Developing empirical relationships to predict grain size and hardness of the weld nugget of friction stir welded AA7075-T6 aluminium alloy joints

Experimental Techniques - High-strength, precipitation-hardening AA7075-T6 alloy is used extensively in aircraft primary structures. Friction stir welding process is an emerging solid state joining...     

A separated law of hardening in strain gradient plasticity

This paper presents a separated law of hardening in plasticity with strain gradient effects. The value of the length parameter ℓ contained in this model was estimated from the experimental data for copper. The project supported by the National Natural Science Foundation of China     

Experimental and numerical study on formability of friction stir welded TWB sheets based on hemispherical dome stretch tests

In order to investigate formability performance and also to obtain guidelines for the stamping process design of friction stir welded TWB (tailor welded blank) sheets, the hemispherical dome stretching test was experimentally performed and the results of the base and friction stir welded samples were compared. Also, in order to better understand the experimental results, numerical analysis was performed. In this work, five automotive sheets, 6111-T4, 5083-H18, 5083-O aluminum alloy, dual-phase steel (DP590) and AZ31 magnesium alloy sheets were considered by (friction stir) welding the same materials. To represent mechanical properties for the numerical analysis, the non-quadratic orthotropic yield function, Yld2000-2d, was utilized for the aluminum alloy and DP590 sheets, while the Cazacu anisotropic/asymmetric yield function was applied for the AZ31 sheet considering different hardening behavior in tension and compression.     

Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals

A Taylor-like polycrystal model is adopted here to investigate the plastic behavior of body centered cubic (b.c.c.) sheet metals under plane-strain compression and the subsequent in-plane biaxial stretching conditions. The <111> pencil glide system is chosen for the slip mechanism for b.c.c. sheet metals. The {110} <111> and {112} <111> slip systems are also considered. Plane-strain compression is used to simulate the cold rolling processes of a low-carbon steel sheet. Based on the polycrystal model, pole figures for the sheet metal after plane-strain compression are obtained and compared with the corresponding experimental results. Also, the simulated plane-strain stress—strain relations are compared with the corresponding experimental results. For the sheet metal subjected to the subsequent in-plane biaxial stretching and shear, plastic potential surfaces are determined at a given small amount of plastic work. With the assumption of the equivalence of the plastic potential and the yield function with normality flow, the yield surfaces based on the simulations for the sheet metal are compared with those based on several phenomenological planar anisotropic yield criteria. The effects of the slip system and the magnitude of plastic work on the shape and size of the yield surfaces are shown. The plastic anisotropy of the sheet metal is investigated in terms of the uniaxial yield stresses in different planar orientations and the corresponding values of the anisotropy parameter R, defined as the ratio of the width plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial tensile loading. The uniaxial yield stresses and the values of R at different planar orientations from the polycrystal model can be fitted well by a yield function recently proposed by Barlat et al. (1997b).     

Modelling and FE-simulation of plastic ductile damage evolution in plates and shells

Summary  The present paper reports on modelling and numerical simulation of thin-walled structures close to failure taking into consideration the effects of both geometrical and physical nonlinearity. The approach accounts for finite displacements and rotations, and the material model adopted includes elastic–plastic behaviour, isotropic and kinematic hardening, and ductile damage. Particular attention is paid to the problems of localised damage, damage progression and final collapse of the structure. Numerical simulation of the nonlinear response of bars, plates and shells to quasistatic monotonic and variable loading illustrates how material damage affects the load-carrying behaviour of structural components. Received 28 November 1999; accepted for publication 29 March 2000     

Constitutive modeling of temperature and strain rate dependent elastoplastic hardening materials using a corotational rate associated with the plastic deformation

In this paper, a constitutive model with a temperature and strain rate dependent flow stress (Bergstrom hardening rule) and modified Armstrong-Frederick kinematic evolution equation for elastoplastic hardening materials is introduced. Based on the multiplicative decomposition of the deformation gradient,new kinematic relations for the elastic and plastic left stretch tensors as well as the plastic deformation-dependent spin tensor are proposed. Also, a closed-form solution has been obtained for the elastic and plastic left stretch tensors for the simple shear problem.To evaluate model validity, results are compared with known experimental data for SUS 304 stainless steel, which shows a good agreement with the results of the proposed theoretical model.Finally, the stress-deformation curve, as predicted by the model, is plotted for the simple shear problem at room and elevated temperatures using the same material properties for AA5754-O aluminium alloy.     

Modelling the flow of power law fluids in a packed bed using a volume-averaged equation of motion

The development of a theoretical model for the prediction of velocity and pressure drop for the flow of a viscous power law fluid through a bed packed with uniform spherical particles is presented. The model is developed by volume averaging the equation of motion. A porous microstructure model based on a cell model is used. Numerical solution of the resulting equation is effected using a penalty Galerkin finite element method. Experimental pressure drop values for dilute solutions of carboxymethylcellulose flowing in narrow tubes packed with uniformly sized spherical particles are compared to theoretical predictions over a range of operating conditions. Overall agreement between experimental and theoretical values is within 15%. The extra pressure drop due to the presence of the wall is incorporated directly into the model through the application of the no-slip boundary condition at the container wall. The extra pressure drop reaches a maximum of about 10% of the bed pressure drop without wall effect. The wall effect increases as the ratio of tube diameter to particle diameter decreases, as the Reynolds number decreases and as the power law index increases.     

