Mechanics of indentation of plastically graded materials—II: Experiments on nanocrystalline alloys with grain size gradients

A systematic study of depth-sensing indentation was performed on nanocrystalline (nc) Ni-W alloys specially synthesized with controlled unidirectional gradients in plastic properties. A yield strength gradient and a roughly constant Young's modulus were achieved in the nc alloys, using electrodeposition techniques. The force vs. displacement response from instrumented indentation experiments matched very well with that predicted from the analysis of Part I of this paper. The experiments also revealed that the pile-up of the graded alloy around the indenter is noticeably higher than that for the two homogeneous reference alloys that constitute the bounding conditions for the graded material. These trends are also consistent with the predictions of the indentation analysis.     

Mechanics of adhesive contact on a power-law graded elastic half-space

We consider adhesive contact between a rigid sphere of radius R and a graded elastic half-space with Young's modulus varying with depth according to a power law E=E₀(z/c₀)^k (0<k<1) while Poisson's ratio ν remaining a constant. Closed-form analytical solutions are established for the critical force, the critical radius of contact area and the critical interfacial stress at pull-off. We highlight that the pull-off force has a simple solution of P_cr=−(k+3)πRΔγ/2 where Δγ is the work of adhesion and make further discussions with respect to three interesting limits: the classical JKR solution when k=0, the Gibson solid when k→1 and ν=0.5, and the strength limit in which the interfacial stress reaches the theoretical strength of adhesion at pull-off.     

An investigation of the step-wise propagation of a mode-II fracture in a poroelastic medium

In this paper we use an eXtended Finite Element Method based model for the simulation of shear fracture in fully saturated porous materials. The fracture is incorporated as a strong discontinuity in the displacement field by exploiting the partition of unity property of finite element shape functions. The pressure is assumed to be continuous across the fracture. However, the pressure gradient, i.e. the fluid flow, can be discontinuous. The failure process is described by the cohesive zone approach and a Tresca fracture condition without dilatancy. We investigate the propagation of a shear fracture under compression asking the question whether or not a Tresca criterion can result in stepwise propagation in a poroelastic medium. In order to evaluate possible numerical artefacts, we also look at the influence of the element size and the magnitude of a time increment. The performance of the X-FEM model and the influence of the pore pressure on the fracture propagation are addressed. Our simulations do not show evidence for step wise progression in mode II failure.     

Finite element simulation of dip coating,I: Newtonian fluids

A finite element simulation of the dip coating process based on a discretization of the continuum with discontinuous pressure elements is presented. The algorithm computes the flow field from natural boundary conditions while an extra condition provided by the existence of free surface is employed to displace the meniscus location towards the actual position. The process is iterative and uses a pseudo-time stepping technique coupled to a cubic spline fitting of the free surface. Numerical predictions exhibit good agreement with experimental data for Newtonian fluids in the case of flat plate dip coating as well as in the case of wire dip coating.     

Stress analysis of functionally graded rotating discs: analytical and numerical solutions

This study deals with stress analysis of annular rotating discs made of functionally graded materials(FGMs).Elasticity modulus and density of the discs are assumed to vary radially according to a power law function,but the material is of constant Poisson’s ratio.A gradient parameter n is chosen between 0 and 1.0.When n = 0,the disc becomes a homogeneous isotropic material.Tangential and radial stress distributions and displacements on the disc are investigated for various gradient parameters n by means of the diverse elasticity modulus and density by using analytical and numerical solutions.Finally,a homogenous tangential stress distribution and the lowest radial stresses along the radius of a rotating disc are approximately obtained for the gradient parameter n = 1.0 compared with the homogeneous,isotropic case n = 0.This means that a disc made of FGMs has the capability of higher angular rotations compared with the homogeneous isotropic disc.     

A finite element formulation for analysis of functionally graded plates and shells

Summary A finite element formulation is derived for the thermoelastic analysis of functionally graded (FG) plates and shells. The power-law distribution model is assumed for the composition of the constitutent materials in the thickness direction. The procedure adopted to derive the finite element formulation contains the analytical through-the-thickness integration inherently. Such formulation accounts for the large gradient of the material properties of FG plates and shells through the thickness without using the Gauss points in the thickness direction. The explicit through-the-thickness integration becomes possible due to the proper decomposition of the material properties into the product of a scalar variable and a constant matrix through the thickness. The nonlinear heat-transfer equation is solved for thermal distribution through the thickness by the Rayleigh-Ritz method. According to the results, the formulation accounts for the nonlinear variation in the stress components through the thickness especially for regions with a variation in martial propperties near the free surfaces.     

