On the Theory of Elasticity of Microinhomogeneous Media with Account for Stochastic Changes in the Connectivity of Constituent Components

The paper proposes a mathematical model aimed at calculating the effective elastic moduli of a micro-inhomogeneous two-component isotropic composite material, which components are connected randomly depending on the level of their relative volumetric contents. A stochastic equation is formulated for the connectivity parameter of the constituent components, according to which, with an increase in the volumetric content of the filler, individual inclusions build the structures of the matrix mixture in the form of interpenetrating frameworks, and then turn into a new binding matrix with individual inclusions from the material of the rest of the old matrix. The algorithm for the numerical solution of this stochastic differential equation is constructed in accordance with the Euler-Maruyama method. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for the random connectivity function of the constituent components of the composite material. A variant of the method aimed at calculating the mathematical expectation of a random connectivity function of the constituent components has been developed and the corresponding differential equation has been obtained for it. It is shown that the numerical solution of this equation and the average value of the production factor function calculated for all realizations of stochastic trajectories give close numerical values. New macroscopic constitutive relations are found for microinhomogeneous materials with a variable microstructure and their effective elastic moduli are calculated. It is noted that the formulas for these effective elastic moduli generalize the known results for isotropic composite materials. The values of the effective elastic moduli, constructed according to the expressions obtained in the paper, lie within the Khashin-Shtrikman range for the lower and upper bounds of the effective elastic moduli of the composite materials. The numerical analysis of the developed models showed a good agreement with the known experimental data.

On the respect to the Hashin-Shtrikman bounds of some analytical methods applying to porous media for estimating elastic moduli

In this work, some popular analytic formulas such as Maxwell (MA), Mori-Tanaka approximation (MTA), and a recent method, named the Polarization approximation (PA) will be applied to estimate the elastic moduli for some porous media. These approximations are simple and robust but can be lack reliability in many cases. The Hashin-Shtrikman (H-S) bounds do not supply an exact value but a range that has been admitted by researchers in material science. Meanwhile, the effective properties by unit cell method using the finite element method (FEM) are considered accurate. Different shapes of void inclusions in two or three dimensions are employed to investigate. Results generated by H-S bounds and FEM will be utilized as references. The comparison suggests that the method constructed from the minimum energy principle PA can give a better estimation in some cases. The discussion gives out some remarks which are helpful for the evaluation of effective elastic moduli. Keywords: Maxwell approximation; polarization approximation; Mori-Tanaka approximation; effective elastic moduli; porous medium.

Nanomechanical Response of Pulsed Tungsten Inert Gas Welded Titanium Alloy by Nanoindentation and Atomic Force Microscopy

Nanohardness and Effective Elastic Moduli were measured for pulsed-Gas Tungsten Arc Welded Ti-5Al-2.5Sn alloy using autogenous mode through nanoindentation and atomic force microscopy. Experiments were conducted using a Berkovich tip on nanoindentor with Berkovich tip and elliptical pile ups were measured using an Atomic Force Microscope. Nanohardness and effective elastic moduli were calculated in the base metal, heat affected zone and fusion zone of the weldments using different approaches namely Oliver–Pharr method, AFM analysis and work of indentation. A significant difference was observed in the nanomechanical response using these approaches which was attributed to the pile up morphology of the nano indents. The presence of residual stress in the weldments also significantly influenced the nanohardness profile across the weld joint. The present research suggested that the work of indentation is most suitable for assessment of nanomechanical properties of Ti-5Al-2.5Sn alloy weldments among the three techniques studied in this investigation.

On the estimates for the elastic moduli of random Voronoi triclinic polycrystals

Our major new results and the previous ones on the bounds for elastic random polycrystals, and most advanced 3D finite element results for random 3D Voronoi polycrystals are resumed and analysed (together for the first time). Recently obtained numerical Dirichlet and Neumann simulation results for the effective elastic moduli of a large 10000-grain-size random Voronoi polycrystal representative volume element (RVE) for a number of triclinic and monoclinic base crystals (Mursheda and Ranganathan, 2017) are compared critically with the bounds on the moduli. Though major parts within the simulation results fall within the bounds of Pham (2011), some Dirichlet upper estimates still lie outside the bounds. Many more RVEs are needed to represent the Voronoi polycrystal on the same RVE-size-level, and larger RVEs are needed for checking the convergence and comparisons with the bounds.