contribution of different crystallites to the intensity of the measured peak. Consequently, the surface grains participate more effectively in diffraction than the grains which are deeper in the sample.

Determination of the Partition Coefficient for the Grinding of a Hard WC-Co-Cr Coating with a Diamond Wheel

□ Determination of the partition ratio is fundamental to a thermo-mechanical approach to the grinding process. To establish the energy balance, accurate measurements and numerical modelling must be combined. In this work, such a method is applied to the grinding of a HVOF WC-Cr-Co hard coating. A double pole grindable thermocouple was developed to measure the temperature in the arc of cut, and its effective length. The heat flux absorbed by the workpiece is found by fitting the numerical solution to the temperature profile measured during the cooling phase, behind the contact. Identical results for the heat flux may be obtained by fitting the results to the analytical solution from moving heat source theory, but the maximum interface temperature is underestimated. A finite element solution, using MSC Marc software allows a closer approach to the maximum temperature measured at the interface during grinding, provided that the thickness of the coating is correctly modelled.

Stored energy and recrystallization process

Stored energy plays a crucial role in recrystallization process. One can distinguish two contributions to this energy. The first one is the elastic energy, connected with residual stresses, i.e., with grain-grain interaction. Another part of the stored energy is due to dislocation density, which is mainly localized inside grains. The latter one is considered as a main driving force of recrystallization. However, the stored energy connected with residual stresses can also have some influence on this process. Both types of energy can be determined experimentally and predicted by deformation models. Taking into account both types of the stored energy, some features of recrystallization textures can be explained.

Experimental Methodology Destined to Establish the Frequency Response Function (FRF) between a Dynamic Force and the Signals Emitted by a Piezoelectric Dynamometer

This paper proposes an experimental method for obtaining the Frequency Response Function (FRF) between a dynamic force and the signals emitted by a piezoelectric dynamometer. This function is known as Transmissibility. In the FRF obtaining stage, different configurations of mounting and excitation have been compared to improve the function quality. The method has been developed with a three components dynamometer fixed on a milling machine. The FRF has two principal applications: it is used to evaluate the measurement system accuracy and to correct the measurements, if necessary. The method has been developed with the purpose of studying the cutting forces in machining process. Furthermore, it has been identified the influence of the parts of the measurement chain in the measuring system response.

Study of Asymmetric Rolling of Titanium by the Finite Elements Method with Implemented Crystalline Model

The goal of this work was to study the asymmetric rolling process using the Finite Element Method (FEM) coupled with the deformation model of polycrystalline material. The Leffers-Wierzbanowski (LW) model was selected to be implemented into FEM. This implementation enables a study of heterogeneous plastic deformation process, like asymmetric rolling, taking into account its crystallographic nature. The asymmetric rolling was realized using two identical rolls, driven by independent motors, rotating with different angular velocities. This enabled to obtain a controlled range of rolling asymmetry. Our aim was to examine the properties of asymmetrically rolled commercially pure titanium (Grade 2).

Three-Dimensional Thermomechanical Simulation of Fine-Pitch High-Count Ball Grid Array Flip-Chip Assemblies

Flip-chip technology is increasingly prevalent in electronics assembly [three-dimensional (3D) system-in-package] and is mainly used at fine pitch for manufacture of megapixel large focal-plane detector arrays. To estimate the reliability of these assemblies, numerical simulations based on finite-element methods appear to be the cheapest approach. However, very large assemblies contain more than one million solder bumps, and the optimization process of such structures through numerical simulations turns out to be a very time-consuming task. In many applications, the interconnection layer of such flip-chip assemblies consists of solder bumps embedded in epoxy filler. For such configurations, we propose an alternative approach, which consists in replacing this heterogeneous interconnection layer by a homogeneous equivalent material (HEM). A micromechanical model for the estimation of its equivalent thermoelastic properties has been developed. The obtained constitutive law of the HEM was then implemented in finite-element software (Abaqus®). Thermomechanical responses of tested assemblies submitted to loads corresponding to manufacturing conditions have been analyzed. The homogenization–localization process allowed estimation of the mean values of stresses and strains in each phase of the interconnection layer. To access more precisely the stress and strain fields in these phases, two models of structural zoom, taking into account the real solder bump geometry, have been tested. The obtained local stress and strain fields corroborate the experimentally observed damage initiation of the solder bumps.