A perturbation analysis of the unstable plastic flow pattern evolution in an aluminum alloy

In the tensile loading of sheet metals made from some polycrystalline aluminum alloys, a single deformation band appears inclined to the elongation axis in the early stage of deformation, and symmetric double bands are observed in the later stage. This evolution of spatial characteristics of such an unstable plastic flow pattern in a polycrystalline aluminum alloy has been analyzed by a perturbation method. A small number of slip modes are taken to describe the tensile strain. A rate-dependent constitutive equation is used for each slip mode to account for the interaction between dislocations and solute atoms in dynamic strain aging. Unconstrained and constrained models are used to impose appropriate loading conditions at the early and later deformation stages, respectively. Both plane-strain and plane-stress cases are considered. It is found out that the change of boundary conditions and material inhomogeneity during the course of plastic deformation are closely related to the evolution of spatial characteristics of shear band (the Portevin–Le Chatelier band) patterns observed in experiments.     

Instability and localization of plastic flow in shear at high strain rates

Shear band formation in a thermal viscoplastic heat conducting material is described in a simple shear test at high strain rate with inertia effects. The classical perturbation method is discussed, and a new relative perturbation method accounting for non-steadiness of plastic flow is presented. They respectively provide instability and localization criteria which are compared. Furthermore both are compared to available nonlinear exact results and to experimental data. The influence of material parameters, initial imperfections, and boundary conditions is described.     

Exponential stress and strain singularity at tip of mode I growing crack in power hardening plastic material

The stress-strain distribution near the tip of a Mode I growing crack in a power hardening plastic material is reconsidered. Two types of asymptotic equations are derived and solved numerically. It is shown that when the crack tip is approached, the stress is singular of the order r^−δ, while the strain is singular of the order r^−nδ, where r is the distance measured from the crack tip. The parameter δ is a constant; it depends on the hardening exponent n being greater than one.     

Mechanical characterisation of aluminium alloy 7449-T7651 at high strain rates and elevated temperatures using split hopkinson bar testing

Experimental Techniques - The mechanical behaviour of a 7 series high-strength aluminium alloy that is mainly used by the aerospace industry is under investigation. Aluminium alloy AA7449-T7651 is...     

A two stage method for structural damage detection using a modal strain energy based index and particle swarm optimization

A two-stage method is proposed here to properly identify the site and extent of multiple damage cases in structural systems. In the first stage, a modal strain energy based index (MSEBI) is presented to precisely locate the eventual damage of a structure. The modal strain energy is calculated using the modal analysis information extracted from a finite element modeling. In the second stage, the extent of actual damage is determined via a particle swarm optimization (PSO) using the first stage results. Two illustrative test examples are considered to assess the performance of the proposed method. Numerical results indicate that the combination of MSEBI and PSO can provide a reliable tool to accurately identify the multiple structural damage.     

A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell

We formulate the continuum field equations and constitutive equations that govern deformation, stress, and electric current flow in a Li-ion half-cell. The model considers mass transport through the system, deformation and stress in the anode and cathode, electrostatic fields, as well as the electrochemical reactions at the electrode/electrolyte interfaces. It extends existing analyses by accounting for the effects of finite strains and plastic flow in the electrodes, and by exploring in detail the role of stress in the electrochemical reactions at the electrode-electrolyte interfaces. In particular, we find that that stress directly influences the rest potential at the interface, so that a term involving stress must be added to the Nernst equation if the stress in the solid is significant. The model is used to predict the variation of stress and electric potential in a model 1-D half-cell, consisting of a thin film of Si on a rigid substrate, a fluid electrolyte layer, and a solid Li cathode. The predicted cycles of stress and potential are shown to be in good agreement with experimental observations.     

Photoelastic technique to investigate the effect of temperature and strain rate on plastic flow in mild steel

Dynamic photoelastic-coating technique was used to observe successive developments of plastic flow in tension at a temperature ranging from ?157°C to 20°C. A type of plastic flow occurred which was determined by a combination of temperature and strain rate. A correlation was found to exist between photoelastic observations and the equation of thermal activation.     

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for: an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations resulting from a piecewise quadratic Peierls potential; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles and lattice friction, resulting in hardening, path dependency and hysteresis. A chief advantage of the present theory is that it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. In particular, no numerical grid is required in calculations. The phase-field representation enables complex geometrical and topological transitions in the dislocation ensemble, including dislocation loop nucleation, bow-out, pinching, and the formation of Orowan loops. The theory also permits the consideration of obstacles of varying strengths and dislocation line-energy anisotropy. The theory predicts a range of behaviors which are in qualitative agreement with observation, including: hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect, whereby reverse yielding gradually eliminates the influence of previous loading; the evolution of the dislocation density under cycling loading, leading to characteristic ‘butterfly’ curves; and others.     