Modelling the slight compressibility of anisotropic soft tissue

In order to avoid the numerical difficulties in locally enforcing the incompressibility constraint using the displacement formulation of the Finite Element Method, slight compressibility is typically assumed when simulating transversely isotropic, soft tissue. The current standard method of accounting for slight compressibility of hyperelastic soft tissue assumes an additive decomposition of the strain-energy function into a volumetric and a deviatoric part. This has been shown, however, to be inconsistent with the linear theory for anisotropic materials. It is further shown here that, under hydrostatic tension or compression, a transversely isotropic cube modelled using this additive split is simply deformed into another cube regardless of the size of the deformation, in contravention of the physics of the problem. A remedy for these defects is proposed here: the trace of the Cauchy stress is assumed linear in both volume change and fibre stretch. The general form of the strain-energy function consistent with this model is obtained and is shown to be a generalisation of the current standard model. A specific example is used to clearly demonstrate the differences in behaviour between the two models in hydrostatic tension and compression.     

Modeling the evolution of stress due to differential shrinkage in powder-processed functionally graded metal–ceramic composites during pressureless sintering

Pressureless sintering of powder-processed functionally graded materials is being pursued to economically produce metal–ceramic composites for a variety of high-temperature (e.g., thermal protection) and energy-absorbing (e.g., armor) applications. During sintering, differential shrinkage induces stresses that can compromise the integrity of the components. Because the strength evolves as the component is sintered, it is important to model how the evolution of the differential shrinkage governs the stress distribution in the component in order to determine when the strength will be exceeded and cracking initiated. In this investigation, a model is proposed that describes the processing/microstructure/property/performance relationship in pressurelessly sintered functionally graded plates and rods. This model can be used to determine appropriate shrinkage rates and gradient architectures for a given component geometry that will prevent the component from cracking during pressureless sintering by balancing the evolution of strength, which is assumed to be a power law function of the porosity, with the evolution of stress. To develop this model, the powder mixture is considered as a three-phase material consisting of voids, metal particles, and ceramic particles. A micromechanical thermal elastic–viscoplastic constitutive model is then proposed to describe the thermomechanical behavior of the composite microstructure. The subsequent evolution of the thermomechanical properties of the matrix material during sintering is assumed to obey a power law relationship with the level of porosity, which is directly related to the shrinkage strain, and was refined to account for the evolving interparticle cohesion of the matrix phase due to sintering. These thermomechanical properties are incorporated into a 2-D thermomechanical finite element analysis to predict the stress distributions and distortions that arise from the evolution of differential shrinkage during the pressureless sintering process. Differential shrinkage results were verified quantitatively through comparison with the shape profile for a pressurelessly sintered functionally graded nickel–alumina composite plate with a cylindrical geometry, and the stress distribution results verified from qualitative observations of the absence or presence of cracking as well as the location in specimens with different gradient architectures. The cracking was mitigated using a reverse gradient at one end of the specimen, and the resulting distortions associated with the shape profile were determined to be no more than 15% reduced from the predictions. The effects of geometry were also studied out-of-plane by transforming the plate into a rod through an increase in thickness, while in-plane effects were studied by comparing the results from the cylindrical specimen with a specimen that has a square cross-sectional geometry. By transforming from a plate to a rod geometry, the stress no longer exceeds critical levels and cracks do not form. The results from the in-plane geometric study indicated that critical stresses were reached in the square geometry at temperatures 100 °C less than in the cylindrical geometry. Additionally, the location of primary cracking was shifted towards the metal-rich end of the specimen, while the stress distribution associated with this shift and the lower temperature for the critical stress resulted in secondary cracking.     

Periodic solutions of rigid body–viscous flow interaction

This paper describes the work on extending the finite element method to cover interactions between a viscous flow and a moving body. The problem configuration of interest is that of an arbitrarily shaped body undergoing a simple harmonic motion in an otherwise undisturbed incompressible fluid. The finite element modelling is based on a primitive variables representation of the Navier-Stokes equations using curved isoparametric elements. The non-linear boundary conditions on the moving body are obtained using Taylor series expansion to approximate the velocities at the fixed finite element grid points. The method of averaging is used to analyse the resulting periodic motion of the fluid. The stability of the periodic solutions is studied by introducing small perturbations and applying Floquet theory. Numerical results are obtained for several example body shapes and compared with published experimental results. Good agreement is obtained for the basic non-linear phenomenon of steady streaming.     