Investigation of turbulence and skin friction modification in particle-laden channel flow using lattice Boltzmann method

Interaction between turbulence and particles is investigated in a channel flow. The fluid motion is calculated using direct numerical simulation (DNS) with a lattice Boltzmann (LB) method, and particles are tracked in a Lagrangian framework through the action of force imposed by the fluid. The particle diameter is smaller than the Kolmogorov length scale, and the point force is used to represent the feedback force of particles on the turbulence. The effects of particles on the turbulence and skin friction coefficient are examined with different particle inertias and mass loadings. Inertial particles suppress intensities of the spanwise and wall-normal components of velocity, and the Reynolds shear stress. It is also found that, relative to the reference particle-free flow, the overall mean skin-friction coefficient is reduced by particles. Changes of near wall turbulent structures such as longer and more regular streamwise low-speed streaks and less ejections and sweeps are the manifestation of drag reduction.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part II: A very high work hardening aluminum alloy (annealed 1100 Al)

Results are presented on the evolution of subsequent yield surfaces with finite deformation in a very high work hardening annealed 1100 aluminum alloy. In Part I [Khan, A.S., Kazmi, R., Stoughton, T., Pandey, A., 2009a. Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part 1: a very low work hardening aluminum alloy (Al-6061–T6511) 25, 1611–1625.] of this paper, similar results are presented for a very low work hardening aluminum alloy. Those results were very different from the present ones, and all the results were for proportional loading paths. The subsequent yield surfaces are determined in tension, free end torsion and combined tension–torsion proportional and non-proportional loading paths, using 10 με deviation from linearity definition of yield. Yield surfaces are also determined after linear, bi-linear, and non-linear unloading paths after finite deformation under tension, free end torsion, and combined tension–torsion loading. The initial yield surface is closer to the von-Mises surface and the subsequent yield surfaces show distortion, expansion, positive cross-effect, and “nose” in the loading direction. Additionally, the subsequent yield surfaces after non-proportional loading paths show shrinkage and compounded distortion. The yield surfaces after unloading depict strong anisotropy, positive cross-effect and exhibits different proportion of distortion in each loading conditions. The Young’s and shear modulus decrease with plastic deformation and this decrease is much less than those reported in the published literature.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: A very low work hardening aluminum alloy (Al6061-T6511)

In the present study, the initial and subsequent yield surfaces in Al 6061-T6511, based on 10 με deviation from linearity definition of yield, are presented. The subsequent yield surfaces are determined during tension, free end torsion, and combined tension–torsion proportional loading paths after reaching different levels of strains. The yield surfaces are also obtained after linear, bi-linear and non-linear unloading paths after finite plastic deformation. The initial yield surface is very close to the von-Mises yield surface and the subsequent yield surfaces undergo translation and distortion. In the case of this low work hardening material, the size of the yield surfaces is smaller and negative cross-effect is observed with finite plastic deformation. The subsequent yield have a usual “nose” in the loading direction and flattened shape in the reverse loading direction; the observed nose is more dominant in the case of tension and combined tension–torsion loading than in torsional loading. The size of the yield surfaces after unloading is smaller than the initial yield surface but larger than subsequent yield surfaces obtained during prior loading, show much smaller cross-effect, and the shape of these yield surfaces depends strongly on the loading and unloading paths. Elastic constants (Young’s and shear moduli) are also measured within each subsequent yield surfaces. Evolution of these constants with finite deformation is also presented. The decrease of the two moduli is found to be much smaller than reported earlier in tension by Cleveland and Ghosh [Cleveland, R.M., Ghosh, A.K., 2002. Inelastic effects on springback in metals. Int. J. Plast. 18, 769–785]. Part-II and III [(Khan et al., 2009a) and (Khan et al., 2009b)] of the papers will include experimental results on annealed 1100 Al (a very high work hardening material) and on both Al alloys (Al6061-T6511 and annealed 1100 Al) in tension- tension stress space, respectively. The results for both cases are quite different than the ones that are presented in this paper.     

Mechanism of buckling development and strain reversal occurrence in elastic–plastic cylindrical shells under axial impact

In order to clarify the mechanism of loading and unloading, buckling growth, and strain-reversal occurrence in elastic–plastic cylindrical shells that are impacted axially, non-linear dynamic equations in incremental form are derived and solved by using the finite difference method. The validity of the developed theory is verified by comparing theoretical and experimental results. From the calculation results, it is found that, at the initial stage of the impact process, the effective stress related to the middle-surface stresses in the part near the impacted end rises with time, and exhibits plastic loading. The radial restraints on the impacted end have obvious influence on the strain reversal occurrence at an early stage of the impact process. The effective modulus controls the post-buckling deformation of the shell, which contrasts with the situation for the bar to be impacted axially.