Reducing cross-flow vibrations of underflow gates: Experiments and numerical studies

An experimental study is combined with numerical modelling to investigate new ways to reduce cross-flow vibrations of hydraulic gates with underflow. A rectangular gate section placed in a flume was given freedom to vibrate in the vertical direction. Horizontal slots in the gate bottom enabled leakage flow through the gate to enter the area directly under the gate which is known to play a key role in most excitation mechanisms.For submerged discharge conditions with small gate openings the vertical dynamic support force was measured in the reduced velocity range 1.5<V_r<10.5 for a gate with and without ventilation slots. The leakage flow significantly reduced vibrations. This attenuation was most profound in the high stiffness region at 2<V_r<3.5.Two-dimensional numerical simulations were performed with the Finite Element Method to assess local velocities and pressures for both gate types. A moving mesh covering both solid and fluid domain allowed free gate movement and two-way fluid–structure interactions. Modelling assumptions and observed numerical effects are discussed and quantified. The simulated added mass in still water is shown to be close to experimental values. The spring stiffness and mass factor were varied to achieve similar response frequencies at the same dry natural frequencies as in the experiment. Although it was not possible to reproduce the vibrations dominated by impinging leading edge vortices (ILEV) at relatively low V_r, the simulations at high V_r showed strong vibrations with movement-induced excitation (MIE). For the latter case, the simulated response reduction of the ventilated gate agrees with the experimental results. The numerical modelling results suggest that the leakage flow diminishes pressure fluctuations close to the trailing edge associated with entrainment from the wake into the recirculation zone directly under the gate that most likely cause the growing oscillations of the ordinary rectangular gate.     

Size-dependent sharp indentation—I: a closed-form expression of the indentation loading curve

In this paper, a closed-form expression of the size-dependent sharp indentation loading curve has been proposed based on dimensional analysis and the finite deformation Taylor-based nonlocal theory (TNT) of plasticity (Int. J. Plasticity 20 (2004) 831). The key issue is to link the results of FEM based on TNT plasticity with those obtained using conventional FEM by taking as the effective strain gradient, η, that presented in the work of Nix and Gao (J. Mech. Phys. Solids 46 (1998) 411), thus avoiding large-scale finite element computations using strain gradient plasticity theories. Two experiments carried out on 316 stainless-steel and pure titanium have been used to verify the effectiveness of the present analytical model; the results demonstrate that the present analytical expression of the size-dependent indentation loading curve corresponds very well to the experimental indentation loading curve. The empirical constant, α, in the Taylor model estimated from the experimental data has the correct order of magnitude. Also, the results presented in this part can be further applied to establish an analytical framework to extract the plastic properties of metallic materials with sharp indentation on a small scale where the size effect caused by geometrically necessary dislocations is significant. This will be discussed in detail in the second part of the paper.     

Stability of crystalline solids—I: Continuum and atomic lattice considerations

Many crystalline materials exhibit solid-to-solid martensitic phase transformations in response to certain changes in temperature or applied load. These martensitic transformations result from a change in the stability of the material's crystal structure. It is, therefore, desirable to have a detailed understanding of the possible modes through which a crystal structure may become unstable. The current work establishes the connections between three crystalline stability criteria: phonon-stability, homogenized-continuum-stability, and the presently introduced Cauchy-Born-stability criterion. Stability with respect to phonon perturbations, which probe all bounded perturbations of a uniformly deformed specimen under “hard-device” loading (i.e., all around displacement type boundary conditions) is hereby called “constrained material stability”. A more general “material stability” criterion, motivated by considering “soft” loading devices, is also introduced. This criterion considers, in addition to all bounded perturbations, all “quasi-uniform” perturbations (i.e., uniform deformations and internal atomic shifts) of a uniformly deformed specimen, and it is recommend as the relevant crystal stability criterion.     

A centrally cracked thin circular disk, Part I: 3D Elastic–plastic finite element analysis

A full field solution, based on small deformation, three-dimensional elastic–plastic finite element analysis of the centrally cracked thin disk under mode I loading has been performed. The solution for the stresses under small-scale yielding and lo!cally fully plastic state has been compared with the HRR plane stress solution. At the outside of the 3D zone, within a distance of rσ_o/J=18, HRR dominance is maintained in the presence of a significant amount of compressive stress along the crack flanks. Ahead of this region, the HRR field overestimate the stresses. These results demonstrate a completely reversed state of stress in the near crack front compared to that in the plane strain case. The combined effect of geometry and finite thickness of the specimen on elastic–plastic crack tip stress field has been explored. To the best of our knowledge, such an attempt in the published literature has not been made yet. For the qualitative assessment of the results some of the field parameters have been compared to the available experimental results of K, gives a fair estimate of the crack opening stress near the crack front at a distance of order 10⁻² in. On the basis of this analysis, the Linear Elastic Fracture Mechanics approach has been adopted in analyzing the fatigue crack extension experiments performed in the disk (Part II).